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Random-effects models

An alternative approach to treating the α i MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaadMgaaeqaaaaa@38AD@ in (1) as fixed constants over time is to treat it as a random variable. Returning to (1) where the intercepts vary due to individual level differences, we have y i t = α i + j = 1 k β k x k i t + ε i t . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaWGPbGaamiDaaqabaGccqGH9aqpcqaHXoqydaWgaaWcbaGaamyAaaqabaGccqGHRaWkdaaeWbqaaiabek7aInaaBaaaleaacaWGRbaabeaakiaadIhadaWgaaWcbaGaam4AaiaadMgacaWG0baabeaakiabgUcaRiabew7aLnaaBaaaleaacaWGPbGaamiDaaqabaaabaGaamOAaiabg2da9iaaigdaaeaacaWGRbaaniabggHiLdGccaGGUaaaaa@4FB9@ Treating α i MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaadMgaaeqaaaaa@38AD@ as a random variable is equivalent to setting the model up as:

y i t = α + j = 1 k β j x j i t + ( α i + λ t + ε i t ) . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaWGPbGaamiDaaqabaGccqGH9aqpcqaHXoqycqGHRaWkdaaeWbqaaiabek7aInaaBaaaleaacaWGQbaabeaakiaadIhadaWgaaWcbaGaamOAaiaadMgacaWG0baabeaaaeaacaWGQbGaeyypa0JaaGymaaqaaiaadUgaa0GaeyyeIuoakiabgUcaRmaabmaabaGaeqySde2aaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaeq4UdW2aaSbaaSqaaiaadshaaeqaaOGaey4kaSIaeqyTdu2aaSbaaSqaaiaadMgacaWG0baabeaaaOGaayjkaiaawMcaaiaac6caaaa@5786@

For simplicity we consider only the case when λ t = 0. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaSbaaSqaaiaadshaaeqaaOGaeyypa0JaaGimaiaac6caaaa@3B49@ Thus, the error term for (11) is ( α i + ε i t ) . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacqaHXoqydaWgaaWcbaGaamyAaaqabaGccqGHRaWkcqaH1oqzdaWgaaWcbaGaamyAaiaadshaaeqaaaGccaGLOaGaayzkaaGaaiOlaaaa@3F97@ We assume that

E ( α i ) = E ( ε i t ) = 0 , E ( α i ε i t ) = 0 , E ( α i α j ) = { σ α 2    if  i = j 0      if  i j ,  and E ( ε i t ε j s ) = { σ ε 2    if  i = j ,   t = s 0      otherwise . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGfbWaaeWaaeaacqaHXoqydaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaacqGH9aqpcaWGfbWaaeWaaeaacqaH1oqzdaWgaaWcbaGaamyAaiaadshaaeqaaaGccaGLOaGaayzkaaGaeyypa0JaaGimaiaacYcaaeaacaWGfbWaaeWaaeaacqaHXoqydaWgaaWcbaGaamyAaaqabaGccqaH1oqzdaWgaaWcbaGaamyAaiaadshaaeqaaaGccaGLOaGaayzkaaGaeyypa0JaaGimaiaacYcaaeaacaWGfbWaaeWaaeaacqaHXoqydaWgaaWcbaGaamyAaaqabaGccqaHXoqydaWgaaWcbaGaamOAaaqabaaakiaawIcacaGLPaaacqGH9aqpdaGabaabaeqabaGaeq4Wdm3aa0baaSqaaiabeg7aHbqaaiaaikdaaaGccaqGGaGaaeiiaiaabccacaqGPbGaaeOzaiaabccacaWGPbGaeyypa0JaamOAaaqaaiaaicdacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabMgacaqGMbGaaeiiaiaadMgacqGHGjsUcaWGQbaaaiaawUhaaiaacYcacaqGGaGaaeyyaiaab6gacaqGKbaabaGaamyramaabmaabaGaeqyTdu2aaSbaaSqaaiaadMgacaWG0baabeaakiabew7aLnaaBaaaleaacaWGQbGaam4CaaqabaaakiaawIcacaGLPaaacqGH9aqpdaGabaabaeqabaGaeq4Wdm3aa0baaSqaaiabew7aLbqaaiaaikdaaaGccaqGGaGaaeiiaiaabccacaqGPbGaaeOzaiaabccacaWGPbGaeyypa0JaamOAaiaacYcacaqGGaGaamiDaiabg2da9iaadohaaeaacaaIWaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGVbGaaeiDaiaabIgacaqGLbGaaeOCaiaabEhacaqGPbGaae4CaiaabwgacaqGUaaaaiaawUhaaaaaaa@9C93@

We also assume that all of the elements of the error term are uncorrelated with the explanatory variables, x j . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGQbaabeaakiaac6caaaa@38C8@

The key econometric issue is that the presence of α i MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaadMgaaeqaaaaa@38AD@ in the error term means that the correlation among the residual of the same cross-sectional unit is not zero; the error terms for one farm, for instance, are correlated with each other. Therefore, the error terms exhibit heteroskedasticity. The appropriate estimation technique is generalized-least-squares, a technique that attempts to adjust the parameter estimates (and their standard error estimates) for heteroskedasticity and autocorrelation. Alternatively one can assume that α i MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaadMgaaeqaaaaa@38AC@ and ε i t MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaSbaaSqaaiaadMgacaWG0baabeaaaaa@39AD@ are normally distributed and use a ML estimator. Hsiao [2003: 35-41] and Cameron and Trivedi [2005: 699-716]offer greater detail on the estimation of the parameters of both the fixed-effects and the random-effects models. It is enough for our purposes to accept that the econometricians have found a number of ways to estimate these parameters.

Random-effects or fixed effect model?

Economists generally prefer to use fixed-effects models. The decision to use fixed-effects or random-effects does not matter when T is large because the two methods will yield the same estimates of the parameters. When the number of individual categories ( N ) is large and the number of time periods ( T ) is small, the choice of which model to use becomes unclear. Hsiao summarized this somewhat arcane issue with the following observations:

If the effects of omitted variables can be appropriately summarized by a random variable and the individual (or time) effects represent the ignorance of the investigator, it does not see reasonable to treat one source of ignorance () as fixed and the other source of ignorance () as random. It appears that one way to unify the fixed-effects and random-effects models is to assume from
the outset that the effects are random. The fixed-effects model is viewed as one in which investigators make inferences conditional on the effects that are in the sample. The random-effects model is viewed as one in which investigators make unconditional or marginal inferences with respect to the population of all effects. There is really no distinction in the “nature (of the effect).” It is up to the investigator to decide whether to make inference with respect to population characteristics or only with respect to the effects that are in the sample. Hsiao [2003: 43]

Needless to say, Hsiao’s advice may well leave many researchers without any idea of whether to use a random-effects or a fixed-effects model. In your own research I suggest that you consult an econometrician for advice .

Questions & Answers

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Source:  OpenStax, Econometrics for honors students. OpenStax CNX. Jul 20, 2010 Download for free at http://cnx.org/content/col11208/1.2
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