Autoregressions in Small Samples

Vector autoregressions (VARs) are a key tool in econometrics, and remain immensely popular in applied econometric work. Many applied papers use ordinary least squares (OLS) to estimate VARs, despite the fact that it is well known that OLS suffers from a bias in small samples: it tends to underestimate persistence. Many corrections have been proposed for OLS to try and correct for this bias, but their use is restricted as one is liable to criticism for having made ad hoc assumptions if one uses them. The same is true of Bayesian inference in VARs, where the choice of prior is influential on your results and controversial.

CEP member Albert Marcet, and Marek Jarocinski, aim to overcome such criticisms and design a widely acceptable correction to OLS in small samples. Their fundamental idea is simple. They use a Bayesian approach, and try and find a way of using a prior that is as transparent as possible, and on which most economists would agree.

To use Bayesian methods one most have a prior about the distribution of the parameters you are trying to estimate. This is where the controversy comes from, as expressing your priors in this way is often very non-transparent. For example, one may have to express a prior about the distribution of the fourth autocorrelation parameter of one of the series. The innovation of the current authors is to allow researchers to express their priors as priors about the data. For example, a researcher could give her prior about the initial growth rates of each of the series in her VAR. The authors argue that this kind of prior is much more transparent than a prior on parameters, and is also much more likely to be similar across different practitioners. For example, if one of your series is real GDP growth, it is likely that most economists would give a prior for its growth rate with mean of around 3%.

To use these kinds of priors – priors on the data – in Bayesian estimation one must eventually transform them into priors on the parameters, as they are required for use in Bayes rule. Thus the authors propose a method of numerically converting these data priors into parameter priors prior to estimation.

They present empirical applications, one on the famous study by Christiano et al. (1999) on the macroeconomic effects of monetary policy shocks. They demonstrate that their method works in this environment and makes a quantitative difference to the results: applying their data driven prior, they find that the estimated effect of monetary policy shocks on output is much higher than originally estimated by Christiano et al.(1999).

In forming their estimator the authors also discuss the relationship between OLS and Bayesian econometrics in VARs, and an apparent puzzle: Even though OLS is known to be biased in small samples, it is still the best estimator for a Bayesian who uses a flat prior and quadratic loss. This means that a classical economist who uses OLS will be unhappy, because he knows OLS is biased, and will apply corrections. But a Bayesian with a flat prior will be happy with OLS, since it is, in his opinion, the best estimator.

The authors show that there is actually no puzzle here. They show that the disagreement boils down to a disagreement about initial conditions, and that for the same initial conditions, classical economists and Bayesians agree. However, they do show that using OLS implies extreme prior beliefs about initial growth rates. They show that in a VAR with a constant term, using OLS (which corresponds with having a flat prior) implies that the researcher believes that the growth rates of the series used in her VAR is very likely to exceed 100% in the first few periods. Since for most economic series, such as GDP, this is a very implausible prior they argue that one should avoid using OLS in any setting with data on GDP or similar series.

For further reading see:

"Autoregressions in Small Samples: Priors about Observables and Initial Conditions"[Link to paper ] (Marek Jarocinski, Albert Marcet), CEP Discussion Paper 1061 July 2011