Time-varying parameter VARs estimated the easy way using kernels

1. Time-varying parameter VARs estimated the easy way using kernels Lecture to MSc Time Series students, Bristol, Spring 2014

2. Simple, univariate, random coefficients model Cogley and Sargent introduced this model into empirical macro, though used very widely before this in empirical biology, stats, finance Random walk process is for parsimony. Includes assumption that the process is bounded, and by the unit circle. Note ln assumption for the shock to the variance of the data.

3. Why are we interested in TVP-VARs? • Well, why should VARs be time-invariant? • They could be mis-specified. • Lucas suggested TVP-VARs arise out of shifting monetary and fiscal policy reactions, in turn derived from policymakers ‘chasing tail’ with mis-specified econometric models. • VARs are key diagnostic tools for certain propagation mechanisms. • If this propagation is highly time-varying in the data, then it had better be thus in our model too. • And if it isn’t, that says something is wrong with our model. • And if it is, then it may suggest time-variation is caused by policy.

4. Example of TVP-VAR Cogley and Sargent estimated just this VAR, with x=unemployment. Kapetanios and Yates estimated it too, in a way that is simpler, and which I’m going to show you.

5. CSP, univariate model of inflation persistence Cogley, Sargent and Primiceri () ‘Inflation-gap persistence’ Rise and then fall in univariate inflation persistence. Profound implications for the memory of a shock, or the unconditional variance. Key part of the debate on the Great Moderation.

6. Why is tv of inflation persistence interesting? • Inflation persistence was a puzzle from the standpoint of early sticky price macro models. • Rationalised as product of i) indexation [CEE] ii) habits in consumption [?]. iii) bad monetary policy [Dittmar, GavinandKydland; Benati] • TV leans more towards iii) • TV in persistence=tv in forecastability • Inflation is a characteristic of all modern economies and monetary macro models, so features of it are very important

7. Estimation: Gibbs-Sampling, and kernels/rolling windows • Two main methods. • Difficult method: Bayesian estimation using Gibbs Sampling. Complex multi-block distribution factored carefully into pieces we can draw from. Breaks down with large VARs. • Easy method: kernel estimation, which is very like rolling windows.

8. Kernel estimation of a tvp-ar(1) If we set K=1 for all periods, we just have OLS for an AR(1) [x’x inv, x’y] If we set K=1 for some periods, then 0 outside this, then we have rolling regression Final equation here is a Normal kernel (ie has the form of the normal density function h is the ‘bandwidth’ = the ‘variance’ of the normal.

9. Illustrating how the kernel changes through the sample Observations more distant from the current t get less weight.

10. Some conditions required for it to work • Rho_t doesn’t move around too suddenly • It’s bounded inside the unit circle. Ie the VAR/AR is always ‘instantaneously’ stationary • The kernel satisfies some regularity conditions.

11. Bandwidth choice for kernel estimation • In principle bandwidth should be data dependent. • Depends on how informative neighbouring observations are about the current one. • Very slow moving series=high bandwidth. • But don’t know how slow moving rho_t is until we have estimated it. • Suggests data-dependent algorithm....

12. Algorithm for data-dependent bandwidth • Start with commonly used bandwidth, eg h=T^0.5. • Generate sequence for rho_hat_t. • Set new bandwidth based on how fast moving you estimate rho_hat_t to be. • Redo estimation. • Iterate until convergence. • In practice, T^0.5 does a great job, so we don’t bother.

13. Sources on TVP-VARs: 1) work by me • Kapetanios and Yates ‘Evolving UK and US macroeconomic dynamics...’ • Giraitis, Kapetanios and Yates ‘Inference on stochastic time varying coefficient models’ • ....`Inference on multivariate stochastic time-varying coefficient models’ • Giraitis, Kapetanios, Theodoridis and Yates ‘Estimating tvp DSGE models using minimum distance methods’

14. Sources on TVP-VARs: 2) work by others • Cogley and Sargent: drifts and volatilities paper • Cogley, Sargent and Primiceri: inflation gap persistence • Benati and Mumtaz: evolving US macro dynamics with identified shocks • Mumtaz and Sander-Plasman: tvp model of real exchange rate dynamics

15. Outputs of the TVP procedure TVP estimation gives us estimated VAR objects [coefficients A and vcov] for every observation in the sample. For each of those objects, we can perform identification just as we did with the fixed coefficient exercise. This gives us ingredients for tvirf to monetary policy shock identified by cholesky factor, in a 3 variable VAR, say, with ir, inf, ygap.

16. Charts on next slide from GKY ‘inference on multivariate….’ • Gives insight into how economy became more RBC-like towards end [LHS] of the sample, as hours work rise strongly then. • Also gives insight into costs of using small dimension VAR.

18. Charts from Kapetanios Yates ‘evolving…’ Show time-variation in AR model for inflation for the US. Note middle chart of top panel. The more such features change, the more they cast doubt on fixed-coefficient models. Also cast doubt on hard wired behavioural explanations, in favour of regime-change explanations.