Editing Autocorrelation (section)

==Regression analysis==

In [[regression analysis]] using [[time series analysis|time series data]], autocorrelation in a variable of interest is typically modeled either with an [[autoregressive model]] (AR), a [[moving average model]] (MA), their combination as an [[autoregressive-moving-average model]] (ARMA), or an extension of the latter called an [[autoregressive integrated moving average model]] (ARIMA). With multiple interrelated data series, [[vector autoregression]] (VAR) or its extensions are used.

In [[ordinary least squares]] (OLS), the adequacy of a model specification can be checked in part by establishing whether there is autocorrelation of the [[errors and residuals in statistics|regression residuals]]. Problematic autocorrelation of the errors, which themselves are unobserved, can generally be detected because it produces autocorrelation in the observable residuals. (Errors are also known as "error terms" in [[econometrics]].) Autocorrelation of the errors violates the ordinary least squares assumption that the error terms are uncorrelated, meaning that the [[Gauss–Markov theorem|Gauss Markov theorem]] does not apply, and that OLS estimators are no longer the Best Linear Unbiased Estimators ([[BLUE]]). While it does not bias the OLS coefficient estimates, the [[Standard error (statistics)|standard errors]] tend to be underestimated (and the [[T-statistics|t-scores]] overestimated) when the autocorrelations of the errors at low lags are positive.

The traditional test for the presence of first-order autocorrelation is the [[Durbin–Watson statistic]] or, if the explanatory variables include a lagged dependent variable, [[Durbin–Watson statistic#Durbin h-statistic|Durbin's h statistic]]. The Durbin-Watson can be linearly mapped however to the Pearson correlation between values and their lags.<ref>{{cite web|url=http://statisticalideas.blogspot.com/2014/05/serial-correlation-techniques.html|work= Statistical Ideas|title= Serial correlation techniques|date = 26 May 2014}}</ref>  A more flexible test, covering autocorrelation of higher orders and applicable whether or not the regressors include lags of the dependent variable, is the [[Breusch–Godfrey test]]. This involves an auxiliary regression, wherein the residuals obtained from estimating the model of interest are regressed on (a) the original regressors and (b) ''k'' lags of the residuals, where 'k' is the order of the test. The simplest version of the [[test statistic]] from this auxiliary regression is ''TR''<sup>2</sup>, where ''T'' is the sample size and ''R''<sup>2</sup> is the [[coefficient of determination]]. Under the null hypothesis of no autocorrelation, this statistic is asymptotically [[Chi-squared distribution|distributed as <math>\chi^2</math>]] with ''k'' degrees of freedom.

Responses to nonzero autocorrelation include [[generalized least squares]] and the [[Newey West|Newey–West HAC estimator]] (Heteroskedasticity and Autocorrelation Consistent).<ref>{{cite book | title = An Introduction to Modern Econometrics Using Stata |first= Christopher F. |last=Baum | publisher = Stata Press | year = 2006 | isbn = 978-1-59718-013-9}}</ref>

In the estimation of a [[moving average model]] (MA), the autocorrelation function is used to determine the appropriate number of lagged error terms to be included. This is based on the fact that for an MA process of order ''q'', we have <math>R(\tau) \neq 0</math>, for <math> \tau = 0,1, \ldots , q</math>, and <math> R(\tau) = 0</math>, for <math>\tau >q</math>.