Inference with Time-Varying Parameter Models using Bayesian Shrinkage

In macroeconomics, predicting future realisations of economic variables is the central issue for policymakers in central banks as well as the investors in businesses. Nowadays, it is common to have a large number of economic variables in the time series data, and selecting the important variables is essential in achieving quality forecasts. Hence, our research interest lies in the variable selection problem within time series data analysis.

One of the key factors to determine the quality of a forecast is the model that is used for inference. The Time-Varying Parameter (TVP) model is an important means of understanding the effect of predictor variables on a given response. This allows the defined data to vary with time. However, the effects can be difficult to estimate and interpret if the number of predictors are large. To perform variable selection techniques in Bayesian inference, we apply continuous shrinkage priors to the TVP model in order to select the important variables over time. In particular, we are interested in three shrinkage priors: Bayesian Lasso, Normal-Gamma and Dirichlet-Laplace. These continuous shrinkage priors have Bayesian hierarchical representations according to the scale mixtures of Normals, which are encouraged to shrinkage by the assumption of sparsity.

Markov Chain Monte Carlo (MCMC) algorithms are used to estimate the TVP models, as the model can be expressed in a state space model. This is particularly useful when we are estimating the time-varying regression coefficients by using the Kalman filter forward and backward sampling steps. In addition, the dynamic variance components are sampled by fitting a stochastic volatility model within the MCMC framework.

To further improve the estimations of the time-varying parameters, such as the time-varying regression coefficients and the stochastic volatilities, we then move on to using the Particle Gibbs (PG) and the Particle Gibbs with Ancestor Sampling (PGAS) algorithms as both allow the joint updates of the two time-varying parameters by using Conditional Particle Filters (PF). However, when the time-varying regression coefficients and the stochastic volatilities are highly correlated, both PG and PGAS algorithms produce poor estimations for these parameters. We make an improvement within the PG framework by marginalising the stochastic volatilities over the time-varying regression coefficients. The marginalised PG updates both parameters in two steps: the first is to update the stochastic volatilities by using the PG with Kalman filter forward step and the second is to update the time-varying regression coefficients by using the Kalman filter backward step.

In order to draw comparisons for shrinkage strength among the shrinkage priors for the TVP model, the aforementioned sampling methods, such as MCMC, marginalised PG and PGAS are applied to equity premium data. We find that both Normal-Gamma and Dirichlet-Laplace have shrinkage by selecting fewer variables compared to Bayesian Lasso. The posterior estimates of the parameters obtained by marginalised PGAS tend to mix better and converge to stationaries faster than the results from MCMC and marginalised PG.

Finally, we compare the forecasting power between the TVP model with stochastic volatility and the TVP model with constant variance with all shrinkage priors. Our findings suggest that by assuming the variance component to be constant, the TVP model with constant variance outperforms the TVP model with stochastic volatility in forecasting for all shrinkage prior set-ups. Both Normal-Gamma and Dirichlet-Laplace produce smaller values of root-mean-square-error and the log predictive score when making one-step-ahead out of sample predictions.