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When you want to use the IV (instrumental variable) estimator, you typically first test if you have a strong instrument.

You do so by regressing the (endogenous) predictor against the instrument. With the regression coefficient, you can calculate the F statistic.

My question is: do you have to include an intercept in this regression model? In all the formulas I find, there is no intercept included.

But this would not be meaningful? Below is the regression of my endogenous variable on the instrument, with no intercept, which leads to a wrong conclusion about the regression coefficient?

Thanks in advance.

enter image description here

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up vote 4 down vote accepted

It is rarely advisable to exclude the intercept unless you have very strong theoretical reasons for doing so. If you perform your first stage regression of the endogenous variable $y_{2}$ on the instrument vector $Z$, $$y_{2} = \alpha + Z' \beta + \epsilon$$ and you omit the constant $\alpha$, your coefficient estimate for the instruments will be $$ \begin{align} E[\beta] &= E[(Z'Z)^{-1}Z'y_{2}] \newline &=E[(Z'Z)^{-1}Z'(\alpha + \beta Z + \epsilon_i)] \newline &= E[(Z'Z)^{-1}Z'\alpha + (Z'Z)^{-1}Z'Z\beta + (Z'Z)^{-1}Z'\epsilon] \newline &= E[(Z'Z)^{-1}Z'\alpha] + \beta \end{align} $$ where the term $(Z'Z)^{-1}Z'\epsilon$ vanishes because $E[Z'\epsilon]=0$ by assumption. Hence your $\beta$ will be biased if $\alpha \neq 0$, which is even true if $\alpha$ is not significantly different from zero. In this case your F-test on the excluded instruments will equally be false.

What is mostly the case in books and articles is that if they specify a first stage like $$y_2 = X'\beta + Z'\pi + \nu$$ it is implicitly assumed that the vector of covariates $X$ includes a constant, i.e. $X = (\alpha, x_1, x_2,..., x_k)$. So long story short: there should be an intercept in both the first and second stage (as well as the $X$ should be the same in both first and second stage, I just omitted them from the bias proof above for simplicity).

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