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From |
"David M. Drukker, StataCorp" <[email protected]> |

To |
[email protected] |

Subject |
RE:st: Part II: An error in the xtabond-description |

Date |
Mon, 23 Aug 2004 19:02:40 -0500 |

Giovanni Bruno <[email protected]> pointed out that there is an issue with > [the formula for] S2 (last formula at page 32 in the xt manual), which > is based on A2 that necessarily uses 1-step residuals. This is not > the 2-step Sargan in Arellano and Bond (1991) paper, formula (10) p. > 282 that employs 2-step residuals. Giovani is correct that equation (10) in Arellano and Bond (1991) uses the two-step residuals to define the Sargan statistic for the two-step estimator. However, in this equation Arellano and Bond (1991) deviated from the common definition of the two-step Sargan statistic as the value of the GMM objective function at its two-step optimum. The definition implemented in -xtabond- is the more common definition of the Sargan-Hansen statistic. To clarify define, A_2 = (\sum_i Z_i'e1_i'e1_i Z_i)^{-1} S_2 = (\sum_i e2_i'Z_i)A_2(\sum_i Z_i' e2) where e1_i are one-step residuals e2_i are two-step residuals Z_i is the matrix of instruments for each panel i. The coefficients beta that minimize S_2 with residuals e2_i are the GMM estimates. This is the Hansen (1982) formulation in which the weight matrix A_2 is a function of the residuals from the previous step, but not the current step. As Mark Schaffer <[email protected]> pointed out, the Sargan statistic with A_2 as function of e2_i still converges to a chi-squared distribution, so there is nothing wrong with equation in Arellano and Bond (1991). However, as Mark also noted, that definition differs from the standard textbook version in which A_2 is a function of e1_i which is discussed in Hayashi(2000) and Davidson and MacKinnon(2004, 1993). Thus, the A_2 as a function of e1_i is well documented. --David [email protected] References Arellano, M. and Bond, S. (1991) "Some tests of specification for panel data: Monte Carlo evidence and an application to Employment Equations", The Review of Economic Studies, 58(2) 277-297. Davidson, R and MacKinnon J. (2004) Econometric Theory and Methods. New York:Oxford University Press. Davidson, R and MacKinnon J. (1993) Estimation and Inference in Econometrics. New York:Oxford University Press. Hansen, L.P. (1982) "Large Sample properties of GMM estimators" Econometrica 50(4) 1029-1054. Hayashi, F. (2000) Econometrics. New York:Princeton University Press. * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

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