What to make of explanatories in time series?

Question

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?

Answer 1

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.

Answer 2

The same intuition as in cross-section regression can be used in time-series regression. It is perfectly valid to try to explain the trend using other variables. The main difference is that it is implicitly assumed that the regressors are random variables. So in regression model:

$$Y_t=\beta_0+X_{t1}\beta_1+...+X_{tk}\beta_k+\varepsilon_t$$

we require $E(\varepsilon_t|X_{t1},...,X_{tk})=0$ instead of $E\varepsilon_t=0$ and $E(\varepsilon_t^2|X_{t1},...,X_{tk})=\sigma^2$ instead of $E\varepsilon_t^2=\sigma^2$.

The practical part of regression stays the same, all the usual statistics and methods apply.

The hard part is to show for which types of random variables, or in this cases stochastic processes $X_{tk}$ we can use classical methods. The usual central limit theorem cannot be applied, since it involves independent random variables. Time series processes are usually not independent. This is where importance of stationarity comes into play. It is shown that for large part of stationary processes the central limit theorem can be applied, so classical regression analysis can be applied.

The main caveat of time-series regression is that it can massively fail when the regressors are not stationary. Then usual regression methods can show that the trend is explained, when in fact it is not. So if you want to explain trend you must check for non-stationarity before proceeding. Otherwise you might arrive at false conclusions.

Answer 3

When you have supporting/causal/helping/right-hand side/exogenous/predictor series, the approach that is preferred is to construct a single equation, multiple-input Transfer Function. One needs to examine possible model residuals for both unspecified/omitted deterministic inputs i.e. do Intervention Detection ala Ruey Tsay 1988 Journal of Forecasting and unspecified stochastic inputs via an ARIMA component. Thus you can explicitly include not only the user-suggested causals (and any needed lags !) but two kinds of omitted structures ( dummies and ARIMA ).

Care should be taken to ensure that the parameters of the final model do not change significantly over time otherwise data segmentation might be in order and that the residuals from the final model can not be proven to have heterogeneous variance.

The trend in the original series may be due to trends in the predictor series or due to Autoregressive dynamics in the series of interest or potentially due to an omitted deterministic series proxied by a steady state constant or even one or more local time trends.

Answer 4

As a less technical point of view, often times it's not very helpful just explaining the trend; that is, to treat time as the predictor of primary interest. The variation of a series over time often imply the underlying effects of other variables, including autoregressive and/or exogenous processes, which is more conceptually relevant to investigate. It follows that if those variables also vary over time, then controlling for the time effect is in fact needed to not fall in the artificially significant relationship as @mpiktas showed.