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I am looking into comparing the predictive accuracy of forecasts of different models against a benchmark model. For this I have looked into the Diebold-Mariano test statistic. However, I am using the modified Harvey, Leybourne and Newbold version of this test and I do not understand the difference between the used T and h in this modification?

The original DM-statistic value is multiplied by the square root of:

[T + 1 - 2h + h(h-1)/T]/T

As far as I can see, the T and h will be equal for h-period forecast and as T being the length of the forecast errors?

Here is a reference: https://www.bankofcanada.ca/wp-content/uploads/2010/02/wp04-2.pdf

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$h$ is the forecast horizon, whereas $T$ is the sample size for the loss differentials.

For example, if I have data $Y_1, ..., Y_{100}$, I could construct a sequence of 2-step-ahead forecasts like this:

  1. Fit my model on $Y_1, ..., Y_{90}$, and derive $\hat{Y}_{92}$.
  2. Fit my model on $Y_1, ..., Y_{91}$, and derive $\hat{Y}_{93}$.

...

  1. Fit my model on $Y_1, ..., Y_{98}$, and derive $\hat{Y}_{100}$.

I then evaluate my loss function for each of $L(Y_i, \hat{Y}_i)$, for $i=92,...,100$. I do the same for my benchmark model and obtain loss differentials $d_{92},...,d_{100}$.

In this example, we have $h=2$ and $T=9$.

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  • $\begingroup$ Okey, so h is sort of the time increments of the total forecasted horizon? And, just a question, if you were to do as the example, in e.g. step 3 through 9, would you use Yhat_92 or Y_92 when estimating e.g. Yhat_94? (Sorry, for the math, I don't know how to use math mode here...) $\endgroup$
    – kristoffer3110
    Commented May 22, 2020 at 17:05
  • $\begingroup$ $h$ is the number of steps ahead for the forecast, yes. When forecasting for $\hat{Y}_{94}$, you would base it on the actual data up to $Y_{92}$, otherwise it would not be a 2-step-ahead forecast (if you use only $Y_1,...,Y_{90}$, it is a 4-step-ahead forecast). $\endgroup$
    – Chris Haug
    Commented May 23, 2020 at 14:51
  • $\begingroup$ That makes sense, thanks for clarifying. $\endgroup$
    – kristoffer3110
    Commented May 23, 2020 at 16:11

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