Transcription of Lecture Notes on Measurement Error
1 Steve PischkeSpring 2007 Lecture Notes on Measurement ErrorThese Notes summarize a variety of simple results on Measurement errorwhich I nd useful. They also provide some references where more completeresults and applications can be Measurement ErrorWe will start with the simplest regressionmodels with one independent variable. For expositional ease we also assumethat both the dependent and the explanatory variable have mean zero. Supposewe wish to estimate the population relationshipy= x+ (1)Unfortunately, we only have data onex=x+u(2)ey=y+v(3) our observed variables are measured with an additive Error . Let s make thefollowing simplifying assumptionsE(u) = 0(4)plim1n(y0u) = 0(5)plim1n(x0u) = 0(6)plim1n( 0u) = 0(7)The Measurement Error in the explanatory variable has mean zero, is uncorre-lated with the true dependent and independent variables and with the equationerror.
2 Also we will start by assuming 2v= 0, there is only measurementerror inx. These assumptions de ne theclassical errors -in-variables (2) into (1):y= (ex u) + =yi= ex+ ( u)(8)The Measurement Error inxbecomes part of the Error term in the regressionequation thus creating an endogeneity bias. Sinceexanduare positively corre-lated (from (2)) we can see that OLS estimation will lead to a negative bias inb if the true is positive and a positive bias if is assess thesize of the biasconsider the OLS-estimator for b =cov(ex;y)var(ex)=cov(x+u; x+ )var(x+u)1andplimb = 2x 2x+ 2u= where 2x 2x+ 2uThe quantity is referred to as reliability or signal-to-total variance ratio. Since0< <1the coe cientb will be biased towards zero. This bias is thereforecalledattenuation biasand is the attenuation factor in this bias isplimb = = (1 ) = 2u 2x+ 2u which again brings out the fact that the bias depends on the sign and size of.
3 In order to gure out what happens to theestimated standard Error rstconsider estimating the residual variance from the regressionb =y b ex=y b (x+u)Add and subtract the true Error =y xfrom this equation and collect = (y x) +y b x b u= + ( b )x b uYou notice that the residual contains two additional sources of variation com-pared to the true Error . The rst is due to the fact thatb is biased towardszero. Unlike in the absence of Measurement Error the termb does not vanishasymptotically. The second term is due to the additional variance introducedby the presence of Measurement Error in the regressor. Note that by assumptionthe three random variables ,x, anduin this equation are uncorrelated. Wetherefore obtain for the estimated variance of the equation errorplimc 2 = 2 + (1 )2 2 2x+ 2 2 2uFor the estimate of the variance ofpn b , call itbs, we haveplimbs=plimc 2 c 2ex= 2 + (1 )2 2 2x+ 2 2 2u 2x+ 2u= 2x 2x+ 2u 2 2x + 2x 2x+ 2u(1 )2 2+ 2u 2x+ 2u 2 2= 2 2x+ (1 )2 2+ 2(1 ) 2= s+ (1 ) 22 The rst term indicates that the true standard Error is underestimated in pro-portion to.
4 Since the second term is positive we cannot sign the overall biasin the estimated standard , the t-statistic will be biased downwards. Thet-ratio converges toplimtpn=plimb plimpbs= q s+ (1 ) 2=p qs+ (1 ) 2which is smaller than = ExtensionsNext, consider Measurement Error in the dependent vari-abley, let 2v>0while 2u= 0. Substitute (3) into (1):ey= x+ +vSincevis uncorrelated withxwe can estimate consistently by OLS in thiscase. Of course, the estimates will be less precise than with perfect to the case where there is Measurement Error only inx. The fact thatmeasurement Error in the dependent variable is more innocuous than measure-ment Error in the independent variable might suggest that we run thereverseregressionofxonythus avoiding the bias from Measurement Error .
5 Unfortu-nately, this does not solve the problem. Reverse (8) to obtainex=1 y 1 +uuandyare uncorrelated by assumption butyis correlated with the equationerror now. So we have cured the regression of errors -in-variables bias butcreated an endogeneity problem instead. Note, however, that this regressionis still useful because andyare negatively correlated so thatd1= is biaseddownwards, implying an upward bias forb r= 1= d1= . Thus the results fromthe standard regression and from the reverse regression will bracket the truecoe cient, plimb < <plimb r. Implicitly, this bracketing result usesthe fact that we know that 2 and 2uhave to be positive. The bounds of thisinterval are obtained whenever one of the two variances is zero. This impliesthat the interval tends to be large when these variances are large.
6 In practice thebracketing result is therefore often not very informative. The bracketing resultextends to multivariate regressions: in the case of two regressors you can run theoriginal as well as two reverse regressions. The results will imply that the true( 1; 2)lies inside the triangular area mapped out by these three regressions,and so forth for more regressors [Klepper and Leamer (1984)].3 Another useful fact to notice is thatdata transformationswill typicallymagnify the Measurement Error problem. Assume you want to estimate therelationshipy= x+ x2+ Under normality the attenuation factor forb will be the square of the attenua-tion factor forb [Griliches (1986)].So what can we do to get consistent estimates of ? If either 2x, 2u, or is known we can make the appropriate adjustmentfor the bias in.
7 Either one of these is su cient as we can estimate 2x+ 2u(=plimvar(ex))consistently. Such information may come fromvalidation studies of our data. In grouped data estimation, regressionon cell means, the sampling Error introduced by the fact that the meansare calculated from a sample can be estimated [Deaton (1985)]. This onlymatters if cell sizes are small; grouped data estimation yields consistentestimates with cell sizes going to in nity (but not with the number of cellsgoing to in nity at constant cell sizes). Any instrumentzcorrelated withxbut uncorrelated withuwill identifythe true coe cient sinceb IV=cov(y;z)cov(ex;z)=cov( x+ ;z)cov(x+u;z)plimb IV= xz xz= In this case it is also possible to get a consistent estimate of the populationR2= 2 2x= 2y.
8 The estimatorcR2=b IVcov(y;ex)var(y)=b IVb rwhich is the product of the IV coe cient and the OLS coe cient from thereverse regression, yieldsplimcR2= 2x 2y=R2 Get better DataOften we are interested in using panel data to eliminate xede ects. How does Measurement Error a ect the xed e ects estimator? Extendthe one variable model in (1) to include a xed e ect:yit= xit+ i+ it(9)4Di erence this to eliminate the xed e ect yit 1= (xit xit 1) + it it 1As before we only observeexit=xit+uit. Using our results from aboveplimb = 2 x 2 x+ 2 uSo we have to gure out how the variance in the changes ofxrelates to thevariance in the levels. 2 x=var(xt) 2cov(xt;xt 1) +var(xt 1)If the process forxtis stationary this simpli es to 2 x= 2 2x 2cov(xt;xt 1)= 2 2x(1 )where is the rst order autocorrelation coe cient inxt.
9 Similarly, de nertobe the autocorrelation coe cient inutso we can writeplimb = 2x(1 ) 2x(1 ) + 2u(1 r)= 11 + 2u(1 r) 2x(1 )In the special case where bothxtandutare uncorrelated over time the attenua-tion bias for the xed e ects estimator simpli es to the original . Fixed e ectsestimation is particularly worrisome whenr= 0, the Measurement Error isjust serially uncorrelated noise, while the signal is highly correlated over this case, di erencing doubles the variance of the Measurement Error while itmight reduce the variance of the signal. In the e ort to eliminate the bias arisingfrom the xed e ect we have introduced additional bias due to Measurement er-ror. Of course, di erencing is highly desirable if the Measurement erroruit=uiis a xed e ect itself.
10 In this case di erencing eliminates the Measurement errorcompletely. In general, di erencing is desirable whenr> . For panel earningsdata 2r[Bound (1994)], [Bound and Krueger (1991)].Sometimes it is reasonable to make speci c assumptions about the behaviorof the Measurement Error over time. For example, if we are willing to assumethatuitis while thex s are correlated then it is possible to identify thetrue even in relatively short panels. The simplest way to think about this isin a four period panel. Form di erences between the third and second periodand instrument these with di erences between the fourth and the rst (u4 u1)0(u3 u2) = 05by the assumption foruit. The long and short di erences forxitwill becorrelated, on the other hand, since thex s are correlated over time.
