The data used in this paper are longitudinal data of three macroeconomic variables, inflation, exchange and monetary policy rates across five continents. The study covers a 10-year period from 2014 to 2023, to provide an insightful relationship between these variables. The data was sourced from Statista and Focus Economic data hubs and are measured in percentages.

This paper will begin with summary statistics after presenting the data.

Summary statistics such as the mean, the standard deviation, skewness and kurtosis as well as the coefficient of variation (CV), among others will be calculated in order to describe the data. In particular, CV will be calculated in order to know which country’s dependent and independent variables are most variable and uncertain. CV is calculated as [21] :

$\text{CV}=\frac{\text{Standard deviation}}{\text{Mean}}\ast 100\%$ (1)

To address the limitations of traditional linear models in capturing non-linearity and group-level heterogeneity, this study proposes a novel application of a Generalized Additive Mixed Model (GAMM) framework. Unlike conventional approaches that assume linearity and fixed relationships across groups, this study incorporates smooth functions and continent-specific random slopes to flexibly model the relationship between inflation, exchange rate, and monetary policy rate across five continents. The novelty lies in combining transformation techniques to address heteroscedasticity, non-linear smooth terms for monetary policy rate, and hierarchical random effects to capture both baseline differences and varying exchange rate sensitivities across regions. This approach provides insights that are not optimally captured by standard linear or mixed models, particularly in a global macroeconomic context.

The model specification and estimation procedures presented in this section are extensions of the analysis conducted in yet to be published research on the effect of exchange and monetary policy rates on inflation and apart from the fixed effects, only random intercepts were modelled. In this extended model, both random intercepts and random slopes are incorporated to account for the continent-level heterogeneity.

According to West et al. (2015) [22] , the general formula of a Linear Mixed Model is expressed as:

$\begin{array}{l}{Y}_{ti}={\beta }_1\ast X_{ti}^{\left(1\right)}+{\beta }_2\ast X_{ti}^{\left(2\right)}+{\beta }_3\ast X_{ti}^{\left(3\right)}+\cdots +{\beta }_p\ast X_{ti}^{\left(p\right)}\\ +u_{1i}\ast Z_{ti}^{\left(1\right)}+\cdots +u_{qi}\ast Z_{ti}^{\left(q\right)}+{\epsilon }_{ti}\\ t=1,\cdots ,n_i\\ i=1,2,\cdots ,m\end{array}$ (2)

where,

$n_i$ = the number of observations for subject $i$

$m$ = number of subjects

$Y_{ti}$ = the $t^{th}$ observation of the $i^{th}$ subject

$X_{ti}^{\left(1\right)}=1$ $\forall i$

$X_{ti}^{\left(2\right)},X_{ti}^{\left(3\right)},\cdots ,X_{ti}^{\left(p\right)}$ = the $t^{th}$ observation value of subject $i$ for the corresponding covariates $X^{\left(2\right)},X^{\left(3\right)},\cdots ,X^{\left(p\right)}$ associated with the fixed effects.

$Z_{ti}^{\left(1\right)},Z_{ti}^{\left(2\right)},\cdots ,Z_{ti}^{\left(q\right)}$ = the $t^{th}$ observation value for subject $i$ for the corresponding covariates $Z^{\left(1\right)},Z^{\left(2\right)},\cdots ,Z^{\left(q\right)}$ associated with random effects; $q\le p$

${\beta }_1,{\beta }_2,\cdots ,{\beta }_p$ = fixed effects

$u_{1i},u_{2i},\cdots ,u_{qi}$ = random effects specific to subject $i$

${\epsilon }_{ti}$ = residual associated with the $t^{th}$ observation on the $i^{th}$ subject

The expression in (2) given the study data and considering a model with both a random intercept and random slope for exchange rate ( $X_{ti}^{\left(2\right)}$) can be written as:

$Y_{ti}={\beta }_1\ast X_{ti}^{\left(1\right)}+{\beta }_2\ast X_{ti}^{\left(2\right)}+{\beta }_3\ast X_{ti}^{\left(3\right)}+u_{1i}\ast Z_{ti}^{\left(1\right)}+u_{2i}\ast Z_{ti}^{\left(2\right)}+{\epsilon }_{ti}$ (3)

$p=k+1;k=2$ (i.e., exchange and monetary policy rates)

$q=2$ (i.e., the random intercept and random slope)

where;

$X_{ti}^{\left(1\right)}=1$ $\forall i$, $X_{ti}^{\left(2\right)}$ = exchange rate

$X_{ti}^{\left(3\right)}$ = monetary policy rate

${\beta }_1$= Global intercept

${\beta }_2$= Fixed effect of Exchange Rate

${\beta }_3$= Fixed effect of Monetary Policy Rate

$u_{1i}$= Continent-specific random intercept

$u_{2i}$= Continent-specific random slope for Exchange Rate

${\epsilon }_{ti}$= Residual error

In (3), $Z_{ti}^{\left(1\right)}$ represents the design matrix for the random effects; linking the random coefficients to the observed data. Specifically, $Z_{ti}^{\left(1\right)}=1$ corresponds to the continent-specific random intercept $\left(u_{1i}\right)$, while $Z_{ti}^{\left(2\right)}=X_{ti}^{\left(2\right)}$ corresponds to the continent-specific random slope on Exchange Rate $\left(u_{2i}\right)$. In general, $Z_i$ represents the design structure of random effects.

To address violations of homoscedasticity observed in the residuals of the extended LMM, a square root transformation was applied to Inflation Rate (the response variable) in order to stabilize variance and reduce heteroscedasticity [23] [24] . $Y_{ti}$ becomes $\sqrt{Y_{ti}}$, then (3) can be described as:

$\sqrt{Y_{ti}}={\beta }_1\ast X_{ti}^{\left(1\right)}+{\beta }_2\ast X_{ti}^{\left(2\right)}+{\beta }_3\ast X_{ti}^{\left(3\right)}+u_{1i}\ast Z_{ti}^{\left(1\right)}+u_{2i}\ast Z_{ti}^{\left(2\right)}+{\epsilon }_{ti}$ (4)

Following the square root transformation of Inflation Rate to correct heteroscedasticity in the extended LMM, diagnostic checks using patterned residual plots of the transformed model revealed violation of the linearity assumption in the relationship between Exchange, Monetary Policy Rates and Inflation. Consequently, a Generalized Additive Mixed Model (GAMM) was employed to model non-linear effects while accounting for random effects (Wood, 2017) [2] which allows a flexible, non-linear relationships between Monetary Policy Rate and Inflation by incorporating smooth functions. Then (4) can be expressed as:

$\sqrt{Y_{ti}}={\beta }_1\ast X_{ti}^{\left(1\right)}+{\beta }_2\ast X_{ti}^{\left(2\right)}+f\left(X_{ti}^{\left(3\right)}\right)+u_{1i}\ast Z_{ti}^{\left(1\right)}+u_{2i}\ast Z_{ti}^{\left(2\right)}+{\epsilon }_{ti}$ (5)

where,

$f\left(X_{ti}^{\left(3\right)}\right)$ = smooth term for Monetary Policy Rate (where non-linear effect is suspected)

In matrix notation the GAMM is described as:

$\sqrt{Y_i}=X_i\beta +Z_i{u}_i+f_i+{\epsilon }_i$; (6)

$i=1,2,\cdots ,m$; ‘ $m$’ = number of subjects (continents)

where:

$\sqrt{Y_i}$ = an $n_i\times 1$ vector of continuous responses represented as follows:

$\sqrt{Y_i}=\left(\begin{array}{c}\sqrt{Y_{1i}}\\ \sqrt{Y_{2i}}\\ ⋮\\ \sqrt{Y_{n_{i}i}}\end{array}\right)$ (7)

$n_i$ = the number of observations for subject $i$

$X_i$ = an $n_i\times p$ fixed effects design matrix represented as follows:

$X_i=\left(\begin{array}{cccc}{X}_{1i}^{\left(1\right)}& X_{1i}^{\left(2\right)}& \cdots & X_{1i}^{\left(p\right)}\\ X_{2i}^{\left(1\right)}& X_{2i}^{\left(2\right)}& \cdots & X_{2i}^{\left(p\right)}\\ ⋮& ⋮& \ddots & ⋮\\ X_{n_{i}i}^{\left(1\right)}& X_{n_{i}i}^{\left(2\right)}& \cdots & X_{n_{i}i}^{\left(p\right)}\end{array}\right)$ (8)

Noting that $p=k+1$ (where $k$ is the number of covariates and $X_{ti}^{\left(1\right)}=1$, $t=1,2,\cdots ,n_i$), $X_i$ can be written as:

$X_i=\left(\begin{array}{cccc}1& X_{1i}^{\left(2\right)}& \cdots & X_{1i}^{\left(p\right)}\\ 1& X_{2i}^{\left(2\right)}& \cdots & X_{2i}^{\left(p\right)}\\ ⋮& ⋮& \ddots & ⋮\\ 1& X_{n_{i}i}^{\left(2\right)}& \cdots & X_{n_{i}i}^{\left(p\right)}\end{array}\right)$ (9)

$\beta$ = a vector of $p$ unknown regression coefficients or fixed effect parameters described as:

$\beta =\left(\begin{array}{c}{\beta }_1\\ {\beta }_2\\ ⋮\\ {\beta }_p\end{array}\right)$ (10)

$Z_i$ = $n_i\times q$ ( $q\le p$) design matrix for the random effects represented as, where $Z_{ti}^{\left(1\right)}=1$ $\forall i$:

$Z_i=\left(\begin{array}{cccc}1& Z_{1i}^{\left(2\right)}& \cdots & Z_{1i}^{\left(q\right)}\\ 1& Z_{2i}^{\left(2\right)}& \cdots & Z_{2i}^{\left(q\right)}\\ ⋮& ⋮& \ddots & ⋮\\ 1& Z_{n_{i}i}^{\left(2\right)}& \cdots & Z_{n_{i}i}^{\left(q\right)}\end{array}\right)$ (11)

$u_i$ = $q\times 1$ vector of random effects represented as:

$u_i=\left(\begin{array}{c}{u}_{1i}\\ u_{2i}\\ ⋮\\ u_{qi}\end{array}\right)$ (12)

$f_i$ = $n_i\times 1$ vector for the smooth term ( $s$) represented as:

$f_i=\left(\begin{array}{c}f\left(X_{1i}\right)\\ f\left(X_{2i}\right)\\ ⋮\\ f\left(X_{n_{i}i}\right)\end{array}\right)$ (13)

To operationalize (13), the smooth term is represented using a basis expansion expressed as:

$f\left(X_{n_{i}i}\right)=\underset{k=1}{\overset{d}{{\displaystyle \sum }}}{b}_k\left(X_{n_{i}i}\right){\gamma }_k$

where;

$b_k\left(X_{n_{i}i}\right)$ = pre specified basis functions

${\gamma }_k$ = unknown coefficients to be estimated

$d$ = basis dimension

The choice of $d$ balances flexibility against overfitting. Larger $d$ allows for greater flexibility but may lead to overfitting unless penalized, while smaller $d$ enforces smoother functions but risks under-fitting.

In practice, $d$ is often guided by rules of thumb or empirical evaluation and the general guide for selecting basis dimension in this context is described in Table 1 .

${\epsilon }_i$ = a vector of residuals represented as:

${\epsilon }_i=\left(\begin{array}{c}{\epsilon }_{1i}\\ {\epsilon }_{2i}\\ ⋮\\ {\epsilon }_{n_{i}i}\end{array}\right)$ (14)

From this study, equation (6) can be written in a matrix form as:

$\left[\begin{array}{c}\sqrt{Y_{1i}}\\ \sqrt{Y_{2i}}\\ ⋮\\ \sqrt{Y_{10i}}\end{array}\right]=\left[\begin{array}{cc}1& X_{1i}^{\left(2\right)}\\ 1& X_{2i}^{\left(2\right)}\\ ⋮& ⋮\\ 1& X_{10i}^{\left(2\right)}\end{array}\right]\left[\begin{array}{c}{\beta }_1\\ {\beta }_2\end{array}\right]+\left[\begin{array}{cc}1& Z_{1i}^{\left(2\right)}\\ 1& Z_{2i}^{\left(2\right)}\\ ⋮& ⋮\\ 1& Z_{10i}^{\left(2\right)}\end{array}\right]\left[\begin{array}{c}{u}_{1i}\\ u_{2i}\end{array}\right]+\left[\begin{array}{c}f\left(X_{1i}^{\left(3\right)}\right)\\ f\left(X_{2i}^{\left(3\right)}\right)\\ ⋮\\ f\left(X_{10i}^{\left(3\right)}\right)\end{array}\right]+\left[\begin{array}{c}{\epsilon }_{1i}\\ {\epsilon }_{2i}\\ ⋮\\ {\epsilon }_{10i}\end{array}\right]$

That is, $\sqrt{Y_i}=X_i\beta +Z_i{u}_i+f_i+{\epsilon }_i$

Note that $q=2$ to accommodate:

Assumptions:

(a) $u_i~N\left(0,D\right)$

(b) ${\epsilon }_i~N\left(0,R_i\right)$

where:

$D=Var\left(u_i\right)=\left(\begin{array}{ccccc}Var\left(u_{1i}\right)& cov\left(u_{1i},u_{2i}\right)& cov\left(u_{1i},u_{3i}\right)& \cdots & cov\left(u_{1i},u_{qi}\right)\\ cov\left(u_{2i},u_{1i}\right)& var\left(u_{2i}\right)& cov\left(u_{2i},u_{3i}\right)& \cdots & cov\left(u_{2i},u_{qi}\right)\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ cov\left(u_{qi},u_{1i}\right)& cov\left(u_{qi},u_{2i}\right)& cov\left(u_{qi},u_{3i}\right)& \cdots & var\left(u_{qi}\right)\end{array}\right)$(15)

And

$R_i=Var\left({\epsilon }_i\right)=\left(\begin{array}{ccccc}Var\left({\epsilon }_{1i}\right)& cov\left({\epsilon }_{1i},{\epsilon }_{2i}\right)& cov\left({\epsilon }_{1i},{\epsilon }_{3i}\right)& \cdots & cov\left({\epsilon }_{1i},{\epsilon }_{n_{i}i}\right)\\ cov\left({\epsilon }_{2i},{\epsilon }_{1i}\right)& var\left({\epsilon }_{2i}\right)& cov\left({\epsilon }_{2i},{\epsilon }_{3i}\right)& \cdots & cov\left({\epsilon }_{2i},{\epsilon }_{n_{i}i}\right)\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ cov\left({\epsilon }_{n_{i}i},{\epsilon }_{1i}\right)& cov\left({\epsilon }_{n_{i}i},{\epsilon }_{2i}\right)& cov\left({\epsilon }_{n_{i}i},{\epsilon }_{3i}\right)& \cdots & var\left({\epsilon }_{n_{i}i}\right)\end{array}\right)$(16)

Assumptions (a) and (b) state that the error components are normal.

A Q-Q plot of residuals and random effects was used to assess the normality assumption of the GAMM. In this plot, residual quantiles are plotted against theoretical normal quantiles. A close alignment with a 45-degree reference line suggests that residuals are approximately normally distributed. Departures from this line may indicate issues such as skewness, outliers, or violations of model assumptions. Normality of residuals is crucial for valid inference in models assuming Gaussian errors [2] [25] .

(c) Independence: An autocorrelation function (ACF) plot of the model residuals was used to evaluate the assumption of independence. The ACF plot displays the correlation of residuals across various lags. The absence of significant autocorrelation is indicated when all correlation values lie within the 95% confidence bounds. Deviations beyond these bounds suggest that residuals are temporally correlated, necessitating adjustments such as incorporating autoregressive correlation structures in the model.

The Durbin-Watson test is used to detect first-order serial correlation in model residuals particularly in grouped longitudinal data and the test statistic is given by (Durbin & Watson, 1950) [26] :

$DW=\frac{{\displaystyle {\sum }_{t=2}^n{\left(e_t-e_{t-1}\right)}^2}}{{\displaystyle {\sum }_{t=1}^n{e}_t^2}}$ (17)

where:

$e_t$ = the residual at time, $t$

$n$ = the total number of residuals

Hypothesis:

H₀: The errors are not autocorrelated (Errors are independent)

H₁: The errors are autocorrelated (Errors are not independent)

Decision rule:

If the test statistic has $p<0.05$, then the null hypothesis of autocorrelation is rejected and if the test statistic has $p>0.05$, then we do not reject the null hypothesis of autocorrelation

(d) Level 1 exogeneity, i.e, $E\left(e_{ti}|X_i\right)=0$ and Level 2 exogeneity, i.e. $E\left(u_i|X_i\right)=0$. Sophia & Skrondal, (2012) [27] say that with the assumption of normality of $e$ and $u$, coupled with the above exogeneity conditions, $e$ and $u$ are independent.

(e) Homoscedasticity: A residual versus fitted values plot was used to assess homoscedasticity in the GAMM. The residuals should be randomly scattered around zero with consistent vertical spread across all levels of fitted values. A uniform spread supports the assumption of constant variance, whereas systematic patterns or increasing/decreasing spread indicate heteroscedasticity [2] .

(f) Model Convergence and Fit: Convergence of the GAMM was assessed through convergence diagnostics provided by the estimation routine. The absence of warnings and stable estimates of smoothing parameters indicated successful convergence. Model fit was evaluated using observed vs. fitted plots and smooth term plots with confidence intervals. A close alignment between fitted and observed values suggests a good fit, while smooth term plots confirmed that non-linear relationships were adequately captured without overfitting [2] .

This formulation goes beyond prior models by capturing latent non-linearity in monetary policy effects and accommodating heterogeneity in the exchange rate-inflation relationship through random slopes. It thus responds to the methodological gap in previous studies that modelled only fixed effects or assumed linearity across regions. The GAMM framework adopted here provides a richer, more flexible structure that aligns better with the underlying data complexities observed in macroeconomic trends.

The Frequentist implementation of the proposed GAMM framework allows us to estimate fixed, smooth, and random components simultaneously using Restricted Maximum Likelihood (REML). This contrasts with simpler LMMs, enabling nuanced understanding of cross-regional variation and non-linear effects.

Generalized Additive Mixed Models (GAMMs) extend the linear mixed model framework by allowing for non-linear relationships between predictors and the response variable through the inclusion of smooth functions. As described by Wood (2017) [2] , the response in a GAMM is expressed as:

$g\left(E\left[Y_i\right]\right)=X_i\beta +Z_i{u}_i+{\displaystyle {\sum }_j{f}_j\left(x_{ij}\right)}$ (18)

In this study, the model includes fixed effects, random intercept and a random slope for the Exchange Rate, allowing for group-specific variation in both the baseline inflation level and the effect of exchange rate changes, and smooth terms for Monetary Policy Rate, where non-linear effects are suspected. The random effects vector is described as follows:

$u_i=\left[\begin{array}{c}{u}_{1i}\\ u_{2i}\end{array}\right]$

where,

$D=\left[\begin{array}{cc}{\sigma }_{u_{1i}}^2& {\sigma }_{u_{1i}{u}_{2i}}\\ {\sigma }_{u_{1i}{u}_{2i}}& {\sigma }_{u_{2i}}^2\end{array}\right]$

$D$ is assumed to be an unstructured covariance matrix, hence;

$Cov\left(u_{1i},u_{2i}\right)={\sigma }_{u_{1i}{u}_{2i}}\ne 0$.

The residuals ${\epsilon }_i~N\left(0,{\sigma }^2{I}_{n_i}\right)$, hence [2] :

$R_i={\sigma }^2{I}_{n_i}$

The marginal variance-covariance matrix for $Y_i$ is described as:

$Var\left(Y_i\right)=V_i=Z_{i}D{Z^{\prime }}_i+R_i$

Importantly, the variance components in $D$ i.e. ( ${\sigma }_{u_{1i}}^2,{\sigma }_{u_{2i}}^2,{\sigma }_{u_{1i}{u}_{2i}}$) and in $R$ i.e. ( ${\sigma }^2$) are not fixed but are estimated directly from the data during model fitting. This ensures that the model captures the true variability across the continents and within observations instead of imposing arbitrary assumptions.

Parameter estimation in GAMMs relies on penalized maximum likelihood, where smoothing parameters and random effects are estimated simultaneously. This is implemented using Restricted Maximum Likelihood (REML) to ensure unbiased estimation of variance and smoothing parameters, particularly effective for small sample sizes [2] .

The estimator for the fixed effects $\beta$ using either the REML is:

$\hat{\beta }={\left({\displaystyle {\sum }_i{X^{\prime }}_i{\hat{V}}_i^{-1}{X}_i}\right)}^{-1}{\displaystyle {\sum }_i{X^{\prime }}_i{\hat{V}}_i^{-1}{y}_i}$ (19)

where:

${\hat{V}}_i=Z_i\hat{D}{Z^{\prime }}_i+{\hat{R}}_i$ (20)

The variance of $\hat{\beta }$, $var\left(\hat{\beta }\right)$ is a $p\times p$ covariance matrix calculated using the formula:

$var\left(\hat{\beta }\right)={\left({\displaystyle {\sum }_i{X^{\prime }}_i{\hat{V}}_i^{-1}{X}_i}\right)}^{-1}$ (21)

Compared to traditional linear mixed models with only random intercepts, the inclusion of both random slopes and smooth terms in GAMMs introduce additional complexity and flexibility, enabling the model to capture group-specific deviations and non-linear effects simultaneously. This results in a variance structure, $V_i$ that is both group-specific and non-exchangeable, reflecting realistic heterogeneity across continents [2] .

The smooth effect of the monetary policy rate on inflation is estimated using basis function expansions, a common approach in GAMM [2] [28] . Specifically, the smooth function $f(.)$, representing a potentially nonlinear effect of a covariate $X$, is approximated as a linear combination of known basis functions described in the context of this study:

$f\left(X_{n_{i}i}^{\left(3\right)}\right)={\displaystyle {\sum }_{k=1}^d{\gamma }_k{b}_k\left(X_{n_{i}i}^{\left(3\right)}\right)}$ (22)

where,

$b_k\left(X_{n_{i}i}^{\left(3\right)}\right)$ = known basic functions

${\gamma }_k$ = unknown basic coefficients to be estimated from the study data

$d$ = the basic dimensions (number of basic functions used)

The coefficients $\gamma ={\left({\gamma }_1,\cdots ,{\gamma }_K\right)}^{\text{T}}$ are estimated by minimizing a Penalized Least Squares (PLS) criterion of the form:

$\text{PLS}={\displaystyle {\sum }_{n_{i}i}{\left(\sqrt{Y_{n_{i}i}}-{\eta }_{n_{i}i}\right)}^2}+\lambda {\displaystyle \int {\left(f^{″}\left(X_{n_{i}i}^{\left(3\right)}\right)\right)}^2\text{d}x}$ (23)

where,

$\sqrt{Y_{n_{i}i}}$= the square root-transformed response

${\eta }_{n_{i}i}=X_{n_{i}i}\beta +Z_{n_{i}i}{u}_i+f\left(X_{n_{i}i}^{\left(3\right)}\right)$ is the linear predictor

$\lambda$= is the smoothing parameter, given by; $\lambda =\frac{{\sigma }_{\epsilon }^2}{{\sigma }_u^2}$

${\left(f^{″}\left(X_{n_{i}i}^{\left(3\right)}\right)\right)}^2\text{d}x$ = the penalty term that ensures smoothness by penalizing excessive curvature

(a) Adjusted R²: This is a refined version of the coefficient of determination (R²) that accounts for the number of predictors in the model. It is especially useful when comparing models with a different number of explanatory variables. The formula is expressed as:

$R_{adj}^2=1-\frac{\left(1-R^2\right)\left(n-1\right)}{n-p-1}$ (24)

where,

$n$ = number of observations

$p$ = number of estimated parameters

$R^2$ = unadjusted coefficient of determination

A higher Adjusted $R^2$ indicates a better-fitting model after accounting for the number of predictors [29] .

(b) K-Fold Cross Validation: K-fold cross-validation is a standard technique to assess the predictive performance of a model by splitting the data into mutually exclusive subsets or folds. The formula is given by:

$\text{CV-MSE}=\frac{1}{K}{\displaystyle {\sum }_{k=1}^{K}MS{E}_k}$ (25)

Where Mean Square Error for each fold $MS{E}_k$ is given by:

$MS{E}_k=\frac{1}{n_k}\underset{i\in foldk}{{\displaystyle \sum }}{\left(y_i-{\hat{y}}_i\right)}^2$

Lower CV-MSE values indicate better model predictive performance [30] .

To strengthen inference and account for parameter uncertainty, a Bayesian version of the GAMM was also implemented through the Gibbs sampler. This Bayesian approach further enhances interpretability by providing credible intervals for all model components, while enabling posterior predictive checks and sensitivity diagnostics which is not feasible under Frequentist analysis alone.

The Bayesian formulation of the generalized additive mixed model (GAMM) applied to the square root-transformed outcome variable $y^{*}=\sqrt{y}$ expressed in (6).

Following the Bayesian approach recommended by Van der Merwe and Botha (1993) [2] [28] [29] [31] , the joint posterior distribution is obtained through Bayes’ Theorem:

$P\left(\theta |y^{*}\right)\propto P\left(y^{*}|\theta \right)P\left(\theta \right)$

Given $\theta =\left({\beta }_i,\gamma ,u,{\sigma }_{\epsilon }^2,D\right)$ the joint likelihood is expressed as:

$p\left(y^{*},u|{\beta }_1,{\beta }_2,\gamma ,{\sigma }_{\epsilon }^2,D\right)=p\left(y^{*}|{\beta }_1,{\beta }_2,\gamma ,u,{\sigma }_{\epsilon }^2\right)\ast p\left(u|D\right)$ (26)

where,

$y^{*}|{\beta }_1,{\beta }_2,\gamma ,u,{\sigma }_{\epsilon }^2~N\left(X\beta +B\gamma +Zu,{\sigma }_{\epsilon }^{2}I\right)$

and

$u~N\left(0,D\right)$

The priors are assumed to be:

The posterior distribution is proportional to:

$\begin{array}{c}P\left(\beta ,\gamma ,u,D,{\sigma }_{\epsilon }^2,\lambda |y^{*}\right)\propto N\left(y^{*}|X\beta +B\gamma +Zu,{\sigma }_{\epsilon }^{2}I\right)N\left(u|0,D\right)p\left(\beta \right)\\ \ast p\left(\gamma |\lambda \right)p\left(D\right)p\left({\sigma }_{\epsilon }^2\right)p\left(\lambda \right)\end{array}$

Using Gibbs Sampling, we iteratively sample from the full conditional posterior distributions of the parameters:

$\begin{array}{l}{\sigma }_{\epsilon }^2|y^{*},{\beta }_1,{\beta }_2,\gamma ,u\\ ~IG\left[\frac{1}{2}n,\frac{1}{2}{\left\{y^{*}-{\displaystyle {\sum }_{i=1}^2{X}_i{\beta }_i}-B\gamma -Zu\right\}}^{\prime }\left\{y^{*}-{\displaystyle {\sum }_{i=1}^2{X}_i{\beta }_i}-B\gamma -Zu\right\}\right]\end{array}$ (27)

$D|u~\text{Inverse-Wishart}\left(v+G,\text{Λ}+{\displaystyle {\sum }_{i=1}^G{u}_i{u^{\prime }}_i}\right)$ (28)

where;

$v$ = prior degrees of freedom, such that $v>q+1$

$G$ = number of continents

$\Lambda$ = the prior scale matrix given as:

$\Lambda =\left[\begin{array}{cc}{\sigma }_{u_{1i}}^2& {\sigma }_{u_{1i}{u}_{2i}}\\ {\sigma }_{u_{1i}{u}_{2i}}& {\sigma }_{u_{2i}}^2\end{array}\right]$

$u|y^{*},{\beta }_1,{\beta }_2,\gamma ,{\sigma }_{\epsilon }^2,D~N\left(\tilde{u},{\tilde{V}}_u\right)$ (29)

where:

${\tilde{V}}_u=\left(\frac{1}{{\sigma }_{\epsilon }^2}{Z}^{\prime }Z+D^{-1}\right),\tilde{u}={\tilde{V}}_u\left(\frac{1}{{\sigma }_{\epsilon }^2}{Z}^{\prime }\left(y^{*}-X\beta -B\gamma \right)\right)$

${\beta }_i|y^{*},\gamma ,u,{\sigma }_{\epsilon }^2~N\left(\tilde{\beta },{\tilde{V}}_{\beta }\right)$ (30)

where:

${\tilde{V}}_{\beta }={\left(\frac{1}{{\sigma }_{\epsilon }^2}{X}^{\prime }X\right)}^{-1},\tilde{\beta }={\tilde{V}}_{\beta }\left(\frac{1}{{\sigma }_{\epsilon }^2}{X}^{\prime }\left(y-B\gamma -Zu\right)\right)$

$u|y^{*},{\beta }_1,{\beta }_2,\gamma ,{\sigma }_{\epsilon }^2,D~N\left(\tilde{\gamma },{\tilde{V}}_{\gamma }\right)$ (31)

with,

${\tilde{V}}_{\gamma }={\left(\frac{1}{{\sigma }_{\epsilon }^2}{B}^{\text{T}}B+\lambda S\right)}^{-1},\tilde{\gamma }={\tilde{V}}_{\gamma }\frac{1}{{\sigma }_{\epsilon }^2}{B}^{\text{T}}\left(y^{*}-X\beta -Zu\right)$

Given;

$\lambda =\text{Gamma}\left({\alpha }_{\lambda },b_{\lambda }\right)$

The Gibbs sampler iterates over these conditional distributions to draw samples from the posterior. After convergence, the posterior samples provide estimates, credible intervals, and inference for all model parameters including fixed effects, smooth terms, random effects, and variance components [2] [28] [29] .

To ensure the validity and stability of the posterior distributions obtained from the Gibbs sampler, convergence diagnostics were performed following established guidelines in Bayesian inference [32] .

(a) Gelman-Rubin Diagnostic: The Gelman–Rubin diagnostic evaluates the relationship between multiple chains by comparing within-chain and between-chain variances.

$\hat{R}=\sqrt{\frac{\hat{V}}{W}}$ (32)

where;

$W$ = within chain variance

$\hat{V}$ = estimated target variance of the posterior

Decision Rule:

(b) Effective Sample Size (ESS): The ESS measures the number of effectively independent draws from the posterior distribution, accounting for autocorrelation. It is estimated using the formula expressed as follows;

$\text{ESS}=\frac{N}{1+2{\displaystyle {\sum }_{k=1}^{\infty }{\rho }_k}}$ (33)

Decision Rule:

(a) Posterior Predictive Checks (PPC): PPC was used to evaluate model fit by comparing replicated data from the posterior to the observed data. If the Bayesian p-value is close to 0.5, the model fits well. On the other hand, if the Bayesian p-value is near 0 or 1 it suggests lack of fit [32] .

(b) K-Fold Cross Validation: This was used to estimate predictive accuracy by partitioning data into training and validation sets. The formula is given as:

Let, the log predictive density be defined as;

$lp{d}_k=\underset{i\in fol{d}_k}{{\displaystyle \sum }}\log \left(\frac{1}{S}\underset{s=1}{\overset{S}{{\displaystyle \sum }}}p\left(y_i|{\theta }^{\left(s\right)}\right)\right)$

Sum across all folds that is the Expected Log Pointwise Predictive Density (ELPD);

$elp{d}_{K\text{-}fold}={\displaystyle {\sum }_{k=1}^{K}lp{d}_k}$ (34)

Higher ELPD = better predictive performance and differences > 4 - 10 can be meaningful depending on data size [33] .

(c) Bayesian R²: This measures the proportion of variance explained by the model and defined by Gelman et al. (2019) [34] as:

$R^2=\frac{Var\left(\hat{y}\right)}{Var\left(\hat{y}\right)+E\left[{\sigma }_{\epsilon }^2\right]}$ (35)

where,

$\hat{y}=X\beta +Sb+Zu$ the model’s posterior mean

$Var\left(\hat{y}\right)$ = the variance of fitted values across observations

${\sigma }_{\epsilon }^2$ = the posterior samples of residual variance.

$R^2$ close to 1 suggests a better model fit that is the higher the $R^2$ the better the model.

The linear mixed regression model involves two fixed effects and two random effects variables; the random coefficient and the random slope for exchange rate. Hence our hypotheses are given by:

H₀: ${\beta }_i=0$ (There is no statistically significant relationship between the fixed effects (exchange rate and monetary policy rate) and the response variable (inflation rate))

H₁: ${\beta }_i\ne 0$ (There is statistically significant relationship between the fixed effects and the response variable)

H₀: $u_{1i}=0$ (There is no variation in baseline Inflation Rate across continents; that is, the intercept is the same for all continents)

H₁: $u_{1i}\ne 0$ (There is variation in baseline Inflation Rate across continents; that is, the intercept differs between continents.)

H₀: $u_{2i}=0$ (There is no variation in the effect of Exchange Rate across continents, that is, the slope is the same for all continents)

H₁: $u_{2i}\ne 0$ (There is variation in the effect of Exchange Rate across continents, that is, the slope differs between continents)

In this study where a GAMM was adopted to estimate the parameters, the standard test statistic described by Wood (2017) [2] for a fixed effect ${\beta }_i$ is given by:

$t_{cal}=\frac{{\hat{\beta }}_i}{SE\left({\hat{\beta }}_I\right)}$ (36)

However, due to the hierarchical and smooth structure of GAMMs, which include both random effects and penalized smooth terms, the null distribution of the test statistic does not follow a standard t-distribution. This is particularly the case when random slopes or complex smooth functions are present. Thus, the degree of freedom associated with the null distribution of the test statistic cannot be approximated as $n-p$ rather an approximate degree of freedom is applied using the Satterthwaite approximation, which accounts for uncertainty in estimating random effects variances. The approximation is given as (Luke, 2017) [35] :

$df=\frac{{\left(Var\left(\hat{\beta }\right)\right)}^2}{{\displaystyle \sum \left(\frac{Var{\left({\hat{\beta }}_i\right)}^2}{n_i}\right)}}$ (37)

where:

$Var\left(\hat{\beta }\right)$ = the estimated variance of $\hat{\beta }$

$n_i$ = the number of observations associated with each variance component ${\hat{\beta }}_i$

$Var\left({\hat{\beta }}_i\right)$ = the variance of the $i^{th}$ component of the variance.

Decision rule:

All analyses will be carried out using R software.

Table 2 presents the dataset from five continents on Inflation Rate (IR), Exchange Rate (ER) and Monetary Policy Rate (MPR) for the period 2014-2023.

Source: Statista and Focus Economics data hub.

The descriptive statistics for exchange rate are provided in Table 3 . The coefficient of variation indicates that Africa’s exchange rate with CV = 31.70% is the most variable and uncertain while North America has the most stable exchange rate with CV = 9.92%.

Similarly, Table 4 shows the summary statistics for monetary policy rate. The result of the coefficient of variation reveals that North America’s monetary policy rate with CV = 55.04% is the most variable and uncertain while Africa has the most stable monetary policy rate with CV = 18.00%.

The summary statistics for inflation rate are displayed in Table 5 . The result of the coefficient of variation suggests that Europe’s inflation rate with CV = 64.05% is the most variable and uncertain while Africa has the most stable inflation rate with CV = 33.95%.

This section presents the results of a GAMM used to investigate the impact of Exchange Rate and Monetary Policy Rate on Inflation, with random effects for continent-specific variations, particularly random slopes for Exchange Rate.

The fixed effects estimates are reported in Table 6 . The intercept is statistically significant ( $p=0.002$), which suggests that the baseline level of inflation, that is, when covariates are zero, is approximately 2.5574.

The estimate of p-value = 0.639 > 0.05 for exchange rate suggests that exchange rate has a positive but statistically insignificant effect on inflation across all continents.

The smooth term for Monetary Policy Rate (interpreted through the basis function coefficient) is significantly negative, which suggests that higher monetary policy rates are associated with lower inflation levels. However, the estimated smooth function reveals that this relationship is non-linear: the effect is strongly negative at lower levels of monetary policy rate, flattens out as the rate increases, and may even become slightly positive at higher levels.

The random effects estimates are presented in Table 7 showing that the random slope variance for Exchange Rate (0.0143) is higher than the random intercept variance (0.0020). This reveals that the effect of exchange rate on inflation varies more substantially across continents than baseline inflation levels do. The residual variance (0.5047) captures the variability in inflation not explained by the model and the estimate suggests that some variation remains unexplained.

The continent-specific random slopes for Exchange Rate are summarized in Table 8 which illustrate heterogeneity in its relationship with inflation. While Europe and South America show a positive relationship between exchange rate and inflation, Africa, Asia, and North America show a negative relationship. This result indicates that the impact of exchange rate changes on inflation is not uniform globally, which underscores the use of a mixed model approach with continent-specific random slopes.

1. Test for Normality of Residuals

As observed from Figure 1 , there is a close alignment with the 45-degree reference line which suggests that residuals are normally distributed.

2. Test for Homoscedasticity

The Homoscedasticity Plot in Figure 2 indicates a uniform spread around zero which confirms the assumption of constant variance (homoscedasticity).

As observed in Figure 3 , at the various lags, the autocorrelation values lie within the 95% confidence bounds. The Durbin–Watson test results are given in Table 9 , which suggests the absence of significant autocorrelation.

In this analysis, the Durbin-Watson statistic is 2.1084, which is very close to 2. This indicates no evidence of autocorrelation in the model residuals. Again, the p-value = 0.6014 > 0.05 which confirms that we do not reject the null hypothesis and conclude that the residuals are not correlated that is, the assumption of independence is satisfied.

The model fit and predictive performance are reported in Table 11 . An adjusted R-squared value of 0.817 means that approximately 81.7% of the variation in inflation is explained by the model, after accounting for the number of predictors. This result reflects a very good model fit which reinforces that the included fixed and random effects capture the majority of the variability in inflation across countries and continents.

The posterior means of the fixed effects are shown in Table 12 , with corresponding 95% confidence intervals in Table 13 . The results suggest that exchange rate remains statistically insignificant, while the smooth term for monetary policy rate has a significant nonlinear effect on inflation. Closer inspection of the smooth function plot represented in Figure 6 , Figure 7 show that while inflation decreases with monetary policy rate at lower levels, the relationship reverses at higher values, producing a curved non-linear pattern.

Bayesian random effects are presented in Table 14 , showing minimal variability in the random slope of exchange rate, moderate continent-level variation in baseline inflation, and substantial variation in the nonlinear monetary policy effect.

The continent-specific random slopes are reported in Table 15 . All confidence intervals for continent-specific slopes cross zero which implies that although there is variation between the continents but the effect of exchange rate on inflation remains insignificant by continent.

The Gelman-Rubin diagnostics are presented in Table 16 summarising the PSRF for each model parameter specifically, Beta_mcmc, sigma2_e_mcmc, u_mcmc, and b_mcmc. A PSRF close to 1 suggests convergence across chains and most parameters show values near 1, indicating convergence. However, values above 1.1 for some parameters (e.g., Beta and u_mcmc) suggest that a few chains may not have fully converged, particularly those with point estimates of 1.24 - 1.29 and upper confidence bounds exceeding 1.5.

The multivariate PSRF values for Beta_mcmc, b_mcmc, and u_mcmc are 1.24, 1.01, and 1.21 respectively. Again, values above 1.1 (especially Beta_mcmc and u_mcmc) raise concerns about convergence for these sets of parameters.

Effective sample sizes (ESS) are reported in Table 17 which shows that most parameters have sufficiently large ESS (e.g., b_mcmc var6–var10 >10,000), suggesting reliable posterior estimates. However, low ESS values (e.g., Beta_mcmc var2 = 23.49, u_mcmc var6 = 21.03) indicate poor mixing in some chains.

Convergence is mostly achieved, but there are a few parameters (notably some random slopes and fixed effect terms) where PSRF and ESS suggest that results should be interpreted with caution.

From Figure 8 the observed and predicted values align reasonably well. Posterior predictive check (PPC) values are presented in Table 18 . PPC p-values which is greater than 0.05 suggest a strong model fit, indicating that the model replicates the observed data well.

The Bayesian R² values are given in Table 19 . The posterior mean (0.828489) with a confidence interval that excludes zero indicates that the model explains approximately 83% of the variance in inflation which suggests strong explanatory power.

Predictive performance assessed through 5-fold Cross-Validation (CV) is reported in Table 20 . The 5-fold CV Root Mean Square Error = 0.394 is relatively low, supporting good predictive performance. Generally, the model demonstrates strong fit and predictive power, with acceptable replication of data patterns.

The comparison between Frequentist and Bayesian GAMMs is summarized in Table 21 ;

1) The Bayesian GAMM offers slightly better predictive performance, as evidenced by lower RMSE value.

2) The Bayesian R² = 0.828 slightly exceeds the Frequentist adjusted R² = 0.817, which suggests slightly higher explanatory power.

3) The residual variance is considerably lower in the Bayesian model which indicates a tighter fit to the data.

4) While the Frequentist model detected greater random slope variability, the Bayesian estimate showed smaller variance and wider uncertainty intervals, which suggests weaker and less consistent evidence of slope variation across continents.

To examine the impact of exchange rate and monetary policy rate on inflation, this study tested the significance of both fixed and random effects using Frequentist and Bayesian GAMMs. The hypotheses focused on whether (i) exchange and monetary policy rates (fixed effects) significantly influence inflation, (ii) baseline inflation levels vary across continents (random intercepts), and (iii) the effect of exchange rate varies by continent (random slopes).

The results from both approaches indicated that the monetary policy rate has a statistically significant effect on inflation, while the exchange rate did not exhibit a statistically significant effect on inflation. These findings align with long-standing macroeconomic theories, such as the Taylor Rule, and reinforce conclusions from prior empirical studies across regions, including those in Africa, Asia and North America which found the monetary policy rate to be a consistently strong determinant of inflation. In contrast, the weak or inconsistent effect of the exchange rate echoes evidence from Europe and Canada, where exchange rate pass-through has diminished over time or is actively mitigated by monetary policy frameworks.

By adopting a hierarchical modelling approach, this study was able to explore continent-specific differences in inflation dynamics, going beyond many existing studies that rely on traditional time-series or panel methods. The random intercept results showed strong evidence of variation in baseline inflation across continents, supporting the literature’s emphasis on structural and institutional differences such as trade exposure, exchange rate regimes, and fiscal stability, as key determinants of regional inflation trends.

In the evidence for random slopes, the variation in the exchange rate’s effect across continents was mixed. The Frequentist GAMM indicated moderate heterogeneity, suggesting that exchange rate influences may differ by continent, particularly in regions with external vulnerabilities like Latin America or Sub-Saharan Africa. In contrast, the Bayesian GAMM provided weaker evidence, with credible intervals including zero, suggesting that any heterogeneity is not statistically robust within that framework. This discrepancy mirrors findings from empirical literature where exchange rate pass-through is shown to be context-dependent, stronger in developing and commodity-exporting economies (e.g., Nigeria, Brazil) and weaker in advanced or inflation-targeting economies (e.g., US, Eurozone).

In terms of model performance, Bayesian GAMM slightly outperformed the Frequentist GAMM in terms of fit and prediction. This result underscores the literature advocating Bayesian hierarchical models for their ability to flexibly capture complex data structures and accommodate uncertainty in parameter estimates, especially in macroeconomic contexts with nested and cross-regional influences.

The hypothesis testing revealed that for the monetary policy rate (fixed effect), the null hypothesis was rejected under both approaches, confirming its robust influence on inflation. Conversely, for the exchange rate (fixed effect), the null hypothesis was not rejected, aligning with empirical studies that found weak or inconsistent global effects.

The result for the random intercepts shows that the null hypothesis was rejected, indicating significant differences in average inflation levels across continents. Similarly, for random slopes, the null hypothesis was rejected in the Frequentist model but weakly supported in the Bayesian model, indicating potential but inconclusive heterogeneity in the exchange rate’s impact on inflation across continents.

These findings affirm the central role of monetary policy as a global anchor of inflation, while also highlighting the conditional and region-specific relevance of exchange rate effects. They further validate the study’s methodological approach, which was deliberately chosen in light of existing literature:

1) Allowing random slopes for the exchange rate to capture regional heterogeneity.

2) Fixing the effect of the monetary policy rate globally, based on its documented consistent impact across diverse contexts.

3) Using both Frequentist (REML-based) and Bayesian (Gibbs Sampling) GAMMs to ensure robustness of inference and improved predictive performance.

By extending the traditional linear mixed model with a more flexible hierarchical structure, this study contributes a richer, more globally sensitive understanding of how exchange rates and monetary policy jointly shape inflation. It responds directly to gaps in the literature, where most prior studies failed to simultaneously capture global patterns and regional variations within a unified statistical framework.

This study has some limitations:

1) The analysis is conducted at the continental level, which may mask important within-continent heterogeneity; future studies could extend this work by incorporating country-level data or refined regional groupings to provide a more detailed assessment of cross-country differences.

2) The findings are associational rather than causal, and the possibility of endogeneity cannot be ruled out. For example, central banks may adjust monetary policy in response to inflation trends which introduces potential reverse causality. In addition, unobserved confounding factors, feedback mechanisms or omitted variable bias may influence the observed relationships. There is a need for future research using causal identification strategies.

3) Furthermore, the study covers a short time span (2014-2023). While this period was selected to reflect recent structural and policy environments, it does not capture longer-term inflation dynamics across different monetary regimes. Extending the time horizon in future research would provide greater statistical power and allow for comparisons across distinct policy eras.