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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 $\bar{D}=0$.

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