# Can I use an exogenous variable from my model as an instrumental variable?

Can I use an exogenous variable specified in my model as an instrument for an endogenous one?

It can be reasoned that there is a relationship between the two.

My conjecture is this:

• Given that the exogenous variable is included in the model, its effect upon the dependent variable is accounted for and, therefore, it will not be contained in the error term. This satisfies the exogeneity requirement for valid instruments.

Is this correct and can it be done?

• "Is this correct...?"---If "this" is using an exogenous regressor already in the model to instrument an endogenous regressor, then, no, that would not be correct. Sep 5, 2020 at 20:26
• @Michael Why is this the case? Sep 6, 2020 at 3:12

## 2 Answers

That would not work.

Let's say regressors are $$x_1$$ and $$x_2$$. $$x_1$$ is endogenous, and $$x_2$$ is exogenous control. You're interested in causal inference for $$x_1$$. What you're proposing is to instrument $$x_1$$ with $$z = x_2$$.

The proposed 2SLS procedure is then the following. In the first stage, you would regress $$x_1$$ on $$z$$ and $$x_2$$ to get $$\hat{x}_1$$ and regress $$x_2$$ on $$z$$ and $$x_2$$ to get $$\hat{x}_2 = x_2$$.

In this case, the regression of $$x_1$$ on $$z$$ and $$x_2$$ would be trivially multi-colinear, and $$\hat{x}_1$$ is just given by regressing $$x_1$$ on $$x_2$$---i.e. $$\hat{x}_1$$ is a scalar multiple $$x_2$$.

So the second stage regression, where you normally regress $$y$$ on $$\hat{x}_1$$ and $$\hat{x}_2$$, is again trivially multi-colinear. You would be regressing $$y$$ on only $$x_2$$---you have lost $$x_1$$, the regressor you're interested in, completely.

Empirically speaking, a variable cannot serve both as control and an instrument.

An instrument $$z$$ channels its exogenous variation through its correlation with $$x_1$$ (notice this statement contains both conditions for a valid instrument). Now if you have a control $$x_2$$ in the regression, then $$z$$ must have some residual variation after controlling for $$x_2$$. Obviously, $$x_2$$ has no variation after controlling for $$x_2$$. This is the problem.

(Even more informally, what you're proposing would be a universal solution for finding instruments. Clearly that cannot be the case.)

Let's say that your model is $$y=\beta_0+\beta_1x_1+\dots+\beta_kx_k+u$$. If $$\text{Cov}(x_j,u)=0$$ for $$j=1,\dots,k-1$$, but $$\text{Cov}(x_k,u)\ne 0$$, you can replace $$x_k$$ with another variable $$z_1$$ if:

• $$z_1$$ is exogenous, therefore $$\text{Cov}(z,u)=0$$;
• $$z_1$$ is partially correlated with $$x_k$$, once the other exogenous variables $$x_1,\dots,x_{k-1}$$ have been netted out, i.e. in $$x_k=\delta_0+\delta_1x_1+\dots+\delta_{k-1}x_{k-1}+\theta_1 z_1+r$$ the coefficient of $$z_1$$ is nonzero (Jeffrey M. Wooldridge, Econometric Analysis of Cross Section and Panel Data, MIT Press, 2010, pp. 89-90).

You can't test the first condition, but you can and should test the second condition (Wooldridge, p. 92).