The Behavior of Forecast Errors From a Nearly Integrated Ar(1) Model as Both Sample Size and Forecast Horizon Become Large

Abstract

We develop asymptotic approximations to the distribution of forecast errors from an estimated AR(1) model with no drift when the true process is nearly I(1) and both the forecast horizon and the sample size are allowed to increase at the same rate. We find that the forecast errors are the sums of two components that are asymptotically independent. The first is asymptotically normal whereas the second is asymptotically nonnormal. This throws doubt on the suitability of a normal approximation to the forecast error distribution. We then perform a Monte Carlo study to quantify further the effects on the forecast errors of sampling variability in the parameter estimates as we allow both forecast horizon and sample size to increase.

Robustness of Multistep Forecasts and Predictive Regressions at Intermediate and Long Horizons

“…In this paper, we propose a local-asymptotic model that builds on the work of Kemp (1999), Valkanov (2003), Torous et al (2004), Chevillon and Hendry (2005) and Hjalmarsson (2011). We prove a new key property of direct multi-step estimators, namely their robustness to misspecification of the serial correlation of the error process.…”

“…Long run forecasting has been studied by Stock (1996), Phillips (1998), Kemp (1999) Finally, the problem of misspecification may arise even for h = 1 if ε t or t exhibits serial correlation and cross correlation. We define their joint autocovariance function…”

Local to unity, long‐horizon forecasting thresholds for model selection in the AR(1)

“…2 Many authors use this specification to develop unit root tests and median unbiased estimation procedures (including Bobkoski, 1983;Chan and Wei, 1987;Phillips, 1987;Stock, 1991;Elliott et al, 1996;Elliott, 1999), while Stock (1996), Phillips (1998), Kemp (1999) and Ng and Vogelsang (2002) use it to analyse forecasting. Stock (1996), Phillips (1998) and Kemp (1999), who consider Case I, consider both one-stepahead and 'long' horizons.…”

“…Stock (1996), Phillips (1998) and Kemp (1999), who consider Case I, consider both one-stepahead and 'long' horizons. For the latter case, they treat the forecast horizon asymptotically as a fraction h of the sample size of the time series (i.e.…”

“…Since Stock (1996), Phillips (1998) and Kemp (1999) each report local to unity, long-horizon asymptotic distributions of Case I forecast errors for RW and OLS, I must only apply their theorems to build the threshold function c*(h) for Case I. I do so in the third section.…”