On the Likelihood of Local Projection Models

TL;DR

This paper derives the likelihood function for local projection models from a stationary vector moving average process, clarifying when treating them as multivariate regressions with correlated errors is justified.

Contribution

It provides an analytical derivation of the likelihood for local projection models, bridging a gap in understanding their statistical properties.

Findings

Likelihood derivation from vector MA process

Validation through numerical experiments

Justification for multivariate regression treatment

Abstract

A local projection model is defined by a set of linear regressions that account for the associations between exogenous variables and an endogenous variable observed at different time points. While it is standard practice to separately estimate individual regressions using the ordinary least squares estimator, some recent studies treat a local projection model as a multivariate regression with correlated errors, i.e., seemingly unrelated regressions, and propose Bayesian and non-Bayesian methods to improve the estimation accuracy. However, it is not clear how and when this way of treatment of local projection models is justified. The primary purpose of this paper is to fill this gap by showing that the likelihood of local projection models can be analytically derived from a stationary vector moving average process. By means of numerical experiments, we confirm that this treatment of local projections is tenable for finite samples.