Nonparametric Lag Selection for Time Series

Abstract

A nonparametric version of the Final Prediction Error (FPE) is analysed for lag selection in nonlinear autoregressive time series under very general conditions including heteroskedasticity. We prove consistency and derive probabilities of incorrect selections that have been previously unavailable. Since it is more likely to over®t (have too many lags) than to under®t (miss some lags), a correction factor is proposed to reduce over®tting and hence increase correct ®tting. For the FPE calculation, the local linear estimator is introduced in addition to the Nadaraya-Watson estimator in order to cover a very broad class of processes. To achieve faster computation, a plug-in bandwidth is suggested for the local linear estimator. Our Monte-Carlo study corroborates that the correction factor generally improves the probability of correct lag selection for both linear and nonlinear processes and that the plug-in bandwidth works at least as well as its commonly used competitor. The proposed methods are applied to the Canadian lynx data and daily returns of DM/US-Dollar exchange rates. nonparametric version of the FPE. While they allowed for heteroskedasticity, which is a well known feature of ®nancial and many other time series, they did not show consistency. In this paper we close this gap and prove consistency of the FPE based lag selection in the presence of heteroskedasticity.More importantly, we derive the probabilities of incorrect lag selection for the nonparametric FPE criteria. Based on these calculated probabilities, which are new to this research area, we conclude that over®tting is more likely than under®tting. Here over®tting occurs if one chooses super¯uous lags in addition to the correct ones, while missing correct lags is called under®tting. Consequently, we suggest a correction of the nonparametric FPE to reduce over®tting and hence increase correct ®tting. Unlike the correction of Vieu (1994), ours incorporates asymptotic analysis. It is also found to substantially increase correct ®tting in our simulation experiments.Such calculations of over-and under®tting probabilities cannot be simply duplicated for cross-validation. One should also note that in some crude sense the general FPE as de®ned in (2.2) is`equivalent' to the cross-validation, i.e. their difference is of higher order (Cheng and Tong, 1992). In the same way, they are both`equivalent' to the data-driven asymptotic FPE de®ned in (3.4). These higher order terms are no longer negligible for the probability calculations. This is why we prefer the data-driven asymptotic FPE to the cross-validation. A second reason is that the plug-in method can be easily applied to the asymptotic FPE and gives a better rate of convergence than the cross-validation method. For such comparisons in density estimation see Jones et al. (1996). Therefore, we doubt cross-validation criteria can perform numerically as well as our asymptotic FPE criteria.The other authors used exclusively the Nadaraya-Watson estimator for their lag selection procedures. However, the Nadaray...

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