I am reading "Tests of Conditional Predictive Ability" by Giacomini and White (2006).

The Null Hypothesis of equal Conditional Predictive Ability is formulated as

$H_0: E[\Delta L_{m,t+\tau}|G_t]=0 \quad a.s.$ (1)

and the authors note that in the case of forecast horizon $\tau=1$ and $G_t = F_t$, where $F_t$ denotes the sigma algebra generated by the history of the process,

$H_0: E[\Delta L_{m,t+1}|F_t]=0$ (2)

this implies that $\Delta L_{m,t+1}$ constitutes a Martingale Difference Series (MDS).

Considering a $q\times1$ vector of $F_t$ measurable functions $h_t$, they state that the MDS property implies

$H_{0,h}: E[h_t \Delta L_{m,t+1}]=0$ (3)

The (global) Alternative Hypothesis is formulated as

$H_{A,h}: E[\bar{Z}_{m,n}^{'}]E[\bar{Z}_{m,n}]\geq\delta>0$ (4)

where $\bar{Z}_{m,n}=n^{-1}\sum_{t=m}^{T-1}h_t \Delta L_{m,t+1}$ (5)

Under Stationarity of ${Z_{m,t}}$ the Null and Alternative are exhaustive from which I conclude that the Null Hypothesis can equivalently be written as

$H_{0,h}: E[\bar{Z}_{m,n}^{'}]E[\bar{Z}_{m,n}]=0$ (6)

I would be grateful if someone could clarify or point to helpful references for the following questions.

(a) How does the MDS property (Equation 2) imply Equation 3?

(b) What is the interpretation of the Alternative Hypothesis, especially the term with the product of expected values of Sample Means?

(c) How does the Null under Stationarity (Equation 6) follow from Equation 3?

(d) How come that the Alternative Hypothesis is formulated in terms of sample rather than population quantities?

Thank you very much in advance.


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