Multistep forecast selection for panel data

Abstract

We develop a new set of model selection methods for direct multistep forecasting of panel data vector autoregressive processes. Model selection is based on minimizing the estimated multistep quadratic forecast risk among candidate models. To attenuate the small sample bias of the least squares estimator, models are fitted using bias-corrected least squares. We provide conditions sufficient for the new selection criteria to be asymptotically efficient as n (cross sections) and T (time series) approach infinity. The new criteria outperform alternative selection methods in an empirical application to forecasting metropolitan statistical area population growth in the US.

Keywords:

• Forecasting
• model selection
• panel data
• misspecification
• bias-correction
• final prediction error
• Mallows criterion

JEL classification:

• C23
• C53

Acknowledgments

The author thanks Qazi Haque, Peter C.B. Phillips, and participants at the 2017 New Zealand Econometric Study Group meeting and the 2018 Econometric Society Australasian Meeting for their helpful feedback and comments.

Notes

1 Due to the bias correction, BCLS does not minimize the in-sample multistep quadratic loss, and thus it should not be characterized as the quadratic loss function forecast.

2 Note that the order of the polynomial trend $(p-1)$ in the panel process coincides with that used in the candidate models (1). Theorem 3.1 below shows that the polynomial trend introduces an $O(p^2{T}^{-1})$ term into the QFR, indicating that overfitting the trend order can entail a large reduction in forecast accuracy. Although not pursued in this article, the results derived herein can inform the development of methods for joint selection of k and $p$.

3 Note that the bias-correction implicitly used to construct the in-sample QL of the BCLS fitting in (13) could also be based on the fitted VAR$(k_{n,T})$. This would simply entail replacing ${\widehat{\mathit{\xi }}}_h(k,k_{n,T})$ with ${\mathbf{J}}_{m{k}_{n,T},\mathit{mk}}^{\prime }{\widehat{\mathit{\xi }}}_h(k_{n,T},k_{n,T})$ in (9). This would ensure that the resultant sequence of QLs exhibits a basin-like shape in $k=1,\dots ,k_{n,T}$.

4 Ing (2007) shows that asymptotic efficiency can be restored by having the penalty on model dimensionality approach infinity at a suitably bounded rate. However, increasing the penalty precludes asymptotic efficiency for infinite order processes unless the parameter space is further restricted.

5 Note, however, that the lag order of the direct multistep regression can be less than the one-step regression when the process is of finite order. See Ing (2004).

6 We thank an anonymous referee for this suggestion.

7 Regional adjustment models usually include regional unemployment rates. We do not include this variable because county (and thus MSA) unemployment is only available from 1990 onwards, and we prefer to use a long time series to construct several recursive forecasts over 2002–2016 based on a longer sample spanning 1969–2016.

8 That is, data spanning 1969–1996 are used to forecast 1997 (h = 1), 1998 (h = 2), 1999 (h = 3), and 2000 (h = 4); data spanning 1969–1997 are used to forecast 1998 (h = 1), 1999 (h = 2), 2000 (h = 3), and 2001 (h = 4), and so on.

9 If the measure of in-sample quadratic loss was nonincreasing in the lag order (as is the case with OLS), we would expect the opposite result. However, the in-sample quadratic loss of the BCLS fitting is not necessarily decreasing in k.

Additional information

Funding

This work was supported in part by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand under grant No. 16-UOA-239.