# Pairwise comparison test for out-of-sample MSEs

In order to compare the out-of-sample forecasting accuracy of two competing models, I am trying to implement the equal accuracy test proposed in this article: http://www.timberlake-consultancy.com/slaurent/pdf/Handbook_volfor.pdf

The difference between the two MSEs is rewritten as:

$$MSE(k,t) - MSE(j,t) = cov(D_t,S_t) + \bar{D}\bar{S},$$

where $$D_t$$ and $$S_t$$ are, respectively, the difference and the sum between model $$k$$ and model $$j$$ residuals. The authors state that the null hypothesis of equal accuracy can be specified as:

$$H_0: cov(D_t,S_t)=0 \cup \bar{D}=0,$$

which can be restated as:

$$H_0: \alpha = 0 \; \cup \; \beta=0,$$

where $$\alpha$$ and $$\beta$$ are the coefficients of the linear regression $$D_t = \alpha + \beta (S_t - \bar{S}) + \epsilon_t.$$ However, I do not understand the rationale behind $$H_0$$ and how I should proceed to implement the test in practice. The two MSEs should be equal only if $$cov(D_t, S_t) = \bar{D}\bar{S}$$ and not if $$cov(D_t,S_t) = 0$$ or $$\bar{D}=0$$.