14.661 F17 Lecture Note: MaC-Metrics

MaC-metrics MIT 14.661 (FALL 2017) A Life’s Identification Challenges • The iconic empirical life-cycle labor supply function looks like this: ln hit = µi + (⇢ where = 1 2 1 r)t + ln wit + uit (1) and uit is a “tacked-on” error • We’re after , the ISE, but it’s not easily captured – Estimation of (1) isn’t shovel-ready: the control variable µi isn’t found in the CPS. A function of the marginal utility of wealth, this variable is negatively correlated with wages, wit – We have limited data on hourly wages; instead, we work with average hourly earnings, AHEit ⌘ hyit . So we’re naively regressing hours it 1 worked on (hours worked) ; The results might not be pretty; rather, they’re pretty negative! B Problems and Solutions • Analysis of covariance (deviations from means) or di↵erencing kills the unobserved fixed e↵ect (whew!) • These transformations also aggravate the bias from our poorly measured wage variable. The bias here is worse than classical attenuation bias: the fact that mismeasured hours appears on both sides of the equation of interest induces a powerful negative term known as “division bias” • We might instead try grouping strategies, as in Angrist (1990, 1991). This approach potentially kills the measurement error as well as the fixed e↵ect. I like that a lot! 1 C Division Bias Details Suppose the labor supply equation of our heart’s desire is ⇤ ln h⇤it = ↵ + ln wit + uit (2) For the purposes of this discussion, we’ll start by assuming we’d be happy to estimate (2) by OLS. The empirical supply function uses AHE with well-measured hours ⇤ wit = yit , h⇤it where yit is annual earnings. This is the hourly wage for those who are paid hourly, and its a notional time price for others. Either way, we assume this correctly-measured AHE is what consumers use to make work decisions. In practice, however, hours are poorly measured: hit = h⇤it where it it , is proportional classical measurement error. Then ln hit = ln h⇤it + ⌘it , where ⌘it = ln it . (3) This implies that ln wit = ln yit ln hit = ln yit ln h⇤it ⇤ ⌘it = ln wit ⌘it (4) Substituting on both sides of (2), we now have ln hit = ↵ + (ln wit + ⌘it ) + uit + ⌘it = ↵ + ln wit + {uit + (1 + )⌘it } (5) Without worrying about the fixed e↵ect, the OVB in OLS estimates of (5) is OV B = ⇤ Cov(ln wit ⌘it , (1 + )⌘it ) 2 ln w = (1 + ) 2 ⌘ 2 ln w which is big-time bad, even compared to the usual m.e. attenuation bias. (Note that 2 ⌘ 2 ln w is one minus the signal-to-noise ratio for log wages.) Analysis of covariance aggravates division bias To kill the fixed e↵ect, you might di↵erence or deviate from means. Suppose you have a two-period panel, so (2) with fixed e↵ects becomes OLS on first di↵s: ln h⇤it = ⇤ ln wit + uit , ⇤ ln wit ⌘it . while the noisy wage becomes ln wit = 2 Assuming m.e. is serially uncorrelated, the variance of ⌘it is 2 ⌘2 . ⇤ = wi⇤ . Then Wages, by contrast, are highly persistent. Suppose, wit ln wit = ⌘it . In other words, the change in wages is pure noise. Then we have OV B = (1 + )2 2 ⌘2 2 ⌘ = (1 + ) so di↵erencing here makes matters substantially worse. Research on measurement error in hours and wages bears this out: measured wage changes are noisy indeed (see, e.g., Bound and Krueger, 1991). See MaCurdy, Thomas E. "An Empirical Model of Labor Supply in a Life-Cycle Setting." Journal of Political Economy 89, no. 6 (1981): 1076-1078. See Angrist, Joshua D. "Grouped-data estimation and testing in simple labor-supply models." Journal of Econometrics 47, no 2-3 (1991): 256-259. See Card, David. "Intertemporal Labour Supply: An Assessment." In Advances in Econometrics: Sixth World Congress, Vol II. Cambridge University Press, 1996. pp. 59-60. 3