PANEL Procedure

Hausman-Taylor Estimation (HTAYLOR Option)

You perform Hausman-Taylor estimation by specifying the HTAYLOR option in the MODEL statement. The Hausman and Taylor (1981) model is a hybrid that combines the consistency of a fixed-effects model with the efficiency and applicability of a random-effects model. One-way random-effects models assume exogeneity of the regressors; that is, they are independent of both the cross-sectional and observation-level errors. When some regressors are correlated with the cross-sectional errors, you can adjust the random-effects model to deal with this form of endogeneity.

Consider the one-way model:

y Subscript i t Baseline equals bold x Subscript 1 i t Baseline bold-italic beta 1 plus bold x Subscript 2 i t Baseline bold-italic beta 2 plus bold z Subscript 1 i Baseline bold-italic gamma 1 plus bold z Subscript 2 i Baseline bold-italic gamma 2 plus nu Subscript i Baseline plus e Subscript i t

The regressors are subdivided so that bold x Subscript 1 i t and bold x Subscript 2 i t vary within cross sections, whereas bold z Subscript 1 i and bold z Subscript 2 i do not and would otherwise be dropped from a fixed-effects model. The subscript 1 denotes variables that are independent of both error terms (exogenous variables), and the subscript 2 denotes variables that are independent of the observation-level errors e Subscript i t but correlated with cross-sectional errors nu Subscript i. The intercept term (if your model has one) is included as part of bold z Subscript 1 i.

The Hausman-Taylor estimator is a two-stage least squares (2SLS) regression on data that are weighted similarly to data for random-effects estimation. The weights are functions of the estimated variance components.

The observation-level variance is estimated from a one-way fixed-effects model fit. Obtain bold y Subscript w, bold upper X Subscript w, and ModifyingAbove bold-italic beta With caret Subscript w from the section One-Way Fixed-Effects Model (FIXONE and FIXONETIME Options). Then ModifyingAbove sigma With caret Subscript e Superscript 2 Baseline equals SSE slash left-parenthesis upper M minus upper N right-parenthesis, where

SSE equals left-parenthesis bold y Subscript w Baseline minus bold upper X Subscript w Baseline ModifyingAbove bold-italic beta With caret Subscript w Baseline right-parenthesis prime left-parenthesis bold y Subscript w Baseline minus bold upper X Subscript w Baseline ModifyingAbove bold-italic beta With caret Subscript w Baseline right-parenthesis

To estimate the cross-sectional error variance, form the mean-residual vector bold r equals bold upper P prime 0 left-parenthesis bold y minus bold upper X Subscript w Baseline ModifyingAbove bold-italic beta With caret Subscript w Baseline right-parenthesis, where bold upper P 0 equals normal d normal i normal a normal g left-parenthesis bold upper J overbar Subscript upper T Sub Subscript i Subscript Baseline right-parenthesis. You can use the mean residuals to obtain intermediate estimates of the coefficients for bold z 1 and bold z 2 via two-stage least squares (2SLS) estimation. At the first stage, use bold x 1 and bold z 1 as instrumental variables to predict bold z 2. At the second stage, regress bold r on both bold z 1 and the predicted bold z 2 to obtain ModifyingAbove bold-italic gamma With caret Subscript 1 Superscript m and ModifyingAbove bold-italic gamma With caret Subscript 2 Superscript m.

To estimate the cross-sectional variance, compute ModifyingAbove sigma With caret Subscript nu Superscript 2 Baseline equals StartSet upper R left-parenthesis nu right-parenthesis slash upper N minus ModifyingAbove sigma With caret Subscript e Superscript 2 Baseline EndSet slash upper T overbar, where upper T overbar equals upper N slash left-parenthesis sigma-summation Underscript i equals 1 Overscript upper N Endscripts upper T Subscript i Superscript negative 1 Baseline right-parenthesis and

upper R left-parenthesis nu right-parenthesis equals left-parenthesis bold r minus bold upper Z 1 ModifyingAbove bold-italic gamma With caret Subscript 1 Superscript m Baseline minus bold upper Z 2 ModifyingAbove bold-italic gamma With caret Subscript 2 Superscript m Baseline right-parenthesis prime left-parenthesis bold r minus bold upper Z 1 ModifyingAbove bold-italic gamma With caret Subscript 1 Superscript m Baseline minus bold upper Z 2 ModifyingAbove bold-italic gamma With caret Subscript 2 Superscript m Baseline right-parenthesis

The design matrices bold upper Z 1 and bold upper Z 2 are formed by stacking the data observations of bold z Subscript 1 i and bold z Subscript 2 i, respectively.

After variance-component estimation, transform the dependent variable into partial deviations: y Subscript i t Superscript asterisk Baseline equals y Subscript i t Baseline minus ModifyingAbove theta With caret Subscript i Baseline y overbar Subscript i period. Likewise, transform the regressors to form bold x Subscript 1 i t Superscript asterisk, bold x Subscript 2 i t Superscript asterisk, bold z Subscript 1 i Superscript asterisk, and bold z Subscript 2 i Superscript asterisk. The partial weights ModifyingAbove theta With caret Subscript i are determined by ModifyingAbove theta With caret Subscript i Baseline equals 1 minus ModifyingAbove sigma With caret Subscript e Baseline slash ModifyingAbove w With caret Subscript i, with ModifyingAbove w With caret Subscript i Superscript 2 Baseline equals upper T Subscript i Baseline ModifyingAbove sigma With caret Subscript nu Superscript 2 Baseline plus ModifyingAbove sigma With caret Subscript e Superscript 2.

Finally, you obtain the Hausman-Taylor estimates by performing 2SLS regression of y Subscript i t Superscript asterisk on bold x Subscript 1 i t Superscript asterisk, bold x Subscript 2 i t Superscript asterisk, bold z Subscript 1 i Superscript asterisk, and bold z Subscript 2 i Superscript asterisk. For the first-stage regression, use the following instruments:

  • bold x overTilde Subscript i t, the deviations from cross-sectional means for all time-varying variables (correlated and uncorrelated) for the ith cross section during time period t

  • left-parenthesis 1 minus ModifyingAbove theta With caret Subscript i Baseline right-parenthesis bold x overbar Subscript 1 i period, where bold x overbar Subscript 1 i period are the means of the time-varying exogenous variables for the ith cross section

  • left-parenthesis 1 minus ModifyingAbove theta With caret Subscript i Baseline right-parenthesis bold z Subscript 1 i

Multiplication by the factor left-parenthesis 1 minus ModifyingAbove theta With caret Subscript i Baseline right-parenthesis is redundant in balanced data but necessary in the unbalanced case to produce accurate instrumentation; see Gardner (1998).

Let k 1 equal the number of regressors in bold x 1, and let g 2 equal the number of regressors in bold z 2. Then the Hausman-Taylor model is identified only if k 1 greater-than-or-equal-to g 2; otherwise, no estimation takes place.

Hausman and Taylor (1981) describe a specification test that compares their model to a fixed-effects model. For a null hypothesis of fixed effects, Hausman’s m statistic is calculated by comparing the parameter estimates and variance matrices for both models, which is identical to how it is calculated for one-way random-effects models; for more information, see the section Hausman Test. However, the number of degrees of freedom of the test is not based on matrix rank but instead is equal to k 1 minus g 2.

Last updated: June 19, 2025