文档介绍：Recitation Notes 5
Konrad Menzel
October 13, 2006
1 Instrumental Variables (continued)
Omitted Variables and the Wald Estimator
Consider a Wald estimator for the Angrist (1991) approach to estimating the intertemporal elasticity of
substitution of labor supply for the regression
hit = α+ βwit + εit
and assume that there are actually aggregate shocks to wages, in a way such that changes in the workers’
marginal utility of wealth generate a linear trend in labor supply, γ0 + γ1t. Therefore, the ”right”
regression we should actually run is
hit = γ0 + γ1t + βwit + εit
Then, using some - arbitary - aggregate binary variable Z1t as an instrumental variable gives - by the
2SLS formula from last week - the Wald estimand
E E
ˆ[hit|Z1t = 1] −[hit|Z1t = 0]
plimN β1 =
E[wit|Z1t = 1] − E[wit|Z1t = 0]
E[wit|Z1t = 1] − E[wit|Z1t = 0] β+ E[tγ+ εit|Z1t = 1] − E[tγ+ εit|Z1t = 0]
=
E[wit|Z1t = 1] − E[wit|Z1t = 0]
Assuming that Cov(Z1t, εit) = 0, this simplifies to
E E
ˆ[t|Z1t = 1] −[t|Z1t = 0]
plimN β1 = β+ γ1 =: β+ ω1γ1
E[wit|Z1t = 1] − E[wit|Z1t = 0]
If . we are looking at a balanced panel for the years 1969 to 1979 and Z1t = 1l{t > 1974}, we can
calculate that
75 + 76 + 77 + 78 + 79 69 + 70 + 71 + 72 + 73 + 74
E[t|Z1 = 1] − E[t|Z1 = 0] = −= 77 − =
t t 5 6
So if there is an increase in wages over time whic