[nerdy] Reply to Hendry and Mizon: we have DSGE models with time-varying parameters and variances

Hendry and Mizon summarised a recent paper of theirs on VoxEU explaining that DSGE models break down in crises because these events involve shifts in the distribution of observeables that fixed-parameter, fixed-variance DSGE models can't articulate.  They tell the story in a way that a lay reader might conclude is catastrophic for microfounded, dynamic macroeconomics, and/or rational expectations.

But it isn't.

Several papers have taken steps to articulate models that have time-varying propagation parameters, and time-varying variances.  And there is a literature connecting these models to empirical macro models that estimate time-varying econometric counterparts.  None of these papers make it into the citations of the VoxEU post, or the original academic paper.

Part of the discussion is about how the equilibrium laws of motion of the economy, got by invoking the law of iterated expectations, in some cases, aren't derivable by these same means with time-varying parameters.  This is well-known.  But DSGE modellers who use time-varying parameters, or time-varying variances, know how to solve such models, at least in cases where there aren't too many things moving around all at once.  Finding expectations that, when used, generate laws of motion whose expectations are equal to what you started with is neat and easy with time-invariant parameters.  But though difficult when they vary, the process of searching for this 'fixed point' as it's called is conceptually the same, and often achievable.

Some examples of time-varying DSGE models:  (i)  Models with 'stochastic volatility' [variances of shocks that move around in continuous and random small steps over time], including Caldera et al, and Fernandez-Villaverde and Rubio-Ramirez.  (ii) Or Markov-regime-switching models [models in which parameters like price-stickiness, or policy parameters, move around randomly through a small set of possibilities]. including Foerster et al.

All these models rest on a degree of time-invariance;  in a stochastic volatility model, the shocks to variances are themselves drawn from a fixed variance.  In a Markov-Switching model, the switches occur with fixed probabilities.  But in principle we could push the time-variation one step further if we really wanted to.  [In fact in the case of Markov-Switching I believe there are examples that do this, though I can't lay my hands on one now].

The article dwells on the notion that from the perspective of agents the mean and the variance of relevant distributions won't be known ahead of time.  But expectations can be calculated provided that the distributions from which this means and variances are drawn are known.

At any rate, the critique is somewhat academic, because many authors have pushed the boundaries of DSGE models by dropping the notion of rational expectations.  Sargent and coauthors have worked out the equilibria for agents who are Bayesian learners, yet doubt the distributions for relevant concepts implied by their models.  Ilut has figured out a simple DSGE model in which agents respond to changes in the degree of ambiguity about the distribution of technology over time.  Models with learning can be simulated recursively, so there is no problem at all shoving through changes in policy or economic parameters, or changes in variances.  I can't find an example that does this, but that's because it's so easy that no-one would get any points for trying to tell anyone else that this was possible on their computer!

A further point to make is that we will rarely be able to say decisively that the distribution has changed.  A common theme in the stochastic volatility literature is that it is hard to distinguish a high probability draw from a new distribution with a larger variance from a low probability draw from the old distribution with a small variance.   Perhaps the post Great Moderation era indicates a shift to a new distribution of macro variables.  Perhaps it just reveals that the time-invariant distribution involved a higher probability of disasters than we thought before the crisis.  The failure of our pre-crisis DSGE models doesn't necessarily indicate we need time-varying ones, just models that generate low probability extreme outcomes (like a crash).  We should guard against jumping too quickly from atheoretic econometric analysis which appears to show distribution-shifts, to concluding that time-varying distribution DSGE models are necessary.

Hendry and Mizon make a number of scathing references to the fact that central banks like the Bank of England operate with these fixed-parameter DSGE models, in apparent oblivion to the fact that distributions are changing all around them.  In the BoE's defence, their modelling staff know the time-varying DSGE and empirical macro literatures well, and some of them have published in these fields, and many of the contributors have presented in the Bank's seminar series.  Further, the staff and the MPC don't follow the models slavishly, or necessary believe literally in the assumption of rational expectations which Hendry and Mizon think (mistakenly in my view) is so problematic.

Strictly speaking, the post criticises 'standard' macro models.  Leaving open that they accept that there are 'non-standard' varieties that are immune from their critiques.  In which case there is no dispute.  But I think this other work deserves a mention.   It illustrates that time-variation in variances isn't catastrophic for rational expectations or DSGE models.  And anyway, who cares, given the strange and wonderful new work on more realistic, non-rational expectations, which most central banks would, I surmise, subscribe to.

[Update:  this post, and the Hendry and Mizon paper, sparked a discussion on econjobrumors.  One of the contributors makes a great point that I hadn't thought of, which is that many DSGE models generate multiple equilibria, which is another class of model that would produce data that might appear to an econometrician to manifest distribution changes, even though the Data Generating Process had not changed at all.]