# Asymptotic distribution of the Sample Average of Covariance Stationary Time Series

The Diebold-Mariano (DM) test is a widely used approach for selecting a prediction method in emprical finance.

It implements the following logic - if we have two prediction methods that yield predictions for $$\widehat{y_{1t}}$$ and $$\widehat{y_{2t}}$$ respectively for the actual outcome in period t $$y_t$$, the respective values of a user-defined loss function are $$L(\widehat{y_{1t}},y_t)$$ and $$L(\widehat{y_{2t}},y_t)$$. $$d_t =L(\widehat{y_{1t}},y_t)-L(\widehat{y_{2t}},y_t)$$ is the loss-differential between the two models. If it is positive in expectation we want to use prediction algorithm 2. If it is negative we prefer prediction method 1.

The main point of the paper is that if $$d_t$$ is covariance stationary and of short memory:

$$\Rightarrow \sqrt{T}(\overline{d} - \mu) \overset{d}{\rightarrow} N(0, 2\pi f_d(0))$$

where $$\overline{d}$$ is the sample mean $$d_t$$, $$\mu = E[d_t]$$ and $$f_d(0)$$ is the spectral density of d evaluated at zero and $$T$$ is the number of observations.

The authors do not prove this result or give a hint at how to derive it. In all versions of the paper they only claim that "standard results may be used to derive the asymptotic distribution".

Although this should be a pretty basic issue I don't see how to prove it.

Hints or a complete proof will be of great help to me.

• handwaved results are the bane of financial econometrics Commented May 28, 2021 at 23:56
• @Snoop, I gave a presentation on this about a month ago and even the lecturer, who had assigned me this paper didn't know, where the result comes from. Commented May 29, 2021 at 0:11