RDP 2012-08: Estimation and Solution of Models with Expectations and Structural Changes 1. Introduction

Standard solution methods for linear rational expectations models, like Blanchard and Kahn (1980), Binder and Pesaran (1995), Uhlig (1995), King and Watson (1998), Klein (2000), Sims (2002) and Anderson (2010), deal with the case where the parameters of the structural model are constant. These methods are at the heart of likelihood-based estimation of such models. In practice, the magnitude of changes in the properties of observable variables is used to help define sub-samples for which a time-invariant structure seems valid, and estimation is then done with these sub-samples.^[1] The analysis of Lubik and Schorfheide (2004), for instance, is based on the assumption that the target inflation rate in the United States – like other structural parameters – stayed constant in the pre-Volcker years, but then possibly shifted in the early 1980s, based on the estimates of steady-state inflation in each of the sub-samples.

Findings of structural instabilities seem to apply to many models of macroeconomic aggregates. While we cannot do justice to the complete literature, one can point to the work of Clarida, Galí and Gertler (2000) who find a significant difference in the way monetary policy was conducted pre- and post-late 1979 in the United States; Ireland (2001) who detects shifts in the discount factor; Inoue and Rossi (2011) who show that the Great Moderation was due to both changes in shock volatilities and policy and private sector parameters; and Stock and Watson (2007) who provide evidence of changes in the variance of shocks to trend inflation.

Our objective in this paper is to develop solutions for linear stochastic models with model-consistent expectations in the presence of structural changes that are possibly foreseen. The solution extends one recently proposed by Cagliarini and Kulish (forthcoming) by providing an econometric representation, namely a state space form, to which the Kalman filter can be applied to construct the likelihood function of the data. As we show below, the reduced-form solution takes the form of a time-varying coefficients VAR, where movements in the coefficients are governed by the nature of the underlying structural changes.

In the basic case we assume that expectations are formed in such a way as to be consistent with whatever structure (model) holds at each point in the sample. This is analysed in Section 3 by looking at cases where the structural changes are either unknown in advance or where there is some foresight about them. In this scenario a second structure (model) will hold at some given future date and, because agents know what that date is, they factor it into the formation of their expectations before the actual date at which the change occurs. Examples of this latter situation could be a change in inflation targets or an announcement about the introduction of a tax. Section 3 also deals with the situation where beliefs about the structural change can be different from the truth (reality). Thus, if one thinks of a single structural change in the sample, expectations may be based on the first period model for some time into the second period. Of course, eventually it seems reasonable to think that beliefs must centre upon the second period model. We simply specify when beliefs agree with this second model and do not model any learning behaviour.

The particular case in which the structure evolves and agents' beliefs are fully aligned with reality coincides with the problem posed by Cagliarini and Kulish (forthcoming). The generalisation of this paper involves allowing a difference between beliefs and reality to exist for a period of time before, after or during a sequence of structural changes. This generalisation is useful for at least two reasons. First, it is capable of capturing the consequences of structural changes that may go temporarily unnoticed as, for example, happened during the US productivity slowdown of the early 1970s. Second, it may be used to capture the impact of policy announcements which are less than perfectly credible.

Many of the issues we address have long been recognised in the literature. In fact, more than half a century ago Marschak (1953) noticed that, in the case of an anticipated structural change, the purely empirical projections of observed past regularities into the future would not be a reliable guide for decision-making, unless past observations were supplemented by some knowledge of the way the structure was expected to change. Since then the technical apparatus has changed a great deal, but these insights are just as powerful today. In the context of estimating and solving dynamic stochastic models with expectations, some knowledge of the structural changes that might have taken place in-sample allows us to increase the number of observations that are usable in estimation, and therefore has the potential of improving the quality of the estimation. But regardless of what may be the situation in a particular application, this paper is the first to provide the tools to accomplish maximum likelihood estimation of dynamic stochastic economies with structural changes under a variety of assumptions regarding expectations formation.

The paper is organised as follows. The next section reviews the Binder and Pesaran (1997) solution procedure for models with forward-looking expectations. As mentioned previously, Section 3 then extends the solution to situations of structural change and derives the likelihood for the implied model. Section 4 introduces two examples. The first is an anticipated credible disinflation program while the second is a temporary fall in trend growth happening alongside a looser monetary policy. Section 5 concludes and Appendix A provides details of the construction of the log likelihood with the Kalman filter.

Footnote

An interesting exception is Cùrdia and Finocchiaro (2005) who estimate a model for Sweden with a monetary regime change. [1]