{smcl}
{* 3feb2007}{...}
{hline}
help for {hi:ivreg28}
{hline}

{title:Extended instrumental variables/2SLS, GMM and AC/HAC, LIML and k-class regression}

{p 4}Full syntax

{p 8 14}{cmd:ivreg28} {it:depvar} [{it:varlist1}]
{cmd:(}{it:varlist2}{cmd:=}{it:varlist_iv}{cmd:)} [{it:weight}]
[{cmd:if} {it:exp}] [{cmd:in} {it:range}]
{bind:[{cmd:,} {cmd:gmm}}
{cmd:bw(}{it:#}{cmd:)}
{cmd:kernel(}{it:string}{cmd:)}
{cmd:liml}
{cmd:fuller(}{it:#}{cmd:)}
{cmd:kclass(}{it:#}{cmd:)}
{cmd:coviv}
{cmd:cue}
{cmd:cueinit}{cmd:(}{it:matrix}{cmd:)} 
{cmdab:cueopt:ions}{cmd:(}{it:string}{cmd:)} 
{cmdab:r:obust}
{cmdab:cl:uster}{cmd:(}{it:varname}{cmd:)}
{cmd:orthog(}{it:varlist_ex}{cmd:)}
{cmd:endog(}{it:varlist_en}{cmd:)}
{cmdab:red:undant(}{it:varlist_ex}{cmd:)}
{cmd:fwl(}{it:varlist}{cmd:)}
{cmdab:sm:all}
{cmdab:noc:onstant} {cmdab:h}ascons}
{cmd:first} {cmd:ffirst} {cmd:noid} {cmd:savefirst} {cmdab:savefp:refix}{cmd:(}{it:prefix}{cmd:)} 
{cmd:rf} {cmd:saverf} {cmdab:saverfp:refix}{cmd:(}{it:prefix}{cmd:)} 
{cmd:nocollin} {cmd:noid}
{cmdab:l:evel}{cmd:(}{it:#}{cmd:)}
{cmdab:nohe:ader}
{cmdab:nofo:oter}
{cmdab:ef:orm}{cmd:(}{it:string}{cmd:)} 
{cmdab:dep:name}{cmd:(}{it:varname}{cmd:)}
{bind:{cmd:plus} ]}

{p 4}Replay syntax

{p 8 14}{cmd:ivreg28}
{bind:[{cmd:,} {cmd:first}}
{cmd:ffirst} {cmd:rf} 
{cmdab:l:evel}{cmd:(}{it:#}{cmd:)}
{cmdab:nohe:ader}
{cmdab:nofo:oter}
{cmdab:ef:orm}{cmd:(}{it:string}{cmd:)} 
{cmdab:dep:name}{cmd:(}{it:varname}{cmd:)}
{cmd:plus} ]}

{p 4}Version syntax

{p 8 14}{cmd:ivreg28}, {cmd:version}

{p}{cmd:ivreg28} may be used with time-series or panel data,
in which case the data must be {cmd:tsset}
before using {cmd:ivreg28}; see help {help tsset}.

{p}All {it:varlists} may contain time-series operators;
see help {help varlist}.

{p}{cmd:by}, {cmd:rolling}, {cmd:statsby}, {cmd:xi},
{cmd:bootstrap} and {cmd:jackknife} are allowed; see help {help prefix}.

{p}{cmd:aweight}s, {cmd:fweight}s, {cmd:iweight}s and {cmd:pweight}s
are allowed; see help {help weights}.

{p}The syntax of {help predict} following {cmd:ivreg28} is

{p 8 16}{cmd:predict} [{it:type}] {it:newvarname} [{cmd:if} {it:exp}]
[{cmd:in} {it:range}] [{cmd:,} {it:statistic}]

{p}where {it:statistic} is

{p 8 23}{cmd:xb}{space 11}fitted values; the default{p_end}
{p 8 23}{cmdab:r:esiduals}{space 4}residuals{p_end}
{p 8 23}{cmd:stdp}{space 9}standard error of the prediction{p_end}

{p}These statistics are available both in and out of sample;
type "{cmd:predict} {it:...} {cmd:if e(sample)} {it:...}"
if wanted only for the estimation sample.

{title:Contents}
{p 2}{help ivreg28##s_description:Description}{p_end}
{p 2}{help ivreg28##s_robust:Calculation of robust, AC, HAC standard errors}{p_end}
{p 2}{help ivreg28##s_gmm:GMM estimation}{p_end}
{p 2}{help ivreg28##s_liml:LIML, k-class and GMM-CUE estimation}{p_end}
{p 2}{help ivreg28##s_sumopt:Summary of robust, HAC, AC, GMM, LIML and CUE options}{p_end}
{p 2}{help ivreg28##s_overid:Testing overidentifying restrictions}{p_end}
{p 2}{help ivreg28##s_endog:Testing subsets of regressors and instruments for endogeneity}{p_end}
{p 2}{help ivreg28##s_relevance:Tests of under- and weak identification and instrument redundancy}{p_end}
{p 2}{help ivreg28##s_first:First stage regressions, identification, and weak-id-robust inference}{p_end}
{p 2}{help ivreg28##s_rf:Reduced form estimates}{p_end}
{p 2}{help ivreg28##s_fwl:Estimating the Frisch-Waugh-Lovell regression}{p_end}
{p 2}{help ivreg28##s_ols:OLS and Heteroskedastic OLS (HOLS) estimation}{p_end}
{p 2}{help ivreg28##s_collin:Collinearities}{p_end}
{p 2}{help ivreg28##s_speed:Speed options: nocollin and noid}{p_end}
{p 2}{help ivreg28##s_small:Small sample corrections}{p_end}
{p 2}{help ivreg28##s_options:Options summary}{p_end}
{p 2}{help ivreg28##s_macros:Remarks and saved results}{p_end}
{p 2}{help ivreg28##s_examples:Examples}{p_end}
{p 2}{help ivreg28##s_refs:References}{p_end}
{p 2}{help ivreg28##s_acknow:Acknowledgements}{p_end}
{p 2}{help ivreg28##s_citation:Authors}{p_end}
{p 2}{help ivreg28##s_citation:Citation of ivreg28}{p_end}

{marker s_description}{title:Description}

{p}{cmd:ivreg28} implements a range of single-equation estimation methods
for the linear regression model: OLS, instrumental
variables (IV, also known as two-stage least squares, 2SLS),
the generalized method of moments (GMM),
limited-information maximum likelihood (LIML), and k-class estimators.
In the language of IV/GMM, {it:varlist1} are the exogenous
regressors or "included instruments",
{it:varlist_iv} are the exogenous variables excluded
from the regression or "excluded instruments",
and {it:varlist2} the endogenous regressors that are being "instrumented".

{p}{cmd:ivreg28} will also estimate linear regression models using
robust (heteroskedastic-consistent),
autocorrelation-consistent (AC) and
heteroskedastic and autocorrelation-consistent (HAC) variance estimates.

{p}{cmd:ivreg28} provides extensions to Stata's official {cmd:ivreg}
and {cmd:newey}.
{cmd:ivreg28} supports the same command syntax as official {cmd:ivreg}
and (almost) all of its options.
The main extensions available are as follows:
two-step feasible GMM estimation ({cmd:gmm} option)
and continuously-updated GMM estimation ({cmd:cue} option);
LIML and k-class estimation;
automatic output overidentification and underidentification test statistics;
C statistic test of exogeneity of subsets of instruments
({cmd:orthog()} option);
endogeneity tests of endogenous regressors
({cmd:endog()} option);
test of instrument redundancy
({cmd:redundant()} option);
kernel-based autocorrelation-consistent (AC)
and heteroskedastic and autocorrelation consistent (HAC) standard errors
and covariance estimation ({cmd:bw(}{it:#}{cmd:)} option),
with user-specified choice of kernel ({cmd:kernel()} option);
default reporting of large-sample statistics
(z and chi-squared rather than t and F);
{cmd:small} option to report small-sample statistics;
first-stage regressions reported with various tests and statistics for
identification and instrument relevance;
{cmd:ffirst} option to report only these identification statistics
and not the first-stage regression results themselves; 
{cmd:nofooter} option to suppress footer of regression output.
{cmd:ivreg28} can also be used for ordinary least squares (OLS) estimation
using the same command syntax as official {cmd:regress} and {cmd:newey}.

{marker s_robust}{dlgtab:Calculation of robust, AC, HAC standard errors}

{p}The standard errors reported by {cmd:ivreg28} can be made consistent
in the presence of a variety of violations of the assumption of i.i.d. errors:
{bind:(1) {cmd:robust}} causes {cmd:ivreg28} to report standard errors that are
robust to the presence of arbitrary heteroskedasticity;
{bind:(2) {cmd:cluster}} standard errors are robust to both
arbitrary heteroskedasticity and arbitrary intra-group correlation;
{bind:(3) {cmd:bw(}{it:#}{cmd:)}} requests AC standard errors that are
robust to arbitrary autocorrelation;
{bind:(4) {cmd:bw(}{it:#}{cmd:)}} combined with {cmd:robust}
requests HAC standard errors that are
robust to both arbitrary heteroskedasticity and arbitrary autocorrelation.

{p}{cmd:ivreg28} allows a variety of options for kernel-based HAC and AC estimation.
The {cmd:bw(}{it:#}{cmd:)} option sets the bandwidth used in the estimation
and {cmd:kernel(}{it:string}{cmd:)} is the kernel used;
the default kernel is the Bartlett kernel,
also known in econometrics as Newey-West (see help {help newey}).
{cmd:ivreg28} can also be used for kernel-based estimation
with panel data, i.e., a cross-section of time series.
Before using {cmd:ivreg28} for kernel-based estimation
of time series or panel data,
the data must be {cmd:tsset}; see help {help tsset}.

{marker s_gmm}{dlgtab:GMM estimation}

{p}When combined with the above options, the {cmd:gmm} option generates
efficient estimates of the coefficients as well as consistent
estimates of the standard errors.
The {cmd:gmm} option implements the two-step efficient
generalized method of moments (GMM) estimator.
The efficient GMM estimator minimizes the GMM criterion function
J=N*g'*W*g, where N is the sample size,
g are the orthogonality or moment conditions
(specifying that all the exogenous variables, or instruments,
in the equation are uncorrelated with the error term)
and W is a weighting matrix.
In two-step efficient GMM, the efficient or optimal weighting matrix
is the inverse of an estimate of the covariance matrix of orthogonality conditions.
The efficiency gains of this estimator relative to the
traditional IV/2SLS estimator derive from the use of the optimal
weighting matrix, the overidentifying restrictions of the model,
and the relaxation of the i.i.d. assumption.
For an exactly-identified model,
the efficient GMM and traditional IV/2SLS estimators coincide,
and under the assumptions of conditional homoskedasticity and independence,
the efficient GMM estimator is the traditional IV/2SLS estimator.
For further details, see Hayashi (2000), pp. 206-13, and 226-27.

{p}The efficient GMM estimators available with {cmd:gmm} correspond
to the above choices for consistent standard errors:
{bind:(1) used} on its own, {cmd:gmm} causes {cmd:ivreg28} to report
coefficient estimates that are efficient in presence of arbitrary heteroskedasticity;
{bind:(2) {cmd:gmm}} combined with {cmd:cluster}
generates coefficient estimates that are efficient in the presence of
arbitrary heteroskedasticity and arbitrary intra-group group correlation;
{bind:(3) {cmd:gmm}} plus {cmd:bw(}{it:#}{cmd:)} requests coefficient estimates that are
efficient in the presence of arbitrary autocorrelation;
{bind:(4) {cmd:gmm}} plus {cmd:bw(}{it:#}{cmd:)} and {cmd:robust}
generates coefficient estimates that are efficient in the presence of
both arbitrary heteroskedasticity and arbitrary autocorrelation.

{marker s_liml}{dlgtab:LIML, k-class and GMM-CUE estimation}

{marker liml}{p} Maximum-likelihood estimation of a single equation of this form
(endogenous RHS variables and excluded instruments)
is known as limited-information maximum likelihood or LIML.
The overidentifying restrictions test
reported after LIML estimation is the Anderson-Rubin (1950) overidentification 
statistic in a homoskedastic context.
LIML, OLS and IV/2SLS are examples of k-class estimators.
LIML is a k-class estimator with k=the LIML eigenvalue lambda;
2SLS is a k-class estimator with k=1;
OLS is a k-class esimator with k=0.
Estimators based on other values of k have been proposed.
Fuller's modified LIML (available with the {cmd:fuller(}{it:#}{cmd:)} option)
sets k = lambda - alpha/(N-L), where lambda is the LIML eigenvalue,
L = number of instruments (included and excluded),
and the Fuller parameter alpha is a user-specified positive constant.
Nagar's bias-adjusted 2SLS estimator can be obtained with the
{cmd:kclass(}{it:#}{cmd:)} option by setting
k = 1 + (L-K)/N, where L-K = number of overidentifying restrictions
and N = the sample size.
For a discussion of LIML and k-class estimators,
see Davidson and MacKinnon (1993, pp. 644-51).

{p} The GMM generalization of the LIML estimator
to the case of possibly heteroskedastic
and autocorrelated disturbances
is the "continuously-updated" GMM estimator or CUE
of Hansen, Heaton and Yaron (1996).
The CUE estimator directly maximizes the GMM objective function
J=N*g'*W(b_cue)*g, where W(b_cue) is an optimal weighting matrix
that depends on the estimated coefficients b_cue.
{cmd:cue} combined with {cmd:robust}, {cmd:cluster}, and/or {cmd:bw},
generates coefficient estimates that are efficient in the presence
of the corresponding deviations from homoskedasticity.
Specifying {cmd:cue} with no other options
is equivalent to the combination of the options {cmd:liml} and {cmd:coviv}.
The CUE estimator requires numerical optimization methods,
and the implementation here uses Stata's {cmd:ml} routine.
The starting values are either IV or two-step efficient GMM
coefficient estimates;
these can be overridden with the {cmd:cueinit} option,
which takes the matrix of starting values b as its argument.
{cmd:cueoptions} passes options to Stata's {cmd:ml}; see help {help ml}.
Estimation with the {cmd:cue} option can be slow and problematic,
and it should be used with caution.

{marker s_sumopt}{dlgtab:Summary of robust, HAC, AC, GMM, LIML and CUE options}

{p}To summarize the robust, HAC, AC, GMM, LIML and CUE options:

{p 4}{cmd:robust} => heteroskedastic-robust SEs{p_end}
{p 4}{cmd:gmm} => heteroskedastic-efficient two-step GMM estimator{p_end}
{p 4}{cmd:robust}+{cmd:gmm} => same as {cmd:gmm}{p_end}
{p 4}{cmd:bw} => autocorrelation-robust SEs{p_end}
{p 4}{cmd:bw}+{cmd:robust} => heteroskedastic and autocorrelation-robust SEs{p_end}
{p 4}{cmd:bw}+{cmd:gmm} => autocorrelation-efficient two-step GMM estimator{p_end}
{p 4}{cmd:bw}+{cmd:robust}+{cmd:gmm} => heteroskedastic and autocorrelation-efficient two-step GMM estimator{p_end}
{p 4}{cmd:liml} => LIML estimation with non-robust SEs{p_end}
{p 4}{cmd:liml}+{cmd:coviv} => LIML estimation with alternative non-robust SEs{p_end}
{p 4}{cmd:liml}+{cmd:robust} => LIML estimation with heteroskedastic-robust SEs{p_end}
{p 4}{cmd:cue} => same as {cmd:liml}+{cmd:coviv}{p_end}
{p 4}{cmd:cue}+{cmd:robust} => heteroskedastic-efficient continuously-updated GMM estimator{p_end}
{p 4}{cmd:cue}+{cmd:bw} => autocorrelation-efficient continuously-updated GMM estimator{p_end}
{p 4 8}{cmd:cue}+{cmd:bw}+{cmd:robust} => heteroskedastic and autocorrelation-efficient continuously updated GMM estimator{p_end}

{p}For further details, see Hayashi (2000), pp. 206-13 and 226-27
(on GMM estimation), Wooldridge (2002), p. 193 (on cluster-robust GMM),
and Hayashi (2000), pp. 406-10 or Cushing and McGarvey (1999)
(on kernel-based covariance estimation).

{marker s_overid}{marker overidtests}{dlgtab:Testing overidentifying restrictions}

{p}The Sargan-Hansen test is a test of overidentifying restrictions.
The joint null hypothesis is that the instruments are valid
instruments, i.e., uncorrelated with the error term,
and that the excluded instruments are correctly excluded from the estimated equation.
Under the null, the test statistic is distributed as chi-squared
in the number of overidentifying restrictions.
A rejection casts doubt on the validity of the instruments.
For the efficient GMM estimator, the test statistic is
Hansen's J statistic, the minimized value of the GMM criterion function.
For the 2SLS estimator, the test statistic is Sargan's statistic,
typically calculated as N*R-squared from a regression of the IV residuals
on the full set of instruments.
Under the assumption of conditional homoskedasticity,
Hansen's J statistic becomes Sargan's statistic.
The J statistic is consistent in the presence of heteroskedasticity
and (for HAC-consistent estimation) autocorrelation;
Sargan's statistic is consistent if the disturbance is homoskedastic
and (for AC-consistent estimation) if it is also autocorrelated.
With {cmd:gmm}, {cmd:robust} and/or {cmd:cluster},
Hansen's J statistic is reported.
In the latter case the statistic allows observations
to be correlated within groups.
For further discussion see e.g. Hayashi (2000, pp. 227-8, 407, 417).

{p}The Sargan statistic can also be calculated after
{cmd:ivreg} or {cmd:ivreg28} by the command {cmd:overid}.
The features of {cmd:ivreg28} that are unavailable in {cmd:overid}
are the J statistic and the C statistic;
the {cmd:overid} options unavailable in {cmd:ivreg28}
are various small-sample and pseudo-F versions of Sargan's statistic
and its close relative, Basmann's statistic.
See help {help overid} (if installed).

{marker s_endog}{dlgtab:Testing subsets of regressors and instruments for endogeneity}

{marker ctest}{p}The C statistic
(also known as a "GMM distance"
or "difference-in-Sargan" statistic)
implemented using the {cmd:orthog} option,
allows a test of a subset of the orthogonality conditions, i.e.,
it is a test of the exogeneity of one or more instruments.
It is defined as
the difference of the Sargan-Hansen statistic
of the equation with the smaller set of instruments
(valid under both the null and alternative hypotheses)
and the equation with the full set of instruments,
i.e., including the instruments whose validity is suspect.
Under the null hypothesis that
both the smaller set of instruments
and the additional, suspect instruments are valid,
the C statistic is distributed as chi-squared
in the number of instruments tested.
Note that failure to reject the null hypothesis
requires that the full set of orthogonality conditions be valid;
the C statistic and the Sargan-Hansen test statistics
for the equations with both the smaller and full set of instruments
should all be small.
The instruments tested may be either excluded or included exogenous variables.
If excluded exogenous variables are being tested,
the equation that does not use these orthogonality conditions
omits the suspect instruments from the excluded instruments.
If included exogenous variables are being tested,
the equation that does not use these orthogonality conditions
treats the suspect instruments as included endogenous variables.
To guarantee that the C statistic is non-negative in finite samples,
the estimated covariance matrix of the full set orthogonality conditions
is used to calculate both Sargan-Hansen statistics
(in the case of simple IV/2SLS, this amounts to using the MSE
from the unrestricted equation to calculate both Sargan statistics).
If estimation is by LIML, the C statistic reported
is now based on the Sargan-Hansen test statistics from
the restricted and unrestricted equation.
For further discussion, see Hayashi (2000), pp. 218-22 and pp. 232-34.

{marker endogtest}{p}Endogeneity tests of one or more endogenous regressors
can implemented using the {cmd:endog} option.
Under the null hypothesis that the specified endogenous regressors
can actually be treated as exogenous, the test statistic is distributed
as chi-squared with degrees of freedom equal to the number of regressors tested.
The endogeneity test implemented by {cmd:ivreg28}, is, like the C statistic,
defined as the difference of two Sargan-Hansen statistics:
one for the equation with the smaller set of instruments,
where the suspect regressor(s) are treated as endogenous,
and one for the equation with the larger set of instruments,
where the suspect regressors are treated as exogenous.
Also like the C statistic, the estimated covariance matrix used
guarantees a non-negative test statistic.
Under conditional homoskedasticity,
this endogeneity test statistic is numerically equal to
a Hausman test statistic; see Hayashi (2000, pp. 233-34).
The endogeneity test statistic can also be calculated after
{cmd:ivreg} or {cmd:ivreg28} by the command {cmd:ivendog}.
Unlike the Durbin-Wu-Hausman tests reported by {cmd:ivendog},
the {cmd:endog} option of {cmd:ivreg28} can report test statistics
that are robust to various violations of conditional homoskedasticity;
the {cmd:ivendog} option unavailable in {cmd:ivreg28}
is the Wu-Hausman F-test version of the endogeneity test.
See help {help ivendog} (if installed).

{marker s_relevance}{dlgtab:Tests of under- and weak identification and instrument redundancy}

{marker cancortest}{p}{cmd:ivreg28} automatically reports tests of
both underidentification and weak identification.
The Anderson (1984) canonical correlations test is
a likelihood-ratio test of whether the equation is identified,
i.e., that the excluded instruments are "relevant",
meaning correlated with the endogenous regressors.
The null hypothesis of the test is that
the matrix of reduced form coefficients has rank=K-1
where K=number of regressors,
i.e, that the equation is underidentified.
Under the null of underidentification,
the statistic is distributed as chi-squared
with degrees of freedom=(L-K+1)
where L=number of instruments (included+excluded).
A rejection of the null indicates that the model is identified.
{it:Important}: a result of rejection of the null
should be treated with caution,
because weak instrument problems may still be present.
See Hall et al. (1996) for a discussion of this test,
and below for discussion of testing for the presence of weak instruments.
Note: the Anderson canonical correlations test assumes the regressors are distributed
as multivariate normal.

{marker cdtest}{p}The test for weak identification automatically reported
by {cmd:ivreg28} is based on the Cragg-Donald (1993) F statistic,
a close relative of the Anderson canonical correlations statistic.
Denoting the minimum eigenvalue of the canonical correlations as CCEV
and the minimum eigenvalue of the Cragg-Donald statistic as CDEV,
CDEV=CCEV/(1-CCEV),
the Anderson LR test statistic is -N*ln(1-CCEV) and
the Cragg-Donald F statistic is CDEV*(N-L)/L2,
where L is the number of instruments and L2 is the number of excluded instruments.
"Weak identification" arises when the excluded instruments are correlated
with the endogeous regressors, but only weakly.
Estimators can perform poorly when instruments are weak,
and different estimators are more robust to weak instruments (e.g., LIML)
than others (e.g., IV);
see, e.g., Stock and Yogo (2002, 2005) for further discussion.
Stock and Yogo (2005) have compiled critical values
for the Cragg-Donald F statistic for
several different estimators (IV, LIML, Fuller-LIML),
several different definitions of "perform poorly" (based on bias and test size),
and a range of configurations (up to 100 excluded instruments
and up to 2 or 3 endogenous regressors,
depending on the estimator).
{cmd:ivreg28} will report the Stock-Yogo critical values
if these are available;
missing values mean that the critical values
haven't been tabulated or aren't applicable.
See Stock and Yogo (2002, 2005) for details.
The critical values reported by {cmd:ivreg28} for
(2-step) GMM are the IV critical values,
and the critical values reported for CUE are the LIML
critical values.
Note that the test statistic and the critical values
assume conditional homoskedasticity and independence.
In the special case of a single endogenous regressor,
a robust test statistic for weak instruments is available
with the {cmd:first} or {cmd:ffirst} options;
see below under {help ivreg28##s_first:First stage regressions}.

{marker redtest}{p}The {cmd:redundant} option allows a test of
whether a subset of excluded instruments is "redundant".
Excluded instruments are redundant if the asymptotic efficiency
of the estimation is not improved by using them.
The test statistic is a likelihood-ratio test
based on the canonical correlations
between the regressors and the instruments with,
and without, the instruments being tested.
Under the null that the specified instruments are redundant,
the statistic is distributed as chi-squared
with degrees of freedom=(#endogenous regressors)*(#instruments tested).
Rejection of the null indicates that
the instruments are not redundant.
See Hall and Peixe (2000) for further discussion of this test.
Note: this test assumes the regressors are distributed
as multivariate normal.

{p}Calculation and reporting of all underidentification
and weak identification statistics
can be supressed with the {cmd:noid} option.

{marker s_first}{dlgtab:First stage regressions, identification, and weak-id-robust inference}

{marker partialr2}{p}The {cmd:first} and {cmd:ffirst} options report
various first-stage results and identification statistics.
Both the Anderson canonical correlations likelihood-ratio test statistic
-N*ln(1-EV)
and its close relative,
the chi-squared version of the Cragg-Donald (1993) test statistic
N*(EV/(1-EV)), are reported;
both are tests of whether the equation is identified (see {help ivreg28##s_relevance:above}).
The first-stage results also include Shea's (1997) "partial R-squared" measure
of instrument relevance that takes
intercorrelations among instruments into account,
the more common form of "partial R-squared"
(a.k.a. the "squared partial correlation" between the excluded
instruments and the endogenous regressor in question),
and the F-test of the excluded instruments
in the corresponding first-stage regression.
When the model has only one endogenous regressor,
(a) the two measures of "partial R-squared" coincide;
(b) the F-stat form of the Cragg-Donald statistic
coincides with the (non-robust) first-stage F-test
of the excluded instruments.
The two partial R-squared measures, the F statistic,
the degrees of freedom of the F statistic,
and the p-value of the F statistic for each endogenous variable
are saved in the matrix e(first).
The first-stage results are always reported with small-sample statistics,
to be consistent with the recommended use of the first-stage F-test as a diagnostic.
If the estimated equation is reported with robust standard errors,
the first-stage F-test is also robust.
Note that in the special case of only one endogenous regressor,
this provides a robust test of weak or underidentification.

{marker wirobust}{p}The first-stage output also includes
two statistics that provide weak-instrument robust inference
for testing the significance of the endogenous regressors
in the structural equation being estimated.
The first statistic is the Anderson-Rubin (1949) test
(not to be confused with the Anderson-Rubin overidentification test for LIML estimation;
see {help ivreg28##s_liml:above}).
The second is the closely related Stock-Wright (2000) S statistic.
The null hypothesis tested in both cases
is that the coefficients of the endogenous
regressors in the structural equation are jointly equal to zero,
and, in addition,
that the overidentifying restrictions are valid.
Both tests are robust to the presence of weak instruments.
The tests are equivalent to estimating
the reduced form of the equation
(with the full set of instruments as regressors)
and testing that the coefficients of the excluded instruments
are jointly equal to zero.
In the form reported by {cmd:ivreg28},
the Anderson-Rubin statistic is a Wald test
and the Stock-Watson statistic is a GMM-distance test.
Both statistics are distributed as chi-squared
with L2 degrees of freedom, where L2=number of excluded instruments.
The traditional F-stat version of the Anderson-Rubin test is also reported.
See Stock and Watson (2000), Dufour (2003), Chernozhukov and Hansen (2005) and Kleibergen (2007)
for further discussion.
For related alternative test statistics that are also robust to weak instruments,
see {help condivreg} and the corresponding discussion
in Moreira and Poi (2003) and Mikusheva and Poi (2006).

{p}The {cmd:savefirst} option requests that
the individual first-stage regressions are saved
for later access using the {cmd:estimates} command.
If saved, they can also be displayed using
{cmd:first} or {cmd:ffirst} and the {cmd:ivreg28} replay syntax.
The regressions are saved with the prefix "_ivreg28_",
unless the user specifies an alternative prefix with the
{cmdab:savefp:refix}{cmd:(}{it:prefix}{cmd:)}
option.

{marker s_rf}{dlgtab:Reduced form estimates}

{p}The {cmd:rf} option requests that
the reduced form estimation of the equation be displayed.
The {cmd:saverf} option requests that
the reduced form estimation is saved
for later access using the {cmd:estimates} command.
If saved, it can also be displayed using the
{cmd:rf} and the {cmd:ivreg28} replay syntax.
The regression is saved with the prefix "_ivreg28_",
unless the user specifies an alternative prefix with the
{cmdab:saverfp:refix}{cmd:(}{it:prefix}{cmd:)}
option.

{marker s_fwl}{dlgtab:Estimating the Frisch-Waugh-Lovell regression}

{marker fwl}{p}The {cmd:fwl(}{it:varlist}{cmd:)} option requests that
the exogenous regressors in {it:varlist} are "partialled out"
from all the other variables
(other regressors and excluded instruments) in the estimation.
If the equation includes a constant,
it is also automatically partialled out as well.
The coefficients corresponding to the regressors in {it:varlist}
are not calculated.
By the Frisch-Waugh-Lovell (FWL) theorem,
the coefficients for the remaining regressors
are the same as those that would be obtained
if the variables were not partialled out.
The {cmd:fwl} option is most useful when using {cmd:cluster}
and #clusters < (#exogenous regressors + #excluded instruments).
In these circumstances,
the covariance matrix of orthogonality conditions S is not of full rank,
and efficient GMM and overidentification tests are infeasible
since the optimal weighting matrix W = {bind:S^-1}
cannot be calculated.
The problem can be addressed by using {cmd:fwl}
to partial out enough exogenous regressors for S to have full rank.
A similar problem arises when the regressors include a variable that is a singleton dummy,
i.e., a variable with one 1 and N-1 zeros or vice versa,
if a robust covariance matrix is requested.
The singleton dummy causes the robust covariance matrix estimator to be
less than full rank.
In this case, partialling-out the variable with the singleton dummy
solves the problem.
Specifying {cmd:fwl(_cons)} will cause just the constant to be partialled-out,
i.e., the equation will be estimated in deviations-from-means form.
Note that variable counts are not adjusted for the partialled-out variables.
This means that the model degrees of freedom
do not include the partialled-out variables,
and any small-sample statistics such as t or F statistics
will be affected.
Also note that after estimation using the {cmd:fwl} option,
the post-estimation {cmd:predict} can be used only to generate residuals,
and that in the current implementation,
{cmd:fwl} is not compatible with instruments (included or excluded)
that use time-series operators.

{marker s_ols}{dlgtab:OLS and Heteroskedastic OLS (HOLS) estimation}

{p}{cmd:ivreg28} also allows straightforward OLS estimation
by using the same syntax as {cmd:regress}, i.e.,
{it:ivreg28 depvar varlist1}.
This can be useful if the user wishes to use one of the
features of {cmd:ivreg28} in OLS regression, e.g., AC or
HAC standard errors.

{p}If the list of endogenous variables {it:varlist2} is empty
but the list of excluded instruments {it:varlist_iv} is not,
and the option {cmd:gmm} is specified,
{cmd:ivreg28} calculates Cragg's "heteroskedastic OLS" (HOLS) estimator,
an estimator that is more efficient than OLS
in the presence of heteroskedasticity of unknown form
(see Davidson and MacKinnon (1993), pp. 599-600).
If the option {cmd:bw(}{it:#}{cmd:)} is specified,
the HOLS estimator is efficient in the presence of
arbitrary autocorrelation;
if both {cmd:bw(}{it:#}{cmd:)} and {cmd:robust} are specified
the HOLS estimator is efficient in the presence of
arbitrary heteroskedasticity and autocorrelation;
and if {cmd:cluster(}{it:varname}{cmd:)} is used,
the HOLS estimator is efficient in the presence of
arbitrary heteroskedasticity and within-group correlation.
The efficiency gains of HOLS derive from the orthogonality conditions
of the excluded instruments listed in {it:varlist_iv}.
If no endogenous variables are specified and {cmd:gmm} is not specified,
{cmd:ivreg28} reports standard OLS coefficients.
The Sargan-Hansen statistic reported
when the list of endogenous variables {it:varlist2} is empty
is a Lagrange multiplier (LM) test
of the hypothesis that the excluded instruments {it:varlist_iv} are
correctly excluded from the restricted model.
If the estimation is LIML, the LM statistic reported
is now based on the Sargan-Hansen test statistics from
the restricted and unrestricted equation.
For more on LM tests, see e.g. Wooldridge (2002), pp. 58-60.
Note that because the approach of the HOLS estimator
has applications beyond heteroskedastic disturbances,
and to avoid confusion concerning the robustness of the estimates,
the estimators presented above as "HOLS"
are described in the output of {cmd:ivreg28}
as "2-Step GMM", "CUE", etc., as appropriate.

{marker s_collin}{dlgtab:Collinearities}

{p}{cmd:ivreg28} checks the lists of included instruments,
excluded instruments, and endogenous regressors
for collinearities and duplicates. If an endogenous regressor is
collinear with the instruments, it is reclassified as exogenous. If any
endogenous regressors are collinear with each other, some are dropped.
If there are any collinearities among the instruments, some are dropped.
In Stata 9+, excluded instruments are dropped before included instruments.
If any variables are dropped, a list of their names are saved
in the macros {cmd:e(collin)} and/or {cmd:e(dups)}.
Lists of the included and excluded instruments
and the endogenous regressors with collinear variables and duplicates removed
are also saved in macros with "1" appended
to the corresponding macro names.

{p}Collinearity checks can be supressed with the {cmd:nocollin} option.

{marker s_speed}{dlgtab:Speed options: nocollin and noid}

{p}Two options are available for speeding execution.
{cmd:nocollin} specifies that the collinearity checks not be performed.
{cmd:noid} suspends calculation and reporting of
the underidentification and weak identification statistics
in the main output.

{marker s_small}{dlgtab:Small sample corrections}

{p}Mean square error = sqrt(RSS/(N-K)) if {cmd:small}, = sqrt(RSS/N) otherwise.

{p}If {cmd:robust} is chosen, the finite sample adjustment
(see {hi:[R] regress}) to the robust variance-covariance matrix
qc = N/(N-K) if {cmd:small}, qc = 1 otherwise.

{p}If {cmd:cluster} is chosen, the finite sample adjustment
qc = (N-1)/(N-K)*M/(M-1) if {cmd:small}, where M=number of clusters,
qc = 1 otherwise.

{p}The Sargan and C (difference-in-Sargan) statistics use
error variance = RSS/N, i.e., there is no small sample correction.

{p}A full discussion of these computations and related topics
can be found in Baum, Schaffer, and Stillman (2003) and Baum, Schaffer and
Stillman (2007). Some features of the program postdate the 2003 article.


{marker s_options}{title:Options summary}

{p 0 4}{cmd:gmm} requests the two-step efficient GMM estimator.
If no endogenous variables are specified, the estimator is Cragg's HOLS estimator.
See help {help ivgmm0} (if installed) for more details.

{p 0 4}{cmd:bw(}{it:#}{cmd:)} impements AC or HAC covariance estimation
with bandwidth equal to {it:#}, where {it:#} is an integer greater than zero.
Specifying {cmd:robust} implements HAC covariance estimation;
omitting it implements AC covariance estimation.

{p 0 4}{cmd:kernel(}{it:string)}{cmd:)} specifies the kernel
to be used for AC and HAC covariance estimation;
the default kernel is Bartlett (also known in econometrics
as Newey-West).  Other kernels available are (abbreviations in parentheses):
Truncated (tru); Parzen (par); Tukey-Hanning (thann); Tukey-Hamming (thamm);
Daniell (dan); Tent (ten); and Quadratic-Spectral (qua or qs).

{p 4 4}Note: in the cases of the Bartlett, Parzen,
and Tukey-Hanning/Hamming kernels, the number of lags used
to construct the kernel estimate equals the bandwidth minus one.
Stata's official {cmd:newey} implements
HAC standard errors based on the Bartlett kernel,
and requires the user to specify
the maximum number of lags used and not the bandwidth;
see help {help newey}.
If these kernels are used with {cmd:bw(1)},
no lags are used and {cmd:ivreg28} will report the usual
Eicker/Huber/White/sandwich variance estimates.

{p 0 4}{cmd:liml} requests the limited-information maximum likelihood estimator.

{p 0 4}{cmd:fuller(}{it:#}{cmd:)} specifies that Fuller's modified LIML estimator
is calculated using the user-supplied Fuller parameter alpha,
a non-negative number.
Alpha=1 has been suggested as a good choice.

{p 0 4}{cmd:kclass(}{it:#}{cmd:)} specifies that a general k-class estimator is calculated
using the user-supplied #, a non-negative number.

{p 0 4}{cmd:coviv} specifies that the matrix used to calculate the
covariance matrix for the LIML or k-class estimator
is based on the 2SLS matrix, i.e., with k=1.
In this case the covariance matrix will differ from that calculated for the 2SLS
estimator only because the estimate of the error variance will differ.
The default is for the covariance matrix to be based on the LIML or k-class matrix.

{p 0 4}{cmd:cue} requests the GMM continuously-updated estimator (CUE).

{p 0 4}{cmd:cueinit(}{it:matrix}{cmd:)} specifies that the starting values
for the CUE estimator use those in a user-supplied matrix b.
If omitted, the default behavior is to use starting values
from IV or 2-step efficient GMM estimation.

{p 0 4}{cmd:cueopt(}{it:string}{cmd:)} passes user-specified options
to Stata's {cmd:ml} routine; see help {help ml}.

{p 0 4}{cmd:robust} specifies that the Eicker/Huber/White/sandwich estimator of
variance is to be used in place of the traditional calculation.  {cmd:robust}
combined with {cmd:cluster()} further allows residuals which are not
independent within cluster (although they must be independent between
clusters).  See {hi:[U] Obtaining robust variance estimates}.

{p 0 4}{cmd:cluster}{cmd:(}{it:varname}{cmd:)} specifies that the observations
are independent across groups (clusters) but not necessarily independent
within groups.  {it:varname} specifies to which group each observation
belongs; e.g., {cmd:cluster(personid)} in data with repeated observations on
individuals.  {cmd:cluster()} can be used with {help pweight}s to produce
estimates for unstratified cluster-sampled data, but see help {help svyreg}
for a command especially designed for survey data.  Specifying {cmd:cluster()}
implies {cmd:robust}.

{p 0 4}{cmd:orthog}{cmd:(}{it:varlist_ex}{cmd:)} requests that a C-statistic
be calculated as a test of the exogeneity of the instruments in {it:varlist_ex}.
These may be either included or excluded exogenous variables.
The standard order condition for identification applies:
the restricted equation that does not use these variables
as exogenous instruments must still be identified.

{p 0 4}{cmd:endog}{cmd:(}{it:varlist_en}{cmd:)} requests that a C-statistic
be calculated as a test of the endogeneity
of the endogenous regressors in {it:varlist_en}.

{p 0 4}{cmd:redundant}{cmd:(}{it:varlist_ex}{cmd:)} requests a likelihood-ratio test
of the redundancy of the instruments in {it:varlist_ex}.
These must be excluded exogenous variables.
The standard order condition for identification applies:
the restricted equation that does not use these variables
as exogenous instrumenst must still be identified.

{p 0 4}{cmd:small} requests that small-sample statistics (F and t-statistics)
be reported instead of large-sample statistics (chi-squared and z-statistics).
Large-sample statistics are the default.
The exception is the statistic for the significance of the regression,
which is always reported as a small-sample F statistic.

{p 0 4}{cmd:noconstant} suppresses the constant term (intercept) in the
regression.  If {cmd:noconstant} is specified, the constant term is excluded
from both the final regression and the first-stage regression.  To include a
constant in the first-stage when {cmd:noconstant} is specified, explicitly
include a variable containing all 1's in {it:varlist_iv}.

{p 0 4}{cmd:first} requests that the full first-stage regression results be displayed,
along with the associated diagnostic and identification statistics.

{p 0 4}{cmd:ffirst} requests the first-stage diagnostic and identification statistics.
The results are saved in various e() macros.

{p 0 4}{cmd:nocollin} suppresses the checks for collinearities
and duplicate variables.

{p 0 4}{cmd:noid} suppresses the calculation and reporting
of underidentification and weak identification statistics.

{p 0 4}{cmd:savefirst} requests that the first-stage regressions results
are saved for later access using the {cmd:estimates} command.
The names under which the first-stage regressions are saved
are the names of the endogenous regressors prefixed by "_ivreg28_".
If these use Stata's time-series operators,
the "." is replaced by a "_".
The maximum number of first-stage estimation results that can be saved
depends on how many other estimation results the user has already saved
and on the maximum supported by Stata (20 for Stata 8.2 and 9.0, 300 for Stata 9.1).

{p 0 4}{cmdab:savefp:refix}{cmd:(}{it:prefix}{cmd:)} requests that
the first-stage regression results be saved using the user-specified prefix
instead of the default "_ivreg28_".

{p 0 4}{cmd:rf} requests that the reduced-form estimation of the equation
be displayed.

{p 0 4}{cmd:saverf} requests that the reduced-form estimation of the equation
be saved for later access using the {cmd:estimates} command.
The estimation is stored under the name of the dependent variable
prefixed by "_ivreg28_".
If this uses Stata's time-series operators,
the "." is replaced by a "_".

{p 0 4}{cmdab:saverfp:refix}{cmd:(}{it:prefix}{cmd:)} requests that
the reduced-form estimation be saved using the user-specified prefix
instead of the default "_ivreg28_".

{p 0 4}{cmd:level(}{it:#}{cmd:)} specifies the confidence level, in percent,
for confidence intervals of the coefficients; see help {help level}.

{p 0 4}{cmd:noheader}, {cmd:eform()}, {cmd:depname()} and {cmd:plus}
are for ado-file writers; see {hi:[R] ivreg} and {hi:[R] regress}.

{p 0 4}{cmd:nofooter} suppresses the display of the footer containing
identification and overidentification statistics,
exogeneity and endogeneity tests,
lists of endogenous variables and instruments, etc.

{p 0 4}{cmd:version} causes {cmd:ivreg28} to display its current version number
and to leave it in the macro {cmd:e(version)}.
It cannot be used with any other options.
and will clear any existing {cmd:e()} saved results.

{marker s_macros}{title:Remarks and saved results}

{p}{cmd:ivreg28} does not report an ANOVA table.
Instead, it reports the RSS and both the centered and uncentered TSS.
It also reports both the centered and uncentered R-squared.
NB: the TSS and R-squared reported by official {cmd:ivreg} is centered
if a constant is included in the regression, and uncentered otherwise.

{p}{cmd:ivreg28} saves the following results in {cmd:e()}:

Scalars
{col 4}{cmd:e(N)}{col 18}Number of observations
{col 4}{cmd:e(yy)}{col 18}Total sum of squares (SS), uncentered (y'y)
{col 4}{cmd:e(yyc)}{col 18}Total SS, centered (y'y - ((1'y)^2)/n)
{col 4}{cmd:e(rss)}{col 18}Residual SS
{col 4}{cmd:e(mss)}{col 18}Model SS =yyc-rss if the eqn has a constant, =yy-rss otherwise
{col 4}{cmd:e(df_m)}{col 18}Model degrees of freedom
{col 4}{cmd:e(df_r)}{col 18}Residual degrees of freedom
{col 4}{cmd:e(r2u)}{col 18}Uncentered R-squared, 1-rss/yy
{col 4}{cmd:e(r2c)}{col 18}Centered R-squared, 1-rss/yyc
{col 4}{cmd:e(r2)}{col 18}Centered R-squared if the eqn has a constant, uncentered otherwise
{col 4}{cmd:e(r2_a)}{col 18}Adjusted R-squared
{col 4}{cmd:e(ll)}{col 18}Log likelihood
{col 4}{cmd:e(rankxx)}{col 18}Rank of the matrix of observations on rhs variables=K
{col 4}{cmd:e(rankzz)}{col 18}Rank of the matrix of observations on instruments=L
{col 4}{cmd:e(rankV)}{col 18}Rank of covariance matrix V of coefficients
{col 4}{cmd:e(rankS)}{col 18}Rank of covariance matrix S of orthogonality conditions
{col 4}{cmd:e(rmse)}{col 18}root mean square error=sqrt(rss/(N-K)) if -small-, =sqrt(rss/N) otherwise
{col 4}{cmd:e(F)}{col 18}F statistic
{col 4}{cmd:e(N_clust)}{col 18}Number of clusters
{col 4}{cmd:e(bw)}{col 18}Bandwidth
{col 4}{cmd:e(lambda)}{col 18}LIML eigenvalue
{col 4}{cmd:e(kclass)}{col 18}k in k-class estimation
{col 4}{cmd:e(fuller)}{col 18}Fuller parameter alpha
{col 4}{cmd:e(sargan)}{col 18}Sargan statistic
{col 4}{cmd:e(sarganp)}{col 18}p-value of Sargan statistic
{col 4}{cmd:e(sargandf)}{col 18}dof of Sargan statistic = degree of overidentification = L-K
{col 4}{cmd:e(j)}{col 18}Hansen J statistic
{col 4}{cmd:e(jp)}{col 18}p-value of Hansen J statistic
{col 4}{cmd:e(jdf)}{col 18}dof of Hansen J statistic = degree of overidentification = L-K
{col 4}{cmd:e(arubin)}{col 18}Anderson-Rubin overidentification LR statistic
{col 4}{cmd:e(arubinp)}{col 18}p-value of Anderson-Rubin overidentification LR statistic
{col 4}{cmd:e(arubindf)}{col 18}dof of A-R overid statistic = degree of overidentification = L-K
{col 4}{cmd:e(idstat)}{col 18}Anderson canonical correlations LR statistic
{col 4}{cmd:e(idp)}{col 18}p-value of Anderson canonical correlations LR statistic
{col 4}{cmd:e(iddf)}{col 18}dof of Anderson canonical correlations LR statistic
{col 4}{cmd:e(cdf)}{col 18}Cragg-Donald F statistic
{col 4}{cmd:e(cdchi2)}{col 18}Cragg-Donald chi-sq statistic
{col 4}{cmd:e(cdchi2p)}{col 18}p-value of Cragg-Donald chi-sq statistic
{col 4}{cmd:e(arf)}{col 18}Anderson-Rubin F-test of significance of endogenous regressors
{col 4}{cmd:e(arfp)}{col 18}p-value of Anderson-Rubin F-test of endogenous regressors
{col 4}{cmd:e(archi2)}{col 18}Anderson-Rubin chi-sq test of significance of endogenous regressors
{col 4}{cmd:e(archi2p)}{col 18}p-value of Anderson-Rubin chi-sq test of endogenous regressors
{col 4}{cmd:e(ardf)}{col 18}degrees of freedom of Anderson-Rubin tests of endogenous regressors
{col 4}{cmd:e(ardf_r)}{col 18}denominator degrees of freedom of AR F-test of endogenous regressors
{col 4}{cmd:e(redstat)}{col 18}LR statistic for instrument redundancy
{col 4}{cmd:e(redp)}{col 18}p-value of LR statistic for instrument redundancy
{col 4}{cmd:e(reddf)}{col 18}dof of LR statistic for instrument redundancy
{col 4}{cmd:e(cstat)}{col 18}C-statistic
{col 4}{cmd:e(cstatp)}{col 18}p-value of C-statistic
{col 4}{cmd:e(cstatdf)}{col 18}Degrees of freedom of C-statistic
{col 4}{cmd:e(cons)}{col 18}1 when equation has a Stata-supplied constant; 0 otherwise
{col 4}{cmd:e(fwlcons)}{col 18}as above but prior to partialling-out (see {cmd:e(fwl)})

Macros
{col 4}{cmd:e(cmd)}{col 18}ivreg28
{col 4}{cmd:e(version)}{col 18}Version number of ivreg28
{col 4}{cmd:e(model)}{col 18}ols, iv, gmm, liml, or kclass
{col 4}{cmd:e(depvar)}{col 18}Name of dependent variable
{col 4}{cmd:e(instd)}{col 18}Instrumented (RHS endogenous) variables
{col 4}{cmd:e(insts)}{col 18}Instruments
{col 4}{cmd:e(inexog)}{col 18}Included instruments (regressors)
{col 4}{cmd:e(exexog)}{col 18}Excluded instruments
{col 4}{cmd:e(collin)}{col 18}Variables dropped because of collinearities
{col 4}{cmd:e(dups)}{col 18}Duplicate variables
{col 4}{cmd:e(ecollin)}{col 18}Endogenous variables reclassified as exogenous because of
{col 20}collinearities with instruments
{col 4}{cmd:e(clist)}{col 18}Instruments tested for orthogonality
{col 4}{cmd:e(redlist)}{col 18}Instruments tested for redundancy
{col 4}{cmd:e(fwl)}{col 18}Partialled-out exogenous regressors
{col 4}{cmd:e(small)}{col 18}small
{col 4}{cmd:e(wtype)}{col 18}weight type
{col 4}{cmd:e(wexp)}{col 18}weight expression
{col 4}{cmd:e(clustvar)}{col 18}Name of cluster variable
{col 4}{cmd:e(vcetype)}{col 18}Covariance estimation method
{col 4}{cmd:e(kernel)}{col 18}Kernel
{col 4}{cmd:e(tvar)}{col 18}Time variable
{col 4}{cmd:e(ivar)}{col 18}Panel variable
{col 4}{cmd:e(firsteqs)}{col 18}Names of stored first-stage equations
{col 4}{cmd:e(rfeq)}{col 18}Name of stored reduced-form equation
{col 4}{cmd:e(predict)}{col 18}Program used to implement predict

Matrices
{col 4}{cmd:e(b)}{col 18}Coefficient vector
{col 4}{cmd:e(V)}{col 18}Variance-covariance matrix of the estimators
{col 4}{cmd:e(S)}{col 18}Covariance matrix of orthogonality conditions
{col 4}{cmd:e(W)}{col 18}GMM weighting matrix (=inverse of S if efficient GMM estimator)
{col 4}{cmd:e(first)}{col 18}First-stage regression results
{col 4}{cmd:e(ccev)}{col 18}Eigenvalues corresponding to the Anderson canonical correlations test
{col 4}{cmd:e(cdev)}{col 18}Eigenvalues corresponding to the Cragg-Donald test

Functions
{col 4}{cmd:e(sample)}{col 18}Marks estimation sample



{marker s_examples}{title:Examples}

{p 8 12}{stata "use http://fmwww.bc.edu/ec-p/data/hayashi/griliches76.dta" : . use http://fmwww.bc.edu/ec-p/data/hayashi/griliches76.dta }{p_end}
{p 8 12}(Wages of Very Young Men, Zvi Griliches, J.Pol.Ec. 1976)

{p 8 12}{stata "xi i.year" : . xi i.year}

{col 0}(Instrumental variables.  Examples follow Hayashi 2000, p. 255.)

{p 8 12}{stata "ivreg28 lw s expr tenure rns smsa _I* (iq=med kww age mrt)" : . ivreg28 lw s expr tenure rns smsa _I* (iq=med kww age mrt)}

{p 8 12}{stata "ivreg28 lw s expr tenure rns smsa _I* (iq=med kww age mrt), small ffirst" : . ivreg28 lw s expr tenure rns smsa _I* (iq=med kww age mrt), small ffirst}

{col 0}(Testing for the presence of heteroskedasticity in IV/GMM estimation)

{p 8 12}{stata "ivhettest, fitlev" : . ivhettest, fitlev}

{col 0}(Two-step GMM efficient in the presence of arbitrary heteroskedasticity)

{p 8 12}{stata "ivreg28 lw s expr tenure rns smsa _I* (iq=med kww age mrt), gmm" : . ivreg28 lw s expr tenure rns smsa _I* (iq=med kww age mrt), gmm}

{p 0}(Continuously-updated GMM (CUE) efficient in the presence of arbitrary heteroskedasticity.  NB: may require 50+ iterations.)

{p 8 12}{stata "ivreg28 lw s expr tenure rns smsa _I* (iq=med kww age mrt), cue robust" : . ivreg28 lw s expr tenure rns smsa _I* (iq=med kww age mrt), cue robust}

{col 0}(Sargan-Basmann tests of overidentifying restrictions for IV estimation)

{p 8 12}{stata "ivreg28 lw s expr tenure rns smsa _I* (iq=med kww age mrt)" : . ivreg28 lw s expr tenure rns smsa _I* (iq=med kww age mrt)}

{p 8 12}{stata "overid, all" : . overid, all}

{col 0}(Tests of exogeneity and endogeneity)

{col 0}(Test the exogeneity of 1 regressor)

{p 8 12}{stata "ivreg28 lw s expr tenure rns smsa _I* (iq=med kww age mrt), gmm orthog(s)" : . ivreg28 lw s expr tenure rns smsa _I* (iq=med kww age mrt), gmm orthog(s)}

{col 0}(Test the exogeneity of 2 excluded instruments)

{p 8 12}{stata "ivreg28 lw s expr tenure rns smsa _I* (iq=med kww age mrt), gmm orthog(age mrt)" : . ivreg28 lw s expr tenure rns smsa _I* (iq=med kww age mrt), gmm orthog(age mrt)}

{col 0}(Frisch-Waugh-Lovell (FWL): equivalence of estimations with and without partialling-out)

{p 8 12}{stata "ivreg28 lw s expr tenure rns _I* (iq=kww age), cluster(year)" : . ivreg28 lw s expr tenure rns smsa _I* (iq=med kww age), cluster(year)}

{p 8 12}{stata "ivreg28 lw s expr tenure rns _I* (iq=kww age), cluster(year) fwl(_I*)" : . ivreg28 lw s expr tenure rns smsa _I* (iq=med kww age), cluster(year) fwl(_I*)}

{col 0}(FWL: efficient GMM with #clusters<#instruments feasible after partialling-out)

{p 8 12}{stata "ivreg28 lw s expr tenure rns _I* (iq=kww age), cluster(year) fwl(_I*) gmm" : . ivreg28 lw s expr tenure rns smsa (iq=med kww age), cluster(year) fwl(_I*) gmm}

{col 0}(Examples following Wooldridge 2002, pp.59, 61)

{p 8 12}{stata "use http://fmwww.bc.edu/ec-p/data/wooldridge/mroz.dta" : . use http://fmwww.bc.edu/ec-p/data/wooldridge/mroz.dta }

{col 0}(Test an excluded instrument for redundancy)

{p 8 12}{stata "ivreg28 lwage exper expersq (educ=age kidslt6 kidsge6), redundant(age)" : . ivreg28 lwage exper expersq (educ=age kidslt6 kidsge6), redundant(age)}

{col 0}(Equivalence of DWH endogeneity test when regressor is endogenous...)

{p 8 12}{stata "ivreg28 lwage exper expersq (educ=age kidslt6 kidsge6)" : . ivreg28 lwage exper expersq (educ=age kidslt6 kidsge6)}

{p 8 12}{stata "ivendog educ" :. ivendog educ}

{col 0}(... endogeneity test using the {cmd:endog} option)

{p 8 12}{stata "ivreg28 lwage exper expersq (educ=age kidslt6 kidsge6), endog(educ)" : . ivreg28 lwage exper expersq educ (educ=age kidslt6 kidsge6), endog(educ)}

{col 0}(...and C-test of exogeneity when regressor is exogenous, using the {cmd:orthog} option)

{p 8 12}{stata "ivreg28 lwage exper expersq educ (=age kidslt6 kidsge6), orthog(educ)" : . ivreg28 lwage exper expersq educ (=age kidslt6 kidsge6), orthog(educ)}

{col 0}(Heteroskedastic Ordinary Least Squares, HOLS)

{p 8 12}{stata "ivreg28 lwage exper expersq educ (=age kidslt6 kidsge6), gmm" : . ivreg28 lwage exper expersq educ (=age kidslt6 kidsge6), gmm}

{col 0}(LIML and k-class estimation using Klein data)

{col 9}{stata "use http://fmwww.bc.edu/repec/bocode/k/kleinI" :. use http://fmwww.bc.edu/repec/bocode/k/kleinI}

{col 0}(LIML estimates of Klein's consumption function)

{p 8 12}{stata "ivreg28 consump L.profit (profit wages = govt taxes trend wagegovt capital1 L.demand), liml" :. ivreg28 consump L.profit (profit wages = govt taxes trend wagegovt capital1 L.demand), liml}

{col 0}(Equivalence of LIML and CUE+homoskedasticity+independence)

{p 8 12}{stata "ivreg28 consump L.profit (profit wages = govt taxes trend wagegovt capital1 L.demand), liml coviv" :. ivreg28 consump L.profit (profit wages = govt taxes trend wagegovt capital1 L.demand), liml coviv}

{p 8 12}{stata "ivreg28 consump L.profit (profit wages = govt taxes trend wagegovt capital1 L.demand), cue" :. ivreg28 consump L.profit (profit wages = govt taxes trend wagegovt capital1 L.demand), cue}

{col 0}(Fuller's modified LIML with alpha=1)

{p 8 12}{stata "ivreg28 consump L.profit (profit wages = govt taxes trend wagegovt capital1 L.demand), fuller(1)" :. ivreg28 consump L.profit (profit wages = govt taxes trend wagegovt capital1 L.demand), fuller(1)}

{col 0}(k-class estimation with Nagar's bias-adjusted IV, k=1+(L-K)/N=1+4/21=1.19)

{p 8 12}{stata "ivreg28 consump L.profit (profit wages = govt taxes trend wagegovt capital1 L.demand), kclass(1.19)" :. ivreg28 consump L.profit (profit wages = govt taxes trend wagegovt capital1 L.demand), kclass(1.19)}

{col 0}(Kernel-based covariance estimation using time-series data)

{p 8 12}{stata "use http://fmwww.bc.edu/ec-p/data/wooldridge/phillips.dta" :. use http://fmwww.bc.edu/ec-p/data/wooldridge/phillips.dta}

{p 8 12}{stata "tsset year, yearly" :. tsset year, yearly}

{col 0}(Autocorrelation-consistent (AC) inference in an OLS Regression)

{p 8 12}{stata "ivreg28 cinf unem, bw(3)" :. ivreg28 cinf unem, bw(3)}

{col 0}(Heteroskedastic and autocorrelation-consistent (HAC) inference in an OLS regression)

{p 8 12}{stata "ivreg28 cinf unem, bw(3) kernel(bartlett) robust small" :. ivreg28 cinf unem, bw(3) kernel(bartlett) robust small}

{p 8 12}{stata "newey cinf unem, lag(2)" :. newey cinf unem, lag(2)}

{col 0}(AC and HAC in IV and GMM estimation)

{p 8 12}{stata "ivreg28 cinf (unem = l(1/3).unem), bw(3)" :. ivreg28 cinf (unem = l(1/3).unem), bw(3)}

{p 8 12}{stata "ivreg28 cinf (unem = l(1/3).unem), bw(3) gmm kernel(thann)" :. ivreg28 cinf (unem = l(1/3).unem), bw(3) gmm kernel(thann)}

{p 8 12}{stata "ivreg28 cinf (unem = l(1/3).unem), bw(3) gmm kernel(qs) robust orthog(l1.unem)" :. ivreg28 cinf (unem = l(1/3).unem), bw(3) gmm kernel(qs) robust orthog(l1.unem)}

{col 0}(Examples using Large N, Small T Panel Data)

{p 8 12}{stata "use http://fmwww.bc.edu/ec-p/data/macro/abdata.dta" : . use http://fmwww.bc.edu/ec-p/data/macro/abdata.dta }{p_end}
{p 8 12}(Layard & Nickell, Unemployment in Britain, Economica 53, 1986, from Ox dist)

{p 8 12}{stata "tsset id year" :. tsset id year}

{col 0}(Autocorrelation-consistent inference in an IV regression)

{p 8 12}{stata "ivreg28 n (w k ys = d.w d.k d.ys d2.w d2.k d2.ys), bw(1) kernel(tru)": . ivreg28 n (w k ys = d.w d.k d.ys d2.w d2.k d2.ys), bw(1) kernel(tru)}

{col 0}(Two-step effic. GMM in the presence of arbitrary heteroskedasticity and autocorrelation)

{p 8 12}{stata "ivreg28 n (w k ys = d.w d.k d.ys d2.w d2.k d2.ys), bw(2) gmm kernel(tru) robust": . ivreg28 n (w k ys = d.w d.k d.ys d2.w d2.k d2.ys), bw(2) gmm kernel(tru) robust}

{col 0}(Two-step effic. GMM in the presence of arbitrary heterosked. and intra-group correlation)

{p 8 12}{stata "ivreg28 n (w k ys = d.w d.k d.ys d2.w d2.k d2.ys), gmm cluster(id)": . ivreg28 n (w k ys = d.w d.k d.ys d2.w d2.k d2.ys), gmm cluster(id)}


{marker s_refs}{title:References}

{p 0 4}Anderson, T.W. 1984. Introduction to Multivariate Statistical Analysis.
2d ed. New York: John Wiley & Sons.

{p 0 4}Anderson, T. W., and H. Rubin. 1949. Estimation of the parameters of a single equation 
in a complete system of stochastic equations. Annals of Mathematical Statistics, Vol. 20, 
pp. 46-63. 

{p 0 4}Anderson, T. W., and H. Rubin. 1950. The asymptotic properties of estimates of the parameters of a single 
equation in a complete system of stochastic equations. Annals of Mathematical Statistics,
Vol. 21, pp. 570-82. 

{p 0 4}Baum, C.F., Schaffer, M.E., and Stillman, S. 2003. Instrumental Variables and GMM:
Estimation and Testing. The Stata Journal, Vol. 3, No. 1, pp. 1-31.
Working paper version: Boston College Department of Economics Working Paper No 545.
{browse "http://ideas.repec.org/p/boc/bocoec/545.html":http://ideas.repec.org/p/boc/bocoec/545.html}

{p 0 4}Baum, C. F., Schaffer, M. E., and Stillman, S. 2007. Enhanced routines for 
instrumental variables/GMM estimation and testing. Unpublished working paper,
forthcoming.

{p 0 4}Chernozhukov, V. and Hansen, C. 2005. The Reduced Form:
A Simple Approach to Inference with Weak Instruments.
Working paper, University of Chicago, Graduate School of Business.

{p 0 4}Cragg, J.G. and Donald, S.G. 1993. Testing Identfiability and Specification in
Instrumental Variables Models. Econometric Theory, Vol. 9, pp. 222-240.

{p 0 4}Cushing, M.J. and McGarvey, M.G. 1999. Covariance Matrix Estimation.
In L. Matyas (ed.), Generalized Methods of Moments Estimation.
Cambridge: Cambridge University Press.

{p 0 4}Davidson, R. and MacKinnon, J. 1993. Estimation and Inference in Econometrics.
1993. New York: Oxford University Press.

{p 0 4}Dufour, J.M.  2003.  Identification, Weak Instruments and Statistical Inference
in Econometrics. Canadian Journal of Economics, Vol. 36, No. 4, pp. 767-808.
Working paper version: CIRANO Working Paper 2003s-49.
http://www.cirano.qc.ca/pdf/publication/2003s-49.pdf

{p 0 4}Hall, A.R. and Peixe, F.P.M.  2000.  A Consistent Method for the Selection of
Relevant Instruments.  Econometric Society World Congress 2000 Contributed papers.
http://econpapers.repec.org/paper/ecmwc2000/0790.htm

{p 0 4}Hall, A.R., Rudebusch, G.D. and Wilcox, D.W.  1996.  Judging Instrument Relevance in
Instrumental Variables Estimation.  International Economic Review, Vol. 37, No. 2, pp. 283-298.

{p 0 4}Hayashi, F. Econometrics. 2000. Princeton: Princeton University Press.

{p 0 4}Hansen, L.P., Heaton, J., and Yaron, A.  1996.  Finite Sample Properties
of Some Alternative GMM Estimators.  Journal of Business and Economic Statistics,
Vol. 14, No. 3, pp. 262-280.

{p 0 4}Kleibergen, F.  2007.  Generalizing Weak Instrument Robust Statistics Towards
Multiple Parameters, Unrestricted Covariance Matrices and Identification Statistics.
Journal of Econometrics, forthcoming.

{p 0 4}Mikusheva, A. and Poi, B.P.  2006.
Tests and confidence sets with correct size when instruments are potentially weak.
The Stata Journal, Vol. 6,  No. 3, pp. 335-347.

{p 0 4}Moreira, M.J. and Poi, B.P.  2003.  Implementing Tests with the Correct Size
in the Simultaneous Equations Model.  The Stata Journal, Vol. 3, No. 1, pp. 57-70.

{p 0 4}Shea, J. 1997.  Instrument Relevance in Multivariate Linear Models:
A Simple Measure.
Review of Economics and Statistics, Vol. 49, No. 2, pp. 348-352.

{p 0 4}Stock, J.H. and Wright, J.H.  2000.  GMM with Weak Identification.
Econometrica, Vol. 68, No. 5, September, pp. 1055-1096.

{p 0 4}Stock, J.H. and Yogo, M.  2005.  Testing for Weak Instruments in Linear IV Regression.
In D.W.K. Andrews and J.H. Stock, eds. Identification and Inference for Econometric Models:
Essays in Honor of Thomas Rothenberg. Cambridge: Cambridge University Press, 2005, pp. 80–108.
Working paper version: NBER Technical Working Paper 284.  http://www.nber.org/papers/T0284.

{p 0 4}Wooldridge, J.M. 2002. Econometric Analysis of Cross Section and Panel Data.
Cambridge, MA: MIT Press.


{marker s_acknow}{title:Acknowledgements}

{p}We would like to thanks various colleagues who helped us along the way, including
David Drukker,
Austin Nichols,
Vince Wiggins,
and, not least, the users of {cmd:ivreg28}
who have provided suggestions,
spotted bugs,
and helped test the package.
We are also grateful to Jim Stock and Moto Yogo for permission to reproduce
their critical values for the Cragg-Donald statistic.

{marker s_citation}{title:Citation of ivreg28}

{p}{cmd:ivreg28} is not an official Stata command. It is a free contribution
to the research community, like a paper. Please cite it as such: {p_end}

{phang}Baum, C.F., Schaffer, M.E., Stillman, S. 2007.
ivreg28: Stata module for extended instrumental variables/2SLS, GMM and AC/HAC, LIML and k-class regression.
{browse "http://ideas.repec.org/c/boc/bocode/s425401.html":http://ideas.repec.org/c/boc/bocode/s425401.html}{p_end}

{title:Authors}

	Christopher F Baum, Boston College, USA
	baum@bc.edu

	Mark E Schaffer, Heriot-Watt University, UK
	m.e.schaffer@hw.ac.uk

	Steven Stillman, Motu Economic and Public Policy Research
	stillman@motu.org.nz


{title:Also see}

{p 1 14}Manual:  {hi:[U] 23 Estimation and post-estimation commands},{p_end}
{p 10 14}{hi:[U] 29 Overview of model estimation in Stata},{p_end}
	  {hi:[R] ivreg}
{p 0 19}On-line:  help for {help ivreg}, {help newey};
{help overid}, {help ivendog}, {help ivhettest}, {help ivreset},
{help xtivreg28}, {help xtoverid},
{help condivreg} (if installed);
{help est}, {help postest};
{help regress}{p_end}
