Different results from lm and gls
r
Solution
It looks like the AR(1) has gobbled up some of the trend - the parameter `phi` is exceedingly large. Essentially the GLS model has an extra model corresponding to the AR(1) part. Hence you have
regression + AR(1) + $\epsilon$
Together regression and AR(1) combine to give the same fitted value as regression from your `lm()` fit, but the fit is just decomposed differently and the interpretation is also different.
GLS estimates the value for parameter $\phi$ (`phi` in the output above), and that is why the other estimates of the coefficients have changed. Instead you could specify the value of $\phi$ via `corAR1(value = myphi)` where `myphi` is your input value for $\phi$. One option might be to fit the `lm()` model, then estimate the $\phi$ from the residuals of that model, then take the estimated value of $\phi$ anbd plug that into the GLS model and fit. That way you end up with a GLS model that includes the autocorrelation and thence the standard errors etc take it into account in the `summary()` output etc.
All that said, such a large AR(1) is more than likely an indicator that something is wrong or that this isn't the right model. I would be sure to check the model fits etc.
Problem
I am trying to fit a linear time series model in R. My first approach was using lm: ``` > m1 = lm(logp~logg, data = data) > summary(m1) Call: lm(formula = logp ~ logg, data = data) Residuals: Min 1Q Median 3Q Max -0.56209 -0.21766 -0.02728 0.20243 0.82112 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 2.14218 0.59651 3.591 0.000556 *** logg -0.57819 0.04931 -11.725 < 2e-16 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 0.2921 on 83 degrees of freedom Multiple R-squared: 0.6236, Adjusted R-squared: 0.619 F-statistic: 137.5 on 1 and 83 DF, p-value: < 2.2e-16 ``` However, I realised that the residuals are autocorrelated, and I want to compensate for that. So I used gls instead: ``` > m2 = gls(logp~logg, data = data, correlation=corAR1(form=~1)) > summary(m2) Generalized least squares fit by REML Model: logp ~ logg Data: data AIC BIC logLik -83.1498 -73.47444 45.5749 Correlation Structure: AR(1) Formula: ~1 Parameter estimate(s): Phi 0.9313839 Coefficients: Value Std.Error t-value p-value (Intercept) 4.82358 1.1435778 4.217972 1e-04 logg -0.35891 0.0925918 -3.876257 2e-04 Correlation: (Intr) logg 0.986 Standardized residuals: Min Q1 Med Q3 Max -1.5206442 -0.7602385 -0.2905489 0.6310135 2.7341294 Residual standard error: 0.3788309 Degrees of freedom: 85 total; 83 residual ``` My understanding is that the parameter estimates should still be the same, but the t-statistics should be different, as shown here. However, I get very different parameter estimates. Why is that? Am I doing something wrong, or am I misunderstanding the statistics? When I compare the fitted values using `m1$fitted.values` and `m2$fitted` they are exactly the same. This leads me to believe that the parameter estimates from gls should be interpreted in a different way than those from lm, but how?