RDP 2014-02: Fiscal Policy and the Inflation Target 7. Expectations

An unusual feature of my simulations is the treatment of expectations. I have assumed that agents base their expectations on small-scale VARs rather than on the full model. Although this assumption is usually inappropriate for analysis of policy rules, it seems a reasonable approximation given the unusual characteristics of this particular question.

As I discuss below, it is not obviously feasible or desirable to conduct the relevant simulations with model-consistent expectations. But even if it were, it is not clear that they would be significantly different. One reason for this is that the inflation-variability frontier does not seem to be noticeably affected by different assumptions about expectations. As noted in footnote 10, frontiers calculated using VAR expectations are very similar to estimates Reifchneider and Williams calculate using model-based expectations, when put on a consistent basis. A more important reason is that fiscal multipliers are likely to be similar. This can be seen in Table 3, which compares FRB/US fiscal multipliers calculated using VAR expectations and calculated using the nonlinear model-consistent version of FRB/US (called ‘PFVER’).[11] The table shows effects on GDP after four quarters. Under VAR expectations, agents implicitly assume that shocks persist as long as similar shocks in the past (though for purposes of constructing Table 3, all that needs to be assumed is that they last at least as long as the multiplier horizon, four quarters). For model-consistent expectations, agents are assumed to know how long a shock will last, so this needs to be specified: I assume that both shocks and zero bound episodes are expected to last eight quarters, the same experiment as Coenen et al (2012, Figure 3, lower left panel). Eight quarters is a bit longer than most zero bound episodes in my simulations (Table 2), but may be representative, given that the positive threshold makes stimulus last slightly longer than time at the zero bound, and that most stimulus occurs in longer-lasting recessions. For the experiments shown in Table 3, multipliers with model-consistent expectations are slightly smaller than those with VAR expectations, but the difference is small. These comparisons suggest that the overall impact of fiscal policy at the zero bound may not be significantly different once expectations become model-consistent.

Table 3: FRB/US Fiscal Multipliers, with Different Assumptions about Expectations
Effect on level of real GDP (per cent deviation from baseline) after four quarters of a change in fiscal variables by 1 per cent of GDP expected to last eight quarters
VAR expectations Model-consistent expectations
Government purchases 0.99 0.94
Reduction in personal tax receipts 0.31 0.27
Transfers 0.42 0.28

Source: author's calculations

The reason multipliers are similar is that these experiments are not historically unusual. Under VAR expectations, the household expects a tax cut (for example) to persist for as long as unusual variations in disposable income have persisted in the past, which is a few years. So there is a moderate increase in permanent income and hence consumption. Under model-consistent expectations, the household expects the tax cut to persist as long as the model predicts the funds rate will remain near zero, which is assumed to be two years in Table 3. That is, model-consistent expectations give households information that is not very surprising. So household behaviour is similar.

Similarity in results, in itself, is not a reason for preferring VAR expectations. That preference reflects both practical and conceptual considerations.

Stochastic simulations of nonlinear models under model-consistent expectations are computationally difficult. Accordingly, previous researchers have used linearised versions of their models. Linearisation can change model properties in unattractive ways (Braun, Körber and Waki 2012). Whereas the nonlinear version of FRB/US with VAR-based expectations has been tested, documented, scrutinised, and successfully used in a wide variety of applications, the properties of the linearised version are less well-established, reducing the confidence that can be placed in them.

It is not even clear that linearisation is feasible for this study given that the fiscal rule is nonlinear. Reifschneider and Williams (2000) include one nonlinear constraint in an otherwise linear model. Coibion et al (2012) have two nonlinear constraints in a smaller model. I am not aware that multiple nonlinear constraints have been successfully included in a large-scale model. In both the Reifschneider and Williams and the Coibion et al exercises the introduction of nonlinear constraints involves a large cost in computational complexity. Moreover, the expectations are not actually ‘model-consistent,’ but ‘model-based’. They represent the model's deterministic solution, not the mean of the stochastic solution. Because the zero bound is asymmetric, these will differ.

In terms of principle, it is ordinarily appropriate to model policy rules using model-consistent expectations. As people gain experience of a rule, their behaviour will adapt to be consistent with it and systematic errors should disappear. However, modelling newly-introduced rules under the assumption of model-based expectations can be misleading. Economic agents have limited information processing capabilities. It is implausible to assume that they quickly learn the structure of the model, when most professional economists do not know it. In particular, new policies may not have perfect credibility. Empirical observation suggests that people often do not perceive a change in policy regime simply because it is announced. For example, changes in monetary policy rules in the early 1980s in the United States and United Kingdom seem to have led to recessions rather than changes in inflation expectations. Similarly, households respond to tax cuts when the policy change is implemented, not when it is enacted or credibly announced (see the numerous references listed in footnote 3 of Auerbach et al (2010)). Accordingly, the use of VAR-based expectations may be more appropriate to describe behaviour until households have had experience of a change in structure.

The use of fiscal stimulus at the zero bound falls in between the extremes of a ‘one-off’ shock and a change to which everyone has adjusted. The difficulty in assuming quick learning is that we are discussing a policy that is applied infrequently. Interest rates only occasionally hit the zero bound and the constraint seriously binds only once every decade or so. To be precise, in simulations with β = 0.7 and a 2 per cent inflation target, a stimulus exceeding 2 per cent of GDP occurs once every 12 years, on average. And the size and duration of stimulus have large variances. So learning from experience of the change in structure seems likely to take a long time, perhaps decades. In the meantime, it seems plausible to assume that households will continue to behave in line with historical correlations. This is a ‘short-run’ solution, but it is long enough to matter. Permanent effects will be different (in character, if not in size), and would be interesting to model, but that is not a reason for neglecting the effects that will occur within our lifetime.


To conduct stochastic simulations, it would be necessary to linearise the model. The approximation errors involved would further change the multipliers, but not in an economically interpretable manner. [11]