## Abstract

We develop a $C_{p}$ statistic for the selection of regression models with

stationary and nonstationary ARIMA error term. We derive the asymptotic

theory of the maximum likelihood estimators and show they are consistent and

asymptotically Gaussian. We also prove that the distribution of the sum of

squares of one step ahead standardized prediction errors, when the

parameters are estimated, differs from the chi-squared distribution by a

term which tends to infinity at a lower rate than $\chi _{n}^{2}$. We

further prove that, in the prediction error decomposition, the term

involving the sum of the variance of one step ahead standardized prediction

errors is convergent. Finally, we provide a small simulation study.

Empirical comparisons of a consistent version of our $C_{p}$ statistic with

BIC and a generalized RIC show that our statistic has superior performance,

particularly for small signal to noise ratios. A new plot of our time series

$C_{p}$ statistic is highly informative about the choice of model.

stationary and nonstationary ARIMA error term. We derive the asymptotic

theory of the maximum likelihood estimators and show they are consistent and

asymptotically Gaussian. We also prove that the distribution of the sum of

squares of one step ahead standardized prediction errors, when the

parameters are estimated, differs from the chi-squared distribution by a

term which tends to infinity at a lower rate than $\chi _{n}^{2}$. We

further prove that, in the prediction error decomposition, the term

involving the sum of the variance of one step ahead standardized prediction

errors is convergent. Finally, we provide a small simulation study.

Empirical comparisons of a consistent version of our $C_{p}$ statistic with

BIC and a generalized RIC show that our statistic has superior performance,

particularly for small signal to noise ratios. A new plot of our time series

$C_{p}$ statistic is highly informative about the choice of model.

Originalsprog | Engelsk |
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Udgivelsessted | Aarhus |

Udgiver | Institut for Økonomi, Aarhus Universitet |

Antal sider | 31 |

Status | Udgivet - 13 nov. 2012 |

Navn | CREATES Research Paper |
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Nummer | 2012-46 |

## Emneord

- AIC, ARMA models, bias correction, BIC, $C_{p}$ plot, generalized RIC, Kalman filter, Kullback-Leibler distance, state-space formulation