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I have a question about multivariate hypothesis testing in out-of-sample evaluations.

Generally, let`s asssume we want to predict three different stock returns in a 1-month ahead forecasting setting. We have collected five different features for linear regression appraoches. Furthermore, we already splitted our data in a initial training set, which grows by each prediction iteration.

Lets assume the following aspects:

  1. we firstly model a multivariate multiple predictive regression
  2. we secondly model a VARX model for the forecasting purpose

Finally, we end up with three time series of prediction error for each model (Error stock A, Error stock B and Error stock C). Thus, we can calculate the means squarred forecasting errors (MSFE) for each model and for each response.

My question is:

How can we jointly test (by simultaneously analyzing all three MSFE), which model is more accuracte in terms of predicting ALL stocks in the 1-month ahead setting?

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