The NSE-20 data used for model estimation consisted of daily observations transformed into log returns. Forecasting was performed using the daily series, while the reported forecast results were presented in monthly aggregated form for ease of interpretation.

The study utilized both simulated and real data. The empirical analysis was based on the Nairobi Securities Exchange 20 Share Index (NSE-20) historical data covering the period from December 1997 to March 2026.The data were obtained from the official Nairobi Securities Exchange. All simulations, model estimation, and forecasting procedures were conducted using R software.

Prior to model estimation, preliminary analysis was performed to determine the appropriate autoregressive order. Partial Autocorrelation Function (PACF) plots were examined, and competing models were compared using the Akaike Information Criterion (AIC). The model with the lowest AIC was selected for subsequent analysis [3].

The nonlinear dynamics of the NSE-20 Share Index were modelled using the two-regime Exponential Smooth Transition Autoregressive (ESTAR) model. The model is expressed as

where

G ( y t − d ; γ , c ) represents the smooth transition continuous function bonded between zero and one.

γ > 0 is responsible for the smoothing parameter.

p is the lag order.

c represents the location parameter.

d represents the delay parameter.

y t − d represents the threshold variable which is taken to be a lagged endogenous variable of y t .

ϕ i 1 and ϕ i 2 are autoregressive parameters of the model

The exponential transition function is defined as:

The transition function is bounded between zero and one, allowing smooth movement between regimes. The delay parameter determines the threshold variable governing regime switching, while the smoothing parameter controls the speed of transition between regimes.

Prior to fitting the ESTAR model, the data were tested for nonlinearity using the Lagrange Multiplier (LM) test. The nested hypothesis testing procedure proposed by Teräsvirta [2] was then employed to determine whether the ESTAR or Logistic Smooth Transition Autoregressive (LSTAR) specification was more appropriate.

Bayesian estimation was adopted to estimate the ESTAR model parameters. Let

where

z ∈ ℝ n × 2 ( k + 1 ) is the design matrix.

ϕ ∈ ℝ 2 ( k + 1 ) is the collection vector of autoregressive parameters.

Assuming ε t ~ i i d   N ( 0 , σ 2 ) ;

denote the vector of unknown model parameters.

Parameter estimation was based on Bayes’ theorem, where the posterior distribution is proportional to the product of the likelihood function and the prior distribution,

The likelihood function was constructed under the assumption that the model errors were independently and normally distributed with mean zero and variance σ 2 . Posterior inference was obtained by combining the likelihood with the specified prior distributions.

Non-informative prior distributions were adopted to minimize subjective influence on posterior estimation. Normal prior distributions with large variances were assigned to the autoregressive coefficients and location parameter, while the variance parameter was assigned an inverse-gamma prior. The smoothing parameter was constrained to positive values throughout estimation to satisfy the ESTAR model assumptions.

The delay parameter was treated as a discrete parameter and estimated jointly with the remaining model parameters during the Bayesian estimation procedure.

Posterior distributions were obtained using the Markov Chain Monte Carlo (MCMC) algorithm. Gibbs sampling was employed to update parameters whose full conditional distributions were available, while the Metropolis-Hastings algorithm was used to sample parameters whose conditional distributions did not have closed-form expressions.

The smoothing parameter was updated using the Metropolis-Hastings algorithm because of its nonlinear role in the transition function. Candidate values were generated from a proposal distribution and accepted according to the Metropolis-Hastings acceptance probability. The delay parameter was estimated by evaluating candidate delay values and updating the parameter according to their posterior probabilities.

To assess convergence of the Markov chains, trace plots, posterior density plots, and autocorrelation function (ACF) plots were examined for all estimated parameters. Posterior summaries, including posterior means, standard deviations, and 95% credible intervals, were computed after discarding the burn-in iterations.

The forecasting performance of the Bayesian ESTAR model was evaluated using both in-sample and out-of-sample forecasts. Model accuracy was assessed using the Root Mean Square Error (RMSE) and the Mean Absolute Error (MAE), where lower values indicated superior forecasting performance.

The forecasting results obtained from the Bayesian ESTAR model were subsequently compared with those of the Bayesian SETAR model to evaluate the effectiveness of modelling gradual regime transitions in forecasting the Nairobi Securities Exchange 20 Share Index.

The descriptive statistics of the Nairobi Securities Exchange 20 Share Index (NSE-20) were computed to summarize the distributional characteristics of the data before model estimation. The statistics provide information on the central tendency, dispersion, and distributional properties of the series and offer preliminary insights into its behavior.

Table 1 shows that the NSE-20 Share Index exhibited considerable variability during the study period, indicating substantial fluctuations in market performance. The differences between the minimum and maximum values suggest pronounced changes in market conditions over time. The distributional measures further provide preliminary evidence that the series may deviate from normality, thereby supporting the use of nonlinear modelling techniques.

Table 1. Descriptive statistics for the NSE-20 share index.

Figure 1 illustrates the evolution of the NSE-20 Share Index over the study period. The series exhibits periods of sustained growth interspersed with episodes of decline, suggesting the presence of changing market regimes. These fluctuations motivate the application of nonlinear time series models capable of accommodating regime-switching behavior.

Figure 1. Time series plot of the NSE20-share index.

Prior to model estimation, the stationarity of the series was assessed using the Zivot-Andrews [4] unit root test, which accounts for possible structural breaks in the data.

The results presented in Table 2 indicate that the transformed series was stationary after accounting for structural breaks. The null hypothesis of a unit root was rejected, confirming that the data satisfied the stationarity assumption required for estimating nonlinear autoregressive models.

Table 2. Zivot-Andrews unit root test results.

The Lagrange Multiplier (LM) test was employed to determine whether the linear autoregressive model adequately described the data or whether a nonlinear STAR model was more appropriate [5].

As shown in Table 3, the null hypothesis of linearity was rejected, providing statistical evidence of nonlinear dynamics in the NSE-20 Share Index. The results therefore justified the use of the STAR modelling framework instead of conventional linear autoregressive models.

Table 3. Lagrange multiplier test.

Following confirmation of nonlinearity, the nested hypothesis testing procedure proposed by Teräsvirta [2] was applied to determine the appropriate transition function (Table 4).

Table 4. Nested hypothesis test for selecting the STAR transition function.

The nested hypothesis tests favoured the Exponential Smooth Transition Autoregressive (ESTAR) specification over the Logistic Smooth Transition Autoregressive (LSTAR) model. This finding suggests that the transition between market regimes occurs gradually rather than abruptly, supporting the adoption of the Bayesian ESTAR model for subsequent analysis.

A set of simulated data of length N = 2000 were generated and analyzed from the model below:

where ε t ~ i i d   N ( 0 , 0.008 2 ) and G ( y t − d ; γ , c ) = 1 − exp [ − 40 ( y t − 2 − 0.08 ) 2 ] . Therefore, the true parameter values are p = 2 , ϕ = ( 0.01 , 1.55 , − 0.88 , 0.03 , − 1.05 , 0.35 ) T , σ 2 = 0.08 2 , d = 2 , γ = 40 , c = 0.08 . The true parameter values were deterministically initialized from Livingston and Nur [6].

Posterior sampling was performed using 20,000 MCMC iterations, of which the first 2,000 iterations were discarded as burn-in. Gibbs sampling was used for parameters with tractable full conditional distributions, while the smoothing parameter was updated using a random-walk Metropolis-Hastings step. Convergence was assessed using trace plots, posterior density plots, and autocorrelation diagnostics.

Posterior inference was obtained using Markov Chain Monte Carlo methods. Posterior means, posterior standard deviations and 95% credible intervals were computed for all model parameters.

Table 5 presents the posterior summaries of the Bayesian ESTAR model parameters. The posterior means and corresponding 95% credible intervals indicate that the model parameters were estimated with satisfactory precision. The estimated smoothing parameter confirms the presence of gradual regime transitions, while the estimated location parameter identifies the transition point between market regimes.

Table 5. Posterior parameter estimates for the Bayesian ESTAR model.

The estimated Bayesian ESTAR model with the estimated parameters is shown in Equation (6) below:

where

The convergence of the Markov chains was evaluated using graphical and numerical diagnostic tools (Figures 2-4).

Figure 2. Trace plot for γ in NSE-20 log returns.

Figure 3. Posterior density plot for γ in the NSE-20 log returns.

Figure 4. ACF plot for γ in the NSE-20 log returns.

The trace plots indicate stable oscillations around constant posterior means with no evidence of systematic trends, suggesting convergence of the Markov chains. The posterior density plots are unimodal and approximately symmetric, while the autocorrelation functions decline rapidly with increasing lag, indicating good mixing of the chains. These diagnostics confirm that the posterior estimates are reliable for statistical inference.

For forecasting evaluation, the NSE-20 data were divided into a training sample covering December 1997 to March 2025 and an out-of-sample testing period covering April 2025 to March 2026. The Bayesian ESTAR and Bayesian SETAR models were estimated using the same training data and evaluated over the same forecast horizon.

Table 6 presents the forecast accuracy of the Bayesian ESTAR model for both the in-sample and out-of-sample periods. As expected, the model achieved lower forecast errors for the in-sample forecasts, with an RMSE of 41.83 and an MAE of 32.45, indicating a good fit to the training data.

Table 6. In-sample and out-sample forecasting performance of ESTAR.

The out-of-sample forecasts produced an RMSE of 57.16 and an MAE of 44.28, reflecting a moderate increase in forecast error when predicting unseen observations. Despite this increase, the relatively small differences between the in-sample and out-of-sample error measures suggest that the Bayesian ESTAR model generalizes well and maintains satisfactory forecasting performance for the NSE-20 share index.

The forecasting performance of the Bayesian ESTAR model was evaluated using both in-sample and out-of-sample forecasts. Forecast accuracy was assessed using the Root Mean Square Error (RMSE) and Mean Absolute Error (MAE).

Table 7 shows that the Bayesian ESTAR model achieved low forecast errors for both the in-sample and out-of-sample periods, indicating satisfactory predictive performance. Figure 5 further demonstrates that the forecasted values closely follow the observed NSE-20 Share Index, confirming that the model effectively captures the nonlinear dynamics and gradual regime transitions of the series.

Table 7. Forecasting performance of the Bayesian ESTAR model.

Figure 5. Actual vs Forecasted NSE-20 share index.