# [R] Coefficient of determination in a regression model with AR(1) residuals

From: Hofert Marius <m_hofert_at_web.de>
Date: Thu, 24 Apr 2008 07:57:53 +0200

Dear R-users,

I used lm() to fit a standard linear regression model to a given data set, which led to a coefficient of determination (R^2) of about 0.96. After checking the residuals I realized that they follow an autoregressive process (AR) of order 1 (and therefore contradicting the i.i.d. assumption of the regression model). I then used gls() [library nlme] to fit a linear regression model with AR(1)-residuals. The residuals look perfect (residual plot, ACF, PACF, QQPlot, Ljung- Box test).
As mentioned on http://en.wikipedia.org/wiki/ Coefficient_of_determination (citation [2008-04-24]: "For cases other than fitting by ordinary least squares, the R^2 statistic can be calculated as above" and later: "Values for R^2 can be calculated for any type of predictive model"), I tried to calculate the standard R^2 for the model with AR(1) residuals. However, I ended up with R^2 larger than 1!
As mentioned on the German wikipedia page (http://de.wikipedia.org/ wiki/Bestimmtheitsmaß), in models fitted using Maximum Likelihood Estimation (MLE), the coefficient of determination does _not_ exist (citation [2008-04-24]: "Bei bestimmten statistischen Modellen, z.B. bei Maximum-Likelihood-Schätzungen, existiert das Bestimmtheitsmaß R^2 nicht"). Any comments on that?

The German Wikipedia page mentions McFadden's pseudo-coefficient of determination, the English Wikipedia page the one of Nagelkerke. I know there are others, too. Is there a general agreement on which "coefficient of determination" (or goodness-of-fit measure in general) to use for a regression model with autocorrelated errors? Is there a possibility to compare (non-graphically) the standard regression model with the model with AR(1) residuals to justify the better fit of the latter?

Best regards.

Marius

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