L__

 

ABSTRACT

DATA ERRORS AND ECONOMIC PARAMETER
ESTIMATION: A CASE STUDY OF
INTERNATIONAL TRADE DATA
By

Tracy W. Murray

It is well known that the existence of observation
errors in the variables leads to least-squares estimators
which are biased and inconsistent. This result is based
upon a set of classical assumptions regarding the error
items; namely, that the errors are independently and
normally distributed with zero mean and constant variance.
However, given the classical assumptions, one can obtain
consistent estimates by alternative methods. The purpose
of this study is to test the validity of these assumptions.
The method requires two sets of observations (21 and 22)

on a single set of events (Z), and is based on the follow—

ing elementary relationships:

where r and s are observation errors. Upon subtracting,

the true value (Z) cancels, leaving

 

 

 

 

Tracy w. Murray

2 - z = Z + r — Z — s

Under the classical assumptions concerning r and s, we have
2 2

: (
r N\O, or S

), s : N(O, o ) and E (rs) = 0.

Consequently,

2 2
(r-s) : N(O, or + as ).

The validity of the classical assumptions can then be tested

by reference to observations on (r-s). These observations

 

are derived from international trade data where the ex-
porting and importing countries independently report trade
by common commodity classifications (SITC). Our results
indicate quite clearly that, for the set of data under in-
vestigation, the classical assumptions cannot be accepted.
Consequently, alternative error hypotheses were specified
and pre-tested with a preliminary sample of 216 observa-
tions. Pre-testing led to the adoption of a specific error
hypothesis, which then passed a test based on a new sample
of 1579 observations. These results have implications for
the estimation of international price and income elasti—

cities.

(II III

DATA ERRORS AND ECONOMIC PARAMETER
ESTIMATION: A CASE STUDY OF

INTERNATIONAL TRADE DATA
By

Tracy W. Murray

A THESI

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Economics

1969

 

 

 

 

DEDICATION:

Judge Robert J. Murray:

A self made professional.

ii

 

ACKNOWLEDGMENT

This thesis is the result of the efforts of people
too numerous to mention. However, a few contributors must
be singled out for their extensive assistance. Professor
Jan Kmenta's contributions far exceed that of a full parti-
cipant in every stage of this research. Without his con-
fidence and encouragement this thesis, in all likelihood,
would not have been written. I would also like to single
out Professor J. Ramsey, whose constant assistance in
solving the many problems which arose during this project
and his careful, critical and patient eye in examining the
many drafts of this thesis have made the result a much
better work. In addition to professional assistance in the
research for this thesis, I would like to thank Professor
M. Kreinin for his unlimited support during my entire pro-
fessional career. I cannot thank him enough. The financial
support for this thesis was provided by the Institute for
International Business and Economic Studies.

Finally, I would like to thank two people who have
not participated directly in the research for this thesis,
but whose presence have been extremely helpful. I would

like to thank Professor John P. Henderson, without whose

iii

"IIII|II.II

 

 

vindication and encouragement, many a graduate student

at Michigan State University would have fallen by the

way. And last, but by no means least, I would like to

thank my wife Kathi for her constant encouragement and

patience with my progress. I hope the rewards of the

future will in some way compensate her for this year of

loneliness.

iv

 

 

Illllll’lll I III... II

 

TABLE OF CONTENTS

DEDICATION .

ACKNOWLEDGMENT.

LIST OF TABLES.

Chapter

I. INTRODUCTION .

I. l The Problem.
I. 2 The Objective of This Study
I. 3 The Plan of the Thesis .

II. THEORETICAL ANALYSIS OF ERRORS IN
ECONOMIC DATA. . . . .

II.l The Errors in Variables Model
II.2 The Method for Testing the
Classical Error Assumptions
III. TESTING THE ERROR DISTRIBUTION .

III.l The Sample
III.2 Specification of the Error

Distribution. . .
III.3 Testing the Error Distribution.
III.A Additional Evidence .
III.5 Conclusion

IV. IMPORT DEMAND ELASTICITIES AND ERRORS IN

TRADE DATA.

IV.l The Import Demand Model and
Errors in Trade Data . .

IV.2 Elasticity Estimates

IV.3 Conclusion

Page
ii
iii
vii

O‘\O\|-‘

'1

 

Chapter Page

V. SUMMARY AND CONCLUSIONS . . . . . . . 7’4
APPENDIX . . . . . . . . . . . . . . 78
REFERENCES. . . . . . . . . . . . . . 81

vi

l/l‘il. I . . _

 

 

Table

LIST OF TABLES

Trade Flows . . . . . .

Summary of Preliminary Test Results for
the Alternative Composite Error
Hypotheses . . . . .

Observations per Trade Flow for the Samples
Used to Test the Composite Error
Hypothesis

Composite Error Hypothesis Test Results.

Tests for Composite Error Independence

Test Results for the Conditional Mean
Parameters . . . .

Estimates of the Adjustment Parameter
Published and Adjusted Elasticity Estimates

Selected Estimates of the Adjustment
Parameter (U. S. Imports)

vii

Page
22

31

A1
A2
45

50
65
68

70

 

 

CHAPTER I

INTRODUCTION

I.l The Problem

In economics decisions are based upon theory and
fact. In the natural sciences, unlike in the social
sciences, theories are hypothesized and tested by the use
of controlled laboratory experiments. In the laboratory
the scientist has always been very conscious of possible
errors in his observations. He has increased the ac-
curacy of measurement by repeating his experiments and
by developing measuring instruments with known degrees
of accuracy. Consequently, estimates of the parameters
of the probability distribution of the errors of observa—
tion can be obtained. These estimates can then be used
to remove error bias from empirical results. On the other
hand, the social scientist has been unable to conduct
laboratory experiments. He has improvised by observing
the real world. But, unfortunately, repeated observations
come at different points in time and are often accompanied
by different sets of conditions. Furthermore, to compound
the problem the observer does not always know the accuracy

of his measuring instruments and, therefore, the accuracy

 

 

of his data. To handle the first problem of changing sit-
uations, the economist has often utilized the reqression
technique. He has, thereby, attempted to quantify the
effects on the objective variable of all relevant factors

which change from observation to observation. This method

 

also attempts to separate quantitatively the effects of

 

each independent variable. But, to date, a satisfactory E}
solution to the second problem, i.e., that of knowing 5‘
the accuracy limits of the observations, has not been found.
Unfortunately, many economists today proceed in their work 1'
5"

by completely ignoring this problem and assuming that the
data they use are accurate descriptions of the facts they I
wish to study.
In On the Accuracy of Economic Observations Morgenstern
says that the first step forward is to stop presenting econ-
omic statistics as if they were free from error. "Perhaps
the greatest step forward that can be taken, even at short
notice, is to insist that economic statistics be only pub-
lished together with an estimate of their error. Even if
only roughly estimated this would produce a wholesome effect"
[22, p. 30“]. There have been other economists, albeit too

few, who have turned their efforts toward the problem of

errors in data. Little, if any, of this effort has been

concentrated on estimating the errors in economic data. In-

stead, the results of the effort has been to examine the

effect of data errors on empirical results. This effort can

 

 

 

 

be divided into two types of investigation; namely, the

empirical approach and the theoretical approach. One
attempt of the empirical approach has been to estimate the
parameters of relatively simple econometric models using
two or more sets of data with different reliability ex—
pectations. In a study by Denton and Kuiper [8] parameter
estimates of several macro models were computed using pre-
liminary data and later using revised data. The parameter
estimates differed significantly. Since neither the true
data nor the true parameters are known, this method is

still inadequate for measuring the errors in variables or
the bias resulting from these errors. The only conclusion
that can be reached is that errors in variables do contri-
bute a significant, but still unknown, bias to empirical
results. The theoretical approach for dealing with ob-
servational errors is based upon certain assumptions about
the errors. This approach can further be divided into a
specialized attempt aimed at removing the bias in a specific
model, and a generalized attempt aimed at removing the bias
in any model using a specified estimating technique, eg.,
least-squares regression or maximum-likelihood. In a
specialized attempt G. Orcutt [25] shows that price elasti-
cities in international trade, estimated by the least-squares
regression technique, are biased toward zero. This result
is based upon the following assumptions regarding the errors

of observations:

 

 

 

l. The errors have zero expectation.

2. The errors are independent from each other.
3. The errors are independent from the variable
measured with error.
A. The errors have constant density.
5. The errors have finite range.
6. The errors exist in the independent variable
only.
M. Kemp [lU] further refines Orcutt's argument by assuming
that errors exist in the dependent variable also. He con-
cludes that the elasticity estimates are biased toward minus
one rather than zero. M. Kemp's conslusion has recently
been shown to hold when the errors are measured as a per-
centage of the quantity of trade [30]. In a generalized
attempt Johnson [13] demonstrates that least-squares ex—
timators of the parameters of linear single equation models
are underestimated in magnitude, i.e., biased toward zero.
CPhese conclusions are well known and follow from the clas-
Esical "Errors in Variables Model" which will be discussed
fi.n Chapter II. It was also demonstrated that the regres—
ssion coefficient estimators are inconsistent. This model
1.8 based upon the following set of assumptions:
1. The errors have zero expectation.
2. The errors have constant variance.
3. The errors are independent from each other.
A. The errors are independent from the variable

measured with error.

 

Both the empirical and the theoretical approaches to

the error problem have their shortcomings. The empirical

approach compares parameter estimates derived from different

sets of data. Since all sets of data contain error, all the

parameter estimates must be suspected of containing some

error bias. Hence, one cannot be certain of the magnitude
or the direction of the error bias. On the other hand, the
theoretical approach does provide a strong hypothesis re-

garding the direction of the bias. Furthermore, it identi—

fies the information which is needed to remove the bias.
But the results are dependent upon the rather imposing list

of untested assumptions. If any of these assumptions are

false, the bias withdraws back to the realm of the unknown.
The solution to the error problem via the empirical

approach requires knowledge of the true value of the para-

Ineters. This could generally be obtained only if the true

Kralue of the variables were known, in which case there would
A solution via the theoretical approach

If

tae no error problem.
Jrequires a valid set of assumptions from which to build.
tLhe assumptions of a particular theoretical method prove to
EDe incorrect, all is not lost. The theoretical method can
lDee altered to incorporate a new set of assumptions. The

Iiéiw conclusions will provide the directional bias plus iden-
tlfihfy the factors which must be known to remove the bias.

I311t, above all, this solution requires a valid set of

a S sumptions .

 

 

 

 

1.2 The Objective of This Study

The purpose of this study is to test the classical
assumptions underlying the "Errors in Variables Model".
To repeat the classical assumptions: the errors are in-
dependently and normally distributed with zero mean and

constant variance:
6 : N(O, 02). E(eieJ)=O, ifij E(eiXJ)=O (1)

where e is the observational error and X the variable
which is measured with error. The first task of this
study will be to test the above assumptions. If any, or
all, of the assumptions are rejected, a new set of assump-
tions about the errors will be specified and tested. Fin—
ally, the results of this testing will be applied to the
problem of estimating international price and income elasti—

cities to determine whether errors in observations have a

significant impact on the estimation of these economic

parameters.

32.3 The Plan of the Thesis

In Chapter II the problems in estimating the para-
nieters of the "Errors in Variables Model" will be examined.
1Tb will be shown that least-squares estimators are biased
Eirid inconsistent. Three alternative methods of estimating
1311a parameters will be discussed. All of these methods are

based upon the classical assumptions about the error dis-

t-71"ibution function. In the second part of this chapter a

 

 

method of testing the classical assumptions is presented.

The method is applicable only to situations in which there
exists two independent sets of observations on a single
set of economic events. Chapter III will be devoted to
applying the method developed in Chapter II to statistics
on international commodity trade flows. In addition to
the classical assumptions, alternative error probability
distribution hypotheses are developed and tested. In
Chapter IV the results of Chapter III are applied to the

estimation of international price and income elasticities.

 

 

 

CHAPTER II

THEORETICAL ANALYSIS OF ERRORS
IN ECONOMIC DATA

*
II.l The Errors in Variables Model

Consider the following bivariate linear model:
Y = a + B X (l)

where X and Y are unobserved. Assume that x and y are ob-

served such that

(2)

where u and v are the observational errors. Note that an

exact relationship between X and Y has been hypothesized.**

 

*This model is presented with more rigor in M.G. Ken-
dall and A. Stuart [15, Chapter 29].

*The model in regression form would be

Y=Q+Bx+d

where d is the stochastic disturbance. Rewriting in terms
of the observed variables, we get

y = a + B (X—u) + (d+V)

_. TNT)»-

 

u~

 

 

Because of this exact relationship, X and Y are both

stochastic or both non-stochastic. In order to estimate
the parameters of this model it is necessary to make
some assumptions about the stochastic elements u and v.
The following classical assumptions are generally made:

the errors are independently and normally distributed

with zero mean and constant variance: {I
2 I
u : N(O,ou ) v : N(O,ov2) E(uka) = o l
E(uiuJ) = E(V1Vj) = O, i # J (3) l .
a

Cov(u,X) = Cov(u,Y) = Cov(v,X) = Cov(v,Y) = 0

It is also necessary to specify whether X and consequently
Y are stochastic. The particular specification examined
below will depend upon the estimation technique. These

techniques are discussed in turn.

II.l.l Estimation Method 1: Least-squares [13]

The problem arises because Xi and Yi are not observed
sand, consequently, the normalleast—squares estimators cannot
tae calculated. However, measures of X1 and Yi do exist;
riamely, x1 and yi. Attempts to estimate a and B using the
Cbbserved values lead to biased and inconsistent estimators

[213, p. 139]. For example, with random X the probability

 

IJnless we have a priori information about d, v or some re—
liationship between d and v, the effects of these two random
\fariables cannot be separated. The regression form is dis-
cElissed in J. Johnston [13].

 

 

lO
limit for the estimator of B is

plim (E) = ——B-—2 7‘ e (A)

l + 0u

3x7

where °u2 is the variance of u and OX2 is the variance of
X.* It is clear that for large samples E(g), in absolute
values, is less than B. This error bias could be removed
if the variance of u were known. For random X, consistent
estimators for a and B are

kw;— (5)

X u

 

(6)

9
II
‘<l
I
113*
NI

II.l.2 'Estimation Method 2: Maximum-likelihood [15, pp.
375—4181.

Assume that X is non-stochastic. As there are (n+4)
IDarameters to be estimated, the maximum-likelihood method
tareaks down [19]. In order to proceed additional infor-

niation is needed. The general assumption which is

 

n
x _
For non-random x, substitute lim '%I 2 (x1 — x)2
n+00 i=1
I‘or'ox2, assuming this limit exists and is finite.

 

Ill

 

 

ail... ._ pwnwfiémim .
1...... .4... swimwypjb. .

. . . ii.-. , in“... , ._.:am.mi.a_...fl;, wflfi
. . a. 1. ._: ...._...flr.r.u.._¢u. .m. .q. r.. .r...)1}..t.‘.

 

 
 

g - “4932...... . mam ..-....w .

 

ll

frequently made is that the ratio of the error variances

is known, i.e.,

2
Cu (7)

A = ———-

O

V

This additional information is sufficient to identify con-

sistent estimators for a and B [15, p. 386].

11.1.3 Estimation Method 3: Instrumental Variables [10,29]
This method depends upon the identification of another
variable (Z) which is independent from the error terms and

highly correlated with X. The estimator for B is

A w-'. z.—Z)
e=c(yiy)(1_- (8)
(xi-x) (Zi-Z)

 

For large samples this becomes

A X,Z -
plim (e) = B 8331x,23 - B . (9)

Thus, B is a consistent estimator of 8, however, the vari-
ance of B is quite large and this could be quite critical
if Cov(X, Z) were near zero. Moreover, there is no way to
determine the degree of inefficiency since Cov(X, Z) is
unknown. This method has the additional shortcomings that:

1. Different choices of instrumental variables

yield different estimates of 8.

 

 

1
t
0!

 

l2

2. Finding variables which are independent from u
and v yet correlated with X is often difficult.

3. Testing for the assumptions of independence from
u and v and correlation with X is impossible

since neither u, v nor X are observable.

II.l.A Estimation Method A: Grouping,0bservations [2,35]
In addition to the classical assumptions given in
equations (3), this method requires the following:
1. the observations can be ranked according to Xi

l-X2|>o where the subscripts stand for

2. 11m |x
n+0!)
groups.

The estimates are obtained by ranking the X1 in ascending

order. Group 1 contains the respective observations of the

k highest Xi and Group 2 contains the k lowest. The two
groups are mutually exclusive so that ksg, The estimators
are
y1‘y2
B = —:——:" (10)
x1‘x2
a = 5-8 2 . (11)

TPhese estimators are consistent but have large variances.
The major problem in this method is ranking according

tie the unobserved Xi' If observations are ranked by the

<>t>served x the errors (ui) will influence the ranking.

1,

 

 

13

Any misranking will lead to bias. A second problem is
encountered when X is a random variable. In this case

assumption 2 is violated.

II.l.5 Summary of Estimation
The main point to be made is that all four of these

methods are dependent upon the validity of the classical
assumptions about the observation errors. Seldom, if

ever, do researchers test these assumptions when conducting
empirical investigations. The primary task of this study
is to test the classical assumptions using economic data.
The methodology underlying these tests will be presented

in Section II.2 below. The results of the tests will be

presented in Chapter III.

II.2 The Method for Testing thnglassical Error Assumptions
One method for testing the classical assumptions about

the observation errors involves two sets of observations

(21 and Z2) on a single set of events (Z), and is based on

‘the following elementary relationships:

+1”

N
II
N

(12)
+8

N
II
N

VVllere r and s are observational errors. Upon subtracting,

1311e true value (Z) cancels, leaving

’9 {4/
l .5”

1.: I, l’

.
hr .‘

$3,

.

,r-
,‘.

L V.

I“.
p
‘ -

u

I

a

. .
‘

~ :1:

 

 

 

 

 

 

l“

= Z+r-Z-s
(13)

Under the classical assumptions concerning r and s, we have

r : N(O,o:) (1A)
2
s : N(0,os) . (15)
Consequently,
2
(r—s) N(O,o ) (16)
where
2 2 2
o = or + as + 2 Cov(r,s) .

Therefore, the validity of the classical assumptions can be
If, upon

tested by reference to observations on (r-s).
testing, equation (16) is rejected then equation (IA) and/or
equation (15) must be rejected and, consequently, the clas-

sical assumptions cannot hold for both r and s. If, on the
crther hand, equation (16) is not rejected, it is possible,

tliough unlikely, that equations (1“) and (15) are both untrue
= E(s) # O This leads to

FRDr example, assume that E(r)
E(r)-E(s) = 0 , and equation (16) is true when the

E(r-s) =
Clléissical assumptions are untrue. This kind of situation

1-55 also possible, though highly unlikely, for the assumptions
Consequently, if the

C31? normality and constant variance.

 

15

classical assumptions are not rejected on the basis of
tests on (r—s), they are likely to hold for r and s
separately.

The full classical assumptions require, in addi—

tion, that r and s be independently generated, i.e.,

E(rirj) = E(Sisj) = O , i # j (17)

Cov(ri,Zi) = Cov(Si,Zi) = O (18)

The corresponding relationships for (r—s) are

E[(ri-si)(rJ-SJ)1 = 0 , i # J (19)

Cov[(ri—si), Zi] = O . (20)

Consider equation (19) first. Upon expanding this term,

we have
E[(ri'si)(rj_sj)] = E(rirj) + E(sisj)_2E(risj)° (21)
(Ionsequently, a test for independence based on (r-s) could
Iaeject equation (21) when, in fact, the classical assumptions
riold for r and s separately. This would occur when there
edcists a correlation between r and s. However, since ri and

£31_ result from different observations on Zi’ it seems de-

1P€Ensible to assume that E(risi)=0. In this case, rejection

C31? equation (21) would indicate that E(rirJ) and/or E(sisj)

511‘63 not zero, hence, the classical assumptions do not hold

 

16

for both r and 8. Of course, if equation (21) is not re-

jected, the classical assumptions are likely to hold for
r and s separately.
Next, consider the hypothesis of r and 3 being in-

dependent from Z. This test involves the hypothesis of

equation (20); namely,
Cov[(ri-si),Zi] = Cov(ri,Zi) — Cov(si,Zi) = O (22)

Rejection of equation (21) on the basis of a test using

(r-s) indicates that r and/or 8 are correlated with Z,

and the classical assumptions must be rejected. Acceptance

of the hypothesis means that the classical assumptions are

valid or that the dependence between r and Z is equal but

opposite in sign to the dependence between 5 and Z.

Since this latter case appears to be unlikely, acceptance

of a test based on (r-s) indicates that the classical

assumptions hold for r and s separately.

SII.2.1 Summary of the Testinngethod

With few reservations results of the tests on (r-s)
vngll lead to similar conclusions about r and s separately.

iff‘ the null hypothesis is rejected, it is jointly untrue

fRDI~ r and s. That is to say, at least one error term,

1? (Dr 5, must violate the null hypothesis. If the null
halpothesis is not rejected then either the null hypothesis

1-53 true for both r and s or both r and s violate the

 

 

v.1-3Illbr'lr

 

 

l7

hYpOthesis but in an offsetting manner. This possibility

of accepting a false hypothesis means that the classical

error assumptions are more likely to be violated than the
tests indicate.

 

CHAPTER III

TESTING THE ERROR DISTRIBUTION

In Section II.2 of the last chapter an introduction
to the methodology underlying the test for the classical
error hypothesis was presented. Equation (13) defined
(r—s) as the difference between two independent obser-
vations of the same event where r and s are the unobserv-
able measurement errors for the two observations. (r—s),
which is observable, will be referred to as the composite
error. In equations (l6), (l9) and (20) the parameters
of the composite error probability distribution function
were specified under the assumption that the classical
error hypothesis is true. To repeat; under the null
hypothesis, (r—s), the composite error, is assumed to be

distributed as follows:
(r—s) : N(0, o2)
E[(ri-si)(ri-sj)1 = O , i # j (1)
Cov[(r—s), Z] = O .

TPlie relationship between tests on (r-s) and tests on r and

is separately was discussed in the last chapter. With few

18

 

19

reservations, results of the composite error tests will
lead to similar conclusions about r and s separately.
For reasons discussed in Section III.2 below, three

primary error hypotheses will be tested in this chapter.

They are as follows:

HNl: (r—s) : N(O,o2) . (2)

This is the classical error hypothesis.

(3)

HN2: (r-s) : N(u,02) .

This hypothesis differs from the classical case only in

the specification of the mean, which is assumed constant

but not necessarily zero.

HN3: (r—s) : N(a+B-E(VT),G2) (4)
where VT is the average of two observations on the

same event. This specification hypothesizes the mean to

vary with the expectation of the average of the observations

can the value of trade.
In addition to these, two subordinate error hypotheses

vvere tested; namely, that the error is distributed as double
etxponential with zero mean and constant variance (HDEl) or
5153 double exponential with constant mean and constant
Kr
ariance (HDE2).
A mathematical formulation for each of these hypotheses

fieES presented in Section III.l along with a discussion of the

 

 

20

sample to be used in testing these hypotheses. In Section

III.2 a preliminary sample is used to conduct a series of

pre-tests on these hypotheses. This effort results in the

specification of a single error hypothesis which is for—

mally tested with a new sample in Section III.3. Finally,

additional cooberating evidence is taken up in Section

III.A. The implications of the accepted error hypothesis

for economic parameter estimation will be examined in

Chapter IV.

111.1 The Sample

As stated in the introductory chapter, only data on
international commodity flows, exports and imports but not
capital flows, will be included in this study, i.e., trade
in manufactured products reported according to the SITC

commodity group classification at the 3-digit level of

aggregation.* These trade flows are reported independently

by the exporting and the importing country, and are broken

ciown by country of destination and by country of origin

I°espective1y. That is to say, a specific trade flow (for

enxample SITC 512) from Country A to Country B will be

Inaported by Country A as exports and by Country B as imports.

These data, however, present some problems. Statis-

tiixzs are published in both value and quantity terms. Quan-

t3113y figures are obtained by deflating the reported value

 

 

* The data source is OECD [2A].

 

 

 

.. . . . . U 1 . H ..... .
4.4..“ V; ,. 0.,“ . . s . ‘ I. J .. .

, V r . . . . . 1.? . ._. _ . .
.1 . .. vs... T. ,..
.11...- $2.. .. _. . a , . .. _. 1.. .

 

 

 

21

of trade using a price index. These deflated figures are

valid only for homogeneous commodity groups, i.e., groups
consisting of identical products which sell for the same

price. For non-homogeneous commodity groups the number of

units traded could increase while the value of trade de-

clined. This could result from a decrease in the number

of high value goods traded and a greater increase in the

number of low value goods traded. Consequently, quantity

figures are non-homogeneous commodity groups are almost

meaningless. Because of this, the study will be limited to

value data.
A second problem arises because many countries report

the value of imports on a different basis than the value of

exports. Imports are reported "including cost, insurance

and freight" (c.i.f.) while exports are reported "freight

on board" (f.o.b.). Since import valuation includes the

cost of shipping and export valuation does not, the true

‘Value of imports c.i.f. will always exceed the true value

of‘exports f.o.b. This problem could be avoided if in-

SIArance and freight costs for every commodity group were

kniown. Since this information is not available, coverage

vvdLlee limited to those countries which report imports and

exports on the same basis. Canada and the United States

I’ealaorts both imports and exports f.o.b. while European

C(DI—Intries report only exports f.o.b. For this reason, only

t33t-"£3.de flows between Canada and the United States and from

 

22

Europe to Canada and the Unites States will be in-
cluded.

The sample observations are obtained from quarterly
statistics covering the 5 year period from 1959 to 1963.
Earlier periods were excluded since both the United States
and Canada reported imports on a c.i.f. basis. In 196A
OECD changed their reporting from quarterly to biannual
periods. Initially attempts were made to include all of
the trade flows listed in Table l for each commodity group

considered. However, some flows were excluded due to an

TABLE l.——Trade flows.

 

 

Code Exporter Importer

1 Austria Canada

2 Belgium-Luxembourg Canada

3 Denmark Canada

A France Canada

5 Germany Canada

6 Italy Canada

7 Netherlands Canada

8 Norway Canada

9 Portugal Canada
10 Sweden Canada
11 Switzerland Canada
12 United Kingdom Canada

13 United States Canada

14 Austria United States
15 Belgium-Luxembourg United States
16 Denmark United States
17 France United States
18 Germany United States
19 Italy United States
20 Netherlands United States
21 Norway United States
22 Portugal United States
23 Sweden United States
2“ Switzerland United States
25 United Kingdom United States

26 Canada United States

23

insufficient number of observations. It was feared that
small sample errors in estimation might affect the test
results unduly. Other flows were excluded because of an
insufficient value of trade. Reported trade is rounded
to the nearest $1,000 unit. Assume that the exporting
country records $510 and the importing country records
$490. The actual difference between the estimates
(composite error) is $20. However, the reported figures
are $1,000 and $0 respectively. This gives a reported
composite error of $1,000vflflxfluis roughly double the re-
corded value of trade in both countries. To minimize this
problem trade flows which averaged less than $1,500 per
quarter were eliminated.

The data used in testing are based upon the following
elementary relationships. Let x1 be the export observation
for one particular trade flow, eg., x2 is Belgium-Luxem-
bourg's observation of their exports of commodity group 512
to Canada. Next let mi be the import observation eg., m2
is Canada's observation of their imports on commodity group

512 from Belgium-Luxembourg. Finally, define TVTi to be

the true value of trade, eg., TVT2 is the value of commodity

group 512 moving from Belgium-Luxembourg to Canada, where

TVTi is unknown. The following relationships hold:

(5)

>4
II

+
i TVTi r1

(6)

3
u
is
.<
+3
+
(I)
H

 

 

2“

where r is the error in the exporter's observation, s is
the observational error made by the importer and i specifies
the trade flow. Upon subtracting, the true (unknown) value

of trade cancels, leaving

x - m = TVT + r - TVT - s

i i i i i

(7)

This term will be called the composite error (CE). There-

fore,

CE =x —m =r -S (8)

ij ij ij
where i specifies the trade flow and j specifies the ob-

servation period.

The estimated value of trade (VT) is

_xi +mi _ l
VTiJ ——-L-2——-1——TVTiJ +§(r1J+ $13). (9)

The remainder of this section will be devoted to math-
ematical specifications of the alternative composite error
ciistributions to be tested in this thesis. The economic
JIAstifications for these hypotheses are discussed in Section
III.2 below.

Prior to these specifications, however, it must be

I>C>inted out that the tests for normality utilize published
S13atistics which are transformed to produce observations

fPomadistribution which is hypothesized to be normal with

 

25

zero mean and unit variance. Since, as will be obvious
below, the raw data are transformed for testing using es—
timated rather than predetermined parameters, the sample
of transformed data is actually distributed, under the
null hypothesis, as "t", not normal, random variables
with approximately 18 degrees of freedom. Relying on "t"
being asymptotically normal for 18 degrees of freedom is
not entirely defensible. As a result, care will be taken
to analyze the test results for the discrepancy between
the normal and "t" distributions. Note at this point that
the "t" distribution is less peaked than the normal dis-
tribution and, consequently, any test rejecting normality
due to the sample distribution being more peaked than the
hypothesized normal distribution will be sufficient evi-
dence for rejecting the "t" distribution as well.

The transformations for each hypothesis are given
below with Z being the final transformed observation

1.1

and n the sample size.*

HNl: The null hypothesis is CE : N(0,oi2), or
Z : N (0,1) (10)
where
Zij = Egii (11)

*Z is actually distributed as "t" with approximately
3-8 degrees of freedom, but will be tested as normal. See
ab ove .

 

 

 

26

 

n
2 1 2
i n.J=1 ij
and Si is the positive square root of 812.
C C 2
HN2° The null hypothesis is CEij' N(ui,oi ), or
Z : N(O,l) (13)
where
Z = CEij' CEi (1”)
ij Si
1‘" 1“ <)
CE = — 2 CE. 15
i nj=l ij
s2=.l_§(CE—EE)2 (16)
l n-l _ ij i
j-l
and Si is the positive square root of 812.
o 0 . 2
HN3' The null hypothesis is CEij' N(ai+8i E(VTij)’Oi )

where E(VTij) is the expected value of VTij which is defined

in equation (9) above. Since E(VTi

will be replaced by VTij

parameters di and Bi

) is not measurable, it
for the purpose of estimating the

The inaccuracy resulting from this

replacement is likely to be negligible because VTiJ is com-

piled using a relatively consistent procedure of reporting

'trade and, consequently, is unlikely to depart significantly

:from its expectation.

Testing this hypothesis necessitated rather sophisti-

<3ated transformations of the data.

First, least-squares

Eastimates of a1 and Bi were calculated. However, as is well

 

27

known, least-squares residuals, even under the null hypo-
theses, have a non-scalar variance-covariance matrix and,
consequently, cannot be used directly to test certain
hypotheses about 6 [31). Fortunately, H. Theil [31,32]
has developed a method to obtain a set of (n—k) observable
"residuals" (g) which, under the null hypothesis are un-

biases, uncorrelated and have constant variance:

E(S) = 0 3(33') = 02:n_k . (17)
In addition, if e is distributed normally, the Theil
"residuals" (8) will be normal. It should be pointed out
that Theil's method provides a set of n-k independent re-
siduals rather than n residual elements as k degrees of
freedom have been lost in the estimations of the cOeffi-
icients. These residuals will be employed for testing

HN3.* The Theil residuals were transformed to obtain

2 : N(0,l) (18)
where
9(-
ii
7.13 = Si (19)

*The Theil residuals were calculated by a computer
LDrogram written by J. Ramsey [27].

 

28

1 n—k

S = (

___ )2
i n—k 3:1

313 (20)

and s1 is the positive square root of 512
For reasons to be given below, Section III.2, the
double exponential distribution will be considered as an
alternative to the normal distribution. The double ex-
ponential distribution is a symmetric distribution which
is more peaked than the normal distribution. The proper-
ties of this distribution are discussed in Appendix A.
The transformations for the double exponential test

are given below with Z being the final transformed ob-

iJ

servations and n the sample size.*

HDEl: The null hypothesis is CE : DE(0,Bi), or
z : DE(O,1) (21)
where
CE
2 = ——J-.1 (22)
13 B
i
and
A l n | |
B = —2 CE . (23)
i nj=l ij

 

*Transforming the raw data using estimated rather than
Ibredetermined parameters yields a sample which is not strictly
Ciouble exponential. However, in conducting these tests the
Eissumption will be made that the transformed sample is ap-
IDI‘oximately double exponential.

 

29

 

HDE2: The null hypothesis is CE : DE(A1B1), or
z : DE(O,1) (211)
where
CE -A
213 = —iE—i (25)
B1
A1 = median (CEij) (26)
and
A l n A
B1 11:13; ICEiJ-Ail (27)

The following is a summary of these five hypotheses:

HNI: The error is normally distributed with zero mean
and constant variance.
HN2: The error is normally distributed with constant

mean and constant variance.

H The error is normally distributed with mean a

N3:
linear function of E(VT) and constant variance.

H The error is double exponentially distributed

DEl:
with zero mean and constant variance.

H The error is double exponentially distributed

DE2:
'with constant mean and constant variance.

III.2 Specification of the Error Distribution

Hypothesis H was tested first. The null hypothesis

N1

   

30

is that the composite error is independently and normally
distributed with zero mean and constant variance. The
sample was drawn from SITC commodity group 653 and in-
cluded 18 trade flows with a total of 216 observations.
It should be noted that the sample consists of observations
from a single commodity group, but several trade flows.
The trade statistics were transformed to obtain obser-
vations which, under the null hypothesis, are normally
distributed with zero mean and unit variance, equations
(10)-(12) above. The parameters used to transform the
raw observations were estimated separately for each trade
flow. Using the transformed sample, the Kolmogorov good-
ness of fit test strongly rejected HN1.* (See Table 2.)
The alternative hypothesis was that the composite error
is not distributed as normal with zero mean and constant
variance.

An examination of the results revealed that the
sample distribution was more peaked than the hypothesized
normal distribution, i.e., the sample had too many ob-
servations near zero mean. Because of this phenomenon,
the more peaked double exponential distribution was con-
sidered as an alternative to the normal distribution.
'The parameters of this distribution were estimated for

:for each trade flow as specified in Appendix A. Through

E

* The Kolmogorov goodness of fit test is discussed
1J1 M. G. Kendall and A. Stuart [15,pp.A52-A61]. See also
IX. W. Birnbaum [A].

f .1. Asthauni~Z~W4MJ|ll I 1.. 1 . I.... [A
. ». LNTHMN CV9?

 

 

])n1..1|((. {1.|1P.

 

31

 

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32

transformation of the raw data, observations were obtained
which have a hypothesized double exponential distribution
with zero mean and unit variance parameter. These trans-
formations are presented in equations (22) and (23) above.
The transformed sample was tested using the Kolmogorov
goodness of fit test. The double exponential distribution,
like the normal distribution, was strongly rejected. (See
Table 2.) Again, further examination revealed the sample
distribution to be more peaked than the hypothesized dis-
tribution. These results indicated that the sample dis-
tribution was extremely peaked around zero mean. This
extreme peakedness of the sample distribution could be due
to incorrect transformation of the raw data. Recall from
before, equation (11), that the transformation is simply
division by the estimated standard deviation. This means
that the transformed observations, which have small mag-
nitudes, could be the result of division by standard de-
viations which are too large, i.e., over-estimated. Mis-
specifying the population mean, as is shown below, will
definitely lead to such a bias in the estimation of the
standard deviation.

Assume that X is distributed as follows:

X : N(u,c2) . (28)

Let S2 be an unbiased estimator of the population variance.

 

33

Then

E(s2) (29)

II
Q

where

(30)

If, however, we estimate the variance under some incorrect

*
specification of the mean, say, u # u , we have

E(s2) = o2 + (fi-u)2 (31)

where

(I)
II
II M3

*2
(Xi—u) . (32)

n 1 1

*
Since (u-u)2>0, 32 will be an estimate of the population

variance which is biased upward. Since the standard de-
viation is the positive square root of the variance, it
will also be biased upward.* Thus, if the composite error
mean is not zero, the estimates of the standard deviations,

under hypothesis H ould be biased upward; and the

N1’ w
transformed observations would be biased toward zero yield-

ing a peaked sample distribution as was observed.

 

* This upward bias is reduced somewhat because, even
under the null hypothesis, S is not an unbiased estimator
of c . For sample sizes from 16 to 20 observations the
ratio of E(S) to o is 0.97. See E. E. Cureton [6].

 

3A

This leads to the following alternative composite

error hypothesis:

H The composite error is independently and nor—

N2:
mally distributed with constant mean and con-

stant variance for each commodity group.

The mathematical specification of H was presented in

N2
equations (l3)—(l6) above.

The realism of this revision of the classical as—
sumption can be further rationalized by examining the
procedure used in compiling international trade statistics.
A given commodity group at the 3-digit level of aggrega-
tion consists of numerous individual commodities. In some
cases the inclusion of a particular product in a given
commodity group is arbitrary. For example, one country
may include tractors in a farm machinery commodity group
while another country may call them transportation equip-
ment. For these two countries the composite error may be
due to the difference in the definition of the reported
commodity group. This is called the misclassification
problem. Another possibility is that the two countries
use different prices for valuing the same commodity. This
would lead to different reported values of trade even if
both correctly recorded the units of trade. In this case,
one country would consistently report a higher value. Con-
sequently, there are reasons to believe that the composite

error may have a non—zero mean.

 

35

Under the assumption of a normal distribution, the
population mean and variance shown in equations (15) and
(16) were estimated for each trade flow. The data were
standardized to obtain observations from a hypothesized
normal distribution with zero expectation and unit variance.
The Kolmogorov goodness of fit test did not reject HN2'
(See Table 2 above.) Even though the normal distribution
was not rejected, the double exponential distribution,
hypothesis HDE2’ was tested after, of course, appropriate
parameter estimation and transformation of the data,
equations (25)-(27) above. This test did not reject the
double exponential distribution. For this sample the
Kolmogorov goodness of fit test did not distinguish be-

tween the normal H and double exponential HDE2 hypo-

N2
theses. Consequently, further testing was necessary.

A specific test was considered to distinguish be—
tween these two distributions. The null hypothesis is
that the population is distributed normally with zero mean
and unit variance with the alternative hypothesis being
a double exponential distribution with zero mean and unit
variance. The zero mean and unit variance conform to the
hypothesized distribution of the transformed observations.
Thus, this test is applied to the transformed sample ob—
servations where the transformations are calculated under

the null hypothesis. The null hypothesis is rejected when

 

 

 

 

36

n

n n —
2 < N 2
)3 IX :lXi - 1n{ku('§) } (33)

|_1
i=1 i 21
where n is the sample size, In the natural logarithm, n =3.
1A16 and ka =l.96 for a 5 percent level of significance.*
This test is referred to as the Comparison Test.

The Comparison test was applied to hypothesis HN2’
that the composite error is independently and normally
distributed with constant mean and constant variance against
the alternative that the distribution is double exponential
with constant mean and constant variance. The null hypo-
thesis was rejected in favor of the double exponential dis—
tribution. (See Table 2.) Consequently, HN2 was rejected.
It should be pointed out that this double test rejection
of the null hypothesis reduces the level of significance

of the test, i.e., H is rejected if either the Kolmogorov

N2
goodness of fit test or the Comparison Test reject the null
hypothesis. A reduction of the level of significance means
that the probability of rejecting a true hypothesis is in—
creased. If the Kolmogorov goodness of fit test and the
Comparison Test are independent and if both are conducted
at the 5 per cent level of significance, the probability

of rejecting H , when it is true, is l — (l-.05)2 = .0975.

N2

 

*

The test is adopted from M. G. Kendall and A. Stuart
[15, p. 169]. The fact that the transformed sample is "t"
rather than normal, under the null hypothesis, strengthens
this test as K2=2.1 for the "t" distribution with 18 degrees
of freedom.

 

37

The acceptance of the double exponential distribution
indicates that the transformed sample is still more peaked
than the normal distribution. As before, this could be due
to a misspecification of the mean. For example, the mean
may change from observation to observations. In this case,
estimates of the population standard deviation under the
incorrect assumption that the mean is constant would be
biased upward; and the transformed observations would be
biased toward zero. Reasons similar to those presented
above could be given in favor of the mean changing with
the value of trade, i.e., the mean being conditional on
the value of trade. If misclassification is a problem,
it is probable that trade in the excluded commodity item
changes with trade in the 3-digit commodity group. This
would lead to a composite error that changes with the
value of trade. Likewise, if valuation is a problem, the
difference between the exporter's reported value of trade
and the importer's reported value of trade would be ex—
pected to change as the quantity of trade changes. Of
course, the value of trade will change with the quantity.
It is also likely that the error varies with the observed
value of trade. In compiling trade statistics the problems
involved in classifying trade declarations by commodity
group, in placing values on the particular items traded
and the basic clerical problems in recording the trade de—

clarations will obviously become more complex. This leads

 

 

 

38

to an alternative revision in the classical error as-
sumptions; namely, the composite error mean is condit-
ional upon (varies with) the expectation of the observed
value of trade. This is hypothesis HN3 and will be
tested under the assumption that the mean is a simple

linear function of the expectation of the observed value

of trade. The function is as follows:

CE = “1+81'EWTij) + e (3“)

ij ij

where i specifies the trade flow, j specifies the ob—

servation and s has the following distribution:
a : N(0,012) E(eksl) = o, kfil Cov[s,E(VT)] a o (35)

where k and l vary over all observations in the 1th sample.
The transformations necessary for testing are given in
equations (19) and (20) above. The transformation para-
meters are estimated separately for each trade flow. The
Kolmogorov goodness of fit test was applied to the sample
distribution under the null hypothesis that the trans—
formed population is normally distributed with zero mean
and unit variance. The test did not reject the null hypo-
thesis. The Comparison Test did not reject the null hypo-
thesis and, thereby, supported the normal distribution.
(See Table 2 above.) Consequently, hypothesis HN3 is ac—
cepted.

The results of the preliminary tests conducted in

 

 

 

39

this section are presented in Table 2 above. As the table
shows, HNl and HN2 were rejected while HN3 was not. These
three hypotheses all specify an error which is normally
distributed with constant variance. They differ only in

the specification of the mean. Both the zero mean (HNl)

and the constant mean (HN2) hypotheses were rejected in
favor of the conditional mean (HN3). However, since the
sample was used in specifying the final accepted hypothesis,
it cannot be used to test the hypothesis. In the next sec-

tion hypothesis HN3 will be tested with a new sample.

III.3 Testing the Error Distribution

The specific hypothesis to be tested is that the
random element of the compositeerror (e) is independently
and normally distributed with zero mean and constant vari—
ance, as defined in equations (3“) and (35) above. The
composite error (CE) is assumed to be a linear function of
the expectation of the observed value of trade, equation
(3A) above. As indicated in Section III.l, the least-
squares residuals (e), under the null hypothesis, will
n93 be distributed like a and, therefore, cannot be used
to test hypotheses about 8. However, as indicated, the
Theil residuals, under the null hypothesis, are indepen-
dently and normally distributed with zero mean and con-
stant variance and, therefore, can be used to test certain

hypotheses about 8 [31,32].* The final transformed

 

*The Theil residuals were calculated by a computer
program written by J. Ramsey [27].

 

 

 

40

Observations are derived from the Theil residuals, as
shown in equations (19) and(20) above, and have a hypo-
thesized distribution which is normal with zero mean and
unit variance. The parameters used to transform the ob-
servations are estimated separately for each trade flow
presented in Table 3, and are applied only to observations
from their respective trade flows. The transformed ob-
servations are then aggregated into four larger samples,
i.e., one sample for each SITC commodity group. Table 3
identifies the samples used in testing hypothesis HN3 and
shows the number of observations per trade flow in each
SITC commodity group. There are a total of 84 trade flow
samples with 1579 observations.

The Kolmogorov goodness of fit test was applied to
the four samples specified in the last paragraph. For
this test the sample is hypothesized to be normal with
zero mean and unit variance. The alternative hypothesis
is that the population is not normal with zero mean and
unit variance. The results of these four tests are pre-
sented in Table A. As the table shows, the null hypothesis
that the composite error is normally distributed with con—
ditional mean a+BE(VT) and constant variance was not re-
jected in any of the tests at the 5 per cent level of

significance. In other words, hypothesis H was accepted.

N3
The next step is to test for serial independence.

Actually, serial independence is a proxy for the more

 

 

 

 

 

Al

‘TABLE 3.—-Observations per trade flow for the samples used
to test the composite error hypothesis.a

 

SITC Commodity Groups

 

 

Trade 512 666 711 863
Flow Organic Pottery Power Developed
Codeb Chemicals Generating Cinemato-
Machinery graphic
Film
1 IVT l9 l8 IVT
2 20 2O 18 19
3 20 20 20 IVT
A 20 20 20 20
5 20 20 20 20
6 20 20 20 20
7 20 20 20 IVT
8 IVT 19 1N0 IVT
9 IVT 19 INO IVT
10 19 20 20 IVT
ll 16 15 16 16
12 20 20 20 16
13 20 19 20 20
1A IVT 19 19 IVT
15 19 19 19 l9
l6 l9 l9 19 19
l7 19 19 19 19
18 19 19 l9 l9
19 19 19 19 19
20 19 19 19 IVT
21 19 19 18 IVT
22 IVT 19 IVT IVT
23 19 19 19 19
2A 15 15 15 1A
25 l9 l9 19 15
26 INO INC 17 17

 

aIVT represents trade flows which were excluded because
INO represents trade

flows which were excluded because of an insufficient number
of observations.

of an insufficient value of trade.

b

See Table 1.

 

 

 

 

 

 

A2

TABLE A.——Composite error hypothesis test results.a

 

 

Commodity Number Kolmogorov Comparison
Group Observations Test Test
512 3A1 Accept Accept
666 A25 Accept Accept
711 387 Accept AcceptC
863 258 Accept Accept

 

aA11 tests are conducted at the 5 percent level of
significance. The null hypothesis is that the composite
error is distributed normally with conditional mean and
constant variance, HN3' The transformed observations are

distributed normally with zero mean and unit variance. The
alternative hypothesis for the Kolmogorov Test is that the
transformed observations are not from a population which
is N(0,l). For the Comparison Test the alternative hypo-
thesis is double exponential, i.e., DE(O,1).

bThe number of observations is less than the original
sample size because of the method of estimating the Theil
Residuals. Two observations from each trade flow are lost.
See H. Theil [31].

0This test rejected normality for the initial sample.
However, after eliminating two outlying observations (from
a sample of A33 observations) the test results were re-
versed to accept normality. An outlier is an observation
which is suspected of being from a different population
than the other observations in the sample. Popular tests
for outliers are based upon one of two sample statistics.
One test accepts the outlier hypothesis if the range of
sample observations is sufficiently large relative to the
estimated standard deviations. See David et. a1. [7]. A
second test accepts the outlier hypothesis if the deviation
of the outlier from the sample mean is sufficiently large
relative to the estimated standard deviation. See Thompson
[33]. The outliers identified in this sample occur in
trade flows 10 and 17 (See Table l.) and are accepted as
outliers at the 1 per cent level of significance using
either of the above tests.

 

 

 

 

 

 

 

 

 

 

 

 

A3

important concept of general independence. That is to
say, in the estimation of stochastic models it is im-
portant to know whether the observations are drawn in-
dependently or, if not, how they are related. Testing
for dependence can be a Herculean task since there are
literally an infinite number of possible relationships
among the observations. Because of this, investigators
usually hypothesize some logical relationship among the
observations and test this relationship. If it is re—
jected, they conclude that the observations are inde-
pendent. It is usually assumed that if economic events
are related, this relationship will be most obvious be-
tween events which occur near one another, i.e., nearness
in the sense of time, geography or some other logical con—
cept. The most commonly hypothesized relationship as—
sumes that events are related through time, i.e., that a
given current event is related to the same event occurring
in other time periods. Furthermore, if events are related
over time, it is most logical that the relationship is
strongest between events occurring in consecutive time
periods, thus, the term serial. The test for independence
of observations will hypothesize such a serial relationship,
i.e., an observation in time period t will be related to an
observation in time period t+1. If this hypothesis is re-
jected the conclusion will be that the observations are

independent.

 

 

 

 

 

 

 

AA

In hypothesis HN3’ the composite error consists of
a predictable element, the conditional mean (ai+ BiE(VTij))’
and a random or stochastic element (Eij)' Serial dependence
assumes a relationship between 61(3) and 61(J+l). But none
of the Eij are observable since they are deviations from
the conditional mean function which must be estimated. The
Durbin—Watson statistic, however, was developed to test
for serial independence in linear regression models.*
Therefore, the Durbin-Watson statistic will be used to

test for serial independence. This test, of course, is

 

based on least-squares (not Theil) residuals.

A serial ranking of the observations exists only
within trade flows. Since there are 8A trade flows in—
cluded in the total sample, there will be 8A tests for
serial independence. The results of these tests, conducted
at the 5 per cent level of significance, are presented in
Table 5. Suffice it to say that there is a lack of con-
clusive evidence for rejecting serial independence; the
independence hypothesis was rejected in A of 8A trade flows
with non-conclusive tests in 13.

The last test is for independence of the residuals
from the value of trade. Of course, acceptance of HN3
clearly indicates a linear relationship between the com—

posite error and the value of trade. However, in

 

*

See C. F. Christ [5, pp. 523—531]. Critical values
for the Durbin-Watson statistic are given in Table B-A,
p. 672.

 

A5

TABLE 5.——Tests for Composite Error Independence.a

 

Serial Independence Independence from the
Tradeb Value of Trade

 

Flow Commodity Group Commodity Group
COde 512 666 711 863 512 666 711 863

 

>3>ZII>3>>II>
fi>3>SU 2|>
>3>3>3>§I>5U
SU>3>SU3>SUW
T133112) 3>

\oooqoxmtwmre
>223>3>3>

3>SUJ>3>fi> n>n>w

3>Zh>3>3>3>3>3>3>3>3>3>3>3>h>3>3>3>23>3>3>3>203>
3>D>3>3>3> 5U3>Z

>n>a>a>sun>a>n>a>a>n>3>n>a>wn>3>a>3>a>a>a>a>a>a>

H

CD
3>Zfi> Zh>3>3>3>3>3> ZZ>3>
ZZS>3> >3>Zh>3>3>3>w3>>3>3>
>W3> SUIUD>3>3>3>3> >SUD>5U
:UbWID >3>3>SU313>3>C1>3>3>SUSU

11>D>3>3>
>3>SUZU

 

aA11 tests are conducted at the 5 per cent level of
significance. Omissions represent trade flows for which
there exists no sample. See Table 3. The entries are:
A: Accept the independence hypothesis.
R: Reject the independence hypothesis.
N: The test, based upon the Durbin-Watson statistic,
is inconclusive.

bSee Table l.

 

 

 

 

 

 

A6

Stochastic models it is important to know whether the sto-
chastic element is related to any of the explanatory vari-
ables and, if so, how it is related. In the case of hypo-
thesis HN3 it becomes important to know if the random
element (a) is related to the value of trade. Tests will
be conducted for the 8A trade flows rather than the ag-
gregated samples used in testing HN3' Combining trade
flows covering different countries and/or different com-
modity groups is inappropriate since, if dependence exists,

the parameters expressing the dependence could differ from

 

one trade flow to another and,consequently,go undetected
in tests combining observations from more than one trade
flow. To complicate the problem, each trade flow has an
insufficient number of observations to use a Contingency
Table and the Chi—square test for independence.

If hypothesis HN3 is true, the composite error will
be randomly distributed about its conditional mean. How—
ever, if e is dependent upon the expectation of the ob-
served value of trade, the composite error can be written

as
CE = d+B-E(VT) + e (36)
where

e = f[E(VT)] + E CovLe,E(VT)] = o (37)

 

 

 

 

 

 

 

A7

and f[E(VT)] is some non-linear function. Rewriting (36),

we have

CE = a + h[E(VT)] +é (38)
where

h[E(VT)] = B-E(VT) + f[E(VT)] Cov[e,E(VT)] = o (39)

This is to say that the linear function (36) is misspecified

since

Cov[s ° E(VT)] # 0 (”0)

 

J. Ramsey has developed a set of tests for certain misspeci—
fications of the general linear regression model [26]. One
of the specification errors considered is an incorrect func-
tional form, i.e., the dependent variable is not a linear
function of the explanatory variables but rather some non-
linear function. Ramsey's test RESET is especially ap-
propriate for testing the type of misspecification hypo-
thesized in equation (36) and is designed to detect non-
linear relationships among the variables [26,28, p. 7].

The null hypothesis is that the linear function is correctly
specified. That is to say, in equation (36) f[E(VT)] is
identically zero and E[e-E(VT)] = 0 . If the null hypothesis
is not rejected, the conclusion is that the errors are ran-
domly distributed about the linear conditional mean function

and, consequently, there exists no dependence between the

 

 

 

 

 

 

A8

random element in the composite error and the expectation
of the observed value of trade.

The results of the misspecification tests for the
8A trade flows are presented in Table 5. Independence
was rejected in 25 trade flows (30 per cent of the sam-
ples). There are more rejections than would be expected
by chance under the null hypothesis which indicates that,
at least for the rejected trade flows, the conditional
mean is not a simple linear function of the expectation

of the observed value of trade and, therefore, the random

 

element of the composite error is not independent from
the value of trade. However, for a majority of the trade
flows (70 per cent) the independence hypothesis was not
rejected.

Further examination of these results reveals that
the rejections are concentrated by commodity group and
by country. Commodity group SITC 666 had only 2 rejec-
tions in 25 trade flows, a result well within the accep-
tance region. SITC groups 711 and 863, however, had re—
jection rates of AA per cent, 10 rejections in 23 trade
flows and 7 in 16 respectively. Two countries, Switzerland
and Norway, had rejections in at least 50 per cent of their
possible trade flows, 6 of 8 and 2 of A respectively. These
mixed results indicate that more effort may be required to
specify the conditional mean functions. However, hypo—

thesis HN3 will not be rejected on the basis of this

 

 

 

 

 

 

 

 

 

 

 

 

 

A9

evidence. To repeat HN3:
CE = a+s-E(VT)+e (Al)
where

e : N(O,o2) E(ekel) = o, k s 1 Cov[e-E(VT)] = 0 (A2)

III.A Additional Evidence

 

The first evidence is obtained through an examination
of estimated parameters of the conditional mean function.
If both parameters (a and B) are zero, hypothesis HN3 re-

duces to the classical error assumptions, i.e., hypothesis

 

HNl' If only the slope parameter (8) is zero, the hypo-
thesis reduces to HN2‘ The "t" statistic will be used to
test the hypothesis that B = 0 and the F-Test will be used
for a = B = 0. The results of these tests are presented
in Table 6.

The classical error assumptions hold only when

a = B = O. This case is indicated by H in Table 6 and

1
occurs in 17 of the 8A trade flows. The most extreme
departure from the classical case considered is hypothesis

H where 8% O. This is indicated by H3 and occurs in

N3
A0 trade flows. The remaining alternative, d # 0 and
B = 0, is hypothesis HN2° This case is denoted by H2
in Table 6 and appears in 27 trade flows.

A closer examination shows that the trade flow

samples of SITC 666 seem to be consistent with a simpli-

fication of the composite error hypothesis to HN2' In

 

 

 

 

 

 

 

 

 

 

 

 

 

 

50

TABLE 6.—-Test results for the conditional mean parameters.a

 

 

Trade SITC SITC SITC SITC
F1°wb 512 666 711 863
Code
1 H3 H1
2 H1 H1 H3 H3
3 H3 H2 H3
u H2 H2 H3 H3
5 H2 H2 H2 H3
6 H1 H1 H2 H3
7 H3 H1 H1
8 H1
9 H1
10 H3 H3 H3
11 H3 H3 H3 H1
12 H3 H3 H2 H3
13 H3 H2 H3 H2
1a H2 H3
15 H1 H1 H3 H2
16 H1 H3 H3 H3
17 H2 H2 H3 H1
18 H3 H3 H2 H3
19 H1 H2 H3 H3
20 H2 H2 H3
21 H2 H1 H3
22 H2
23 H2 H2 H2 H3
2A H2 H2 H1 H3
25 H3 H2 H3 H3
26 H2 H3

 

aAll tests were conducted at the 5 per cent level of
significance. The results of tests on the estimated para-
meters of the conditional mean function are as follows:

H1:
H :

b

2

The hypothesis a= B: 0 is accepted using the F-
test. Hypothesis HN3 reduces to hypothesis HN

This indicates the remaining alternative, dfiONl
and B=O. Hypothesis HN3 reduces to hypothesis
HN2'

The hypothesis 8=0 is rejected using the t-test.
Hypothesis HN3 does not reduce to any of the

alternative hypotheses considered.

See Table l.

 

 

 

 

51

only 6 of the 25 trade flows was 8 7‘ o. For SITC 512
this was the case in 8 of 20 trade flows. For SITC 711
and 863 the predominent evidence is toward rejection of
such a simplification of the composite error distribution.
For comparison purposes, hypothesis HN2 was fully tested
using the sample presented in Table 3. The results were

as expected. For SITC 666 hypothesis H was not re—

N2
jected. Nor was it rejected for SITC 512, but for both
SITC 711 and 863 HN2 was rejected. Consequently, under
HN3’ an examination of the parameters of the conditional

mean functions seemsto indicate when the hypothesis can

be reduced to HN2 or to HNl'

III.5 Conclusion

As a result of a test using a preliminary sample
consisting of 216 observations, the classical assumptions
concerning the errors in observing the value of trade are
rejected. Consequently, a series of alternative hypotheses
were specified and tested using the same sample with the
resulting acceptance of hypothesis HN3’ that the error is
normally distributed with constant variance and a mean
which is a linear function of the expectation of the ob-
served value of trade. Since the preliminary sample was
employed in specifying hypothesis HN3’ a new sample con-
sisting of 1579 observations was drawn to test formally
this hypothesis. Four independent tests, one for each

commodity group in the new sample, were conducted on

 

 

 

52

hypothesis HN3 with no test rejecting the hypothesis.

Independence tests were also conducted using the new

sample without sufficient evidence to reject the formal
specification of hypothesis HN3'

As an outgrowth of these results, the parameters

of the conditional mean function of hypothesis HN3 were
examined for significance. In those cases where B=O,

hypothesis HN3 reduces to HN2; and in those cases where
d=B=0, hypothesis HN3 reduces to HNl' This examination
revealed that for commodity groups SITC 512 and 666 8:0
in a majority of the trade flows. For SITC 711 and 863

this was not the case. As a result, hypothesis H was

N2
formally tested using the new sample with results as ex-
pected. For commodity groups SITC 512 and 666, hypothesis
HN2 was not rejected. However, for SITC 711 and 863, HN2
was rejected.

In the next chapter the problem of estimating in-
ternational price and income elasticities when the value

of trade is measured with error will be considered.

 

 

 

 

 

 

 

CHAPTER IV

IMPORT DEMAND ELASTICITIES
AND ERRORS IN TRADE DATA

It has long been known that the effectiveness of
international trade policy is highly dependent upon the
value of import and export price and income elasticities.
Most empirical estimates of price elasticities using
samples which pre-date WW II were quite low indicating
inelastic demand. These results lead to the conclusion
that devaluation of a currency would not improve the de-
valuating country's balance of payments. However, those
in policy deciding positions, and most of the profession,
could not agree with this conclusion. In a pathbreaking
article in 1950 Orcutt [25] showed that elasticity es-
timates are biased downward, toward zero, when there
exists errors in the price variable. This finding was
quickly championed by those who felt that devaluation
was the correct policy to improve a country's balance of
payments. In 1961 Kemp [1A], updating Orcutt's analysis,
demonstrated that since the quantity variable is obtained
by deflating the value of trade by the price of imports,

there exists errors in the quantity of trade variable as

53

 

5A

well. As a result, import price elasticities are biased
toward minus one rather than zero. This result favored
the devaluation "pessimists".

This controversy has been diminished more recently
with new empirical estimates of import demand elasticities
using post-war data. [1,12,18]. These recent studies yield
substantially higher estimates indicating that the import
demand function is quite elastic with respect to both price
and income. Furthermore, these studies have been relative-
ly consistent in their findings. It is safe to say that
the concensus today favors the devaluation "optimists".

It should be pointed out that most estimates of im-
port price elasticities cover broadly aggregated commodity
groups, i.e., dividing total imports into 5 or 6 commodity
groups. More recently there has been emphasis placed on
elasticities for much narrower definitions of commodity
groups. Both the U. S. Tariff Commission and the Office
of the Special Representative for Trade Negotiations have
been concerned with the effect of tariff reductions on
individual commodities. Recent and future trade liberali-
zation policies will change the price competitiveness of
specific U. S. industries. These changes could bring
economic hardship to firms involved as consumers shift
from domestic to foreign sources of supply. The alleviation
of such hardship will require the reallocation of resources

to other areas. Accurate price elasticity estimates are

 

 

 

55

necessary to identify in advance the industries involved
and to quantify the nagnitude of the necessary reallo-
cation. Consequently, efficient trade policy requires
accurate elasticity estimates at the micro— as well as the
macro—level.

This chapter will be concerned with the effect that
errors in estimating the value of trade have on estimated
import demand elasticities. The recent estimates of
Houthakker and Magee [l2] and Kreinin [18] will be examined
in light of the conclusions of Chapter III concerning the
errors in trade data. In Section IV.1 the model used in
both of these studies is modified so that elasticity es-
timates can be adjusted to account for errors in the value
of trade statistics. These adjustments require the es-
timation of an adjustment parameter. In Section IV.2 the
adjustment parameter will be estimated for each commodity
group covered in the studies mentioned above. The ad-
justed elasticity estimates will also appear in this section.
Finally, the implications for estimating import demand elas-
ticities for more narrowly defined commodity groups will be

discussed.

IV.1 The Import Demand Model and Errors in Trade Data
The studies by Houthakker and Magee [l2] and Kreinin
[18] estimated the following import demand function:

los(M) = 80 + Bllos(Y) + B2log(P) + u (1)

 

 

 

 

 

 

 

 

 

 

 

 

56

where M is an index of the quantity of imports,
Y is an index of real gross national product,
and P is the price of imports index relative to the
domestic wholesale price index.
Since we have information about the errors in the value of
imports in current value, this function must be trans-
formed to fit our information. Noting that the quantity
indices are constructed by deflating the value of imports
index by the price of imports index, equation (1) can be

rewritten as
V Pm
108(?;) = 80 + 81108(Y) + B210g(§g) + u (2)

where V is the value of imports index

and Pm and Pd refer to the import and domestic price
indexes respectively.

Transforming equation (2) to isolate log(V) as the dependent

variable, we have

log(V) = 50 + Bllog(Y) + (B2+l)los(Pm)

— 82102(Pd) + u (3)

Since V is an index it can be represented by the current
value divided by its value during the base period.
Furthermore, the base period value is a constant and can,
therefore, be incorporated with the constant term. Conse—

quently, we have

*
log(V*) = B

0 + Bllog(Y) + (82+1)log(Pm)

- B2log(Pd) + u (A)

 

 

 

 

 

57

where V* is the current value of imports,
80* is 60 + log(VBP) where VBP is the value of im-
ports during the base period,
81 is the income elasticity of demand for imports,
B2 is the price elasticity of demand for imports.
(The model assumes that an increase in domestic
prices has the same effect as an equal percentage
decline in the price of imports.)

In some cases the independent variables were lagged.

However, in this event the above equation can be modified to
‘ P

m
los<Vt*) = 80* + Bllog(Yt_l) + 8210g(P t'1)

 

dt—1

+ log(Pm ) + u . (5)
t

Ball and Marwah tried several different lag structures in
their estimates of the import demand function and concluded
"It was found that the estimates were not highly sensitive
to changes in the lag structure . . ." [1, p. 396].
Kreinin [18] also found this to be the case. Upon esti—
mating the price and income elasticities, first, with no
lag and, second, with a one quarter lag, he found the
estimates to lie well within one standard error of each
other. As a result of the seemingly unimportance of the
lag structure, equation (A) will be used for both the lagged
and unlagged models.

In equation (A) the investigators have assumed that

the variables are measured without error. However, the

 

 

 

 

 

 

 

 

58

results of Chapter III show that V* is not free from error.

In Chapter III it was concluded that
(r-s) = a0 + d1 - E(VT) + e (6)

where r is the error in observing the value of exports,
s is the error in observing the value of imports,
VT is the estimated value of trade (Chapter III,
equation ll)

and e is independently and normally distributed with
zero mean and constant varianceo

To allocate the error between r and s, it will be assumed

that

r = aOr + alr ° E(VT) + er (7)
and

s = dos + ”15 - E(VT) + as (8)
where

o‘Or ‘ 0‘0s = 0‘0

0Llr - oLls - 0"1 (9)

Er — as = 5

Since s=m-TVT, we can rewrite equation (8) as

m—TVT = + a ° E(VT) + as (10)

o‘Os ls
where m is the observed value of imports
and TVT is the unknown true value of imports°

After assuming that E(VT) = TVT, we have

m = + (als + 1) - TVT + as (11)

0‘Os

0]."

 

 

 

 

 

 

 

 

 

59

m=ao*+al*0TVT+€*. (12)

This model is a classical errors in variables model since
TVT is unknown. We do, however, have observations on TVT
reported by both the exporter (x) and the importer (m),

where
x = TVT + r (13)
m = TVT + s . (14)

Using the average of these observations in place of TVT for

estimating the parameters of equation (12) above gives

* * *
do + a1 (VT) + e

where (15)

VT = %(x+m) = TVT + %(r+s) ,

From Chapter II, Section 11.1, the least-squares estimator
for al*, under the classical assumptions, is biased as
follows:
* d *
m1) = __11__ (16)
Var(§(r+s))

Var(TVT)

Assuming that r and s are independent,

Var(%(r+s)) = % (o 2 + c 2)

r s . <17)

 

 

 

 

60

Furthermore,

Var(r—s) = or? + 052 = 1: Var(%(r+s)) (18)

Since (r-s) is simply the difference between x and m, Var
(r-s) can be estimated. If it is also assumed that %(r+s)

is independent from TVT, we have

Var(VT) = Var(TVT) + Var(%(r+s)) (19)
or

Var(TVT) = Var(VT) — % Var(r—s) (20)

where both terms of the right side of equation (20) can be
estimated from sample information. Substituting equations

(17), (18) and (20) into equation (16) yields

5(-
a

E(al ) — . (21)
Var(r—s)

l +
u Var (VT) - Var(r-s)

 

Consequently, the errors in variables bias can be removed
from the estimated parameters of equation (12) above.

As was stated before, in estimating import demand
elasticities researchers have assumed that the value of
trade is measured without error, i.e., they have used m
as though it were equal to TVT, as the dependent variable.
Their estimates have, therefore, measured the responsive-
ness of the observed quantity of trade rather than the

true quantity to changes in the relevant independent

,1

 

 

 

 

61

variable. However, upon substituting equation (12) into

the import demand model, equation (A), we have

* * * *
Log(dO +d -TVT+e ) = so +81Log(Y) + (82+l)log(Pm)

l

— 6210g(Pd) + u . (22)

Isolating TVT as the dependent variable poses somewhat
of a problem.* However, if equation (12) can be approxi-

mated by

 

 

*Isolating TVT can be accomplished by rewriting
equation (22) as .

do + dl TVT + e = BOY PM Pd e
B B (8 +1) -8
_ o 1 2 2 u * *
or TVT — a—; Y PM Pd e - do - e

l

Elasticities can be estimated by differentiating this
equation with respect to the relevant independent vari-
able, dividing through by TVT and multiplying by the
same independent variable. For example, the income
elasticity, in expectation, is

E(B )=E3TVT_ B1 = 81
1TVT 381 TVT 1"“— “OT—0*—
- _T__¥__
do + dl TVT

* *
Since do and d1 can be estimated separately, the bias
in 81 can be removed by inserting a value for TVT, say,
the mean of the observed value of trade.

Elasticity bias due to errors in observing the
value of trade was examined using this approach with

 

 

 

 

 

 

 

 

 

 

62

 

results similar to those presented in the body of this
chapter. The results of examining the first entry in
Table 8 will be discussed in this note.

Equation (12) was estimated for the finished
manufactures commodity group with the following re—
sults:

m = —u.u186 + 1.015 TVT R2 = .9988

(9.2896) (.0060)

Correcting for the bias due to using the observed VT
rather than TVT yields

*
E(A * _ Var(r—s) = 0‘l
0‘1 ) ' -Var TVT - Var r—s 1.00l5 '

 

Because of the insignificant magnitude of the errors
in variables bias, do and d1* will not be corrected.
To account for the error in observing the value of
trade in the elasticity estimates, we have

6
Mel ) = 3 , .
TVT 1 0L0
'fl—A’r
dO + dl TVT

This elasticity will be evaluated for TVT equal to

the sample mean of the observed value of trade ($1M20).
Applying this correction to the first entry in Table 8
gives

 

* *
Allowing do and d1 to vary over two standard devi-
ations from their respective point estimates yields
minimum and maximum adjusted elasticity estimates of
2.58 and 2.66 respectively. These results are identical
to those presented in Table 8 in the text.

 

 

 

 

 

 

 

63
'
m = 6(TVT)Ye€ (23)

'
where e is a random disturbance, isolating TVT is
straightforward.* Upon substituting equation (23)

into equation (22), we have

1
it-
Log(6(TVT)Ye€ = 50 + Bllog(Y) + (82 + 1) log(PM)

 

- leog(Pd) + u (2“)
Rewriting (2M) gives
a 81 (82+l)
T =— ._ +—
Log( VT) Y + Ylog(Y) Y log(PM)
B2 1
- 7Vlog(Pd) + u (25)
where
* 9 I
a = 80 _ log(5) u = u - e

Subtracting the log(Pm) from both sides of equation (25)

yields

*
log(QM) = d* + 81*log(Y) + 82 log(PM>

+ s *1og(Pd) + u* (26)

3

 

*

Of course, care must be taken to see that equation
(23) is a good approximation to equation (12). If not,
some other method of isolating TVT as the dependent vari-
able must be used.

 

 

 

64

where
a* = 2
Y
81
81* = 17 is the income elasticity of imports
(82+1)
82* = {——1T—— — l} is the elasticity of imports
with respect to the price of imports
B
83* = 1% is the elasticity of imports with respect
to a change in the domestic wholesale price index {6
I
and u* = 3 .
Y 5

Consequently, it is possible to adjust the published im—

port elasticity estimates of 81 and B2 to account for errors

 

in measuring the value of imports provided that equation (23)
can be used to approximate the relationship between the ob-
served value of imports and the expected true value of

imports.

IV.2. Elasticity Estimates

As stated earlier in this chapter, the estimates
published by Houthakker and Magee [12] and Kreinin [18]
will be examined. Since the sample incorporated in the
above studies differed from that used in Chapter III, a
new sample of U. S. imports was drawn which conforms with
the samples used in these elasticity studies. The sample
consists of 36 quarterly observations covering the period
of 1959 to 1967 for each commodity group presented in
Table 7 below. Since the data source was the same as that
used in Chapter III, the errors in observation associated
with this new sample are assumed to have the same charac-

teristics as those examined in Chapter III; namely, the

 

 

 

 

 

 

65

TABLE 7.—-Estimates of the adjustment parameter.

 

 

 

Power Linear
Commodity ya Function Function
Group (s )
Y R2 R2
Finished
Manufactures 1.003 .9985 .9988
(.0067)
SITC 5 and 7 1.025 .9977 .9984
(.0082)
SITC 6 and 8 .972 .9980 .9977
(.0073)
SITC 5, 6, 7 .997 .9987 .9988
and 8 (.0060)
a b

See equation (23). See equation (12).

errors are independently and normally distributed with
constant variance and mean linearly conditional upon the
estimated value of trade.

In Section IV.1, equation (26) above, it was seen
that published elasticity estimates could be adjusted to
account for errors in the observed value of imports. The
adjustment required only the estimated value of Y from
equation (23) provided that this equation is a good approxi-
mation of equation (12). To compare this approximation
the R2 statistic is calculated for both of these equations
for each commodity group. They are presented in Table 7.
A casual examination shows these two equations to fit the
data almost identically. Consequently, it appears de—
fensible to use equation (23) as an approximation to

equation (12) for the purpose of adjusting those import

 

 

 

 

66

elasticities estimated from the model presented in
Section IV.l above.

Equation (26) above shows the relationship between
the published elasticity estimates for 81 and 82 and the
adjusted elasticities, 81*, 82* and 83*. Note that there
are now three parameters instead of two. This is due to
the fact that a change in the import price effects both
the quantity variable, which is measured with error, and
the relative price variable. Consequently, the adjusted

import price elasticity will differ from the domestic

 

price elasticity.*

The point estimates of y and their standard errors
for the four commodity groups covered in the elasticity
studies are presented in Table 7.** An examination of
these estimates shows that only two of the four are signi-
ficantly different from unity. This is important because
for y = 1, the elasticity estimates are unaffected by er-

rors in import data. In this case equation (23) reduces to

m = y TVT e6 (27)

V
where e : N(O, o2)

 

* 9(-

*
If y is unity, B2 = 83 = 82.

xx

Since the errors in variables bias is negligible,
estimates of y will not be adjusted for this bias. See
footnote page 61.

 

 

 

67

or,
E(m) = 5TVT

Consequently, as will be shown below, the percentage
change in observed imports equals the percentage change
in the estimated value of trade. Writing the percentage

changes yields the following:

E(m ) - E(m ) 6 TVT - 6 TVT
l O = l 0 (28)

E(mo) TVTO

 

Consequently, in this case, even though imports are ob—
served with error, the percentage change in imports is
not.* Furthermore, for those commodity groups in which
9 is significantly different from unity (in a statistical
sense), the magnitude of ; is very close to unity. This
indicates that errors in trade data may not substantially
effect the estimation of import demand elasticities, at
least for these commodity groups.

The published elasticity estimates are presented in
Table 8 together with their adjusted estimates. For each
elasticity parameter two adjusted estimates were calculated:

(1) a minimum estimate was obtained by setting y equal to

 

*Since the true value of trade is unknown, its value
has been estimated from both the importer's and exporter's
estimate. (See Chapter III, equation (11).) The validity
of the above statement depends upon this estimate of the
true, but unknown, value of trade.

 

 

 

68

TABLE 8.—-Published and adjusted elasticity estimates.

 

 

 

Authorsa Commodity Parameterb PublishedC Adjustedb
Group Elasticity Elasticity
Estimates Estimates
Minimum Maximum
H-M Finished Y 2.63 2.58 2.66
Manufac- P —H.05 -3.95 —M.10
tures m
Pd —u.05 -3.97 -4.09
K Finished Y 2.35 2.30 2.37
Manufac— Pm -N.72 -4.61 —U.78
tures Pd —u.72 —u.63 -u.77
K SITC 5 Pm -2.0 -l.88 -1.97
and 7 Pd -2.0 -1.92 —1.98
K SITC 6 P -5.6 -5.67 -5.88
and 8 Pm 6 66 8
d —5. -5. -5. 3
K SITC 5, 6
7 and 8 ’ Pm -U.2 -4.15 -u.3l
Pd -u.2 -4.l6 —u.29
a

H—M: Houthakker and Magee [12].
K: Kreinin [18].

Y : Income elasticity of demand for imports.

P ' Elasticity of demand for imports with respect

to the price of imports.

Pd : Elasticity of demand for imports with respect
to the domestic price level.

Pm = Pd is a constraint imposed by the authors.

dThe minimum adjusted elasticity estimate is obtained
by adding two standard errors to the point estimate of y
and the maximum from subtracting two standard errors from
the point estimate of Y . See Table 7.

 

 

69

its point estimate plus two standard errors and (2) a
maximum estimate by letting Y equal Y minus two standard
errors. This gives roughly the 95 per cent confidence
interval for the adjusted point estimate of the published
import demand elasticity. It is quite obvious from Table
8 that the adjusted elasticity range is insignificantly
small when compared with the magnitude of the published
estimate.

Furthermore, for those published estimates where

the standard errors of the point estimate were reported,

 

the entire range of the adjusted estimate falls well
within one standard error of the point estimate.

Even though elasticity estimates for different
commodity definitions will not be examined, it is useful
to know if the conclusions of this section can be gener-
alized to different levels of aggregation. For this pur-
pose, the adjustment parameter, Y in equation (23) above,
was estimated for U. S. imports of various SITC 3-digit
commodity group trade flows selected from those included
in Table 3, Chapter 111. These estimates are presented
in Table 9 below. It should be pointed out that there
are approximately 150 3—digit commodity groups in the
SITC 5, 6, 7 and 8 aggregation examined in Table 8 above.
The R2 statistics for both the linear and approximating
power functions, equation (12) and (23) respectively, are

also presented in Table 9.

 

 

 

 

 

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.mmfin moanmfipm> :H whoppo opp pom oopmznom coon p0: o>m£ moquHumm mmmsem

o

 

 

 

 

 

 

 

 

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Amm.v “mo.av
Ho. Ho. so. so. am. om.m mm
Ama.v Azm.v Amo.v Aoa.v
as. mm. mm. we. mm. mo.H mm. mm. mo.H mm. mm. wm.a mm
Azm.v Aqm.v A©H.v Amm.v
mm. mm. mm. mm. ma. mH.H mm. mm. mm. mm. mm. oo.H 2m
Amm.v Amfl.v “mo.v Amm.v
mm. mm. mo.a am. mm. mH.H mm. mm. Hm. om. so. mm.a mm
Amo.v Amm.v
am. am. mH.H 0:. NH. as. Hm
AoH.V Asa.v
um. mm. mo.H as. cs. mo.H om
Ana.v Ams.v Amo.v Ama.v
MW m:. as. mm. :0. ma. 2w. om. om. so.a mm. mm. om.H ma
:3 Am: :3 :2: -
mm. om. as. am. 3:. em.a as. me. mm. me. so. ms.a ma
Aoa.v Ame.v “mo.v Amo.v
ma. we. oo.fi ma. Hm. mo.a am. mm. mm. :m. mm. :m.H NH
Anfi.v Amm.v
E. 3. am. m3. 3. 3. S
Amfl.v Anfl.v AmH.V
mm. mw. om.H m. as. em. He. we. Ho.H ma
can 30m A>mv sag 30s A>mv gal 31m A>nv sag 30m A>wv
p . 6 U o r w .. o n a . o
> > > > 6600
o m 0 1m 0 m 0 30am
m o m oopmge
mow oeHm Has osHm mom ueHm mfim oeHm
.AmuLOQEa .m .Dv wbmuoEMLmd ucoeumSnUm ecu mo moumEHumm nooomammll.m mqm¢e

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

71

A comparison of the R2's indicates that the power

function is a relatively good approximation to the linear
function. Furthermore, in a majority or the cases where
the R2's differed by .05 or more, the R2 for the power
function was larger than for the linear function. An
examination of the estimates of the adjustment parameter
(Y) reveals substantial variation among the estimates.
They range from .07 to 2.30. In those cases where Y<1
elasticity estimates based upon this data would be biased
toward zero whereas for Y>l the magnitude of the elasti—
city estimate is biased upward. Hence, on a priori
grounds we cannot predict the direction of the bias with-

out some information about the error term.

IV.3 Conclusion

The most significant conclusion of this chapter is
that the import demand elasticity estimates reported by
Houthakker and Magee [12] and Kreinin [18] are not biased
due to errors in the estimated value of imports. However,
this conclusion cannot be generalized to elasticity es-
timates covering different commodity definitions. The
information presented in Tables 8 and 9 indicates that
there exists significant variation in the error bias of
elasticity estimates covering different levels of aggre-
gation, different commodity groups and/or different trad-
ing partners. Without some a priori information concerning

the errors in the value of imports, one cannot specify the

 

 

 

 

72

direction or the magnitude of this bias in estimating
import demand elasticities.

It should be mentioned that throughout this exam—
ination of error bias, it has been assumed that the other
variables in the model have been measured without error.
Consequently, the adjusted estimates presented in Table
8 are not claimed to be unbiased estimates of the re- 7”
spective import demand parameters. They are only claimed
to be free from bias due to errors in measuring the value

of trade.

 

 

Finally, these results are relevant only for the g
data under investigation and the economic model examined.
Different data could, of course, have different error
characteristics. The model is an important consideration
for two reasons. First: in elasticity studies it is
important only to correctly observe the percentage charge.
The magnitude of the observations is irrelevant, as was
shown in Section IV.2 above. In another model it may be
important to correctly observe the magnitude of the vari-
ables. Second: as was shown in Chapter II, Section 11.1,
even under the classical assumptions concerning the obser-
vational error, least—squares estimators of the regression
coefficients are biased when the independent variable is
measured with error. The bias depends upon the variances
of the error and of the independent variable. In the

model examined in this study the variable measured with

 

 

 

73

error happened to be the dependent, not the independent,
variable. In other models, studying other economic con-
cepts, the value of trade may enter as an independent
variable, in which case the error variance also becomes
relevant. As a consequence, these results are not meant

to be generalized beyond the scope of this study.

 

 

 

CHAPTER V

SUMMARY AND CONCLUSIONS

The central problem of this thesis is concerned
with obtaining single—equation least-squares estimators
of economic parameters when the variables are measured
with error. In general, it is necessary to make a priori

assumptions about the error terms. The most commonly

 

made assumptions are that each error term is independently
and normally distributed with zero mean and constant
variance. These assumptions are generally referred to
as the classical assumptions.

Starting from this point in Chapter II, a method
is presented which can be used to test the above assump-
tions concerning the error term. The method of testing
requires two independent sets of observations (x and m)
on a single set of events (TVT), and is based on the
following elementary relationships:

X=TVT+r (1)

m=TVT+s (2)

where r and s are the observation errors. Upon substracting,

7A

 

 

 

75

the true value (TVT) cancels, leaving

x - m = TVT + r — TVT - s

= r - s. (3)

Under the classical assumptions concerning r and s, we

have
r N(O, Or2)
s : N(O, 632) (u)
E(rs) = 0.
Consequently,
(r—s) : N(O, o 2 + 0 2). (5)

1" S

The validity of the classical assumptions can then be
tested by reference to observations on (r-s).

In Chapter III this method is applied to selected
international trade flows. The value of trade flowing
from one country to another is reported independently by
the exporter and by the importer according to common in-
ternational commodity groups (SITC). For a given com-
modity group and a specific exporter and importer, the
difference between these reported figures is one ob—
servation from the hypothesized distribution (r-s) speci-
fied above. Using a sample drawn in this manner, our
results indicate quite clearly that, for the set of data

under investigation, the classical assumptions must be

 

 

 

76

rejected. Consequently, alternative error distribution
hypotheses were specified and pre-tested using a pre-
liminary sample of 216 observations. This pre-testing
led to the adoption of the following error hypothesis:
the error is independently and normally distributed with
constant variance and mean linearly conditional upon the
estimated value of trade. Using a new sample of 1579
observations, this hypothesis was subjected to four in—
dependent tests. In each test the hypothesis was not
rejected.

In Chapter IV the results of this testing are ape
plied to the problem of estimating import demand elasti-
cities. The examination of a model used frequently for
estimating international price and income elasticities
revealed that published elasticity estimates could be
adjusted to account forerrorsin observing the value of
trade. Estimates of U. S. import price and income
elasticities for highly aggregated commodity groups,
which were published in two recent studies, were adjusted
for errors in observing the value of imports. Since the
estimates did not differ significantly from the published
estimates, it was concluded that errors in observing the
value of imports did not significantly effect the esti-
mation of import demand elasticities for these highly
aggregated commodity definitions. Further examination

revealed that this conclusion cannot be generalized to

 

 

77

elasticity estimates for commodity groups defined at lower
levels of aggregation. For these commodity groups there
appears to be substantial variation in the bias in elasti-
city estimates due to errors in observing the value of
trade.

To restate the major conclusions: (1) For the data
under investigation, the classical assumptions concerning F“
the errors in observations must be rejected. As a result,
an alternative error distribution was hypothesized, tested

and accepted. The alternative hypothesis is that the

 

errors are independently and normally distributed with

1......

constant variance and mean linearly conditional upon the
estimated value of trade. (2) Estimates of import demand
elasticities for highly aggregated commodity definitions
are not significantly effected by errors in observing the
value of imports. However, this conclusion cannot be
generalized to elasticity estimates for more narrowly

defined commodity groups.

 

 

APPENDIX A

THE DOUBLE EXPONENTIAL DISTRIBUTION

 

EM MMS‘ \I We.

 

 

THE DOUBLE EXPONENTIAL DISTRIBUTION

In general form the double exponential probability

distribution function (DE(A,B)) is

_I

f(X) = 2B

mpg-

 

an «a

where A is the mean and 2B2 is the variance. The stan-
dardized distribution is obtained by substituting t for

x where

ml

to get

— - e -|t|. - m<t<m

f(t) g

The maximum-likelihood estimators for A and B are
derived separately. Consider the distribution function

for B=l.

f((x)= % e — IX-AI.

Writing the maximum-likelihood function in natural log-

arithms (1n),

n
E

lnL(X/A) = - n°ln2 - i=1

IXi-AI.

79

 

 

 

 

 

,1‘
I
H
A
1..
l

   

80

This function is maximized by minimizing the second term

on the right. This yields

A = median (Xi)’

The estimator for B is derived from the likelihood function

 

 

2: Xi'A
lnL(X|A,B) = - n ° ln2 - n ° lnB _i=l I B I F
For B > O, the likelihood function becomes r
n
- O 1 z -
1nL(X|A,B) — - n-ln2 - n lnB - 8 i=1 |xi Al.
I
:
Maximizing
BlnL n l n
3B ="""L7‘2 z lXi'Al =0
B B i=1
or
n

 

 

 

 

REFERENCES

 

 

10.

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e-q

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_..._ "k'

 

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