4

Forecast Combination and

Encompassing

Michael P. Clements and David I. Harvey

Abstract

Forecast combination is often found to improve forecast accuracy. This chapter considers different
types of forecast combination and tests of forecast encompassing. The latter indicate when a com-
bination is more accurate than an individual forecastex post, in a range of circumstances: when the
forecasts themselves are the objects of interest; when the forecasts are derived from models with
unknown parameters; and when the forecast models are nested. We consider forecast encom-
passing tests which are framed in terms of the model’s estimated parameters and recognize that
parameter estimation uncertainty affects forecast accuracy, as well as conditonal tests of encom-
passing. We also look at the conditions under which forecast encompassing can be established
irrespective of the form of the loss function.

4.1 Introduction 169
4.2 Historical development 171
4.2.1 Forecast combination 171
4.2.2 Forecast encompassing 176
4.3 Model-based forecasts 180
4.4 Nested model comparisons 185
4.5 Conditional tests of forecast encompassing 187
4.5.1 Quantile forecasts 190
4.6 Loss functions and forecast combination 192
4.7 Conclusions 194

4.1 Introduction

In this chapter we consider the closely related topics of forecast combination
and tests of forecast encompassing. A test for forecast encompassing is just one
of the many tests of predictive ability that might be considered. For example,
one might test whether a set of forecasts are unbiased, or efficient (e.g., Mincer
and Zarnowitz, 1969; Figlewski and Wachtel, 1981; Zarnowitz, 1985; Keane and
Runkle, 1990) or as accurate given some loss function as a rival set of forecasts.
Assuming a squared-error loss function, the approach of Granger and Newbold
(1977) (sometimes known as the Morgan–Granger–Newbold test in recognition of
Morgan, 1939) is one such test of whether differences between rival forecasts can

169