Fixed effects estimator
Encyclopedia
In econometrics
and statistics
, a fixed effects model is a statistical model
that represents the observed quantities in terms of explanatory variables that are treated as if the quantities were non-random. This is in contrast to random effects models and mixed model
s in which either all or some of the explanatory variables are treated as if they arise from the random causes. Often the same structure of model, which is usually a linear regression
model, can be treated as any of the three types depending on the analyst's viewpoint, although there may be a natural choice in any given situation.
In panel data
analysis, the term fixed effects estimator (also known as the within estimator) is used to refer to an estimator
for the coefficient
s in the regression model. If we assume fixed effects, we impose time independent effects for each entity that are possibly correlated with the regressors.
There are two common assumptions made about the individual specific effect, the random effects assumption and the fixed effects assumption. The random effects assumption (made in a random effects model) is that the individual specific effects are uncorrelated with the independent variables. The fixed effect assumption is that the individual specific effect is correlated with the independent variables. If the random effects assumption holds, the random effects model is more efficient
than the fixed effects model. However, if this assumption does not hold (i.e., if the Durbin–Watson test fails), the random effects model is not consistent
.
where
is the dependent variable observed for individual 
at time
is the time-variant regressor, 
is the time-invariant
regressor,
is the unobserved individual effect, and 
is
the error term.
could represent motivation, ability, genetics
(micro data) or historical factors and institutional factors (country-level data).
The two main methods of dealing with
are to make the random effects or fixed effects assumption:
1. Random effects (RE): Assume
is independent of 
or 
. (In some biostatistical applications, 
are predetermined, 
and 
would be called the population effects or "fixed effects", and the individual effect or "random effect" 
is often denoted 
.)
2. Fixed effects (FE): Assume
is not independent of 
. (There is no equivalent conceptualization in biostatistics; a predetermined 
cannot vary, and so cannot be probabilistically associated with the random 
.)
To get rid of individual effect
a differencing or within
transformation (time arranging) is applied to the data and then
is
estimated via Ordinary Least Squares (OLS). The most common differencing
methods are:
1. Fixed effects (FE) model:
where
and 
.
Econometrics
Econometrics has been defined as "the application of mathematics and statistical methods to economic data" and described as the branch of economics "that aims to give empirical content to economic relations." More precisely, it is "the quantitative analysis of actual economic phenomena based on...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, a fixed effects model is a statistical model
Statistical model
A statistical model is a formalization of relationships between variables in the form of mathematical equations. A statistical model describes how one or more random variables are related to one or more random variables. The model is statistical as the variables are not deterministically but...
that represents the observed quantities in terms of explanatory variables that are treated as if the quantities were non-random. This is in contrast to random effects models and mixed model
Mixed model
A mixed model is a statistical model containing both fixed effects and random effects, that is mixed effects. These models are useful in a wide variety of disciplines in the physical, biological and social sciences....
s in which either all or some of the explanatory variables are treated as if they arise from the random causes. Often the same structure of model, which is usually a linear regression
Linear regression
In statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple regression...
model, can be treated as any of the three types depending on the analyst's viewpoint, although there may be a natural choice in any given situation.
In panel data
Panel data
In statistics and econometrics, the term panel data refers to multi-dimensional data. Panel data contains observations on multiple phenomena observed over multiple time periods for the same firms or individuals....
analysis, the term fixed effects estimator (also known as the within estimator) is used to refer to an estimator
Estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule and its result are distinguished....
for the coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
s in the regression model. If we assume fixed effects, we impose time independent effects for each entity that are possibly correlated with the regressors.
Qualitative description
Such models assist in controlling for unobserved heterogeneity when this heterogeneity is constant over time and correlated with independent variables. This constant can be removed from the data through differencing, for example by taking a first difference which will remove any time invariant components of the model.There are two common assumptions made about the individual specific effect, the random effects assumption and the fixed effects assumption. The random effects assumption (made in a random effects model) is that the individual specific effects are uncorrelated with the independent variables. The fixed effect assumption is that the individual specific effect is correlated with the independent variables. If the random effects assumption holds, the random effects model is more efficient
Efficiency (statistics)
In statistics, an efficient estimator is an estimator that estimates the quantity of interest in some “best possible” manner. The notion of “best possible” relies upon the choice of a particular loss function — the function which quantifies the relative degree of undesirability of estimation errors...
than the fixed effects model. However, if this assumption does not hold (i.e., if the Durbin–Watson test fails), the random effects model is not consistent
Consistency (statistics)
In statistics, consistency of procedures such as confidence intervals or hypothesis tests involves their behaviour as the number of items in the data-set to which they are applied increases indefinitely...
.
Quantitative description
Formally the model iswhere
is the dependent variable observed for individual at time is the time-variant regressor, is the time-invariantregressor,
is the unobserved individual effect, and isthe error term.
could represent motivation, ability, genetics(micro data) or historical factors and institutional factors (country-level data).
The two main methods of dealing with
are to make the random effects or fixed effects assumption:1. Random effects (RE): Assume
is independent of or . (In some biostatistical applications, are predetermined, and would be called the population effects or "fixed effects", and the individual effect or "random effect" is often denoted .)2. Fixed effects (FE): Assume
is not independent of . (There is no equivalent conceptualization in biostatistics; a predetermined cannot vary, and so cannot be probabilistically associated with the random .)To get rid of individual effect
a differencing or withintransformation (time arranging) is applied to the data and then
isestimated via Ordinary Least Squares (OLS). The most common differencing
methods are:
1. Fixed effects (FE) model:
where
and .-
where
and
2. First difference (FD) model:
-
where
and
3. Long difference (LD) model:
-
where
and
Another common approach to removing the individual effect is to add a dummy
variable for each individual
. This is numerically, but not
computationally, equivalent to the fixed effect model and only works if
the number of time observations per individual, is much larger than the
number of individuals in the panel.
A common misconception about fixed effect models is that it is impossible to
estimate
the coefficient on the time-invariant regressor. One can estimate 
using Instrumental Variables
techniques.
Let
We can't use OLS to estimate
from this equation because 
is
correlated with
(i.e. there is a problem with endogeneity from
our FE assumption). If there are available instruments one can use IV
estimation to estimate
or use the Hausman–Taylor method.
Equality of Fixed Effects (FE) and First Differences (FD) estimators when T=2
The fixed effects estimator is:
Since each
can be re-written as 
, we'll re-write the line as:
Thus the equality is established.
Hausman–Taylor method
Need to have more than one time-variant regressor (
) and time-invariant
regressor (
) and at least one 
and one 
that are uncorrelated with
.
Partition the
and 
variables such that 
where 
and 
are uncorrelated with 
. Need 
.
Estimating
via OLS on 
using 
and 
as instruments yields a consistent estimate.
Testing FE vs. RE
We can test whether a fixed or random effects model is appropriate using a Hausman testHausman testThe Hausman test or Hausman specification test is a statistical test in econometrics named after Jerry A. Hausman. The test evaluates the significance of an estimator versus an alternative estimator...
.
-
: 
-
: 
If
is true, both 
and 
are
consistent, but only
is efficient. If 
is true,
is consistent and 
is not.
-
where 
The Hausman test is a specification test so a large test statistic might be
indication that there might be Errors in Variables (EIV) or our model is
misspecified. If the FE assumption is true, we should find that
.
A simple heuristic is that if
there could be EIV.
Steps in Fixed Effects Model for sample data
- Calculate group and grand means
- Calculate k=number of groups, n=number of observations per group, N=total number of observations (k x n)
- Calculate SS-total (or total variance) as: (Each score - Grand mean)^2 then summed
- Calculate SS-treat (or treatment effect) as: (Each group mean- Grand mean)^2 then summed x n
- Calculate SS-error (or error effect) as (Each score - Its group mean)^2 then summed
- Calculate df-total: N-1, df-treat: k-1 and df-error k(n-1)
- Calculate Mean Square MS-treat: SS-treat/df-treat, then MS-error: SS-error/df-error
- Calculate obtained f value: MS-treat/MS-error
- Use F-table or probability function, to look up critical f value with a certain significance level
- Conclude as to whether treatment effect significantly effects the variable of interest
External links
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