 # Christopher Dougherty EC220 - Introduction to econometrics (chapter 11) Slideshow: model c assumptions Original citation: Dougherty, C. (2012) EC220 -

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1 ASSUMPTIONS FOR MODEL C C.1The model is linear in parameters and correctly specified Y =  1 +  2 X 2 + … +  k X k + u C.2The time series for the regressors are weakly persistent C.3There does not exist an exact linear relationship among the regressors C.4The disturbance term has zero expectation C.5The disturbance term is homoscedastic ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA Assumptions C.1, C.3, C.4, C.5, and C.8, and the consequences of their violations are the same as those for Model B and they will not be discussed further here.

ASSUMPTIONS FOR MODEL C C.1The model is linear in parameters and correctly specified Y =  1 +  2 X 2 + … +  k X k + u C.2The time series for the regressors are weakly persistent C.3There does not exist an exact linear relationship among the regressors C.4The disturbance term has zero expectation C.5The disturbance term is homoscedastic ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA Assumption C.2 replaces the Model B assumption that the values of the regressors in the observations in the sample are drawn randomly from fixed populations. 2

ASSUMPTIONS FOR MODEL C C.1The model is linear in parameters and correctly specified Y =  1 +  2 X 2 + … +  k X k + u C.2The time series for the regressors are weakly persistent C.3There does not exist an exact linear relationship among the regressors C.4The disturbance term has zero expectation C.5The disturbance term is homoscedastic ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA The Model B assumption is wholly unrealistic in a time series setting because the values of many time series variables are correlated, often strongly so, from one time period to the next. 3

ASSUMPTIONS FOR MODEL C C.1The model is linear in parameters and correctly specified Y =  1 +  2 X 2 + … +  k X k + u C.2The time series for the regressors are weakly persistent C.3There does not exist an exact linear relationship among the regressors C.4The disturbance term has zero expectation C.5The disturbance term is homoscedastic ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA A discussion of the meaning of ‘weakly persistent’ and the importance of this assumption leads us to relatively advanced technical issues and will be deferred to Chapter 13. 4

ASSUMPTIONS FOR MODEL C C.6The values of the disturbance term have independent distributions u t is distributed independently of u t' for t' ≠ t C.7The disturbance term is distributed independently of the regressors u t is distributed independently of X jt' for all t' (including t ) and j C.8The disturbance term has a normal distribution STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B Assumption C.6 is rarely an issue with cross-sectional data. When observations are generated randomly, there is no reason to suppose that there should be any connection between the value of the disturbance term in one observation and its value in any other. 5

ASSUMPTIONS FOR MODEL C C.6The values of the disturbance term have independent distributions u t is distributed independently of u t' for t' ≠ t C.7The disturbance term is distributed independently of the regressors u t is distributed independently of X jt' for all t' (including t ) and j C.8The disturbance term has a normal distribution STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B However, with time series data it often happens that the value of the disturbance term in one observation is correlated with its value in the next. The reasons for this and its consequences are discussed in the next slideshow but one. 6

ASSUMPTIONS FOR MODEL C C.6The values of the disturbance term have independent distributions u t is distributed independently of u t' for t' ≠ t C.7The disturbance term is distributed independently of the regressors u t is distributed independently of X jt' for all t' (including t ) and j C.8The disturbance term has a normal distribution STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B Assumption C.7, like Assumption B.7, requires that the disturbance term be distributed of the values of all of the regressors in all of the observations. 7

ASSUMPTIONS FOR MODEL C C.6The values of the disturbance term have independent distributions u t is distributed independently of u t' for t' ≠ t C.7The disturbance term is distributed independently of the regressors u t is distributed independently of X jt' for all t' (including t ) and j C.7'The disturbance term has zero conditional expectation E(u t │values of all the regressors in all observations ) = 0 STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B As with Assumption B.7, it can be expressed in the weaker form of a conditional expectation. 8

STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B We have already seen, in the context of cross-sectional data, how violations of Assumption B.7 can lead to inconsistent estimates (measurement error bias in Chapter 8 and simultaneous equations bias in Chapter 9). ASSUMPTIONS FOR MODEL C C.6The values of the disturbance term have independent distributions u t is distributed independently of u t' for t' ≠ t C.7The disturbance term is distributed independently of the regressors u t is distributed independently of X jt' for all t' (including t ) and j C.8The disturbance term has a normal distribution 9

STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B The assumption is of even greater practical importance in the context of time series regressions. It will be discussed in detail in the next slideshow. ASSUMPTIONS FOR MODEL C C.6The values of the disturbance term have independent distributions u t is distributed independently of u t' for t' ≠ t C.7The disturbance term is distributed independently of the regressors u t is distributed independently of X jt' for all t' (including t ) and j C.8The disturbance term has a normal distribution 10

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 11.1 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/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 course EC212 Introduction to Econometrics http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course 20 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 11.07.25

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