I have modeled a stationary time series with another related stationary time series. I'm having a problem with cyclicality in the residuals and I don't know how to fix it.
Here is my model:
$\text{TS}_1(t) = \beta_0 + \beta_1\cdot \text{TS}_2(t) + \epsilon$
Here is a chart of the residuals:
There is clearly a strong trend. I tried to take out the trend by adding a 15 day lagged variable. The new residuals are looking a lot better, but there still looks like there is some kind of trend or abnormality (they don't look random to me).
Here is the model with the Lag:
$\text{TS}_1(t) = \beta_0 + \beta_1\cdot \text{TS}_2(t) + \beta_2\cdot(\text{TS}_2(t-15)) + \epsilon$
I haven't ever done anything like this before. I know adding a lagged variable in AR models can remove seasonality, but I don't know if that applies to the errors on a time series regressed on a different time series.
Is adding a lagged variable to the model the appropriate way of removing trends in the residuals? What tests can I run (other than just looking at the chart) to decide of the trend is still an issue? I ran the Durbin-Watson test (both models failed), but I don't know if the test applies when modeling one time series from another.