Inference in Direct Multi-Step and Long Horizon Forecasting Regressions
Abstract
This paper proposes and evaluates a new method for inference in direct multi-step and long horizon forecasting regressions. A direct-multi step forecast is a regression of a vector of variables observed at a future horizon on variables observed in the current time period. A long horizon regression is a regression of the cumulative sum of a vector of variables observed through a future horizon on variables observed in the current period. Direct multi-step forecasts and long horizon regressions have a variety of applications in empirical macroeconomics and finance. Examples include local projections and tests of return predictability.The residuals from both direct multi-step and long horizon forecasting regressions are serially correlated, and can be expressed as a vector moving average (VMA) process of the one step ahead forecast residuals. The proposed estimator imposes the VMA structure on the serially correlated residuals to estimate the covariance matrix of the OLS estimates of direct multi-step and long horizon forecasting regressions. The parameters governing the VMA process are estimated using OLS regressions. In modeling the covariance matrix, I present a unified framework for inference in both direct multi-step and long horizon forecasts, by taking advantage of the fact that error terms in a long horizon forecast are equal to the sum of the error terms from direct multi-step forecasts.
This results in substantially more accurate and efficient estimates of the covariance matrix, relative to existing methods that do not impose any structure on the autocorrelation process of the residuals. A simulation study comparing the proposed estimator with the most commonly used alternative illustrates the benefits of the approach. The proposed estimator has a lower root mean squared error and size tests closer to its nominal values.