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Christopher Dougherty EC220 - Introduction to econometrics (chapter 14) Slideshow: fixed effects regressions: within-groups and first- differences methods Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 14). [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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FIXED EFFECTS REGRESSIONS: WITHIN-GROUPS METHOD The two main approaches to the fitting of models using panel data are known, for reasons that will be explained in a moment, as fixed effects regressions, discussed in this slideshow, and random effects regressions, discussed in a later one. Fixed effects estimation (within-groups method) 1

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Three versions of the fixed effects approach will be described. In the first two, the model is manipulated in such a way that the unobserved effect is eliminated. Fixed effects estimation (within-groups method) FIXED EFFECTS REGRESSIONS: WITHIN-GROUPS METHOD 2

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In the first version, the mean values of the variables in the observations on a given individual are calculated by averaging the observations for that individual. The unobserved effect i is unaffected because it is the same for all observations for that individual. Fixed effects estimation (within-groups method) FIXED EFFECTS REGRESSIONS: WITHIN-GROUPS METHOD 3

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If the second equation is subtracted from the first, the unobserved effect disappears. Fixed effects estimation (within-groups method) FIXED EFFECTS REGRESSIONS: WITHIN-GROUPS METHOD 4

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This is known as the ‘within-groups’ method because the model is explaining the variations about the mean of the dependent variable in terms of the variations about the means of the explanatory variables for the group of observations relating to a given individual. Fixed effects estimation (within-groups method) FIXED EFFECTS REGRESSIONS: WITHIN-GROUPS METHOD 5

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The possibility of tackling unobserved heterogeneity bias in this way is a major attraction of panel data for researchers. Fixed effects estimation (within-groups method) FIXED EFFECTS REGRESSIONS: WITHIN-GROUPS METHOD 6

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However, there are some prices to pay. First, the intercept 1 and any X variable that remains constant for each individual will drop out of the model. Fixed effects estimation (within-groups method) FIXED EFFECTS REGRESSIONS: WITHIN-GROUPS METHOD 7

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The elimination of the intercept may not matter, but the loss of the unchanging explanatory variables may be frustrating. Fixed effects estimation (within-groups method) 8 FIXED EFFECTS REGRESSIONS: WITHIN-GROUPS METHOD

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For example, if one is fitting an earnings function to data for a sample of individuals who have completed their schooling, the schooling variable will disappear. Fixed effects estimation (within-groups method) FIXED EFFECTS REGRESSIONS: WITHIN-GROUPS METHOD 9

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Note that this happens even if, as would usually be the case, different individuals have different years of schooling. For each individual, the deviation in schooling in year t from the mean for that individual will be 0. Fixed effects estimation (within-groups method) FIXED EFFECTS REGRESSIONS: WITHIN-GROUPS METHOD 10

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Thus if the object of the exercise were to obtain an estimate of the returns to schooling untainted by unobserved heterogeneity bias, one ends up with no estimate at all. Fixed effects estimation (within-groups method) 11 FIXED EFFECTS REGRESSIONS: WITHIN-GROUPS METHOD

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A second problem with the fixed effects approach is that the dependent variables are likely to have much smaller variances than in the original specification. Now they are measured as deviations from the individual mean, rather than as absolute amounts. Fixed effects estimation (within-groups method) 12 FIXED EFFECTS REGRESSIONS: WITHIN-GROUPS METHOD

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This is likely to have an adverse effect on the precision of the estimates of the coefficients. It is also likely to aggravate measurement error bias if the explanatory variables are subject to measurement error. Fixed effects estimation (within-groups method) FIXED EFFECTS REGRESSIONS: WITHIN-GROUPS METHOD 13

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A third problem is that the manipulation involves the loss of n degrees of freedom. The reason for this will be explained later in the next slideshow. Fixed effects estimation (within-groups method) FIXED EFFECTS REGRESSIONS: WITHIN-GROUPS METHOD 14

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In a second version of the fixed effects approach, the first-differences method, the unobserved effect is eliminated by subtracting the observation for the previous time period from the observation for the current time period, for all time periods. Fixed effects estimation (first-differences method) FIXED EFFECTS REGRESSIONS: FIRST-DIFFERENCES METHOD 15

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For individual i in time period t the model may be written as shown. Fixed effects estimation (first-differences method) FIXED EFFECTS REGRESSIONS: FIRST-DIFFERENCES METHOD 16

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For the previous time period, the relationship is given by the second equation. Fixed effects estimation (first-differences method) FIXED EFFECTS REGRESSIONS: FIRST-DIFFERENCES METHOD 17

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Subtracting the second equation from the first, one obtains the third, rewritten as the fourth, and again the unobserved heterogeneity has disappeared. Fixed effects estimation (first-differences method) FIXED EFFECTS REGRESSIONS: FIRST-DIFFERENCES METHOD 18

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Note that the error term is now ( it – it–1 ). Its previous value was ( it-1 – it–2 ). Thus the differencing gives rise to moving average autocorrelation if it satisfies the regression model assumptions. Fixed effects estimation (first-differences method) FIXED EFFECTS REGRESSIONS: FIRST-DIFFERENCES METHOD 19

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However, if it is subject to AR(1) autocorrelation and is close to 1, taking first differences may approximately solve the problem. Fixed effects estimation (first-differences method) FIXED EFFECTS REGRESSIONS: FIRST-DIFFERENCES METHOD 20 if is close to 1

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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 14.2 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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