# Dynamic Factor Models, Factor-Augmented Vector Autoregressions, and Structural Vector Autoregressions in Macroeconomics

## Abstract

This chapter provides an overview of and user’s guide to dynamic factor models (DFMs), their estimation, and their uses in empirical macroeconomics. It also surveys recent developments in methods for identifying and estimating SVARs, an area that has seen important developments over the past 15 years. The chapter begins by introducing DFMs and the associated statistical tools, both parametric (state-space forms) and nonparametric (principal components and related methods). After reviewing two mature applications of DFMs, forecasting and macroeconomic monitoring, the chapter lays out the use of DFMs for analysis of structural shocks, a special case of which is factor-augmented vector autoregressions (FAVARs). A main focus of the chapter is how to extend methods for identifying shocks in structural vector autoregression (SVAR) to structural DFMs. The chapter provides a unification of SVARs, FAVARs, and structural DFMs and shows both in theory and through an empirical application to oil shocks how the same identification strategies can be applied to each type of model.

## Dynamic Factor Models

At some point I might take some notes here but for now I think I can simply leverage this stuff out of the box (e.g. using the R package dfms) without worrying too much about the core details.

## Structural Vector Autoregressions

If shocks were observed, you could simply run OLS of the outcome variable on the current and past shock values, and you’d be good to go. If the shock is a true shock, it’s uncorrelated with other variables, so there’s no omitted variable bias. The coefficients will be unbiased and represent the impulse response of the outcome to the shock:

If a time series of shocks were observed, it would be straightforward to estimate the effect of that shock, say , on a macro variable by regressing on current and past values of . Because the shock is uncorrelated with the other shocks to the economy, that regression would have no omitted variable bias. The population coefficients of that regression would be the dynamic causal effect of that shock on the dependent variable, also called the structural impulse response function (SIRF). The cumulative sum of those population coefficients would be the cumulative causal effect of that shock over time, called the cumulative SIRF.

Since the shocks are typically not in fact observed, the whole point of SVARs is to leverage the assumption that the forecast errors / innovations of the VAR fully represent the space of structural shocks to uncover the impact of those structural shocks on relevant variables of interest. Economists use the term “span the shocks,” but all this means in practice is that the VAR innovations are linear combinations of all the structural shocks in the economy and vice versa. Said this way, this seems like a fairly strong assumption, but accepting it at least in principle lets us do some very interesting things.

The premise of SVARs is that the space of the innovations to a vector of time series variables —that is, the one step ahead forecast errors of based on a population projection of onto its past values—spans the space of the structural shocks. Said differently, in population the econometrician is assumed to be as good at one step ahead forecasting of the economy as an agent who directly observes the structural shocks in real time. The task of identifying the structural shock of interest thus reduces to the task of finding the linear combination of the innovations that is the structural shock.

However, if one is only interested in the effect of a subset of shock, only those shocks need to be spanned by the VAR innovations. For example, if I’m only interested in the effect of a single shock, then only that shock needs to be a linear combination of the VAR innovations in the system. This is a helpful distinction in theory, though I wonder in practice to what extent results would ever depend on this being true vs the broader assumption of full spanning needing to be true.

In practical terms, this tends to mean that you need at least as many variables as shocks you want to identify. Adding more variables than that helps to ensure the structural shocks of interest are in fact fully spanned, but the math works as soon as you have enough variables / reduced form equations in the VAR.

Forecast error variance decomposition (FEVD): the breakdown of how important each shock is in explaining the variation in some outcome variable. FEVDs are calculated as the relative size of the variance of a shock to the overall variance in the forecast errors in the outcome variable over future periods (the h-step ahead forecast errors). This it is calculated separately for each horizon (h), shock (j), and outcome variable triplet (i). In a way, you can think about the FEVD as simply using the idea of a system of equations to say that the innovations of one variable can be explained by the structural shocks within the system, explicitly calculating the relative contributions.

Historical decomposition (HD): These are literally just the moving average representation of the time series. Thus once you have the impulse response function you effectively also have the historical decomposition. Another way to think about this is that you can project the demeaned observed variable on current and past shocks and that result is the contribution of the shocks to that variable.

## References

@baiPrincipalComponentsEstimation2013