RDP 2003-12: The Real-Time Forecasting Performance of Phillips Curves Appendix C: Implications of Our Forecast Combination Results for Model Specification

In Section 4, when real-time combination tests involving two sets of forecasts showed that neither fully encompassed the other, we used the rolling weights estimated from these tests to form a new, combined set of forecasts. An alternative response, however, has been proposed in the literature. This approach, due to Chong and Hendry (1986), argues that the failure of either model to encompass the other should be taken to indicate that one should adapt one of the models, building into it elements from the other, until it does fully encompass the latter. It is therefore interesting to ask: do our real-time forecast combination results in Section 4 have implications for the optimal specifications of our Phillips curves?^[37]

The main option that forecasters in our setting might investigate would be adding extra (possibly longer) lags of inflation to their Phillips curves in each period, given the strong relative real-time performance of our AR models of inflation. However, the scope for such re-specification is, in practise, limited. To begin with, the general Phillips curve specification used in Gruen et al (2002) already allowed for long lags of inflation, as part of an information set strictly richer than that available to our AR models.^[38] Moreover, a real-time aspect of the specification search process for our Phillips curve-based models makes the use of encompassing tests of forecast performance, for assessing model mis-specification, problematic.

Specifically, for each data vintage our procedure for estimating potential output creates a variable, the output gap, which (partly by construction) is found to be a highly significant explanator of inflation, but which is itself influenced by the specification of that vintage's Phillips curve. This makes it difficult to amend our real-time Phillips curve specifications to encompass our AR models – say, by adding additional lags of inflation – since they involve a variable, the output gap, whose construction in each period is itself intertwined with their specification.

Footnotes

That said, the residuals shown in Figure 7 of Gruen et al (2002) for our final vintage Phillips curve display few of the traditional signs of misbehaviour relating to equation mis-specification. For example, they exhibit no significant autocorrelation (at least after excluding from consideration the period associated with the first OPEC oil price shock in early 1974). [37]

Note, however, that we do impose a constraint of long-run verticality on our Phillips curves, which we do not correspondingly impose in our AR models. [38]