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y-y^* =\beta (\pi-\pi^e)
\pi = \frac{\beta(k-1)y^*}{\omega}
b=(X'X)^{-1}X'Y \\
=[(X^*+u)'(X^*+u)]^{-1}(X^*+u)'( X^* \beta + \epsilon +v ) \\
=( (X^*)'(X^*) +u'u )^{-1}( (X^*)'(X^*) \beta+ u'u \beta -u'u \beta ) \\
=\beta [I -( (X^*)'(X^*) +u'u )^{-1}u'u ]
y= \beta x+ \epsilon - \beta u+v
y=y^*+v,x=x^*+u