Time variation in an optimal asymmetric preference monetary policy model

In a similar framework where central bank preferences depend on output gap volatility instead of unemployment volatility, Surico (2007) finds that output gap volatility is not important in characterizing inflation bias during the Great Moderation period of 1982:4–2002:3.

The linex function was introduced by Varian (1974) in the context of Bayesian econometric analysis. More recently, Nobay and Peel (2003) introduced it in the optimal monetary policy analysis.

For instance, recent empirical evidence [among others, Atkeson and Ohanian (2001), Smets and Wouters (2007), Ball and Mazumder (2011)] suggests that the slope of the Phillips curve has decreased in recent decades.

See Surico (2007, p. 317) for an excellent review of anecdotal evidence found in the literature of the changing political pressures faced by the Fed over the post-war period.

In particular, this occurs when ϕ_t is the only time-varying parameter and follows the same AR(1) process as the one describing c_t. This can be seen by looking at the first-order condition of the central bank’s optimization problem [see the first order equation on Ruge-Murcia (2003, p. 1380)] and the fact that ϕ_t, being a central bank preference parameter, belongs to the central bank’s information set.

Ball and Mazumder (2011) impose this type of restriction in the estimation of the slope of a backward-looking Phillips curve. Moreover, they impose that the Phillips curve slope follows a random walk process, whereas the coefficient associated with the conditional unemployment volatility is assumed to follow an AR(1) process in this paper.

The exact timing for the Great Moderation period is somewhat debatable. Regarding its beginning, Surico (2007) suggests the fourth quarter of 1982 whereas Smets and Wouters (2007) consider the first quarter of 1984. We choose a starting point between these two. We also choose the second quarter of 2007 as the end of the Great Moderation since after this date the rate of unemployment has shown a sharp positive increase. Currently the rate of unemployment is twice as big as it was in the second quarter of 2007 and is about the same as those seen in the early 1980’s. That being said, the estimation results from the Great Moderation period are not sensitive to slight changes in its dating.

The interested reader may check the working paper version of the paper, Cassou and Vázquez (2012), for a comparison of the estimation results under both the stationary and nonstationary specifications of the natural rate of unemployment process. The empirical results are robust across the two specifications and justifies our inclusion of only one here.

An advantage of estimating the conditional variance in a first step is that we use the whole sample to estimate it. This feature is important when we perform a sensitivity analysis across subsamples below. For instance, if the estimated conditional variance were estimated using data from the Great Moderation period, this estimated conditional variance is likely to be biased since it only considers data from a period which featured low volatility.

Of lesser note is that we also included in our likelihood function which Ruge-Murcia (2003) did not. So our likelihoods differ from his by this factor. This omission does not invalidate Ruge-Murcia’s likelihood ratio statistics because the missing factor in both likelihood values cancel.

As Table 2 shows, in the sample period from 1960:1 to 2011:2, this non-negativity constraint was binding. We did estimate the model without this constraint and the estimated value was –0.06 with a standard error of 0.10. Furthermore, the likelihood values are only different in the second decimal place. From these results, we conclude that the constraint is supported by the data.

The characterization of w_t requires three additional parameters describing the variance of the Choleski decomposition together with ρ_c. But because the restriction imposed by the ratio of variances, r, there are only three more parameters in the time variation model than in the Ruge-Murcia model.

The fact that the model empirical fit improves by reducing the variance ratio r during the Great Moderation period is suggestive. That is, r may change over time. In this paper, however, we consider the variance ratio r as a constant parameter during the whole post-war period (1961–2011) and the alternative values assumed for r helps us to investigate whether the test results of the asymmetric central banker preferences are affected by the alternative values assumed for r.