The FE
and FD estimator provide consistent estimators but not for the coefficient
time-invariant regressor (not identified).The FE estimator eliminates anything
that is time-invariant from the model.
This is
may be high price to pay for allowing the
-variables to be correlated with the individual specific
heterogeneity \({{u}_{i}}\) .
For
example, we may interested in the effect of time-invariant variables (like
gender) on a person’s wage.It’s possible to derive IV estimators that can be
considered to be in between a FE and RE approach.
Hausman-Taylor
(1981) estimator is an IV estimator that enables the coefficients of
time-invariant to be estimated.
The key
step is to distinguish between regressors uncorrelated with
and those potentially
correlated with \({{u}_{i}}\)
.
The
method additionally distinguishes between time-varying and time-invariant
regressor.
The
individual effects model is then written as;
\({{y}_{it}}={{\beta }_{0}}+{{\beta }_{1}}x_{1it}^{'}+{{\beta
}_{2}}x_{2it}^{'}+{{\alpha }_{1}}w_{1it}^{'}+{{\alpha }_{2}}w_{2it}^{'}+{{u}_{i}}+{{\varepsilon
}_{it}}\). (1)
where;
\(x_{1it}^{'}\)= \({{k}_{1}}\) variables
(exogenous) that are time varying and uncorrelated with \({{u}_{i}}\) .
\(x_{2it}^{'}\) = \({{k}_{2}}\) variables (endogenous) that are time varying and correlated
with. \({{u}_{i}}\)
\(w_{1it}^{'}\) = \({{l}_{1}}\) variables
(exogenous) that are time-invariant and uncorrelated with \({{u}_{i}}\)
\(w_{2it}^{'}\) = \({{l}_{2}}\) variables (endogenous) that are time-invariant and
correlated with \({{u}_{i}}\)
The
assumptions;
\(E\left( {{u}_{i}}|{{x}_{1it}},{{w}_{1it}}
\right)=0\)
but \(E\left(
{{u}_{i}}|{{x}_{2it}},{{w}_{2it}} \right)\ne 0\)
\(Var\left(
{{u}_{i}}|{{x}_{1it}},{{x}_{2it}},{{w}_{1it}},{{w}_{2it}} \right)=\sigma
_{u}^{2}\)
\(Cov\left( {{u}_{i}},{{\varepsilon
}_{it}}|{{x}_{1it}},{{x}_{2it}},{{w}_{1it}},{{w}_{2it}} \right)=0\)
\(Corr\left( {{u}_{i}}+{{\varepsilon
}_{it}},{{u}_{i}}+{{\varepsilon
}_{is}}|{{x}_{1it}},{{x}_{2it}},{{w}_{1it}},{{w}_{2it}} \right)=\rho =\sigma
_{u}^{2}/{{\sigma }^{2}}\)
The
estimation of Eq(1) will make the OLS and GLS not convergent because some
variables are correlated with the unobserved effects (random effects).
FE
estimator does not allow for estimating \({{\alpha }_{1}}\) and \({{\alpha }_{2}}\) parameter.RE
cannot be used: the correlation among variables associated to \({{\beta }_{2}}\) and \({{\alpha }_{2}}\) parameters
and the individual effects, \({{u}_{i}}\) produce not
consistent estimates.
The Hausman-Taylor
(H-T) method is based on the RE transformation that leads to the model
\({{\tilde{y}}_{it}}={{\beta }_{1}}\tilde{x}_{1it}^{'}+{{\beta
}_{2}}\tilde{x}_{2it}^{'}+{{\alpha }_{1}}\tilde{w}_{1it}^{'}+{{\alpha
}_{2}}\tilde{w}_{2it}^{'}+{{\tilde{u}}_{i}}+{{\tilde{\varepsilon }}_{it}}\) (2)
where,
for example \(\tilde{x}_{1it}^{'}={{x}_{1it}}-{{\hat{\theta
}}_{i}}{{\bar{x}}_{1i}}\) .
Steps on
H-T;
1.
Regress the model by OLS by using differences from the
“temporal” mean;
\(\left( {{y}_{it}}-{{{\bar{y}}}_{i}}
\right)={{\beta }_{0}}+{{\left( {{x}_{1it}}-{{{\bar{x}}}_{1i}} \right)}^{\prime
}}{{\beta }_{1}}+{{\left( {{x}_{2it}}-{{{\bar{x}}}_{2i}} \right)}^{\prime
}}{{\beta }_{2}}+\left( {{{\bar{\varepsilon }}}_{it}}-{{{\bar{\varepsilon
}}}_{i}} \right)\) (3)
2.
(a) From Step 1, use the residual
, \(\tilde{\varepsilon
}\)
to compute the “intra-group” temporal
mean of the residuals;
\(\bar{\varepsilon
}=\frac{\sum\limits_{t=1}^{T}{{{{\tilde{\varepsilon }}}_{it}}}}{T}\) (4)
and stack
them into vector \({\bar{e}}'=\left(
\left( \overbrace{{{{\bar{\varepsilon }}}_{1}},{{{\bar{\varepsilon
}}}_{1}},..{{{\bar{\varepsilon }}}_{1}}}^{T} \right),...\left(
{{{\bar{\varepsilon }}}_{n}},{{{\bar{\varepsilon }}}_{n}},..{{{\bar{\varepsilon
}}}_{n}} \right) \right)\)
(b) Do a regression;
\({{{w}'}_{2it}}={{\beta }_{0}}+{{{w}'}_{1it}}{{\beta
}_{1}}+{{{x}'}_{1it}}{{\beta }_{2}}+{{e}_{it}}\) (IV) (5)
(c) Use
the predicted value\({{\hat{{w}'}}_{2it}}\)
from (b) in the big matrix \(\text{W=}\left(
\text{W}_{\text{1}}^{\text{*}}\text{,\hat{W}}_{\text{2}}^{\text{*}} \right)\)
, where matrices \({{\text{W}}_{k}}\) are formed using the \({{{w}'}_{ki}}\) for
each group \(i\) .
(d) Regress;
\({{\bar{\varepsilon
}}_{it}}={{\alpha }_{0}}+{{\alpha }_{1}}\text{W}_{1it}^{*}+{{\alpha
}_{2}}\text{\hat{W}}_{2it}^{*}+{{\vartheta }_{it}}\) (7)
(e) Note:
we just did a 2SLS regression.
3.
From Step 1, estimate \(\sigma _{\varepsilon }^{2}\) from the regression.
From Step
2, estimate \(\sigma
_{u}^{2}\) from
the RE model, and use the estimate of \({{\sigma }^{*2}}\) from
the 2SLS regression.
Since
\({{\sigma
}^{*2}}=\sigma _{u}^{2}+\frac{\sigma _{\varepsilon }^{2}}{T}\)
then an
estimate of \({{\sigma
}_{u}}\) is
\(\sigma _{u}^{2}={{\sigma }^{*2}}-\frac{\sigma _{\varepsilon }^{2}}{T}\)
4.
We need weights to computed the FGLS.
Let
\(\hat{\theta }=\sqrt{\frac{\hat{\sigma
}_{\varepsilon }^{2}}{\hat{\sigma }_{\varepsilon }^{2}+T\hat{\sigma }_{u}^{2}}}\)
then, for each group \(i\) , let;
\({{V}^{*}}=\left[ {{x}_{1it}},{{x}_{2it}},{{w}_{1i}},{{w}_{2i}}
\right]-\hat{\theta }\left[ {{x}_{1it}},{{x}_{2it}},{{w}_{1i}},{{w}_{2i}}
\right]\) (8)
\({{y}^{*}}={{y}_{it}}-\hat{\theta }{{y}_{it}}\)$ (9)
\({{{z}'}_{it}}=\left[ {{\left( {{x}_{1it}}-{{x}_{1i}}
\right)}^{\prime }},{{\left( {{x}_{2it}}-{{x}_{2i}} \right)}^{\prime
}},{{{{w}'}}_{1i}},{{{{\bar{x}}'}}_{1i}} \right]\) (10)
be the
new weighted data and \({z}'\)
the matrix of instruments, then do a
2SLS regression of \({{y}^{*}}\)
on \({{V}^{*}}\) with
instruments \({z}'\)
:
(a)
Regress \({{V}^{*}}\)
on \({z}'\), then generate the predicted values \({{\hat{V}}^{*}}\) .
(b)
Regress \({{y}^{*}}\) on the
predicted values \({{\hat{V}}^{*}}\)
to get \({{\left(
{\hat{\beta }}',{\hat{\alpha }}' \right)}^{\prime }}\) .
5.
To get the variance of\({{\left( {\hat{\beta }}',{\hat{\alpha }}'
\right)}^{\prime }}\) , one should not use the residual of the 2SLS
regression, because it is not convergent. See Greene Ch8 eq(8.8).
The steps
in H-T suggest to estimate the Eq(2) by instrumental variables using the
following variables as a instruments set
in Eq(9): \(\left(
{{x}_{1it}}-{{x}_{1i}} \right)\) , \(\left( {{x}_{2it}}-{{x}_{2i}} \right)\) and \({{w}_{1i}}\) , \({{\bar{x}}_{1i}}\) ;
(a)
\({{\tilde{x}}_{2it}}\) is
instrumented by its deviation from individual means, \(\left( {{x}_{2it}}-{{x}_{2i}} \right)\)
(b)
\({{\tilde{w}}_{2it}}\) is
instrumented by the individual average of\({{\tilde{x}}_{1it}}\) ,\(\left( {{{\bar{x}}}_{1it}} \right)\)
(c)
\({{\tilde{x}}_{1it}}\)is
instrumented by its deviation from individual means, \(\left( {{x}_{1it}}-{{x}_{1i}} \right)\).
(d)
\({{\tilde{w}}_{1i}}\) is instrumented by \({{w}_{1i}}\) .
The H-T
estimator is based on upon an IV estimator which uses both the within and
between transformation of the strictly exogenous variables as instruments.
1.
The exogenous variables serve as their own IVs.
2.
The within transformation of the exogenous
individual-and-time varying variables serve as IVs for the endogenous
individual-and-time varying variables.
3.
Individual means of the exogenous individual-and-time
varying variables are used as IVs for endogenous time-invariant regressor.
If the
model is identified in the sense that there are at least many time-varying
exogenous regressor \({{x}_{1it}}\)
as there are individual-time invariant
endogenous regressor \({{w}_{2it}}\)
or \({{k}_{1}}\ge
{{l}_{2}}\) , then the H-T
estimator is more efficient than FE.
If the
model is under-identified where \({{k}_{1}}\le {{l}_{2}}\) , then one cannot estimate \({{\alpha }_{1}}\) and \({{\alpha }_{2}}\) , parameters and the H-T estimator of \({{\beta }_{1}}\) and \({{\beta }_{2}}\) are identical to FE.
The
resulting estimator of H-T allows us to estimate the effect of time-invariant
variables, even though the time-varying regressors are correlated with \({{u}_{i}}\).
The trick
is to use the time averages of those time-varying regressors that are
uncorrelated with \({{u}_{i}}\)
as instruments for the time-invariant regressors.
This
require that sufficient time-varying variables are included that have no
correlation with \({{u}_{i}}\).
The
strong advantage of the H-T approach is that one does not have to use external
instruments. With sufficient assumptions, instruments can be derived within the
model.
ESTIMATION WITH STATA
To
estimate the H-T model, we use again Paneldata01.dta.
From the
data, the wage in log (= lwage) is assumed to be a function of week worked (=wks), lives in south
area (south),lives in metropolitan area (smsa), marital status (ms),
year of education (=ed), a quadratic of work experience (=exp, exp2),
working in manufacturing (=ind), wage set be a union contract (=union),
blue collar (=occ) ,gender for female (=fem) and workers is
African American (=blk).
Let we
use the xtsum command to show within variability and which variables are time
invariant.
xtsum lwage
exp exp2 wks ms union occ south smsa ind fem blk ed
|
|
|
|
We check
the correlation between exogenous variables (= south, smsa,ind,
occ, fem and blk) and the endogenous time-invariant
variable (=ed)
pwcorr south
smsa ind occ fem blk ed,star(0.05)
The results
indicate that although fem appear to be weak instrument, the remaining
instruments are probably sufficiently correlated to identified the coefficient
on ed.
Weak IVs
lead to inconsistent estimates of the endogenous variables because there is not
enough information to identify the parameter and cause serious size distortion
in any test performed (Stock, Wright & Yogo, 2002).
Lets we
check again another correlation between exogenous variables (= south, smsa,ind,
occ, fem and blk) and endogenous time-variant variable (=wks,
ms,exp,exp2,union)
pwcorr south
smsa ind occ fem blk wks ms exp exp2 union,star(0.05)
Now, the
H-T model we want to estimate will become;
\({{y}_{it}}={{\beta
}_{0}}+{{\beta }_{1}}x_{1it}^{'}+{{\beta }_{2}}x_{2it}^{'}+{{\alpha
}_{1}}w_{1it}^{'}+{{\alpha }_{2}}w_{2it}^{'}+{{u}_{i}}+{{\varepsilon }_{it}}\) (11)
where;
\({{y}_{it}}\) = lwage
\(x_{1it}^{'}\) = south, smsa, ind, occ .
\(x_{2it}^{'}\) = wks, ms, exp, exp2, union
\(w_{1it}^{'}\) = fem, blk
\(w_{2it}^{'}\) = ed
Before we
estimate the H-T, lets we first estimate the FE and RE estimator;
xtreg lwage
south smsa ind occ fem blk wks ms exp exp2 union ed, fe
xtreg lwage
south smsa ind occ fem blk wks ms exp exp2 union ed, re
Now,
perform the Hausman test to choose between FE and RE model;
quiet xtreg lwage south smsa ind occ fem blk wks ms exp exp2 union ed,
fe
estimates store fe
quiet xtreg lwage south smsa ind occ fem blk wks ms exp exp2 union ed,
re
estimates store re
hausman fe re,sigmamore
The
results show that \({{\chi
}^{2}}\left( 9 \right)\), has \(\rho =0.000\).
This
leads to strong rejection of the hull hypothesis that RE provide consistent
estimates.
That
means, the FE model is more preferable.
The
problem now, we cannot retrieve the value of coefficients for fem, blk and
ed because the FE method drop the variables which is time-invariant.
To estimate the Eq(11) by H-T estimation with
endogenous variables is wks, ms,exp,exp2,union and ed;
xthtaylor
lwage south smsa ind occ fem blk wks ms exp exp2 union ed, endog(exp exp2 wks
ms union ed)
The
estimated of \({{\sigma
}_{u}}=0.9418\) and \({{\sigma }_{\varepsilon }}=0.1518\) indicating that a large fraction of the total
error variance is attributed to \({{u}_{i}}\) .
The \(z\) -statistics
indicate that several the coefficients may not be significantly different from
zero.
The
coefficient on time-invariant variables fem and blk have
relatively large standard errors whereas coefficient on ed is relatively
small.
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