Central Limit Theorems When Data Are Dependent: Addressing the Pedagogical Gaps

Abstract

Although dependence in financial data is pervasive, standard doctoral-level econometrics texts do not make clear that the common central limit theorems (CLTs) contained therein fail when applied to dependent data. More advanced books that are clear in their CLT assumptions do not contain any worked examples of CLTs that apply to dependent data. We address these pedagogical gaps by discussing dependence in financial data and dependence assumptions in CLTs and by giving a worked example of the application of a CLT for dependent data to the case of the derivation of the asymptotic distribution of the sample variance of a Gaussian AR(1). We also provide code and the results for a Monte-Carlo simulation used to check the results of the derivation.

Keywords: Central Limit Theorem, Dependent Data, Gaussian AR(1), Monte-Carlo, Asymptotic Distributions

JEL Classification: A23, C12, C22, G12