christopher dougherty ec220 - introduction to econometrics (chapter 8) slideshow: measurement error...
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Christopher Dougherty
EC220 - Introduction to econometrics (chapter 8)Slideshow: measurement error
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 8). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/134/
Available in LSE Learning Resources Online: May 2012
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MEASUREMENT ERROR
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In this sequence we will investigate the consequences of measurement errors in the variables in a regression model. To keep the analysis simple, we will confine it to the simple regression model.
MEASUREMENT ERROR
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We will start with measurement errors in the explanatory variable. Suppose that Y is determined by a variable Z, but Z is subject to measurement error, w. We will denote the measured explanatory variable X.
MEASUREMENT ERROR
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Substituting for Z from the second equation, we can rewrite the model as shown.
MEASUREMENT ERROR
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We are thus able to express Y as a linear function of the observable variable X, with the disturbance term being a compound of the disturbance term in the original model and the measurement error.
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However if we fit this model using OLS, Assumption B.7 will be violated. X has a random component, the measurement error w.
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And w is also one of the components of the compound disturbance term. Hence u is not distributed independently of X.
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We will demonstrate that the OLS estimator of the slope coefficient is inconsistent and that in large samples it is biased downwards if 2 is positive, and upwards if 2 is negative.
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MEASUREMENT ERROR
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We begin by writing down the OLS estimator and substituting for Y from the true model. In this case there are alternative versions of the true model. The analysis is simpler if you use the equation relating Y to X.
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Simplifying, we decompose the slope coefficient into the true value and an error term as usual.
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We have reached this point many times before. We would like to investigate whether b2 is biased. This means taking the expectation of the error term.
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However, it is not possible to obtain a closed-form expression for the expectation of the error term. Both its numerator and its denominator are functions of w and there are no expected value rules that can allow us to simplify.
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As a second-best measure, we take plims and investigate what would happen in large samples. The plim quotient rule allows us to write the plim of the error term as the plim of the numerator divided by the plim of the denominator, provided that these plims exist.
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For the plims to exist, the numerator and denominator must both be divided by n. Otherwise they would increase indefinitely with the sample size.
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Having divided by n, it can be shown that the plim of the numerator is cov(X, u) and the plim of the denominator is var(X).
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We can decompose both the numerator and the denominator of the error term. We will start by substituting for X and u in the numerator.
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We expand the expression using the first covariance rule.
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If we assume that Z, v, and w are distributed indepndently of each other, the first 3 terms are 0. The last term gives us –2w
2.
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We next expand the denominator of the error term. The first two terms are variances. The covariance is 0 if we assume w is distributed independently of Z.
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0),cov(2)var()var()var()var( 22 wZwZwZwZX
Thus in large samples, b2 is biased towards 0 and the size of the bias depends on the relative sizes of the variances of w and Z.
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Since b2 is an inconsistent estimator, it is safe to assume that it is biased in finite samples as well.
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If our assumptions concerning Z, v, and w are incorrect, b2 would almost certainly still be a biased estimator, but the expression for the bias would be more complicated.
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A further consequence of the violation of Assumption B.7 is that the standard errors, t tests and F test are invalid.
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Measurement error in the dependent variable has less serious consequences. Suppose that the true dependent variable is Q, that the measured variable is Y, and that the measurement error is r.
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We can rewrite the model in terms of the observable variables by substituting for Q from the second equation.
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In this case the presence of the measurement error does not lead to a violation of Assumption B.7. If v satisfies that assumption in the original model, u will satisfy it in the revised one, unless for some strange reason r is not distributed independently of X.
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The standard errors and tests will remain valid. However the standard errors will tend to be larger than they would have been if there had been no measurement error, reflecting the fact that the variances of the coefficients are larger.
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Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author.
The content of this slideshow comes from Section 8.4 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school courseEC212 Introduction to Econometrics http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspxor the University of London International Programmes distance learning course20 Elements of Econometricswww.londoninternational.ac.uk/lse.
11.07.24