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520 Chapter 12 Instrumental Variables Regression

Consider the IV regression model in Equation (12.12) with a single X and a single W:

Yi = b0 + b1Xi + b2Wi + ui. (12.22)

We replace IV Regression Assumption #1 in Key Concept 12.4 [which states that E(ui ͉ Wi) = 04
with the assumption that, conditional on Wi, the mean of ui does not depend on Zi:

E(ui 0 Wi, Zi) = E(ui 0 Wi). (12.23)

Following Appendix 7.2, we further assume that E(ui ͉ Wi) is linear in Wi, so
E(ui ͉ Wi) = g0 + g2Wi, where g0 and g2 are coefficients. Letting ei = ui - E(ui ͉ Wi, Zi)
and applying the algebra of Equation (7.25) to Equation (12.22), we obtain

Yi = d0 + b1Xi + d2Wi + ei, (12.24)

where d0 = b0 + g0 and d2 = b2 + g2. Now E(ei 0 Wi, Zi) = E3ui - E(ui 0 Wi, Zi) 0 Wi, Zi4 =
E(ui 0 Wi, Zi) - E(ui 0 Wi, Zi) = 0, which in turn implies corr(Zi, ei) = 0. Thus IV Regression

Assumption #1 and the instrument exogeneity requirement (condition #2 in Key Concept 12.3)
both hold for Equation (12.24) with error term ei, Thus, if IV Regression Assumption #1 is
replaced by conditional mean independence in Equation (12.23), the original IV regression
assumptions in Key Concept 12.4 apply to the modified regression in Equation (12.24).

Because the IV regression assumptions of Key Concept 12.4 hold for Equation (12.24),
all the methods of inference (both for weak and strong instruments) discussed in this chap-
ter apply to Equation (12.24). In particular, if the instruments are strong, the coefficients
in Equation (12.24) will be estimated consistently by TSLS and TSLS tests, and confidence
intervals will be valid.

Just as in OLS with control variables, in general the TSLS coefficient on the control
variable W does not have a causal interpretation. TSLS consistently estimates d2 in Equa-
tion (12.24), but d2 is the sum of b2, the direct causal effect of W, and g2, which reflects the
correlation between W and the omitted factors in ui for which W controls.

In the cigarette consumption regressions in Table 12.1, it is tempting to interpret the
coefficient on the 10-year change in log income as the income elasticity of demand. If, how-
ever, income growth is correlated with increases in education and if more education reduces
smoking, income growth would have its own causal effect (b2, the income elasticity) plus an
effect arising from its correlation with education (g2). If the latter effect is negative (g2 6 0),
the income coefficients in Table 12.1 (which estimate d2 = b2 + g2) would underestimate
the income elasticity, but if the conditional mean independence assumption in Equation
(12.23) holds, the TSLS estimator of the price elasticity is consistent.