# Where do the coefficients from regression of the instrument on VAR innovation come from in this setting?

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?

Thanks!