Structural VAR Analysis

Kilian and Lutkepohl's Structural Vector Autoregressive Analysis is now out. The back-cover blurbs below are not hyperbole. Indeed Harald Uhlig's is an understatement in certain respects -- to his list of important modern topics covered I would certainly add the "external instrument" approach. For more on that, beyond K&L, which went to press some time ago, see Stock and Watson's masterful 2018 external-instrument survey and extension, just now released as an NBER working paper. (Ungated K&L draft here; ungated S&W draft here.)

Regression on Term Structures

An important insight regarding use of dynamic Nelson Siegel (DNS) and related term-structure modeling strategies (see here and here) is that they facilitate regression on an entire term structure.  Regressing something on a curve might initially sound strange, or ill-posed.  The insight, of course, is that DNS distills curves into level, slope, and curvature factors; hence if you know the factors, you know the whole curve.  And those factors can be estimated and included in regressions, effectively enabling regression on a curve.

In a stimulating new paper, “The Time-Varying Effects of Conventional and Unconventional Monetary Policy: Results from a New Identification Procedure”, Atsushi Inoue and Barbara Rossi put that insight to very good use. They use DNS yield curve factors to explore the effects of monetary policy during the Great Recession.  That monetary policy is often dubbed "unconventional" insofar as it involved the entire yield curve, not just a very short "policy rate".

I recently saw Atsushi present it at NBER-NSF and Barbara present it at Penn's econometrics seminar.  It was posted today, here.

On the Identification of Network Connectedness

I want to clarify an aspect of the Diebold-Yilmaz framework (e.g., here or here).  It is simply a method for summarizing and visualizing dynamic network connectedness, based on a variance decomposition matrix.  The variance decomposition is not a part of our technology; rather, it is the key input to our technology.  Calculation of a variance decomposition of course requires an identified model.  We have nothing new to say about that; numerous models/identifications have appeared over the years, and it's your choice (but you will of course have to defend your choice). 

For certain reasons (e.g., comparatively easy extension to high dimensions) Yilmaz and I generally use a vector-autoregressive model and Koop-Pesaran-Shin "generalized identification".  Again, however, if you don't find that appealing, you can use whatever model and identification scheme you want.  As long as you can supply a credible / defensible variance decomposition matrix, the network summarization / visualization technology can then take over.

Impulse Responses From Smooth Local Projections

Check out Barnichon-Brownlees (2017) (BB).  As proposed and developed in Jorda (2005), they estimate impulse-response functions (IRF's) directly by projecting outcomes on estimates of structural shocks at various horizons, as opposed to inverting a fitted autoregression.  The BB enhancement relative to Jorda is the effective incorporation of a smoothness prior in IRF estimation.  (Notice that the traditional approach of inverting a low-ordered autoregression automatically promotes IRF smoothness.)  In my view, smoothness is a natural IRF shrinkage direction, and BB convincingly show that it's likely to enhance estimation efficiency relative to Jorda's original approach. I always liked the idea of attempting to go after IRF's directly, and Jorda/BB seems appealing.

Mixed-Frequency High-Dimensional Time Series

Notice that high dimensions and mixed frequencies go together in time series. (If you're looking at a huge number of series, it's highly unlikely that all will be measured at the same frequency, unless you arbitrarily exclude all frequencies but one.) So high-dim MIDAS vector autoregression (VAR) will play a big role moving forward. The MIDAS literature is starting to go multivariate, with MIDAS VAR's appearing; see Ghysels (2015, in press) and Mikosch and Neuwirth (2016 w.p.). 

But the multivariate MIDAS literature is still low-dim rather than high-dim. Next steps will be: 

(1) move to high-dim VAR estimation by using regularization methods (e.g. LASSO variants), 

(2) allow for many observational frequencies (five or six, say), 

(3) allow for the "rough edges" that will invariably arise at the beginning and end of the sample, and 

(4) visualize results using network graphics.

Shrinking VAR's Toward Theory: Supplanting the Minnesota Prior?

A recent post, On Bayesian DSGE Modeling with Hard and Soft Restrictions, ended with: "A related issue is whether 'theory priors' will supplant others, like the 'Minnesota prior'. I'll save that for a later post." This is that later post. Its title refers to Ingram and Whiteman's 1994 classic, entitled "Supplanting the 'Minnesota' Prior: Forecasting Macroeconomic Time Series Using Real Business Cycle Model Priors."

So, shrinking VAR's using DSGE theory priors improves VAR forecasts. Sounds like a victory for economics, with the headline "Using Economic Theory Improves Economic Forecasts!" We'd all like that. We all want that.

But the "victory" is misleading, and more than a little hollow. Lots of shrinkage directions improve forecasts. Indeed almost all shrinkage directions improve forecasts. Real victory would require theory-inspired priors to deliver clear extra improvement relative to other shrinkage directions, but they usually don't. In particular, the Minnesota prior, centered on a simple vector random walk, remains competitive. (See Del Negro and Schorfheide (2004) and Del Negro and Schorfheide (2007).) Sometimes theory priors beat the Minnesota prior by a little, sometimes they lose by a little. It depends on the dataset, the variable, the forecast horizon, etc.

The bottom line: Theory priors seem to be roughly as good as anything else, including the Minnesota prior, but certainly they've not led us to anything resembling wonderful new forecasting success. This seems at best a small forecasting victory for theory priors, but perhaps a victory nonetheless, particularly given the obvious appeal of using a theory prior for Bayesian VAR forecasting that coheres with the theory model used for policy analysis.