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I'm working with some time series data (n=40) and trying to fit an OLS with AutoRegressive Errors to model the relationship between my dependent variable and a couple of predictors over time. I'm working in R.

Dependent variable looks like this:

Spending Time Series

Unit root tests suggest that there is indeed a root process.

I'm working through the process of fitting an OLS, inspecting the residuals and the (P)ACF plots, and then using the sarima function to add AR errors onto the model, fine turning the p, d, and q values in response to the model diagnostics.

Eventually, I am able to end up with a model producing output plots looking nice and healthy like this:

post-estimation plot for OLS with AR Errors (1,0,0)

...with the AR term statistically significant in the model. There are other possible specifications with similar-looking plots and significance values, but they have higher AIC/AICc/BIC scores.

At no point however am I differencing the series.

My question is therefore: are my results (and the post-estimation plots) completely false and spurious (as I am using a non-differenced, non-stationary time series)? Or is it indeed possible to use ARIMA terms in an AR Error framework to 'fix' the spurious regression problem?

My confusion comes from the fact that the post-estimation plots would suggest that the autocorrelation has been dealt with. But I imagine that this could be a completely different issue (or non-issue) compared to non-stationarity.

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