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Consider a regression that includes a “troublesome explanator,” like x i MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaDaaaleaacaWGPbaabaGaey4fIOcaaaaa@38FB@ in (9). Assume that there exists a variable z i MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaBaaaleaacaWGPbaabeaaaaa@380D@ (or set of variables) that (1) is correlated with the “troublesome explanator,” (2) is uncorrelated with the error term—like ε i MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaSbaaSqaaiaadMgaaeqaaaaa@38B5@ in (9), and (3) is not one of the explanatory variables in the equation to be estimated. Greene (1990: 300) offers the following example of such a variable. Self-reported income tends to be a very “noisy” variable because sometimes people forget to report minor sources of income and sometimes they deliberately misreport their income. If the regression you are estimating uses income as explanatory variable of consumption, OLS will yield biased estimates. On the other hand, the number of checks written in a month by the household head might serve as an instrumental variable. Clearly, the number of checks written might well be positively correlated with income and there is no reason to assume that it is correlated with the error term in the consumptionequation. Greene suggested this example in 1990 when most people paid their bills with checks. Currently it would not be such a good example because of the development of electronic payment of bills.

It is usually fairly easy to identify instances when IV estimation methods are appropriate. This is especially true when one of the explanatory variables is possibly an endogenous variable. The real problem arises in finding an instrumental variable or a set of instrumental variables. However, assuming you have one or more instrumental variables, the IV method follows the same steps as described above for TSLS. In the first stage you estimate a regression of the “troublesome variable” as a function of the instruments and the exogenous variables in the equation—i.e., you estimate the reduced form equation. In the second stage you use OLS to estimate the original equation with the value of the “troublesome variable” predicted by the first stage regression substituted for the actual values of the “troublesome variable.”

In a sense TSLS is a IV estimation. The exogenous variables not in a particular regression play the role of the instruments. Thus, in the IV estimation of (1), the weather index is the instrument. In the estimation of (2) the price of corn and the income level are the IVs. Thus, in a fully specified model, the exogenous variables excluded from the regression play the role of instrumental variables. In other situations the choice of an appropriate instrument can be very difficult. The selection process demands creativity both in finding the instrument and in defending the choice.

The use either of IV or TSLS comes at a cost. First, the OLS estimators are more precise (i.e., have a smaller standard error) than the TSLS or IV estimators. Second, selecting invalid or weak instruments can create results that are not meaningful. So how does one know if they have chosen a good set of instruments? There is no easy answer to this question. Murray (2006a: 116-117) discusses some possible tests of the validity of an instrumental variable. In the end, however, the “success” of your instrument may depend more on how convincing your justifications are than any statistical test. Some economists, like Steven Levitt, make a living coming up with and justifying the use of some very creative instrumental variables. Murray (2006a) offers a detailed discussion of IV and should be read by any student planning to make use either of TSLS or IV regression estimators.

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Source:  OpenStax, Econometrics for honors students. OpenStax CNX. Jul 20, 2010 Download for free at http://cnx.org/content/col11208/1.2
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