As mentioned earlier, a growing body of literature has revealed that forecast perfor- mance is often not stable over time (see, e.g. Stock and Watson (2007, 2010), Chan et al (2012), Chan (2013), D'Agostino et al (2013), Clark and Ravazzolo (2014), Chan (2015) or Chiu et al (2015)). With this literature in mind, we investigate the intertemporal forecast performance of time varying AR and VAR models under both Gaussian and Student's-t distributions over time by plotting the cumulative sums of log predictive likelihoods. Since they do not provide the best forecasts of any variables over any time horizons we exclude the Markov switching models from this intertemporal analysis. The univariate and multivariate ination results are in Figures 3.6 and 3.7, the output results are in Figures 3.8 and 3.9 and the interest rate results are in Figures 3.10 and
3.11.
Overall, from a holistic macroeconomic modelling perspective, it can be seen that the TVP-VAR-SVt and the simple rolling window AR-t models respectively provide the most accurate multivariate and univariate forecasts. More generally, a few patterns in the forecast performance of all series are worth discussing. First, when comparing Gaussian and fat-tail models, with but one exception in the interest rate forecasts, the fat-tail models produce superior forecasts across all variables. This shows that mod- els with fat-tails produce better forecasts as compared to their Gaussian counterparts. Next, when comparing models with and without stochastic volatility, the models with stochastic volatility produce superior forecasts across all variables. This shows that models with stochastic volatility produce better forecasts as compared to their xed counterparts. Finally, when comparing models with and without time varying parame- ters, the TVP-AR and TVP-VAR models consistently produce superior forecasts across all variables. This shows that models with stochastic volatility and fat-tails produce better forecasts as compared to their xed Gaussian counterparts.
It is also worth discussing some interesting features of the forecast performance of individual variables. First, when looking at the ination results, it's noticeable that before the year 2000 the Gaussian and fat-tail models produce similar forecasts. After 2000 there is a divergence with fat-tail models clearly outperforming the Gaussian counterparts. This break is likely due to the introduction of the goods and services tax (GST). A dierent pattern emerges in the RGDP forecasts results. Specically, rather than a divergence in forecast performance following 2000 there is almost no evidence of a break with dierence between the fat-tailed model and the Gaussian model remaining relatively consistent over the majority of the sample period. A noticeable break does occur in 2006 however, when comparing the multivariate TVP-VAR-SVt and TVP- VAR-SV models. A similar result is found in the multivariate interest rate forecast results in which accounting for fat-tails improves the forecastability of interest rates after the 2007/08 GFC period.
3.6 Conclusion
We assess whether modelling structural change and fat-tailed events can improve the forecast accuracy of key Australian macroeconomic variables: real GDP growth, CPI in- ation and a short-term interest rate taken to be the 90 day Bank Accepted Bills/Negotiable Certicates of Deposit. Methodologically, we incorporate time variation and fat-tails into traditionally Gaussian, xed coecients multivariate and univariate autoregressive models. The class of univariate autoregressive (AR) and multivariate vector autoregres- sive (VAR) models allow for time variation via two sources: (1) in the models coe- cients, (2) in the variance of the shocks. For the multivariate models we consider a third source of time variation via the covariance terms. In addition to accounting for time variation within the coecients and volatilities, all models are estimated under both Gaussian and Student-t error distributions. Adding fat-tails to various models allows increases the likelihood of extreme events and may lead to faster adaptation to expan- sions and/or recessions. For completeness, we also consider the forecast performance of non-linear regime switching as well as rolling-window ARs and VARs.
The results yield four important ndings. First, fat-tailed models consistently out- perform their Gaussian counterparts. Second, time varying parameters and stochastic volatility improves forecast performance across all variables relative to a constant pa- rameter benchmark. Third, stochastic volatility models under a Student's-t distribution are found to generate more accurate density forecasts as compared to the same models under a Gaussian specication. Taken together these results suggest that both struc- tural instabilities and fat-tail events are important features in modelling Australian macroeconomic variables. Finally, when comparing the forecast accuracy of univariate and multivariate models the simple rolling window autoregression with fat-tails pro- duces the most accurate output growth forecasts, whilst the time varying parameter vector autoregression with stochastic volatility and fat-tails produces the best interest and ination forecasts. Nonetheless, from a holistic macroeconomic modelling perspec- tive, the vector autoregression with the proposed modelling features provides important information for central bankers policy decisions.
We note that we have only provided an out of sample study of the proposed mod- elling features. For future research it would be useful analyze the in sample t by incorporating structural instabilities and fat-tails into general equilibrium models of the Australian economy. For instance, the New Keynesian model of Australia devel- oped by Jääskelä and Nimark (2011) could be extended by allowing for time varying Student's-t distributed disturbances within both aggregate demand and supply shocks.