On the Bias and MSE of the IV Estimator Under Weak Identification
AbstractIn this paper we provide further results on the properties of the IV estimator in the presence of weak instruments. We begin by formalizing the notion of weak identification within the local-to-zero asymptotic framework of Staiger and Stock (1997), and deriving explicit analytical formulae for the asymptotic bias and mean square error (MSE) of the IV estimator. These results generalize earlier findings by Staiger and Stock (1997), who give an approximate measure for the asymptotic bias of the two-stage least squares (2SLS) estimator relative to that of the OLS estimator. Because our analytical formulae for bias and MSE are complex functionals of confluent hypergeometric functions, we also derive approximations for these formulae which are based on an expansion that allows the number of instruments to grow to infinity while keeping the population analogue of the first stage F-statistic fixed. In addition, we provide a series of regression results that show this expansion to give excellent approximations for the bias and MSE functions in general. These approximations allow us to make several interesting additional observations. For example, when the approximation method is applied to the bias, the lead term of the expansion, when appropriately standardized by the asymptotic bias of the OLS estimator, is exactly the relative bias measure given in Staiger and Stock (1997) in the case where there is only one endogenous regressor. In addition, the lead term of the MSE expansion is the square of the lead term of the bias expansion, implying that the variance component of the MSE is of a lower order relative to the bias component in a scenario where the number of instruments used is taken to be large while the population analogue of the first stage F-statistic is kept constant. One feature of our approach which ties our findings to the earlier IV literature is that our results apply not only to the weak instrument case asymptotically, but also to the finite sample case with fixed (possibly good) instruments and Gaussian errors, since our formulae correspond to the exact bias and MSE functionals when a fixed instrument/Gaussian model is assumed.
Download InfoIf you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
Bibliographic InfoPaper provided by Econometric Society in its series Econometric Society World Congress 2000 Contributed Papers with number 1622.
Date of creation: 01 Aug 2000
Date of revision:
Contact details of provider:
Phone: 1 212 998 3820
Fax: 1 212 995 4487
Web page: http://www.econometricsociety.org/pastmeetings.asp
More information through EDIRC
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Hillier, Grant H & Kinal, Terrence W & Srivastava, V K, 1984. "On the Moments of Ordinary Least Squares and Instrumental Variables Estimators in a General Structural Equation," Econometrica, Econometric Society, vol. 52(1), pages 185-202, January.
- Ullah, Aman, 1974. "On the sampling distribution of improved estimators for coefficients in linear regression," Journal of Econometrics, Elsevier, vol. 2(2), pages 143-150, July.
- Hahn, Jinyong & Kuersteiner, Guido, 2002. "Discontinuities of weak instrument limiting distributions," Economics Letters, Elsevier, vol. 75(3), pages 325-331, May.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Christopher F. Baum).
If references are entirely missing, you can add them using this form.