# How to use an instrumental variable to estimate the parameter?

I have the following linear model of log wages (w) explained using years of schooling (S), years of experience and its square $$(E,E^2)$$ and 3 dummy variables indicating whether the individual was black (B), lived in the south (Sth), and lived in a metropolitan area (Sm). $$w_i=\alpha+\beta_1S_i+\beta_2E_i+\beta_3E_i^2+\beta_4B_i+\beta_5Sth_i+\beta_6Sm_i+\epsilon_i$$ where $$\epsilon_i$$is an unobserved error term and $$i$$ indexes individuals. A reduced form model for schooling is written as $$S_i=\delta+z_i'\pi+v_i$$ where $$z_i$$ is a $$L\times 1$$vector including the instrument (lived near college) and all the exogenous variables in the first regression equation, $$\pi$$ is a $$L\times 1$$vector of unknown parameters, $$\delta$$ is an unknown scalar parameter, and $$v_i$$ is an error term. Assuming that $$S_i$$ is endogenous, how can I use the reduced form equation in conjunction with the first regression equation to identify the parameter $$\beta_1$$? How does the use of instrumental variable differ from the use of OLS estimator in this case?

I am not an econometrician so this will be a very informal explanation how to get the Instrumental Variable (IV) estimates. Since $$S_i$$ is endogenous,

$$\text{cov}(S_i,\epsilon_i)\neq0$$

We can think of it as having two components:

$$S_i = v_i + u_i$$

Suppose, $$\text{cov}(u_i,\epsilon_i) =0$$ and $$\text{cov}(v_i,\epsilon_i)\neq0$$. $$v_i$$ is the endogenous component of $$S_i$$. What we can do is find a set of variables (including instruments) that are uncorrelated with $$\epsilon_i$$, this would be the $$u_i$$ component.

$$S_i = u_i + v_i = \delta+z_i'\pi+v_i$$

We can use ordinary least squares (OLS) to estimate the following:

$$S_i = \hat{\delta} + z_i'\hat{\pi}_i + \hat{v}_i$$

Let $$\hat{S}_i$$ be the predicted values:

$$\hat{S}_i = \hat{\delta} + z_i'\hat{p_i}$$

Notice that the predicted values $$\hat{S}_i$$ have netted out the endogenous component. As such, as long as your instruments were chosen correctly, $$\hat{S}_i$$ should no longer be correlated with $$\epsilon_i$$. Finally, run the following (OLS) regression to get the IV estimates:

$$w_i=\alpha+\beta_1\hat{S}_i+\beta_2E_i+\beta_3E_i^2+\beta_4B_i+\beta_5Sth_i+\beta_6Sm_i+\epsilon_i$$

The overall method is referred to as Two-stage Least Squares (2SLS). This could also be done using Generalized Method of Moments (GMM). As long as the instruments are appropriate you will now have consistent estimates of $$\beta_1$$.

Reduced form is a regression of dependent variable on instrument directly without using some two stage approach.

Consider the following example of endogenous system

Second Stage: $$Y = \alpha + \beta X + \epsilon$$

First Stage: $$X = \mu + \gamma Z+ \eta$$

Where $$Y$$ is dependent variable $$X$$ endogenous regressor and $$Z$$ is your instrument.

One option is to estimate this with some two stage approach as previous post suggests where you would first estimate predicted $$\hat{X}$$ and then use the $$\hat{X}$$ in second stage instead of actual $$X$$ - however, this is not reduced form.

A reduced form is when you use the instrument directly in stead of the endogenous variable so in this case reduced form would be:

$$Y= \kappa + \omega Z + \xi$$

You can work from this general case to your specific one.

I would recommend looking at the handbook Mostly Harmless Econometrics form Angrist and Pischke. They discuss reduced form in Ch4 instrumental variables in action.

The difference between OLS and IV estimator (in matrix notation) is as follows:

OLS: $$\hat{\beta} = (\sum_i X_i X_i^{\prime} )^{-1}\sum_i X_i y_i$$ - This estimator is biased in the presence of endogeneity

IV: $$\hat{\beta}_{2SLS} = (x^{\prime} Z(Z^{\prime}Z)^{-1}x)^{-1}x'Z(Z^{\prime}Z)^{-1}y$$ - (here $$x$$ is given by $$x= Z \gamma + \eta$$) this estimator has still bias but the bias declines if the first stage is strong - has high F-statistics as in approximation the bias term gets multiplied by $$1/(F+1)$$ where F is the test statistics so bias goes to zero as $$F$$ goes to infinity and in practice its very close to zero for $$F>10$$.

• Unless you give some precision on the difference between $x$ and $X$, there is a risk of confusion between the $\beta$ which appears in your first equation and $\hat{\beta}$, Commented Nov 27, 2019 at 12:50
• @Bertrand you are right I will correct that
– 1muflon1
Commented Nov 27, 2019 at 12:51
• I appreciate the mathematical rigour in your answer. The only point I want to raise is your comment on 2SLS not being reduced form. $S_i=\delta+z_i'\pi+v_i$ is in fact a reduced form equation. The OP asked how to use this reduced form equation in conjunction with the structural equation to estimate the model. The way the question is worded leads me to believe the OP is looking for 2SLS. Commented Nov 27, 2019 at 15:21
• @lunar_props oh I might have misunderstood English is my second language I had an impression OP is looking for just estimation of reduced form because that’s sometimes done as well
– 1muflon1
Commented Nov 27, 2019 at 15:22
• @1muflon1 Your answer is fine and at the end of the day the 2SLS and IV estimates are equivalent. Commented Nov 27, 2019 at 15:33