A Quantile-based Test of Protection for Sale Model
Susumu Imai, Queen’s Universityy
Hajime Katayama, University of Sydneyz
Kala Krishna, Pennsylvania State University and NBERx

March 8, 2009

Abstract
This paper proposes a new test of the Protection for Sale (PFS) model by Grossman andHelpman (1994). Unlike existing methods in the literature, our approach does not require
any data on political organization. We use quantile and quantile IV regressions to do sousing the data from Gawande and Bandyopadhyay (2000). Surprisingly, the results do notprovide any evidence favoring the PFS model. We also explain why previous work may have
inadvertently found support for it.
JEL Classi…cation: F13, F14
Keywords: Protection for Sale; Lobbying; Political Economy; Quantile Regression

We are grateful to Pol Antras, Kishore Gawande, Penny Goldberg, Elhanan Helpman, Pravin Krishna, Giovanni Maggi, Xenia Matschke, Mark Melitz, James Tybout, Steve Yeaple, and the participants in seminars at
Princeton University, Cornell/PSU Macro Conference, 2008 Econometric Society Summer Meetings, NBER ITI
Spring Meeting for helpful comments. All errors are ours.
ye-mail:imais@econ.queensu.ca.
ze-mail:h.katayama@econ.usyd.edu.au.
xe-mail:kmk4@psu.edu.

11Introduction

There has been much interest in the political economy aspects of trade policy recently. In part,
this has been triggered by the theoretical framework in the Grossman and Helpman (1994)"Protection for Sale" (PFS) model. Empirical studies such as Goldberg and Maggi (1999)(hereafter GM) and Gawande and Bandyopadhyay (2000) (hereafter GB) have used US dataand shown that as predicted by the PFS framework, protection is positively related to the importpenetration ratio for politically unorganized industries, but negatively for organized ones.1
In these studies, the key explanatory variable is the dummy variable indicating whether
the industry is politically organized. Its construction requires the classi…cation of industries
into politically organized and unorganized ones. For this purpose GM and GB used data on
contributions along with some simple rules.2 GM classify an industry as politically organized
if its Political Action Committees’(PAC) contribution is greater than a pre-speci…ed thresholdlevel. GB’s classi…cation rule is based on the idea that if industries are organized, then industries
with higher import penetration ratios are likely to make higher campaign contributions. Several
questions naturally arise about their classi…cation rules. First, are their rules consistent with
the PFS model? Second, do their rules correctly distinguish between organized and unorganizedindustries? These issues are of vital importance, because testing and structural estimation of the
PFS model requires political organization to be correctly classi…ed and in a manner consistent
with the PFS model.
The objective of this paper is two-fold. First, we show that their classi…cation rules may not
1

Subsequently, Mitra et al. (2002) and McCalman (2004) used Turkish and Australian data respectively, andprovided similar evidence.
2
They could not just say that organized industries were those that made campaign contributions, as is assumed
in the PFS model, because in the US data all industries make Political Action Committees’(PAC) contributions.
Thus, all industries should be classi…ed as politically organized. But in this case, the PFS model predicts theequilibrium level of protection should be increasing in the inverse import penetration ratio for all industries while
in fact, it is decreasing! Moreover, as lobbying e¤orts could cancel out, protection levels could be quite low. In
the small country case, for example, if all industries are taken to be organized, and all agents own some of at
most one factor, the equilibrium tari¤ equals the optimal one, namely zero.

2

be consistent with the PFS model and consequently their parameter estimates may be biased.
We argue that using a cuto¤ level of contributions may not be advisable as contributions caneasily be very small despite the industry being organized. We provide a simple numerical exampleof the PFS model where the level of the industry’s contribution varies greatly depending on its
import penetration. We show that organized industries may make very small contributions if
their import penetration is high. This implies that using a particular threshold of campaign
contribution as a device to distinguish between organized and unorganized industries, as is donein GM, results in mis-classi…cation and is inconsistent with the PFS model. Furthermore, in our
numerical example, import penetration and the level of equilibrium campaign contributions arenegatively correlated. This is exactly the opposite of the relationship assumed by GB and otherpapers using their approach to classify industries as organized or not, casting their procedure
into doubt as well. We then argue that there is no instrument for the classi…cation errors.
Thus, we have a problem in implementing the usual tests of the PFS model. To deal with
it we propose a new test of the PFS model. The test is based not on the well-known andextensively-examined prediction of the PFS but on other implications past studies have notexplored. Importantly, our test does not require classi…cation of industries as organized or not;nor does it require data on contributions made to political parties, data which is available for
the US but is not usually available for other countries. Our approach relies on the relationshipbetween observables (i.e., the protection measure, import penetration, and import demand elasticity) implied by the PFS model and thus it is entirely consistent with the PFS framework. In
particular, we exploit the following prediction of the PFS model: politically organized industries
should have higher protection than unorganized ones given the inverse import penetration ratio
and other control variables. This suggests that industries with higher protection are more likely
to be politically organized, and thus for these industries, we should expect a positive relationshipbetween the inverse import penetration ratio and the protection measure. Thus, in a quantile3

regression, we should see this relationship hold for the higher quantiles of the protection measure conditional on the inverse import penetration ratio and other controls.3 This prediction istested on the data used in GB. Contrary to much of the literature, our new test does not provide
empirical support for the PFS model.4
Below, we review the PFS model, explain and implement our test. We also explain why
previous work may have inadvertently found support for it.

2

The PFS Model and Its Estimation

2.1

The PFS Model

The exposition in this section relies heavily on Grossman and Helpman (1994). There is acontinuum of individuals, each of in…nitesimal size. Each individual has preferences that arelinear in the consumption of the numeraire good and are additively separable across all goods.
On the production side, there is perfect competition in a speci…c factor setting: each good is
produced by a factor speci…c to the industry, ki in industry i, and a mobile factor, labor, L.
Thus, each speci…c factor is the residual claimant in its industry. Some industries are organized,
and being organized or not is exogenous to the model. Tari¤ revenue is redistributed to allagents in a lump sum manner. Owners of the speci…c factors in organized industries make up
the lobby group which can make contributions to the government to in‡uence policy if it raises
their total welfare. Government cares about both the contributions made to it and social welfare
and puts a relative weight ofon social welfare, W (p) where p is the domestic price and equals

the tari¤ vector plus the world price p .5
3Note that a quantile regression approach does not involve ordering the endgenous variable and runningseparate regressions for each quantile. Instead it allows parameter estimates to di¤er across quantiles while
conditioning on explanatory variables. IV quantile regression further deals with endogeneity.
4
To our knowledge, this is also the …rst use of quantile IV techniques which are quite new in econometrics (seeChernozhukov and Hansen (2005)) in trade.
5
We use bold letters for vectors.

4

The timing of the game is as follows: …rst, lobbies simultaneously bid contribution functions
that specify the contributions made contingent on the trade policy adopted (which determines
domestic prices). The government then chooses what to do to maximize its own objectivefunction. In this way, the government is the common agent all principals (organized lobbies)
are trying to in‡uence. Such games are known to have a continuum of equilibria. By restricting
agents to bids that are “truthful” so that their bids have the same curvature as their welfare, aunique equilibrium can be obtained.6 The equilibrium outcome in this unique equilibrium is as if
the government was maximizing a weighted social welfare function with a greater weight on thewelfare of organized industries. In other words, equilibrium tari¤s can be found by maximizing

G(p) = W (p) +P

Wj (p),

j J0

where J0 is the set of politically organized industries.
In their model, the welfare of the lobby group in industry j is

Wj (p) =wherej (pj ) + lj +Nj[T (p) + S(p)] ,Nj (pj ) is producer surplus in industry j; lj is labor income of the owners of the speci…cfactors employed in industry j, wage is unity, Nj =N =j is the fraction of agents who own thespeci…c factor j, while T (p) + S(p) is the sum of tari¤ revenue and consumer surplus in theeconomy. Maximizing G(p) gives, after some manipulation:xj (pj )(Ij

L ) + (pjpj )m0j (pj )( +where Ij is unity if j is organized and zero otherwise,

L ) = 0,(1)

L (assuming that each individual owns

6
For a detailed discussion of this concept, see Bernheim and Whinston (1986). Imai et al. (2008) provide a
new elementary proof of their result.

5

at most one speci…c factor) corresponds to the fraction of the population that owns the speci…ccapital of organized industries, zj = xj (pj )=mj (pj ) where xj (pj ) and mj (pj ) denote the supply
and imports of industry j; while ej =that (pjm0j (pj )pj =mj (pj ). Rewriting equation (1) using the factpj ) = tj pj where tj is the tari¤ rate gives:tj=
1 + tj

Ij

L

+

L

zj
ej

.

This is the basis of the key estimating equation, which we call the protection equation:

tj
zj
zj
=
+ Ij .
1 + tjej
ejThis equation provides the well known prediction of the PFS model:= 1= ( +2.2

L ) > 0, and

(2)

=[

L= (

+

L )] < 0,+ > 0.7

A Problem in Estimation — the Classi…cation of Industries

To make equation (2) estimable, an error termj is added in a linear fashion:

tj
zj
zj
=
+ Ij + j .
1 + tjejej(3)

To allow for the fact that a signi…cant fraction of industries have zero protection in the data,
equation (3) can be modi…ed as follows:tj= M ax
1 + tj7zj
zj+ Ij + j ; 0 .
ej
ej

(4)

This holds as long as there are some agents who do not own any speci…c capital of organized industries,

L < 1.

6

To test the key prediction (i.e.,< 0,> 0 and+> 0), equations (3) and (4) have been

estimated in a number of previous studies.
Although data on the measure of trade protection, the import penetration ratio, and theimport-demand elasticities are often available, it is harder to de…ne whether an industry ispolitically organized or not. To deal with this problem, GM used data on campaign contributions
at the three-digit SIC industry level. An industry is classi…ed as politically organized if thecampaign contribution exceeds a speci…ed threshold level. GB use an alternative approach.
They run a regression where the dependent variable is the log of the corporate PAC spending per
contributing …rm relative to value added and the regressors include the interaction of the importpenetration from …ve countries into the sub-industry and the two-digit SIC dummies. Then,
industries are classi…ed as politically organized if any of the coe¢ cients on its …ve interaction
terms are found to be positive. This procedure is based on the notion that in organized industries,
an increase in contributions would likely occur when import penetration increased.
Both of these procedures are questionable. We o¤er a formal argument that claims: (1)when industries are misclassi…ed, only under very strong assumptions can we consistently estimate parameters in the protection equation; (2) both of the above classi…cation approaches areinconsistent with the PFS model and result in mis-classi…cation of industries, with the likely
outcome being inconsistent parameter estimates.
To see the …rst claim, let

j be classi…cation error;

j = IjIj0 where Ij is the true politicalorganization dummy and Ij0 is the political organization dummy used for estimation. Then, thefollowing equation is essentially estimated as the protection equation:

tj
zj
zj
=
+ Ij0 +
1 + tj
ej
ej

where

j

j zj =ej + jj,is the composite error term. This suggests that we could …nd in-7

struments for zj =ej (i.e., variables that are correlated with zj =ej but not correlated with
only ifjj)

is mean zero and independent of zj =ej ; otherwise, instruments for zj =ej would be

unavailable, as any variable correlated with zj =ej will be correlated withj zj =ej and hencej.

Importantly, as we will show below, the classi…cation schemes used by GM and GB may resultin classi…cation error that is not mean zero and/or independent of zj =ej , thereby making theirinstruments invalid.8
Next, we discuss the second claim. Given the model and the menu auction equilibrium of the
PFS model, it is easy to verify that the equilibrium campaign contribution schedule should besuch that government welfare in equilibrium should equal the maximized value of the government
objective function when industry i is not making any contributions at all. Thus, the equilibrium
campaign contribution can be expressed as follows:9

E

Bi (p ) ="E

W (p ) +P

E#

Wj (p ) + W (p(i)) +j J0 ;j6=iP

Wj (p(i)) = Hi (p(i))

Hi (pE ),j J0 ;j6=iwhere Bi (pE ) is the campaign contribution of industry i at the equilibrium domestic pricevector pE , and p(i) is the vector of domestic price chosen by the government when industry i
is not making any contributions. Since10 Hi (p) = W (p) +P

Wj (p), it can be seen thatj J0 ;j6=iequilibrium contributions are essentially the di¤erence in the value of the function Hi (p) : RN
! R between p(i) and pE .
Let p(t) be a path from pE to p(i) as t goes from zero to unity. Since the line integral ispath independent, we can choose this path as desired. In particular, we can choose it so thathip(t) = pE + t p(i) pE so that p(t = 0) = pE , p(t = 1) = p(i), and Dp(t) = p(i) pE .
If their instruments for zj =ej are correlated with j , so are their instruments for Ij0 ; they used the same
instruments as those for zj =ej .
9As the equilibrium bids of a lobby group equal its welfare of the lobby group less a constant, the constants
will cancel out in the expression.
10Note that H has to be indexed by i.
88Hence,E

Hi (p(i))

Hi (p ) =Z1

dHi (p(t))
dt =
dt

0

Z1

DHi (p(t)) Dp(t)dt,(5)0where DHi (p(t)) is the vector of partial derivatives of the real valued function Hi (:) with respect
to the vector p and Dp(t) is the vector of the derivatives of p with respect to t anddenotestheir dot product.
The vector p(i) must take the same form as pE (the domestic price chosen by the government
when industry i is making contributions) but withpl (i) plI(l 2 J0 fig)=pl (i)+ L( Li ) zliel

L being replaced by; pl (i) =1

L

i . Thus,pl
I(l2J0 fig) ( L+ Lii ) zl;el

where I is an indicator function. Note the analogy with equation (1). This equation allows us
to …nd p (i) from the data.
Now using the line integral de…ned in equation (5) and substituting for DHi (p(t)) = @Hi (p)=@pj
hi
and for Dp(t) = p(i) pE , we get
E

Bi (p ) =Z1 X

Li ) (pj (t)+ [I (j 2 L

fig)

( L

X

pE
j g

j

0

=

j

f( +

fpj (i)+ [I (j 2 LThus, depending on

i,

,

Z1

f ( +

fig)

( L

0

pj )

@mj (pj (t))@pji )] xj (pj (t))gfpj (i)

pE
j gdt

(pj (t) pj )pj (t)zj (t)ej (t)

Li)1i )]gxj (pj (t)) dt.

E
L , xj (:), and zj =ej , Bi (p ) can be small even for politically organizedindustries. This is evident from a numerical example. We assume there are 400 industries(N = 400), of which 200 are politically organized (Np = 200). We set pi = 2:0,
L = 0:5,i =L =N , and xi = 10000.

= 50:0,We also set zi =ei = i=1000 for industries i = 1; :::; Np

9which are politically organized and zNp +i =eNp +i = i=1000 for industries Np + i = Np + 1; :::; N
which are not politically organized.
Figure 1depicts the equilibrium campaign contributions for politically organized industries in
the above example11 . Notice that these contributions vary from 0 to 40 depending on the valueof z=e. This illustrates the possibility that GM’s classi…cation based on a threshold of campaign
contribution could well mis-classify industries with low campaign contribution and low z=e(high import penetration and/or high e) as politically unorganized. Hence, classifying politicalorganization based on a uniform threshold, as done by GM and others, leads to classi…cation
error, which is not independent of zj =ej .
Figure 1 also shows that the equilibrium campaign contributions increase with z=e for politically organized industries12 . In other words, for politically organized industries, campaign
contributions are negatively correlated with import penetration. This is the opposite of therelationship used by GB to classify political organization. Our example therefore suggests thatthe correct organized industries may be the ones which GB classi…ed as unorganized and viceversa, i.e., I = 1

2.3

IGB where IGB is the politically organization dummy by GB.

Another Look at GB and GM

Mis-classi…cation on the part of GB has an important implication for the interpretation of theirparameter estimates: although their estimates seem consistent with the PFS predictions (i.e.,
GB < 0,GB > 0, andGB + GB > 0), they are not, given the correct political organization

dummy. This can be easily seen by noticing that when I = 1
11

IGB is the political organization

We did not plot the campaign contributions of politically unorganized industries becase they obviously arezero.
12
The positive relationship between campaign contributions and z=e in the simulated model is in line with the
PFS model; it predicts that for politically organized industries, protection is positively related to z=e: Hence,
campaign contributions and z=e are likely to be positively related as long as greater campaign contributions tend
to result in higher protection. However, a negative relationship between them is con…rmed in the data; in the data
used in GB and the data by Facchini et al. (2006) (who reconstructed the GM dataset). In this data, log(z=e)
and log of campaign contributions per dollar of value added are found to be negatively correlated. We present
these relationships in Figures 2 and 3, respectively.

10dummy, the protection equation should betj= ( GB +1 + tjThis implies b =

GB +

GB

GB )

> 0, b =

zj
ej

GB

GB (1

IGB )

zj
+ j.
ej

< 0, and b + b =GB< 0, which is clearly

inconsistent with the PFS framework. In this way, classi…cation error could have led GB to
inadvertently conclude that the data supported the PFS model.
We next argue how the approach of GM might be giving a false positive coe¢ cient estimate
for z=e for the organized industries due to classi…cation error. In our example, the correlations
between protection, contributions and z=e are as in the data and though the true model isclearly inconsistent with the PFS framework, estimation of the protection equation using GM’s
classi…cation approach provides results in support of the PFS model!
To show this, we generate protection levels that are decreasing in z=e for organized industries
as well as for unorganized ones, which is consistent with our quantile estimation results but
inconsistent with the PFS model. Speci…cally, we use the following equation:

tj
= max
1 + tjwhere ( 0 ;ones, and

1 ) = (0:5;
j

0+

1

zj
+ j ; 0:0
ej

2:5) for organized industries, ( 0 ;1 ) = (0:05;

0:25) for unorganized

N (0; 0:02), zj =ej = j=2000, j = 1; :::; 200 for both. Organized industries havehigher protection levels but for both organized and unorganized industries, protection falls with
z=e: The total number of industries as well as the number of organized industries are set tobe the same as the ones used earlier. As observed in the actual data, we set the campaign
contributions to be positively correlated with the import penetration ratio13 . Note that thepositive correlation is consistent with a plausible scenario where greater protection requires
13

See footenote 12.

11greater campaign contributions. We normalize the campaign contributions to be equal to theprotection measure t=(1 + t) and classify industries to be politically organized if the campaign
contributions exceed the threshold of 0:25. This results in about 50% of the organized industries
being wrongly classi…ed as unorganized.
Using simulated data from the above exercise on protection and z=e, we estimate the protection equation by OLS14 and then obtain b =0:95 ( 9:87) and b = 3:14 (14:28) wheret-statistics are in parentheses. The results are clearly in support of the PFS model even though
the simulated model is not. The reason for the result is simple. On the one hand, the simulation
setup makes z=e negatively correlated with protection. On the other hand, I
correlated with the protection measure since I

z=e is positively

z=e is positive only for organized industries

with high protection. This is because only the organized industries with low z=e are classi…ed
correctly and such industries tend to have high protection.

3

A Proposed Approach

3.1

Quantile Regression

Equation (4) and the restrictions on the coe¢ cients have at least two implications. First, z=e
has a negative e¤ect on the level of protection for unorganized industries while it has a positivee¤ect for organized ones. Second, given z=e, organized industries have higher protection. These
implications lead to the following claim: given z=e, high-protection industries are more likely to
be organized and thus the e¤ect of an increase in z=e on protection tends to be that of organizedindustries.
The logic of this argument is illustrated in Figure 4 where the distribution of t= (1 + t) is
plotted for given z=e. The variation of t= (1 + t) given z=e occurs for two reasons. First, because
14

Both z=e and the political organization are constructed to be exogenous. Since the classi…caton error iscorrelated with z=e and mean nonzero, we cannot correct for the bias by any instruments as discussed earlier.

12some industries are organized while others are not and these two behave di¤erently, and second,
because of the error term. As a result, the distribution of t= (1 + t) comes from a mixture of two
distributions, namely those for the politically organized industries and those for the unorganized.
These two distributions for some given values of z=e are plotted in Figure 4. The two dashedlines give the conditional expectations of t= (1 + t) for the organized and unorganized industries
as a function of z=e. In line with the PFS model, the two lines start at the same vertical intercept
point and the line for the organized industries is increasing while the other is decreasing in z=e.
For each z=e, if we look at the industries with high t= (1 + t), they tend to be the politicallyorganized ones. Thus, at high quantiles, the relationship between t= (1 + t) and z=e should bethat for organized industries, i.e., should be increasing as depicted by the solid line labelled the90th quantile in Figure 4.
The relevant proposition (Proposition 1) can be found in Appendix 1. The proposition
essentially states that in the quantile regression of t=(1 + t) on z=e, the coe¢ cient on z=e should
be close to+ > 0 at the quantiles close to= 1. To examine this, we use quantile regression(Koenker and Bassett, 1978) and estimate the following equation:

QT ( jZ) =

where

( )+

( ) Z=10000,(6)denotes quantile, T = t=(1 + t), Z = z=e, and QT ( jZ) is the conditional -th quantile

function of T . If the PFS model is correct,( ) converges to ( + ) > 0 as

approaches its

highest level of unity from below.

3.2

IV Quantile Regression

In the quantile regression, Z is assumed to be an exogenous variable. However, Z is likely to beendogenous as discussed in the literature (e.g., Tre‡er, 1993) and hence the parameter estimates13

of the quantile regression are likely to be inconsistent. It is therefore important to allow for the
potential endogeneity of Z. We formally show that even in the presence of this endogeneity,
the main prediction of the PFS model in terms of our quantile approach does not change. The
relevant proposition (proposition 2), an analogue of proposition 1, is presented in Appendix 1.
To test the prediction in the presence of possible endogeneity of Z, we estimate the following
equation by using IV quantile regression (Chernozhukov and Hansen, 2004a; 2004b; 2005; 2006):

P (T

( )+

( ) Z=10000jW1 ) = ,(7)

where W1 is a set of instrumental variables.
Importantly, nowhere in equations (6) and (7) is the political organization dummy present;
these equations involve only variables that are readily available. This way our approach does
not require classi…cation of industries in any manner and as a result, we can avoid any biases
due to mis-classi…cation.
An issue that we need to deal with is the endogeneity of political organization. We do so bycontrolling for capital-labor ratios, which is essentially equivalent to allowing the capital-laborratio to be a determining factor for the probability of political organization. This is motivatedby Mitra (1999) who provides a theory of endogenous lobby formation. His model predicts thatamong others, industries with higher levels of capital stock are more likely to be politicallyorganized.
Even after controlling for the capital-labor ratio, there still could remain a correlation between the error term of the equation determining political organization and the error term ofequation (4). Since our method is not subject to classi…cation error, one of the main sourcesof correlation between the error terms in the two equations in GM and other studies, we are14

less subject to this criticism15 . Moreover, as long as the error term of the equation determining
political organization and that of the protection equation are positively correlated, or as longas the negative correlation is not too strong, our quantile IV procedure will still be consistent.
This is because only when the negative correlation in the errors is very strong (large positive
shocks in protection are correlated with shocks that make an industry unorganized) could themost protected industries be unorganized ones. Plausible scenarios actually would suggest the
opposite.

4

Estimation

4.1

Data

We use part of the data used in Gawande and Bandyopadhyay (2000).16 The data consist of
242 four-digit SIC industries in the United States. In the dataset, the extent of protection, t;
is measured by the nontari¤ barrier (NTB) coverage ratio. z is measured as the inverse of theratio of total imports to consumption scaled by 10; 000. e is derived from Shiells et al. (1986)
and corrected for measurement error by GB. See GB for more details along with the sample
statistics of the variables. Of particular note about the data is that 114 of 242 industries (47%)
have zero protection, suggesting the potential importance of the corner solution outcome of T
in the quantile regression.
We …rst examine the relationship between the protection measure T and Z using linear
regression (OLS). Column 1 row OLS/IV of Table 1 shows that they are negatively and insignificantly correlated in the data. After we control for two exogenous variables used in GB, tari¤s onintermediate goods (IN T ERM T AR) and NTB coverage of intermediate goods (IN T ERM T B),
15

In those studies, classi…cation error enters both the disturbance term of the equation determining the politicalorganization and the disturbance term of the protection equation. Thus, classi…cation error necessarily resulted
in correlation between the disturbance terms.
16
We are grateful to Kishore Gawande for kindly providing us with the data.

15

T and Z remain negatively correlated (Column 2 row OLS/IV). To address the potential endogeneity of Z we use IV estimation. We use three instruments that are found in GB to
be strongly correlated with Z: the fraction of employees classi…ed as scientists and engineers(SCIEN T IST S), the fraction of employees classi…ed as managerial (M AN AGERS), and crossprice elasticity of imports (CROSSELI). Again, we …nd that T and Z are negatively related(Column 3 row OLS/IV). Controlling further for the capital labor ratio does not change thenegative relationship (Column 4 row OLS/IV). Next, we examine their relationship at various
quantiles, with special attention to high quantiles.

4.2

Quantile Regression

Column 1 of Table 1 presents the estimation results of equation (6)17 . The results do not appear
to provide any supporting evidence for the PFS model; the null hypothesis that( )0 cannotbe rejected at high quantiles (in fact, at all quantiles) in favor of the one-sided alternative that( ) > 0. Moreover, the point estimates indicate that thecontrary to the PFS prediction and decrease asand( ) are all negative at high quantiles

goes from 0:4 to 0:9.

are estimated to be zero at the 0:1-0:4 quantiles, suggesting that the corner solution(T = 0) greatly a¤ects the estimates at lower quantiles. From this evidence, it is conjectured
that the existence of corners also a¤ects the estimates at the mean. Thus, …ndings based on thelinear model (i.e., equation (3)) in GB, Bombardini (2005), and others are likely to be subject
to bias due to the corner solution problem. In contrast, our method does not su¤er from the
problem, since the focus is mainly on the higher quantiles where the e¤ect of corner solution isminimal18 .
Following GB, we also control for IN T ERM T AR and IN T ERM N T B. As Column 2 of17

We used stata for all the estimation exercises in this paper except for the IV quantile regression.
Of course, this advantage comes with a cost. That is, the quantile approach does not allow us to estimate
the structural parameters and separately.
18

16

Table 1 shows, our main …ndings do not change; the null hypothesis that( )0 cannot berejected at high quantiles, and at the 0.7th quantile the estimate is statistically signi…cant at 5%
level.

andare found to be zero at the 0:1 and 0:2 quantiles, again suggesting the importanceof corner solutions.

4.3

IV Quantile Regression

Column 3 of Table 1 presents the estimation results of equation (7)19 . As in the quantile regression, we cannot reject the null hypothesis that (0:9)

0 in favor of the one-sided alternative.

The point estimates are not favorable for the PFS model, either; even after correcting for theendogeneity of Z, the estimate ofat the highest quantile is not positive as required by the PFS

model. As presented in Column 4 of Table 1, qualitatively similar results are obtained when we
further control for the capital labor ratio. For a robustness check, we also use a varied set of instruments: (1) 17 GB’s instruments available to us, IN T ERM T AR and IN T ERM N T B, andtheir squared terms, (2) instruments in (1) plus their interaction terms, (3) SCIEN T IST S only,(4) M AN AGERS only, and (5) CROSSELI only. We also examine speci…cations with/without
capital-labor ratios. Our main …ndings appear to be robust; regardless of which instrument we
use and whether we control for capital-labor ratios, the null hypothesis at the highest quantile
cannot be rejected. Moreover, the point estimates of

4.4( ) are all negative at high quantiles20 .

Alternative Speci…cation

For a further robustness check, we examine a di¤erent model speci…cation. Note that by moving
ej to the left hand side of the equation, equation (2) can be re-expressed as:tj
ej = zj + Ij zj :1 + tj19

All the IV quantile regression estimation is done by using a MATLAB code written by Christian Hansen(available at http://faculty.chicagogsb.edu/christian.hansen/research).
20
For all these results, see Imai et al. (2008).

17

This provides a basis of an alternative model for our quantile-based test: for quantile regression,

QT e ( jz) =( )+( ) z=10000;(8)

where Te = te=(1 + t), QT e ( jz) is the conditional -th quantile function of Te ; for IV quantile
regression,
P (Te

( )+

( ) z=10000jW2 ) = ;(9)where W2 is a set of instrumental variables. As the dependent variable now involves elasticity,
we exclude CROSSELI from the set of instrumental variables used earlier.
As presented in Table 2, the results resemble those presented before; point estimates ofat high quantiles are all negative. Our main results therefore do not seem to be driven by themodel speci…cation. We also examine the robustness of our results to a varied sets of instruments.
Though we tried hard to get the point estimate ofat high quantiles to be positive, we were able

to do so in only one case. When the set of instruments included all GB’s instruments available tous, except several elasticity-related variables, their squared terms, and their interaction terms,
could we get the point estimate ofat 0:9 quantile to be positive21 . However, even in this case,the estimate was not signi…cant even at the 10% level and hence this could not be seen as strong
support of the PFS model.
In Figure 5 and 6, we plot the relationship between the inverse import penetration ratio andthe protection measure. In both speci…cations, with and without the elasticity on the RHS, therelationship is negative, especially if we look at high quantiles of the protection measure, forall values of inverse import penetration ratio. In both …gures we can see that the relationshipbetween the inverse import penetration ratio and the protection measure is quite di¤erent fromthe one shown in Figure 4.
21

The results are available on request.

18

5

Discussion

There are several possible explanations for our results. The …rst possibility is heteroskedasticity.
If the error term has higher variance when the industry is unorganized, i.e.,"j = wj + (1Ij ) j ;(10)then unorganized industries would have error terms with much higher variance. As a result,unorganized industries would dominate in high quantiles as well as in low quantiles, whereas theorganized industries would be found mostly around the median. Hence, at high quantiles, thenegative quantile regression coe¢ cients correspond to , which is negative, and not

+

> 0.

One might think that this could explain the presence of negative slope coe¢ cients in the higher
quantiles. While this possibility cannot be completely ruled out, it is hard to reconcile withthe fact that almost all industries have positive campaign contributions and both GM and GB
report that more than half of the industries are organized, so that it is reasonable to think thata signi…cant fraction of the industries are likely to be organized. If this is so, then it is surprising
to …nd that the slope coe¢ cients of the quantile regressions are negative at almost all quantiles
except for the zeros at low quantiles, which comes from the corners. Could hetroskedasticity interms of z=e account for our results? Simple forms of this clearly cannot. If the variance of theerror term increases in z=e, one could actually get a false positive in favor of the PFS model. If
the variance of the error term decreases in z=e, then one could obtain the reverse pattern withthe slope coe¢ cient rising for lower quantiles. We …nd neither of these patterns in the data.
Second, the small sample may make it di¢ cult for our approach to provide evidence favoring
the PFS model. This problem can be overcome by using more disaggregated data, althoughsuch an exercise is beyond the scope of the current paper.
Third, note that when classi…cation error is considered, our results are consistent with part19

of GB’s results and not inconsistent with those of GM. If political organization were correctly
assigned in GB as argued earlier, then GB might also have found no support for the PFS
model. Recall that in our example where we computed the relationship between the equilibrium
campaign contribution and z=e for organized industries, it was positive instead of negative.
If the positive relationship holds in reality, we argued that the industries that were originallyclassi…ed as organized should be classi…ed as unorganized and vice versa. Then, the true resultsof the GB estimation should be b > 0, b < 0 and b + b < 0, part of which (i.e., b + b < 0) is
indeed consistent with our quantile and quantile IV results (i.e.,( ) < 0 for= 0:9). We also

argued that mis-classi…cation due to the GM’s approach could result in evidence favoring the
PFS model even when the true model is inconsistent with the PFS framework. This suggeststhat the GM’s results are not inconsistent with our results against the PFS model.
It is worth explaining why we chose to take a quantile (IV) approach rather than someother approach, even though it does not provide estimates of the structural parameters. Givencurrent techniques, there may be another way to satisfactorily estimate the model that does
not require classi…cation ex ante of industries into the two groups. This would involve theestimation of GM setup but with organization treated as unobservable.22 The issue in thiscase would be identi…cation. The exclusion restriction for identi…cation would require that atleast one exogenous variable that determines z=e (i.e., instruments for z=e) does not enter inthe political organization equation, and thus does not in‡uence tari¤s directly. But such an
instrument is likely to be hard to …nd.
22

This is equivalent to a switching regression approach where the outcome of the switching regression is notobservable. One may also think of this as an unobserved heterogeneity model where the unobserved types areallowed to be endogenous and are estimated in addition to the tari¤ equation.

20

6Conclusion

In this paper, we proposed and implemented a new test of the PFS model that does not require
data on political organizations. To our surprise, the …ndings so far are not supportive of the
PFS model. Clearly, more work is needed on this. One fruitful research avenue might be tolook at countries other than the United States using our approach as it does not require data on
political organization. Another research avenue is to use more disaggregated data so that ourapproach can provide statistically more clear-cut evidence. Finally, other predictions of the PFS
model such as those on equilibrium contribution levels predicted by the PFS model relative toactual contributions need to be tested, and we hope to do so in future work.

References
Bernheim, B. Douglas and Michael D. Whinston (1986) “Menu Auctions, resource Allocation, and Economic In‡uence,” Quarterly Journal of Economics, 101(1), 1-31.
Bombardini, Matilde (2005) “Firm Heterogeneity and Lobby Participation,”forthcoming in
Journal of International Economics.
Chernozhukov, Victor and Christian Hansen (2004a) “Instrumental Variable Quantile Regression,” MIT Department of Economics Working Paper No. 06-19.
Chernozhukov, Victor and Christian Hansen (2004b) “The E¤ects of 401(k) Participation on the Wealth Distribution: An Instrumental Quantile Regression Analysis,” Review
of Economics and Statistics, 86(3), 735-751.
Chernozhukov, Victor and Christian Hansen (2005) “An IV Model of Quantile Treatment E¤ects,” Econometrica, 73(1); 245

21

261

Chernozhukov, Victor and Christian Hansen (2006) “Instrumental Quantile Regression
Inference for Structural and Treatment E¤ect Models,” Journal of Econometrics, 132(2),491-525.
Facchini, Giovanni, Johannes V. Biesebroeck, and Gerald Willmann (2006) “Protection for Sale with Imperfect Rent Capturing,” Canadian Journal of Economics, 39(3),845-873.
Gawande, Kishore and Usree Bandyopadhyay (2000) “Is Protection for Sale? Evidence
on the Grossman-Helpman Theory of Endogenous Protection,” Review of Economics andStatistics, 82(1), 139-152.
Goldberg, Pinelopi K. and Giovanni Maggi (1999) “Protection for Sale: An EmpiricalInvestigation,” American Economic Review, 89(5), 1135-1155.
Grossman, Gene and Elhanan Helpman (1994) “Protection for Sale,”American Economic
Review, 84(4), 833-50.
Koenker, Roger W., and Gilbert W. Bassett (1978) “Regression Quantiles,” Econometrica, 46(1), 22-50.
Imai, Susumu, Hajime Katayama, and Kala Krishna (2008) “A Quantile-based Test ofProtection for Sale Model,” NBER Working Paper, 13900, National Bureau of Economic
Research, Inc.
McCalman, Phillip (2004) “Protection for Sale and Trade Liberalization: An Empirical Investigation,” Review of International Economics, 12(1), 81-94.
Mitra, Devashish (1999) “Endogenous Lobby Formation and Endogenous Protection: A Longrun Model of Trade Policy Determination,”American Economic Review, 89(5), 1116-1134.

22

Mitra, Devashish, Dimitrios D. Thomakos, and Mehmet A. Ulubasoglu (2002) “‘Protection for Sale’in a Developing Country: Democracy vs. Dictatorship,” Review of Economics and Statistics, 84(3), 497-508.
Shiells, Clinton R., Robert M. Stern, and Alan V. Deardor¤ (1986) “Estimates of the
Elasticities of Substitution between Imports and Home Goods for the United States,”
Weltwirtschaftliches Archiv, 122(3), 497-519.
Tre‡er, Daniel (1993) “Trade Liberalization and the Theory of Endogenous Protection: An
Econometric Study of U.S. Import Policy,”Journal of Political Economy, 101(1), 138-160.

23

Appendix 1: Quantile Regression
Proposition 1 (Quantile Regression) Assume that (1) Zj is bounded below by a positive
number, i.e. there exists Z > 0 such that Zj

Z, (2)

j has a smooth density function which

has support that is bounded from above and below, (3)j is independent of both Zj and and Ij ,and (4)> 0. Then, forsu¢ ciently close to 1,

quantile conditional on Zj can be expressedas
QT ( jZj ) = F

10+ ( + )Zj

(11)

where
0

Proof. For any 0 <P Tj

P (Ij = 0):
P (Ij = 1)

=

(12)

< 1, for any T > 0,

T jZj = PjT

Zj P (Ij = 0) + PjT( + )Zj P (Ij = 1) .

(13)

Let
T =F

1

0

+ ( + )Zj

(14)

where
0

P (Ij = 0)
, or
P (Ij = 1)=

From equation (15), we can see that for
1, we have

= P (Ij = 0) +

% 1,

0

P (Ij = 1) .

(15)

0 % 1 as well. Hence, forsu¢ ciently close to

0 close enough to 1 such that

F

1

0

+ Zj

F

1

24

0

+ Z>F

1

(1) .

Hence,
T =F

1

P

T

0

+ ( + )Zj > F

1

Zj

1

(1) + Zj

and
j

P

F

j

(1) = 1

which results inP

jT

Zj = 1.

(16)

Substituting equations (14), (15), and (16) into (13), we obtain

P Tj

Therefore, for

T jZj

F

1

0

= P (Ij = 0) + Pj

P (Ij = 1)

= P (Ij = 0) +P (Ij = 0) = .

su¢ ciently close to 1,

QT ( jZj ) = T = F

10+ ( + )Zj .

We make two remarks on the assumptions. First, we assume thatj has bounded support(assumption 2). This assumption is reasonable since the protection measure is usually derived
from the NTB coverage ratio (e.g., Goldberg and Maggi, 1999; Gawande and Bandyopadhyay,2000) and therefore it is clearly bounded above and below. Second, we assume thatjisindependent of both Zj and and Ij (assumption 3). This is rather a strong assumption and will
be relaxed next. In particular, we allow Zj to be correlated with

25

j.

Assume the model is as follows:

Tj=

Zj + j if Ij = 0

Tj= ( + ) Zj + j if Ij = 1

where Zj = g (Wj ; vj ) and Wj is an instrument vector and vj is a random variable independentof Wj . We will show that( ) ! ( + ) > 0 as

% 1.

Let us de…ne uj as follows:j = E [ j jvj ] + uj ;ujj

E [ j jvj ] ,where uj is assumed to be i.i.d. distributed. For the sake of simplicity, we assume that both uj
and E [ j jvj ] are uniformly bounded, hence so is

j . Furthermore,

Tj = max Tj ; 0 .

Then, for Ij = 0 the model satis…es the assumptions A1-A5 of Chernozhukov and Hansen (2006).
Similarly for Ij = 1. Therefore, from Theorem 1 of Chernozhukov and Hansen (2006), it follows
that
P T

F

1

( ) + Zj jWj =for Ij = 0,

and
P T

F

1

( ) + ( + ) Zj jWj =for Ij = 1.

Proposition 2 (Quantile IV) Assume that Zj is bounded below by a positive number, i.e.

26there exists Z > 0 such that Zj

Z. Then, for

P T

1

F

0

su¢ ciently close to 1,+ ( + ) Zj jWj = ;

where
0

P (Ij = 0):
P (Ij = 1)=

Proof.
0

P (Ij = 0)
, or
P (Ij = 1)=

= P (Ij = 0) +0

P (Ij = 1) .

Then,

P Tj
= P

j +

+P

1

F
Zj

j +(

0

+ ( + ) Zj jWj

F

1

0

+ ) Zj+ ( + ) Zj jWj P (Ij = 0)
1

F0

+ ( + ) Zj jWj P (Ij = 1)

= P

j

F

1

0

+ Zj jWj P (Ij = 0) + P

= P

j

F

1

0

+ Zj jWj P (Ij = 0) +From the de…nition of

0 , forsu¢ ciently close to 1, we have0

0 % 1 as well.

% 1,

j

F

1

0

+ Z>F

Because1(1) .

Hence,
P

j

F

1

0

+ Zj jWj = 1.

27

0

jWj P (Ij = 1)

P (Ij = 1) :0 close enough to 1 such thatF

1

is uniformly bounded, forTherefore,

P Tj

It follows that for

F

1

0

+ ( + ) Zj jWj = P (Ij = 0) +0

P (Ij = 1) = .

su¢ ciently close to 1,

P T

F

1

0

+ ( + ) Zj jWj = .

28

29
No

No

K=L

0:123 (0:194)

0:225 (0:208)

0:145 (0:082)

0:125 (0:063)

0:082 (0:075)

0:032 (0:035)

0:020 (0:007)

0:099 (0:011)

0:000 (0:000)

0:000 (0:000)( )0:027 (0:025)

No

Yes

0:079 (0:070)

0:029 (0:049)

0:043 (0:027)

0:033 (0:026)

0:019 (0:026)

0:021 (0:019)

0:018 (0:018)

0:000 (0:009)

0:000 (0:004)

( )

(3)

2:131 (1:533)

5:780 (7:446)

4:970 (7:401)

3:890 (6:624)

8:500 (13:08)

7:180 (14:49)

7:160 (12:05)

17:81 (23:37)

0:000 (0:000)

0:000 (0:000)( )0:056 (0:027)

Yes

Yes

0:090 (0:152)

0:057 (0:077)

0:056 (0:037)

0:075 (0:038)

0:040 (0:043)

0:036 (0:026)

0:032 (0:019)

0:027 (0:019)

0:000 (0:015)

( )

(4)

1:826 (0:451)

10:90 (13:81)

7:360 (11:50)

3:610 (6:416)

4:150 (8:596)

8:320 (18:37)

4:740 (9:130)

4:810 (9:368)

11:90 (14:83)

0:000 (0:000)( )are controlled for, respectively. For details of these variables, see Gawande and Bandyopadhyay (2000).

by 200 bootstrap resampling. GB Controls and K=L indicate whether IN T ERM T AR and IN T ERM N T B are controlled for and whether capital-labor ratios

quantile regressions (equation (7)) and the IV estimation method. Standard errors are in parentheses. For IV quantile regressions, standard errors are calculated

Note: Columns (1) and (2) provide the estimation results of quantile regressions (equation (6)) and OLS, and columns (3) and (4) the estimation results of IV

Yes

0:044 (0:018)

0:001 (0:050)

0:046 (0:029)

0:044 (0:023)

0:053 (0:025)

0:026 (0:010)

0:029 (0:002)

0:026 (0:001)

0:000 (0:000)

0:000 (0:000)

( )

No

0:165 (0:219)

0:505 (0:343)

0:258 (0:241)

0:126 (0:123)

0:045 (0:103)

0:003 (0:025)

0:000 (0:000)

0:000 (0:000)

0:000 (0:000)

0:000 (0:000)

( )

(2)

GB Controls

0:085 (0:009)

OLS/IV

0:028 (0:015)

0:6

0:308 (0:040)

0:002 (0:003)

0:5

0:9

0:000 (0:000)

0:4

0:157 (0:036)

0:000 (0:000)

0:3

0:8

0:000 (0:000)

0:2

0:077 (0:018)

0:000 (0:000)

0:1

0:7

( )

(quantile)(1)

Table 1: Quantile Regression and IV Quantile Regression Results

30
No

No

K=L

0:072 (0:247)

0:350 (0:531)

0:220 (0:135)

0:176 (0:140)

0:099 (0:101)

0:052 (0:060)

0:035 (0:012)

0:019 (0:013)

0:000 (0:000)

0:000 (0:000)( )0:004 (0:060)

No

Yes

0:133 (0:123)

0:018 (0:091)

0:012 (0:076)

0:014 (0:066)

0:015 (0:046)

0:043 (0:031)

0:026 (0:022)

0:000 (0:018)

0:000 (0:006)

( )

(3)

5:599 (3:274)

8:770 (6:242)

12:59 (12:08)

9:150 (17:33)

22:65 (23:72)

17:51 (22:92)

18:33 (21:26)

25:00 (23:64)

0:000 (6:696)

0:000 (0:478)( )0:044 (0:078)

Yes

Yes

0:067 (0:243)

0:030 (0:114)

0:055 (0:101)

0:104 (0:085)

0:070 (0:059)

0:067 (0:038)

0:047 (0:030)

0:051 (0:034)

0:000 (0:025)

( )

(4)

5:041 (3:739)

10:70 (12:98)

13:55 (15:83)

8:850 (18:90)

14:95 (21:59)

11:45 (22:82)

8:200 (20:54)

18:30 (18:50)

3:800 (11:16)

0:000 (2:896)( )are controlled for, respectively. For details of these variables, see Gawande and Bandyopadhyay (2000).

by 200 bootstrap resampling. GB Controls and K=L indicate whether IN T ERM T AR and IN T ERM N T B are controlled for and whether capital-labor ratios

quantile regressions (equation (9)) and the IV estimation method. Standard errors are in parentheses. For IV quantile regressions, standard errors are calculated

Note: Columns (1) and (2) provide the estimation results of quantile regressions (equation (8)) and OLS, and columns (3) and (4) the estimation results of IV

Yes

0:068 (0:027)

0:029 (0:103)

0:068 (0:037)

0:071 (0:045)

0:070 (0:032)

0:063 (0:017)

0:058 (0:003)

0:044 (0:003)

0:000 (0:000)

0:000 (0:000)

( )

No

0:119 (0:279)

0:645 (0:704)

0:335 (0:405)

0:189 (0:188)

0:065 (0:153)

0:004 (0:041)

0:000 (0:000)

0:000 (0:000)

0:000 (0:000)

0:000 (0:000)

( )

(2)

GB Controls

0:126 (0:014)

OLS/IV

0:046 (0:021)

0:6

0:454 (0:061)

0:002 (0:005)

0:5

0:9

0:000 (0:000)

0:4

0:236 (0:048)

0:000 (0:000)

0:3

0:8

0:000 (0:000)

0:2

0:133 (0:025)

0:000 (0:000)

0:1

0:7

( )

(quantile)(1)

Table 2: Quantile Regression and IV Quantile Regression Results: An Alternative Speci…cation