The role of seasonality in modeling agricultural markets is well recognized. However, traditional approaches to account for seasonality assume that seasonal pattern is constant, even though some evidence of changing seasonal pattern exists in the literature. This paper seeks to explore the impact of incorporating changing seasonal pattern into poultry market modeling. Keywords: seasonality, trigonometric variable, seasonal frequency.seasonality, trigonometric variable, seasonal frequency., Livestock Production/Industries, Marketing,

Ji, Wen

English

Research Papers in Economics

 
   
 
 
 
 
Changing Seasonal Patterns in the Poultry Market 
 
 
 
 
 
 
Wen Ji 
 
 
 
Department of Agricultural and Resource Economics 
North Carolina State University 
Raleigh, NC 27695 
Tel. (919) 515-3006 
Email: wji@unity.ncsu.edu 
 
May 6, 2000 
To be presented at American Agricultural Economics Associations annual meeting July 
30 ~  Aug 2, 2000, Tampa, Florida 
 
 
ABSTRACT: The role of seasonality in modeling agricultural markets is well 
recognized. However, traditional approaches to account for seasonality assume that 
seasonal pattern is constant, even though some evidence of changing seasonal pattern 
exists in the literature. This paper seeks to explore the impact of incorporating changing 
seasonal pattern into poultry market modeling.  
 
Keywords: seasonality, trigonometric variable, seasonal frequency. 
 
 
 
 
 
 
 
Copyright 2000 by Wen Ji. All rights reserved. Readers may make verbatim copies of this document for 
non- commercial purposes by any means, provided that this copyright notice appears on all such copies. 
   1
Changing Seasonal Patterns in the Poultry Market 
 
 
1. Introduction 
 
Many agricultural markets exhibit characteristics of significant seasonality both in 
prices and quantities. The traditional approach for accounting for seasonality in modeling 
demand relationships has been to use dummy variables. Examples include Malone an 
Reece (1976), Haidacher et al. (1982), Wohlgenant and Hahn(1982), Martinez et al. 
(1986), Brester and Schroeder(1985) etc. Another approach to capture seasonal 
differences is to use harmonic analysis. This approach makes use of the periodic 
properties of trigonometric variables to characterize seasonality, e.g., Kesavan and Buhr 
(1995). 
Both of these approaches assume a priori a constant seasonal pattern. Even though 
some evidence of changing seasonal patterns exists in the poultry market literature, there 
has been  little empirical testing of this assumption. For example, Lasley et al. (1985) 
noted that turkey consumption for the first three quarters of the year changed much more 
than for the fourth quarter between 1960 and 1980. Witzig (1977) suggested that seasonal 
patterns for broiler prices in the early 1970's were quite different from those in earlier 
years. Schrimper (1998) indicated that the seasonal pattern of turkey and broiler 
consumption and prices seems to exhibit significant variability.  
This paper explores the impact of incorporating changing seasonal pattern into 
poultry market modeling. In particular, procedures provided by Arnade and Pick (1998) 
are followed  in creating a variable that consists of an interaction term between a trend   2
variable and a trigonometric variable representing a particular frequency. This variable 
can then be used as an exogenous variable in the economic model to detect and account 
for  changes in the seasonal cycle. 
The rest of the paper is organized as follows. Section 2 estimates two versions of an 
inverse demand equation. Model 1 assumes constant seasonality, while model 2 includes 
variables to capture changing seasonal pattern. Section 3 presents the empirical results.   
The last section contains a summary and conclusions. 
  
2. Model  Specification 
 
In Arnade and Pick (1998), a simple method was developed to test and account for a 
changing seasonal pattern using harmonic analysis. They used a variable consisting of an 
interaction term between a trend variable and a trigonometric variable representing a 
particular seasonal frequency. They show that this variable can be used as an explanatory 
variable to detect changes in the seasonal cycle. In this paper, inverse demand equations 
for broilers and turkey will be estimated using similar procedure. 
 
      The basic model is specified as  
t s
s
s r
r
r t t
j
t j j t i t i t i g b f b tr b m b p b q b p b a p ε + + + + + + + + = ∑ ∑ ∑
= = =
−
6
1
, 8
6
1
, 7 6 5
4
2
, , 1 1 , 0 0 ,     , 
  i=1, 2           ( 3 )  
where  t i p ,  is the log of own price for either broilers or turkey at time t,  t j p , is the log of 
the price of substitutes representing substitution effects. We consider turkey and broiler   3
as substitutes for each other as well as beef and pork.  t m is the log of per capita monthly 
income,   t tr  is a time trend variable. r f and  s g  are the trigonometric variables, where 
r f = )
12
2
cos( tr
rπ , and  s g = )
12
2
sin( tr
sπ . The elements of  r f and  s g  are cyclical processes 
at the seasonal frequencies  )
12
2
( π r  and  )
12
2
( π s . The coefficients  ) , ( , 8 , 7 s r b b  represent the 
contribution of each cycle to the seasonal processes. The one-period lagged price ( 1 , − t i p ) 
reflects the partial adjustment of price.  
The above specification does not consider varying seasonality. We will refer to 
this as model 1. In order to evaluate the effect of non-constant seasonality, consider the 
following (Chow, 1983) 
) sin( ) cos( 2 1 • + • β β         ( 4 )  
The dot inside the parentheses represents the arguments of the function. The 
amplitude of the function is  
2 / 1 2
2
2
1 ) ( β β +          ( 5 )  
According to Arnade and Pick (1998), there are two kinds of seasonal trends that 
can be monitored by using an interaction term between a trend variable and the 
trigonometric variable.  One is amplitude shift and the other is phase shift.  
Introducing the interaction term into model 1: 
) sin( * * ) sin( ) cos( * * ) cos( 2 2 1 1 • + • + • + • tr tr α β α β  (6) 
where tr is a variable that exhibits a trend effect over the time period. If either  1 α  and/or 
2 α  are significant, then from the definition of amplitude in eq. (5), we can conclude that 
the amplitude of the seasonal cycle is changing over time.   4
The test for phase shift is more complex. Such a shift changes the starting and 
ending points or the location of a seasonal cycle. The location of a cycle at a particular 
frequency depends on a weighted average of the cosine and sine variables. These weights 
are determined by the parameters of the trigonometric variables in the estimated model. If 
the location of the seasonal cycle is displaced by an amount τ , then  ) tan(τ can be shown 
to be proportional to  2 1 / β β . Thus, changes in the relative coefficients of the 
trigonometric variables can represent a phase shift in the season. If there is no phase shift, 
then the tangent of displacement τ without the interaction term equals the tangent of 
displacement in the presence of the interaction term. Thus, when no phase shift occurs in 
the seasonal cycle, the following holds: 
1
2
1 1
2 2
*
*
β
β
β α
β α
=
+
+
trend
trend
  ,          ( 7 )  
which is equivalent to the restriction  1 1 2 2 / β α β α = . To test for phase shift, significance 
of the restriction can be tested. If imposing the restriction does not significantly change 
the fit of the model, then the hypothesis of no phase shift can not be rejected. 
         Our technique will be as follows. 
         Step 1. The demand equation (3) is estimated without considering seasonal trend.  
         Step 2. For the frequencies that are shown to be significant in step 1, we use the 
above method to test for amplitude and phase shift. Amplitude test was performed by 
setting both interaction terms equal to zero and testing against the unrestricted model. 
The phase test was done using Wald test.   5
Step 3. For the frequencies that exhibit changing seasonal patterns, we include the 
interaction terms in the model. This is referred to as model 2. Model 2 is estimated to 
determine how the estimated coefficients change. 
 
3. Empirical  Results 
 
Monthly data for broiler and turkey consumption and price for the period January 
1976 through December 1995 are obtained from Poultry Year Book (USDA). Price data 
for beef and pork are obtained from Red Meat Yearbook (USDA). Monthly CPI and 
monthly per capita disposable income are obtained from Federal Reserve Bank. The 
retail prices were deflated by monthly CPI ( 1982-1984 =100).  
Estimated results are reported in table 1 and table 2.  
Estimation of model 1 shows that most of the trigonometric variables are significant 
for turkey. These include  5 4 3 2 5 1 , , , , , g g g g f f . The Broiler equation has 2 significant 
trigonometric variables, these are  1 f and  2 g . This indicates that turkey shows stronger 
evidence of seasonality than broiler. 
The amplitude and phase shift tests for these frequencies are given in table 1. The 
significance levels for both tests are 5%. For turkey, we reject the null hypothesis of no 
amplitude shift at all the frequencies considered. For broiler, we reject the null hypothesis 
of no amplitude shift at  π
6
2
, but could not reject the null at  π
6
1
. Phase shift test results 
show that there is no much evidence of phase shift for both turkey and broiler. We could   6
not reject the null hypothesis of no phase shift at all frequencies considered except for 
turkey at  π
6
1
. 
The test results show that turkey shows much stronger evidence of changing seasonal 
pattern than broiler.  
Next we compare the estimated coefficients from the two models. Bear in mind that 
the only difference between the two models is that model 1 does not allow for changing 
seasonal pattern, while model 2 includes interaction terms to capture varying seasonality.  
Price flexibility and income elasticity results are given in table 2. 
From table 2, we find that own price elasticity for turkey is very sensitive to the two 
specifications. Without considering changing seasonal patterns, this estimate is 0.0079, 
which is insignificant. But in model 2 that allows for changing seasonal pattern, the 
estimate is –0.0473, and is significant. For income elasticity of turkey, the estimate 
changes from –6.6599 in model 1 to 2.1870 in model 2. The coefficient estimates for beef 
and pork are also different for the two models, though insignificant in both cases.  
When we compare the estimated coefficients for broiler, we do not find significant 
differences between the two models. Own price elasticity is –0.0956 in model 1, and  -
0.0921 in model 2. Income elasticities are also quite close in the 2 models: 3.7390 in 
model 1 and 3.8005 in model 2. The coefficients for beef and pork are significant in the 
broiler equation.  
In sum, the coefficients in turkey equation are quite sensitive to the two specifications 
and the coefficients in broiler equation are not. This is not too surprising considering the 
fact that turkey consumption shows  stronger evidence of variation in seasonal pattern 
than broiler. Notice that the estimated coefficients for beef and pork price are not   7
significant in the turkey equation, but significant in the broiler equation. On the contrary, 
fewer trigonometric variables and interactions are significant in the broiler equation than 
those in the turkey equation. This means that, seasonal variation in turkey consumption is 
a very important factor in explaining variations in consumption demand relations. Thus, it 
is misleading to ignore the fact that the seasonal pattern is not constant. Even though 
seasonality also exists for broiler, the degree of variation in seasonal pattern is not as 
great as for turkey. Variation in consumption demand relations is mostly explained by 
other factors like income, substitution effects etc. Thus ignoring the changing seasonal 
pattern will not have much effect on estimation.  
 
4. Conclusion 
 
The role of seasonality in modeling agricultural markets is well recognized. However, 
traditional approaches to account for seasonality assume that seasonal pattern is constant, 
even though seasonal pattern may be changing over time.  
This paper provides empirical evidence that the seasonal pattern for both turkey and 
broiler consumption is not constant. This implies that economic models that allow for 
changing seasonal patterns are more appropriate. The results of using Arnade and Pick 
(1998) method to account for changing seasonality in demand equations show that, in the 
case of high variation in seasonal pattern, like turkey consumption, the price and income 
elasticities are quite sensitive to whether or not the model allows for changing 
seasonality. This implies that in estimating poultry (as well as perhaps other agricultural   8
products) demand equations, if data shows strong evidence of changing seasonality, the 
selected model  should be able to account for this variation. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   9
Table1. Amplitude and Phase shift test results 
          Amplitude
1     Phase
2       
Turkey       Broiler    Turkey      Broiler 
π 6
1  12.40
*         0.94    6.15 (0.01) 
*   0.86 (0.35) 
π 6
2  10.45
*         5.09
*    1.51 (0.22)   0.27 (0.60) 
π 6
3  5.83
*     0.01  (0.93) 
π 6
4  5.33
*     0.03  (0.86) 
π 6
5  4.28
*     0.43  (0.51) 
 
1 Amplitude test statistics are F-statistics with corresponding degrees of freedom. 
2  Phase shift test statistics are 
2 χ -statistics with 1 degree of freedom. 
p-values are in parentheses for phase shift tests. 
* statistically significant at 5% level. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   10 
 
 
Table2. Sensitivity of coefficient estimates 
                 Turkey       Broiler   
              Model 1           Model 2                      Model 1          Model 2 
intercept               -0.1554       -0.6590       intercept                  -2.95
*      -2.8749
* 
1 − t p (turkey)        0.8366
*       0.8168
*     1 − t p (broiler)           0.7938
*       0.7995
* 
t q (turkey)        0.0079        -0.0473
*     t q (broiler)             -0.0956
*      -0.0921
* 
t p (broiler)        0.1185
*       0.1236
*     t p (turkey)             -0.0677         -0.0589 
t p (beef)       -0.0060        0.0093        t p (beef)                  0.0852
*       0.0772
* 
t p (pork)       -0.0037      -0.0069        t p (pork)                  0.1117
*       0.1038
* 
trend          0.0003  -0.0002       trend                       -0.0005
*      -0.0004
* 
income elasticity  -6.6599       2.1870         income elasticity     3.7390
*       3.8005
* 
 
* statistically significant at 5% level. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   11 
References 
 
1.     Arnade, C., Pick, D. (1998)  Seasonality and unit roots: the demand for fruits. 
Agricultural Economics 18: 53-62 
2.  Chow, G.C.(1983) Econometrics. McGraw Hill, New York, NY. 
3.  Kesavan, T., Buhr, B. (1995) Price determination and dynamic adjustments: an 
inverse demand system approach to meat products in the United States. Empirical 
Economics 20: 681-698. 
5. Schrimper, R. (1998) Seasonality of broiler and turkey consumption and prices. 
Presented at 1998 Northeastern Agricultural and Resource Economics Association 
Meetings, Ithaca, New York. 
6.  Schrimper, R. (1998) Seasonal variability of poultry processing. Presented at 1998 
Southern Agricultural Economics Association Meetings, Little Rock, Arkansas. 
7.  Wohlgenant, M. L., Hahn, W.F. (1982) Dynamic adjustment in monthly consumer 
demands for meat. American Journal of Agricultural Economics 82: 553-557. 
 
 
 
 
 
 
 
 
 

Dynamic adjustment in monthly consumer demands for meat.

CHANGING SEASONAL PATTERNS IN THE POULTRY MARKET

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