Beliefs counterfactual simulation

Definition

A class of counterfactual exercises in Markov-switching DSGE (MS-DSGE) models in which the analyst modifies agents’ beliefs about the set of possible future regimes — by introducing a hypothetical regime, changing transition probabilities, or altering the perceived characteristics of an existing regime — without necessarily changing the regime that is actually in place. Because the rational-expectations solution of an MS-DSGE model depends on agents’ expectations about all possible regimes (through the transition matrix), changing beliefs alone changes the equilibrium law of motion even if the realized regime path is held fixed.

Intuition

In a standard (non-switching) counterfactual, one asks “what would have happened if the central bank had followed a different rule?” This requires changing the actual policy. In a beliefs counterfactual, one asks “what would have happened if agents had believed a different regime was possible?” — the Fed need not actually change its behavior. For example, if agents in the 1970s had believed that an extremely hawkish Chairman (the “Eagle” regime) would be appointed with high probability, their inflation expectations would have been lower, and through the expectations channel, actual inflation would have decreased — even without the Eagle regime ever materializing.

Formal notation

In the MS-DSGE solution $S_t=T(x{i}_t,theta,H)S_{t-1}+R(x{i}_t,theta,H)Q(x{i}_t^{v}o)ep{s}_t$, the matrices T and R depend on the transition matrix H, which encodes agents’ beliefs. A beliefs counterfactual modifies H (or expands the regime set from m to m+1 regimes) to produce new solution matrices $T_{c}f,R_{c}f$, then simulates the model using the historical shocks under these new matrices. The key distinction from a standard counterfactual: the realized regime sequence $x{i}_t$ may be held at its estimated values while only H is changed.

Variants

• Hypothetical regime introduction: Bianchi (2013) introduces an “Eagle” regime (more hawkish than Hawk) that never occurs but whose presence in the transition matrix changes equilibrium dynamics.
• Transition probability counterfactual: changing H to make a particular regime more likely in the future, without changing the current regime.
• Belief commitment counterfactual: asking what happens if agents believe the central bank has committed to never returning to a particular regime.

Comparison

• vs standard regime-fixed counterfactual: a regime-fixed counterfactual (e.g., “what if the Hawk regime had prevailed throughout?”) changes both the actual regime and beliefs. A beliefs counterfactual separates the two channels.
• vs Lucas critique arguments: beliefs counterfactuals are explicitly Lucas-critique robust because they work through the expectation channel of the structural model.
• vs communication/forward guidance: beliefs counterfactuals formalize the mechanism through which central bank communication (forward guidance, inflation targeting announcements) affects outcomes via the expectations channel.

When to use

• Analyzing the role of central bank credibility and commitment.
• Evaluating the impact of forward guidance or regime announcements.
• Decomposing the effects of a regime change into the direct effect (different policy parameters) and the beliefs effect (changed expectations about future regimes).

Known limitations

• Requires a fully specified MS-DSGE model with a well-defined solution across the modified transition matrix — the hypothetical regime must satisfy determinacy conditions.
• The hypothetical regime parameters are not identified from data (they are chosen by the analyst), so the counterfactual results depend on the analyst’s assumptions.
• Assumes rational expectations about the (modified) transition matrix, which may overstate agents’ actual information about regime transitions.

Open problems

• Combining beliefs counterfactuals with learning (agents gradually update their beliefs about the transition matrix).
• Extending to nonlinear MS-DSGE models where higher-order approximations are used.
• Quantifying the credibility channel in the post-2008 ZLB environment.

Key papers

• regime-switches-agents-beliefs-post-world — introduces the concept and applies it to the Great Inflation and Great Recession

My understanding

This is a powerful tool unique to the MS-DSGE framework. For the CRE project, it suggests a natural extension: beliefs counterfactuals for cap rates. For instance, one could ask “what would cap rates have been if agents in 2010 had believed the Fed would return to an active regime sooner?” This would separate the direct effect of monetary policy on discount rates from the indirect effect operating through expectations about future regimes — a decomposition that would be informative for understanding CRE price dynamics around regime transitions.