HISTORIC NOTE

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not reflect current scientific knowledge
or recommendations. These texts
represent the historic publishing
record of the Institute for Food and
Agricultural Sciences and should be
used only to trace the historic work of
the Institute and its staff. Current IFAS
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site maintained by the Florida
Cooperative Extension Service.

Copyright 2005, Board of Trustees, University
of Florida

DECEMBER 1980

TECHNICAL REPORT
80-2

-

An Analysis of Weekly F.O.B.

Prices for Fresh Limes

(Jo

AN ANALYSIS OF WEEKLY F.O.B. PRICES FOR

FRESH LIMES

J. Scott Shonkwiler and Robert
Assistant Professors

L. Degner

Y:-: A.a.... .. : r

University of Florida
IFAS, Food and Resource Economics Department
Gainesville, Florida 32611

The Florida Agricultural Market Research Center is
a service of
The Food and Resource Economics Department
of the
Institute of Food and Agricultural Sciences

The purpose of this Center is to provide timely, applied research

on current and emerging marketing problems affecting Florida's agri-

cultural and marine industries. The Center seeks to provide research

and information to production, marketing, and processing firms, groups

and organizations concerned with improving and expanding markets for

Florida agricultural and marine products.

The Center is staffed by a basic group of economists trained in

agriculture and marketing. In addition, cooperating personnel from

other IFAS units provide a wide range of expertise which can be applied

as determined by the requirements of individual projects.

^-a)

LT 1

TABLE OF CONTENTS

LIST OF TABLES . . . . .

LIST OF FIGURES . . . . .

INTRODUCTION . . . . .

The Florida Lime Market . . .

Causal Analysis an Transfer Function Identifi

Seasonality of the Lime Price Equation .

SUMMARY . . . . .

FOOTNOTES . . . . .

REFERENCES . . . ... . .

. .

catii

. .

LIST OF TABLES

Table

1 Filter models . .

2 Residual cross correlations

3 Causality tests . .

4 Estimated transfer function

5 Florida lime price equation

. . . . .

. . . . .

. . . . .

model . . ... .

. . . . .

Page

ii

iv

1

3

4

15

25

26

27

Page

7

9

10

14

18

LIST OF FIGURES
Figure Page

1 Average weekly F.O.B. lime prices and shipments 1976/77
through 1979/80 seasons . . . . 2

2 Seasonal variation of FLSt parameter and 95 percent
confidence interval . . . . . 19

3 Seasonal variation of FLSt-1 parameter and 95 percent
confidence interval . . . . ... 20

4 Seasonal variation of FLSt-2 parameter and 95 percent
confidence interval . . . . . 21

5 Seasonal variation of FLSt-3 parameter and 95 percent
confidence interval . . . . 22

6 Seasonal. variation of sum of parameters forFLS FLS. I,
FLS_ 2and FLSt 3 and 95 percent confidence interval for
thi sum . . . . . 24

AN ANALYSIS OF WEEKLY F.O.B. PRICES FOR
FRESH LIMES

J. Scott Shonkwiler and Robert L. Degner

INTRODUCTION

Many recent studies have analyzed the supply and marketing of fresh

Florida limes (Degner and Mathis; Mathis; Degner and Rooks; Ward and De;

Degner, Shonkwiler and Cubenas [A, B],; Pagoulatos, Shonkwiler and Degner).

A concern common to most of these studies is that of depressed prices

resulting from large seasonal supplies during the summer months. Another

concern focuses on the effect of Mexican lime exports to the U.S. and

corresponding effects on Florida fresh lime prices and future production.

To date, however, no study has integrated these two concerns so that a

single model can address them simultaneously.

The objective of this study is to provide a systematic means for

testing causal relationships and for specifying distributed lag formulations

within the context of a model for weekly lime prices. Our discussion pro-

ceeds as follows: We first review the general market forces affecting

Florida's F.O.B. lime prices and state our hypotheses. Next, causal

relationships are investigated via the transfer function approach. The

magnitude and statistical significance of seasonal parameters are then

investigated, and finally, we discuss the implications of our analysis.

J. SCOTT SHONKWILER and ROBERT L. DEGNER are assistant professors of
food and resource economics, University of Florida.

The Florida Lime Market

Weekly fresh lime prices typically exhibit substantial variation

during the annual April through March marketing season (Figure 1). The

lime production cycle explains much of the volatility. Very few limes

are produced during the winter and early spring, but large quantities

are available during late spring and most of the summer. Thus, it is

hypothesized that lime quantities, or shipments, significantly affect

weekly prices.

Because the shelf life of limes is of sufficient duration, we postu-

late that inventories in the marketing channel probably affect weekly

prices. Unfortunately there are no data available on the volume of

unsold limes held by wholesalers and retailers. Therefore previous levels

of lime shipments are used as a proxy.

Lime imports, primarily 'Persian' limes from Mexico, have increased

dramatically in the past five years. These imports are indistinguishable

from limes produced in Florida and are essentially perfect substitutes.

Thus, they are hypothesized to have a significant, adverse affect on

Florida lime prices.

There is a widespread belief in the industry that consumer demand

for fresh limes varies seasonally, and is affected by special holidays.

Seasonal demand parameters, explored by Ward and De, are further investi-

gated and refined. Models specifying no seasonality and models with

seasonality are estimated.

Other domestically produced limes and lemons are commonly believed to

have significant negative effects on Florida fresh lime prices, but the

preponderance of evidence indicates otherwise. Ward and De obtained a

weak relationship between lime prices and lemon supplies and other

FIGURE 1

AVERAGE WEEKLY F.O.B. LIME PRICES AND SHIPMENTS
1976/77 THROUGH 1979/80 SEASONS

$10

Lime Prices
($/10 lb. carton)
( .. )

Lime Revenues
($100,000)
( ..)

I
.11
/
~I ..
'I
'I

I *s= ~
'I
I'
~1

35 1000 cartons

Lime Shipments
30 (50 lb. units)
( )

40
Week of Season

30

studies have found only a marginal relationship (Degner, Shonkwiler and

Cubenas; Pagoulatos, Shonkwiler and Degner). Although fresh limes are

also produced commercially in California, their lime production is pri-:.

marily comprised of the seeded 'Mexican' variety which is viewed by many

retailers as inferior to the 'Persian' variety because of its tendency to

quickly turn yellow, resulting in decreased marketability (Degner and

Mathis).

Finally factors such as population, consumer disposable income, and

long term advertising and promotion programs, while important in an extended

analysis, are subsumed under a trend term over the four year period studied.

Thus we focus our attention on how Florida fresh lime prices are related to

Florida and Mexico lime shipments, seasonal factors, and a secular trend

term.

Causal Analysis and Transfer Function Identification

Four years of weekly data on lime prices and shipments were analyzed

for the 1976-77 through 1979-80 April to March seasons (Federal-State

Market News Service). First differences of the logarithms of the variables

of interest were used to provide stationary series for the Mexican imports

(MLS) and Florida F.O.B. price (FPR) and shipments (FLS) variables.1

For a traditional economic analysis, prior information and/or experimen-

tation is used to discriminate between endogenous and exogenous variables as

well as to specify the form and length of lagged responses. While theory: and

observation can suggest the general nature of such relationships, the exact

nature and timing of causal relationships may be unknown in an emoirical

study (Bessler aniSchrader). The transfer function approach proposes that

the data be given the opportunity to provide this information.

The transfer function approach can employ the Granger notion of

causality to discern interrelationships between two series, Specifically

Granger says that"Yt is causing Xt if we are better able to predict Xt

using all available information than if the information apart from Yt had

been used." (p.428). This prediction oriented definition of causality is

conceptually straightforward, but it is subject to the major deficiency

that rarely is it possible to include all relevant information in predicting

Xt. A simpler definition is proposed which states that Yt causes Xt when
it can improve the prediction of Xt compared to the prediction of Xt taking

into account the past history of Xt alone,

Causality may be unidirectional or have feedback, and maybe instan-

taneous or delayed. To indicate unidirectional causality we write Xt_j Yt

for j > 0 (j = .0 implies an instantaneous relationship). Feedback occurs

when current, lagged, and future Y causes and is caused by current, lagged

and future X.

Haugh has suggested a method to elicit the existence of Granger

causality between two time series. Assume that the two stationary series

may be represented by the univariate models:

(1) G(L)Xt =ut

(2) H(L)Yt = vt

where G(L) and H(L), the filters, are invertible polynomials in the lag

operator L, and ut, vt are white noise processes (innovations) having
2 2 2
variances 0 v, respectively By construction. each individual noise

process is not autocorrelated and represents that part of the series which
cannot be explained by past information. To assess Granger causality in

a systematic manner, Haugh suggests examining the cross correlations

between the two residual series.

To implement this technique for eliciting causality univariate

models were identified and fitted to the three time series. Using the

Box-Jenkins approach auto regressive models of order five (AR5) were

fitted to the Mexican imports (MLS) and Florida shipments (FLS) series,

and a first order auto regressive model (ARI) was used to filter the

Florida lime price series (FPR). The estimated models are presented in

Table 1. The chi-squared statistics were calculated under the null hy-

pothesis that the residuals from these models are mutually uncorrelated.

The associated a-levels indicate that the null hypothesis cannot be re-

jected at a reasonable level of significance for any of the estimated

filter models. It is assumed that each residual series is white noise.

This assumption then implies that the cross correlations between the.

series are not confounded by the effects of autocorrelation in the in-

dividual series.

The cross correlation between the residuals is denoted at lag K as

(3) uv(K) = E(ut-k,')

[E(ut)2E(v)2] 1/2

If for a positive value of k the cross correlation is significantly different

than zero we say that Xt leads (causes) Yt' and conversely for a negative

value of k we say that Yt leads (causes) Xt.

Of course, u and v are not observed but are replaced by their estimated

values form (1) and (2). The sample counterpart to the lefthand side of (4)

then is ruv(k). Under the null hypothesis that X and Y are independent

series, Haugh has shown that the r(k) are asympototically normally and

independently distributed with mean zero and standard deviation T -1/2, where

T is the sample size. Once the residual cross correlation estimates have

been calculated, statistical tests of significance for individual

Table 1.--Filter Models

Florida Lime Shipments

(1 + .355B + .08882
(.069) (.073)

- .011B3 -
(.073)

U2 = .108

.1584 -
(.073)

.176B)FLSt = a
(.070)

X2(19) = 16.24

Florida Lime Price

(1 .392B)FPRt = a2t
(.064)
32 = .023

X2(19) = 15.7

Mexican Lime Shipments

(1 + .742B + .476B2 + .443B3 + .412B4 + .236B5)MLSt = a3t

(.068) (.081)

(.082) (.081) (.069)

o2 = .260 X (19) = 24.3

a = .641

a = .677

a = .185

estimates are obtained by the criterion that sample cross correlations

exceed their approximate standard deviations by a factor of two. That is

(4) r^(k) 1 2T-1/2

indicates a significant cross correlation. Individual significant cross

correlations may then be used to detect causal directions at specific lags.

Overall tests of unidirectional causality have been suggested

(Pierce) using the following statistics

m
(5) Q = T [r^^(k)]2 indicates X leads Y at the a level
k=l
2
of significance if it exceeds X m, and
-m 2
(6) Q = T 2 [r^(k)] indicates Y leads X at the a level
y*x uv
k=-1
of significance if it exceeds x2, m.

The calculated residual cross correlations are presented in Table 2.

Table 3 summarizes the results of the statistical tests which follow from

the proceeding discussion. The null hypothesis of no instantaneous

causality between the series may be rejected with high degrees of confidence

for the Florida lime shipment-Florida lime price relationship and the

Florida lime shipment-Mexican import relationship. On the other hand there

appears to be no instantaneous relationship between Florida lime prices

and Mexican shipments.

The unidirectional causality tests show that the null hypothesis of

one series not causing the other may be strongly rejected in one case and

marginally rejected in another. The clear rejection implies that Mexican

imports lead (or cause ) Florida lime prices. Additionally at the .112

level we can reject the hypothesis that Florida lime prices do not cause

Table 2.--Residual cross correlations

Current residuals
Current and Florida lime Florida lime Mexican lime
Lagged Residuals shipments (alt) price (a2t) shipments (a2t)

-.184

-.179

-.032

-.017

-.060

-.110

-.022

alt

alt-I
alt-2

alt-3
al t-4

alt-5

alt-6
a2t

a2t-'1

a2t-2
a2t-3

a2t-4
a2t-5

a2t-6

a3t

a3t-1

a3t-2

a3t-3

a3t-4

a3t-5
a3t-6

-.184

-.106

-.012

-.093

-.013

-.022

-.019

.268

-.041

-.007

-.033

.106

.041

.068

.268

.010

-.082

-.019

-.031

-.012

.001

.023

.145

.051

.013

.027

-.092

-.060

.023

-.099

-.215

-.169

-.060

.039

.005

Table 3.--Causality tests.

Instantaneous Causality

Hypothesis

alt a2t

a1t i a3t

a2t / a3t

Unidirectional Causality

Hypothesis

alt a2t

alt Aa3t

a2t /-alit

a2t / a3t

a3t /+alt
a3t /*a2t

a
208 observations.

b
6 degrees of freedom.

Calculated correlation

-.184

.268

.023

Calculated X2b

10.30

1.72

4.38

7.72

4.24

18.64

a-level

.008

.001

.746

a-level

.112

.944

.625

.259

.645

.005

Florida lime prices. However from Table 2, the cross correlation between

alt-land a2t reveals that we may reject at the .01 level the hypothesis

that lime shipments lagged one period do not cause current Florida lime

prices. Thus these results indicate instantaneous and unidirectional!

causality from the two shipment series to the price series. Further

instantaneous causality exists between the two shipment series, but it

appears that neither series leads the other.

It should be noted that several difficulties with this method of

identifying causal relationships have been noted. It has been shown

by Sims (1977) that the chi-square tests for unidirectional causality

are biased toward acceptance of the null hypothesis. Additionally, Feige

and Pearce have pointed out that the causality tests may be highly con-

ditioned by the filters used to obtain the whitened noise processes u

and v. Nevertheless, this procedure provides a systematic means for

permitting the data themselves to suggest patterns of interrelationship

and generates the major results necessary for specifying the transfer

function.

The causal relationships detected via cross-correlating the residuals

of the filtered series readily lend themselves to transfer function analysis.

Suppose that Xt leads Yt, but not conversely, then as Sims (1972) has

noted we can properly write Yt as a distributed lag on Xt. Mathematically

(7) Yt = V(L)Xt + nt

where nt is some (complex) noise process. The weights VO, VIL, V2L2...

are called the impulse response parameters of the system. As noted by

Zeller and Palm, this expression has as its counterpart the final form

representation of a dynamic econometric model.

In determining the order of V(L) the residual cross correlations of

the filtered series are analyzed to suggest the so-called dynamic shock

model which is of the form

(8) vt = W(L)ut + Y(L)at

where ut and vt are the previously defined noise process, 'F(L) is a
polynomial in the lag operator L of the same degree as W(L) and at is
the dynamic shock model error process (Haugh and Box). By construction,
Cov (ut, us) = 0 for all t f s; i.e., the ut are orthogonal to each other.
Thus, each individual W. represents the (bivariate) regression coefficient

relating vt to ut-k, i.e.
-a1
(9) Wi p uv(1) .

Once the order of the dynamic shock model is determined and the

appropriate elements of W(L) estimated, the impulse response model is
obtained by replacing the observed error processes by the filtered series:

(10) H(L)Yt = W(L)G(L)Xt + Y(L)at

or

(11) Yt = H(L)-1W(L)G(L)Xt + H(L)-1Y(L)at

which may in general be estimated by a non-linear least squares or
maximum likelihood algorithm.
Given the nature of the cross correlations between a2t and alt, a3t
the general form of the dynamic shock model is hypothesized to be

(12) a2t = (W0 W1B) alt + (-W2B2 W3B3)a3t + nt.

The estimated parameters are calculated to be

W0 = -.0849

W1 =
Wi =

W2 =

W3 =

.0826

.0639

.0503

The noise processes in the dynamic shock model may be replaced by

their associated filter models which were presented in Table 1. This

substitution yields

(13) (1 .392B)FPRt

By carrying out the indicated

the resulting remainders, the

(14)

FPRt = (-.085 -

.122B3 -

= (-.0849- .0826B)(1 + .355B + .088B2

.011B3 .158B4 .176B5)FLSt + (-.0639B2

- .0503B3)(1 + .742B + .476B2 + .443B3

+ .412B4 + .236B5)MLSt + nt.

multiplications and divisions and truncating

impulse response function is identified as

.146B .094B2 .043B3)FLSt + (-.064B2

.106B4 .097B5 .087B6 .070B7 .039B8)MLSt

+ nt.

where nt is some (complex) noise process.

The noise component may be ignored at the expense of incurring some

inefficiency in estimation. Thus the sample data was used to estimate

the model given by expression (14) yielding the results in Table 4. There

is substantial agreement between the identified transfer function model

(expression 14) and the estimated model (Table 4). In fact 95 percent

confidence intervals on the estimated parameters include each of the point

estimates of the corresponding identified transfer function's parameters.

Table* 4.--Estimated transfer function model.

Standard error

.030

,032

.033

.031

.021

.025

.026

.026

.026

.024

.020

Variable

FLSt

FLSt-1

FLSt-2

FLSt-3

MLSt-2

MLSt_3

MLSt-4

MLSt-5

MLSt-6

MLSt-7

MLSt-8

Parameter

-.133

-.187

-.108

-.034

-.071

-.134

-.148

-.116

-.093

-.062

-.034

= .0203

= 1.61

Before attempting to correct for the error process on the hypothesized

transfer or dynamic regression model, we consider ways of accommodating

possible seasonal influences in the following section.

Seasonality of the Lime Price Equation

There is evidence of strong seasonal effects associated with the

Florida lime price- quantity relationship (Ward and De). Seasonal

influences may be ascribed to varying levels of consumer demand as affected

by weather patterns or holidays, changes in marketing methods as production

levels vary, or changes in size or quality of the fruit during the season.

In particular we are concerned with identifying whether these changes

shift the price-quantity curve or alter its slope, or both. To investigate

such seasonal effects a flexible technique is required.

Ward and De incorporate seasonality via slope and intercept interaction

by using a sine wave having a 52 week period. While this may be justified

by the strong annual lime production cycle, this technique appears unneces-

sarily restrictive because it requires the seasonal effects to be smooth,

continuous and symmetric. An alternative approach is to assume that

seasonal effects may be captured by piecewise linear segments termed splines.

By making the linear segments small enough, even highly curvilinear re-

lationships may be closely approximated. The method used in this study

is outlined below.

Let s represent the seasonal unit of observation and S represent the

length of season so that s = 1, ..., S. Those points or values of s where

the slopes of the splines may change will be termed nodes and designated

by ni with i = 1, ..., k. The general form of the seasonal linear spline

function may be written (Robb)

B(s) = a0 + b0s + (a1 + bl(s n1))d1 + (a2 + b2(s n2))d2 + ...

where an individual d. = 1 if s > ni, otherwise d. = 0.

By imposing the constraint that the piecewise linear sections be connected
at the node points the following expression can be derived
k
B(s) = a0 + Z bi Z(s)i
i=l

where

Z(s)i = (s n )di s(S ni)
S

Let Z(s)t represent the vector (1, Z(s)1t, ..., Z(s)kt) and note that s = t
when t < S. Partition the design matrix X into [X1 !X2} where X, is the
T x I submatrix of variables to be investigated for seasonal effects.
Then the T x k matrix Z which incorporates the seasonal splines has as
its tth element Xlt 8 Z(s)t.
Estimation proceeds via the model

Y = [Z :X2] 1 +
2

and each of S seasonal parameters for the variables in X1, say the first,
is recovered as

B(s)1 = Z(s) "11

l1k

and the variance of B(s), is given as

V[B(s)1] = Z(s)Cov(11... 3lk)Z'(s).

To implement this technique, decisions must be made concerning the

number and placement of the nodes. It was decided that four nodes

placed evenly apart and corresponding to spring, summer, fall and winter

quarters should be detailed enough to investigate seasonal changes.

Both intercept and slope shifters for the Florida shipments variable were

included in the full model. However the intercept splines Sl-S3 were

dropped from this model due to their high level of insignificance

(F3,179 = .0105). The estimated model's residuals were then analyzed

and an autoregressive error structure of first and fifth orders AR(1,5)

was imposed to whiten the residual series. This estimated model appears

in Table 5.

Despite the large number of regressors in the estimatedmodel note

that its calculated variance is substantially below that of the filter

model presented in Table 1. Although a number of coefficients on the

Florida shipments variables and their transformations are not different

than zero at customary levels of significance, the total effect of all

seasonal forces may result in an individual seasonal coefficient being

significant during certain parts of the season. The seasonal patterns of

the calculated parameters are presented in Figures 2 through 5 along with

their corresponding 95 percent confidence intervals.

Examination of Figures 2 through 5 points out the relative insensitivity

of Florida price to current and lagged Florida lime shipments. Note-that from

the 24th week no coefficient is significant at the .05 level. The

seasonal coefficients on FLSt and FLSt_1 follow nearly the same pattern

and depict a significant inverse relationship between price and quantity

during a 20 week interval near the first of the season. The seasonal

coefficient on FLSt_2 is negative and significant for two short periods

during the first half of the season.

Table 5.--Florida lime price equation.

Variable

Intercept

FLSt

FLSt*S1

FLSt*S2

FLSt*S3

FLSt-l1

FLSt-1*S1

FLSt- *S2

FLStI*S3

FLSt-2

FLSt-2*Sl

FLSt-2*S2

FLSt-2*S3

FLSt-3

FLSt_3*S1

FLSt-3*S2

FLSt-3*S3

MLSt-2

MLSt- 3

MLSt-4

MLSt-5

MLSt-6

MLSt-7

MLSt-8

S

Parameter

-.0365

-.03536

.0457

-.0375

.0107

-.0594

.0571

-.0323

.00127

-.09.11

-.00856

.0229

-.0234

-.069.7

-,042

.0327

-.0109

-.0589

-.119

-.118

-.0788

-.0584

-.0366

-.0244

.00183

Standard error

.026

.055

.017

.020

.016

.069

.020

.022

.019

.075

.020

.022

.021

.072

.020

.020

.018

.020

.026

.029

.030

,029

.026

.019

'2 .2 .00078
= .0171 19) = 20.5 a=.364

FIGURE 2

SEASONAL VARIATION OF FLSt PARAMETER
AND 95 PERCENT CONFIDENCE INTERVAL

20 25

35 40

Parameter
Value

.2.

-.3

-.4

-.5

I I I I 1 -

5 10

45
Week

_~'LL''-L

FIGURE 3

SEASONAL VARIATION OF FLSt_ PARAMETER
AND 95 PERCENT CONFIDENCE INTERVAL

40 45 50

Week

Parameter
Value

.2

.1 ,

0 .

-.1.

-.2.

-.3 .

-.4

-.5.

5 10 15 20
5 10 15 20

g I m

- -

FIGURE 4

SEASONAL VARIATION OF FLSt-2 PARAMETER
AND 95 PERCENT CONFIDENCE INTERVAL

- -Y -

S S I

30 35 40

45 50

Week

Parameter
Value

.2-

.1 -

0.

-.1 -

-.2 .

-.4

-.5

_`LL''I

,..~r

. *

FIGURE 5

SEASONAL VARIATION OF FLSt-3 PARAMETER
AND 95 PERCENT CONFIDENCE INTERVAL

4' *. .

5 10 15 20 25 30 35 40 45 50
Week

Parameter
Value

0 1-

-.1 .

-.2

-.3

-.5

o

Figure 5 shows that the seasonal coefficient on FLS t-3 is not

significantly different than zero at the .05 level over the entire

season. Note that a larger alpha level would reveal a significant

positive effect centered around the 13th week.

Clearly these results show that the effects of current and lagged

Florida shipments on current price are highly variable and not subject to

straightforward interpretation. Because the price-quantity model is

hypothesized to be neither a supply or demand curve, there was no a priori

expectation as to the pattern of the seasonal coefficients. Indeed we

can conclude that with 95 percent confidence there is an inverse relation-

ship between price and some quantity variables only during the first half

of the year, otherwise there is no significant (a= .05) effect.

The total effect of current and lagged Florida shipments can be found

by summing the seasonal coefficients found in Figures 2 through 5. These

results are presented in Figure 6. Again the same general pattern of

a significantly negative long run response to Florida shipments is seen

during the first half of the season, and no significant relationship

appears thereafter.

Additional characteristics of the estimated model stem from the other

estimated coefficients. The lagged Mexican shipments variables are generally

significantly negative at the .05 level. The cumulative effect of lagged

Mexican shipments yields a .494 coefficient. This suggests that a 10 percent

increase in Mexican shipments at all lags reduces Florida price by almost

5 percent. Clearly this is a much stronger effect than that of cumulative

Florida shipments for all but about 10 weeks during the season.

In completing the discussion of the estimated model, it is seen that

the intercept term is negative. Because all variables are specified as

first differences, this coefficient represents a secular trend effect and

FIGURE 6

SEASONAL VARIATION OF SUM OF PARAMETERS FOR
FLSt, FLSt-1, FLSt2 AND FLSt-3 and 95 PERCENT
CONFIDENCE INTERVAL FOR THIS SUM

Sum of
Parameters

0

-.2

-.4

-.6

-.8

-1.0

-1.2

5 10 15 20 25 30 35 40 45 50

Week

_ __

suggests (ceteris paribus) prices have been trending downward. On the

other hand, the coefficient for the variable denoting week (s) is signifi-

cantly positive. This suggests (ceteris paribus) that over the season

fresh lime prices tend to increase. Finally the calculated-X2 statistic

for the regression model implies that there is no pattern to the rcesilduals

at any conventional level of significance.

Implications

The previous discussion has quantified the relationship of fresh

Florida lime prices to current and lagged Florida lime shipments and lagged

Mexican lime shipments. Mention of price flexibilities or implied demand

elasticities has been avoided for several reasons. First the hypothesized

model does not follow closely the theoretical constructs which guide

specification of demand curves. Secondly, the market is not fully modeled

because the demand for processing limes has not been accounted for. Un-

fortunately, weekly data on processing lime use is proorietory and un-

available. Thirdly, some of the seasonal coefficients on current Florida

shipments show a (weakly) positive magnitude. This may suggest that over

part of the season a demand curve has not been identified, but rather a

hybrid supply/demand curve has been estimated.

These apparent shortcomings in the analysis presented do not preclude

meaningful interpretation of the results, however. An immediate consequence

of this study is the implication that the Ward and DE findings may be

misleading. During the period they analyzed, 52 percent of limes harvested

went to processing use, yet no mention of this important outlet is made.

Further, their calculation of tremendouly high implied elasticites of

demand during much of the season appears unrealistic given the nature of

the product and the weekly observation units. Finally, their demand model

is incomplete because it does not take into account Mexican lime imports

as important substitutes.

Our results imply that during the May to August peak harvest season

prices are significantly, inversely related toFlorida fresh lime shipments

Because the total effect of these increased shipments depresses price

by a less than proportionate amount, we agree with Ward and De that iprorating

is not an economically acceptable solution to the relatively low prices which

occur during this period. If some of this production could be spread more

evenly across the season, however, revenues would be increased.

A major implication of this study stems from the analysis of lagged

Mexican lime shipments. Florida lime producers apparently face a strong

competitor. The estimated distributed lag on Mexican shipments shows that

shipments three and four weeks prior have a substantial impact on Florida

lime prices. By monitoring Mexican lime shipments, Florida producers may

be able to avoid some price erosion by either accelerating or delaying

marketing when a particularly large Mexican shipment enters the United

States. Additionally, the case against prorating is further strengthened

in light of the fact that reduced Florida supplies may cause it to lose

marketing share or market channels.

SUMMARY

This study investigated the response of Florida lime prices to levels

of Florida and Mexican lime shipments. The lag structures on the lime

shipments variables were discerned using a transfer function or dynamic

regression approach.

The analysis offered a systematic way of relating prices and quantities.

Further a flexible transformation was introduced which permitted the

coefficients on the Florida shipments variables to vary seasonally. The

estimated model then yieled seasonal effects which either depicted a

significant ( a= .05 ) inverse relationship between Florida shipments

and price, or no significant relationship at all during a large part of

the season. The effects of Mexican lime shipments on Florida lime prices

are inverse and substantial.

This study represents an extension of the Ward and De report by

incorporating the Mexican data. Further the use of unrestricted lag

patterns, less severely restricted seasonal components., and presentation

of statistics on the reliability of the seasonal coefficients make the

results more general and complete.

FOOTNOTES

1 Stationarity implies that the series possesses a finite, time

invariant mean and variance.

2 For example, if G(L) is of degree k then the left hand side of

expression 1 may be written

(1 GIL G2L2 GkLk)X = Xt GXt_1 G2Xt2 -

... GkXt-k

where G. is the coefficient on the i lag of X .
1 t

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