indirect effect through πt = qk-,t1 (θt, θP , yt, Ik,t). To accomplish this, we exploit our knowledge
of qk-,t1 (θt, θP , yt , Ik,t) to write:
fk,s (θt, Ik,t,θP ,πt,νk,t) = fkfs (θt,Ik,t,θP ,yt,νk,t)lyt.q-1 (θt,θp ,yt,Ik,t)=πt
where, on the right-hand side, we set yt so that the corresponding implied value of πt matches
its value on the left-hand side. This does not necessarily require qk-,t1 (θt, θP, yt, Ik,t) to be
invertible with respect to yt , since we only need one suitable value of yt for each given
(θt , θP , Ik,t , πt) and do not necessarily require a one-to-one mapping. By construction, the
support of the distribution of yt conditional on θt , θP , Ik,t , is sufficiently large to guarantee
the existence of at least one solution because, for a fixed θt , Ik,t , θP , variations in πt are
entirely due to yt . We present a more formal discussion of our identification strategy in
Section 3.3 of the Web appendix.
In our empirical analysis, we make further parametric assumptions regarding fk,s and qk,t ,
which open the way to a more convenient estimation strategy to account for endogeneity.
The idea is to assume that the function qk,t (θt, θP, yt, πt) is parametrically specified and
additively separable in πt , so that its identification follows under standard instrumental
variables conditions. Next, we replace Ik,t by its value given by the policy function in the
technology
θk,t+1 = fk,s (θt, qk,t (θt,θP,yt,πt) ,θP,πt, νk,t) .
Eliminating Ik,t solves the endogeneity problem because the two disturbances πt and νk,t are
now independent of all explanatory variables, by assumption if the πt are serially indepen-
dent. Identification is secured by assuming that fk,s is parametric and additively separable
in νk,t (whose conditional mean is zero) and by assuming a parametric form for fπt (πt), the
density of πt . We can then write:
E [θk,t+ι∖θt,θP,yt] = У fk,s (θt,qk,t (θt,θp,yt,πt) ,θp,πt, 0) f∏t (πt) dπt ≡ fk,s (θt,θp,yt,β) .
The right-hand is now known up to a vector of parameters β which will be (at least) locally
identified if it happens that ∂fk,s (θt,θP,yt,β) /∂β evaluated at the true value of β is a
vector function of θt, θP , yt that is linearly independent. Section 4.2.5 below describes the
specific functional forms used in our empirical analysis, and relaxes the assumption of serial
independence of the πt .
21
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