I have an enquiry regarding the Granger Causality analysis. It is said that it is performed to check whether “X causes Y”, or to put it differently, whether X contains any predictive information with regards to Y and it mainly builds two regression models (one nested to other).
The first model (unrestricted) regresses Y against lagged values of Y and lagged values of X while the second model (restricted) regresses Y against lagged values of Y. It then uses a nested F test to compare the two models and draw conclusions.
Thus I am wondering, what’s the difference between this and a time series linear regression model which uses lagged values of the same variables for making predictions. In case a significant predictor is found, can one use the respective unrestricted model for prediction purposes?
Here is a "picture" of what I need to do (it may be easier for you to realize which point I fail to understand):
I have a DJIA closing values time series and I also have several different sentiment time series extracted from Twitter (for instance positive to negative tweets ratio). I need to assess whether integrating such sentiment time series in a predictive model, improves the prediction accuracy. Firstly I “stationarize” my time series and following I conduct a Granger causality analysis so as to determine whether these carry any predictive information about the DJIA closing values.
Following, I need to build a predictive model. Based on my understanding at the moment, an ARIMAX model would be appropriate for this task. So I thought I should deploy a predictive model which will include both past DJIA values and some of the sentiment time series as predictors. My question is how to build such a model and how is this going to be different from the unrestricted model I will have already used in the GCA.
Moreover I am not sure how to decide on which of the time series should I use as predictors and how to decide on the lags I need to include for both the AR and the MA part.
Thank you in advance for your help.