The asymptotic variance in constructing confidence intervals for the stationary mean

Abstract

In this study, using Monte Carlo simulations, we evaluate three alternative methods for constructing
confidence intervals for the population mean in the case of a stationary first order
autoregressive process, AR(1), with parameter φ. Differentiating the three methodologies with
respect to the way of estimating the asymptotic variance, we infer that in constructing confidence intervals we have to avoid the use of the observations of the time series under consideration for
the estimation of the autovariance and the autocorrelation coefficients. Instead, it is preferable
to identify the series according to Box-Jenkins and then use the asymptotic variance derived
from the corresponding ARMA model after the substitution of the OLS parameter and error
variance estimates. It is worth mentioning that using the asymptotic variance, for small samples
and in the case of an AR(1) with positive φ values, the expected actual confidence levels are
larger as compared to the corresponding nominal ones, indicating a potential area for future research.