Having worked mostly with cross sectional data so far and very very recently browsing, scanning stumbling through a bunch of introductory time series literature I wonder what which role explanatory variables are playing in time series analysis.
I would like to explain a trend instead of de-trending.
Most of what I read as an introduction assumes that the series is stemming from some stochastic process. I read about AR(p) and MA processes as well as ARIMA modelling. Wanting to deal with more information than only autoregressive processes I found VAR / VECM and ran some examples, but still I wonder if there is some case that is related closer to what explanatories do in cross sections.
The motivation behind this is that decomposition of my series shows that the trend is the major contributor while remainder and seasonal effect hardly play a role. I would like to explain this trend.
Can / should I regress my series on multiple different series? Intuitively I would use gls because of serial correlation (I am not so sure about the cor structure). I heard about spurious regression and understand that this is a pitfall, nevertheless I am looking for a way to explain a trend.
Is this completely wrong or uncommon? Or have I just missed the right chapter so far?
Best Answer
Based upon the comments that you've offered to the responses, you need to be aware of spurious causation. Any variable with a time trend is going to be correlated with another variable that also has a time trend. For example, my weight from birth to age 27 is going to be highly correlated with your weight from birth to age 27. Obviously, my weight isn't caused by your weight. If it was, I'd ask that you go to the gym more frequently, please.
As you are familiar with cross-section data, I'll give you an omitted variables explanation. Let my weight be $x_t$ and your weight be $y_t$, where $$\begin{align*}x_t &= \alpha_0 + \alpha_1 t + \epsilon_t \text{ and} \\ y_t &= \beta_0 + \beta_1 t + \eta_t.\end{align*}$$
Then the regression $$\begin{equation*}y_t = \gamma_0 + \gamma_1 x_t + \nu_t\end{equation*}$$ has an omitted variable---the time trend---that is correlated with the included variable, $x_t$. Hence, the coefficient $\gamma_1$ will be biased (in this case, it will be positive, as our weights grow over time).
When you are performing time series analysis, you need to be sure that your variables are stationary or you'll get these spurious causation results. An exception would be integrated series, but I'd refer you to time series texts to hear more about that.