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It can be shown that the 2SLS is of the above form. The advantage of 2SLS estimators over other IV estimators is that 2SLS can easily combine multiple instrumental variables, and it also makes including control variables easier.

Some people use the word "IV estimator" to refer to any estimator that uses instrumental variables. To them, IV estimators contain 2SLS, LIML, k-class estimators, and others, so 2SLS is a special case of IV. For example, the title of Bekker's (1994, Econometrica) paper is "Alternative approximations to the distribution of instrumental variable estimators".

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More traditional people mean by IV the particular instrumental variable estimator $(Z'X)^{-1}Z'y$ for the exactly identified case ($Z$ = instrument matrix, $X$ = regressor matrix, $y$ = regressand vector), and 2SLS is a generalization of IV to the overidentified case. But, as Paul says, 2SLS can be expressed as an IV estimator of this second sense because it is $(\hat{X}'X)^{-1} \hat{X}'y$, where $\hat{X} = Z(Z'Z)^{-1}Z'X$ is the instrument matrix.

It seems to me that the sentence "2sls is predicting the endogenous variable" means the first stage regression of the endogenous regressor on the instrumental variables (to get $\hat{X}$). The expression "instrumental variables are similar to proxy variables" looks more casual. Proxy variables (e.g., IQ for ability) can be used to solve the endogeneity problem. Instrumental variables are another way of solving the endogeneity problem. In that sense they are "similar".

Generally 2SLS is referred to as IV estimation for models with more than one instrument and with only one endogenous explanatory variable.You can also use two stage least squares estimation for a model with one instrumental variable. It can be shown that IV estimation equals 2SLS estimation when there is one endogenous and one instrumental variable.Finally 2SLS can be used for models with multiple endogenous explanatory variables as long as we have the same amount of instruments as endogenous variables.

In other words, the true underlying effect of control and treat on the outcome y is 1 and the estimated value of y when all other parameters are at 0 is 5. A simple ordinary least squares model (i.e. M1

One solution here is to use an instrumental variable estimator for the affected treatment variable and employ a 2SLS regression. There are a lot of econometrics texts on what this is doing along with ample statistical notation and theoretical discussion, but here is how someone more interested in the application should think about this.

The first model is the OLS model that showed a clear downward bias in the coefficient size for the treatment when the treatment is correlated with the error term. The true effect of the treatment on the response variable y is 1 but the OLS coefficient for the treatment is only .511. The first-stage model attempts to remove the variation in the treatment that is correlated with the error term by regressing the treatment variable on the control variable and the instrumental variable that is correlated with the treatment but not the error term. This results in fitted values for the treatment (treat_hat) that are substituted for the endogenous treatment variable in the second-stage model. This second-stage model is identical in form to the OLS model, but only with a treatment variable where the sources of endogeneity have been stripped from the variable. The coefficient for this fitted treatment variable approaches 1, which is what the true effect is from the data-generating process.

The goal for this post is to offer something more accessible to my future students in quantitative public policy analysis on how to deal with endogeneity in important treatment variables. There are a number of approaches here but instrumental varables and 2SLS are particularly attractive. Econometrics textbooks can make this seem daunting but students who learn more by application than by notation will find these tools relatively straightforward.

Regressors and instruments for ivreg() are most easilyspecified in a formula with two parts on the right-hand side, forexample, y ~ x1 + x2 | x1 + z1 + z2, where x1and x2 are, respectively, exogenous and endogenousexplanatory variables, and x1, z1, andz2 are instrumental variables. Both components on theright-hand side of the model formula include an implied intercept,unless, as in a linear model estimated by lm(), theintercept is explicitly excluded via -1. Exogenousexplanatory variables, such as x1 in the example, must beincluded among the instruments. A worked example is describedimmediately below. As listing exogenous variables in both parts on theright-hand side of the formula may become tedious if there are many ofthem, an additional convenience option is to use a three-part right sidelike y ~ x1 | x2 | z1 + z2, listing the exogenous,endogenous, and instrumental variables (for the endogenous variablesonly), respectively.

A standard wage equation uses a semi-logarithmic linear regressionfor wage, estimated by ordinary least squares (OLS), withyears of education as the primary explanatory variable,adjusting for a quadratic term in labor-market experience,as well as for factors coding ethnicity, residence in acity (smsa), and residence in the U.S.south:

Equivalently, the same model can also be specified slightly moreconcisely using three parts on the right-hand side indicating theexogenous variables, the endogenous variables, and the additionalinstrumental variables only (in addition to the exogenousvariables).

Thus, using two-stage least squares to estimate the regression yieldsa much larger coefficient for the returns to schooling, namely 13.3% peryear. Notice as well that the standard errors of the coefficients arelarger for 2SLS estimation than for OLS, and that, partly as aconsequence, evidence for the effects of ethnicity and thequadratic component of experience is now weak. Thesedifferences are brought out more clearly when showing coefficients andstandard errors side by side, e.g., using thecompareCoefs() function from the carpackage or the msummary() function from the modelsummarypackage:

Three-stage least-squares (3SLS) estimates are obtained using reg3. reg3 sometimes reports a negative R2 and model sum of squares. The discussion below focuses on 2SLS/IV; the issues for 3SLS are the same.

MSS is negative because RSS is greater than TSS. RSS is greater than TSS because ybar is a better predictor of y (in the sum-of-squares sense) than Xb!

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where u^ is a vector of the residuals from the MIIV-2SLS equation, V is the matrix of values of all MIIVs for the equation, and N is the sample size. TSargan approximates a chi square variate in large samples. Its degrees of freedom equals the difference in the number of right hand side covariates in the L2O equation and the number of MIIVs for that equation.

However, if you interpret instrumental variable regression as a two-stage linear regression, it is easy to show why it could end up being negative. Namely, suppose the endogenous variables ($\mathbf{X}$) are regressed on the exogenous variables ($\mathbf{Z}$), and the predicted values ($\hat{\mathbf{X}}$) are then used as covariates in the second stage:

Since $\hat{\mathbf{X}} \neq \mathbf{X}$, the error that is minimized in the second stage is not the same as the error used to calculate the residual sum of squares. Consequently, the residual sum of squares need not be less than the total sum of squares anymore. (And more importantly, the $R^2$ has become meaningless.)

The idea behind two-stage least squares is to find variation in the outcome that can be traced to variation in the instruments.The exclusion restriction implies that any such variation is operating through the treatments, so we can back out the effects of the treatments on the outcomes once we know the effects of the instruments on the treatments.Formally, the 2SLS estimator of $\theta$ is as follows: e24fc04721