the contribution of γM(3) is minuscule in explaining aggregate employment volatility in CW’s
simulated data, while it is large and economically significant in our findings. Simply put, the
reported values of neither R2 nor λ change when adding the nonlinear term in CW’s comment,
while they change substantially in our setting (λ falls and R2 rises).
The first column in Table 1 is based on Table 1a in CW (2003). It is apparent from their
table that the R2 reported when estimating (2) is the same with or without the third moment of
gaps: 0.90. Similarly, the estimated value of the non-linear parameter γ, even though statistically
significant,5 is economically irrelevant, as the adjustment speed varies by less than 0.013 over the
relevant range of gaps.6 By contrast, in the corresponding exercise in Table 3 of CEH, reported in
the second column of Table 1 here, the R2 increases by 0.15 when adding a non-linear parameter
and the variation of the speed of adjustment over the relevant range is more than ten times as large
as that in CW’s model.
The economic irrelevance of the non-linearities estimated by Cooper and Willis is even more
striking in the 2002 version of their comment, where they used a more realistic value for the first
order correlation of productivity shocks (0.81 at an annual level).7 There they report anR2 of 0.97,
both for the model with and without the non-linear parameter,8 and the adjustment speed implied
by their non-linear model varies only by 0.005 over the relevant range of values taken by the gap.
5The statistical significance they find possibly reflects the fact that they use time series with 1000 observations in
their simulations, while CEH’s estimates are based on 35 observations.
6Where the ‘relevant range’ is defined as μG ± 2σG, with μG and σG denoting the mean and standard deviation of
the cross-section of static gaps, respectively. A tedious but straightforward calculation from first principles shows that
2 _ ∑k≥ 0 dk 2
σ G (1 - α)2 σ ’
with:
dk = λ+ρ 1(1 - λ) k + [1
Gλ
λ+ρ-1
ρk,
where G = (1 - δ) / (1 - δρ), λ denotes the speed of adjustment in the partial adjustment representation of the quadratic
adjustment cost model, δ denotes the discount rate that results in this model when calculating the (correct) dynamic
target as a present value of future static targets, σε denotes the standard deviation of firm-specific productivity shocks
and α is defined on p. 23 in CW (2002).
7As we pointed out the errors in the first and second versions of CW’s critique, they reacted by looking for a new
parameter configurations and new model specifications that might help their case. Their lack of success, despite two
major revisions of their original comment, possibly is the best evidence of the robustness of our findings.
8This is for the benchmark with high adjustment costs. For the benchmark with low adjustment costs, both values
reported for R2 are 1.00.