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T2W14_P1_measurement_error[1].pdf

T2W14_P1_measurement_error[1].pdf - Measurement Error and...

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Unformatted text preview: Measurement Error and IV Edmund Wright (based on Wooldridge 9.3 & 15.3) February 2, 2015 An explanatory variable x is endogenous if cov(x, u) 6= 0. Measurement error can be a source of endogeneity. We will focus on measurement error today, and how it can be mitigated using instrumental variables. 1 How does measurement error lead to endogeneity? Suppose the model y = β0 + β1 x∗1 + u, (1) satises the rst four Gauss-Markov assumptions. We know from last term that if we were to estimate this equation using OLS we would get unbiased and consistent estimators for β0 and β1 . But suppose while we can observe y , we can't actually observe x∗1 . And so we can't regress y on x∗1 . What we can observe is a measure of x∗1 , which we can call x1 , which contains random error. So x1 = x∗1 + e1 , where e1 is the measurement error. An example of this situation might be where the variable of interest, x∗1 , is actual income, but what we can observe, x1 , is reported income, which will not always be accurate. So we can't observe x∗1 , but we can observe x1 . Suppose we replace x∗1 with x1 in the regression equation, and estimate the parameters by running an OLS regression of y on x1 . That is, we estimate y = β0 + β1 x1 + v. Will our results be unbiased and consistent? Note that given (1), the error term in this equation we are estimating is equal to v = u − β1 e 1 . Whether our results are unbiased and consistent depends on whether this is correlated with x1 . This question depends, of course, on what assumptions we make. The usual assumptions are these: 1 • E [e1 ] = 0. • Cov(x1 , u) = 0. (Which, given that (1) satisfying the G-M assumptions implies x∗1 is uncorrelated with u, means we need that e1 is uncorrelated with u.) • Cov(x∗1 , e1 ) = 0. The variable of interest x∗1 is uncorrelated with the mea- surement error. This is known as the assumption. classical errors-in-variables (CEV) Under the CEV assumption, we have that Cov(x1 , e1 ) = Cov(x∗1 + e1 , e1 ) = Cov(e1 , e1 ) = σe21 , where σe21 is the variance of the measurement error. And so the covariance between v and the observed variable x1 is Cov(x1 , v) = Cov(x1 , u − β1 e1 ) = Cov(x1 , −β1 e1 ) = −β1 Cov(x1 , e1 ) = −β1 σe21 . And so as long as the measurement error has positive variance, we have an endogenity problem: our explanatory variable is correlated with the error term in our estimated equation, which of course means our OLS estimators will be biased and inconsistent. 2 What kind of inconsistency? Attenuation bias. Recall from last term that the probability limit of the estimator for β1 is (plugging in the variables for this case) plim(βˆ1 ) = β1 + Cov(x1 , v) , V ar(x1 ) which given that Cov(x1 , v) = −β1 σe21 and Cov(x∗1 , e1 ) = 0 is equal to plim(βˆ1 ) = = −β1 σe21 V ar(x∗1 + e1 ) −β1 σe21 β1 + 2 σx∗ + σe21 β1 + 1 = β1 (1 − σe21 ) σx2∗ + σe21 1 = β1 σx2∗ 1 σx2∗ + σe21 . 1 The term multiplying β1 is less than 1, and so the probability limit of the estimator will always be closer to zero than the true parameter. This is called attenuation bias (note: to 'attenuate' is to reduce), and in the case of a positive 2 parameter it means the parameter will be underestimated. Note also that if the variance of the measurement error is large compared to the variance of the variable of interest, the bias will be large, and if it is small compared to the variance of the variable of interest, the bias will be small. 3 Mitigating with IV Measurement error leads to endogeneity, and IV is common strategy for dealing with endogeneity. How in this case? For an instrumental variable to be useful, it must be correlated with x1 , but uncorrelated with the error term v . As in this case v = u − β1 e1 , the IV should as usual be uncorrelated with the error term u from the original model (1), but also uncorrelated with the measurement error e1 . One possibility for an instrumental variable is a second measurement of x∗1 , say z1 = x∗1 + a1 , where a1 is the measurement error for this measurement. If the measurement errors e1 and a1 are uncorrelated - that is, both x1 and z1 mismeasure x∗1 , but their errors are uncorrelated - then z1 satises the requirements for a good instrument. An example of where this might be possible is where we have household income reported separately by two people from the household. An alternative is to just use some other other exogenous variable as an IV. As long as this variable is not correlated with the measurement error (and as usual is not correlated with y after controlling for x1 ) then it may be used as an instrument. 4 Example exam question: Summer 2014, A2 b-d (b) The model is P LSic = αc + βIic + ic , but the equation that is estimated is LSic = αc + βIic + vic , where Iic = IicP + IicT , and so vic = ic − βIicT . We can think of IicT as the measurement error in this question. For the OLS estimator to be consistent we would need Cov(I, v) = 0, but, if the model satises the Gauss-Markov assumptions, and additionally cov(I, ) = 0, cov(I p , I T ) = 0, in fact in this case Cov(I, v) = Cov(I,  − βI T ) = Cov(I, −βI T ) = Cov(I p + I T , −βI T ) = Cov(I T , −βI T ) = −βσI2T which, if I T has positive variance, is not equal to zero. 3 (c) (Not necessary to answer this question, but: The reason this might work is that education is unlikely to be correlated with the measurement error, in this case transitory income, but likely to be correlated with the variable of interest, in this case permanent income, and so with total income.) Recall that the instrument educ is relevant if cov(educ, I) 6= 0. Recall that the two-stage-least-squares method of instrumental variable regression involves doing a rst stage regression of Iic on educi , that is estimating the equation Iic = π0 + π1 educi + ui . The coecient on educi in this equation, π1 , is equal to cov(educi , Iic )/var(educi ), and so to test whether cov(educi , Iic ) 6= 0 we can do a t-test of the null hypothesis H0 : π1 = 0 (irrelevant) against the alternative hypothesis H1 : π1 6= 0 (relevant). Rule of thumb (you should try √and remember this) is that we have suciently relevant instrument if t-stat > 10. (d) While educi is perhaps unlikely to be correlated with the measurement error, it suers from the problem of being likely to aect the dependent variable, life satisfaction in other ways, that are not through income. And so might well be correlated with the error term, which would mean it is not a valid instrument. 4 ...
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