# #StackBounty: #regression #econometrics #covariance #residuals #covariance-matrix Covariance matrix of the residuals in the linear regr…

### Bounty: 50

I estimate the linear regression model:

$$Y = Xbeta + varepsilon$$

where $$y$$ is an ($$n times 1$$) dependent variable vector, $$X$$ is an ($$n times p$$) matrix of independent variables, $$beta$$ is a ($$p times 1$$) vector of the regression coefficients, and $$varepsilon$$ is an ($$n times 1$$) vector of random errors.

I want to estimate the covariance matrix of the residuals. To do so I use the following formula:

$$Cov(varepsilon) = sigma^2 (I-H)$$

where I estimate $$sigma^2$$ with $$hat{sigma}^2 = frac{e’e}{n-p}$$ and where $$I$$ is an identity matrix and $$H = X(X’X)^{-1}X$$ is a hat matrix.

However, in some source I saw that the covariance matrix of the residuals is estimated in other way.
The residuals are assumed to following $$AR(1)$$ process:

$$varepsilon_t = rho varepsilon_{t-1} + eta_t$$

where $$E(eta) = 0$$ and $$Var({eta}) = sigma^2_{0}I$$.

The covariance matrix is estimated as follows

$$Cov(varepsilon) = sigma^2 begin{bmatrix} 1 & rho & rho^2 & … & rho^{n-1}\ rho & 1 & rho & … & rho^{n-2} \ … & … & … & … & … \ rho^{n-1} & rho^{n-2} & … & … & 1 end{bmatrix}$$

where $$sigma^2 = frac{1}{1-rho}sigma^2_0$$

My question is are there two different specifications of the covariance matrix of residuals or these are somehow connected with each other?

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