8.6 CONCLUSION

Univariate models such as threshold autoregressions or smooth threshold autoregressions have been motivated as nonlinear extensions to the linear AR specification. While such models can sometimes capture nonlinear dynamics, their parameters are often poorly estimated and effectively fitted to very few episodes in the data. This means that forecasts from such models sometimes become quite extreme, rendering estimation error particularly important.3

Another class of models, Markov switching models, have the potential to be more robust to this limitation of the nonlinear models as they limit the range of forecasts based on the regimes identified in the historical sample. Moreover, these models have proved easy to use to generate multiperiod forecasts under the assumption that state transitions are driven by a first-order homogeneous Markov chain. The general evidence, consistent with what we find here, is that Markov switching models can be quite successful for variables that display clear regime switching behavior such as interest rates, but produce less accurate forecasts for variables whose regimes are either historically unique, and thus do not repeat over time, or have fewer regime shifts so that the regime-specific parameters cannot be estimated with sufficient precision.

Some authors (e.g., Teräsvirta, 2006) argue that forecasts from individual nonlinear models can be combined with forecasts from other nonlinear models or with forecasts from simple, linear models. This is a promising way to handle the robustness issues arising due to the effect that estimation error has on nonlinear models in particular. We discuss this point further in chapter 14. Density forecasting is another area for which nonlinear univariate models have shown some promise. We discuss this in chapter 13.

Vector Autoregressions

Since their introduction to econometrics by Sims (1980), vector autoregressions S(VARs) have become a workhorse for forecasting macroeconomic and financial time series. VARs generate dynamic forecasts in a way that ensures consistency across different equations and forecast horizons. They are frequently used to forecast macroeconomic variables such as inflation and GDP growth and are integral to the calculation of stock prices in the log-linearized present value model of Campbell and Shiller (1988).

Sims (1980) argued that the large-scale macroeconomic forecasting models used at the time of his study were built on shaky foundations, including identifying restrictions that were difficult to justify. Instead he proposed to use unrestricted VARs of much lower dimension, the idea being that the data would help identify any patterns that might be helpful for prediction purposes.

Section 9.1 introduces vector autoregressions. Section 9.2 discusses classical estimation of VARs and their use in forecasting. We cover Bayesian approaches to estimation and forecasting with VARs, including models with time-varying parameters and large-dimensional VARs, in section 9.3. Section 9.4 discusses dynamic stochastic general equilibrium (DSGE) models which embed VARs in the context of a macroeconomic model that imposes restrictions from agents’ optimizing and forward-looking behavior. Section 9.5 covers the use of conditional forecasts, while section 9.6 provides empirical examples and Section 9.7 concludes.