Christopher Dougherty EC220 - Introduction to econometrics (chapter 8) Slideshow: model b: properties of the regression coefficients Original citation:

Christopher Dougherty

EC220 - Introduction to econometrics (chapter 8)Slideshow: model b: properties of the regression coefficients

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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We will now look at the properties of the OLS regression estimators with the assumptions of Model B. We will do this within the context of the simple regression model. We will start by demonstrating unbiasedness.

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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS

We have seen that the slope coefficient can be decomposed into the true value plus a weighted linear combination of the values of the disturbance term in the sample, where the weights depend on the observations on X.

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We now take expectations. 2 is just a constant, so it is unaffected.

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We have now used the first expectation rule to rewrite the expectation of the linear combination as the sum of the expectations of its components.

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iinnnnii uaEuaEuaEuauaEuaE ...... 1111


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In Model A, the values of X were nonstochastic. This meant that the ai terms were also nonstochastic and could therefore be taken out of the expectations as factors. E(ui) = 0 for all i, and hence we proved unbiasedness.

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Model A:

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We cannot do this with Model B because we are assuming that the values of X are generated randomly (from a defined population).


Model A:

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Instead we appeal to Assumption B.7. We saw in the Review chapter that if X and Y are two independent random variables, the expectation of the product of functions of them can be decomposed as the product of the expectations of the functions.


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Under Assumption B.7, ui is distributed independently of every value of X in the sample. It is therefore distributed independently of ai. So if X and u are independent, we can make use of the decomposition.


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Since E(ui) = 0 for all I, under Assumption B.4, we have proved unbiasedness, assuming E(ai) exists. For this to be the case, there must be some variation in X in the sample (Assumption B.3). Otherwise the denominator of the expression for ai would be zero.


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The next property, efficiency, we will take for granted. The Gauss–Markov theorem assures that the OLS estimators are BLUE (best linear unbiased estimators), provided that the regression model assumptions are valid.

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We will prove consistency. We have decomposed the limiting value of the estimator of the slope coefficient into the true value and the limiting value of the error term.

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We divide the numerator and the denominator of the error term by n. We do this because they then have finite probability limits and we can make use of the plim quotient rule shown. If we did not divide by n, they would increase without limit as n increases.

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We have now made use of the plim quotient rule.

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Given the regression model assumptions, it can be shown that the denominator converges on the variance of X as the sample size becomes large, using a Law of Large Numbers. It can also be shown that the numerator converges on the covariance of X and u.

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Assumption B.7 states that the covariance of X and u is zero. Hence we have proved consistency, provided that the regression model assumptions are valid.

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Finally, a note on Assumption B.8, that the disturbance term has a normal distribution. The justification is that it is reasonable to suppose that the disturbance term is jointly generated by a number of minor random factors.


ASSUMPTIONS FOR MODEL B

B.5 The disturbance term is homoscedastic.

B.6 The values of the disturbance term have independent distributions

B.7 The disturbance term is distributed independently of the regressors

B.8 The disturbance term has a normal distribution

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A central limit theorem states that the combination of these factors should approximately have a normal distribution, even if the individual factors do not.

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If the disturbance term has a normal distribution, the regression coefficients also have normal distributions. This follows from the fact that a linear combination of normal distributions is also normal.

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What happens if we have reason to believe that the assumption is not valid? The central limit theorem comes into the frame a second time.

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The random component of a regression coefficient is a linear combination of the values of the disturbance term in the sample.

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By a central limit theorem, it follows that the combination will have an approximately normal distribution, even if the individual values of the disturbance term do not, provided that the sample is large enough.

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Hence asymptotically (in large samples) it ought to be safe to assume that the regression coefficients have normal distributions, even if Assumption B.8 is invalid, provided that the other regression model assumptions are satisfied.

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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 8.3 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

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Christopher Dougherty EC220 - Introduction to econometrics (chapter 8) Slideshow: model b: properties of the regression coefficients Original citation: