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An Analysis of Weekly F.O.B.
Prices for Fresh Limes
AN ANALYSIS OF WEEKLY F.O.B. PRICES FOR
J. Scott Shonkwiler and Robert
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
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.
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 . . . ... . .
LIST OF TABLES
1 Filter models . .
2 Residual cross correlations
3 Causality tests . .
4 Estimated transfer function
5 Florida lime price equation
. . . . .
. . . . .
. . . . .
model . . ... .
. . . . .
LIST OF FIGURES
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
J. Scott Shonkwiler and Robert L. Degner
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
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
AVERAGE WEEKLY F.O.B. LIME PRICES AND SHIPMENTS
1976/77 THROUGH 1979/80 SEASONS
($/10 lb. carton)
( .. )
I *s= ~
35 1000 cartons
30 (50 lb. units)
Week of Season
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
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
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-
The cross correlation between the residuals is denoted at lag K as
(3) uv(K) = E(ut-k,')
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
- .011B3 -
U2 = .108
.176B)FLSt = a
X2(19) = 16.24
Florida Lime Price
(1 .392B)FPRt = a2t
32 = .023
X2(19) = 15.7
Mexican Lime Shipments
(1 + .742B + .476B2 + .443B3 + .412B4 + .236B5)MLSt = a3t
(.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
(5) Q = T [r^^(k)]2 indicates X leads Y at the a level
of significance if it exceeds X m, and
(6) Q = T 2 [r^(k)] indicates Y leads X at the a level
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 and Florida lime Florida lime Mexican lime
Lagged Residuals shipments (alt) price (a2t) shipments (a2t)
Table 3.--Causality tests.
a1t i a3t
a2t / a3t
a2t / a3t
6 degrees of freedom.
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
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.
(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
(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
The noise processes in the dynamic shock model may be replaced by
their associated filter models which were presented in Table 1. This
(13) (1 .392B)FPRt
By carrying out the indicated
the resulting remainders, the
FPRt = (-.085 -
= (-.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
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.
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
B(s) = a0 + Z bi Z(s)i
Z(s)i = (s n )di s(S ni)
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 +
and each of S seasonal parameters for the variables in X1, say the first,
is recovered as
B(s)1 = Z(s) "11
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.
'2 .2 .00078
= .0171 19) = 20.5 a=.364
SEASONAL VARIATION OF FLSt PARAMETER
AND 95 PERCENT CONFIDENCE INTERVAL
I I I I 1 -
SEASONAL VARIATION OF FLSt_ PARAMETER
AND 95 PERCENT CONFIDENCE INTERVAL
40 45 50
5 10 15 20
5 10 15 20
g I m
SEASONAL VARIATION OF FLSt-2 PARAMETER
AND 95 PERCENT CONFIDENCE INTERVAL
- -Y -
S S I
30 35 40
SEASONAL VARIATION OF FLSt-3 PARAMETER
AND 95 PERCENT CONFIDENCE INTERVAL
4' *. .
5 10 15 20 25 30 35 40 45 50
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
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
SEASONAL VARIATION OF SUM OF PARAMETERS FOR
FLSt, FLSt-1, FLSt2 AND FLSt-3 and 95 PERCENT
CONFIDENCE INTERVAL FOR THIS SUM
5 10 15 20 25 30 35 40 45 50
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.
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.
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
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.
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 -
where G. is the coefficient on the i lag of X .
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