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st: New package, -swain- : Correct small sample chi-square overidentification test

From   John Antonakis <>
Subject   st: New package, -swain- : Correct small sample chi-square overidentification test
Date   Wed, 20 Mar 2013 13:11:53 +0100


With the usual thanks to Kit Baum, a new package -swain- is available on SSC. This package should be interesting to those who estimate structural equation models via -sem- (using maximum likelihood). It might also be useful to those estimating models via two or three-stage least squares, which can be also estimated with -sem-.*

Here is a description of -swain-: Correct small sample chi-square overidentification test after -sem-

      swain corrects the chi-square test of fit for structural
      equation models (with or without latent variables). The
      chi-square statistic is asymptotically correct; however, it does
      not behave as expected in small samples (Kenny & McCoach, 2003)
      and/or when the model is complex (Curren, Bollen, Paxton & Kirby,
      2002). Thus, particularly in situations where the ratio of the
      number of parameters estimated to sample size is relatively
      small, the chi-square test will tend to overreject correctly
      specified models. To obtain a closer approximation to the
      distribution of the chi-square statistic, Swain (1975) developed
      a correction; this scaling factor, which converges to 1
      asymptotically, is multiplied with the chi-square statistic. The
      resulting correction better approximates the chi-square
      distribution resulting in more appropriate Type 1 reject error.

To install swain, simply type -ssc install swain- from the Stata command line.

*How to estimate instrumental variable models via -sem-.

E.g. 1

ivregress 2sls y (x = z1 z2) z3

can be estimated in -sem- as:

sem (x <- z1 z2 z3) (y <- x z3) , cov(e.x*e.y)
cov(e.x*e.y) allows cross equations disturbances of x and y to correlate (and it the Hausman test).

E.g. 2

reg3 (y = x1 z3) (x = z1 z2 z3 x2) (x2 = z4 z3)

can be estimated in -sem- as:

sem (y <- x1 z3) (x <- z1 z2 z3 x2) (x2 <- z4 z3), cov(e._OEn, unstructured)

The cov option above allows all cross-equation disturbances of endogenous variables correlate. Thus, the Hausman test is the Wald test:

test (_b[cov(e.y,e.x1):_cons] = 0) (_b[cov(e.y,e.x2):_cons]=0) (_b[cov(e.x1,e.x2):_cons]=0)

Note, the Hansen-Sargan overidentification statistic in 2sls is the chi-square test of fit in sem (reported as the "LR test of model vs. saturated model" on the bottom of the output). In small sample size situations where there are many parameters to be estimated, -swain- will correct this overidentification statistic.

Another advantage of using -sem- is that there are score tests (modification indices or Langrange Multiplier tests) available after estimation (-estat mindices-) that will identify model constraints that are inconsistent with the data.



John Antonakis
Professor of Organizational Behavior
Director, Ph.D. Program in Management

Faculty of Business and Economics
University of Lausanne
Internef #618
CH-1015 Lausanne-Dorigny
Tel ++41 (0)21 692-3438
Fax ++41 (0)21 692-3305

Associate Editor
The Leadership Quarterly

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