I am trying to solve a problem of dimentionality reduction on a Matrix of predictors X(136x481). I found that PCA does not a good job in my case because it create components that explain just the variance of the Matrix X without taking into consideration Y. Considering that the final goal is to estimate the rolling coefficients to explain the changes in Y, I though that maybe PLS Regression as a preliminary step whould have worked way better than PCA.
Therefore I run the PLS regression and I obtained 126 components that explain 96% of the variance. Now considering that not all of them are significant, I was thinking to run a rolling stepwise selction that captures which of the previously estimated components are significant within each regression window in order to get a new Matrix of predictors that I will use to estimate the coefficients in a rolling fashion.
Does the above procedure make any sense? Am I missing something?
Thank you for your help.
I will explain you now better the rationale behind these procedure. Basically, the vector of Y is a vertor of return generated by n-factors(the columns of X). I do not know, however, which are the factors that capture the return on each period (136 monthly returns). Therefore, I decided that to be sure to include a Comprehensive pool of factors for the Whole time series, the best approach would have been to include the largest number of factors available. Now, the returns on the vector Y comes from exposures (columns of X) that change very often through time, which means that at each period, one factor that previously contributed now may not contribute. The first problem generated by this approach is therefore that I need to find a way to select the relevant variables in a rolling framework( that is, within each regression window) in a way that are both statistically significant and not correlated to a level that creates multicollinearity problems.
Summing up, I have to create a new Matrix X that contains a smaller number of factors but that are highly significant for the Y and at the same time not correlated. After that I have to run a rolling OLS to estimate the coefficients of this new Matrix, to estimate the rolling exposures (betas) on these components but this is behind the scope of the question.
Coming back to the problem, I thought to use PLS regression as you said just to obtain the scores to be used in the stepwise. The rolling stepwise regression I performed after basically select within each regression window the scores that are significant (pval<=.05 ) for the Y values every time the window slides. I did this process mainly because the Scores obtained from PLS are combinations of the original variables which as I said, may be significant in one period but not significant in another. Could you please explain me why this selection process based on adding/dropping the scores of PLS would not yield good results?
edit I now understand that basically once you set up the problem od dimentionality reduction with PC you have to use all of them. But if I wantto accomplish both dimentionality reduction and statistic significance in a rolling fashion how can I do it behind the stepwise procedure? I am asking this because stepwise still generates matrix of predictors that are significant but highly correlated