Testing for a common latent variable in a linear regression: Or how to "fix" a bad variable by adding multiple proxies for it
We analyse models in which additional "controls" or proxies are included in a regression. This might occur intentionally if there is significant measurement error in a key regressor or if a key variable is not measured at all. We develop a test of the hypothesis that a subset of the regressors are all proxying for the same latent variable and we show how an estimate of the structural coefficient might be obtained more efficiently than is available in the current literature. We apply the procedure to the determinants of sleep among young South Africans. We show that the income variable in the time use survey is badly measured. Nevertheless the measured impact of income on sleep is significant and amounts to 35 minutes per day between children with the median income and those in the topmost income bracket. Including a variety of asset proxies increases the estimated size of the coefficient enormously. The specification tests indicate that some of the asset proxies, however, have independent effects. Access to electricity, in particular, is not simply proxying for income. Instead it seems to be capturing access to various forms of entertainment, such as television. Even when this independent effect is properly accounted for, the size of the income coefficient is still 40% to 100% larger than in the specifications without the proxies.
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- Martin Browning & S¯ren Leth-Petersen, 2003. "Imputing consumption from income and wealth information," Economic Journal, Royal Economic Society, vol. 113(488), pages F282-F301, 06.
- Martin Wittenberg, 2005. "The school day in South Africa," SALDRU/CSSR Working Papers 113, Southern Africa Labour and Development Research Unit, University of Cape Town.
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"Sleep and the Allocation of Time,"
NBER Working Papers
2988, National Bureau of Economic Research, Inc.
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- Darren Lubotsky & Martin Wittenberg, 2006.
"Interpretation of Regressions with Multiple Proxies,"
The Review of Economics and Statistics,
MIT Press, vol. 88(3), pages 549-562, August.
- Darren Lubotsky & Martin Wittenberg, 2001. "Interpretation of Regressions with Multiple Proxies," Econometrics 0110005, EconWPA.
- Roland G. Fryer & Paul S. Heaton & Steven D. Levitt & Kevin M. Murphy, 2005. "Measuring the Impact of Crack Cocaine," NBER Working Papers 11318, National Bureau of Economic Research, Inc.
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