The Application of Garch Family Models for Agricultural Crop Products in Amhara Region, Ethiopia

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1 DOI: htt://dx.doi.org/ /star.v3i4.7 ISSN: 6-75(rint) and (Online) Science, Technology and Arts Research Journal Sci. Technol. Arts Res. J., Oct-Dec 014, 3(4): Journal Homeage: htt:// The Alication of Garch Family Models for Agricultural Cro roducts in Amhara Region, Ethioia Belayneh Debasu Kelkay 1* and Emmanuel G/Yohannes 1 Deartment of Statistics, Arba Minch University,.O. Box: 1, Arba Minch, Ethioia Ethioian Civil Service University,.O. Box: 5648, Addis Ababa, Ethioia Abstract In the recent ast, the rice of general commodities has increased in Ethioia as well as in the world. The main objective of this study is to identify and analyze the factors that affect the average monthly rice volatility of ulses (bean and ea) in Amhara National Regional State over the eriod of December 001 to June 01 GC. The return series considered exhibited tyical characteristics of financial time series such as volatility clustering, letokurtic distributions and asymmetric effect and thus, can suitably modeled using GARCH family models. Among such models entertained in this study, ARMA(4,4)-EGARCH(,3) with GED for bean and ARMA(1,0)-EGARCH(1,) with student-t for ea were chosen to be the best fit models. From the results, exchange and general and food inflation s were found to be an increasing effect on rice volatility of bean and ea. On the other hand, rainfall was found to have a stabilizing effect on the rice volatility of these cros. Moreover, saving interest has a decreasing effect on the rice volatility of bean. The results also revealed that rice volatility has seasonal variation. The asymmetric terms were found to be significant in all GARCH models considered. Thus, rice volatility tends to over-react in resonse to bad news as comared to good news. Furthermore, the significance of the EGARCH terms rovides strong evidence of volatility sillover from one eriod to another. Coyright@014 STAR Journal. All Rights Reserved. INTRODUCTION In Ethioia, agriculture accounts for almost 41% of the gross domestic roduct (GD), 80% of exorts, and 80% of the labor force. Many other economic activities deend on agriculture, including marketing, rocessing, and exort of agricultural roducts. roduction is overwhelmingly by small-scale farmers and enterrises and a large art of commodity exorts are rovided by the small agricultural cash-cro sector. rincial cros include coffee, ulses (e.g., bean and ea), oilseeds, cereals, otatoes, sugarcane and vegetables. Exorts are almost entirely agricultural commodities, and coffee is the largest foreign exchange earner. In 005/006 Ethioia s coffee exorts reresented 0.9% of the world exort, and oil seeds and flowers each reresent 0.5% (IMF, 009). Agricultural households in develoing countries face a variety of risks. The most visible manifestation of these risks is high food rice instability, which, because of its inherent economic and olitical imlications, has attracted the attention of almost all actors in food olicy making over the ast few decades. However, all actors agree on one oint, i.e. the direct consequences of rice instability on consumers, roducers, as well as on overall economic growth. For oor consumers, consequences of rice instability are severe. Since a large share of their income is sent on food, an unusual rice increase forces them to cut down food intake, take their children out of school, or, Original Research A eer-reviewed Official International Journal of Wollega University, Ethioia Article Information Article History: Received : Revised : Acceted : Keywords: rice volatility Bean and ea Amhara Region Garch family models *Corresonding Author: Belayneh Debasu belaynehd@gmail.com in extreme cases, simly to starve. Even when such rice shocks are temorary, they can have long term economic imacts in terms of nutritional well-being, labor roductivity, and survival chances (Hoddinott, 006; Myers, 1993). Variability in food rices can also have imortant effects even if average rices remain constant. This might haen if fluctuations in food roduction become more common or larger but average roduction remains the same. This would lead to more frequent and larger rice changes, which might be redictable or unredictable. If the increased variability were largely redictable, this would cause fewer roblems than if the changes were unredictable. However, rice changes are generally less redictable than might be imagined. Unstable rices for stale foods are likely to have larger negative effects than unstable rices for other agricultural commodities because stale foods are imortant for both oor farmers and oor consumers. On the consumer side, stale foods account for a large share of the exenditures of the oor. On the roducer side, they are the most widely lanted cros in develoing countries, esecially on smallholdings (FAO, 011). It is crucial to examine the attern of rice volatility and identify its determinant on cereal cros. According to 49

2 Jordaan et al. (007), the accu measurement of the stochastic comonent in rices may contribute to the decision maker being able to make more informed decisions when choosing one cro over another. It may also contribute to olicy decisions regarding the ossible imlementation of commodity rice stabilization rograms. Examining the underlying causes of ulse rice volatility has great role for managing rice instability for roducers, consumers, whole sellers and agricultural rice olicy reforms for the country as well. In Amhara National Regional State (ANRS), agriculture contributed to about 55.8% of the total regional GD. The main field cros in the Amhara region are cereals (wheat, barley, teff, sorghum, maize, etc), ulses (field ea, chickea, bean, etc) and oil cros (sesame, rae seed, sunflower, etc). Cereals account for more than 80 % of cultivated land and 85 % of total cro roduction. About 33 % of the livestock and 5-30 % of cro roduction in Ethioia are from the Amhara region (BoFED, 011). As many studies indicated rice volatility of agricultural commodities has a negative imact on the economy of the country through income instability for roducers, consumers and whole sellers and also leads to a major decline in future outut if the rice changes are unredictable and erratic. Therefore, this study was an attemt to identify the attern of average monthly rice volatility of ulse seed (ea and bean) in Amhara Region by develoing aroriate time series models that can fit financial data. Therefore, this study has attemted to address the following roblems (1) is there volatility in the rice of some selected agricultural cros roducts (cereal and ulse seed)? () which agricultural commodities under consideration have highly volatile rices? (3) which model is a good fit to data on rice of agricultural cro roducts? The main objective of this study is to identify and analyze the factors that affect the rice volatility of bean and ea seeds in Amhara Regional. Secifically, this study tries to address the following key issues (1) to fit and select an aroriate GARCH family models for the rice volatility of ulse seeds (bean and ea), () to assess the attern of their rice volatility and (3) to estimate and forecast the rice volatility of bean and ea seeds. MATERIALS AND METHODS Source and Tye of Data To assess the average monthly rice volatility and its determinants on certain ulse seeds (beans and ea), the data were obtained from Central Statistical Agency (CSA), National Bank of Ethioia (NBE) and National Metrological Agency of Ethioia, on monthly basis from December 001 to June 01 G.C. Average monthly rice of ulse seed (bean and ea) is used as deendent variables. Exchange, saving interest, lending interest, general inflation, food inflation, non-food inflation, average temerature (in degree Celsius) and average rain fall (in mm) are used as indeendent variables. Since the data are not seasonally adjusted also seasonal dummies are used. (G) ARCH Models The Box-Jenkins time series model such as Autoregressive (AR), Moving Average (MA) and ARMA are often very useful in modeling general time series data. However, they all require the assumtion of homoskedasticity (or constant variance) for the error term in the model. Autoregressive Conditional Heteroskedasticity (ARCH), sea GARCH, TGARCH and EGARCH models have been emloyed in this study to investigate the attern of rice volatility and its determinants. Model Secification: Stationarity and Unit-Root roblem A given series is said to be stationary if its mean and variance are constant overtime and the value of the covariance between any two time eriods deends only on the distance or ga or lag between the two time eriods and not the actual time at which the covariance is comuted. Generally the concet of stationarity can be summarized by the following conditions. A time series {y t} is said to be stationary if: E(y t) = E(y t-s) = μ, E(y t-μ) = E(y t-s-μ) = σ y, E(y t-μ) (y t-s-μ) = E(y t-j-μ) (y t-j-s-μ) = γ(s), where μ, σ y and γ(s) are all time invariant. The assumtion of stationarity is somewhat unrealistic for most macro economic variables. A non-stationary rocess arises when at least one of the conditionsfor stationarity does not hold. Let us consider an autoregressive rocess of order one (AR (1) rocess): y t=ρy t-1+ε t, [1] where ε t denotes a serially uncorrelated white noise error term with a mean of zero and a constant variance. Non-stationarity can originate from various sources but the most imortant one is the resence of so-called unit roots. Equation (1) is said to be a unit root rocess when ρ= 1. If a variable is stationary in level, i.e. without running any differencing, then the variable is said to be integd of order zero, denoted by I(0). Similarly, if it becomes stationary by differencing d times, then the variable is said to be integd of order d, written as I(d), d= 1,, 3,. Unit-root test hels to detect whether a variable is stationary or not. It also rovides the order of integration at which the variable can be stationary. Let t, t= 1,, 3 be the rice of a commodity at time eriod t (t in days, months, etc). Instead of analyzing t, which often dislays unit-root behavior and thus cannot be modeled as stationary, we often analyze log- returns on t (Fryzlewicz, 007): Y t = log t log t-1 = log t t 1 = log 1 + t t 1 t. The series y t, log- return series, dislays many of the tyical characteristics in financial time series such as volatility, clustering and letokurtosis. 50

3 The Mean Model ARMA Model Autoregressive moving average (ARMA) modeling is a secific subset of univariate modeling in which a time series is exressed in terms of ast values of itself lus current and lagged values of a white noise error term. ARMA (, q) mean model (Box-Jenkins, 1976) is given by: y t=φ o+ q Φiy t-i j =1 θ jεt-j+εt, [] Where y t is average monthly log return rice of selected cros at time t, Φ 0 is constant mean, Φ 1, Φ,.,Φ are autoregressive arameters, ε t, ε t-1, are white noise error with mean zero and variance and σ t and θ 1, θ,., θq are moving average arameters. ARIMA Model Autoregressive Integd Moving Average (ARIMA) model was introduced by Box and Jenkins in 1960s for forecasting a variable. ARIMA models consist of unit-root non-stationary time series which can be made stationary by the order of integration d. The general form of ARIMA (, d, q) is written as: asymmetric ARCH models, was introduced. The most oular model roosed to cature the asymmetric effects is Nelson s (1991) exonential GARCH, or EGARCH model. The ARMA(,q)-EGARCH (,) model is given as: q y t = Φ o+ Φiy t-i j =1 θjεt-j + εt, ln(σ t )=α 0+ R ε t i α ε t i i + λ σ i t i + σ j =1 βjln(σ t-j) [7] t i In this model secification, β 1, β,. β are the GARCH arameters that measure the imact of ast volatility on the current volatility. TGARCH rocess The TGARCH model with mean and conditional variance equations is given as: σ t =α 0+ y t = Φ o+ αiε t-i+ λ q Φiy t-i j =1 θjεt-j + εt, idt-iε t-i+ β j =1 jσ t-j, [8] where d t-i= 1 if ε t-i 0, and d t-i= 0 otherwise. The TGARCH model allows a resonse of volatility to news with different coefficients for good and bad news. Δ d ψ (B)Y t = Φ o+θ q(b)ε t, [3] Where ψ (B) = 1-Φ 1B- -Φ B, Θ q(b) = 1-θ 1B- - θ qb q,δ = 1-B, d is the order of integration and B is the backward shift oerator. ARCH Model The autoregressive conditional hetroskedasticity model for the variance of the errors, denoted by ARCH (), was roosed by Engle (198). The conditional variance is given by: ε t = σ tυ t and σ t=α 0+ αiε t-i, [4] where υ t is IID normal residual with mean zero and unit variance and σ t is the conditional variance of the residuals at time t, i.e., σ t =Var (ε t ε t-1, ε t-,.). We imose the non-negativity constraints α 0,α i>0 i = 1,,.,. GARCH Model ARCH model was generalized by Bollerslev (1986) as GARCH(,) which allows the conditional variance to be deendent uon revious own lags. Then ARMA(,q) - GARCH(,) model is given by: y t = Φ o+ σ t=α 0+ q Φiy t-i j =1 θjεt-j + εt, αiε t-i+ j =1 βjσ t-j [5] Restrictions: α 0>0, α i 0, β j 0 for,,, and j=1,,,. The conditional variance equation of GARCH(,) with exlanatory variables for wheat seed is given by: σ t = α 0 + αiε t-i + j =1 βσ t-j + γ X t, [6] where X t = (x 1t, x t,.., x kt) is a vector of exlanatory variables and γ = (γ 1, γ,.., γ k) is a vector of regression coefficients of the exlanatory variables. EGARCH rocess In order to cature ossible asymmetry exhibited by financial time series, a new class of models, termed the In this study, the general inflation, food inflation, non-food inflation, exchange, saving interest, lending interest, temerature, rain fall and monthly seasonal dummies were introduced into the conditional variance equation as indeendent variables in order to determine the imact of these variables on the volatility of average monthly rice returns under consideration. The conditional variance equation of GARCH(,) with exlanatory variables for each cereal cros and ulse seeds is given by: σ t =α 0+ αiε t-i+ j =1 βσ t-j + γ X t, [9] where X t = (x 1t, x t,.., x kt) is a vector of exlanatory variables and γ = (γ 1, γ,.., γ k) is a vector of regression coefficients of the exlanatory variables. Assuming the resence of asymmetric effect on the GARCH family model, the conditional variance equations for EGARCH (,) and TGARCH(,) with exlanatory variables are given by: ln(σ t )=α 0+ σ t = α 0 + R ε t i α ε t i i + λ σ i + t i σ j =1 βjln(σ t-j)+γ X t,[10] t i αiε t-i + λ idt-iε t-i + j =1 βjσ t-j+ γ X t. [11] Assumtions of the Models a. The exected value of the error term is zero, i.e. E[ε t]=0 b. The variance of the error terms is conditionally hetroskedastic. c. Error terms are indeendent having normal or student-t or GED distribution with mean zero and variance σ t. d. There is no serial autocorrelation among successive error terms. e. No severe multicollinearity exists among exlanatory variables. rocedures for Model Building Testing for the resence of Unit Root A test of stationarity (or non-stationarity) that has become widely oular over the ast several years is the unit root test. There is a major roblem with regression that involves non- stationary variables as the standard 51

4 errors roduced are biased. Due to such bias, conventional criteria used to judge whether there is a casual relationshi between the variables are unreliable. Such a regression is what we call surious regression. It is therefore very imortant to be able to detect the resence of unit roots in time series. For these tests, the null hyothesis is that the time series has a unit root. The widely used unit-root tests are Augmented Dickey Fuller (ADF) test (Dickey and Fuller, 1979) and hillis erron () test (hillis and erron, 1987). The Augmented Dickey Fuller (ADF) Test The ADF test is comarable with the simle DF test, but is augmented by adding lagged values of the first difference of the deendent variable as additional reressors which are required to account for ossible occurrence of autocorrelation. Consider the AR () model: y t = μ + αy t 1 + whereα = -(1- i= ψ i y t + ε t, [1] Φ i j =i. i= ) and ψ i = Φ j If the null hyothesis H 0: α = 0 is not rejected, then we need to difference the data to make it stationary or we need to ut a time trend in the regression model to correct for the variables deterministic trend. The hillis and erron () Test An imortant assumtion of the DF test is that the error terms ε t are indeendently and identically distributed. The ADF test adjusts the DF test to take care of ossible serial correlation in the error terms by adding lagged difference terms of the deendent variable. hillis and erron use nonarametric statistical methods to take care of the serial correlation in the error terms without adding lagged difference terms. For details see erron and Ng (1996) and Nabeya and erron (1994). Testing ARCH Effects The Box-Jenkins (1976) aroach is based on the assumtion that the residuals are homoskedastic (remain constant over time) for ARMA or ARIMA model. But in financial data, ARCH effect is commonly found (Cotter and Stevenson, 007, Asteriou and Hall, 007). According to Tsay (005), there are two available methods to test for ARCH effects. (i) Ljung-Box Test: It was develoed by Box and ierce (1970) and modified by Ljung and Box (1978) and tests the joint significances of serial correlation in the standardized and squared standardized residuals for the first k lags instead of testing individual significance. They suggested testing the hyothesis: H 0: ρ 1 = ρ =. = ρ k = 0 H 1: not all ρ j = 0 whereρ j is the ACF at lag j = 1, k. They suggested the statistic: (k) = n(n+) k d j j =1, n j where n denotes the length of the series after any differencing and d j denotes the squared residual. (ii) Lagrange Multilier (LM) Test: This test was suggested by Engle (198) and used to test the significance of serial correlation in the squared residuals for the first q lags. ε t =γ 0+γ 1ε t 1 +.+γ qε t q [13] The null hyothesis is that, γ 0 = γ 1 =. = γ q = 0. The test statistic n.r is distributed as chi-square with q degrees of freedom, where R is the coefficient of determination from equation (13) and n is number of observations. The rejection of the null hyothesis indicates the resence of ARCH () effects. Test of Normality When dealing with GARCH family models, the data is first tested for normality (i.e. whether the returns follow a normal distribution). The test is named after Jarque and Bera (198). H 0: the observations come from a normal distribution. The test statistic is: JB = n 6 *(S + (k 3) ), 4 where n is the number of observations, S is the samle skewness and K is the samle kurtosis. Under the null hyothesis, the Jarque-Bera statistic is distributed as chi-square distribution with two degrees of freedom. Model Order Selection in GARCH Family Model A model selection criterion considers the best aroximating model from a set of cometing models.an imortant ractical roblem is the determination of the ARCH order and the GARCH order for a articular series. Since GARCH models can be treated as ARMA models for squared residuals, traditional model selection criteria such as the Akaike information criterion (AIC) roosed by Akaike (1974) and the Schwartz Bayesian information criterion (SBIC) roosed by Schwartz (1989) can be emloyed to identify the otimal lag secification for the model. These criteria are comuted using the loglikelihood estimates. Given the criterion values of two or more models, the model having minimum AIC or BIC is most reresentative of the true model and, on this account, may be interreted as the best aroximating model among those being considered (Dayton, 003). The formal exressions for the above criteria in terms of the log- likelihood are: AIC = -ln(l) + K [14] BIC = -ln(l) + K.ln(n) [15] where n = number of observations K = number of arameters estimated L = value of the likelihood function (log L(σ t )) The main reason for referring the use of a model selection rocedure such as BIC in comarison to traditional significance tests is the fact that a single holistic decision can be made concerning the model that is best suorted by the data in contrast to what is usually a series of ossibly conflicting significance tests. Model arameter Estimation Under the resence of ARCH effects, the OLS estimation is not efficient since volatility models used are non-linear in conditional variance though linear in mean. As many studies indicated, the commonly used method known as the maximum likelihood estimation has been emloyed in GARCH family model. Financial time series data ossess volatility clustering and letokurtosis characteristics which lead to the use of different distributional assumtions for residuals such as: - 5

5 Normal, Student-t and GED. Thus, in this study the Gaussian (Normal), Student-t distribution and the GED were considered for GARCH family model arameter estimation and the aroriate distributions for the residuals were identified based on robust estimation. The estimation of conditional volatility models are tyically erformed by MLE rocedures in Bollerselv and Wooldridge (199). Maximum likelihood method follows the following stes: 1. Secify the aroriate equations for the conditional mean and the variance.. Secify the log-likelihood function of the model to maximize. 3. Use regression to get initial guesses for the mean arameters from mean equation. 4. Choose some initial guesses for the conditional variance arameters. 5. Secify a convergence criterion. Maximization of the likelihood function of the model analytically in terms of its arameter is imossible because of non-linearity of GARCH family models. Model Adequacy Checking After a GARCH family model has been fit to the data, the adequacy of the fit has been evaluated using a number of grahical and statistical diagnostics. The followings are the methods for model adequacy checking that were used in this study:- 1. The ACFs of the residuals should be indicative of a white noise rocess.. The standardized residuals should be normally distributed. This was checked through Jarque-Bera test. 3. The Ljung-Box test is one of the widely used tests for the aroriateness of the fitted model; to test whether the model of the mean is aroriately secified and to test for the remaining ARCH effects 4. Evaluating the erformance of different forecasting models: the most widely used statistical evaluation measures are MAE, RMSE, MAE and Theils- U Inequality Coefficient (TU). These are alied to measure the forecasting accuracy of the ARCH- GARCH model in this study. rediction using GARCH Family Models An imortant task of modeling conditional volatility is to gene forecasts for both the future value of a financial time series as well as its conditional volatility. Conditional variance forecasts from GARCH family models are obtained with similar aroach to forecasts from ARMA models by iterating with the conditional exectations oerator. In other words, when the estimation of the unknown arameters is done, estimates of the standard deviation series can be calculated recursively via the definition of the Conditional variance for the GARCH (, ) family rocess which hels to examine the ast behavior of average monthly domestic rice volatilities of the series under consideration. Statistical Analysis The return series were constructed for each of the rices to allow a market wide measure of volatility to be examined. The data analysis is carried out using EViews 7 and STATA 11 software. RESULTS Figure 1 is a lot of average monthly rice trend of ulses. It can be observed that monthly rices show an increasing trend over the study eriod. The emirical result shows that the average monthly rice for bean and ea are and with standard deviation and 3.491, resective of their order (Table 1). In the case of log return series, the coefficients of kurtosis exceed three, indicating that the log return series are eaked relative to the normal distribution (that is, letokurtic). Moreover, the series exhibit ositive skewness. The Jarque-Bera test of normality rejects the normality of all the series under consideration. Table dislays summary statistics for each of the exlanatory variables. The samle mean (SD) was estimated to be about (3.186) for exchange in birr, (0.7596) for saving interest, (0.744) for lending interest. Moreover, mean (SD) was estimated to be about 15.8 (14.87) for general inflation, 19.1 (1.04) for food inflation, (9.089) for non-food inflation, (1.449) for average monthly temerature and (3.3705) for average monthly rain fall Average monthly rice of bean Average monthly rice of ea Figure 1: Average monthly rice trend of Bean and ea 53

6 statistics Table 1: Summary results for average monthly rice and Log-return series for Bean and ea Exchange Statistics Average monthly rice Log-return series bean ea bean ea Mean Median Maximum Minimum Std. Dev skewness kurtosis Jarque-Bera value Saving Interest Table : Summary results for covariates Lending interest General inflation Food inflation Non-food inflation Temerature (in 0 c) Rain fall (in mm) Mean Median Minimum Maximum St. dev Tests of Stationarity Before considering volatility models, the first logical ste is to check the stationarity of the average monthly rice using ADF test and unit root test. In ADF test, the null hyothesis of unit root is rejected if the test statistic is less than the critical value or the -value is less than the level of significance (α=0.05). As can be seen from the table 3, the null hyothesis of unit root would not be rejected, that is, there is a unit root roblem in each of the series indicating that each average monthly rice series is non-stationary. Table 3: ADF unit root test at level for average monthly rices The table 4 shows that all the t-statistics are less than the critical values. These indicate that the null hyothesis of unit root would be rejected in all of the four cases. Hence the log return series are stationary. All the variables excet saving interest are nonstationary at level. However, excet saving interest all the variables are stationary after first difference as shown in Table 5, imlying that all exlanatory variables are integd of order one. rices Test Statistics 1% critical value 5% critical value 10% critical value -value Bean ea Table 4: ADF unit root test at level for average monthly rice of log-return series Log-returns Test Statistics 1% critical value 5% critical value 10% critical value -value Bean * ea * * Statistically significant Table 5: ADF unit root test of the first difference of exlanatory variables Exlanatory variable ADF test statistic 1% critical value 5% critical value 10% critical value -value Exchange * Lending interest * General inflation * Food inflation * N-food inflation * Temerature * Rain fall * * Statistically significant Estimation of Mean Equation In the secification of the mean equation, lower order ARMA models are often considered, say, the twenty five combinations of AR (0-4) and MA (0-4). Otimal lag length was selected based on the minimum BIC rovided that no serial autocorrelation exists in the residuals from the secified mean model. The resence of autocorrelation in the residuals was tasted using the Lagrange Multilier (LM) test for each of the mean equations considered. Only models with no remaining serial correlations are considered as candidate models. Among the candidate mean models for the rice return series of bean, ARMA (4, 4) has the smallest BIC and exhibits no serial autocorrelation. 54

7 Similarly, ARMA (1, 0) has found to have the smallest BIC for the return series of ea. The fitted mean equations are shown in Tables 6 and 7. Table 6: ARMA (4, 4) mean equation for average monthly rice return series of Bean Variable Coefficient Std. Error t-statistic rob. C AR(1) AR() AR(3) AR(4) MA(1) MA() MA(3) MA(4) Table 7: ARMA (1, 0) mean equation for average monthly rice return series of ea Variable Coefficient Std. Error t-statistic rob. C AR(1) Testing for ARCH Effects The ARCH LM test hels to test the hyothesis that there is no ARCH effect u to lag. Table 8 shows the results of ARCH LM test for lags 1, and 3 for monthly rice return series. The test for the null hyothesis of no ARCH effects using Engle LM test and F-test confirmed the resence of ARCH (1) effects in the residuals from mean equations for bean and ea average monthly rice returns. These results indicate that the resective log return series are volatile and need to be modeled using GARCH family models. Otimal Order Selection and arameter Estimation of GARCH Family Model The otimal lag for GARCH family models has to be determined rior to the construction of the final model to investigate the determinants of monthly rice volatility. Since there is a consensus that GARCH(1,1) family model is the most convenient secification in the financial literature (Bollerslev et al., 199 and Lee and Hansen, 1994), the GARCH(1,1) model is comared to various higher-order models of volatilities based on the minimum AIC and BIC. After testing for different orders of and of GARCH family, it was found that EGARCH(1,3) under Normal distributional assumtion for residuals, EGARCH(,1) under Student-t distributional assumtion for residuals and EGARCH(,3) under GED distributional assumtion for residuals for the rice volatility of bean and EGARCH(1,1) under Normal distributional assumtion for residuals, EGARCH(1,) under Student-t distributional assumtion for residuals and EGARCH(,1) under GED distributional assumtions for residual for the rice volatility of ea were found to be the best models to describe the data as they ossess minimum BIC. The summary results are dislayed in Table 9. Moreover, to select the aroriate error distribution for selected asymmetric GARCH class models among normal, Student-t and GED distributions, the four forecast accuracy statistics: RMSE, MAE, MAE and Theil Inequality coefficient were alied using in-samle forecast. The results show that ARMA(4,4)-EGARCH (,3) model with GED for residuals and ARMA(1,0)- EGARCH (1,) model with student-t for residuals for bean and ea, resectively erform best as comared to others as they ossess the smallest forecast error measures in the majority of the statistics considered. The arameters in the mean and variance equations are estimated using the maximum likelihood (ML) method. The results are shown in Table 10. Table 8: ARCH LM test summary statistics Item ARCH() X statistic -value F-statistic -value BIC ARCH(1) Bean ARCH() ARCH(3) ARCH(1) ea ARCH() ARCH(3) Table 9: Otimal lag selected based on BIC under different distributional assumtions of residuals Variable Model Error Distribution BIC Asymmetric term (α=5%) ARMA(4,4)-EGARCH(1,3) Normal significant bean ARMA(4,4)-EGARCH(,1) Student-t Not significant ARMA(4,4)-EGARCH(,3) GED significant ARMA(1,0)-EGARCH(,) Normal Significant ea ARMA(1,0)-EGARCH(1,) Student-t significant ARMA(1,0)-EGARCH(,1) GED Not significant 55

8 Table 10: ML arameter estimates of the volatility models for Wheat, Bean and ea arameter Bean Mean (-value) Variance (-value) Mean (-value) Variance (-value) Constant (0.000)** 5.089(0.000)** (0.000)** (0.0017)** AR(1) (0.0001)** (0.08)* AR() (0.0009)** AR(3) 0.071(0.0074)** AR(4) 0.013(0.0000)** MA(1) -0.08(0.0031)** MA() -0.83(0.361) MA(3) 0.71(0.0010)** MA(4) (0.0064)** ARCH (-1) 0.961(0.000)** (0.0019)** ARCH (-) (0.003)** Asymmetric (-1) (0.001)** (0.0000)** Asymmetric (-) 0.816(0.000)** EGARCH (-1) (0.018)* -0.51(0.0405)* EGARCH (-) (0.000)** 0.411(0.094)* EGARCH (-3) 0.110(0.000)** Exchange (0.0351)* 1.345(0.000)** Saving interest (0.036)* (0.6) Lending interest (0.050).6144(0.586) General inflation (0.00)** 0.480(0.08)* Food inflation 0.113(0.007)** 0.899(0.0000)** N-Food inflation 0.49(0.041)* (0.950) Temerature (0.647) 0.673(0.0038)** Rain fall (0.0359)* (0.0135)* February (0.0108)* (0.0033)** March -0.11(0.006)** (0.000)** Aril (0.001)** 0.330(0.836) May (0.4479) 0.576(0.751) June 1.078(0.156).574(0.1407) July (0.000)**.3735(0.0000)** August (0.091)* (0.0100)* Setember (0.9399).717(0.0070)** October (0.00)**.1119(0.7) November (0.490).5899(0.1374) December (0.0816) 1.437(0.0067)** * Significance at the 5% level and ** significance at the 1% level. ea DISCUSSION Monthly rice Return Series for Bean From the results, exchange, general inflation, food inflation and non-food inflation have ositive and significant effect on the rice volatility of bean. An increase in exchange, general inflation, food inflation and non-food inflation leads to increase in the volatility of average monthly rice of bean. In contrast, saving interest and average rainfall had significant negative effect. The rainfall result is in line with the findings by Alisher (01). From the observed results of seasonal dummies, rices in February, July and August have an increasing significant effect, while March, Aril and October have decreasing effect. The results indicate that EGARCH (-1), EGARCH (-) and EGARCH (-3) terms are ositive and statistically significant at the 5% level. The ositive coefficient of the EGARCH(-1), EGARCH(-) and EGARCH(-3) terms show that the 1-, - and 3- month lagged rice volatility of bean leads to an increase in current month volatility. Also, 1- and - month lagged shocks (ARCH (-1) and ARCH (-) terms) of the average monthly rice of bean have statistically significant effect. Similarly, the asymmetric term was ositive and statistically significant at the 1% level of significance. Thus, bad news had larger imact on the rice volatility than good news. Monthly rice Return Series for ea The results of ea also indicate that exchange, general inflation, food inflation and temerature are ositively significant, while rainfall negatively affects rice volatility of ea. The rices in July, August, Setember and December have a ositive significant effect. On the other hand, rices in February and March affect the rice volatility of ea negatively. The EGARCH (-1) term has a negative effect on the current rice volatility of ea. This result is not in line with the findings by Greene (003). And 1- month lagged shock (ARCH (-1) term) of the average rice of ea had a ositive significant effect. Likewise, the asymmetric (-1) term was ositively significant at the 1% level. 56

9 Checking Adequacy of Fitted Models Various diagnostic tests were erformed to check the aroriateness of the fitted models. The Ljung-Box (k) test indicates that autocorrelations in the standardized residuals are not significantly different from zero for the first 3 lags for bean and ea return series, indicating that the residuals are uncorrelated (white noise). The tests for the remaining ARCH effect at time lag 1, and 3of squared residuals shows no remaining ARCH effect as the -values from both chi-square and F tests are greater than 5%. The results reveal that the coefficients of skewness were and and the coefficients of kurtosis were.6791 and.7501 for bean and ea, resectively. The Jarque-Bera test statistics were insignificant in all cases imlying that the residuals were aroximately normally distributed. Thus, the volatility models fitted for average monthly rices were good fit for the data. In-samle Forecast of Average Monthly rice Volatility Using EGARCH Fitted Models Using the fitted volatility models for average monthly rice of wheat, bean and ea, the volatility of rices (using variance as a volatility measure) was forecasted. The dynamic in-samle forecasts are resented in Figure and Figure 3. It can be observed that high rice volatility was observed during for bean. Also high rice volatility values were observed during the year 008 and for ea. Moreover, it can be seen that the average monthly rice of ea shows more volatility (in articular around 008) as comared to the other monthly rice series Forecast of Variance Figure : In samle forecast of average monthly rice volatility for Bean Forecast of Variance Figure 3: In samle forecast of average monthly rice volatility for ea CONCLUSIONS This study considered the average monthly rice volatility and its determinants ulse (bean and ea) from December 001 to June 01 G.C in Amhara National Regional State (ANRS). From the emirical results, it can be concluded that average rice return series of, bean and ea show the characteristics of financial time series such as volatility clustering, letokurtic distributions and asymmetric effect. This justifies the use of the GARCH family models. The forecast erformances of the models were evaluated using the MAE, MAE, RMSE and Theil inequality coefficient. Asymmetric EGARCH model with GED and Student t distributional assumtion for residuals was found to be the best fit model. That is, ARMA(4,4)- EGARCH(,3) model with GED for bean and ARMA(1,0)- EGARCH(1,) model with student-t for ea were found to be the best fit models for average monthly rice of log return series. Monthly average rice volatility of bean had a significant ositive relationshi with exchange, general inflation and food inflation. Thus, an increase in exchange, general inflation and food inflation ush u the average rice volatility of bean. Also inflation of non-food items had a ositive significant effect on the average rice volatility of bean. On the other hand, rice volatility of bean had a negative relationshi with saving interest and rainfall. The volatility in the average rice of ea had a significant ositive relationshi 57

10 with exchange, general inflation, food inflation and temerature. Rainfall was negatively affecting the volatility of average rice of ea. Some of the monthly dummies were found to be significant. This indicates that rice volatility has seasonal attern. In all the series considered, the asymmetric term (s) was (were) found to be ositive and significant. This is an indication that unanticiated increase in rices had larger imact on rice volatility than unanticiated decrease of the same. Moreover, the EGARCH terms were significant in all volatility models considered. This is a strong evidence of the resence of volatility sillover from one eriod (month in our case) to another. ACKNOWLEDGEMENTS We are very gful to Arba Minch University for financial suort to my M.Sc. studies at Gondar University through the whole academic years. My sincere thanks go to the National Bank of Ethioia, Central Statistical Agency and National Metrological Agency for their cooeration at data collection stage. REFERENCE Akaike, H. (1974). Anew Look at Statistical Model Identification. IEEE Transactions on Automatic Control AC 19(6): Alisher Mirzabaev (01). Climate volatility and change in central Asia: Economic imacts and adatation. ANRS-BoFED (011). Amhara National Regional State- Bureau of Finance and Economic. Asteriou, D. and Hall, S.G. (007). Alied Econometrics: A Modern Aroach Using Eviews and Microfit: Revised Edition, Hamshire, algrave Macmillan. Bollerslev, T. and J.M. Wooldridge (199), uasimaximum likelihood estimation and inference in dynamic models with time-varying covariances, Econometric Reviews 11: Bollerslev, T., Chou, R. Y and Kroner, K.F. (199). ARCH Modeling in Finance. A Review of Theory and Emirical Evidence. Journal of Economics 5: Bollersslev. T. and Taylor (1986). Generalized Auto regressive Conditional Heteroskedasticity. Journal of Econometrics 31: Box, G.E.. and Jenkins, G.M. (1976). Time Series Analysis, Forecasting and Control. Revised Edition, Holden Day. Dayton, C.M. (003). Model Comarison Using Information Measures. Journal of Modern Alied Statistical Methods (): Dickey, D.A. and Fuller, W.A. (1979). Distributions of the Estimators for Autoregressive Time Series with a Unit Root. Journal of the American Statistical Association 74: Engle, R.F. (198). Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation. Journal of Econometrics 50: FAO (011). Resonding to Global Food rice Volatility and its Imact on Food Security. Fryzlewicz,. (007). Lecture Notes: Financial time series ARCH and GARCH models, Deartment of Mathematics, University of Bristol, Bristol BS8 1TW, UK..z.fryzlewicz@bristol.ac.uk, htt:// bris.ac.uk /~mazf/. Hoddinott, J. (006). Shocks and their consequences across and within households in rural Zimbabwe. Journal of Develoment Studies 4(): IMF (009). The Federal Democratic Reublic of Ethioia: Selected Issues Series, International Monetary Fund Country Reort, No. 08/59,. 35f. Jordaan, H., Jooste, B.A. and Alemu, Z.G. (007). Measuring the rice Volatility of Certain Field Cros in South Africa: Using the ARCH/GARCH Aroach. Journal of agriculture 46. Lee, S.W., Hansen, B.E. (1994). Asymtotic roerties of the maximum likelihood estimator and test of the stability of arameters of the GARCH and IGARCH models. Econometric Theory 10: 9-5. Ljung, G. and Box, G. E.. (1978): On a Measure of Lack of Fit in Time Series Models, Biometrika, 66, Nabeya, S. and erron,. (1994). Local asymtotic distribution related to the AR(1) model with deendent errors. Journal of Econometrics 6(): Nelson, D.B. (1991). Conditional Heteroskedasticity in Asset Returns. A New Aroach Econometrician 59(): erron. and Ng. S. (1996). Useful modifications to some unit root tests with deendent errors and their local asymtotic roerties. Review of Economic Studies 63: hillis, C.B. and erron,. (1987). Testing for a Unit Root in Time Series Regression. Biometrics 75: Shewartz, G. W. (1989). Why Does Stock Market Volatility Change over Time? Journal of Finance 44(5): Tsay, R.S. (005). Analysis of Financial Time Series, nd Edition. John Wiley and Sons, New York. 58