Journal of Statistical and Econometric Methods

Asymptotic relationship between sample mean and sample variance for autoregressive processes of order 1

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  • Abstract

    Autoregressive processes of order 1 (or AR(1) processes) have been extensively used in econometrics and time series literature. Noting that an early important result concerning the sample mean 𝑈 and variance 𝑆 of independent normally distributed random variables 𝑈 with equal means and variances is that 𝑈 and 𝑆 are independent, the present article investigates whether this result can be extended to AR(1) non-stationary processes as the sample size becomes very large. To this end, a property called “asymptotic stationarity” is used for algebraic calculations. A result for asymptotic independence concerning the sample mean and variance is then adequately derived for these types of processes.

    Mathematics Subject Classification: 62E20; 62M10
    Keywords: Autoregressive process; Asymptotic stationarity; Asymptotic independence