supplement to “labor heterogeneity and the pattern … · supplement to “labor heterogeneity...

SUPPLEMENT TO “LABOR HETEROGENEITY AND THEPATTERN OF TRADE”

ALFONSO CEBREROS

1. Realtionship between Theory and the Estimation

Framework

The estimation framework of section 4 is given byxijz = �i + �j + �z +Wjz� + g (sz, µj ,�j) + "ijz

It is reasonable to wonder whether this specification is theoretically justified.As Deardorff [1984] points out

“The major obstacle to the testing of trade theories has beenthe difficulty of constructing tests that all would agree weretheoretically sound. The intuitive content of most trade the-ories is quite simple and straightforward. But empirical testsof the theories are often faulted on the grounds that they testpropositions that do not derive rigorously from the theories.”

In this appendix I provide an informal derivation of a relationship betweentrade flows and importer, exporter, and industry characteristics such as theone specified in the estimation framework. This derivation clearly lays outthe way in which the empirical framework is and is not linked to the theory.

Assume that preferences are given by a two-tier CES structure:

U =

ˆ 1

0↵ (z) ln [Q (z)] dz

Q (z) =

✓ˆ!z

2⌦z

q (!z)��1� d!z

◆ �

��1

with´ 10 ↵ (z) dz = 1. Here, the first tier is a Cobb-Douglas utility index

over consumption bundles Q (z) from different sectors, indexed z 2 [0, 1],and the second tier is a CES aggregator over different varieties within eachsector. The parameter � > 1, is the elasticity of substitution across varieties,assumed common across sectors, and ⌦z is the set of available varieties insector z.

Date: October 2014.1

SUPPLEMENT TO “LABOR HETEROGENEITY AND THE PATTERN OF TRADE” 2

This preference structure leads to the following expression for the expendi-tures of country i on a variety from sector z

ei (!z) = Dizpi (!z)1�� .

Here, pi (!z) is the price paid in i for a variety in sector z, and Diz cap-tures the strength of demand in country i for varieties in sector z.1Given theabove expression for expenditures, trade volumes are given by the followingexpression

Xijz = MjzDizp1��ijz ,

where Xijz are the imports of country i, from country j, in sector z; pijz isthe price paid by consumers in i for a variety from country j in sector z, andMjz is the mass of firms in the exporting country in industry z.

On the production side, assume that final goods are produced by monopo-listically competitive firms. The unique factor of production for these firmsis a sector specific composite input that is assumed to be non-traded andproduced competitively by a constant returns to scale technology. Given thedemand structure outlined above, optimal pricing by final good producersresults in a constant markup over marginal cost:

pijz =

✓�

� � 1

◆⌧ijzcjz,

where cjz is the cost of the sector specific composite in country j, and ⌧ijzare the trade barriers faced by country j in sector z when servicing demandfrom country i.

Finally, assume that ⌧1��ijz = (Tij · Tiz) e

�uijz , where uijz ⇠ N

�0,�2

�are

i.i.d. unobserved/unmeasured trade barriers. Given this assumption, andthe optimal pricing strategy of final good producers, trade flows may beexpressed as

Xijz = MjzDizp1��ijz

= MjzDiz

✓�

� � 1

◆1��

c1��jz (Tij · Tiz)

�1 euijz .

Taking logs on both sides of this expression yields

log (Xijz) = log

"✓�

� � 1

◆1��#+[log (Diz)� log (Tiz)]�log (Tij)+log

⇣Mjzc

1��jz

⌘+uijz,

which suggests the expression

log (Xijz) = �iz + �ij + log

⇣Mjzc

1��jz

⌘+ uijz,

1The strength of demand Diz

depends on: (a) the CES ideal price index for sector z

in country i, Piz

=

⇣´!z2⌦z

pi

(!z

)

1�� d!z

⌘ 11�� ; (b) the share of income spent on goods

from sector z, ↵ (z), and (c) the income of country i.


where �iz is an importer-industry specific term, and �ij is an importer-exporter specific term.

Notice that this expression has the following implication for the distributionof relative exports across sectors:

Elog

✓X

ijz

Xikz

◆� log

✓X

ijz

0

Xikz

0

◆�= (� � 1)

log

✓cjz

0

ckz

0

◆� log

✓cjz

ckz

◆�+

log

✓M

jz

Mkz

◆� log

✓M

jz

0

Mkz

0

◆�.

This expression relates the distribution of relative exports to two terms: thefirst, which refers only to relative costs of production, is tied to comparativeadvantage, while the second term comes from the endogenous adjustment inthe entry and exit of firms to achieve equilibrium.

In two-country models of international trade it is not too hard to show that

log

✓cjz0

ckz0

◆� log

✓cjzckz

◆> 0 ) log

✓Mjz

Mkz

◆� log

✓Mjz0

Mkz0

◆> 0.

That is, there is relatively more entry into the comparative advantage sec-tors.2 Thus, in a two-country model of international trade both forces pointin the same direction. In general, the intuition that the relative mass offirms Mjz/Mkz should be decreasing in relative costs of production cjz/ckzis not easy to extend to a multi-country model, specially when countries areasymmetric. In a multi-country setup, third country effects crucially affectproduction patterns.3 Therefore, the desired result is not generally validand must either be assumed, or derived under stringent conditions. Sincethe equilibrium distribution of firms across countries and sectors is such thatthere remains no further possibility for profitable entry, assuming that therelative mass of firms Mjz/Mkz is decreasing in relative costs cjz/ckz mayprove innocuous to the extent that it is not unreasonable to expect that en-try is relatively higher in those sectors where a country enjoys a comparativeadvantage. Here I will proceed under the assumption that this is indeed thecase.

A fully specified general equilibrium model will provide a mapping fromexporter and industry characteristics to the mass of firms and the cost of theindustry specific composite input. That is,

log

⇣Mjzc

1��jz

⌘= f (characteristics of exporter j, characterisitics of sector z) .

For example, in the model of Romalis [2004], cjz is given by

cjz = (!j)z

2See, for example, Romalis [2004]. Although this author considers a multi-countrymodel, in his setup the world is divided into North and South countries with all countrieshomogeneous within each block. Thus, the model is essentially a two-country model ofinternational trade. See also Bernard, Redding, and Schott [2007].

3See Behrens et. al. [2009].


where !j is the relative price of skilled to unskilled labor, and z is the costshare of skilled labor in the sector (i.e. z is the sector’s skill intensity).Because of transport costs, there is a failure of factor price equalization(FPE) and in equilibrium !j = !

⇣Lsj/L

uj

⌘, where Ls/Lu is the relative

endowment of skilled to unskilled labor. In this way, Romalis is able toderive the three way relationship between trade, factor endowments, andfactor intensities necessary to test the trade implications of a standard factorproportions model.

It should be clear that the most important part in deriving the estimatingequation of interest is the way in which the term log

⇣Mjzc

1��jz

⌘is modeled,

since this is the term that is germane to the theory of comparative advantage.I do not specify a full general equilibrium model, but given the previousdiscussion, here I have chosen to model the term of interest as

log

⇣Mjzc

1��jz

⌘= Wjz� + g (sz, µj ,�j) + ✏ijz,

so that the expression for trade flows is given aslog (Xijz) = �iz + �ij +Wjz� + g (sz, µj ,�j) + "ijz,

where "ijz = ✏ijz + uijz.

This specification is parsimonious in the way in which alternative sources ofcomparative advantage affect trade flows (i.e. Wjz� controls linearly for alter-native source of comparative advantage), while allowing for more flexibilityin the way in which the distribution of talent affects the pattern of trade. Ofcourse, this is only a heuristic derivation, and it is worthwhile to point outthat in a fully specified model log

⇣Mjzc

1��jz

⌘might be a complicated, highly

nonlinear object. Thus, the specification that is being proposed should be in-terpreted with care, and estimated coefficients should be interpreted as thosefor the best approximation to the true CDF E

hlog

⇣Mjzc

1��jz

⌘|Xi

within agiven class of functions.

The above derivation suggests that the estimating equation should haveimporter-industry and importer-exporter fixed effects, rather than separateindustry, importer, and exporter fixed effects. The reason why I adopt thelatter specification instead of the former, is that the computational costsof including importer-industry and importer-exporter fixed effects is toohigh.4Therefore, by replacing importer-industry and exporter-importer fixedeffects with separate industry, importer, and exporter fixed effects, the ex-pression relating trade flows to exporter and industry characteristics becomes

4Estimating importer-industry and importer-exporter fixed effects would require closeto 19,000 dummy variables given the number of exporters and industries included in thesample, while including separate importer, exporter, and industry fixed effects requiresabout 300 dummy variables only.


log (Xijz) = �i + �j + �z +Wjz� + g (sz, µj ,�j) + "ijz.

2. Construction of Skill Distribution

In this section I describe in detail the construction of the skill distribution foreach exporter in the sample, and carefully document the properties of thesedistributions. Due to the novelty of the construction of the distribution ofskills using test scores rather than educational attainment, it is of interest inits own right to carefully study the statistical properties of these distributionsand the cross country comparison amongst them.5

Table 1 summarizes the IALS data.6The last three columns in table 1 displaythe correlations across the three dimensions of literacy tested by the IALS.As can be observed, all of these dimensions of literacy are highly correlated.

To gain further insight into the relationship between these different dimen-sions of literacy, I consider simple regressions of one of these variables on theother two7:

Prose = 33.58⇤⇤⇤(0.35)

+ 0.63⇤⇤⇤(0.004)

Document + 0.23⇤⇤⇤(0.004)

Quantitative�R2

= 0.88�

Document = �8.61⇤⇤⇤(0.32)

+ 0.46⇤⇤⇤(0.003)

Prose + 0.56⇤⇤⇤(0.003)

Quantitative�R2

= 0.93�

Quantitative = 10.56⇤⇤⇤(0.37)

+ 0.23⇤⇤⇤(0.004)

Prose + 0.75⇤⇤⇤(0.003)

Document�R2

= 0.9�

These regressions suggest that any of these three dimensions of literacy iswell explained by the other two, and that there is a positive, and statistically

5In the literature, “skill” is just a short-hand for the ability of workers to generate out-put. We are thus interested in the distribution of workplace productivity in the workforce,and it might be cause for concern whether the variable that has been denoted as skill isin fact the relevant measure of labor productivity of who’s distribution we care about.That is, it could well be the case that the labor productivity that matters for productionis a = f (s), not s directly, and the object of interest would be the distribution of a inthe population, not of s. I will be particularly interested in comparisons of the first twomoments of the distribution of s. In this case, if E [s

i

] � E [sj

], then E [ai

] � E [aj

] forany monotone increasing function f (·). On the otherhand, if Var (s

i

) � Var (sj

), thenVar (a

i

) � Var (aj

) iff f (·) is such thatE [(f (s

i

) + f (sj

)) (f (si

)� f (sj

))] � (E [f (si

)] + E [f (sj

)]) (E [f (si

)]� E [f (sj

)]) .

There is little that can be done to address this kind of misspecification, and I proceed underthe assumption that f (·) satisfies the conditions stated above so that observed productivitydifferences s are informative about the determinants of the pattern of trade, and aboutthe importance of labor heterogeneity as a determinant of comparative advantage.

6µ denotes mean, � denotes standard deviation, and ⇢ denotes correlation.7*** denotes statistical significance at the 0.1 percent level.


Country µprose

µdoc.

µquant.

�prose

�doc.

�quant.

⇢prose,doc

⇢prose,quant

⇢doc,quant

Belgium 282.60 287.85 292.16 51.52 49.99 54.74 0.92 0.89 0.93Canada 259.97 256.92 259.65 64.19 72.03 63.45 0.92 0.91 0.93Chile 212.96 211.97 200.04 54.14 54.32 68.36 0.95 0.94 0.96

Czech Rep. 272.01 285.70 301.28 40.23 49.98 50.74 0.89 0.88 0.93Denmark 276.61 295.66 299.66 35.27 44.56 43.07 0.93 0.88 0.93Finland 290.20 290.99 287.45 47.46 52.33 46.02 0.93 0.87 0.93Germany 275.41 284.50 291.78 44.93 44.57 42.69 0.91 0.90 0.92Hungary 239.80 245.78 267.00 42.43 53.09 54.19 0.87 0.87 0.88Ireland 264.48 257.68 262.81 55.34 57.26 61.58 0.94 0.93 0.96Italy 254.53 247.12 256.66 58.67 58.68 60.17 0.93 0.92 0.96

Netherlands 281.77 284.59 286.69 44.52 46.90 47.32 0.94 0.90 0.95New Zealand 282.06 275.12 276.48 50.97 54.09 52.89 0.94 0.91 0.95

Norway 290.44 298.02 298.53 43.55 51.47 47.55 0.93 0.89 0.94Poland 228.83 222.99 234.22 58.68 70.48 67.62 0.93 0.92 0.97Slovenia 229.72 231.93 242.29 57.56 64.79 66.33 0.94 0.93 0.96Sweden 293.92 298.93 299.91 53.68 54.18 54.34 0.93 0.88 0.97

Switzerland 266.47 272.67 279.65 53.13 59.43 58.53 0.91 0.90 0.92UK 266.16 265.15 268.05 59.93 64.28 64.79 0.96 0.93 0.96USA 260.24 254.23 261.80 71.85 74.19 71.93 0.94 0.93 0.95

Pooled Sample 264.40 265.94 270.56 57.90 63.50 63.44 0.93 0.91 0.95Table 1. IALS Summary Statistics

significant, relationship between any of these three dimensions of literacy andthe other two.

The sample correlations presented in table 16, and the results from simpleregression analysis, suggest that these three dimensions of literacy in factcontain much redundant information. Therefore, it seems appropriate toconsolidate these different literacy scores into a single variable which I willcall “skill”. I define the variable “skill” as

Skill = !pProse + !dDocument + !qQuantitative,

where the weights (!p,!d,!q) are chosen through principal component anal-ysis.8

8From an initial set of m correlated variables, principal component analysis (PCA)creates uncorrelated indices or components, where each component is a linear weightedcombination of the initial variables. Being uncorrelated, the indices measure differentdimensions of the data. The components are ordered so that the first component (PC1)

explains the largest possible amount of variation in the original data.


The weights (!p,!d,!q) are the weights corresponding to the first componentfrom PCA performed on the IALS survey data.9 PCA is particularly appro-priate when the original variables are highly correlated -suggesting a certiandegree of redundancy in the information contained by these variables-, whichis the case here as can be verified in Table 16. Table 17 presents summarystatistics for the skill distribution constructed in this fashion.10

Country µ � median 1

stqu. 3

rdqu. min max

Belgium 287.6 50.64 295.5 263.1 322 29.81 413.5Canada 258.8 65.01 269.2 219.9 305.9 14.18 445.1Chile 207.6 58.74 213.5 169.6 250 45.77 380.5

Czech Rep. 287.6 46.072 292.7 260.1 318.9 70.41 475.7Denmark 291.7 40.29 296.8 268.8 320.9 109.5 397.8Finland 289.6 47.37 296 264.6 322 61.86 418.6Germany 283.8 42.74 285 257.8 313.8 102.3 419Hungary 251.9 48.39 255.7 223.4 284.2 102.2 431.1Ireland 261.6 57.06 268.4 227.9 302.2 54.13 403.5Italy 252.8 57.94 262.9 220.6 295.1 48.26 395

Netherlands 284.4 45.23 291 260.3 315.7 56.53 417.3New Zealand 277.8 51.51 283 250.6 277.8 42.51 412

Norway 295.9 46.57 303.4 271 328.2 71.01 410.1Poland 228.6 64.84 239.3 191.9 274.9 24.77 381.4Slovenia 234.9 62.05 244.2 197.9 279.6 44.71 407.9Sweden 297.6 52.74 303.2 269.5 333.8 50.76 421.9

Switzerland 273.2 55.52 283.5 250.2 310.2 58.75 404.2UK 266.5 62.04 274 231.4 311.2 19.74 470USA 258.7 71.13 271.5 218.7 310.9 40.93 437.9

Table 2. Summary Statistics for Skill Distribution

Brown et. al [2007] discuss the robust features of literacy attainment surveys.One of their main findings is that one of the most robust features arising fromthe different literacy attainment surveys currently available, is the apparenttrade-off between mean and standard deviation, that is, there is a negativerelationship between mean scores and their dispersion. Regression analysisconfirms that there is a statistically significant negative relationship between

9Actually, since the weights from principal component analysis do not sum to one -the sum of their squares does -, I normalize the weights so that they sum to unity. This,additionally, makes sure that the change of basis that is involved in PCA does not affectthe range of potentially observable skill levels (i.e. the skill variable remains in the range[0, 500]).

10Weights are chosen independently across countries. That is, I perform a principalcomponents analysis on the data of each country individually. The weights across the threedimensions of literacy are roughly equal, with quantitative literacy typically receiving aslightly higher weight, and the weights on these three variables are roughly the same acrosscountries.


these two variables. It is estimated that a 10 percent decrease in the standarddeviation of the distribution of skills results in a 4 percent decrease in themean skill level.11

I also perform kernel density estimation to obtain densities for the skill dis-tribution. The estimation is performed using the “Normal Reference Rule”(see Silverman [1998] and Wasserman [2006] for details). Selected results arepresented in Figures 8.2-8.5.

0 100 200 300 400

0.000

0.001

0.002

0.003

0.004

0.005

0.006

N = 3045 Bandwidth = 12.46

Density

Figure 2.1. USA

100 200 300 400

0.000

0.002

0.004

0.006

0.008

N = 4140 Bandwidth = 7.621

Density

Figure 2.2. Switzerland

100 150 200 250 300 350 400

0.000

0.002

0.004

0.006

0.008

0.010

N = 3026 Bandwidth = 7.041

Density

Figure 2.3. Denmark

100 200 300 400

0.000

0.002

0.004

0.006

0.008

0.010

N = 2062 Bandwidth = 8.163

Density

Figure 2.4. GermanyFigure 2.5. Kernel Density Estimates of Skill Distribution

The most salient features arising from this construction of the skill distri-bution are: (i) There is an apparent trade-off between mean and standarddeviation; (ii) The skill distribution is relatively “bell shaped” although witha slightly positive bias (i.e. the left tail is longer than the right tail); (iii) For

11log (µ) = 7.1621

(0.4535)

⇤⇤⇤ � 0.3962(0.1140)

⇤⇤log (�)

�R2

= 0.4156�.


four of the nineteen countries, the skill distribution is bimodal. This suggeststhe possibility that for these countries the overall skill distribution is in factthe mixture of the skill distribution for two separate populations, and (iv)The countries with the most dispersed skill distribution are Anglo-Saxoncountries, while the countries with the highest mean skill levels are typicallyScandinavian countries. This confirms results presented in Bomardini et. al.[2009] and Grossman and Maggi [2000].

In what follows I discuss some further statistical properties of the skill dis-tributions. In particular, I perform a battery of hypotheses tests whose aimis to further our understanding of the characteristics of this novel construc-tion of the distribution of skills at the country level and of the cross-countrydifferences that prevail for these distributions.

I start by considering the issue of symmetry. Some authors, such as Gross-man and Maggi [2004], make use of this assumption, and while not crucialin its own right for the aims of this paper, the extent of asymmetry maywell be important in other applications or in alternative mechanisms linkingworker heterogeneity to comparative advantage. Table 28 presents summarystatistics for the the degree of skewness of these skill distributions.12 Theseresults confirm that in the sample, the skill distribution has a long left-tailas suggested by the kernel density estimates. However, the estimate forPearson’s index of skewness is in all cases well within the interval (�1, 1),thus suggesting that the extent of asymmetry might not be severe enough toreject the null hypothesis of a symmetric distribution.

Formally testing the null of symmetry is not typically a hypothesis test thatis easily implemented, with some tests proposing test statistics with limitingdistributions in which neither the test statistic nor its limiting distributionare readily calculated (see, for example, Csorgo and Heathcoate [1987] orAki [1993]). There is no default or standard test for symmetry commonly

12Recall that skewness is a measure of the asymmetry of the probability distributionof a real-valued random variable. The most commonly used measure of skewness is thecoefficient of skewness defined by

� =

E⇥(x� µ)3

⇤

�3.

Symmetrical distributions can be shown to have a skewness value of zero, while distri-butions skewed to the left can be shown to have a negative skew value, and distributionsskewed to the right have a positive skew value.

An alternative measure of skewness is given by

S = 3⇥✓

mean � medianstandard deviation

◆.

This measure is known as Pearson’s index of skewness. If S 2 [�1, 1], then the distri-bution is said to be symmetric. If S < �1, then the distribution is said to be skewed tothe left, and if S > 1, then the distribution is said to be skewed to the right.


used in the literature, and here I test the null of symmetry

H0 : f (✓ � x) = f (✓ + x) about some unknown ✓

H1 : f (✓ � x) 6= f (✓ + x) ,

based on the results in Cabilo and Masaro [1996].13Testing H0 at the fivepercent level, I find that the null of symmetry can be rejected for all 19countries under consideration.

Coefficient of Skewness Pearson’s Index of SkewnessBelgium -1.035 -0.47Canada -0.60 -0.48Chile -0.26 -0.30

Czech Rep. -0.65 -0.33Denmark -0.81 -0.38Finland -0.79 -0.41Germany -0.39 -0.08Hungary -0.38 -0.24Ireland -0.62 -0.36Italy -0.82 -0.52

Netherlands -0.91 -0.44New Zealand -0.77 -0.30

Norway -0.98 -0.48Poland -0.55 -0.50Slovenia -0.52 -0.45Sweden -0.87 -0.32

Switzerland -1.10 -0.56UK -0.77 -0.36USA -0.58 -0.54

Table 3. Skewness of Skill Distributions

Next, it is of interest to investigate whether there are true cross-countrydifferences in the distribution of skills. While the number of participants ineach country is relatively large in absolute terms, it is small compared to thecountry’s population. Thus, it might be cause for concern that the observed

13These authors propose a test for symmetry based on the test statistic

Tn

=

pn�¯X �m

�

s

where ¯X is the sample mean, m is the sample median, and s is the sample standarddeviation. That is, T

n

=

⇣pn

3

⌘S, so that the test statistic is a monotone transformation

of Pearson’s index of skewness.For a sample size of n, they propose a ↵�level test of H0 based on the decision rule:

reject Ho

if |Tn

| � c1�↵/2,

where the critical value c1�↵/2 can be found in their Table 2.


differences in skill distributions are the result of sampling variability, ratherthan true underlying differences in the distribution of skills across countries.

I now adress this issue in a series of hypotheses tests. I start by testing thenull

H0 : Fi (z) = Fj (z) 8zH1 : Fi (z) 6= Fj (z) for some z,

for two countries i 6= j. That is, I test that null of equal distributions for agiven pair of exporters.

To test H0 I use the Rank-sum Test of Mann and Whitney (see Mood et.al. [1974] for details). There are 171 possible pairwise comparisons. TestingH0 at the five percent level results in 159 rejections of the null of identicaldistributions. While these results suggest that it is safe to assume thatskill distributions vary across countries, these differences may be subtle, andit might be cause for concern whether there is no variation in particularmoments of interest.

Next, I consider the following hypotheses testsH0 : µi = µj

H1 : µi 6= µj

andH0 : �i = �j

H1 : �i 6= �j .

That is, I test whether there is cross-country variation in the first two mo-ments of the distribution of skills. To test the first hypothesis I use Welch’st-test.14. Testing the null of equal means at the five percent level results in162 (out of the 171 possible pairwise comparisons). To test the null that

14Welch’s t�test is a variation on a Student t�test for the case in which it cannot beassumed that the two populations under consideration share the same variance, and thesamples differ in size. The test statistic is given by

t =µ̂i

� µ̂j

sµ̂i�µ̂j

where

sµ̂i�µ̂j =

ss2i

ni

+

s2j

nj

,

and the µ̂’s are sample means, s is an unbiased estimator of the variance, and n issample size. For hypothesis testing, the distribution of the test statistic is approximatedas a Student’s t distribution with degrees of freedom

d.f. =

�s2i

/ni

+ s2j

/nj

�2

(s2i

/ni

)

2 / (ni

� 1) +

�s2j

/nj

�2/ (n

j

� 1)

.


standard deviations do not vary across country pairs, I use the Levene test(see Brown and Forsythe [1974]). Testing H0 at the five percent level, I canreject the null of equal variances of the skill distributions in 142 of the 171possible cases to consider.

Finally, I consider comparisons of skill distributions in the spirit of the no-tions defining skill abundance and skill diversity in Costinot and Vogel [2010]and Grossman and Maggi [2004]. In particular, I look at the empirical cdf’sof exporter pairs to define whether the bilateral difference in skill endow-ments is one that can be classified as a difference in the abundance of skillsor a difference in the diversity of these. I define skill abundance in terms offirst-order stochastic dominance: the distribution of exporter i is said to bemore skill abundant than that of exporter j, if ˆFi first-order stochasticallydominates ˆFj , denoted by Fi ⌫A Fj . A case of skill abundance is depicted infigure 8.6, where the distribution of skills in Denmark is shown to first-orderstochastically dominate that of Chile. On the other hand, exporter i is saidto be more skill diverse than exporter j, denoted Fi ⌫D Fj , if : (a) the ecdf’scross each other at most once, and (b) ˆFj first-order stochastically dominatesˆFi to the left of the crossing point, and ˆFi first-order stochastically domi-nates ˆFj to the right of the crossing point. Figure 8.7 depicts a case of twoexporters who differ in terms of skill diversity defined in this way.

Figure 2.6. Denmark⌫AChile Figure 2.7. Canada⌫DHungary

Using these definitions of skill abundance and skill diversity I look at allthe possible exporter pairs in the sample to see which relationship holds.Out of the 171 pairwise comparisons possible, 43 correspond to cases whereone distribution is said to be more skill diverse than the other; in only onecase is no relation defined as the empirical cdf’s cross each other more thanonce, and the rest are cases in which it is said that one distribution is moreskill abundant than the other. For the cases in which a skill abundancerelation is said to hold, I can formally test whether one distribution first-order stochastically dominates the other. It seems of interest to formally test


this hypothesis as, by far, bilateral relationships where the main differenceis in terms of skill abundance are the most pervasive. 15

To test the nullH0 : G (x) F (x) 8xH1 : G (x) > F (x) for some x

(i.e.that G first-order stochastically dominates F ) I use the test proposed byBarrett and Donald [2003].16 In all cases in which comparison of the ecdf’sindicated the existence of a skill abundance relationship, I test H0 at thefive percent level, and in all cases the null of stochastic dominance cannotbe rejected.

Together, these set of results hel us understand the extent of cross-countrydifferences in skill distributions and suggest that these observed differencesare not driven by sampling variability, bur rather reflect true differences inthe underlying distributions.

References

[1] Aki, Sigeo. (1993). “On nonparametric tests for symmetry in Rm.” Annals of TheInstitute of Statistical Mathematics.

[2] Barrett, Garry F. and Stephen G. Donald. (2003). “Consistent Tests for StochasticDominance.” Econometrica.

[3] Cabilio, Paul and Joe Masaro. (1996). “A Simple Test of Symmetry about an UnknownMedian.” The Canadian Journal of Statistics.

[4] Costinot, Arnaud and Jonathan Vogel. (2010). “Matching and Inequality in the WorldEconomy.” Journal of Political Economy.

[5] Csorgo, Sandor and C.R. Heathcote. (1987). “Testing for Symmetry.” Biometrika.[6] Deardorff, Alan V. (1984). “Testing Trade Theories and Predicting Trade Flows.”

in Ronald W. Jones and Peter B. Kenen, ed., Handbook of International Economicsvolume 1.

[7] Grossman, Gene M. and Giovanni Maggi. (2000). “Diversity and Trade.” AmericanEconomic Review.

15Formally testing the null hypothesis defining the skill diversity relationship is beyondthe scope of standard testing procedures as the crossing point of the distributions is notknown a priori and must be estimated.

16The test statistic for testing the hypothesis H0 against H1 is given by

ˆS =

✓nm

n+m

◆ 12

sup

z

⇣ˆGm

(z)� ˆFn

(z)⌘,

where ˆGm

and ˆFn

are the empirical cdf’s when the sample sizes are m and n, respectively.Barret and Donald [2003] show that, when testing for first-order stochastic dominance,

p�values can be computed as exp

⇢�2

⇣ˆS⌘2

�. These p�values are justified asymptoti-

cally by Proposition 1 and equation (3) in their paper. This testing procedure is similarto the more familiar Kolmogorov-Smirnov test, and is a consitent test for the completeset of restrictions implied by stochastic dominance (i.e. it tests the relevant inequality atall points in the support of the distributions).


[8] Kennedy, Peter. (2008). A Guide to Econometrics. Sixth edition. Wiley-Blackwell.[9] Mood, Alexander. Franklin A. Graybill and Duane C. Boes. (1974). Introduction to

the Theory of Statistics. McGraw-Hill International Editions, Statistics Series, ThirdEdition.

[10] Romalis, John. (2004). “Factor Proportions and the Structure of Commodity Trade.”American Economic Review.

[11] Silverman, B.W. (1998). Density Estimation for Statistics and Data Analysis. Mono-graphs on Statistics and Applied Probability 26.

[12] Theil, Henri. (1971). Principles of Econometrics. John Wiley & Sons, Inc.[13] Wasserman, Larry. (2006). All of Nonparametric Statistics. Springer Texts in Statis-

tics.

Princeton University

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