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I'm looking at two time series and I would like to determine how they move together. They both are however stationary. Would it make sense to test for cointegration? Linear relationship between both series has a very low R^2 and correlation but I think that there might be a lag or something and that they move together.

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No. Cointegration is a property of nonstationary series. Let's call the variables $y_t$ and $x_t$. Your best bet is to fit models with both lags of $y_t$ and $x_t$:

$$y_t = b_0 + b_1 y_{t-1} + ... + b_n y_{t-n} + b_{n+1} x_{t-1} + ... + b_{2n} x_{t-n} + \varepsilon_t$$

and then do likelihood ratio test on the various models, or look at adjusted $R^2$, AIC or hold out forecast performance to pick a set of plausible models. You could then pick the 'best' model, or just average the results from the top models.

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  • $\begingroup$ If you also write an analogous equation for $x_t$ and consider the two equations together, you will have a VAR model. There is a large literature on VAR models out there. $\endgroup$ – Richard Hardy Jan 30 '15 at 10:55

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