Insurance within the firm

B Appendix: Covariance estimation

For each firm in the sample we obtain a consistent estimate of ∆ε_jγ as the residual from
the IV regression (8). For an unbalanced sample of firms observed for at most T periods,

( ∆εjτ
tejT-i

...

∖ ^δ¾1

If the ∆εj_t observation is missing, it is replaced by zero. Conformably with △ ε₁■, define
with d_j a vector of 0-1 dummy variables. The dummy is 0 if the observation for ∆εj_t is
missing, 1 otherwise. All the autocovariances of the type E (∆ε_ja∆εj) are consistently
estimated by the sample analogs collected in the following autocovariance matrix:

F          F

^c = £△ -△ ε₁-./∑ ^dd

J=I           j=l

where F is the number of firms always present in the data set and ./ denotes an element-
by-element division.

Define with m the vector of all the distinct elements of C, i.e. m = vech (C). Since C
is a symmetric matrix, the number of distinct elements in it is ʃ(ʃ+¹). Conformably with
m, define m_j = vech (△ ε₁△ ε',j), and D =vech ^∑j=ι djdj). The standard errors of the
ɪɪɪɪɪ autocovariances can be retrieved by the variance-covariance matrix of C, i.e.:

M

^v = ∑ [⁽m - ^{m) (}m - ^m)^z. * ^dι^dj] .^/dD
J=I

The standard errors of the estimated moments are simply the square roots of the ele-
ments in the main diagonal of V. A similar strategy is used to obtain an estimate of V
(and corresponding standard errors) for workers’ earnings.

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