Balanced Variable Addition in Linear Models

Abstract

This paper studies what happens when we move from a short regression to a long regression in a setting where both regressions are subject to misspecification. In this setup, the leastsquares estimator in the long regression may have larger inconsistency than the least-squares estimator in the short regression. We provide a simple interpretation for the comparison of the inconsistencies and study under which conditions the additional regressors in the long regression represent a "balanced addition" to the short regression. 11 12 β 2 Proof. This follows directly from (9) and (10), using the fact that E(X ) = 0.The next result gives the inconsistencies ofβ 1u andβ 1r when X and δ are random.

Bias of OLS Estimators due to Exclusion of Relevant Variables and Inclusion of Irrelevant Variables

“…The second result of this paper is a reiteration of the results in Clarke () and De Luca et al . () that inclusion of some omitted variables will not necessarily reduce the magnitude of OVB when some other relevant variables remain omitted. Moreover, we cannot derive sufficient conditions for bias reduction using sign restrictions only.…”

“…This rather surprising result has been highlighted in Clarke () and De Luca et al . (). It arises from the fact that both the short and the long regressions remain mis‐specified, i.e.…”

“…But if, as seems quite realistic, the long regression also suffers from the problem of omitted variables, then it is no longer possible to ensure bias reduction unambiguously by inclusion of omitted variables. One can easily generalize this to see that, in a multivariate setting, inclusion of omitted variables will not necessarily lead to a reduction in OVB if some variables remain omitted, a result that was discussed in Clarke () and has been highlighted in more recent work (De Luca et al ., )…”

Estimating Poverty for Refugee Populations: Can Cross-Survey Imputation Methods Substitute for Data Scarcity?

“…larger inconsistency (De Luca, Magnus, and Peracchi, 2018). Also note that the standard errors around the true poverty estimates are larger than those for the imputation-based estimates, since the latter are model-based; see Dang et al (2019) for more discussion.…”

On Using the t-Ratio as a Diagnostic

“…This, however, is only true when we compare the restricted model with an unrestricted model which coincides with the DGP. If, which is much more likely, we compare two models one of which is small (the restricted model) and the other is somewhat larger (the unrestricted model), but both are smaller than the DGP, then the estimator from the unrestricted model is also biased and, in fact, this bias may be larger than the bias from the restricted model; see (De Luca et al 2018).…”