Insurance within the firm

ʌɛ jt —  ^uj t + Δ^'jt

∆ω_ijt — bu_{j t} + b∆v_{j t} + A(L,p)ξ_{ij t} + A(L,p)∆μ_ijt

where, from equations (8) and (9), ∆ε_jt ≡ A(L,p)∆y_j∙t — ∆z^fj_tθ and ∆ω⅛j∙t ≡ A(L,p')∆wijt —
∆a'_ijtλ — ∆z^,_jtbθ.

Under our hypotheses, the serial correlation properties of ∆εj_t are well defined: since
it follows an MA(1) process (an assumption confirmed by the empirical analysis below),
autocorrelations at the second or higher order are all zero. On the other hand, the serial
correlation properties of ∆ω_ljt depend on the order p of the lag polynomial A(L,p). In gen-
eral, ∆ωijt will follow an MA(p + 1) process. The restrictions on the variance-covariance
matrix of ∆sj_t are standard and are reported below in the simple case of covariance sta-
tionarity:

This simple structure has the obvious advantage that one can identify the variance
of the transitory shock and that of the permanent shock to firm performance using only
information on the variance and the first-order autocovariances of ∆εjt- From equation
(12) one can immediately recover σ²_a and σ².⁷ Measurement error makes this identification
strategy no longer operational; however, as we show later, given the administrative nature
of our data, it is reasonable to assume that measurement error is negligible both at firm and
at worker level. It is straightforward to show that the presence of classical measurement
error in firm data increases the estimate of σ² but has no effect on that of σ²_a .⁸

'In fact — E (∆ε_j∙t∆εjt-ʃ) identifies the variance of the transitory shock σ^ with ∣τ∣ = 1 for all t, while
E [∆ε_j∙( (∆εjt+ι + ∆ε;t + ∆εjt-ι)] identifies the variance of the permanent shock ≤ for all t. If there is
covariance non-stationarity, then the expressions —E (∆ε_j∙_s∆ε_j∙_s+ι) and E [∆εj_s (∆ε_j∙_s+ι + ∆ε;_s + ∆ε_j∙_s-ι)]
identify the variance of the transitory shock and the variance of the permanent shock at time s, respectively.

⁸If a classical measurement error ∏t ~ i.i.d. (θ, σ^) is present, it is straightforward to show that
— E (∆ε_j∙t∆ε_j∙t-_τ) = σ^ + σf for ∣τ∣ = 1, but E [∆εj( (∆ε_j∙t-1 + ∆ε;t + ∆εjt+ι)] still identifies the variance

10

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