random effects estimation random effects regressions when the observed variables of interest are...

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.

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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.

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The first condition is that it is possible to treat each of the unobserved Zp variables as being drawn randomly from a given distribution.

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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.

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We have dealt with the unobserved effect by subsuming it into a compound disturbance term uit.

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The second condition is that the Zp variables are distributed independently of all of the Xj variables.

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If this is not the case, , and hence u, will not be uncorrelated with the Xj 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.

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

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First, we will check the other regression model assumptions. Given our assumption that it satisfies the assumptions, we can see that uit satisfies the assumption of zero expected value.

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Here we are assuming without loss of generality that E(i) = 0, any nonzero component being absorbed by the intercept, 1.

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uit 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.)

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uit will also be distributed independently of the values of Xj, since both i and it are assumed to satisfy this condition.

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Individual Time u

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However there is a problem with the assumption that its value in any observation be generated independently of its value in all other observations.


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Individual Time u

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For all the observations relating to a given individual, i will have the same value, reflecting the unchanging unobserved characteristics of the individual.


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Individual Time u

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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.


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Individual Time u

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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.


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Individual Time u

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The same is true for the observations relating to individual 2, and for all other individuals in the sample.


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Individual Time u

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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 .

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Individual Time u

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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.

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Individual Time u

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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.



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Individual Time u

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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.



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Individual Time u

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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.



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Individual Time u

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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.



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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.


NLSY 1980–1996Dependent variable logarithm of hourly earnings

OLS Fixed effects Random effects

Married 0.184 0.106 – 0.134 –(0.007) (0.012) (0.010)

Soon-to-be- 0.096 0.045 –0.061 0.060 –0.075married (0.009) (0.010) (0.008) (0.009)(0.007)

Single – – –0.106 – –0.134(0.012)

(0.010)

R2 0.358 0.268 0.268 0.346 0.346

n 20,343 20,343 20,343 20,343 20,343

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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 14.3 of C. Dougherty,

Introduction to Econometrics, fourth edition 2011, Oxford University Press.

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http://www.oup.com/uk/orc/bin/9780199567089/.

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or the University of London International Programmes distance learning course

20 Elements of Econometrics

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11.07.25

http://www.oup.com/uk/orc/bin/9780199567089/

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