Tests for over-identification in a single structural equation: what is what?

There is a large number of tests for over-identification for a single structural equation and there seems to be confusion about what test is what and who suggested what. This note tries to clarify this.

Consider a structural equation of the form

\[{y_1} = {Y_2}\beta + {Z_1}\gamma + u \ \ \ \ (1)\]

with reduced form

\[\left[ {{y_1},{Y_2}} \right] = {Z_1}\Phi + {Z_2}\Pi + \left[ {{v_1},{V_2}} \right] \ \ \ \ (2)\]

and denote by \(\hat \beta\) the TSLS estimator of \(\beta\), and by \(\hat u\) the TSLS vector of residuals of the regression of \({y_1}\) on \(\left[ {{Y_2},{Z_1}} \right]\) with instruments \({Z_2}\). Also write \(Z = \left[ {{Z_1},{Z_2}} \right]\) and define the usual projection matrix \({M_A} = {I_T} - A{\left( {A'A} \right)^{ - 1}}A'\) for any full column rank matrix \(A\). Notice \(y_1\) is \(T \times 1\), \(Y_2\) is \(T \times n-1\), \(Z_1\) is \(T \times k_{1}\) and \(Z_2\) is \(T \times k_2\). The dimension of all other terms in (1) and (2) can be easily obtained.

If one assumes that the rows of \(\left[ {{v_1},{V_2}} \right]\) are i.i.d. with zero mean vector and covariance matrix $$ then the structural variance is \({\sigma ^2} = \left( {1, - \beta '} \right)\Omega \left( {1, - \beta '} \right)'\). The TSLS estimator of the structural variance is denoted by \({\hat \sigma ^2} = \hat u'\hat u/T\). However one can estimate this also as \({\tilde \sigma ^2} = \left( {1, - \hat \beta '} \right)\hat \Omega \left( {1, - \hat \beta '} \right)'\) where \(\hat \Omega = {T^{ - 1}}\left[ {{y_1},{Y_2}} \right]'{M_Z}\left[ {{y_1},{Y_2}} \right]\).

Anderson-Rubin test

This is the first test for over-identification and was proposed by Anderson and Rubin (1949) and is based on the statistic

\[\mathop {\min }\limits_\beta \frac{{\left( {{y_1} - {Y_2}\beta } \right)'\left( {{M_{{Z_1}}} - {M_Z}} \right)\left( {{y_1} - {Y_2}\beta } \right)}}{{\left( {{y_1} - {Y_2}\beta } \right)'{M_Z}\left( {{y_1} - {Y_2}\beta } \right)/T}} = \frac{{\left( {{y_1} - {Y_2}{{\hat \beta }_{LIML}}} \right)'\left( {{M_{{Z_1}}} - {M_Z}} \right)\left( {{y_1} - {Y_2}{{\hat \beta }_{LIML}}} \right)}}{{\left( {{y_1} - {Y_2}{{\hat \beta }_{LIML}}} \right)'{M_Z}\left( {{y_1} - {Y_2}{{\hat \beta }_{LIML}}} \right)/T}} \ \ \ \ (3)\]

where \({\hat \beta _{LIML}}\) is the LIML estimator of the structural equation in (1) with reduced form (2)

Sargan Test

This test was proposed by Sargan (1958) and uses as test statistic \(T{R^2}\) where \({R^2}\) is the coefficient of determination of the regression of \(\hat u\) on \(Z = \left( {{Z_1},{Z_2}} \right)\) which is, apart from a constant of proportionality,

\[\displaystyle\frac{{\hat u'\left( {{M_{{Z_1}}} - {M_Z}} \right)\hat u}}{{{{\hat \sigma }^2}}}. \ \ \ \ (4)\]

This test is generalized to the J-test by Hansen (1982). Hausman (1983) has also suggested a test having this structure but does not refer to Sargan’s work.

Basmann Test

Basmann (1960) suggested a modification of the Anderson-Rubin test which uses the TSLS estimator instead of the LIML estimator. This is based on the statistic

\[\frac{{\left( {{y_1} - {Y_2}\hat \beta } \right)'\left( {{M_{{Z_1}}} - {M_Z}} \right)\left( {{y_1} - {Y_2}\hat \beta } \right)}}{{\left( {{y_1} - {Y_2}\hat \beta } \right)'{M_Z}\left( {{y_1} - {Y_2}\hat \beta } \right)/T}}. \ \ \ \ (5)\]

It is worth rewriting this statistic to bring up some similarity and difference with Sargan statistic. Using the fact that \(\hat u = {M_{{Z_1}}}\left( {{y_1} - {Y_2}\hat \beta } \right)\) and that \(\left( {{M_{{Z_1}}} - {M_Z}} \right){M_{{Z_1}}} = \left( {{M_{{Z_1}}} - {M_Z}} \right)\) the numerators of Sargan and Basmann statistic are equal

\[\left( {{y_1} - {Y_2}\hat \beta } \right)'\left( {{M_{{Z_1}}} - {M_Z}} \right)\left( {{y_1} - {Y_2}\hat \beta } \right) = \hat u'\left( {{M_{{Z_1}}} - {M_Z}} \right)\hat u, \ \ \ \ (6)\]

however, the denominator of Basmann test statistic is

\[\begin{gathered} \left( {{y_1} - {Y_2}\hat \beta } \right){M_Z}\left( {{y_1} - {Y_2}\hat \beta } \right)/T = \left( {1, - \hat \beta '} \right)\left( {{y_1},{Y_2}} \right)'{M_Z}\left( {{y_1},{Y_2}} \right)\left( \begin{gathered} 1 \hfill \\ - \hat \beta \hfill \\\end{gathered} \right)/T \\ = \left( {1, - \hat \beta '} \right)\hat \Omega \left( \begin{gathered} 1 \hfill \\ - \hat \beta \hfill \\\end{gathered} \right). \\\end{gathered}\ \ \ \ (7)\]

Thus, Basmann and Sargan tests use alternative estimators of the structural variance.

Byron and Wegge test statistic

Hwang (1980) has shown that the test statistics suggested by Byron (1974) and Wegge (1978) are in fact the same and shows that they can be written as

\[\displaystyle\frac{{\hat \delta '{Z_2}{M_2}{Z_2}\hat \delta }}{{{{\hat \sigma }^2}}} \ \ \ \ (8)\]

where \(\hat \delta = {\left( {{Z_2}'{A_2}{Z_2}} \right)^{ - 1}}Z'{A_2}y\), \({M_2} = {A_2} - {A_2}{Y_2}{\left( {{Y_2}'{A_2}{Y_2}} \right)^{ - 1}}{Y_2}'{A_2}\) and \({A_2} = {M_{{Z_1}}} - {M_Z}\). Simple algebraic transformations show that

\[\hat \delta '{Z_2}{M_2}{Z_2}\hat \delta = \left( {{y_1} - {Y_2}\hat \beta } \right)'\left( {{M_{{Z_1}}} - {M_Z}} \right)\left( {{y_1} - {Y_2}\hat \beta } \right) = \hat u'\left( {{M_{{Z_1}}} - {M_Z}} \right)\hat u \ \ \ \ (9)\]

so that the Byron and Wegge test is identical to the Sargan test. Neither Byron nor Wegge nor Hwang refer to Sargan’s work.

There are also some modern equivalent of these classical tests. These apply under more general conditions including for example heteroskedasticity.

Reference

• 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 21: 570-482.

• Basmann, R. L. (1960). “On Finite Sample Distributions of Generalized Classical Linear Identifiability Test Statistics.” Journal of the American Statistical Association 55: 650-659.

• Byron, R. P. (1974). “Testing Structural Specification Using the Unrestricted Reduced Form.” Econometrica 42: 869-883.

• Hansen, L. P. (1982). “Large Sample Properties of Generalized Method of Moments Estimators.” Econometrica 40: 1029-1054.

• Hausman, J. A. (1983). “Specification and Estimation of Simultaneous Equation Models” in. Handbook of Econometrics, Volume. Z. Griliches and M. D. Intriligator. Amsterdam, North-Holland Publishing Company: 391-448.

• Hwang, H.-S. (1980). “A Comparison of Tests of Overidentifying Restrictions.” Econometrica 48 (7): 1821-1825.

• Sargan, J. D. (1958). “The Estimation of Economic Relationships using Instrumental Variables.” Econometrica 26: 393-415

• Wegge, L. L. (1978). “Constrained Indirect Least Squares Estimators.” Econometrica 46: 435-499.