# Computing design matrix from covariance matrix

Suppose I have the following regression model:

$$Y = b_1 \times T + b_2 \times Z + b_3 \times T\times Z + \epsilon$$

where T is a randomly assigned treatment condition and Z is some covariate. I want to test the hypothesis that $$b_1=0$$, using

$$s^2_{b_1}=\sigma^2 (X'X)^{-1}$$

However, the design matrix X has missing data. The missingness is a function of Z. But, I do have a corrected variance/covariance matrix $$\Sigma$$ (using the Pearson-Lawley correction formula). I know that there's a relationship between $$X'X$$ and $$\Sigma$$. If I remember right, it's

$$\Sigma = \Big[X-\frac{1}{n}ee'X\Big]'\Big[X-\frac{1}{n}ee'X\Big]\frac{1}{n}$$

where e is a vector of ones. But, alas, I have a corrected version of $$\Sigma$$, not $$X$$. Any way to go from $$\Sigma$$ to $$X$$ (or $$X'X$$)?

• I think I can guess most of it, but could you please describe X, e, $\Sigma$, and $\sigma$? – eric_kernfeld Oct 4 '16 at 1:11
• X is the design matrix, e is a vector of ones, Sigma is the variance/covariance matrix, and sigma is the residual variance. – dfife Oct 11 '16 at 16:19
• The thread at stats.stackexchange.com/questions/107597 appears to provide a full answer. – whuber May 14 '19 at 18:16

It is impossible to estimate regression coefficients from the covariance matrix of the covariates alone: you also need something resembling $X^TY$ or $Cov(X, Y)$. To see why, suppose you have three i.i.d. standard normal vectors $x_1, x_2, x_3$ and you want to run regressions $x_1 \approx \beta_{1\sim2} x_2 + \beta_{1\sim3} x_3$ and $x_2 \approx \beta_{2\sim2} x_2 + \beta_{2\sim3} x_3$. The optimal coefficients are quite different -- all zero in the first case versus $(1, 0)$ in the second case -- but the $\Sigma$'s would be identical.