Proxy and instrumental variable methods in a regression model with one of the regressors missing

Rabi N. Bhattacharya; Dilip K. Bhattacharyya

doi:10.1006/jmva.1993.1074

Proxy and instrumental variable methods in a regression model with one of the regressors missing

Rabi N. Bhattacharya, Dilip K. Bhattacharyya

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

Suppose Y has a linear regression on X₁, X₂, but observations are only available on (Y, X₁). If large scale data on (X₁, X₂) are available, which do not include Y, and if the regression of X₂, given X₁, is nonlinear, then one may estimate the regression coefficients of Y by using the proxy g(X₁) {colon equals} E(X₂|X₁) for X₂, or an instrument φ(X₁) which is uncorrelated with X₂. Both methods provide estimators which are asymptotically normal around the true parameter values under appropriate assumptions. A computation of the optimal instrument is provided, and the asymptotic relative efficienties of the two types of estimators compared.

Original language	English (US)
Pages (from-to)	123-138
Number of pages	16
Journal	Journal of Multivariate Analysis
Volume	47
Issue number	1
DOIs	https://doi.org/10.1006/jmva.1993.1074
State	Published - Oct 1993
Externally published	Yes

Keywords

Efficiency
Linear regression
Nonlinear regression
Optimal instrument
Proxy

ASJC Scopus subject areas

Statistics and Probability
Numerical Analysis
Statistics, Probability and Uncertainty

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10.1006/jmva.1993.1074

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N2 - Suppose Y has a linear regression on X1, X2, but observations are only available on (Y, X1). If large scale data on (X1, X2) are available, which do not include Y, and if the regression of X2, given X1, is nonlinear, then one may estimate the regression coefficients of Y by using the proxy g(X1) {colon equals} E(X2|X1) for X2, or an instrument φ(X1) which is uncorrelated with X2. Both methods provide estimators which are asymptotically normal around the true parameter values under appropriate assumptions. A computation of the optimal instrument is provided, and the asymptotic relative efficienties of the two types of estimators compared.

AB - Suppose Y has a linear regression on X1, X2, but observations are only available on (Y, X1). If large scale data on (X1, X2) are available, which do not include Y, and if the regression of X2, given X1, is nonlinear, then one may estimate the regression coefficients of Y by using the proxy g(X1) {colon equals} E(X2|X1) for X2, or an instrument φ(X1) which is uncorrelated with X2. Both methods provide estimators which are asymptotically normal around the true parameter values under appropriate assumptions. A computation of the optimal instrument is provided, and the asymptotic relative efficienties of the two types of estimators compared.

KW - Efficiency

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KW - Nonlinear regression

KW - Optimal instrument

KW - Proxy

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Proxy and instrumental variable methods in a regression model with one of the regressors missing

Abstract

Keywords

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