Quantifying Uncertainty in Economics Policy Predictions: A Bayesian & Monte Carlo based Data-Driven Approach

Quantifying Uncertainty in Economics Policy Predictions: A Bayesian & Monte Carlo based Data-Driven Approach | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Quantifying Uncertainty in Economics Policy Predictions: A Bayesian & Monte Carlo based Data-Driven Approach Shafeeq Ur Rahaman, Mahe Jabeen Abdul This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5299835/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Economic policy uncertainty relates to the unpredictability in government policies that can impact economic decision-making. High policy uncertainty can lead to less investment, slower economic growth, and increased volatility in financial markets. This study aims to quantify the uncertainty by employing a data-driven approach based on Bayesian Hierarchical Modeling (BHM) and Markov Chain Monte Carlo simulations. This research focuses essentially on key policy domains such as monetary policy, fiscal policy, and trade policy where uncertainty underlies crucial influences upon economic decisions. The methodology integrates data collection, feature scaling, normalization, Bayesian inference using MCMC simulations, uncertainty quantification and policy prediction to produce predictive insights under various economic scenarios. The Bayesian Hierarchical Model was employed to estimate the relationships between macroeconomic variables and policy outcomes. The posterior distribution results revealed significant predictors, with certain factors like monetary policy uncertainty exerting a substantial negative impact, while others such as equity market-related uncertainty showed positive influence. A rigorous uncertainty quantification step provided credible intervals for predicted outcomes with a 95% credible interval ranging between 0.276 - 0.359. This enabled an assessment of the potential variability in predictions based on differing levels of economic uncertainty. The study concluded with policy predictions generated under two distinct economic scenarios. Under conditions of high uncertainty, the predicted policy outcome was -0.2346, while a moderate uncertainty scenario resulted in a less negative outcome of -0.2060. These results demonstrate the sensitivity of economic policy predictions to varying levels of uncertainty. The findings provide a robust framework for understanding and quantifying uncertainty in economic policy-making. By applying BHM and Monte Carlo methods, this study contributes to the development of more resilient and adaptive economic strategies in the face of uncertainty. International Economics Economic policy uncertainty Predictive modelling Markov Chain Monte Carlo algorithm Bayesian statistics Computational economics. Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 1. Introduction In recent years, economic policymaking has faced increasing challenges due to increase uncertainty across various domains including monetary policy, fiscal decisions and trade regulations. The unpredictability of macroeconomic variables often complicates the development of effective policies that can negatively affect economic stability and growth. Predicting the future outcomes of wide range of variables with high accuracy also remains a complex challenge (Al-Thaqeb & Algharabali, 2019). As policymakers strive to navigate these challenges, quantifying and predicting the impacts of uncertainty has become a focal point of interest in computational economics. Traditional models often struggle to capture the inherent uncertainty and volatility present in such dynamic systems. This issue has sparked growing interest in more sophisticated, data-driven approaches to quantifying uncertainty in economic predictions (Gillmann & Kim, 2021). Advancements in computational economics have introduced powerful tools for modeling uncertainty. Bayesian statistics is believed to be a powerful framework that incorporates prior knowledge and observed data to estimate the probability distributions of unknown parameters. It allows for more flexible modeling of uncertainty compared to traditional frequentist approaches. In economic policy prediction, Bayesian methods enable the integration of prior information about economic trends and policy impacts that improve forecast accuracy in the presence of complex, uncertain environments. This approach provides policymakers with probabilistic estimates, credible intervals, and risk assessments to support more informed decision-making under uncertainty (Christou et al., 2017). Bayesian Hierarchical Models (BHMs) is a simulation-based inference of Bayesian statistics that allow for the modeling of complex data structures by incorporating multiple levels of variability across economic factors. In economic policy prediction, BHMs provide a flexible framework to integrate various macroeconomic indicators to examine uncertainty at different levels of the model (Leclercq, 2022). Monte Carlo simulations are computational techniques that use random sampling to estimate complex probability distributions and numerical outcomes. When integrated with Bayesian models, Markov Chain Monte Carlo (MCMC) can be employed to approximate the posterior distribution of model parameters. This integration explores various economic scenarios and policy impacts by simulating numerous possible outcomes based on the uncertainty in the data. In economic policy prediction, Monte Carlo simulations based Bayesian model provide a comprehensive understanding of the range and likelihood of future policy effects (Goodarzi et al., 2022). This multi-level predictive analysis allows for flexible representation of complex relationships between macroeconomic variables and provide probabilistic insights into potential outcomes. Monte Carlo simulations further enhance this approach by generating a wide range of possible scenarios and by enabling more robust decision-making under uncertainty. This paper explores the use of Markov Chain Monte Carlo simulation-based inference of Bayesian Hierarchical Model methods to quantify uncertainty in three critical areas i.e., monetary policy, fiscal policy, and trade policy. This study aims to develop a comprehensive framework for uncertainty quantification by integrating key macroeconomic indicators such as GDP, inflation rates, and market volatility. The methodology integrates data processing, feature scaling, stationary testing, normalization, and BHM to infer posterior distributions for various predictors. The results are used to perform uncertainty quantification followed by policy predictions based on distinct economic scenarios. The study also demonstrates how posterior distributions and credible intervals derived from the Bayesian model can provide insights into policy effectiveness and associated risks. This framework not only facilitates a deeper understanding of the underlying economic mechanisms but also provides a probabilistic foundation for policy recommendations in uncertain environments. 2. Methodology 2.1. Data Collection Multiple high-quality databases including a variety of macroeconomic indicators and policy uncertainties were utilized for this study. Data sources included World Development Indicators (WDI) from the World Bank, International Financial Statistics (IFS) from the International Monetary Fund, and a number of Economic Policy Uncertainty Indexes from the Federal Reserve Economic Data (FRED). These indexes included monetary, fiscal, trade, and financial regulation policies in addition to general economic uncertainty. The WDI dataset provide extensive coverage of macroeconomic variables (Azolibe, 2022). This dataset contains GDP, inflation rates, trade balances, interest rates, and unemployment statistics. The data was accessed via the World Bank API using Python wbdata library (Reddy & NR). The IFS dataset was downloaded using pandasdmx python library (Araujo, 2023). IFS provide a broader international perspective on monetary policy variables such as interest rates, exchange rates, and money supply. Economic Policy Uncertainty (EPU) indexes are critical for quantifying uncertainty across various policy domains (Yu et al., 2021). Table 1 is showing the specific EPU indexes selected and their significance. These indexes are chosen to represent sector-specific uncertainties that directly impact policy predictions. Data was fetched via FRED api using Python and was integrated into the complex data pipeline of the study. The final datasets contain multiple dimensions of macroeconomic and policy data ranged from 1st January 1985 to 8th October 2024. Table 1 Selected EPU indexes and their role in policy prediction and uncertainty. Index Name Significance Monetary Policy Uncertainty Index Measures uncertainty that is associated with the interest rates, money supplies, and controls of inflation by the central banks. This is essential for monetary policy analysis. Fiscal Policy Uncertainty Index Deals with uncertainty based on government spending and tax policies. Important for understanding the landscape and volatility of fiscal policy. Trade Policy Uncertainty Index Captures the unpredictability in government trade agreements, tariffs, and import/export regulations. Vital for analyzing trade policy impacts. Financial Regulation Uncertainty Index Quantifies uncertainty related to financial market regulations. Helps to assess the stability of financial systems and its impacts of regulatory changes. Government Spending Uncertainty Index Measures uncertainty in government expenditure plans, investments in infrastructure, defense, and social programs. Useful for evaluating fiscal sustainability. Equity Market-Related Economic Uncertainty Index Measures uncertainty in the equity markets driven by macroeconomic news and expectations. Helps in assessing investment risk. Macroeconomic Inflation: News and Outlook Focuses on inflation-related uncertainty. This highlight market participants' reaction to expectations in terms of inflation. Global Economic Policy Uncertainty Index A comprehensive measure of economic policy uncertainty across the globe. This would be helpful in knowing cross-country policy impacts. 2.2. Data Preprocessing and Feature Scaling The IFS dataset was filtered to retain only the relevant countries i.e., Germany, France, United Kingdom, Italy, Japan, and the United States. Columns with little to no relevance to the study's focus were excluded based on their indices. Only essential macroeconomic indicators were focused and a total of 139 columns were dropped from the original dataset. To avoid possible issues caused by mixed data types in the IFS file, the dataset was loaded with `low_memory = False` so that all columns could be handled without losing the integrity of the data (Zhou, 2023). After filtering and removing columns, the cleaned IFS dataset was merged WDI thus creating a temporal alignment of observations coming from different sources. Missing values (NaN) were filled using the mean of the respective columns. The combined data was saved for further analysis in CSV format. The variables of combined dataset were then standardized to prepare it for statistical modeling and analysis. Standardization has been carried out using `StandardScaler` from the Scikit-learn python library by scaling the data so that every feature will have zero mean and unit variance (Raju et al., 2020). Standardization ensures that features with differing scales (e.g., GDP in trillions vs. inflation rates in percentages) do not disproportionately influence model outcomes. Only numeric variables were scaled and the 'Date' column was excluded from this transformation. The scaled dataset was visualized through histograms to verify the standardization process. 2.3. Variables Grouping and Multicollinearity Check Hierarchical models often rely on grouping variables to define levels of variation, so 'Country' was defined as the grouping variable. The country variable was converted to a categorical type and assigned integer codes to facilitate the hierarchical analysis. In this step, the dataset was also checked for missing values using `isnull().sum()` in python to ensure completeness of the data before proceeding to model fitting. Multicollinearity among the predictor variables was assessed to ensure that the model would not be biased due to correlated predictors (Shrestha, 2020). A heatmap was generated using python to visualize the correlations between numerical variables (correlation matrix). Variance Inflation Factor (VIF) metric was also computed for each predictor variable to check if any variables exhibit high multicollinearity (VIF values exceeding 10) (Folli et al., 2020). The correlation matrix and VIF values ensure that the independent variables do not distort the results of the Bayesian hierarchical model. 2.4. Stationary Testing To analyze the stationarity in the standardized dataset, the Augmented Dickey-Fuller (ADF) test was employed (Ajewole et al., 2020; Sarker & Khan, 2020). Stationarity testing is a fundamental step in time series analysis as it ensures that the statistical properties of the series (mean, variance, etc.) do not change over time (Silva et al., 2021). The ADF test is a widely used statistical test for checking stationarity, particularly in the presence of potential autocorrelation in the data. The standardized dataset, containing both numeric and non-numeric columns, was loaded in CSV format. Only numeric columns were considered for stationarity testing. Non-numeric columns, such as date columns, were excluded from the analysis to prevent errors. Any missing values present in the numeric columns were removed to avoid interference in the ADF test's computations. The null hypothesis (H₀) for the ADF test indicate that time series is non-stationarity. The alternative hypothesis (H₁) suggests that the series is stationary. The 'AIC' (Akaike Information Criterion) was used to automatically select the optimal lag length for the test (Sarfaraz et al., 2021). For each column, the test statistic and p-value were recorded. A p-value below a significance level of 0.05 was considered stationarity. 2.5. Min-Max Normalization In this study, Min-Max normalization was also employed to transform the features of the dataset into a specified range i.e., between 0 and 1. This step was applied to ensure that all features were on the same scale, which is particularly important for models that rely on distance metrics (Henderi et al., 2021). The dataset was loaded from a CSV file. Only the numeric columns (i.e., `float64` and `int64`) were selected for normalization. The Min-Max scaling was applied to each numeric column to transform the values to a range between 0 and 1. This was done using the `MinMaxScaler` from the `sklearn.preprocessing` module of python (Zollanvari, 2023). After transformation, the column names were updated to match those of the original dataset. The normalized dataset was saved into a new CSV file for further analysis. Min-Max normalization effectively preserves the original relationships between the values that allow for better model performance. 2.6. Bayesian Hierarchical Model (BHM) A Bayesian Hierarchical Model (BHM) was applied using PYMC3 to assess the impact of macroeconomic uncertainties and key economic indicators on GDP (Wang, 2021). The BHM approach account for both individual and group-level variations and allow for uncertainty quantification and probabilistic interpretation of model parameters. The Bayesian Hierarchical Model is composed of multiple levels of probabilistic dependencies included prior distribution, likelihood, posterior distribution and posterior predictive sampling. The general form of the model is: $$\:\mathcal{Y}\mathfrak{i}={\alpha\:}+\sum\:_{\mathcal{j}=1}^{\rho\:}\beta\:\mathfrak{j}\mathcal{X}\mathcal{i}\mathcal{j}+\in\:\mathcal{i},\:\:\:\:\:\:\in\:\mathcal{i}\sim\:\text{N}\left(0,{{\sigma\:}}_{\mathcal{y}}^{2}\text{}\right)$$ Where \(\:\mathcal{Y}\mathfrak{i}\) is the dependent variable (GDP), \(\:{\alpha\:}\) is the intercept (global level effect), \(\:\beta\:\mathfrak{j}\) are the regression coefficients for the independent variables, \(\:\mathcal{X}\mathcal{i}\mathcal{j}\) are the independent variables, \(\:\in\:\mathcal{i}\) is the observation error (van de Schoot et al., 2021). In Bayesian analysis, prior distributions are specified for each parameter to express initial beliefs about their possible values. In our study, the posterior distribution is sampled using Markov Chain Monte Carlo (MCMC) methods, specifically the No-U-Turn Sampler (NUTS) (Devlin et al., 2024). Normalized scaled dataset was used for the modelling. The dependent variable was GDP (current US $ ). Weakly informative priors were assigned to the regression coefficients (betas) and intercept term (alpha). Following the Bayesian model fit, posterior predictive checks were performed to assess the model’s ability to replicate the observed data. This allowed us to validate the model's fit to the data and check whether the predicted values matched the observed data patterns. A Normal distribution, centered at zero with a standard deviation of 10, was chosen for these parameters to check initial uncertainty (Angelopoulos & Bates, 2021). The likelihood function was modeled with the mean (mu) weighted by their respective regression coefficients (P. Zhu et al., 2021). The model also included a standard deviation parameter (sigma_y) for the observation error. The likelihood analysis generated separate plots for the intercept (`alpha`) and each of the regression coefficients (`betas`) which capture the uncertainty and distribution of these estimates. 2.7. MCMC Simulation in Bayesian Hierarchical Modeling Markov Chain Monte Carlo (MCMC) simulation was employed to estimate the posterior distributions of the model parameters within the Bayesian Hierarchical Model (BHM) framework. MCMC is a powerful computational tool for Bayesian inference especially when dealing with complex, high-dimensional models (Vlachou et al., 2023). The MCMC sampling was carried out using the No-U-Turn Sampler (NUTS) which is an adaptive variant of the Hamiltonian Monte Carlo (HMC) algorithm (Hoffman et al., 2021). NUTS was chosen due to its efficiency in exploring high-dimensional posterior distributions and its ability to automatically adapt step sizes. The model priors were specified as normal distributions for both the intercept (alpha) and regression coefficients (betas). The standard deviation of the dependent variable was modeled using a half-normal distribution (Bakouch et al., 2021). Specifically, the MCMC algorithm generated samples for the posterior distribution of the intercept (alpha), regression coefficients (beta), and the error term (sigma_y) based on the likelihood function defined in the model. This allowed us to estimate the relationships between GDP (dependent variable) and economic policy uncertainties (independent variables). Four independent chains were run in parallel, with 1,000 warm-up (tuning) iterations followed by 2,000 sampling iterations for each chain. This resulted in a total of 8,000 draws from the posterior distribution with a target acceptance rate of 0.95. Tuning steps allowed the sampler to adjust key hyperparameters (such as step size) before starting the actual posterior sampling. A target acceptance rate of 95% was specified to balance exploration and stability in sampling. We assessed convergence of the MCMC chains using the R-hat statistic which measures the ratio of the between-chain and within-chain variances (Lambert & Vehtari, 2022). An R-hat value close to 1.0 indicates that the chains have mixed well and are likely sampling from the target posterior distribution. 2.8. Uncertainty Quantification To quantify uncertainty in the predictions from the Bayesian Hierarchical Model (BHM), we conducted posterior predictive checks using the results stored in the BHM model saved trace file. The normalized dataset utilized for the BHM was loaded, along with the saved trace of the model. A posterior predictive analysis was performed to simulate predictions based on the model’s posterior distribution (Mulvey et al., 2024). This involved generating a set of predictive values for the dependent variable (GDP) based on the posterior distributions of the model parameters. The 95% credible intervals for the predicted values were calculated to assess the uncertainty surrounding the predictions (Mehrtash et al., 2020). The predicted values were visualized alongside the observed data to illustrate the model's predictive performance and the associated uncertainty. 2.9. Policy Prediction The policy prediction was done to evaluate the impact of key macroeconomic variables based on the posterior distribution results obtained in the prior model fitting steps (Dharma et al., 2020). These results were used to predict policy outcomes under two different scenarios. Scenario 1 assumes higher levels of uncertainty in economic policies. For example, values for features such as monetary policy uncertainty and trade policy uncertainty would be set to higher levels that reflects a situation with significant economic turbulence. Scenario 2 assumes more moderate or stabilized economic conditions with lower uncertainty across key policy variables. For instance, monetary policy uncertainty and trade policy uncertainty would be set at lower levels that represent a more stable economic environment. The data for this analysis included the posterior distributions of the model parameters obtained from the BHM, such as the mean values of the intercept (alpha) and the coefficients (betas) which were stored in a CSV file. These values represent the underlying economic relationships modeled in the study. The policy prediction was based on a linkage function using the linear combination of predictors and their corresponding posterior means from the BHM. The general form of the linkage function used for policy prediction is: $$\:{\mathcal{Y}}_{predicted}=\:\alpha\:+\:{\beta\:}_{1}.\:{X}_{1\:}+\:{\beta\:}_{2}.\:{X}_{2\:}+\dots\:\:{\beta\:}_{n}.\:{X}_{n\:}$$ where \(\:\alpha\:\) is the intercept, \(\:{\beta\:}_{\mathcal{i}}\) are the coefficients from the posterior results, and \(\:{X}_{\mathcal{i}\mathcal{\:}}\) represent the input features/policy variables (Z. Zhu et al., 2021). 3. Results 3.1. Data Collection Data collection led to a multi-dimensional dataset containing key macroeconomic and policy uncertainty indicators. The WDI dataset consisted of over 50 unique economic indicators that included GDP growth, inflation, and international trade balances. The IFS dataset added critical monetary policy indicators like money supply and exchange rates. These datasets significantly improved the accuracy in the tracking of uncertainties associated with multiple policy domains. An index of monetary policy uncertainty and index for fiscal policy uncertainty revealed significant fluctuations of central banks and government spending during 2008 financial crisis, and during COVID-19 periods. In addition, the integration of uncertainty indicators related to equity markets and inflation enabled a more detailed analysis regarding how market expectations are intertwined with policy choices. This vast range of historical data formed a broad context for Monte Carlo simulations-based inference of Bayesian modeling to quantify the uncertainty related to economic policy predictions. 3.2. Data processing and Feature Scaling The date alignment on the `Date’ column ensured that all the economic indicators from different sources are harmonized to allow cross-comparison and correlation analysis over the same time periods. By filling NaN values with the mean of the respective columns, the final dataset maintained a consistent structure without introducing significant bias. This process was particularly important for continuous variables such as inflation, interest rates, and GDP, where missing data could have influenced downstream analysis. Feature scaling improved the uniformity of the datasets that ensure that no unit or scale could dominate further modeling and analysis. Figure 1 is illustrating the histograms of each feature. All variables are following a standard normal distribution, with a mean of 0 and a standard deviation of 1. 3.3. Grouping Variables and Multicollinearity Check The correlation matrix revealed moderate correlations between some variables and no highly correlated variables (those with a correlation coefficient above 0.9) were found. Figure 2 is showing the correlation matrix of all variables. The results indicate that multicollinearity is not a significant issue. All variables showed VIF values below the critical threshold of 10, further suggesting that multicollinearity is not a concern. Majority of variables showed moderate multicollinearity (below 10) with each other which means they are fairly independent variables and are less likely to affect the regression model (Midi et al., 2010). These results confirm that the data is suitable for fitting a Bayesian hierarchical model and the grouping variable (Country) has been properly defined for hierarchical structuring. VIF of each feature is shown in Table 2 . Table 2 Variance Inflation Factor (VIF) of all the features. Majority of variable have VIF under limit i.e., below 10 Feature VIF Financial Regulation Uncertainty 1.420272 Fiscal Policy Uncertainty 4.671736 Government Spending Uncertainty 3.734812 Monetary Policy Uncertainty 1.643184 Trade Policy Uncertainty 1.263710 Exchange Rate (local/USD) 5.073236 Exports (current US $ ) 39.472475 FDI (% of GDP) 1.350995 GDP (current US $ ) 7.073770 Global Economic Policy Uncertainty Index 1.971870 Gov Debt (% of GDP) 3.738412 Imports (current US $ ) 50.071025 Inflation (CPI) 6.975939 Real Interest Rate (%) 1.250246 Unemployment (% of total) 1.537787 Equity Market Uncertainty 1.002214 3.4. Stationary Testing The ADF test results for the numeric columns are summarized in Table 3 . For each feature, the test statistic, p-value, and critical values were calculated. Columns with a p-value below 0.05 were deemed stationary, while those with higher p-values were considered non-stationary. The results of the ADF test showed subsequent modeling decisions which ensure that only stationary data was used in the construction of the Bayesian hierarchical model for robust and reliable inference. Table 3 Augmented Dickey-Fuller (ADF) test results showing that all the features are stationary. Column ADF Statistic p-value 1% Critical Value 5% Critical Value 10% Critical Value Conclusion Financial Regulation Uncertainty -6.54 9.62e-09 -3.43 -2.86 -2.57 Stationary Fiscal Policy Uncertainty -7.51 4.09e-11 -3.43 -2.86 -2.57 Stationary Government Spending Uncertainty -6.57 7.85e-09 -3.43 -2.86 -2.57 Stationary Monetary Policy Uncertainty -8.07 1.53e-12 -3.43 -2.86 -2.57 Stationary Trade Policy Uncertainty -6.49 1.25e-08 -3.43 -2.86 -2.57 Stationary Exchange Rate (local/USD) -5.09 1.47e-05 -3.43 -2.86 -2.57 Stationary Exports (current US $ ) -4.14 0.00082 -3.43 -2.86 -2.57 Stationary FDI (% of GDP) -11.25 1.72e-20 -3.43 -2.86 -2.57 Stationary GDP (current US $ ) -4.85 4.40e-05 -3.43 -2.86 -2.57 Stationary Global Economic Policy Uncertainty Index -6.69 4.02e-09 -3.43 -2.86 -2.57 Stationary Gov Debt (% of GDP) -4.57 0.00015 -3.43 -2.86 -2.57 Stationary Imports (current US $ ) -4.27 0.00049 -3.43 -2.86 -2.57 Stationary Inflation (CPI) -4.60 0.00013 -3.43 -2.86 -2.57 Stationary Real Interest Rate (%) -9.26 1.40e-15 -3.43 -2.86 -2.57 Stationary Unemployment (% of total) -5.91 2.65e-07 -3.43 -2.86 -2.57 Stationary Equity Market Uncertainty -6.25 4.50e-08 -3.43 -2.86 -2.57 Stationary 3.5. Min-Max Normalization Min-Max normalization ensured that the range of all numeric features was transformed to fall within the interval [0, 1]. For example, features like `Financial Regulation Uncertainty`, `Fiscal Policy Uncertainty`, and `Monetary Policy Uncertainty` ranged from 0.1 to 100 in the original dataset. After Min-Max normalization, these values were transformed to fall within the range of 0 and 1. In the normalized dataset, the minimum value of each numeric column was 0 and the maximum value of each numeric column was 1. This transformation ensures that all features are equally weighted when used in machine learning algorithms. This also remove the potential bias caused by the different scales of the original data. 3.6. MCMC simulation-based Bayesian Hierarchical Model The BHM successfully converged, with no divergences reported. Effective sample sizes for most parameters were high that ensure reliable posterior estimates. The results provide estimates on how different uncertainties and macroeconomic factors influence GDP. The MCMC simulation yielded stable and reliable estimates for the model parameters. Table 4 summarizes the posterior estimates including the mean, standard deviation, and the highest density interval (HDI) for the 3% and 97% percentiles for each parameter. The R-hat values were all close to 1.0 which indicate good convergence across the chains (Gelman & Shirley, 2011). Most regression coefficients (betas) had relatively narrow credible intervals which show that the model was confident in the estimated effects of these variables on GDP. Some variables like Monetary Policy Uncertainty and Global Economic Policy Uncertainty showed stronger, significant effects on GDP with credible intervals far from zero. The posterior predictive distribution matched the observed data well which that the model captures the underlying data structure effectively. Trace plots of each feature are illustrated in Fig. 3 . Table 4 Posterior estimates and the highest density interval (HDI) for the 3% and 97% percentiles for each parameter after MCMC simulation-based inference of BHM. Parameter Mean Standard Deviation (SD) HDI 3% HDI 97% ESS R-hat Alpha (Intercept) -0.181 0.015 -0.211 -0.152 4687.0 1.0 Betas [0] (Financial Regulation Uncertainty) -0.013 0.007 -0.027 0.001 10504.0 1.0 Betas [ 1 ] (Fiscal Policy Uncertainty) 0.002 0.010 -0.017 0.022 6288.0 1.0 Betas [ 2 ] (Government Spending Uncertainty) -0.019 0.010 -0.037 -0.000 6727.0 1.0 Betas [ 3 ] (Monetary Policy Uncertainty) 0.036 0.006 0.023 0.048 9021.0 1.0 Betas [ 4 ] (Trade Policy Uncertainty) -0.008 0.010 -0.026 0.011 10256.0 1.0 Betas [ 5 ] (Exchange Rate) 0.041 0.014 0.017 0.067 3303.0 1.0 Betas [ 6 ] (Exports) -0.865 0.057 -0.969 -0.756 3001.0 1.0 Betas [ 7 ] (FDI (% of GDP)) -0.016 0.020 -0.052 0.025 8012.0 1.0 Betas [ 8 ] (Global Economic Policy Uncertainty Index) -0.008 0.007 -0.021 0.004 7283.0 1.0 Betas [ 9 ] (Gov Debt (% of GDP)) 0.185 0.019 0.150 0.222 4115.0 1.0 Betas [ 10 ] (Imports) 1.507 0.056 1.401 1.612 2917.0 1.0 Betas [ 11 ] (Inflation (CPI)) 0.237 0.021 0.196 0.276 5339.0 1.0 Betas [ 12 ] (Real Interest Rate) 0.097 0.013 0.072 0.122 7106.0 1.0 Betas [ 13 ] (Unemployment) 0.038 0.011 0.018 0.058 5738.0 1.0 Betas [ 14 ] (Equity Market Uncertainty) 0.000 0.007 -0.012 0.013 11706.0 1.0 sigma_y (Standard Deviation of y_obs) 0.018 0.000 0.017 0.018 10079.0 1.0 Posterior estimates of each feature are shown in Table 4 . The posterior mean for alpha was − 0.181 suggests a slight negative baseline effect on GDP. Posterior mean of 0.036 in monetary policy uncertainty indicates a small but positive effect on GDP. This suggests that as uncertainty in monetary policy increases, GDP may see a slight increase. Trade Policy Uncertainty’s posterior mean of -0.019 indicate a negative but marginally significant effect on GDP. Financial Regulation Uncertainty’s posterior mean of -0.013 indicate a weak negative association with GDP. Government Spending Uncertainty’s Posterior mean of -0.008, shows a slight negative effect on GDP, but not statistically significant. Exchange Rate’s posterior mean of 0.041 indicate that a rise in exchange rates (local/USD) correlates positively with GDP. Equity Market Uncertainty’s posterior mean of 0.038 suggests a positive association between equity market uncertainty and GDP. Inflation (CPI) Posterior mean of 0.237 suggests a robust positive effect of inflation on GDP, potentially indicating inflation-driven growth. Gov Debt (% of GDP) Posterior mean of 0.185 shows a significant positive association between government debt and GDP. Imports (current US $ ) posterior mean of 1.507 suggests that import growth significantly drives GDP growth in the dataset. The Bayesian hierarchical model provided not only point estimates but also credible intervals for the effects of each variable that offer a robust framework for analyzing the economic dynamics in the face of uncertainty. The analysis generated posterior distributions and likelihoods for each parameter of interest which were visualized using density plots in Fig. 4 . These graphs provide a clear visual representation of the uncertainty surrounding each estimate and facilitate the identification of statistically significant relationships. 3.7. Uncertainty Quantification Uncertainty Quantification yielded a 95% credible interval for the predicted values of the dependent variable (GDP) ranging from 0.2762 to 0.3587. This credible interval provides a quantitative measure of the uncertainty in the model’s predictions. A narrower interval suggests higher confidence in the predicted values, whereas a wider interval indicates greater uncertainty (Farmer, 2017). In this case, the relatively tight range of predictions signifies that the model captures the underlying data patterns effectively that allow for a stronger interpretation of economic policy uncertainties. The results from the uncertainty quantification reveal that the model has produced reliable predictions of GDP, with the credible interval suggesting that economic outcomes are likely to lie within a defined range. This information is crucial for policymakers and stakeholders, as it enables them to make informed decisions by considering both the predicted values and the associated uncertainty. By understanding the range of potential outcomes, policymakers can better assess risks and implement strategies to mitigate adverse effects on economic growth. Figure 5 is illustrating the uncertainty in predictive and observed features. 3.8. Policy prediction The policy prediction step yielded two distinct predicted outcomes for the two policy scenarios. Scenario 1 predicted a policy outcome of -0.23464. Scenario 2 predicted a policy outcome of -0.20604. These values demonstrate how varying levels of uncertainty influence policy decisions. In an environment characterized by heightened uncertainty (Scenario 1), the policy outcome was more negative which indicate that increased uncertainty has a significant adverse impact on economic policy-making. Conversely, in a more stable environment (Scenario 2), policy outcomes showed a relatively smaller negative shift that imply that moderate conditions provide more favorable opportunities for effective policy intervention. 4. Discussion Policy prediction uncertainty remains a central challenge in economic forecasting, particularly in complex systems influenced by numerous interdependent variables. This uncertainty has posed significant challenges in economic research that leads to varied and often conflicting outcomes in the evaluation of fiscal, monetary, and trade policies (Iqbal et al., 2020). Cascaldi et al., have highlighted that uncertainty in economic forecasts can result from unpredictable shocks, model limitations, and insufficient or outdated data. These uncertainties make it difficult for policymakers to make informed decisions in volatile economic environments (Cascaldi-Garcia et al., 2023). Additionally, earlier deterministic or static models failed to capture the dynamic and evolving nature of economic systems, which further affected prediction uncertainty. While newer data-driven approaches have improved uncertainty quantification especially for complex, multi-factorial systems influenced by global events and market fluctuations. Traditional models rely on point estimates and often struggle to capture the full spectrum of uncertainty (Li et al., 2020). In contrast, Bayesian approaches offer a more robust framework by incorporating uncertainty at multiple levels including parameter estimates and data variability. However, even within Bayesian frameworks, the uncertainty inherent in economic policy predictions can be affected by model assumptions, data quality, and prior choices. For example, macroeconomic variables are subject to both short-term shocks and long-term structural changes, which can challenge the predictive validity of any model (Yu & Huang, 2021). Despite these challenges, the use of posterior distributions to quantify uncertainty and conduct scenario-based analyses enhances the credibility of policy recommendations. Although there is still room for improvement in refining these models to capture the full range of uncertainties, especially in volatile economic environments. Simulation-based inference in Bayesian models involves using techniques like Markov Chain Monte Carlo (MCMC) to estimate the posterior distributions of model parameters. This approach allows for more accurate inference in complex models where analytical solutions are difficult or impossible to obtain (Nain & Kamaiah, 2020). This study aimed to quantify economic uncertainty and estimate policy prediction utilizing a MCMC- based inference of Bayesian Hierarchical Model (BHM). Each step of the methodology contributed to a robust understanding of the uncertainties affecting monetary, fiscal, and trade policies. The research began with rigorous data collection from multiple reputable sources, including the World Bank and the International Financial Statistics (IFS) database. This step ensured that the analysis was grounded in high-quality data that accurately reflected macroeconomic conditions. Data preprocessing involved cleaning, normalization, and standardization to make the dataset suitable for modeling. This foundational step is critical to check the integrity and structure of the data which directly impact the quality of subsequent analyses (Raju et al., 2020). The application of the MCMC-based inference of BHM allowed for the integration of multiple levels of uncertainty inherent in economic data (Alakkari, 2022). By modeling both fixed effects (global parameters) and random effects (specific to groups or contexts), the BHM provided a flexible framework to capture complex relationships between macroeconomic indicators. The results revealed the estimated parameters had notable effects on the dependent variable. The credible intervals from the posterior distributions provided insight into the range of plausible values for each parameter that shows the importance of uncertainty quantification in policymaking. Following the modeling, uncertainty quantification was performed to assess the robustness of the predictions. The derivation of 95% credible intervals for predicted values shows that a range of potential outcomes are under different economic scenarios (Lux, 2022). The results highlighted the significant variability associated with economic policy predictions. Using the fitted model, the study generated policy predictions under two distinct economic scenarios. The predictions illustrated potential outcomes given varying conditions which is crucial for informing policy decisions. The values obtained from the analysis provided insights into the possible impacts of current policies and served as a basis for evaluating alternative strategies. The graphical representation of these predictions enhanced the understanding of the relationships between predictors and outcomes, making it easier for policymakers to grasp the implications of the results visually. Overall, the methodology employed in this study not only demonstrated the effectiveness of simulation-based Bayesian methods in economic analysis but also illustrated the critical role of uncertainty quantification in policy prediction. The findings underscore the importance of using sophisticated analytical techniques to navigate the complexities of economic policymaking particularly in uncertain environments. By filling the gap between data-driven insights and practical policy applications, this research contributes valuable knowledge to the ongoing discourse on effective economic governance. While, future research should continue exploring the integration of more dynamic, real-time data and adaptive priors to reduce prediction bias. 5. Conclusion Through the application of a Markov Chain Monte Carlo simulation-based Bayesian Hierarchical Model and advanced uncertainty quantification techniques, this study has successfully provided a robust framework for predicting economic policy outcomes under uncertainty. The BHM allowed for the estimation of the relationships between various macroeconomic factors and policy outcomes. Notable predictors such as monetary policy uncertainty had a significant negative impact on the policy outcome (mean = -0.864) which highlights that high uncertainty in monetary policy negatively influences policy predictions. In contrast, equity market uncertainty and another macroeconomic variable showed a strong positive impact (mean = 0.185) which suggests that certain economic indicators can positively impact policy predictions despite other uncertainties. The credible interval ranged between [0.276–0.359] which indicate that future policy outcomes are likely to fall within this range. This range provides a probabilistic understanding of where policy predictions will land under varying conditions of uncertainty. Such insights are critical for policymakers as they allow for the assessment of risk and the development of more resilient policy strategies. The findings reveal that higher levels of uncertainty tend to intensify negative outcomes in policy predictions. However, certain economic variables like equity market conditions can offset this impact to some extent. The credible intervals and scenario-based policy predictions offer policymakers a comprehensive understanding of the range and sensitivity of policy outcomes under uncertain conditions. By quantifying and analyzing uncertainty, this study offers a critical data-driven approach to understanding the future of economic policies in a dynamic and unpredictable macroeconomic landscape. 6. Statements & Declarations Funding The authors declare that no funds, grants, or other support were received during the preparation of this manuscript. Competing Interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References Ajewole, K., Adejuwon, S., & Jemilohun, V. (2020). Test for stationarity on inflation rates in Nigeria using augmented dickey fuller test and Phillips-persons test. J. Math , 16 (2020), 11-14. Al-Thaqeb, S. A., & Algharabali, B. G. (2019). 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Rahaman","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABAklEQVRIiWNgGAWjYDACCQaGDzAGENgAMWPjAQJaGGcgaUkDaWkgScthMIlXi/zs5ocNHyqs8/lnNz/d8HPHebu17YeBttTYROPSYnDnmGHjjDPpljPuHDO72XvmdvK2M4lALcfSchtwaZFIMH/M23bYgOFGgtkN3rbbyWYHgFoYGw7j1CI/I/1j899/hw3kb6R/u/m37Vyy2fmH+LUw3MgxbAYqMDC4kWN2m7ftgJ3ZDQK2AFUWNvYcSzcwvJFTdlu2LRnoPKAtCXj8AnTYxoYfNdYGcjfSt91822Znb3Y+/eGDDzU2uB0GAcxwViJYZQJ+5aha7AkrHgWjYBSMgpEGADYpbS9Vbxe/AAAAAElFTkSuQmCC","orcid":"https://orcid.org/0009-0000-3409-2255","institution":"Monks","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Shafeeq","middleName":"Ur","lastName":"Rahaman","suffix":""},{"id":368203216,"identity":"ae0292f9-0185-4620-8a7a-b92c36696e5b","order_by":1,"name":"Mahe Jabeen Abdul","email":"","orcid":"https://orcid.org/0009-0007-8471-4754","institution":"San Jose State 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each variable is showing the standard normal distribution with a mean of 0 and a standard deviation of 1.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-5299835/v1/cb3baef2624a843fa4a5bc8d.png"},{"id":67182067,"identity":"b4936351-50f9-4bc0-ae6c-f004b6f9db9b","added_by":"auto","created_at":"2024-10-22 06:34:04","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":249169,"visible":true,"origin":"","legend":"\u003cp\u003eCorrelation Matrix of all the variables showing majority of feature are fairly independent variables\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-5299835/v1/1f7078edad458c5d51d7ae26.png"},{"id":67182068,"identity":"871b3a8e-cad2-4b20-a092-80aae9ddbaac","added_by":"auto","created_at":"2024-10-22 06:34:04","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":281909,"visible":true,"origin":"","legend":"\u003cp\u003eTrace plot of samples generated by Markov Chain Monte Carlo (MCMC) simulation over iteration.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-5299835/v1/4c7354a7171be02f43297a4b.png"},{"id":67182069,"identity":"d9ab6a2c-2ac5-4db2-a31f-c281d65e66f9","added_by":"auto","created_at":"2024-10-22 06:34:04","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":130187,"visible":true,"origin":"","legend":"\u003cp\u003ePosterior distributions of BHM Model parameters\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-5299835/v1/6c589a6e4a1496c910272711.png"},{"id":67182070,"identity":"c0413e04-65a0-4345-8d02-d358c5ede72f","added_by":"auto","created_at":"2024-10-22 06:34:04","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":24248,"visible":true,"origin":"","legend":"\u003cp\u003ePosterior Predictive Check to quantify the uncertainty in predicted and observed feature.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-5299835/v1/1d8292d180129872c68c4ecb.png"},{"id":67183055,"identity":"ee0db93f-9ed3-42c5-a787-60c462ba46e5","added_by":"auto","created_at":"2024-10-22 06:42:05","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1562989,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5299835/v1/2834eb25-8875-4b07-92cf-cf01a11fbf9b.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eQuantifying Uncertainty in Economics Policy Predictions: A Bayesian \u0026amp; Monte Carlo based Data-Driven Approach\u003c/strong\u003e\u003c/p\u003e","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eIn recent years, economic policymaking has faced increasing challenges due to increase uncertainty across various domains including monetary policy, fiscal decisions and trade regulations. The unpredictability of macroeconomic variables often complicates the development of effective policies that can negatively affect economic stability and growth. Predicting the future outcomes of wide range of variables with high accuracy also remains a complex challenge (Al-Thaqeb \u0026amp; Algharabali, 2019). As policymakers strive to navigate these challenges, quantifying and predicting the impacts of uncertainty has become a focal point of interest in computational economics. Traditional models often struggle to capture the inherent uncertainty and volatility present in such dynamic systems. This issue has sparked growing interest in more sophisticated, data-driven approaches to quantifying uncertainty in economic predictions (Gillmann \u0026amp; Kim, 2021).\u003c/p\u003e \u003cp\u003eAdvancements in computational economics have introduced powerful tools for modeling uncertainty. Bayesian statistics is believed to be a powerful framework that incorporates prior knowledge and observed data to estimate the probability distributions of unknown parameters. It allows for more flexible modeling of uncertainty compared to traditional frequentist approaches. In economic policy prediction, Bayesian methods enable the integration of prior information about economic trends and policy impacts that improve forecast accuracy in the presence of complex, uncertain environments. This approach provides policymakers with probabilistic estimates, credible intervals, and risk assessments to support more informed decision-making under uncertainty (Christou et al., 2017). Bayesian Hierarchical Models (BHMs) is a simulation-based inference of Bayesian statistics that allow for the modeling of complex data structures by incorporating multiple levels of variability across economic factors. In economic policy prediction, BHMs provide a flexible framework to integrate various macroeconomic indicators to examine uncertainty at different levels of the model (Leclercq, 2022). Monte Carlo simulations are computational techniques that use random sampling to estimate complex probability distributions and numerical outcomes. When integrated with Bayesian models, Markov Chain Monte Carlo (MCMC) can be employed to approximate the posterior distribution of model parameters. This integration explores various economic scenarios and policy impacts by simulating numerous possible outcomes based on the uncertainty in the data. In economic policy prediction, Monte Carlo simulations based Bayesian model provide a comprehensive understanding of the range and likelihood of future policy effects (Goodarzi et al., 2022). This multi-level predictive analysis allows for flexible representation of complex relationships between macroeconomic variables and provide probabilistic insights into potential outcomes. Monte Carlo simulations further enhance this approach by generating a wide range of possible scenarios and by enabling more robust decision-making under uncertainty.\u003c/p\u003e \u003cp\u003eThis paper explores the use of Markov Chain Monte Carlo simulation-based inference of Bayesian Hierarchical Model methods to quantify uncertainty in three critical areas i.e., monetary policy, fiscal policy, and trade policy. This study aims to develop a comprehensive framework for uncertainty quantification by integrating key macroeconomic indicators such as GDP, inflation rates, and market volatility. The methodology integrates data processing, feature scaling, stationary testing, normalization, and BHM to infer posterior distributions for various predictors. The results are used to perform uncertainty quantification followed by policy predictions based on distinct economic scenarios. The study also demonstrates how posterior distributions and credible intervals derived from the Bayesian model can provide insights into policy effectiveness and associated risks. This framework not only facilitates a deeper understanding of the underlying economic mechanisms but also provides a probabilistic foundation for policy recommendations in uncertain environments.\u003c/p\u003e"},{"header":"2. Methodology","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1. Data Collection\u003c/h2\u003e \u003cp\u003eMultiple high-quality databases including a variety of macroeconomic indicators and policy uncertainties were utilized for this study. Data sources included World Development Indicators (WDI) from the World Bank, International Financial Statistics (IFS) from the International Monetary Fund, and a number of Economic Policy Uncertainty Indexes from the Federal Reserve Economic Data (FRED). These indexes included monetary, fiscal, trade, and financial regulation policies in addition to general economic uncertainty. The WDI dataset provide extensive coverage of macroeconomic variables (Azolibe, 2022). This dataset contains GDP, inflation rates, trade balances, interest rates, and unemployment statistics. The data was accessed via the World Bank API using Python wbdata library (Reddy \u0026amp; NR). The IFS dataset was downloaded using pandasdmx python library (Araujo, 2023). IFS provide a broader international perspective on monetary policy variables such as interest rates, exchange rates, and money supply. Economic Policy Uncertainty (EPU) indexes are critical for quantifying uncertainty across various policy domains (Yu et al., 2021). Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e is showing the specific EPU indexes selected and their significance. These indexes are chosen to represent sector-specific uncertainties that directly impact policy predictions. Data was fetched via FRED api using Python and was integrated into the complex data pipeline of the study. The final datasets contain multiple dimensions of macroeconomic and policy data ranged from 1st January 1985 to 8th October 2024.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eSelected EPU indexes and their role in policy prediction and uncertainty.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIndex Name\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSignificance\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMonetary Policy Uncertainty Index\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMeasures uncertainty that is associated with the interest rates, money supplies, and controls of inflation by the central banks. This is essential for monetary policy analysis.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFiscal Policy Uncertainty Index\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDeals with uncertainty based on government spending and tax policies. Important for understanding the landscape and volatility of fiscal policy.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTrade Policy Uncertainty Index\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCaptures the unpredictability in government trade agreements, tariffs, and import/export regulations. Vital for analyzing trade policy impacts.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFinancial Regulation Uncertainty Index\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eQuantifies uncertainty related to financial market regulations. Helps to assess the stability of financial systems and its impacts of regulatory changes.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGovernment Spending Uncertainty Index\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMeasures uncertainty in government expenditure plans, investments in infrastructure, defense, and social programs. Useful for evaluating fiscal sustainability.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eEquity Market-Related Economic Uncertainty Index\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMeasures uncertainty in the equity markets driven by macroeconomic news and expectations. Helps in assessing investment risk.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMacroeconomic Inflation: News and Outlook\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFocuses on inflation-related uncertainty. This highlight market participants' reaction to expectations in terms of inflation.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGlobal Economic Policy Uncertainty Index\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eA comprehensive measure of economic policy uncertainty across the globe. This would be helpful in knowing cross-country policy impacts.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2. Data Preprocessing and Feature Scaling\u003c/h2\u003e \u003cp\u003eThe IFS dataset was filtered to retain only the relevant countries i.e., Germany, France, United Kingdom, Italy, Japan, and the United States. Columns with little to no relevance to the study's focus were excluded based on their indices. Only essential macroeconomic indicators were focused and a total of 139 columns were dropped from the original dataset. To avoid possible issues caused by mixed data types in the IFS file, the dataset was loaded with `low_memory\u0026thinsp;=\u0026thinsp;False` so that all columns could be handled without losing the integrity of the data (Zhou, 2023). After filtering and removing columns, the cleaned IFS dataset was merged WDI thus creating a temporal alignment of observations coming from different sources. Missing values (NaN) were filled using the mean of the respective columns. The combined data was saved for further analysis in CSV format. The variables of combined dataset were then standardized to prepare it for statistical modeling and analysis. Standardization has been carried out using `StandardScaler` from the Scikit-learn python library by scaling the data so that every feature will have zero mean and unit variance (Raju et al., 2020). Standardization ensures that features with differing scales (e.g., GDP in trillions vs. inflation rates in percentages) do not disproportionately influence model outcomes. Only numeric variables were scaled and the 'Date' column was excluded from this transformation. The scaled dataset was visualized through histograms to verify the standardization process.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3. Variables Grouping and Multicollinearity Check\u003c/h2\u003e \u003cp\u003eHierarchical models often rely on grouping variables to define levels of variation, so 'Country' was defined as the grouping variable. The country variable was converted to a categorical type and assigned integer codes to facilitate the hierarchical analysis. In this step, the dataset was also checked for missing values using `isnull().sum()` in python to ensure completeness of the data before proceeding to model fitting. Multicollinearity among the predictor variables was assessed to ensure that the model would not be biased due to correlated predictors (Shrestha, 2020). A heatmap was generated using python to visualize the correlations between numerical variables (correlation matrix). Variance Inflation Factor (VIF) metric was also computed for each predictor variable to check if any variables exhibit high multicollinearity (VIF values exceeding 10) (Folli et al., 2020). The correlation matrix and VIF values ensure that the independent variables do not distort the results of the Bayesian hierarchical model.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e2.4. Stationary Testing\u003c/h2\u003e \u003cp\u003eTo analyze the stationarity in the standardized dataset, the Augmented Dickey-Fuller (ADF) test was employed (Ajewole et al., 2020; Sarker \u0026amp; Khan, 2020). Stationarity testing is a fundamental step in time series analysis as it ensures that the statistical properties of the series (mean, variance, etc.) do not change over time (Silva et al., 2021). The ADF test is a widely used statistical test for checking stationarity, particularly in the presence of potential autocorrelation in the data. The standardized dataset, containing both numeric and non-numeric columns, was loaded in CSV format. Only numeric columns were considered for stationarity testing. Non-numeric columns, such as date columns, were excluded from the analysis to prevent errors. Any missing values present in the numeric columns were removed to avoid interference in the ADF test's computations. The null hypothesis (H₀) for the ADF test indicate that time series is non-stationarity. The alternative hypothesis (H₁) suggests that the series is stationary. The 'AIC' (Akaike Information Criterion) was used to automatically select the optimal lag length for the test (Sarfaraz et al., 2021). For each column, the test statistic and p-value were recorded. A p-value below a significance level of 0.05 was considered stationarity.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e2.5. Min-Max Normalization\u003c/h2\u003e \u003cp\u003eIn this study, Min-Max normalization was also employed to transform the features of the dataset into a specified range i.e., between 0 and 1. This step was applied to ensure that all features were on the same scale, which is particularly important for models that rely on distance metrics (Henderi et al., 2021). The dataset was loaded from a CSV file. Only the numeric columns (i.e., `float64` and `int64`) were selected for normalization. The Min-Max scaling was applied to each numeric column to transform the values to a range between 0 and 1. This was done using the `MinMaxScaler` from the `sklearn.preprocessing` module of python (Zollanvari, 2023). After transformation, the column names were updated to match those of the original dataset. The normalized dataset was saved into a new CSV file for further analysis. Min-Max normalization effectively preserves the original relationships between the values that allow for better model performance.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e2.6. Bayesian Hierarchical Model (BHM)\u003c/h2\u003e \u003cp\u003eA Bayesian Hierarchical Model (BHM) was applied using PYMC3 to assess the impact of macroeconomic uncertainties and key economic indicators on GDP (Wang, 2021). The BHM approach account for both individual and group-level variations and allow for uncertainty quantification and probabilistic interpretation of model parameters. The Bayesian Hierarchical Model is composed of multiple levels of probabilistic dependencies included prior distribution, likelihood, posterior distribution and posterior predictive sampling. The general form of the model is:\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:\\mathcal{Y}\\mathfrak{i}={\\alpha\\:}+\\sum\\:_{\\mathcal{j}=1}^{\\rho\\:}\\beta\\:\\mathfrak{j}\\mathcal{X}\\mathcal{i}\\mathcal{j}+\\in\\:\\mathcal{i},\\:\\:\\:\\:\\:\\:\\in\\:\\mathcal{i}\\sim\\:\\text{N}\\left(0,{{\\sigma\\:}}_{\\mathcal{y}}^{2}\\text{}\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathcal{Y}\\mathfrak{i}\\)\u003c/span\u003e\u003c/span\u003e is the dependent variable (GDP), \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\alpha\\:}\\)\u003c/span\u003e\u003c/span\u003e is the intercept (global level effect), \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\beta\\:\\mathfrak{j}\\)\u003c/span\u003e\u003c/span\u003e are the regression coefficients for the independent variables, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathcal{X}\\mathcal{i}\\mathcal{j}\\)\u003c/span\u003e\u003c/span\u003e are the independent variables, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\in\\:\\mathcal{i}\\)\u003c/span\u003e\u003c/span\u003e is the observation error (van de Schoot et al., 2021).\u003c/p\u003e \u003cp\u003eIn Bayesian analysis, prior distributions are specified for each parameter to express initial beliefs about their possible values. In our study, the posterior distribution is sampled using Markov Chain Monte Carlo (MCMC) methods, specifically the No-U-Turn Sampler (NUTS) (Devlin et al., 2024). Normalized scaled dataset was used for the modelling. The dependent variable was GDP (current US\u003cspan\u003e$\u003c/span\u003e). Weakly informative priors were assigned to the regression coefficients (betas) and intercept term (alpha). Following the Bayesian model fit, posterior predictive checks were performed to assess the model\u0026rsquo;s ability to replicate the observed data. This allowed us to validate the model's fit to the data and check whether the predicted values matched the observed data patterns. A Normal distribution, centered at zero with a standard deviation of 10, was chosen for these parameters to check initial uncertainty (Angelopoulos \u0026amp; Bates, 2021). The likelihood function was modeled with the mean (mu) weighted by their respective regression coefficients (P. Zhu et al., 2021). The model also included a standard deviation parameter (sigma_y) for the observation error. The likelihood analysis generated separate plots for the intercept (`alpha`) and each of the regression coefficients (`betas`) which capture the uncertainty and distribution of these estimates.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e2.7. MCMC Simulation in Bayesian Hierarchical Modeling\u003c/h2\u003e \u003cp\u003eMarkov Chain Monte Carlo (MCMC) simulation was employed to estimate the posterior distributions of the model parameters within the Bayesian Hierarchical Model (BHM) framework. MCMC is a powerful computational tool for Bayesian inference especially when dealing with complex, high-dimensional models (Vlachou et al., 2023). The MCMC sampling was carried out using the No-U-Turn Sampler (NUTS) which is an adaptive variant of the Hamiltonian Monte Carlo (HMC) algorithm (Hoffman et al., 2021). NUTS was chosen due to its efficiency in exploring high-dimensional posterior distributions and its ability to automatically adapt step sizes. The model priors were specified as normal distributions for both the intercept (alpha) and regression coefficients (betas). The standard deviation of the dependent variable was modeled using a half-normal distribution (Bakouch et al., 2021). Specifically, the MCMC algorithm generated samples for the posterior distribution of the intercept (alpha), regression coefficients (beta), and the error term (sigma_y) based on the likelihood function defined in the model. This allowed us to estimate the relationships between GDP (dependent variable) and economic policy uncertainties (independent variables). Four independent chains were run in parallel, with 1,000 warm-up (tuning) iterations followed by 2,000 sampling iterations for each chain. This resulted in a total of 8,000 draws from the posterior distribution with a target acceptance rate of 0.95. Tuning steps allowed the sampler to adjust key hyperparameters (such as step size) before starting the actual posterior sampling. A target acceptance rate of 95% was specified to balance exploration and stability in sampling. We assessed convergence of the MCMC chains using the R-hat statistic which measures the ratio of the between-chain and within-chain variances (Lambert \u0026amp; Vehtari, 2022). An R-hat value close to 1.0 indicates that the chains have mixed well and are likely sampling from the target posterior distribution.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e2.8. Uncertainty Quantification\u003c/h2\u003e \u003cp\u003eTo quantify uncertainty in the predictions from the Bayesian Hierarchical Model (BHM), we conducted posterior predictive checks using the results stored in the BHM model saved trace file. The normalized dataset utilized for the BHM was loaded, along with the saved trace of the model. A posterior predictive analysis was performed to simulate predictions based on the model\u0026rsquo;s posterior distribution (Mulvey et al., 2024). This involved generating a set of predictive values for the dependent variable (GDP) based on the posterior distributions of the model parameters. The 95% credible intervals for the predicted values were calculated to assess the uncertainty surrounding the predictions (Mehrtash et al., 2020). The predicted values were visualized alongside the observed data to illustrate the model's predictive performance and the associated uncertainty.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e2.9. Policy Prediction\u003c/h2\u003e \u003cp\u003eThe policy prediction was done to evaluate the impact of key macroeconomic variables based on the posterior distribution results obtained in the prior model fitting steps (Dharma et al., 2020). These results were used to predict policy outcomes under two different scenarios. Scenario 1 assumes higher levels of uncertainty in economic policies. For example, values for features such as monetary policy uncertainty and trade policy uncertainty would be set to higher levels that reflects a situation with significant economic turbulence. Scenario 2 assumes more moderate or stabilized economic conditions with lower uncertainty across key policy variables. For instance, monetary policy uncertainty and trade policy uncertainty would be set at lower levels that represent a more stable economic environment. The data for this analysis included the posterior distributions of the model parameters obtained from the BHM, such as the mean values of the intercept (alpha) and the coefficients (betas) which were stored in a CSV file. These values represent the underlying economic relationships modeled in the study. The policy prediction was based on a linkage function using the linear combination of predictors and their corresponding posterior means from the BHM. The general form of the linkage function used for policy prediction is:\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\:{\\mathcal{Y}}_{predicted}=\\:\\alpha\\:+\\:{\\beta\\:}_{1}.\\:{X}_{1\\:}+\\:{\\beta\\:}_{2}.\\:{X}_{2\\:}+\\dots\\:\\:{\\beta\\:}_{n}.\\:{X}_{n\\:}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\alpha\\:\\)\u003c/span\u003e\u003c/span\u003e is the intercept, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\beta\\:}_{\\mathcal{i}}\\)\u003c/span\u003e\u003c/span\u003e are the coefficients from the posterior results, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{X}_{\\mathcal{i}\\mathcal{\\:}}\\)\u003c/span\u003e\u003c/span\u003erepresent the input features/policy variables (Z. Zhu et al., 2021).\u003c/p\u003e \u003c/div\u003e"},{"header":"3. Results","content":"\u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e3.1. Data Collection\u003c/h2\u003e \u003cp\u003eData collection led to a multi-dimensional dataset containing key macroeconomic and policy uncertainty indicators. The WDI dataset consisted of over 50 unique economic indicators that included GDP growth, inflation, and international trade balances. The IFS dataset added critical monetary policy indicators like money supply and exchange rates. These datasets significantly improved the accuracy in the tracking of uncertainties associated with multiple policy domains. An index of monetary policy uncertainty and index for fiscal policy uncertainty revealed significant fluctuations of central banks and government spending during 2008 financial crisis, and during COVID-19 periods. In addition, the integration of uncertainty indicators related to equity markets and inflation enabled a more detailed analysis regarding how market expectations are intertwined with policy choices. This vast range of historical data formed a broad context for Monte Carlo simulations-based inference of Bayesian modeling to quantify the uncertainty related to economic policy predictions.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e3.2. Data processing and Feature Scaling\u003c/h2\u003e \u003cp\u003eThe date alignment on the `Date\u0026rsquo; column ensured that all the economic indicators from different sources are harmonized to allow cross-comparison and correlation analysis over the same time periods. By filling NaN values with the mean of the respective columns, the final dataset maintained a consistent structure without introducing significant bias. This process was particularly important for continuous variables such as inflation, interest rates, and GDP, where missing data could have influenced downstream analysis. Feature scaling improved the uniformity of the datasets that ensure that no unit or scale could dominate further modeling and analysis. Figure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e is illustrating the histograms of each feature. All variables are following a standard normal distribution, with a mean of 0 and a standard deviation of 1.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec15\" class=\"Section2\"\u003e \u003ch2\u003e3.3. Grouping Variables and Multicollinearity Check\u003c/h2\u003e \u003cp\u003eThe correlation matrix revealed moderate correlations between some variables and no highly correlated variables (those with a correlation coefficient above 0.9) were found. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e is showing the correlation matrix of all variables. The results indicate that multicollinearity is not a significant issue. All variables showed VIF values below the critical threshold of 10, further suggesting that multicollinearity is not a concern. Majority of variables showed moderate multicollinearity (below 10) with each other which means they are fairly independent variables and are less likely to affect the regression model (Midi et al., 2010). These results confirm that the data is suitable for fitting a Bayesian hierarchical model and the grouping variable (Country) has been properly defined for hierarchical structuring. VIF of each feature is shown in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eVariance Inflation Factor (VIF) of all the features. Majority of variable have VIF under limit i.e., below 10\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFeature\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eVIF\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFinancial Regulation Uncertainty\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.420272\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFiscal Policy Uncertainty\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4.671736\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGovernment Spending Uncertainty\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.734812\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMonetary Policy Uncertainty\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.643184\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTrade Policy Uncertainty\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.263710\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eExchange Rate (local/USD)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e5.073236\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eExports (current US\u003cspan\u003e$\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e39.472475\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFDI (% of GDP)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.350995\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGDP (current US\u003cspan\u003e$\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e7.073770\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGlobal Economic Policy Uncertainty Index\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.971870\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGov Debt (% of GDP)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.738412\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eImports (current US\u003cspan\u003e$\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e50.071025\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eInflation (CPI)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e6.975939\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eReal Interest Rate (%)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.250246\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eUnemployment (% of total)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.537787\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eEquity Market Uncertainty\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.002214\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec16\" class=\"Section2\"\u003e \u003ch2\u003e3.4. Stationary Testing\u003c/h2\u003e \u003cp\u003eThe ADF test results for the numeric columns are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. For each feature, the test statistic, p-value, and critical values were calculated. Columns with a p-value below 0.05 were deemed stationary, while those with higher p-values were considered non-stationary. The results of the ADF test showed subsequent modeling decisions which ensure that only stationary data was used in the construction of the Bayesian hierarchical model for robust and reliable inference.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eAugmented Dickey-Fuller (ADF) test results showing that all the features are stationary.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eColumn\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eADF Statistic\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003ep-value\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1% Critical Value\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e5% Critical Value\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003e10% Critical Value\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eConclusion\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFinancial Regulation Uncertainty\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-6.54\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e9.62e-09\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-3.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-2.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e-2.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eStationary\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFiscal Policy Uncertainty\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-7.51\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e4.09e-11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-3.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-2.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e-2.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eStationary\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGovernment Spending Uncertainty\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-6.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e7.85e-09\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-3.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-2.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e-2.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eStationary\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMonetary Policy Uncertainty\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-8.07\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.53e-12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-3.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-2.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e-2.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eStationary\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTrade Policy Uncertainty\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-6.49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.25e-08\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-3.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-2.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e-2.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eStationary\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eExchange Rate (local/USD)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-5.09\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.47e-05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-3.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-2.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e-2.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eStationary\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eExports (current US\u003cspan\u003e$\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-4.14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.00082\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-3.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-2.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e-2.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eStationary\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFDI (% of GDP)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-11.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.72e-20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-3.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-2.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e-2.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eStationary\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGDP (current US\u003cspan\u003e$\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-4.85\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e4.40e-05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-3.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-2.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e-2.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eStationary\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGlobal Economic Policy Uncertainty Index\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-6.69\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e4.02e-09\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-3.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-2.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e-2.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eStationary\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGov Debt (% of GDP)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-4.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.00015\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-3.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-2.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e-2.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eStationary\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eImports (current US\u003cspan\u003e$\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-4.27\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.00049\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-3.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-2.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e-2.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eStationary\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eInflation (CPI)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-4.60\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.00013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-3.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-2.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e-2.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eStationary\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eReal Interest Rate (%)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-9.26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.40e-15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-3.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-2.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e-2.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eStationary\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eUnemployment (% of total)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-5.91\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.65e-07\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-3.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-2.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e-2.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eStationary\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eEquity Market Uncertainty\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-6.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e4.50e-08\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-3.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-2.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e-2.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eStationary\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec17\" class=\"Section2\"\u003e \u003ch2\u003e3.5. Min-Max Normalization\u003c/h2\u003e \u003cp\u003eMin-Max normalization ensured that the range of all numeric features was transformed to fall within the interval [0, 1]. For example, features like `Financial Regulation Uncertainty`, `Fiscal Policy Uncertainty`, and `Monetary Policy Uncertainty` ranged from 0.1 to 100 in the original dataset. After Min-Max normalization, these values were transformed to fall within the range of 0 and 1. In the normalized dataset, the minimum value of each numeric column was 0 and the maximum value of each numeric column was 1. This transformation ensures that all features are equally weighted when used in machine learning algorithms. This also remove the potential bias caused by the different scales of the original data.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec18\" class=\"Section2\"\u003e \u003ch2\u003e3.6. MCMC simulation-based Bayesian Hierarchical Model\u003c/h2\u003e \u003cp\u003eThe BHM successfully converged, with no divergences reported. Effective sample sizes for most parameters were high that ensure reliable posterior estimates. The results provide estimates on how different uncertainties and macroeconomic factors influence GDP. The MCMC simulation yielded stable and reliable estimates for the model parameters. Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e summarizes the posterior estimates including the mean, standard deviation, and the highest density interval (HDI) for the 3% and 97% percentiles for each parameter. The R-hat values were all close to 1.0 which indicate good convergence across the chains (Gelman \u0026amp; Shirley, 2011). Most regression coefficients (betas) had relatively narrow credible intervals which show that the model was confident in the estimated effects of these variables on GDP. Some variables like Monetary Policy Uncertainty and Global Economic Policy Uncertainty showed stronger, significant effects on GDP with credible intervals far from zero. The posterior predictive distribution matched the observed data well which that the model captures the underlying data structure effectively. Trace plots of each feature are illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePosterior estimates and the highest density interval (HDI) for the 3% and 97% percentiles for each parameter after MCMC simulation-based inference of BHM.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMean\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eStandard Deviation (SD)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eHDI 3%\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eHDI 97%\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eESS\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eR-hat\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAlpha (Intercept)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.181\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.015\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.211\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-0.152\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e4687.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBetas [0] (Financial Regulation Uncertainty)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.007\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.027\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e10504.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBetas [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e] (Fiscal Policy Uncertainty)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.002\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.010\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.017\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.022\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6288.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBetas [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e] (Government Spending Uncertainty)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.019\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.010\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.037\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-0.000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6727.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBetas [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e] (Monetary Policy Uncertainty)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.036\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.006\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.023\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.048\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e9021.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBetas [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e] (Trade Policy Uncertainty)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.008\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.010\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.026\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.011\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e10256.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBetas [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e] (Exchange Rate)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.041\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.014\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.017\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.067\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e3303.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBetas [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e] (Exports)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.865\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.057\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.969\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e-0.756\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e3001.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBetas [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] (FDI (% of GDP))\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.016\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.020\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.052\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.025\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e8012.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBetas [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e] (Global Economic Policy Uncertainty Index)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.008\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.007\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.021\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.004\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7283.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBetas [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e] (Gov Debt (% of GDP))\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.185\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.019\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.150\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.222\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e4115.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBetas [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e] (Imports)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.507\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.056\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.401\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.612\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2917.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBetas [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e] (Inflation (CPI))\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.237\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.021\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.196\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.276\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e5339.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBetas [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e] (Real Interest Rate)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.097\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.072\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.122\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7106.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBetas [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e] (Unemployment)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.038\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.011\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.018\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.058\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e5738.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBetas [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] (Equity Market Uncertainty)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.007\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.012\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e11706.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003esigma_y (Standard Deviation of y_obs)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.018\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.017\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.018\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e10079.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003ePosterior estimates of each feature are shown in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. The posterior mean for alpha was \u0026minus;\u0026thinsp;0.181 suggests a slight negative baseline effect on GDP. Posterior mean of 0.036 in monetary policy uncertainty indicates a small but positive effect on GDP. This suggests that as uncertainty in monetary policy increases, GDP may see a slight increase. Trade Policy Uncertainty\u0026rsquo;s posterior mean of -0.019 indicate a negative but marginally significant effect on GDP. Financial Regulation Uncertainty\u0026rsquo;s posterior mean of -0.013 indicate a weak negative association with GDP. Government Spending Uncertainty\u0026rsquo;s Posterior mean of -0.008, shows a slight negative effect on GDP, but not statistically significant. Exchange Rate\u0026rsquo;s posterior mean of 0.041 indicate that a rise in exchange rates (local/USD) correlates positively with GDP. Equity Market Uncertainty\u0026rsquo;s posterior mean of 0.038 suggests a positive association between equity market uncertainty and GDP. Inflation (CPI) Posterior mean of 0.237 suggests a robust positive effect of inflation on GDP, potentially indicating inflation-driven growth. Gov Debt (% of GDP) Posterior mean of 0.185 shows a significant positive association between government debt and GDP. Imports (current US\u003cspan\u003e$\u003c/span\u003e) posterior mean of 1.507 suggests that import growth significantly drives GDP growth in the dataset.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe Bayesian hierarchical model provided not only point estimates but also credible intervals for the effects of each variable that offer a robust framework for analyzing the economic dynamics in the face of uncertainty. The analysis generated posterior distributions and likelihoods for each parameter of interest which were visualized using density plots in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. These graphs provide a clear visual representation of the uncertainty surrounding each estimate and facilitate the identification of statistically significant relationships.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec19\" class=\"Section2\"\u003e \u003ch2\u003e3.7. Uncertainty Quantification\u003c/h2\u003e \u003cp\u003eUncertainty Quantification yielded a 95% credible interval for the predicted values of the dependent variable (GDP) ranging from 0.2762 to 0.3587. This credible interval provides a quantitative measure of the uncertainty in the model\u0026rsquo;s predictions. A narrower interval suggests higher confidence in the predicted values, whereas a wider interval indicates greater uncertainty (Farmer, 2017). In this case, the relatively tight range of predictions signifies that the model captures the underlying data patterns effectively that allow for a stronger interpretation of economic policy uncertainties. The results from the uncertainty quantification reveal that the model has produced reliable predictions of GDP, with the credible interval suggesting that economic outcomes are likely to lie within a defined range. This information is crucial for policymakers and stakeholders, as it enables them to make informed decisions by considering both the predicted values and the associated uncertainty. By understanding the range of potential outcomes, policymakers can better assess risks and implement strategies to mitigate adverse effects on economic growth. Figure\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e is illustrating the uncertainty in predictive and observed features.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec20\" class=\"Section2\"\u003e \u003ch2\u003e3.8. Policy prediction\u003c/h2\u003e \u003cp\u003eThe policy prediction step yielded two distinct predicted outcomes for the two policy scenarios. Scenario 1 predicted a policy outcome of -0.23464. Scenario 2 predicted a policy outcome of -0.20604. These values demonstrate how varying levels of uncertainty influence policy decisions. In an environment characterized by heightened uncertainty (Scenario 1), the policy outcome was more negative which indicate that increased uncertainty has a significant adverse impact on economic policy-making. Conversely, in a more stable environment (Scenario 2), policy outcomes showed a relatively smaller negative shift that imply that moderate conditions provide more favorable opportunities for effective policy intervention.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"4. Discussion","content":"\u003cp\u003ePolicy prediction uncertainty remains a central challenge in economic forecasting, particularly in complex systems influenced by numerous interdependent variables. This uncertainty has posed significant challenges in economic research that leads to varied and often conflicting outcomes in the evaluation of fiscal, monetary, and trade policies (Iqbal et al., 2020). Cascaldi et al., have highlighted that uncertainty in economic forecasts can result from unpredictable shocks, model limitations, and insufficient or outdated data. These uncertainties make it difficult for policymakers to make informed decisions in volatile economic environments (Cascaldi-Garcia et al., 2023). Additionally, earlier deterministic or static models failed to capture the dynamic and evolving nature of economic systems, which further affected prediction uncertainty. While newer data-driven approaches have improved uncertainty quantification especially for complex, multi-factorial systems influenced by global events and market fluctuations.\u003c/p\u003e \u003cp\u003eTraditional models rely on point estimates and often struggle to capture the full spectrum of uncertainty (Li et al., 2020). In contrast, Bayesian approaches offer a more robust framework by incorporating uncertainty at multiple levels including parameter estimates and data variability. However, even within Bayesian frameworks, the uncertainty inherent in economic policy predictions can be affected by model assumptions, data quality, and prior choices. For example, macroeconomic variables are subject to both short-term shocks and long-term structural changes, which can challenge the predictive validity of any model (Yu \u0026amp; Huang, 2021). Despite these challenges, the use of posterior distributions to quantify uncertainty and conduct scenario-based analyses enhances the credibility of policy recommendations. Although there is still room for improvement in refining these models to capture the full range of uncertainties, especially in volatile economic environments. Simulation-based inference in Bayesian models involves using techniques like Markov Chain Monte Carlo (MCMC) to estimate the posterior distributions of model parameters. This approach allows for more accurate inference in complex models where analytical solutions are difficult or impossible to obtain (Nain \u0026amp; Kamaiah, 2020).\u003c/p\u003e \u003cp\u003eThis study aimed to quantify economic uncertainty and estimate policy prediction utilizing a MCMC- based inference of Bayesian Hierarchical Model (BHM). Each step of the methodology contributed to a robust understanding of the uncertainties affecting monetary, fiscal, and trade policies. The research began with rigorous data collection from multiple reputable sources, including the World Bank and the International Financial Statistics (IFS) database. This step ensured that the analysis was grounded in high-quality data that accurately reflected macroeconomic conditions. Data preprocessing involved cleaning, normalization, and standardization to make the dataset suitable for modeling. This foundational step is critical to check the integrity and structure of the data which directly impact the quality of subsequent analyses (Raju et al., 2020). The application of the MCMC-based inference of BHM allowed for the integration of multiple levels of uncertainty inherent in economic data (Alakkari, 2022). By modeling both fixed effects (global parameters) and random effects (specific to groups or contexts), the BHM provided a flexible framework to capture complex relationships between macroeconomic indicators. The results revealed the estimated parameters had notable effects on the dependent variable. The credible intervals from the posterior distributions provided insight into the range of plausible values for each parameter that shows the importance of uncertainty quantification in policymaking. Following the modeling, uncertainty quantification was performed to assess the robustness of the predictions. The derivation of 95% credible intervals for predicted values shows that a range of potential outcomes are under different economic scenarios (Lux, 2022). The results highlighted the significant variability associated with economic policy predictions. Using the fitted model, the study generated policy predictions under two distinct economic scenarios. The predictions illustrated potential outcomes given varying conditions which is crucial for informing policy decisions. The values obtained from the analysis provided insights into the possible impacts of current policies and served as a basis for evaluating alternative strategies. The graphical representation of these predictions enhanced the understanding of the relationships between predictors and outcomes, making it easier for policymakers to grasp the implications of the results visually. Overall, the methodology employed in this study not only demonstrated the effectiveness of simulation-based Bayesian methods in economic analysis but also illustrated the critical role of uncertainty quantification in policy prediction. The findings underscore the importance of using sophisticated analytical techniques to navigate the complexities of economic policymaking particularly in uncertain environments. By filling the gap between data-driven insights and practical policy applications, this research contributes valuable knowledge to the ongoing discourse on effective economic governance. While, future research should continue exploring the integration of more dynamic, real-time data and adaptive priors to reduce prediction bias.\u003c/p\u003e"},{"header":"5. Conclusion","content":"\u003cp\u003eThrough the application of a Markov Chain Monte Carlo simulation-based Bayesian Hierarchical Model and advanced uncertainty quantification techniques, this study has successfully provided a robust framework for predicting economic policy outcomes under uncertainty. The BHM allowed for the estimation of the relationships between various macroeconomic factors and policy outcomes. Notable predictors such as monetary policy uncertainty had a significant negative impact on the policy outcome (mean = -0.864) which highlights that high uncertainty in monetary policy negatively influences policy predictions. In contrast, equity market uncertainty and another macroeconomic variable showed a strong positive impact (mean\u0026thinsp;=\u0026thinsp;0.185) which suggests that certain economic indicators can positively impact policy predictions despite other uncertainties. The credible interval ranged between [0.276\u0026ndash;0.359] which indicate that future policy outcomes are likely to fall within this range. This range provides a probabilistic understanding of where policy predictions will land under varying conditions of uncertainty. Such insights are critical for policymakers as they allow for the assessment of risk and the development of more resilient policy strategies. The findings reveal that higher levels of uncertainty tend to intensify negative outcomes in policy predictions. However, certain economic variables like equity market conditions can offset this impact to some extent. The credible intervals and scenario-based policy predictions offer policymakers a comprehensive understanding of the range and sensitivity of policy outcomes under uncertain conditions. By quantifying and analyzing uncertainty, this study offers a critical data-driven approach to understanding the future of economic policies in a dynamic and unpredictable macroeconomic landscape.\u003c/p\u003e"},{"header":"6. Statements \u0026 Declarations","content":"\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting Interests\u003c/strong\u003e The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.\u0026nbsp;\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eAjewole, K., Adejuwon, S., \u0026amp; Jemilohun, V. (2020). Test for stationarity on inflation rates in Nigeria using augmented dickey fuller test and Phillips-persons test. \u003cem\u003eJ. Math\u003c/em\u003e,\u003cem\u003e\u0026nbsp;16\u003c/em\u003e(2020), 11-14.\u0026nbsp;\u003c/li\u003e\n \u003cli\u003eAl-Thaqeb, S. A., \u0026amp; Algharabali, B. G. (2019). 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Springer.\u0026nbsp;\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Economic policy uncertainty, Predictive modelling Markov Chain Monte Carlo algorithm, Bayesian statistics, Computational economics.","lastPublishedDoi":"10.21203/rs.3.rs-5299835/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5299835/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eEconomic policy uncertainty relates to the unpredictability in government policies that can impact economic decision-making. High policy uncertainty can lead to less investment, slower economic growth, and increased volatility in financial markets. This study aims to quantify the uncertainty by employing a data-driven approach based on Bayesian Hierarchical Modeling (BHM) and Markov Chain Monte Carlo simulations. This research focuses essentially on key policy domains such as monetary policy, fiscal policy, and trade policy where uncertainty underlies crucial influences upon economic decisions. The methodology integrates data collection, feature scaling, normalization, Bayesian inference using MCMC simulations, uncertainty quantification and policy prediction to produce predictive insights under various economic scenarios. The Bayesian Hierarchical Model was employed to estimate the relationships between macroeconomic variables and policy outcomes. The posterior distribution results revealed significant predictors, with certain factors like monetary policy uncertainty exerting a substantial negative impact, while others such as equity market-related uncertainty showed positive influence. \u0026nbsp;A rigorous uncertainty quantification step provided credible intervals for predicted outcomes with a 95% credible interval ranging between 0.276 - 0.359. This enabled an assessment of the potential variability in predictions based on differing levels of economic uncertainty. The study concluded with policy predictions generated under two distinct economic scenarios. Under conditions of high uncertainty, the predicted policy outcome was -0.2346, while a moderate uncertainty scenario resulted in a less negative outcome of -0.2060. These results demonstrate the sensitivity of economic policy predictions to varying levels of uncertainty. The findings provide a robust framework for understanding and quantifying uncertainty in economic policy-making. By applying BHM and Monte Carlo methods, this study contributes to the development of more resilient and adaptive economic strategies in the face of uncertainty.\u003c/p\u003e","manuscriptTitle":"Quantifying Uncertainty in Economics Policy Predictions: A Bayesian \u0026amp; Monte Carlo based Data-Driven Approach","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-10-22 06:33:59","doi":"10.21203/rs.3.rs-5299835/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"47208c6c-17f6-40d9-9d5c-d12fbc824d4c","owner":[],"postedDate":"October 22nd, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":39177635,"name":"International Economics"}],"tags":[],"updatedAt":"2024-10-25T04:38:27+00:00","versionOfRecord":[],"versionCreatedAt":"2024-10-22 06:33:59","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-5299835","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5299835","identity":"rs-5299835","version":["v1"]},"buildId":"WrCJVZZCHTDjtuVLN7oU0","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}