# Two or more time series. What is the best way to test whether one of them is leading and by what time period?

I am trying to prove/disprove that one time series is leading trend for the other ones. Two time series are (probably) independent and the movements are caused by some (let's assume unknown) common factors. What method would be most appropriate? I also want to find the leading period? Data is quarterly.

You could try cross-correlation analysis with R for example. Cross-correlation at lag h measures temporal dependency of two time series (x{t+h},y{t}) at lag h. If h<0 and cross-correlation is statistically significant then you could say that x series leads y series by h time units. For example approval of building permit and finishing of housing project might have several months lag.

In R cross-correlation can be calculated as follows:

ccf(x series,y series,lags to show)

Look spikes in the graph produced by this function/object call.

EDIT:

Raw time series must often be pre-whitened before ccf - analysis should be done. Pre-whitening can be done this way:

1) Create an arima model for series x{t} and save residuals.
2) Use previous model to filter series y{t} so that you get residuals.
3) Do ccf- analysis for the residual series.

Why pre-whiten?

Autocorrelation structures and need for differencing might demand that ccf - analysis should be done for the residual series and not to raw pre-filtered series.