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Christopher Dougherty EC220 - Introduction to econometrics (chapter 8) Slideshow: stochastic regressors and assumptions for model b Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 8). [Teaching Resource] © 2012 The Author This version available at: Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms.

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1 Until now we have assumed that the explanatory variables in a regression model are nonstochastic, that is, that they do not have random components. We have done this to simplify the analysis of the properties of the regression estimators. ASSUMPTIONS FOR MODEL B B.1The model is linear in parameters and correctly specified. Y = 1 + 2 X 2 + … + k X k + u B.2The values of the regressors are drawn randomly from fixed populations B.3There does not exist an exact linear relationship among the regressors B.4The disturbance term has zero expectation STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B

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ASSUMPTIONS FOR MODEL B B.1The model is linear in parameters and correctly specified. Y = 1 + 2 X 2 + … + k X k + u B.2The values of the regressors are drawn randomly from fixed populations B.3There does not exist an exact linear relationship among the regressors B.4The disturbance term has zero expectation STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B We will now progress to Model B and the case of stochastic regressors, where the values of the regressors are assumed to be drawn randomly from defined populations. This is a much more realistic framework for regressions with cross-sectional data. 2

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ASSUMPTIONS FOR MODEL B B.1The model is linear in parameters and correctly specified. Y = 1 + 2 X 2 + … + k X k + u B.2The values of the regressors are drawn randomly from fixed populations B.3There does not exist an exact linear relationship among the regressors B.4The disturbance term has zero expectation STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B We will begin by re-stating the regression model assumptions and we will then review the properties of the modified model. 3

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ASSUMPTIONS FOR MODEL B B.1The model is linear in parameters and correctly specified. Y = 1 + 2 X 2 + … + k X k + u B.2The values of the regressors are drawn randomly from fixed populations B.3There does not exist an exact linear relationship among the regressors B.4The disturbance term has zero expectation STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B Assumption B.1 is the same as A.1. 4

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ASSUMPTIONS FOR MODEL B B.1The model is linear in parameters and correctly specified. Y = 1 + 2 X 2 + … + k X k + u B.2The values of the regressors are drawn randomly from fixed populations B.3There does not exist an exact linear relationship among the regressors B.4The disturbance term has zero expectation STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B The values of the regressors in the observations in the sample are drawn randomly from fixed populations with finite means and finite population variances. In Model A we simply assumed that their values were fixed and given, with no further explanation. 5

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ASSUMPTIONS FOR MODEL B B.1The model is linear in parameters and correctly specified. Y = 1 + 2 X 2 + … + k X k + u B.2The values of the regressors are drawn randomly from fixed populations B.3There does not exist an exact linear relationship among the regressors B.4The disturbance term has zero expectation STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B Note that we do not assume that the regressors are independent of each other. On the contrary, we allow their populations to have joint probability distributions and as a consequence their sample values may be correlated. 6

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ASSUMPTIONS FOR MODEL B B.1The model is linear in parameters and correctly specified. Y = 1 + 2 X 2 + … + k X k + u B.2The values of the regressors are drawn randomly from fixed populations B.3There does not exist an exact linear relationship among the regressors B.4The disturbance term has zero expectation STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B This is the counterpart of Assumption A.2. Note that if there is a constant in the model, the assumption includes a requirement that there be some variation in each of the regressors. 7

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ASSUMPTIONS FOR MODEL B B.1The model is linear in parameters and correctly specified. Y = 1 + 2 X 2 + … + k X k + u B.2The values of the regressors are drawn randomly from fixed populations B.3There does not exist an exact linear relationship among the regressors B.4The disturbance term has zero expectation STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B Strictly speaking, this is a population assumption. However, as a practical matter, it must also be satisfied by the data in a sample. Otherwise there would be exact multicollinearity and it would be impossible to obtain estimates of the parameters of the model. 8

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ASSUMPTIONS FOR MODEL B B.1The model is linear in parameters and correctly specified. Y = 1 + 2 X 2 + … + k X k + u B.2The values of the regressors are drawn randomly from fixed populations B.3There does not exist an exact linear relationship among the regressors B.4The disturbance term has zero expectation STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B Note that if there is a constant in the model, the assumption includes a requirement that there be some variation in each of the regressors. 9

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ASSUMPTIONS FOR MODEL B B.1The model is linear in parameters and correctly specified. Y = 1 + 2 X 2 + … + k X k + u B.2The values of the regressors are drawn randomly from fixed populations B.3There does not exist an exact linear relationship among the regressors B.4The disturbance term has zero expectation STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B This is the same as Assumption A.3. 10

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This is the same as Assumption A.4. ASSUMPTIONS FOR MODEL B B.5The disturbance term is homoscedastic. B.6The values of the disturbance term have independent distributions B.7The disturbance term is distributed independently of the regressors B.8The disturbance term has a normal distribution STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B 11

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This is the same as Assumption A.5. STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B ASSUMPTIONS FOR MODEL B B.5The disturbance term is homoscedastic. B.6The values of the disturbance term have independent distributions B.7The disturbance term is distributed independently of the regressors B.8The disturbance term has a normal distribution 12

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A new assumption. In Model A, the fact that the regressors had fixed nonstochastic values meant that it was impossible for there to be a distributional relationship between them and the disturbance term. Now that they have random components, the possibility arises. STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B ASSUMPTIONS FOR MODEL B B.5The disturbance term is homoscedastic. B.6The values of the disturbance term have independent distributions B.7The disturbance term is distributed independently of the regressors B.8The disturbance term has a normal distribution 13

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We will see that a violation of this assumption has adverse consequences for the properties of the regression coefficients. Note that, as stated, it means that u i is distributed independently of the value of every regressor in every observation, not just observation i. STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B ASSUMPTIONS FOR MODEL B B.5The disturbance term is homoscedastic. B.6The values of the disturbance term have independent distributions B.7The disturbance term is distributed independently of the regressors B.8The disturbance term has a normal distribution 14

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In practice with cross-sectional data it is usually reasonable to suppose that u i is distributed independently of X ji' for all i' ≠ i and so the condition reduces to u i being distributed independently of X ji. STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B ASSUMPTIONS FOR MODEL B B.5The disturbance term is homoscedastic. B.6The values of the disturbance term have independent distributions B.7The disturbance term is distributed independently of the regressors B.8The disturbance term has a normal distribution 15

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ASSUMPTIONS FOR MODEL B B.5The disturbance term is homoscedastic. B.6The values of the disturbance term have independent distributions B.7The disturbance term is distributed independently of the regressors B.7'The disturbance term has zero conditional expectation E(u i │ values of all the regressors in all observations ) = 0 STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B The condition can be weakened to the disturbance term having zero expectation conditional on the values of all the regressors in all the observations. 16

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Again, in practice with cross-sectional data it is usually reasonable to suppose that it is sufficient to condition on the values of the regressors in observation i only. ASSUMPTIONS FOR MODEL B B.5The disturbance term is homoscedastic. B.6The values of the disturbance term have independent distributions B.7The disturbance term is distributed independently of the regressors B.7'The disturbance term has zero conditional expectation E(u i │ values of all the regressors in all observations ) = 0 STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B 17

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As with Model A, we need this assumption when we perform tests. STOCHASTIC REGRESSORS AND ASSUMPTIONS FOR MODEL B ASSUMPTIONS FOR MODEL B B.5The disturbance term is homoscedastic. B.6The values of the disturbance term have independent distributions B.7The disturbance term is distributed independently of the regressors B.8The disturbance term has a normal distribution 18

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Copyright Christopher Dougherty 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 8.1 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 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 or the University of London International Programmes distance learning course 20 Elements of Econometrics

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