IV in R
This section is currently being written, especially the testing part.
Contents
Introduction
In this Section we will demonstrate how to use instrumental variables (IV) estimation (or better Two-Stage-Least Squares, TSLS) to estimate the parameters in a linear regression model. If you want some more theoretical background on why we may need to use these techniques you may want to refer to any decent Econometrics textbook, or perhaps to this page.
Here we will be very short on the problem setup and big on the implementation! When you estimate a linear regression model, say
$y = \alpha_0 + \alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3 + u$
the most crucial of all assumptions you got to make is that the explanatory variables $x_1$ to $x_3$ are uncorrelated to the error term $u$. Of course, the error term $u$ is unobservable and hence it is impossible to empirically test this assumption (notwithstanding a related test introduced below) and the applied econometrician ought to think very carefully whether there may be any reason that makes it likely that this assumption might be breached. The seasoned econometrician would immediately rattle down simultaneity bias, measurement error and omitted relevant variables as the three most common causes for this to happen.
In some such situations you can actually fix the problem, e.g. by including the relevant variable into the model, but in others that is impossible and you need to accept that there is a high probability that, say, $x_3$ is correlated with $u$.
Implementation in R
The R Package needed is the AER package that we already recommended for use in the context of estimating robust standard errors. Included in that package is a function called ivreg which we will use. We explain how to use it by walking through an example.
Example
We will use the Women's Wages dataset to illustrate the use of the IV regression. The dependent variable which we use here is the log wage lwage and we are interested in whether the years of education, educ, has a positive influence on this log wage (here we mirror the analysis in Wooldridge's Example 15.1). An extremely simple model would be to estimate the following OLS regression which models lwage as a function of a constant and educ.
reg_ex1 <- lm(lwage~educ,data=mydata)
print(summary(reg_ex1))
which delivers
Call:
lm(formula = lwage ~ educ, data = mydata)
Residuals:
Min 1Q Median 3Q Max
-3.10256 -0.31473 0.06434 0.40081 2.10029
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.1852 0.1852 -1.000 0.318
educ 0.1086 0.0144 7.545 2.76e-13 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.68 on 426 degrees of freedom
(325 observations deleted due to missingness)
Multiple R-squared: 0.1179, Adjusted R-squared: 0.1158
F-statistic: 56.93 on 1 and 426 DF, p-value: 2.761e-13
This seems to indicate that every additional year of education increases the wage by almost 11% (recall the interpretation of a coefficient in a log-lin model!). The issue with this sort of model is that education is most likely to be correlated with individual characteristics that are important for the person's wage, but not modelled (and hence captured by the error term).
What we need is an instrument that meets the conditions outlined above and here and as in Wooldridge's example we use the father's education as an instrument. The way to do this is as follows:
reg_iv1 <- ivreg(lwage~educ|fatheduc,data=mydata)
print(summary(reg_iv1))
The ivreg function works very similar to the lm command (as usual use ?ivreg to get more detailed help). In fact the only difference is the specification of the instrument |fatheduc. The instruments follow the model specification. Behind the vertical lines we find the instrument used to instrument the educ variable[1].
The result is
Call:
ivreg(formula = lwage ~ educ | fatheduc, data = mydata)
Residuals:
Min 1Q Median 3Q Max
-3.0870 -0.3393 0.0525 0.4042 2.0677
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.44110 0.44610 0.989 0.3233
educ 0.05917 0.03514 1.684 0.0929 .
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.6894 on 426 degrees of freedom
Multiple R-Squared: 0.09344, Adjusted R-squared: 0.09131
Wald test: 2.835 on 1 and 426 DF, p-value: 0.09294
Clearly, the effect of an additional year of education, has significantly dropped and is now only marginally significant. It is, of course, often a feature of IV estimation that the estimated standard errors are significantly smaller than the OLS estimators. The size of the standard error depends a lot on the strength of the relation between the endogenous explanatory variables which we can be checked by looking at the Rsquared of the regression of educ on fatheduc[2].
In order to illustrate the full functionality of the ivreg procedure we re-estimate the model with extra explanatory variables and more instruments than endogenous variables which means that really we are applying a 2SLS estimation (This is the example estimated in Wooldridge's Example 15.5):
reg_iv1 <- ivreg(lwage~educ+exper+expersq|fatheduc+motheduc+exper+expersq,data=mydata) print(summary(reg_iv1))
Before the vertical line we can see the model that is to be estimeted, lwage~educ+exper+expersq. All the action is after the vertical line. First we see the instrumental variables used to instrument educ, fatheduc+motheduc; this is followed by all the explanatory variables that are considered exogenous, exper+expersq.
When you have a model with a lot of variables this way of calling an IV estimation can be quite unwieldy as you have to replicate all the exogenous variables (in red). A slightly different, more economical way of asking R to do the same thing is as follows
reg_iv1 <- ivreg(lwage~educ+exper+expersq|.-educ+fatheduc+motheduc,data=mydata) print(summary(reg_iv1))
After the vertical line you are basically telling R which variable to remove from the instrument set (the endogenous variable, .-educ) and which to add (+fatheduc+motheduc). Make sure you don't forget the decimal point straight after the vertical line when you use this way of specifying the instruments. What you get is the following
Call:
ivreg(formula = lwage ~ educ + age + exper + expersq | . - educ +
fatheduc, data = mydata)
Residuals:
Min 1Q Median 3Q Max
-3.09354 -0.32798 0.05094 0.37402 2.35375
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.0513505 0.4936538 -0.104 0.91720
educ 0.0701490 0.0346051 2.027 0.04328 *
age -0.0002287 0.0049140 -0.047 0.96290
exper 0.0436778 0.0134180 3.255 0.00122 **
expersq -0.0008790 0.0004064 -2.163 0.03111 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.6727 on 423 degrees of freedom
Multiple R-Squared: 0.143, Adjusted R-squared: 0.1349
Wald test: 6.225 on 4 and 423 DF, p-value: 7.106e-05
One feature of IV estimations is that in general it is an inferior estimator of [math]\mathbf{\beta}[/math] if all explanatory variables are exogenous. In that case, assuming that all other Gauss-Markov assumptions are met, the OLS estimator is the BLUE estimator. In other words, IV estimators have larger standard errors for the coefficient estimates. Therefore, one would really like to avoid having to rely on IV estimators, unless, of course, they are the only estimators that deliver consistent estimates.
For this reason any application of IV, should be accompanied by evidence that establishes that it was necessary. Once that is established, one should also establish that the instruments chosen meet the necessary requirements (of being correlated with the endogenous variable and being exogenous to the regression error term).
Testing for exogeneity
The null hypothesis we want to test is that the potentially endogenous variable, here educ is exogenous, i.e. unrelated to the regresison error. If that was the case an IV estimation was not required. The procedure described is as in Wooldridge's textbook in Chapter 15.
Estimate [math]\mathbf{y}=\mathbf{X\beta }+\mathbf{\varepsilon}[/math] by OLS and save the residuals [math]\widehat{\mathbf{\varepsilon}}[/math].
Estimate the first stage regression(s)
[math]\mathbf{x}_{j}=\mathbf{Z\gamma }_{j}\mathbf{+v}_{j}[/math]
by OLS for all [math]\widetilde{k}[/math] elements in [math]\mathbf{X}[/math] that are possibly endogenous and save [math]\widehat{\mathbf{v}}_{j}[/math]. Collect these in the [math]\left( n\times \widetilde{k}\right) [/math] matrix [math]\widehat{\mathbf{V}}[/math].
Estimate the auxilliary regression
[math]\widehat{\mathbf{y }}=\mathbf{X\beta }_{0}+\widehat{\mathbf{V}}% \mathbf{\delta }_{1}+\mathbf{u}[/math]
and test the following hypothesis
[math]\begin{aligned} H_{0} &:&\mathbf{\delta }_{1}=0~~\mathbf{X}\text{ is exogenous} \\ H_{A} &:&\mathbf{\delta }_{1}\neq 0~~\mathbf{X}\text{ is endogenous} \end{aligned}[/math]
using either a t or F-test depending on how many columns we have in [math]\widehat{\mathbf{V}}[/math] .
Implementing this test does not require anything else but the application of OLS regressions. In the following excerpt we assume that the dependent variable is contained in vector y, the elements in [math]X[/math] that are assumed to be exogenous are contained in x1, those elements that are suspected that they may be endogenous are in x2 and the instrument matrix is saved in z. As before, it is assumed that z should contain all elements of x1.
The code also uses the OLSest function for the step 3 regression. However, that could easily be circumvented as for the regressions in Step 1 and 2.
x = [x1 x2]; % Combine to one matrix x
xxi = inv(x'*x);
b = xxi*x'*y; % Step 1: OLS estimator
res = y - x*b; % Step 1: saved residuals
zzi = inv(z'*z); % Step 2: inv(Z'Z) which is used in Step 2
gam = zzi*z'*x2; % Step 2: Estimate OLS coefficients of step 2 regressions
% This works even if we have more than one element in x2
% we get as many columns of gam as we have elements in x2
vhat = x2 - z*gam; % Step 2: residuals (has as many columns as in x2
[b,bse,res,n,rss,r2] = OLSest(res,[x vhat],0); % Step 3 regression
teststat = size(res,1)*r2; % Step 3: Calculate nR^2 test stat
pval = 1 - chi2cdf(teststat,size(x2,2)); % Step 3: Calculate p-valueA function that implements this test can be found here.
Sargan test for instrument validity
One crucial property of instruments is that they ought to be uncorrelated to the regression error terms [math]\mathbf{\varepsilon}[/math]. Instrument endogeneity is set as the null hypothesis of this test with the alternative hypothesis being that the instruments are endogenous.
Estimate the regression model by IV and save [math]\widehat{\mathbf{\varepsilon }}% _{IV}=\mathbf{y}-\mathbf{X}\widehat{\mathbf{\beta }}_{IV}[/math]
Regress
[math]\widehat{\mathbf{\varepsilon }}_{IV}=\mathbf{Z\gamma +u}[/math]
Calculate [math]LM=nR^{2}[/math] from the auxilliary regresion in step 2. [math]LM[/math] is (under [math]H_{0}[/math]) [math]\chi ^{2}[/math] distributed with [math]\left( p-k\right) [/math] degrees of freedom.
MATLAB implementation of this test relies on the availability of the IV parameter estimates. They can be calculated as indicated above. In this section you can find a function called IVest that can deliver the required IV residuals by calling:
[biv,bseiv,resiv,r2iv] = IVest(y,x,z);The third output are the IV residuals (refer to IVest for details) which can then be used as the dependent variable in the second step regression:
[b,bse,res,n,rss,r2] = OLSest(resiv,z,0); % Step 2: calculate Step 2 regression
teststat = size(resiv,1)*r2; % Step 3: Calculates the nR^2 test statistic
pval = 1 - chi2cdf(teststat,(size(z,2)-size(x,2))); % Step 3: Calculate p-valueIt should be noted that this test is only applicable for an over-identified case when the z contains more columns than x. A function that implements this test can be found here.
Instrument relevance
The last instrument property that is required is that the instruments are correlated to the potentially endogenous variables. This is tested using a standard OLS regression that uses the endogenous variables as the dependent variable and all instrument variables (i.e. z) as the explanatory variables. We then need to check whether the restriction that all (non-constant) variables in z are relevant (F-test). If they are relevant, then the instruments are relevant. This is fact exactly what the Step 2 regressions of the Hausmann test do.
