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In a simple regression context: $$ y = \alpha + \beta x + e $$ We can estimate beta from: $$ \hat{\beta} = \frac{cov(x,y)}{var(x)} = \rho_{xy} \frac{\sigma_y}{\sigma_x} $$ This last decomposition is useful in empirical cases as it allows to calculate correlation separately from standard deviation. In my use case, for example, it is reccomended to use a longer window to calculate correlation and a smaller window to calculate standard deviations, since correlations are stronger over longer periods of time.

In a multiple regression case, we use: $$ \hat{\beta} = (X'X)^{-1}X'Y $$ I am not able to separate this correlation/standard deviations concept using the formula above. It seems that it would be possible to make something similar. I am getting a bit lost on the matrix algebra. If anyone can provide me some intuition on where to go or what book to look to I would be very very happy.

Tomas da Nobrega
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    It is not possible to use pairwise covariances to calculate the parameters of multivariate regression. – Tim Dec 08 '23 at 19:29
  • I want to calculate parameters using correlation matrices and not pairwise – Tomas da Nobrega Dec 08 '23 at 20:03
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    Although the duplicate answers the question you have asked, your allusions to "windows" and separate calculations (estimates?) of correlations and SDs suggests you might not be asking the right question. The reference to "correlations being stronger" is especially puzzling, given that you are using them to estimate covariances and covariances are variances. – whuber Dec 08 '23 at 20:48
  • In finance, it is common to calculate rolling window regressions. This happens if we believe that the relationship of variables changes over time. For example, 50 years of data and parameters calculated with 5-year rolling window. Some literature (example Frazzini and Perdersen 2014) suggests that we get better estimates by calculating correlations and variances separately. In a CAPM setting, correlation between market return and asset return appear to be more persistent that volatilities. They calculate correlation with a 5y rolling window and volatility with 1y rolling window. – Tomas da Nobrega Dec 08 '23 at 21:35
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    So my question was really about extending that to a multiple regressions case. I will just estimate both separately and the compute a covariance matrix and proceed with your wonderful explanation. Thank you very much @whuber – Tomas da Nobrega Dec 08 '23 at 21:43

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