instruments

In statistics, an instrumental variable (IV, or instrument) can be used in regression analysis to produce a consistency (statistics)consistent estimator when the explanatory variables (covariates) are correlationcorrelated with the errors and residuals in statisticserror terms. This can be caused by endogeneity (economics)endogeneity, by omitted covariates, or by measurement errors in the covariates. In this situation, ordinary linear regression produces biased and inconsistent estimates. However, if an ''instrument'' is available, consistent estimates may still be obtained. An instrument is a variable that does not itself belong in the regression, that is correlated with the suspect explanatory variable, and that is uncorrelated with the error term.There are three main requirements for using an IV:

The instrument must be correlated with the model's predicting variable.

The instrument cannot be correlated with residuals in the second stage model (that is, the instrument cannot suffer from the same problem as the original predicting variable).

The instrument must act on the outcome only through the predicting variable, not directly.

Mathematics - For the linear regressionregression equation $y\_i\; =\; x\_i\; \backslash beta\; +\; \backslash epsilon\_i,$ the least squaresordinary least squares (OLS) estimator for $\backslash beta$ is: $\backslash widehat\; \_\; =\; \backslash frac\; =\; \backslash frac\; =\backslash beta\; +\; \backslash frac\; .$ When ''x'' and $\backslash epsilon$ are correlationuncorrelated, the second term goes to zero in the limit and the estimator is consistency (statistics)consistent. If they are correlated, however, the estimator is bias (statistics)biased.An instrumental variable is one that is correlated with the independent variable but not with the errors and residuals in statisticserror term. The estimator is:$\backslash widehat\; \_\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash beta\; +\; \backslash frac\; .$ When ''z'' and $\backslash epsilon$ are uncorrelated, the final term vanishes in the limit providing a consistent estimator. Note that when ''x'' is uncorrelated with the error term, ''x'' is itself an instrument. In that case the OLS estimator is a type of IV estimator.The approach above generalizes in a straightforward way to a regression with multiple explanatory variables. Suppose ''X'' is the ''T x K'' matrix of explanatory variables resulting from ''T'' observations on ''K'' variables. Let ''Z'' be a ''T x K'' matrix of instruments. Then,:$\backslash widehat\; \_\; =\; (Z\text{'}X)^\; Z\text{'}Y\; =\; (Z\text{'}X)^\; Z\text{'}(X\backslash beta+\backslash epsilon)\; =\; \backslash beta\; +\; (Z\text{'}X)^\; Z\text{'}\backslash epsilon.$ One computational method often used for implementing the technique is two-stage least-squares (2SLS). One advantage of this approach is that it can efficiently combine information from multiple instruments for ''over-identified'' regressions: where there are fewer covariates than instruments. Under the 2SLS approach, in a first stage, each endogenous covariate (predictor variable) is regressed on all valid instruments, including the full set of exogenous covariates in the main regression. Since the instruments are exogenous, these approximations of the endogenous covariates will not be correlated with the error term. So, intuitively they provide a way to analyze the relationship between the outcome variable and the endogenous covariates.In the second stage, the regression of interest is estimated as usual, ''except'' that in this each endogenous covariate is replaced with its approximation estimated in the first stage. The slope estimator thus obtained is unbiased. A small correction must be made to the sum-of-squared residuals in the second-stage fitted model in order that the associated standard errors be computed correctly.'Stage 1' $\backslash widehat\; =\; Z(Z\text{'}Z)^\; Z\text{'}X$'Stage 2' $\backslash widehat\; \_\; =\; (\backslash widehat\; \text{'}\backslash widehat\; )^\; \backslash widehat\; \text{'}Y$Mathematically, this estimator is identical to the single stage estimator presented above when the number of instruments is the same as the number of covariates.

Applications and problems - The use of the instrumental variables estimation technique often provides a useful, convenient and ethical alternative to the classical randomized experiment. In the randomized experiment, exogenous variation in treatment is provided by the random assignment of participants to the treatment and control conditions, causing the investigator to deny the treatment to the control participants. Using IVE, participants can be permitted to self-select into treatment and control, and the investigator can subsequently tease out the exogenous component of the treatment variation using the instrument. Of course, one does not get anything for nothing -- the IVE technique is only as good as the instruments it employs.The technique is useful for solving the errors in variables problem and for the recovery of structural parameters from simultaneous equations models such as supply and demand. Unfortunately, there is no way to prove that the independent variables are not correlated with the error term, since the error is by definition unobservable. Consequently, one problem is in the selection and defense of suitable instruments. Good instruments are often created by exogenous policy changes (i.e., the cancellation of federal student aid scholarship program), geographic differences in the application of standards (i.e., different states implement different passing standards for a common exam) or generic randomness (e.g., the Vietnam Draft Lottery) have led to exogenous disruptions in the values of the construct being measured by the selected instrument.Another problem is caused by the selection of "weak" instruments. These are instruments that are very poor predictors of the endogenous question predictor in the first-stage equation. In this latter case, the prediction of the question predictor by the instrument will be poor and the obtained predicted values will have very little variation. Consequently, they are unlikely to have much success in predicting the ultimate outcome when they are used to replace the question predictor in the second-stage equation.

References -

emlab.berkeley.edu - Chapter from Daniel McFadden's !textbookCategory:StatisticsCat egory:Econometrics DEBUG REDIRECT (instrumental variable)

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