Two variables: Which moves first?

Say I have a time series of two variables, $X$ and $Y$, and want to find out which one of these two moves first. One example is the one of GDP growth and investment growth, where we usually tell the story that at a business cycle frequency, investment moves "earlier" than production or consumption.

I dont want to do eyeballing econometrics ("this thingy here clearly moved first"). What is the standard way of dealing with these things in the literature? What methodology can be used? Answers that allow for more than two variables are very welcome.

• How much prior information do you have? Can you express these priors as probability distributions? – EnergyNumbers Jul 17 '15 at 9:32
• @EnergyNumbers Thinking about them as GDP (growth) and investment (growth) makes most sense. I have no prior information besides my time series. – FooBar Jul 17 '15 at 9:36
• No prior information at all on the extent to which investment has any influence on growth, or vice versa? That's surprising. In that case, I guess the answer will be Grainger Causality, but I haven't really done much econometrics for years, so will leave that for someone else – EnergyNumbers Jul 17 '15 at 10:28
• @EnergyNumbers I'd like to do the analysis without having to deal with the critique "your results comes from your prior (information)". But if it's that important, I think a sign restriction would still be acceptable. – FooBar Jul 17 '15 at 11:15

"Who moves first" can be conveniently detached from any causal inference, since there may be some third variable influencing both. Certainly one could build a logical and reasonable argument that links increases in investment to increases in GDP (or vice versa), but it is not necessary.

Then, examining cross-correlation coefficients on the levels or on an appropriate transformation (like first differences, or first log-differences) is a simple and valid way to examine the issue, as long as the chosen transformation will provide stationary series.

Specifically, one can examine (sample) ${\rm Corr} (X_t, Y_{t})$, ${\rm Corr} (X_t, Y_{t-k})$ and ${\rm Corr} (X_{t}, Y_{t+k})$, where, again, $X$ and $Y$ can represent the chosen transformation and not necessarily the levels of the variables. Also, what calendar time length will $t$ represent should also be decided according to the phenomenon under study. In turn, the distance represented by $k$ is a matter of experimentation, but it is advisable to start with $k=1$.

If there is a temporal order, then the contemporaneous cross-correlation ${\rm Corr} (X_t, Y_{t})$ should be negligible. Then, the magnitude and sign of the other two cross-correlations should provide statistical evidence as to "who moves first".

• Thank god! I was really disturbed by the comment on causality, as I had exactly in my intuition that ""Who moves first" can be conveniently detached from any causal inference" – FooBar Jul 17 '15 at 16:22

Granger Causality is exactly what you are looking for.

Don't be fooled: Granger Causality does not imply causality. To say that $X$ Granger-causes $Y$ merely means that lagged values of $X$ add some predictive power when predicting $Y$ as compared to a univariate autoregression of $Y$.

Is there some sort of causation between $X$ and $Y$? For example, $X$ increasing by 1 unit in period $t$ caused an increase in $Y$ in period $t+1$?

If this is the case you can create lagged variables and look at their coefficients and statistical significance. You may run into serial correlation problems running OLS which is another issue altogether.

Model $Y_t = \beta_1 Y_{t-1} + \beta_2 Y_{t-2} + \ldots + \beta_n Y_{t-n} + \beta_{n+1} X_t + \beta_{n+2} X_{t-1} + \ldots + \beta_{n+m} X_{t-m}$

You can run a similar model treating $X_t$ as the dependent variable. Most likely $m$ and $n$ need not be too large. And for your purposes it may be appropiate to use $n=0$.

• Given such a model, is there a unique criterion? Is the "first moving variable" the one with the smallest $R^2$? – FooBar Jul 17 '15 at 11:27
• Not really. All you really care about is interpreting the first lag. All the other lags are for purposes of soaking up additional effects. For example, the effect of a change of $X_t$ on $Y_{t+1} might be$\beta$, and then that change in period t might persist and effect$Y_{t+2}$by a factor of$\beta^2$assuming$\beta \in (0,1)\$. I would run a model for X and a model for Y and compare the first lags. If there isn't significant collinearity the coefficient of the one period lagged variable that actually does move first will be significantly higher than its counterpart. – hipHopMetropolisHastings Jul 17 '15 at 11:37
• Do you have some reference for this, be it a textbook or a paper where it is actually applied? – FooBar Jul 17 '15 at 11:38
• I'm not sure about your specific problem, it is just how I would do it. For a reference to understanding endogeneity I'd check out Verbeek -Guide to Modern Econometrics Ch. 5. Maybe where he looks at a simultaneous model. Verbeek is graduate level, for undergraduate level I bet there would be something in Wooldridge (title escapes me). – hipHopMetropolisHastings Jul 17 '15 at 11:51