I am studying VAR identification strategy with instrumental variable. The main clam is that the fitted value ($\Pi$) of a regression of the instrument ($w_t$) on the innovations ($\upsilon_t$) identifies the structural shock up to a constant.

I am just missing the following bit from the entire proof: where does the equation for the fitted value $\Pi$ come from?

$\Pi = E(w_t \upsilon_t') \sum_{\upsilon \upsilon}^{-1} \upsilon_t$ (1)

where $\sum_{\upsilon \upsilon}= \sum_{\upsilon \upsilon}=H \sum_{\epsilon \epsilon} H'= HDH'$

The usual OLS derivation for $\beta$ is $\hat \beta = (X'X)^{-1}X'Y$. Given that we are regressing an instrument on the VAR innovation, shouldn't the result be $\Pi=(\upsilon'\upsilon)^{-1}\upsilon'w_t$?

Where is equation (1) coming from?

Any idea?




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