I am investigating forecast optimality. Diebold (2017, p. 334, list item d) indicates that one of the desirable properties of a good forecast is
Optimal forecasts have $h$-step-ahead errors with variances that are nondecreasing in $h$ <...>.
I would like to test this for my forecasts. I have $H$ series of forecasts:
- a series of 1-step-ahead forecasts,
- a series of 2-step-ahead forecasts,
- ...
- a series of $H$-step-ahead forecasts
with their corresponding series of forecast errors:
- a series of 1-step-ahead forecast errors (a vector $e_1$) with theoretical variance $\sigma_1^2$ and estimated variance $\hat\sigma_1^2$,
- a series of 2-step-ahead forecast errors (a vector $e_2$) with theoretical variance $\sigma_2^2$ and estimated variance $\hat\sigma_2^2$,
- ...
- a series of $H$-step-ahead forecast errors (a vector $e_H$) with theoretical variance $\sigma_H^2$ and estimated variance $\hat\sigma_H^2$.
Each forecast error series corresponds to the same target series. Therefore, the shocks that occur during the lifetime of a forecast$^1$ overlap across series$^2$, making all $H$ forecast error series dependent.
Question: How may I test whether the variances of the forecast errors are nondecreasing w.r.t. the forecast horizon, i.e. $\sigma_1^2 \leq \dots \leq \sigma_H^2$?
References
- Diebold "Forecasting in Economics, Business, Finance and Beyond" (version of 1st August 2017)
$^1$ The lifetime is 1 time period for 1-step-ahead forecast, ..., $H$ time periods for $H$-step-ahead forecast.
$^2$ For a given target, there is 1 shock that is common to all $H$ horizons, ..., $H-1$ shocks that are common to the longest two horizons.
Related older question: "Comparing variances of forecast errors".
