∣τI > 1 are uncorrelated with the current unexplained component of value added growth, if
this is an MA(I) process as in (10). Thus in estimation only ∆εjt-1 and ∆εjε∣1 are used
as instruments. Equations (16)-(17) can be used to identify the Hrst parameter of interest
bυ with one overidentiHcation restriction. This can be tested with standard methods.
IdentiHcation of bu proceeds along similar lines. Start from (14), subtract bu∆εjt on both
sides and multiply both sides by the term (∆ε∙g+ɪ + ∆εjt + ∆εjt-ι). Taking expectations
it yields the moment condition:
E [(∆ε.7^+1 + t + -i) (ɪht — buδ^t)] = 0 (18)
Equation (18) identiHes the second parameter of interest bu. Similarly to the moment
conditions (16) and (17), the intuition for this is that after Hltering the unexplained com-
ponent of earnings growth ∆ω1-jt by the unexplained component of value added growth
∆εjt (weighted by a factor bu, the extent of permanent insurance), what is left is uncor-
related with an MA(2) term centered in ∆εjt with unity coefficients.10 Thus one can use
(∆εjt+1 + ∆εjt + ∆εjt-ι) as an instrument. By identical logic, any other MA term that
contains (∆εjt+1 + ∆εjt + ∆εjt-ι) is a valid instrument. For instance, ^=_q ∆εj∣+к (for
any q > 2) is a valid instrument as well. It follows that the model can be tested via
these additional overidentifying restrictions. In the empirical analysis, we use a set of three
instruments (corresponding to q = 1,2,3). This gives us two overidentifying restrictions.
Note that in (18) and (16)-(17) different instruments identify different parameters, and
that instruments that are valid in one equation are not valid in the other. Also, note
that if we had shocks to value added in levels (i.e., estimates of ε-jt), we could have many
more instruments available to estimate bu in (18). In fact, ε∙g+τ (τ > 1) will all be valid
instruments, as can be easily checked. Finally, the moment conditions derived above are
10To see why this is so, consider equation (14) and rewrite it as:
ʌ^ʊt — K^'- jt + [(ðʃʊ ðn) ʌg;t + ^⅛'t]
In an OLS regression of ∆u⅛∙t on ∆εjt the latter is endogenous because correlated with the error term (the
term in square brackets) via ∆vjt. However, the variable (∆εjt+ι + ∆ε;t + ∆εjt-ι) is a valid instrument,
because correlated with ∆εjt (via the permanent component uj) and uncorrelated with the error term,
as (∆εjt+1 + ∆εjt + ∆εjt-ι) equals (ujt+1 + Ut + ujt-ι) + (¾t+1 — vjt_a) as can be checked after some
algebra.
13