Y = XB + e
B的估计值为beta,残差写成epsilon
Var(beta) = Var[(X'X)^(-1)X'Y]
=E[(X'X)^(-1)X'YY'X(X'X)^(-1)]-E(beta)E(beta)'
=E[(X'X)^(-1)X'YY'X(X'X)^(-1)]-BB'
=E[(X'X)^(-1)X'(XB+epsilon)(B'X'+epsilon')X(X'X)^(-1)]-BB'
=E[(X'X)^(-1)X'XBB'X'X(X'X)^(-1)]+
E[(X'X)^(-1)X'(XBepsilon')X(X'X)^(-1)]+
E[(X'X)^(-1)X'(epsilonB'X'X(X'X)^(-1)]+
E[(X'X)^(-1)X'epsilon epsilon'X(X'X)^(-1)]-BB'
在外生性假设和球形扰动假设下，上式
=BB'-BB'+0+0+sigma^2(X'X)^(-1)
于是B的估计值beta的方差协方差矩阵为sigma^2(X'X)^(-1)，sigma是扰动的方差
