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Given a multivariate regression, how can I test if each element in the coefficient matrix is statistically significant? Would doing a t-test be right? $$\mathbf{Y}=\mathbf{X}\mathbf{B}+\mathbf{E},$$ where $\mathbf{Y}, \mathbf{X}$ and $\mathbf{E} \in R^{n\times m}$ $(n>m)$ and $\mathbf{B}\in R^{m \times m}$. You can treat each column of $\mathbf{Y}$ and $\mathbf{X}$ as a time series.

I found this where it goes through multilinear regression (p.43) and also goes through inferences about the coefficient (p.75). But it considers each row of the coefficient in the test. My question is about testing each element of the coefficient to see if it is significant.

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  • $\begingroup$ How can you test it if Y is univariate? (It is a time series!) $\endgroup$
    – Michael M
    Commented Apr 25, 2023 at 15:49
  • $\begingroup$ Each $\textbf{column}$ can be treated as a time series; rows representing the date points and columns are the variables. @MichaelM $\endgroup$
    – statwoman
    Commented Apr 25, 2023 at 15:52
  • $\begingroup$ Yes. How would you test if Y would only have one column, i.e., would represent only one time series? $\endgroup$
    – Michael M
    Commented Apr 25, 2023 at 15:57
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    $\begingroup$ I believe a t-test would work for multiple linear regression. $t=\frac{\hat{B}_j}{se(\hat{B}_j)}$ @MichaelM $\endgroup$
    – statwoman
    Commented Apr 25, 2023 at 16:10

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If I am understanding this correctly, a t-test with Bonferroni correction to adjust for multiple comparisons should work.

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  • $\begingroup$ Could you elaborate? $\endgroup$
    – statwoman
    Commented May 5, 2023 at 14:51

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