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I have a time series data y (response variable) which I want to predict using 3 independent time series data x1, x2 and x3. My goal is to build a multivariate regression model for which I want to identify useful lags of x1, x2 and x3 (also possibly lags of y) that might be useful predictors for y. Snapshot of y, x1, x2 and x3 respectively are shown below enter image description here My questions are-

  1. If I determine cross correlation between y and x1 (without taking into account x2 and x3), y and x2 (without taking into account x1 and x3), y and x3 (without taking into account x1 and x2) separately, would that give me useful insight about how many lags of x1, x2 and x3 I should use in my regression model? What would be a better approach? Is there any way to take into account other independent variables while performing cross correation between two time series?

  2. Since there is more than one independent variable, would prewhitening the independent variables before determining cross-correlation give me correct result? Or do I have to use spectral analysis?

  3. independent variable x1 and x2 are not stationary. Because of that I have to make the data stationary before determining the ARMA model. Since for determining CCF, the same filter has be applied on y, should I apply the same steps that I took to make x1 stationary on y too? Or do I have to make y stationary by other separate steps? Or do I just apply the ARMA model filter?

  4. Is it advisable to use Granger causality here do determine if y has causal relationship with x1, x2 and x3?

  5. Another question is could you give me insight from looking at the data if y, x1, x2 and x3 have trend? Is there any test to find out trend? if my time series have trend, do I have to detrend them before determining the regression model?

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  • $\begingroup$ 5. Do you want a regression model with autoregressive errors or just a regression model? $\endgroup$ – Tommaso Guerrini Nov 10 '16 at 19:57
  • $\begingroup$ Actually until now I was building just a regression model. But some insight into a model with autegression error will also be helpful $\endgroup$ – orpia Nov 10 '16 at 21:10
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  1. The cross correlation between the observed series is of little value . You should use single pre-whitening to aid the initial identification. Make sure you account for anomalies in the individual x's before you form the arima filter

  2. use pre-whitening

  3. make each variable stationary and use the filter for each of the x's separaetely to estimate the ccf

  4. no need to use Granger here ,, you are trying to identify an initial model

  5. absolutely no detrending is needed to create the arima filters unless they are needed in concert with the arima model

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  • $\begingroup$ Thank you for your answer. For question 3. I am still confused, do I have to make the response variable y stationary too? Also Suppose I am differencing x1 to make it stationary before modeling with with ARMA (1,1), so effectively it is ARIMA(1,1,1). Then since I have to use the same filter on y, do I difference y too before applying ARMA (1,1) on it? Or do I have to make y stationary first using some other procedure? Or do I just apply ARMA (1,1) on raw y? $\endgroup$ – orpia Nov 11 '16 at 3:44
  • $\begingroup$ Use onlinecourses.science.psu.edu/stat510/node/75 . The filter that is used is does not include any differencing operators as each of the series wlll already been suitably differenced by their individual differencing operator.. There are exceptional cases where it is beneficial not to difference at all and allow the transfer model to fully handle the relationship. $\endgroup$ – IrishStat Nov 11 '16 at 11:15

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