I have two daily temperature data sets from stations near each other (y:blue, x1:red).
As expected they co-vary and the errors are auto-correlated. I need to account for the auto-correlation in order to compare coefficients across potential break points in one of the series. The auto.arima function indicates a (3,1,0) order but what I don't understand is why this has such a large impact on the predictor coefficient x1 compared to ordinary linear regression (lm: 0.81 and arima: 0.12).
Call: arima(x = y, order = c(0, 0, 0), xreg = x1) Coefficients: intercept x1 4.6747 0.8164 s.e. 0.2601 0.0153
Series: y ARIMA(3,1,0) Coefficients: ar1 ar2 ar3 x1 -0.3140 -0.2201 -0.1434 0.1237 s.e. 0.0204 0.0212 0.0206 0.0199 sigma^2 estimated as 0.5846: log likelihood=-4055.19 AIC=8120.38 AICc=8120.39 BIC=8153.51
I would have expected the coefficients for x1 to be fairly similar. As a result forecasting/predicting from new data show a poor result (x1:red, predicted y:green). Is there a simple explanation for this?