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Testing for a common latent variable in a linear regression

Wittenberg, Martin (2007): Testing for a common latent variable in a linear regression.

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Abstract

We present a test of the hypothesis that a subset of the regressors are all proxying for the same latent variable. This issue will be of interest in cases where there are several correlated measures of elusive concepts such as misgovernance or corruption; in analyses where key variables such as income are not measured at all and one is forced to rely on various proxies; and where the key regressors are badly measured and one is trying to extract a stronger signal from the regression by adding additional proxies as suggested by Lubotsky and Wittenberg (2006).

We apply this test in three contexts, each characterised by a different estimation challenge arising from data limitations. We reexamine Mauro's (1995) use of three institutional quality measures in his study of corruption and growth. Here several variables, each potentially measured with error, may all be proxies for a single factor: the quality of governance. Our test suggests that the latent variable is driven primarily by the “red tape” measure, rather than the “corruption” variable on which Mauro focuses.

Secondly, we look at the correlates of body mass among black South African women. The key variable of interest, namely “wealth” is not measured at all. Consequently we construct an index from a series of asset variables as suggested by Filmer and Pritchett (2001). Our test shows that some assets have independent impacts on the dependent variable. Once this is recognised the “asset index” comes apart.

Finally we analyse the determinants of sleep among young South Africans. The income variable in the survey is badly measured and we supplement it with asset proxies. The test again suggests that some assets are not proxying for the badly measured income variable. We can nevertheless get a substantially stronger signal on the income variable.

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