# Identification and Estimation of Dynamic Causal Effects in Macroeconomics Using External Instruments

*James H. Stock, Mark W. Watson*
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## Abstract

External sources of as-if randomness — that is, external instruments — can be used to identify the dynamic causal effects of macroeconomic shocks. One method is a one-step instrumental variables regression (local projections – IV); a more efficient two-step method involves a vector autoregression. We show that, under a restrictive instrument validity condition, the one-step method is valid even if the vector autoregression is not invertible, so comparing the two estimates provides a test of invertibility. If, however, lagged endogenous variables are needed as control variables in the one-step method, then the conditions for validity of the two methods are the same.

Makes the point that there can be more shocks than economics variables in your system of equations:

If we collect all such structural shocks and measurement error together in the $m×1$ vector $ε_{t}$, the $n×1$ vector of macroeconomic variables $Y_{t}$ can be written in terms of current and past $ε_{t}$ … In general, the number of shocks plus measurement errors, m, can exceed the number of observed variables, n.

The unit effect normalization – fixing the shock variable such that a one unit movement corresponds to a unit movement in the impulse variable of interest. This solves the scale ambiguity issues the arises from the fact that the true shock is unobserved. It also underpins the local projections approach, as it enables regressions in terms of observables.

Proves that local projections with instrumental variables can be used in situations when the equivalent VAR would be invertible:

we provide conditions for instrument validity for LP-IV, and show that under those conditions LP-IV can estimate dynamic causal effects without assuming invertibility, that is, without assuming that the structural shocks can be recovered from current and lagged values of the observed data. … The structural moving average $Θ(L)$ in (5) is said to be invertible if et can be linearly determined from current and lagged values of $Y_{t}$: … In the linear models of this article, condition (24) is equivalent to saying that $Θ(L)_{−1}$exists.

Invertibility implies that the VAR is fully specced out. In other words, the system contains all the data you need to uncover the true shocks and you don’t need to augment it with anything, nor would you benefit from doing so:

under invertibility, a forecaster using a VAR would find no value in augmenting her system with data on the true macroeconomic shocks, were they magically to become available. … a forecaster using a VAR who magically stumbled upon the history of true shocks would have no interest in adding those shocks to her forecasting equations. … If invertibility holds, then knowledge of the past true shocks would not improve the VAR forecast. If instead those forecasts were improved by adding the shocks to the regression – infeasible, of course, but a thought experiment – then the VAR has omitted some variables, and that omission is an indication of the failure of the invertibility assumption.

Covers the omitted variable bias interpretation of invertibility as discussed in @nakamuraIdentificationMacroeconomics2018:

One interpretation provided in the literature on invertibility is that

invertibility implies that there are no omitted variables in the VAR(e.g. Fernandez-Villaverde et al., 2007): because invertibility implies that the spans of $ε_{T}$ and $ν_{t}$ are the same, there is no forecasting gain from adding past shocks to the VAR.

Invertibility as meaning structural shocks can be expressed as linear combination of current and past values of the observed data:

The structural MA representation $Y_{t}=D(L)ε_{t}$ represents $Y_{t}$ in terms of current and past values of the structural shocks $ε_{t}$. The moving average is said to be invertible if $ε_{t}$ can be expressed as a distributed lag of current and past values of the observed data $Y_{t}$. SVARs typically assume $ε_{t}=H_{−1}η_{t}=H_{−1}A(L)Y_{t}$, so an SVAR typically imposes invertibility.

Finally hit me that the “moving average” representation simply means representing current variables as a linear combination of current and past innovations / shocks, while the “VAR” representation means representing residuals / shocks as a linear combination of current and past observed data.

You can get the moving average representation of a VAR by regressing the outcome variables on its lags, then regress the outcome variables on the residuals from the first regression. This yield the MA coefficients. It is also therefore an easy way to invert a VAR without literally inverting the coefficient matrix.

Solutions to omitted variable bias in the context of a VAR:

- Include large number of variables via DFM, FAVAR, BVAR etc
- instrumental variables estimation