Recovering Multiple Regression Coefficients from Covariance Information

Summary

The document asks whether the coefficients in a regression with two predictors can be recovered without the underlying observations, given each predictor’s separate regression coefficient, the predictors’ variances and correlation, and the outcome variance. The answer expresses the multiple regression coefficients through the predictors’ cross-product matrix and their cross-products with the outcome. Predictor covariance information supplies the matrix, while the univariate slopes and predictor variances imply the predictor–outcome covariances.

The response further states that knowing the full covariance matrix of both predictors and the outcome determines the coefficients. It adds a claim about the joint distribution under normality, but the regression coefficient calculation itself depends on the covariance structure and does not require normality. The note offers a compact identification argument rather than a worked numerical example, and the question’s listed inputs do not explicitly include all predictor–outcome covariances, which must be inferred from the univariate regressions.

Key ideas

• Multiple regression coefficients can be obtained from the predictor covariance matrix and predictor–outcome covariances.
• Predictor variances and their correlation determine the predictors’ covariance matrix.
• Univariate regression slopes and predictor variances imply each predictor’s covariance with the outcome.
• A numerical example is not provided, and the derivation assumes the required covariance terms can be recovered.

Tags

# How to obtain bivariate regression coefficients from two univariate regression coefficients?


# How to obtain bivariate regression coefficients from two univariate regression coefficients?












Let's assume that we want to obtain the coefficients of the following bivariate regression: $Y=\beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon$

However, we don't have access to the data $(X_1,X_2,Y)$. The only information available are:

- Coefficients of the univariate regression $Y=\gamma_0 + \gamma_1 X_1 + \varepsilon$

- Coefficients of the univariate regression $Y=\lambda_0 + \lambda_1 X_2 + \nu$

- Variances of $X_1$ and $X_2$, plus correlation between these two

- Variance of $Y$

How can we obtain $\beta_1$ and $\beta_2$ from this information?

## Answer by Arshdeep (score 0)

https://quant.stackexchange.com/a/78882

Betas are $(x'x)^{-1}(x'y)$. You know $(x'x)^{-1}$ from variances and correlation. You know $x'y$ from the correlation between each x and y (imply from regression coefficient).

Edit for clarity, x is a matrix of the data of the final regression.

Final edit: If you know the covariance matrix of x1,x2 and y, then you know the joint distribution completely and therefore you know the new coefficients completely. They are in this case, deterministic and one need not calculate the coefficients to realise this. This is ofc assuming normality

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.