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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=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@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

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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