# Why does linear regression with ARIMA errors have large impact on predictor coefficient

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?

• The first case regresses levels on levels; the second case regresses differences on differences. – The Laconic Nov 30 '18 at 13:04
• This is unrelated to your question, but both of your time series seem to display some seasonality (?) which is not captured by the ARIMA model. Did you try plotting each series against things like day of week (1-7), month (1-12)? – Isabella Ghement Nov 30 '18 at 13:52
• @TheLaconic I'm not sure what you mean by 'regressing levels on levels' but regarding the 'I' differencing, it makes little difference if I use AR(3) or AR(1) for that matter. – RSA-Robert Dec 3 '18 at 13:15
• @IsabellaGhement although they both are seasonal and a seasonal component does reduce the prediction error in a linear mixed model, I am only interested in the impact on the predictor coefficient. The way I understand it, the coefficient should remain the heaviest weight for the predictions regardless of how the errors are modelled, given the close linear relationship between the two series. – RSA-Robert Dec 3 '18 at 13:33