Random effects estimation RANDOM EFFECTS REGRESSIONS When the observed variables of interest are constant for each individual, a fixed effects regression is not an effective tool because such variables cannot be included. 1

Random effects estimation In this section, we will consider an alternative approach, known as a random effects regression that may, subject to two conditions, provide a solution to this problem. RANDOM EFFECTS REGRESSIONS 2

Random effects estimation The first condition is that it is possible to treat each of the unobserved Z p variables as being drawn randomly from a given distribution. RANDOM EFFECTS REGRESSIONS 3

Random effects estimation If this is the case, the  i may be treated as random variables (hence the name of this approach) drawn from a given distribution and we may rewrite the model as shown. RANDOM EFFECTS REGRESSIONS 4

Random effects estimation We have dealt with the unobserved effect by subsuming it into a compound disturbance term u it. RANDOM EFFECTS REGRESSIONS 5

Random effects estimation The second condition is that the Z p variables are distributed independently of all of the X j variables. RANDOM EFFECTS REGRESSIONS 6

Random effects estimation If this is not the case, , and hence u, will not be uncorrelated with the X j variables and the random effects estimation will be biased and inconsistent. We would have to use fixed effects estimation instead, even if the first condition seems to be satisfied. RANDOM EFFECTS REGRESSIONS 7

Random effects estimation If the two conditions are satisfied, we may use this as our regression specification, but there is a complication. u it will be subject to a special form of autocorrelation and we will have to use an estimation technique that takes account of it. RANDOM EFFECTS REGRESSIONS 8

Random effects estimation First, we will check the other regression model assumptions. Given our assumption that  it satisfies the assumptions, we can see that u it satisfies the assumption of zero expected value. RANDOM EFFECTS REGRESSIONS 9

Random effects estimation Here we are assuming without loss of generality that E(  i ) = 0, any nonzero component being absorbed by the intercept,  1. RANDOM EFFECTS REGRESSIONS 10

Random effects estimation u it will satisfy the condition that it should have constant variance. Its variance is equal to the sum of the variances of  i and  it. (The covariance between  i and  it is 0 on the assumption that  i is distributed independently of  it.) RANDOM EFFECTS REGRESSIONS 11

Random effects estimation u it will also be distributed independently of the values of X j, since both  i and  it are assumed to satisfy this condition. RANDOM EFFECTS REGRESSIONS 12

Random effects estimation Individual Time u 11  1 +   1 +   1 +   2 +   2 +   2 +  23 However there is a problem with the assumption that its value in any observation be generated independently of its value in all other observations. RANDOM EFFECTS REGRESSIONS 13

Random effects estimation Individual Time u 11  1 +   1 +   1 +   2 +   2 +   2 +  23 For all the observations relating to a given individual,  i will have the same value, reflecting the unchanging unobserved characteristics of the individual. RANDOM EFFECTS REGRESSIONS 14

Random effects estimation Individual Time u 11  1 +   1 +   1 +   2 +   2 +   2 +  23 This is illustrated in the table above, which shows the disturbance terms for the first two individuals in a data set, assuming that there are 3 time periods. RANDOM EFFECTS REGRESSIONS 15

Random effects estimation Individual Time u 11  1 +   1 +   1 +   2 +   2 +   2 +  23 The disturbance terms for individual 1 are independent of those for individual 2 because  1 amd  2 are generated independently. However the disturbance terms for the observations relating to individual 1 are correlated because they contain the common component  1. RANDOM EFFECTS REGRESSIONS 16

Random effects estimation Individual Time u 11  1 +   1 +   1 +   2 +   2 +   2 +  23 The same is true for the observations relating to individual 2, and for all other individuals in the sample. RANDOM EFFECTS REGRESSIONS 17

Random effects estimation Individual Time u 11  1 +   1 +   1 +   2 +   2 +   2 +  23 The covariance of the disturbance terms in periods t and t' for individual i is decomposed above. The terms involving  are all 0 because  is assumed to be generated completely randomly. However the first term is not 0. It is the population variance of . RANDOM EFFECTS REGRESSIONS 18

Random effects estimation Individual Time u 11  1 +   1 +   1 +   2 +   2 +   2 +  23 OLS remains unbiased and consistent, despite the violation of the regression model assumption, but it is inefficient because it is possible to derive estimators with smaller variances. In addition, the standard errors are computed wrongly. RANDOM EFFECTS REGRESSIONS 19

Random effects estimation Individual Time u 11  1 +   1 +   1 +   2 +   2 +   2 +  23 We have encountered a problem of the violation of this regression model assumption once before, in the case of autocorrelated disturbance terms in a time series model. RANDOM EFFECTS REGRESSIONS 20

Random effects estimation Individual Time u 11  1 +   1 +   1 +   2 +   2 +   2 +  23 The solution then was to transform the model so that the transformed disturbance term satisfied the regression model assumption, and a similar procedure is adopted in the present case. RANDOM EFFECTS REGRESSIONS 21

Random effects estimation Individual Time u 11  1 +   1 +   1 +   2 +   2 +   2 +  23 However, while the transformation in the case of autocorrelation was very straightforward, in the present case it is more complex and will not be discussed further here. Nevertheless it is easy to fit a random effects model using regression applications such as Stata. RANDOM EFFECTS REGRESSIONS 22

Random effects estimation Individual Time u 11  1 +   1 +   1 +   2 +   2 +   2 +  23 Random effects estimation uses a procedure known as feasible generalized least squares. It yields consistent estimates of the coefficients and therefore depends on n being sufficiently large. For small n its properties are unknown. RANDOM EFFECTS REGRESSIONS 23

The table shows the results of performing random effects regressions as well as OLS and fixed effects regressions. In the next slideshow we consider how we should choose the appropriate estimation approach. RANDOM EFFECTS REGRESSIONS NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R n 20,343 20,343 20,343 20,343 20,343 24

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 14.3 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 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