Modeling Volatility of Price of Some Selected Agricultural
Products in Ethiopia: ARIMA-GARCH Applications
Yegnanew Alem Shiferaw,
Hawassa University, School of Mathematical and Statistical Sciences,
Main Campus,
Hawassa, Ethiopia,
Email: [email protected]
ABSTRACT
Currently change of price in agricultural products is a global problem. GARCH models were
employed to capture the log-return price volatility under the study. GARCH ( 1,1), GARCH (
1,2) and GARCH (2,1 ) models were fitted models for volatility of the log-returns of price of
Cereal, pulse and oil crops respectively.
Keywords: GARCH, ARCH, Price volatility, forcasting
1. INTRODUCTION
1.1 Background of the Study
A classical technique requires one to assume that the error terms have a constant
variance. This assumption is often not very plausible. Most conventional financial time series
models assume that the conditional variance of an asset returns is constant overtime.
However, many researchers such as Hageman (1978) and Lau et.al. (1990) have found that
the significantly non-normality of disturbances of stock returns. It is well known that the
presence of heteroscedasticity in the disturbances of an otherwise properly specified linear
model leads to consistent but inefficient parameters estimates and inconsistent covariance
matrix estimates. As a result, faulty inferences will be drawn when testing statistical
hypotheses in the presence of heteroscedasticity. Violations of homoscedasticity can yield
hypothesis tests that fail to keep false rejections at the nominal level, or confidence intervals
that are either too narrow or too wide. As a matter of fact, financial markets react “nervously”
in the presence of political disorders, economic crises, war or fear of war, or in the event of a
major natural catastrophe or man-made disaster, which is believed to threaten human society.
During such stress periods the prices of financial assets usually fluctuate strongly.
In statistical terms, the conditional variance given the past, i.e.,
Var[ X t | X t 1 , X t 2 , . . .] is not constant over time and the underlying stochastic process
(Xt) is conditionally heteroscedastic. Econometricians would also say that the volatility
1/ 2
 t  Var[ X t || X t 1, X t  2, ... changes over time (Daniel, 2005). This question often


arises in financial applications where the dependent variable is the return on a price and the
variance of the return represents the risk level of those returns. This motivates the
development of alternative volatility models which can capture the characteristics of
1
volatility more accurately. Such models are ARCH and GARCH models. ARCH and
GARCH models, which stand for autoregressive conditional heteroskedasticity and
generalized autoregressive conditional heteroscedasticity, have become widespread tools for
dealing with time series heteroskedastic models such as ARCH and GARCH. The goal of
such models is to provide a volatility measure like a standard deviation that can be used in
financial decisions concerning risk analysis, portfolio selection and derivative pricing.
(Engle, 1982). In this paper we need to employ time series econometric models to explore the
nature of domestic price volatility for some selected agricultural commodities in Ethiopia by
developing separate GARCH models with Box-Jenkins model for conditional mean
specification.
1.2 Statement of the Problem
Concern over the degree of commodity price fluctuations or volatility has attracted
increasing attention in recent economic and financial literature and has been recognized as
one of the most important economic phenomena (Engle, 1982). It has been argued that price
volatility reduces welfare and competition by increasing consumer costs (Zheng, Kinnucan
and Thompson, 2008). Apergis and Rezitis (2011) noted that price volatility leads both
producers and consumers to uncertainty and risk and thus volatility of commodity prices has
been studied to some extent. Commodity prices in general are volatile and in particular
agricultural commodity prices are renowned for their continuously volatile nature (Newbery,
1989). Even though cereal market policies in Ethiopia have undergone dramatic changes
over the past several years (Rashid, 2007), the desirable outcome of any policy measure or
other system of intervention to prevent the markets from going into market price volatility,
however, remained very unsatisfactory. Findings on (Rashid, 2007) indicated that price
volatility in markets of major cereals crops remains high in the country. Besides, agricultural
food price volatility in financial assets is still an area in which little empirical attention has
been paid in Ethiopia. Hence it appears worthwhile to devote effort to modeling agricultural
commodity prices with GARCH models. As a result the following research questions will be
addressed by this study.




Is there domestic market price volatility on the selected agricultural commodities?
Which of the selected commodity is highly volatile in price?
Is there any model that fit the domestic price volatility?
Which model is highly fit to forecast the domestic price volatility?
1.3
Objective of the Study
General Objective: the main objective of this paper is to model and examine the country level
variance effects of some selected agricultural commodity price returns using GARCH
models.
2
Specific Objectives:





To develop appropriate model for domestic price volatility for some selected consumer products
under consideration.
To identify the pattern of domestic price volatility of consumer products under study
To compare the pattern of price volatility in domestic markets for the consumer products
To make prediction on the domestic price volatility for these products.
To make suggestions for policy makers
1.4 1 Significance of the Study
To study the pattern of Crop domestic price volatility and its determinants are widely recognized as an
important input into food balance sheets in mitigating price instability and risks, food insecurity , to make
food policy decisions and strategic plans, granting of licenses for private firms to import or export.
Similar effects of domestic price instability can occur at production levels, investment and income
stability of consumers, whole sellers, producers and as well as at the macro level of the country.
Especially, since domestic food prices are much more volatile than the corresponding global prices
(FAO, 2008), it has been found highly significant with the following founding’s of the study.
 Contributes to the identify the pattern of domestic price volatility for the purpose of being able to
make more informed decisions when choosing one crop over another and to regulate its movement.
 Assist in the identification of the underlying pressures on the domestic price volatility of the
agricultural commodities market especially for the purpose of setting monetary policy in a price
stabilization–targeting regime.
 Contributes to examine the health of agricultural commodities markets for the crops under
consideration.
 Furthermore, the result of this study will be used as a basis to other researchers for further
investigations.
1.5 Organization of the Paper
This chapter is divided into four sections. The first section gives introduction, the second section gives
data and methodology, the third section presents about results and discussion and the fourth section is
concluding remarks.
2. DATA AND METHODOLOGY
2.1 Data
The data relevant to this study has taken from secondary source recorded data by the central statistical
agency. The time series data used in this analysis consists of the monthly commodity prices of selected
products with the sample covers from May 2001 to April 2011 G.C. and has a total of 120 observations.
All the prices are in Ethiopian birr. Commodities under Study are: Cereal crops, Pulse crops and Oil
crops.
Variables of the Study: the variables of the study in this research was the price average monthly domestic
closing price returns (Yt) and conditional variance ( ) at time t for each selected commodity as
dependent variables.
Dependent variables: Average monthly domestic closing price returns (Yt) for mean equation and
conditional variance ( ) at time t for each selected commodity under consideration.
Independent variable are: -Past shock of dependent variable (
)
-Past conditional variance (
2.2 Statistical Model Specifications
)
3
This study will attempt to model the volatility of monthly commodity price return using the
ARCH/GARCH model. Let’s briefly present the conditional variance models as follows.
2.2.1 Conditional Variance (Volatility) Model
ARCH Models: Autoregressive conditionally heteroscedastic (ARCH) models, introduced by Engle
(1982). The ARCH (q) regression model can be Expressed as ut2 in terms of past values of ut2.
q
That is, ut2 =  + 1ut-12 + … + qut-q2 =    i ut i
2
i 1
GARCH (Generalized ARCH) Models: is a model that is mainly used to model volatility (Bollerslev,
1986). It generalizes the ARCH model in the same sort of way that an ARMA model generalizes an MA
model. The GARCH (p, q) model can be expressed as:
q
h
t
p
    i ut i   
2
i 1
2.2.2
j 1
j
h
t j
with the constraints ω>0, αi>0, i=1, . . . , q and βj > 0, j=1,. . . ,p
Basic Procedures for GARCH Family Model Building
Testing Stationary in Time Series: if a time series is stationary, it’s mean, variance and auto-covariance
(at various lags) remain the same no matter at what time we measure them.
Dickey Fuller (DF) Test: there are many tests available to determine whether the series is a stationary or
non-stationary, but the most common test used is the Dickey-Fuller test. The test is basically focuses on
determining value ρ, whether it is equal to one, or it is less than one. Dickey-Fuller model can be
expressed as:
where
and
. The hypotheses are as follows:
The series is non-stationary or contain a unit root:
The series is stationary:
Testing for ARCH effects: according to Tsay (2005) the LM test was employed. The test statistic is
defined as Obs.R2 and follows a chi-square distribution with q degrees of freedom (Engle, 1982). If the
value of test statistic is greater than the critical value from the Chi-square distribution indicates the
evidence of ARCH(q) effects.
Order Selection for GARCH Family Model:
2
ARCH Effect: If an ARCH effect is found to be significant, the PACFs of  t are useful to determine the
order of ARCH model. This is due to the expectation that

2
t
is linearly related to

2
t 1,
, . . . ,  t q in a
2

2
t 2
manner similar to the AR (q) model.
GARCH Effect: In the presence of several competing models, with different number of parameters to
select the model with appropriate order, both AIC and BIC employed. In general a desirable model is one
that minimizes the AIC or the BIC and on the significance tests for each parameter. The formal
expressions for the above criteria in terms of the log-likelihood are:
AIC= -2ln(loglikelihood) +2r
BIC = -2ln(loglikelihood) + (r +r(lnN))
Estimation of GARCH Models: parameter estimation of all of the above models can be achieved most
simply using maximum likelihood Estimation (MLE), since the GARCH model is non-linear. Even if the
true t are not conditionally Gaussian, consistent parameter estimates could be obtained by maximizing
this log-likelihood function.
4
Model Adequacy Checking: when a model has been fitted to a time series, it is advisable to check that the
model really does provide an adequate description of the data. The properties of standardized residuals
are employed to define the best fit data models. When the model fits the data well under normality
assumption: the histogram of standardized residuals should be approximately symmetric, the normal
probability plot should be a straight line and the time plot of residuals should exhibit random variation.
3. RESULTS AND DISCUSSION
3.1 Stationary in Time Series: Unit Root Test
Table 3.1: Dickey-Fuller Test for Unit Root for Raw Prices
The null hypothesis assumes that the data set is non-stationarity such that one can reject the null
Commodities
Test Statistic
1% Cr.Value
5% Cr.Value
10% Cr.Value
Cereal
Pulse
Oil
-0.849
-0.808
-0.401
-3.534
-3.534
-3.534
-2.904
-2.904
-2.904
-2.587
-2.587
-2.587
MacKinnon
value
0.8044
0.8166
0.9099
app.
p-
hypothesis if the t-statistic is less than the critical value(s). Table 3.1 presents as usual raw prices are nonstationarity. Because of their corresponding p-values from DF test statistic were greater than 5% level of
significance to test null hypothesis of non-stationarity was not rejected. Thus, there is no evidence to
reject null hypothesis of non-stationarity at 5% level of significance. If a time series is non-stationary, it
is necessary to look for possible transformation that might induce stationarity. In practice,
econometricians usually transform financial prices into log-return series forms. The log return series is
p

 , where y is the log-return series of the real raw data of prices, p and
given by y  log  t
t
t
t

pt 1 

p are raw real price series at times t and (t-1). Return prices (in natural logarithm) are stationary (see
t 1
Table 3.2 below). This is because of p-values corresponding to each commodity were less than 1%, 5%
and 10% level of significance, implying that null hypothesis of non-stationarity was rejected at 1%, 5%
and 10% level of significance respectively.
Table 3.2: Dickey-Fuller Test for Unit Root for log-Return Prices
It is a typical property of financial log-return series to exhibit skewness and kurtosis (See Table 3.3
below).
Table 3.3: General Summary Statistics for the Return Series
Commodities
Cereal
Cereal
Pulse
Pulse
Oil
Oil
Skewness
Test Statistic
-5.262558
-11.277
0.282812
-9.660
-0.742198
-9.549
Kurtosis
1% Cr.Value
8.13856
-3.535
5.275928
-3.535
6.944698
-3.535
Jarque-Bera
5% Cr.Value
4541.052
-2.904
19.24923
-2.904
62.17423
-2.904
P-value
10% Cr.Value
0.010425
-2.587
0.0366
-2.587
0.0000000
-2.587
MacKinnon
value
0.0000
0.0000
0.0000
app.
p-
Table 3.3 presents summary statistics for the price of log-return series for Ethiopian selected
commodities. The skewness and kurtosis respectively measure the asymmetry and peakedness of the
probability distribution of returns. The two parameters are expected to be zero and three for normal
distribution. The Jarque-Bera statistic with skewness and kurtosis are used to signify the distribution
characteristics of log-return series. The data shows positively and negatively skewed this implied that the
log-return series distribution has significantly fatter tails than does the normal distribution. The more
5
kurtosis statistic which is equal to 8.13856, 5.13856 and 6.944698 indicates the Leptokurtic
characteristics of the return distribution. The implication of non normality is supported by the Jarque Bera test statistic which point out the null hypothesis of normal distribution is rejected. The above facts
clearly pointed out that the monthly log-return series is not normally distributed. Moreover, the return
series of Cereal, Pulse and Oil crops have kurtosis and excess skewness are hints for conditional
heteroscedasticity
.
3.2 Modeling of Volatility
This stage consists of the following four stages: testing for ARCH effects; specifying a volatility model,
carrying out a simultaneous estimation of the mean and volatility equations; diagnostic tests and
forecasting.
3.2.1 Testing for ARCH Effects: since the P-value less than 0.05, the null hypothesis of no ARCH effects
is rejected. This indicates the evidence of ARCH (q) effects. Table 3.4 shows the results of ARCHLM
test indicates that there is an ARCH effects in each of the three cases. This results that the log-return price
series are volatile and need to be modeled using ARCH or GARCH models.
Table 3.4: ARCH LM test summary statistics
Commodities
Cereal
Pulse
Oil
Obs*R-squared
44.469
38.632
41.233
t-Statistic
4.382
2.994
5.452
P-Value
0.000
0.003
0.001
3.2.2 Specifying a Volatility Model
The series presented ARCH effect at least until lag 15, and the estimation residuals became white noises
after the estimation of the models. As can be seen ARCH(2) for cereal, ARCH(1) for pulse and ARCH(2)
for oil prices are identified as significant (best fit and reasonable lag structures). In this study both AIC
and BIC were employed to select an appropriate GARCH model for the sample of the data available.
Table 3.5 displays the summaries of the AIC and BIC of different GARCH models. As it is mentioned
earlier the smaller the rank sum the better model. Therefore GARCH (1,1), GARCH(1,2) and
GARCH(2,1) models are the best volatility models for the prices of cereal, pulse and oil crops
respectively.
Table 3.5.1: GARCH model selection for the monthly log-returns Cereal Crops using AIC and BIC
Statistics
AIC
BIC
Rank Sum
GARCH(1,1)
-154.5444
(1)
-144.869
(1)
2
GARCH(1,2)
-150.509
(3)
-140.8336
(3)
6
GARCH(2,1)
-152.0709
(2)
-142.3955
(2)
4
GARCH(2,2)
-150.2998
(4)
-140.6244
(4)
8
Table 3.5.2: GARCH model selection for the monthly log-returns Pulse Crops using AIC and BIC
Statistics
AIC
BIC
Rank Sum
GARCH(1,1)
-173.194
(2)
-163.5187
(2)
4
GARCH(1,2)
-173.9287
(1)
-164.2534
(1)
2
GARCH(2,1)
-169.3289
(3)
-159.6536
(3)
6
GARCH(2,2)
-167.9358
(4)
-158.2605
(4)
8
Table 3.5.2: GARCH model selection for the monthly log-returns Oil Crops using AIC and BIC
Statistics
AIC
GARCH(1,1)
-381.4066
(2)
GARCH(1,2)
-202.8639
(3)
GARCH(2,1)
-385.1804
(1)
GARCH(2,2)
-189.5158
(4)
6
BIC
Rank Sum
-375.3306
(2)
4
-193.1885
(3)
6
-379.1043
(1)
2
-179.8404
(4)
8
3.2.3 Simultaneous
Estimation of the Mean and Volatility Equations
Here it is tried to simultaneously model the mean and variance of the selected agricultural commodity
prices by considering GARCH models for conditional variance. Table 3.6 shows the final refined results
of the estimated parameters of both the mean and volatility equations. The estimates of GARCH (1,1)
model for cereal crop shows that all the coefficients of mean and variance equation are statistically
significant at both 1% and 5% level of significance. Estimates of GARCH(1,2) model for pulse crops
shows that all the coefficients of mean and variance equation are statistically significant at 5% level of
significance except 1 . In a similar manner, estimates of GARCH (2, 1) model for oil crops shows that

in the variance equation the values of
 , , 
0
1
1
and

2
are statistically significant at 5% level of
significance.
Table 3.6: Maximum Likelihood Estimates of GARCH (1,1), GARCH(1,2) and GARCH(2,1) models
Selected
Commodities
Mean Equation

AR(2) 
MA(1) 
AR(1)
Cereal
Pulse
Oil
Coef.
-1.0275
Std.Err
.049844
P>|z|
0.000
-
-
-
1.0216
.0651777
0.000
.00001
8.10e-06
0.0033
338.98
127.98
0.008
.37228
.2020074
0.005
.48219
.21526
-0.495
1
Coef
Std.Er
.9900
.01987
P>|z|
0.000
Coef.
.06716
Std.Err
.1380262
P>|z|
0.626
.14236
.1143012
0.087
.12764
.1397535
0.361
.0014
.0008552
0.010
0.025
.8003
.3820417
0.036
0.124
0.03
-
-
-
2
-.046
.13782
0.737
1
Variance Equation





0
1
2
.53841
.1589807
0.001
.32016
.23540
0.007
0.036
0.036
0.026
-
-
-
-
-
-
0.462
0.1457
0.040
1
2
3.2.4
Model Adequacy Testing
Model adequacy checking in the standardized residuals is performed for the models with significant
parameters which have been explained in the previous section (Tsay, 2002 and Gouricroux, 1997). These
models include GARCH (1,1), GARCH(1,2) and GARCH(2,1) models.
Table 3.7: Model Adequacy Checking in the Squared Normalized Residuals
Commodities
Cereal
Pulse
Oil
Models
GARCH(1,1)
GARCH(1,2)
GARCH(2,1)
Skewness
0.417369
0.103571
-0.583691
Kurtosis
4.405180
4.734611
5.913047
Jarque-Bera
9.127012
11.53978
35.38155
As compared to the statistics from the original returns series (see Table 3.7), the model adequacy
test’s result in table above points out that the coefficients of skewness and kurtosis measures
slightly reduces in absolute values for GARCH(1,2) and GARCH(2,1) estimated models. The
diagnostic checking based on the distribution of standardized residuals indicates the capability of
GARCH type models to capture the asymmetry and fat tailed characteristics in residual
distribution as well as the squared return autocorrelation in the selected Ethiopian commodities.
7
GARCH models can capture the non-normality characteristics of return series to some extent, the
excess skewness and kurtosis statistics are still exposed.
4. CONCLUSIONS AND RECOMMENDATIONS
4.1 Conclusions
Little study has been done on the domestic price volatility of consumer products in Ethiopia. This study
aims to examine the volatility of some selected agricultural products over a period of 2001-2011. In order
to contribute to the literatures of Ethiopian financial market, this study investigates the estimation and
forecast ability of univariate GARCH models for conditional variance for selected agricultural products.
The main findings of the study are as follows:
 Like other asset prices, commodity prices volatility can be modeled by GARCH family models.
 Return prices show persistent volatility across the samples.
 It is observed that GARCH(1,1), GARCH(1,2) and GARCH(2,1) are the most appropriate fitted
models to use one has to evaluate the volatility of the log-return price of Cereal, pulse and oil
crops respectively. Prices volatility is persistent in all three categories (Cereal, pulse and oil
crops) of selected consumer goods.
 Regarding the forecasting capability, after obtaining the three satisfactory GARCH models, the
forecast process is analyzed. This study tried to illustrate the proper procedure for testing the
relative forecast accuracy of different GARCH models based on root mean square error (RMSE)
with an application to three different commodity prices. The analysis favour the GARCH (1,1),
GARCH(1,2) and GARCH(2,1) models for cereal, pulse and oil crops respectively. Findings of
this study could give good insight to forecast volatility of price of agricultural products in
Ethiopia.
4.2 Recommendations
This study has provided several important insights into the volatility of prices of some selected
agricultural commodities. An in-depth understanding of domestic price volatility is essential for the
society as well as policy makers; policy makers should address the importance of considering the subnational factors in formulating the national commodity prices. The data that we have used in this study is
a period of almost for eleven years and this is limited data to apply GARCH models. Hence the findings
should be treated with great precaution.
References
Apergis, N. and Rezitis, A. (2011) Food Price Volatility and Macroeconomic Factors: Evidence from
GARCH and GARCH-X Estimates, Journal of Agricultural and Applied Economics, 43:1, 95–110.
Bollersler T. (1986), Generalized Autoregressive Conditional Heteroscedasticity. Journal of Econometrics,
31:307-327.
Daniel S., (2005) Estimation in Conditional Heteroscedastic Time Series Models, ISBN 3-540-21135-7
Springer Berlin Heidelberg New York.
Engle, R. F. (1982) Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United
Kingdom Inflation. Econometrica. 50:4, 987-1008.
FAO (2008), Food Outlook, Global Market Analysis
Hageman, R.L. (1978) More Evidence on the Distribution of Security Returns. The Journal of Finance,
33,1213-1221
Lau, A., Lau, H. and Wingender, J., (1990) The Distribution of Stock Returns: New Evidence Against the
Stable Model, Journal of Business and Economic Statistic, 8,217-223.
Newbery D.M. (1989) The Theory of Food Price Stabilization. The Economic Journal, 99, 1065-1082.
Rashid, S. and Meron, A. (2007) Cereal Price Instability in Ethiopia: An Analysis of Sources and Policy
Options, Paper Prepared for the Agricultural Economics Association for Africa, Accra Ghana
Tsay L.S., (2005) Analysis of Financial Time Series, 2nd ed., Hoboken, N.J: Wiley
Zheng, Y., Kinnucan, H. W., & Thompson, H. (2008) News and Volatility of Food Prices. Applied
Economics, 40, 1629-1635.
8