Comments on “Common Drifting Volatility in Large Bayesian VARs”

1. Comments on “Common Drifting Volatility in Large Bayesian VARs” Keith Ord November 16, 2012

2. Models: A Perspective “A model is a metaphor of limited applicability, not the thing itself” [page 54] “I will remember that I didn’t make the [business] world , and it doesn’t satisfy my equations.” [page 198] Emanuel Berman, Models. Behaving. Badly. Free Press, New York, 2011.

3. Model Building • “Realistic” models often have too many parameters for effective estimation – we run into numerical difficulties, unstable solutions or both • Stochastic Volatility [SV] has considerable appeal as a modeling mechanism but numerical problems currently restrict its use to small Bayesian VAR systems • CCM have made ingenious use of a Kronecker structure to enable much larger systems to be estimated and demonstrate that there is little if any loss in efficiency relative to a full SV. • At the same time CSV improves considerably on B-VAR models without an SV component.

4. Related Literature • Reducing the number of parameters is often viewed with alarm, yet the procedure has a distinguished history: • Stein’s 1961 classic shrinkage of a multivariate mean vector • Miller and Williams ( IJF 2004) showed that Stein-type shrinkage of seasonal factors produces improved forecasts • Traditional seasonal adjustment methods (e.g. X-12 ARIMA) may also benefit from such shrinkage (Findley et al., IJF 2004) • Stock and Watson’s work (e.g. JASA, 2002) on dynamic factor analysis allows regression models to accommodate a large number of predictors • CCM’s work may be viewed in this tradition, but recognizing that the ‘shrinkage’ to a single common SV is necessary to achieve the Kronecker structure. Question: Pajor (2006, op. cit.) suggests that Tsay’s SV scheme does better than regular SV. Did you explore this issue with your data?

5. Is There Another Way? • Anderson and Moore (1979) introduced an innovations framework for a multivariate state-space scheme: • This scheme, discussed for example in Hyndman et al (2008, Chapter 17) readily maps into a VAR (or VARMA) scheme and has the advantage that updates do not require the Kalman filter. • As with other MV schemes, the system can quickly get parameter-heavy and various methods exist for reducing the dimensionality of the parameter space. • But what about SV?

6. Innovations (C)SV? • In the univariate case we would consider: • Here, u(.) may take various forms such as the log of the squared error, with a small constant added when near-zero errors are a risk. So perhaps it is better described as a GARCH approach • In the MV case we could proceed directly with • Or, we might consider a CSV form: Question: Is CSV superior?

7. References • Anderson, B.D.O. and Moore, J.B. (1979) Optimal Filtering. Englewood Cliffs: Prentice-Hall • Findley, D.F., Wills, K.C. and Monsell, B.C. (2004) Seasonal adjustment perspectives on damping seasonal factors: Shrinkage estimators for the X12-ARIMA program. International Journal of Forecasting, 20, 5521-556. • Hyndman, R.J., Koehler, A.B., Ord, J.K. and Snyder, R.D. (2008), Forecasting with Exponential Smoothing: The State Space Approach, New York and Berlin: Springer. • Miller, D.M. and Williams, D. (2004) Damping seasonal factors: Shrinkage estimators for the X12-ARIMA program. International Journal of Forecasting, 20, 529 – 549. • Stock, J. H. & Watson, M. W. (2002). Forecasting using principal components from a large number of predictors. Journal of the American Statistical Association, 97, 1167-1179.