Is there a way to use the covariance matrix to find coefficients for multiple regression?

Question

For simple linear regression, the regression coefficient is calculable directly from the variance-covariance matrix $C$, by $$ C_{d, e}\over C_{e,e} $$ where $d$ is the dependent variable's index, and $e$ is the explanatory variable's index.

If one only has the covariance matrix, is it possible to calculate the coefficients for a model with multiple explanatory variables?

ETA: For two explanatory variables, it appears that $$\beta_1 = \frac{Cov(y,x_1)var(x_2) - Cov(y,x_2)Cov(x_1,x_2)}{var(x_1)var(x_2) - Cov(x_1,x_2)^2} $$ and analogously for $\beta_2$. I'm not immediately seeing how to extend this to three or more variables.

Answer 1

Yes, the covariance matrix of all the variables--explanatory and response--contains the information needed to find all the coefficients, provided an intercept (constant) term is included in the model. (Although the covariances provide no information about the constant term, it can be found from the means of the data.)

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Analysis

Let the data for the explanatory variables be arranged as $n$-dimensional column vectors $x_1, x_2, \ldots, x_p$ with covariance matrix $C_X$ and the response variable be the column vector $y$, considered to be a realization of a random variable $Y$. The ordinary least squares estimates $\hat\beta$ of the coefficients in the model

$$\mathbb{E}(Y) = \alpha + X\beta$$

are obtained by assembling the $p+1$ column vectors $X_0 = (1, 1, \ldots, 1)^\prime, X_1, \ldots, X_p$ into an $n \times p+1$ array $X$ and solving the system of linear equations

$$X^\prime X \hat\beta = X^\prime y.$$

It is equivalent to the system

$$\frac{1}{n}X^\prime X \hat\beta = \frac{1}{n}X^\prime y.$$

Gaussian elimination will solve this system. It proceeds by adjoining the $p+1\times p+1$ matrix $\frac{1}{n}X^\prime X$ and the $p+1$-vector $\frac{1}{n}X^\prime y$ into a $p+1 \times p+2$ array $A$ and row-reducing it.

The first step will inspect $\frac{1}{n}(X^\prime X)_{11} = \frac{1}{n}X_0^\prime X_0 = 1$. Finding this to be nonzero, it proceeds to subtract appropriate multiples of the first row of $A$ from the remaining rows in order to zero out the remaining entries in its first column. These multiples will be $\frac{1}{n}X_0^\prime X_i = \overline X_i$ and the number subtracted from the entry $A_{i+1,j+1} = X_i^\prime X_j$ will equal $\overline X_i \overline X_j$. This is just the formula for the covariance of $X_i$ and $X_j$. Moreover, the number left in the $i+1, p+2$ position equals $\frac{1}{n}X_i^\prime y - \overline{X_i}\overline{y}$, the covariance of $X_i$ with $y$.

Thus, after the first step of Gaussian elimination the system is reduced to solving

$$C_X\hat{\beta} = (\text{Cov}(X_i, y))^\prime$$

and obviously--since all the coefficients are covariances--that solution can be found from the covariance matrix of all the variables.

(When $C_X$ is invertible the solution can be written $C_X^{-1}(\text{Cov}(X_i, y))^\prime$. The formulas given in the question are special cases of this when $p=1$ and $p=2$. Writing out such formulas explicitly will become more and more complex as $p$ grows. Moreover, they are inferior for numerical computation, which is best carried out by solving the system of equations rather than by inverting the matrix $C_X$.)

The constant term will be the difference between the mean of $y$ and the mean values predicted from the estimates, $X\hat{\beta}$.

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Example

To illustrate, the following R code creates some data, computes their covariances, and obtains the least squares coefficient estimates solely from that information. It compares them to the estimates obtained from the least-squares estimator lm.

#
# 1. Generate some data.
#
n <- 10        # Data set size
p <- 2         # Number of regressors
set.seed(17)
z <- matrix(rnorm(n*(p+1)), nrow=n, dimnames=list(NULL, paste0("x", 1:(p+1))))
y <- z[, p+1]
x <- z[, -(p+1), drop=FALSE]; 
#
# 2. Find the OLS coefficients from the covariances only.
#
a <- cov(x)
b <- cov(x,y)
beta.hat <- solve(a, b)[, 1]  # Coefficients from the covariance matrix
#
# 2a. Find the intercept from the means and coefficients.
#
y.bar <- mean(y)
x.bar <- colMeans(x)
intercept <- y.bar - x.bar %*% beta.hat  

The output shows agreement between the two methods:

(rbind(`From covariances` = c(`(Intercept)`=intercept, beta.hat),
       `From data via OLS` = coef(lm(y ~ x))))

                  (Intercept)        x1        x2
From covariances     0.946155 -0.424551 -1.006675
From data via OLS    0.946155 -0.424551 -1.006675

Answer 2

The answer above is complete and great. The following aims to complement the answer with a different perspective. I also want to bring attention to the fact that using covariances to estimate coefficients you are implicitly assuming there exists a constant factor on you regression, even if you don't calculate it.

Consider the model: $$ Y = 1\alpha + X^T\beta + e $$

Where $1$ is a vector of ones and $X^T$ is a matrix of observable variables. There are $N$ rows.

We can use Frisch-Waugh-Lovell theorem. Consider the projection matrix: $$ M = I - 1(1^T1)^{-1}1^T $$

Then, by FWL, $\beta$ vector can then be retrieved by estimating with the orthogonal components: $$ MY = MX^T \beta + Me $$ $$ (I - 1(1^T1)^{-1}1^T)Y = (I - 1(1^T1)^{-1}1^T)X^T \beta + Me $$ Now you need to realize that $(1^T 1)^{-1} = 1/N$ and $1^T Y$ just sums all values, for example: $$ (1^T 1)^{-1} 1^T Y = \frac{1}{N} \sum_i^N y_i = E(Y) $$ The above expression can be rewritten as:

$$ (Y - 1E(Y)) = (X^T - 1 E(X^T)) \beta + Me $$ That is, this projection matrix $M$ is demeaning $Y$ and $X$. Consider writing the above expression as: $$ \tilde{Y} = \tilde{X}^T \beta + \tilde{e} $$ And we know that OLS estimates are calculated with: $$ \beta = (\tilde{X}'\tilde{X})^{-1} \tilde{X}' \tilde{Y} $$ $$ \beta = [(X^T - 1 E(X^T))' (X^T - 1 E(X^T)]^{-1} (X^T - 1 E(X^T))' (Y - 1E(Y)) $$ $$ \beta = Cov(X)^{-1} Cov(X,Y) $$