i.e.:
ʌɛ jt — uj t + Δ^'jt
(10)
(11)
∆ωijt — buj t + b∆vj t + A(L,p)ξij t + A(L,p)∆μijt
where, from equations (8) and (9), ∆εjt ≡ A(L,p)∆yj∙t — ∆zfjtθ and ∆ω⅛j∙t ≡ A(L,p')∆wijt —
∆a'ijtλ — ∆z,jtbθ.
Under our hypotheses, the serial correlation properties of ∆εjt 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 ∆ωljt 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 ∆sjt are standard and are reported below in the simple case of covariance sta-
tionarity:
|
⅛ + 2σ≡ |
for |
τ — 0 | |
|
E (∆εjt∆εjt_— — < |
—σl |
for |
∣τ I — 1 |
|
0 4 |
for |
∣τI > 1 |
(12)
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 σ2a 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 σ2a .8
'In fact — E (∆εj∙t∆εjt-ʃ) identifies the variance of the transitory shock σ^ with ∣τ∣ = 1 for all t, while
E [∆εj∙( (∆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 (∆εj∙s∆εj∙s+ι) and E [∆εjs (∆εj∙s+ι + ∆ε;s + ∆εj∙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 (∆εj∙t∆εj∙t-τ) = σ^ + σf for ∣τ∣ = 1, but E [∆εj( (∆εj∙t-1 + ∆ε;t + ∆εjt+ι)] still identifies the variance
10