Insurance within the firm

shocks (“transitory full insurance”, and E (AsjtA^v_j't) = b_uσ²_a)∙

4 Identification strategy

Without further restrictions, from equation (13) we cannot separately identify b_u and b_υ,
nor can we gauge whether b_u = b_υ = b. To see how identification of the relevant parameters
is achieved, start from the general case b_u = b_υ in (11):

∆<^ωij = = ^b_uUjt T ^b_υΔ^,Vjt T $ijt                             ⁽¹4⁾

where ⅛_z∙_t = A(L,p)ξ_ij_t T A(Lμp)Δμ_ij∙_t. Subtract b_υ∆sj_t from both sides to obtain:

∆ωijt — b_υ∆εjt = (b_u, — b_υ)u_jt T A[jt                         (15)

Multiply both sides by ∆εjt-1 and ∆sj_t+ι, respectively, and take expectations to yield the
two moment conditions:

^e [ʌɛjt+1 (∆ω-ij — ^{— b}_υ∆εjt)] = 0                         ⁽¹⁶⁾

^E tʌɛji-1 (∆^ωvt ^{— b}υ∆εj)^] = ^{0                      (17)}

Intuitively, equations (16) and (17) tell us that once one filters the unexplained compo-
nent of earnings growth ∆ω-ij_t by the unexplained component of value added growth ∆sj_t
(weighted by a factor b_υ, the extent of transitory insurance), what is left is uncorrelated
with the past and future unexplained component of value added growth. In an OLS re-
gression of ∆ω-ij_t on ∆sj_t the latter is obviously endogenous because correlated with the
right hand side of equation (15) via Uj_t∙⁹ However, the first lag and lead of ∆sj_t will be
valid instruments, because correlated with ∆sj_t (via the transitory component) and uncor-
related with the error term. At least in principle, all the variables ∆sj_t-_τ (with ∣τ∣ ≥ 1)
are uncorrelated with the error term. However, the instruments that satisfy ∆sj_t-_τ with

⁹It is worth noting that OLS estimation provides unbiased and consistent estimation if b_u = b_υ = b. Thus
an exogeneity test for ∆ε_j∙_t can be used to check whether b,_i = b_υ = b.

12

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