Appendix 3.B
Kolmogorov-Smirnov test.
To test for differences in all moments of the conditional distributions (showed in fig B1 and
B2) we can use the Kolmogrov-Smirnov test of first order stochastic dominance. For this we
refer to the two conditional cumulative distribution functions of productivity, F and G. F
corresponds to the of firms engaged in the international activity we are interested, for
example firms that both import and export ,and G the comparison group, firms the exhibit
an IE Index equal to zero. First-order stochastic dominance of F with respect to G is defined
as: F(z)-G(z)≤0 uniformly in z∈¾ with strict inequality for some z.
Therefore to perform the full test, we first refer to the two-sided Kolmogorov-Smirnov
statistics to reject the hypothesis that both distributions are identical.
In this case the null and the alternative hypotheses are:
H0: F(z)-G(z)=0 ∀ z∈¾ versus H1: F(z)-G(z)≠0 for some z∈¾ (A)
Then subsequently , the one-sided test of stochastic dominance of F(z) with respect to G(z)
will be:
H0: F(z)-G(z)≤0 ∀ z∈¾ versus H1: F(z)-G(z)>0 for some z∈¾ (B)
Rejection of the null hypothesis in (A) and not rejection of the null in (B) imply that the
distribution of F lies to the right of G. In this case, F is said to stochastically dominate G.
Table B1 displays the values of the KS statistics and the corresponding probability levels49.
We can see that in the first two cases, the distributions F(IE=1) and F(SAME=1) are
stochastically dominating the distributions G(IE=0) and G(SAME=0) respectively as it also
appear from the figures B1 and B2a. In the case of the F(NORTH=1) and G(NORTH=0) we
cannot reject the hypothesis of equality of distributions.
49 We are aware that limiting distribution of the Kolmogorov-Smirnov statistics is only known under
independence of observations (see Girma, Kneller and Pisu 2003). But instead of performing the analysis
year by year, we choose to test the stochastic dominance on the distribution of the firm level averages
productivity index conditional on the respective average values of the trade indexes.
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