Even if the two models show a different level of forecast accuracy, the next question is whether the two forecasts are statistically significantly different from one another. The Diebold–Mariano Test of Forecast Differences and the Harvey, Leybourne, and Newbold Test allow us to determine if the errors are statistically significant. The null hypothesis tested is that there is no significant difference between the two forecasts.
Finally, sometimes the exact forecast accuracy is not in question. Rather, it is the ability to predict directional change that is critical. The Pesaran–Timmerman tests for whether a model can adequately predict and track the directional changes over time. The null hypothesis tested is that the forecast does not track directional changes in the data.
The following provides some insights into how the Akaike Information Criterion (AIC), Bayes Information Criterion (BIC), and Hannan–Quinn (HQ) are derived, using an ordinary least squares (OLS) approach in multiple regression, which assumes that the errors are normally distributed, and the log-likelihood function is maximized (see Chapter 12 for more details on multiple regressions solved using maximum likelihood approaches).