TL;DR: Does non-stationarity matter for trend-based time series models (i.e., time-series regressions that are not auto-regressive)?
I came across a question from a third-party provider in which we’re provided with two trend based time-series models - both for sales. One is a single-variable regression that forecasts Sales. The second is a single-variable regression that forecasts ln(Sales). In both models, the independent variable is time (e.g., 1, 2, 3,…, 12 for each quarter of the last three years).
The question asks whether the log-linear transformation improves the results for autocorrelation and non-stationarity. I understand the relevance of autocorrelation here. But regarding non-stationarity - is this even a relevant concern for linear regressions which aren’t autoregressive? If so, why?
(Am intentionally not posting the question verbatim so as not to have the post removed.)
Yes, stationarity is *always* relevant when you are attempting to perform linear regression with time series data. There are some subtleties (I imagine in the regression you’re discussing, folks would be exploring if the y variable is stationary with trend), but generally if you have non stationary variables and no cointegration, then you have violated the classical linear assumptions when attempting to specify the model.
If stationarity is relevant for non-autoregressive models, then how is it ever statistically sound to model something like ln(sales) as a function of time, given that (assuming a constant growth rate) the regression would be an upward sloped line (i.e., non-stationary mean)?
Because there are other variations of stationarity, such as stationary with trend. In other words, testing whether or not the series is stationary AFTER removing a time trend. I imagine that’s what is happening in this question. Taking natural logs is helpful when you have an exponential series, since doing so reduces it to a linear series. And if it has a time trend, you would detrend it first and then check if the residual series is stationary.
Another thing maybe worth mentioning is that ln(sales) better models the linear trend, meaning values are not substantially above/below the trend line (serial correlation in errors).