ARFIMA, ARIMA And ECM Models Forecasting Of Wholesale Price Of Mustard In Sri Ganganagar District Of Rajasthan Of India

Transcription

1 ARFIMA, ARIMA And ECM Models Forecasting Of Wholesale Price Of Mustard In Sri Ganganagar District Of Rajasthan Of India Dr. Richard Kwasi Bannor, Institute of Agribusiness Management, SK Rajasthan Agricultural University, India Mada Melkamu, PhD Agricultural Economics Scholar, College or Agriculture Swami Keshwanand Agricultural University, Bikaner Abstract This study explored modelling and forecasting of wholesale mustard monthly prices in Sri Ganganagar district of Rajasthan using Autoregressive Integrated Moving Average Model (ARIMA), Autoregressive Fractionally Integrated Moving- Average Model (ARFIMA) and Error Correction Model (ECM). Based on minimum AIC and BIC values, ARFIMA (1/2, 0.309, 1) was selected as the best fit model for forecasting of mustard prices whereas ARIMA (1, 1, 1) was selected for ARIMA modelling. The result shows the mean absolute percentage error of ARIMA (1, 1, 1), ARFIMA ( 2, 0.449,1) and ECM based on predictions from January 2005 to June 2015 are 4.45 percent, 4.90 percent and 8.11 percent respectively. Whereas the mean absolute percentage error of ARIMA (1, 1, 1), ARFIMA ( 2, 0.449,1) and ECM based on predictions from January to June 2015 are 6.60 percent, 6.79 percent and 7.35 percent respectively The result therefore shows that among the three models of univariate ARIMA and ARFIMA and multivariate ECM, the best model fit for forecasting of wholesale mustard prices in Sri Ganganagar District of Rajasthan is ARIMA model Keywords: ARFIMA, ECM, ARIMA, Mustard, Rajasthan, Forecasting, GPH test, Impulse response function. Introduction The total oil seeds produced in all India in was million metric tonnes. Among the states, Madhya Pradesh, Rajasthan and Maharashtra contribute 9.26, 6.20 and 5.01 million metric tonnes respectively (Anonymous, 2013). Rajasthan contributes about percent of the total oilseeds produced in the country whereas Madhya Pradesh contributes highest percentage of oilseeds in the country representing about percent of the total oilseeds produced in production year. Rajasthan remains the highest producer of mustard-rape seed in the country with production of 3.65 million metric tonnes in the production year which is represented by percent of the total production in the country (Anonymous, 2013). The production height achieved by the state could be sustained and even improved if there is enough information on the prices of mustard of the various markets in the state. Thus, market information and intelligence are crucial to enable farmers and traders to make informed decisions about what to grow, when to harvest, to which markets produce should be sent to sell and whether to store it or not (Burark et al, 2013). That is to say, the dissemination of complete and accurate marketing information is a key to achieving both operational and pricing efficiency in the mustard marketing system (Bhardwaj, 2011). That notwithstanding, there is no dedicated organization performing task of collection and dissemination of market information to the stakeholders in public domain (Anonymous, 2014). Even though, the most important marketing intelligence need of the farmer is price intelligence. It is in light of the aforementioned reasons, why price forecasting is taken up by Institute of Agribusiness Management, S K Rajasthan Agricultural University, and Bikaner under the aegis of NCAP, ICAR, New Delhi, under the project of Network Project on Market Intelligence for various crops in Rajasthan of which mustard is one (Anonymous, 2014). In order to contribute to ascertain which forecasting model will be best fit for forecasting of mustard prices, this research was undertaken to forecast the prices of mustard in Sri Ganganagar district which is one of the highest producer of mustard in Rajasthan and India as a whole using Autoregressive Integrated Moving Average Model (ARIMA), Autoregressive Fractionally Integrated Moving-Average Model (ARFIMA) and Error Correction Model (ECM). In addition, this research became necessary because researchers have argued that, some econometric models are better in forecasting than other models not based on the particular commodity. However the researcher is of the view that, each forecasting model is of essence and should be judged on how best it forecast a particular commodity at a point in time. Thus, one model might be best for forecasting of a particular commodity at a point in time but cannot be best for forecasting of other agricultural commodity. ARIMA and ECM was selected because Mishra et al, 2015; Kwasi and Sharma (2015) have argued that multivariate time series Open Access Journals Blue Ocean Research Journals 1

2 models are better fit models in forecasting of agricultural commodities because it allows for inclusion of other exogenous variables such as rainfall, prices of other markets etc that have effect on the forecasted time series. Also, Paul (2014) and Amadeh et al., (2013) argued, long memory forecasting models (ARFIMA), is and will be best fit for modelling time series data with long memory. A series with long memory is characterised by their ability to remember events in the long history of the data and their ability to make decisions on the basis of such memories. Hence ARFIMA was selected for the forecasting mustard price series because it shows existence of long memory. Thus, this paper aims to obtain better forecasting accuracy based on several forecasting methods and also seeks to show that several forecasting models (univeriate, multivariate, hybrids, linear and non linear models) should be considered when it comes to forecasting of agricultural commodity prices and the best fit model selected for that particular commodity and at that particular point in time. Furthermore, this paper will add to the little existing literature on forecasting of agricultural commodity prices in Rajasthan. Methodology Sources of data The secondary data used for this study was sourced from AGMARKNET database (from This database is under the directorate of Marketing and Inspection of the Ministry of Agriculture of Government of India. Prices data of one (1) major production and marketing centre in Rajasthan state namely Sri Ganganagar was sourced. The data covers monthly wholesale mustard prices from January 2005 to June Method of Data Analysis The data analytical techniques that were used in this study comprised of unit root test, Geweke Porter-Hudak (GPH) test, Autoregressive Integrated Moving Average model (ARIMA), Autoregressive Fractionally Integrated Moving-Average Model (ARFIMA) and Error Correction Models (ECM) were used. Unit root test: The unit root test approach was shaped by Kwasi, 2015 Bannor and Sharma 2015; Kwasi and Sharma 2015; Mafimisebi et al., 2014; Kwasi and Kobina, According to these authors, citing Juselius, (2006), a stationary series or series with no unit root is one with a mean value which will not vary with the sampling period. In contrast, a non-stationary series will exhibit a time varying mean. Before forecasting prices of mustard in Sri Ganganagar market, it is essential to test for unit root and identify the order of stationarity, denoted as I(0) or I(1). This is necessary to avoid spurious and misleading regression estimates. The framework of ADF methods is based on analysis of the following model Here, p t is the mustard price series being investigated for stationarity, is first difference operator, T is time trend variable, μ t represents zero- mean, serially uncorrelated, random disturbances, k is the lag length; α,β, γ and δ k are the coefficient vectors. Unit root tests were conducted on the β parameters to determine whether or not the price series are closely identified as being I(1) or I(0) process. Box Jenkins model identification for ARIMA modelling This approach was adopted from Kwasi and Sharma (2015) and Kwasi and Kobina Firstly the data was examined to check for the most appropriate class of ARIMA processes through selecting the order of the consecutive and seasonal differencing required to make series stationary, as well as specifying the order of the regular and seasonal ARIMA model necessary to adequately represent the time series model. The Autocorrelation function (ACF) and the Partial Autocorrelation function (PACF) are the most important elements of time series analysis and forecasting. The ACF measures the amount of linear dependence between observations in a time series that are separated by a lag k. The PACF plot helps to determine how many auto regressive terms are necessary to reveal one or more of the following characteristics: time lags where high correlations appear, seasonality of the series, trend either in the mean level or in the variance of the series. Model Parameter Estimation The Box and Jenkins model includes autoregressive and moving average parameters as well as differencing in the formulation of the model. The three types of parameters in the model are: the autoregressive parameters (p), the number of differencing passes (d) and moving average parameters(q). Box Jenkins model are summarized as ARIMA (p,d,q). For example, model described as ARIMA (0,1,0) means that this contains 0 autoregressive (p) parameter and 1 moving average (q) parameter for the times series data after it was differenced once to attain stationarity. Oder Of Autoregressive Process (P) Specifically, for an AR (1) process, the autocorrelation function should have an exponentially decreasing appearance. However, higher- order AR processes are of- Open Access Journals Blue Ocean Research Journals 2

3 ten a mixture of exponentially decreasing and damped sinusoidal components (Kwasi and Sharma, 2015). Order of Moving-Average process (q) The autocorrelation function of an MA series cut of sharply whereas as for AR series, the autocorrelation function exhibits an exponential decay. The autocorrelation function of a MA (q) process becomes zero at lag q+1 and greater, therefore we examine the autocorrelation function to see where it becomes zero. We do this by placing the 95% confidence interval for the autocorrelation function on the autocorrelation plot. In ARIMA model, the future value of a variable is assumed to be a linear function of several past observations and random errors. That is, the underlying process that generate the time series has the form: where yt and ε t are the actual value and random error at time period t, respectively; ø i (i=1, 2,, p) and θ j (j=0, 1, 2,, q) are model parameters. The integers p and q are often referred to as orders of the model. Random errors, ε t, are assumed to be independently and identically distributed with a mean of zero and a constant variance of σ 2. Estimate long memory in mustard price series via Geweke Porter-Hudak (GPH) approach for AR- FIMA modelling The GPH method uses nonparametric methods, thus a spectral regression estimator to evaluate d without explicit specification of the 'short memory' (ARMA) parameters of the series. The series was differenced so that the resulting d estimate will fall in the [-0.5, 0.5] interval. A choice was made of the number of harmonic ordinates to be included in the spectral regression. The regression slope estimate is an estimate of the slope of the series' power spectrum in the vicinity of the zero frequency; if too few ordinates are included, the slope is calculated from a small sample. If too many are included, medium and high-frequency components of the spectrum will contaminate the estimate. A choice of root (T), or power = 0.8, was employed. To evaluate the robustness of the GPH estimate, a range of power values (from ) is commonly calculated as well. Two estimates of the d coefficient s standard error are commonly employed: the regression standard error, giving rise to a standard t -test, and an asymptotic standard error, based upon the theoretical variance of the log periodogram of. The statistic based upon that standard error has a standard normal distribution under the null. Autoregressive Fractionally Integrated Moving- Average Model Forecasting. According to Bannor and Melkamu (2015), the concept of fractional integration is often referred to as defining a time series with longrange dependence, or long memory. Any pure ARIMA stationary time series can be considered a short memory series. An AR (p) model has infinite memory, as all past values of are embedded in, but the effect of past values of the disturbance process follows a geometric lag, damping off to near-zero values quickly. A MA (q) model has a memory of exactly q periods, so that the effect of the moving average component quickly dies off. The model of an autoregressive fractionally integrated moving average process of a time series of order (p,d,q), denoted by ARFIMA (p,d,q), with mean μ, may be written using operator notation as:...3 Where L is the backward-shift operator, 4 is the fractional differencing operator defined by..5 With ( ) denoting the gamma (generalized factorial) function. The parameter d is allowed to assume any real value. The arbitrary restriction of d to integer values gives rise to the standard autoregressive integrated moving average (ARIMA) model. The stochastic process is both stationary and invertible if all roots of and Open Access Journals Blue Ocean Research Journals 3

4 lie outside the unit circle and. The process is non stationary for, as it possesses infinite variance. Assuming that d [0, 0.5), Hosking (1981) showed that the autocorrelation function, ρ( ), of an ARFIMA process is proportional to k 2d 1 as k. Consequently, the autocorrelations of the ARFIMA process decay hyperbolically to zero as k in contrast to the faster, geometric decay of a stationary ARMA process. For d (0, 0.5), diverges as n, and the ARFIMA process is said to exhibit long memory, or long-range positive dependence. The process is said to exhibit intermediate memory (anti-persistence), or longrange negative dependence, for d ( 0.5, 0). The process exhibits short memory for d = 0, corresponding to stationary and invertible ARMA modeling. For d [0.5, 1) the process is mean reverting, even though it is not covariance stationary, as there is no longrun impact of an innovation on future values of the process. If a series exhibits long memory, it is neither stationary (I(0)) nor is it a unit root (I(1)) process; it is an I(d ) process, with d a real number. A series exhibiting long memory, or persistence, has an autocorrelation function that damps hyperbolically, more slowly than the geometric damping exhibited by short memory (ARMA) processes. Thus, it may be predictable at long horizons. Multivariate ECM forecasting Specifications To deal with forecasting which involves two or more time series, the researcher went beyond the univariate ARIMA and ARFIMA models. Error Correction model was used in forecasting wholesale mustard prices in Sri Ganganagar. The model included model included mustard market arrivals in Sri Ganganagar and wholesale mustard prices from January 2005 to June Other market arrivals and prices were used in modelling of both VAR and ECM forecasting such Kota, Alwar, Bikaner, Tonk, and Bharatpur. However, the best forecasting model based on forecasting accuracy was combination of Sri Ganganagar prices and arrivals using the ECM model. Thus arrivals of mustard in the market have effect on the price of wholesale mustard prices hence included in the model to increase the forecasting of accuracy. The researcher understands that the wholesale prices of mustard in Sri Ganganagar could be influenced by multiple time series variables of which market arrivals of the mustard is one. By considering multiple time series jointly for an analysis, it utilizes the additional information in determining the dynamic relationships over time among the series (Stock and Watson 2011). Testing for Johansen co-integration (trace and eigen value tests): If mustard market price and arrival series are individually stationary at same order, the Johansen co-integration model can be used to estimate the long run co-integrating vector using a Vector Auto regression (VAR) model of the form: 6 Where p t is a nx1vector containing the series of interest (the three variable series) at time (t), is the first difference operator and are nxn matrix of parameters on the ith and kth lag of Ig is the identity matrix of dimension g, is constant term, μ t is nx1 white noise error vector. Throughout, p is restricted to be (at most) integrated of order one, denoted I(1), where I(j) variable requires jth differencing to make it stationary. Equation (8) tests the co-integrating relationship between stationary series. Johansen and Juselius (1990) and Juselius (2007) derived two maximum likelihood statistics for testing the rank of Π, and for identifying possible co-integration as the following equations show: Open Access Journals Blue Ocean Research Journals 4

5 Where r is the co-integration number of pair-wise vector, λ t is ith eigen value of matrix Π. T is the number of observations. The λ trace is not a dependent test, but a series of tests corresponding to different r -value. The λ max tests each eigen value separately. This model was used to test for; (1) integration between various mustard market price and arrivals series of Sri Ganganagar. Test for Granger-causality: After undertaking cointegration analysis of the long run linkages of the various variables, and having identified they are linked, an analysis of statistical causation was conducted. The causality test uses an error correction model (ECM) of the following form; Where m and n are number of lags determined by Akaike Information Criterion (AIC).If the null hypothesis that say Tonk mustard market prices in Rajasthan j do not Granger cause Bharatpur mustard market prices in Rajasthan i is rejected (by a suitable F-test) that σ h = 0 for h = 1, 2.n and β=0, this indicates Sri Ganganagar mustard market price j Granger-cause Sri Ganganagar mustard market arrivals i. Impulse Response Function Impulse response function is a shock to the ECM model used in the analysis. Impulse responses identify the responsiveness of the dependent variable which is (endogenous variables) in the models when a shock is put to the error term. A simplified model of impulse response function for mustard prices against arrivals in Sri Ganganagar can be written as:..10 Where is error term or shock or impulse. Hence the model will give us the effect on the unrestricted VAR system when a unit shock is applied to variables. Forecasting Accuracy of ARIMA, ARFIMA and ECM models Measures of forecast accuracy are based on forecast errors. The difference between the actual value and the forecasted value gives you the forecast error. The researcher employed Mean Absolute Percent Error (MAPE) as a measure of accuracy for the models. It indicates the measured error as a percent of actual values. MAPE is simplified by Kwasi and Sharma (2015) as follows: MAPE = 100 [ A t F t / A t ] / T..11 Where A t =Actual Values, F t =Forecasted Values, T=number of time periods in months Open Access Journals Blue Ocean Research Journals 5

6 Results and Discussion Figure 1: Trend of wholesaler mustard prices (Rs/q) in Sri Ganganagar Price in Rupees per quintal of mustard Trend of wholsale prices of mustard in Sri Ganganagar 2005m1 2010m1 2015m1 Months/ Years Figure one shows the trend of wholesale mustard prices (Rs/q) in Sri Ganganagar over the years of study i.e. January 2005 to June The figure shows the price of mustard has been increasing over the years from 2005 even though there are ups and downs movements at certain months which depict seasonality of the prices in a year. Variables Price Level 1(0) Sri Ganganagar ADF Statistics CV= Table 1: Unit Root Testing Philips-Perron Test CV= Test Statistics Test Statistics Prices Arrivals H o : variables are not stationary or has unit root H 1 : Variables are stationary or does not have unit root NB: If the absolute value of ADF, PP statistics is less than their 5% critical value we accept null hypothesis. It is also when the MacKinnon approximate p-value for Z(t) is insignificant. Unit root testing was done to determine stationarity of the price and arrivals series. The result shows that at level, arrivals of mustard are stationary whereas that of prices are not. Table 2: Unit Root Testing at First Difference Variables First Difference 1(1) ADF Statistics Philips-Perron test CV= CV= Test Statistics Test Statistics Prices Unit root test was done on the non stationary price series data. Augmented Dickey Fuller (ADF) and Phillips Perron (PP) tests show that at first difference, the price series become stationary. The stationary series was used in the analysis. Open Access Journals Blue Ocean Research Journals 6

7 Table 3: Diagnostic Checking of ARIMA models ARIMA Model(p,d,q) Schwarz Bayesian Criterion(BIC) Ljung Box Q statistics R 2 (%) 1,1, ,1,1* 10.31* ,2, ,1, ,1, ,1, ,1, ,1, ,1, Source: Author s computation *= Most preferred according to the test Ljung box Q statistics; Ho: No autocorrelation H1: There is autocorrelation NB; Reject null hypothesis when p value is less than 5% Maximum Absolute Percentage Error (MAPE) Table 3 shows various diagnostic tests done to check the adequacy of the models for ARIMA forecasting. The BIC test was conducted because we were considering several ARIMA models as shown. ARIMA (1, 1, 1) had the lowest BIC value of In addition, the Q statistics was used to test for serial correlations in the models above. All the models as shown in table 3 had no autocorrelation in the series; which means, the model is fit for forecasting. R square was also used to test the goodness of fit of the model. However according to Kwasi and Sharma (2015) time series usually have strong trends and seasonal hence the R square is normally high making it difficult to judge the usefulness of a model by just looking at the high R square. Table 4: Forecasted monthly mustard prices using ARIMA (1, 1,1) Month Year Forecasted prices(rs/q) Lower Boundary Upper Boundary July August September October November December Source: Authors computation based on data series Table 4 shows monthly forecasted wholesale mustard prices from July 2015 to December Open Access Journals Blue Ocean Research Journals 7

8 Figure 2: ARIMA model (1,1,1) Figure 2 gives a graphical representation of how the ARIMA(1, 1, 1) fit in forecasting wholesale mustard prices from January 2005 to December Table 5: Estimate long memory in a price series via Geweke Porter-Hudak (GPH) approach for ARFIMA forecasting Power Est. d Std. Err t(h o :d=0) P>/t/ Asy. Std Err. z(h o : d=0) P>/z/ H o : There is short memory in the series, H 1 : There is long memory in the series. NB: If the estimated d value is more than 0, we reject null hypothesis. It is also when the p values are significant. To use Autoregressive Fractionally Integrated Moving- Average Model (ARFIMA) forecasting model, the researcher used Geweke Porter-Hudak (GPH) approach to ascertain whether the series have long memory or otherwise. According to the results from the table 5, the mustard price series have long memory since the estimated d is not equal to zero and the test statistics is also significant at 1 percent. It indicates long memory model i.e. ARFIMA can be used for forecasting of the price series. The existence of the long memory feature in this series is an indication of the fact that if there is a shock to the mustard prices in Sri Ganganagar APMC, the effect of this shock will last a long time and finally disappear after several periods of time. Table 6: Parameter estimates of ARFIMA ( p,d,q) Model ARFIMA (p,d,q) Model Parameter Coefficient Probability 2, 0.449,1 d * AR ** MA * 2,0.489,2 d * AR * MA * 3,0.492,1 d * AR * MA * 3,0.493,3 d * AR * MA * Open Access Journals Blue Ocean Research Journals 8

9 Source: Authors computation, * siginficant at 1% ** siginficant at 5% and *** Significant at 10% The results from table 6 shows the various ARFIMA significant again signifies the series have long memory models estimated. The results shows that all the d and hence ARFIMA is a good model for forecasting the price the various AR and MA were significant. The d being series. Table 7: Log likelihood and AIC values of different ARFIMA models Market Parameter ARFIMA* (2, 0.449,1) ARFIMA (2,0.489,2) ARFIMA (3,0.492,1) ARFIMA (3,0.493,3) Sri Ganganagar Log likelihood * AIC * BIC * Source: Authors computation, * selected ARFIMA model for forecasting based on least AIC figure Table 7 shows the log likelihood, AIC and BIC of the various ARFIMA models tested. Four (4) ARFIMA models were tested to select the best fit model for the forecasting of mustard prices. Based on the minimum AIC and BIC values, ARFIMA (2, 0.449,1) was selected as the best fit model for forecasting of mustard prices. Table 8: Forecasted monthly wholesale monthly mustard prices using ARFIMA (2, 0.449,1) Month Year Forecasted prices(rs/q) Lower Boundary Upper Boundary June July August September October November December Source: Authors Computation based on data series Table 8 shows out of sample forecasted mustard prices in the coming months of 2015 in Sri Ganganagar. The forecasted values are the point predicted values in the months (from June to December 2015). The upper and lower boundary values indicates, the researcher expectation of mustard prices not to fall below the lower boundary or go above the upper boundary for the coming four months. Figure 3: Forecasted out of sample mustard prices (Rs/q) from Jan 2005-December m1 2010m1 2015m1 t sriganganagarprice xb prediction, dynamic(tm(2015m6)) Source: Authors Computation based on data series Open Access Journals Blue Ocean Research Journals 9

10 Table 9: Co-integration results for market pairs Sri Ganganagar prices/arrivals Trace Statistics 5% Critical No. lags Rank Remarks Value Prices Arrivals 0.24* Rank 1 Co-integration Source: Author s computation At rank 1: Ho: There is (1) co-integration of the variables at rank 1 H1: There is no 1 co-integration of the variables at rank 1. NB: We accept null hypothesis when trace statistics or max statistics is less than the 5% Critical value at rank 1. Table 9 shows cointegration test between wholesale mustard prices and arrivals in Sri Ganganagar. The result indicates that, there is one cointegrating rank between prices arrivals of mustard. Thus, in the long the run, wholesale mustard prices and arrivals move together. Table 10: Error Correction Model (ECM) results for wholesale mustard prices and arrivals in Sri Ganganagar Sri Ganganagar P Value Error Correction Term ity Remarks Short run model/causality Short un Causal- prices/arrivals Prob>Chi Direction Prices Arrivals ns ns - No Short run Arrivals Prices * Unidirectional Short run Source: Author s computation, A B=A causes B,*=1% Sign. Ho: No short run causality running from variable A to B H1: Short run causality running from A to B or variable A causes changes in variable B in the short run NB: Reject null hypothesis when the Prob> chi value is < 5% The result indicates that, there is short run causality between prices and arrivals of mustard. Mustard arrivals do Granger cause prices but mustard prices do not Granger cause mustard arrivals in the short run. Table11: Forecasted monthly wholesale mustard prices (Rs/q) using ECM Month Year Forecasted prices(rs/q) Lower Boundary Upper Boundary July August September October November December Source: Authors computation based on data series The result from table 11 shows the forecasted prices from July 2015 to December 2015 using the ECM model. The researcher expects the prices of mustard not to move beyond 5000 Indian rupees per quintal in the coming months based on the ECM model upper boundary results. Open Access Journals Blue Ocean Research Journals 10

11 Figure 4: Graph showing forecasted wholesale mustard prices (Rs/q) in Sri Ganganagar Forecast for sriganganagarprice m6 2015m8 2015m m12 95% CI forecast Figure four (4) shows the forecasted wholesale prices in Sri Ganganagar in the coming months up to December 2015 graphically using ECM. Figure 5: Impulse Response Function 200 vec1, sriganganagararrival, sriganganagarprice vec1, sriganganagarprice, sriganganagarprice step Graphs by irfname, impulse variable, and response variable Impulse response function is a shock to ECM model used in the analysis. Positive shocks are shocks that affect the wholesale mustard market prices in Sri Ganganagar positively i.e. a decrease of mustard price or arrivals in Sri Ganganagar market (Bannor, 2015). The graph indicate that an orthogonalized shock to the average mustard arrivals in Sri Ganganagar has a permanent effect on the average mustard price in Sri Ganganagar. According to this model, unexpected mustard price shocks that are local to the Sri Ganganagar market will also have permanent effect on the mustard prices in Ganganagar. Open Access Journals Blue Ocean Research Journals 11

12 Table 12: Sample forecast of monthly mustard prices using ARIMA (1, 1, 1), ARFIMA (2, 0.449, 1) and ECM from January 2015-June 2015 Month Year Actual In sample forecasted prices (Rs/q) Values ARIMA(1,1,1) ARFIMA(2,0.449,1) ECM January February March April May June MAPE( ) 4.45% 4.90% 8.11% MAPE (Jan-June,2015) 6.60% 6.79% 7.35% Source: Authors Computation based on data series Table 12 indicates the in sample wholesale mustard prices in Sri Ganganagar using the selected ARIMA (1, 1, 1), ARFIMA ( 2, 0.449,1) and ECM models. The result shows about 4.45 percent, 4.90 percent and 8.11 percent of forecasting error was shown by ARIMA (1, 1, 1), ARFIMA ( 2, 0.449,1) and ECM respectively. Thus, the mean absolute percentage error of ARIMA (1, 1, 1), ARFIMA (2, 0.449, 1) and ECM based on forecasted wholesale prices from January 2005 to June 2015 are 4.45 percent, 4.90 percent and 8.11 percent respectively. Whereas the mean absolute percentage error of ARIMA (1,1,1), ARFIMA ( 2, 0.449,1) and ECM based on predictions from January to June 2015 are 6.60 percent, 6.79 percent and 7.35 percent respectively A minimal forecasting error of a model indicates the model is good for forecasting of the mustard prices. These values show the mean uncertainty in each model s predictions or forecasting. Whether these values represent an acceptable amount of uncertainty or otherwise, depends on the degree of risk you are willing to accept (: Kwasi and Sharma, 2015; Kwasi and Kobina, 2014). The result therefore clearly shows that among the three models of univariate ARIMA and ARFIMA and multivariate ECM, the best model fit is ARIMA. This result disagrees with Kwasi and Sharma (2015) who argued that multivariate Vector Autoregressive model has high forecasting accuracy than ARIMA. The results also disagree with Amadeh et al., (2013) who argued ARFIMA forecasting model is better in forecasting than ARIMA when the series exhibits long memory. Conclusions and Recommendations Three forecasting models namely ARIMA, ARFIMA and ECM models were used to forecast the wholesale prices of mustard in Sri Ganganagar district of Rajasthan of India. The results show that ARIMA model is the best fit model for forecasting of mustard prices. That notwithstanding, the study has also shown that long memory model i.e. ARFIMA model can also be used to forecast mustard wholesale prices successfully with minimum forecasting error of less than 5 percent. Multivariate ECM model showed good results but forecasting error was higher than ARIMA and ARFIMA models. The results show that, there is no one specific model that can be said to be best fit for forecasting of all crops. It is therefore suggested that a number of models should be used when forecasting prices of agricultural commodities and the best fit model selected. In addition, this finding has provided insight on how to forecast prices of agricultural commodities which could be useful to farmers, wholesalers, retailers, consumers, government agencies and other stakeholders in mustard trading especially in Rajasthan. Furthermore, further studies should be done to determine mustard market efficiency in the district and the state as whole to complement the forecasting of prices. References [1] Amadeh, H., Amini, A., and Effati, F. (2013). ARIMA and ARFIMA Prediction of Persian Gulf gas-oil F.O.B. Investment Knowledge, 2, 7. [2] Anonymous (2014). Retrieved from /110414/india-oilseeds-and-products-annual-apr- 2014/ on 26/07/2015. [3] Anonymous (2014). Retrieved from LibraryManager/upload/Guar%20Forecast% pdf on 20/08/2015. [4] Anonymous (2013). Pocket book on agricultural statistics Ministry of Agriculture, Department of Agriculture and Cooperation; Directorate of Statistics, New Delhi. [5] Bannor, R. K., and Melkamu, M. (2015). Forecasting wholesale price of cluster bean using the Autoregressive Fractionally Integrated Moving-Average Model: the case of Sri Ganganagar of Rajasthan in India. Journal of Business Management and Economics, 3(8), Open Access Journals Blue Ocean Research Journals 12

13 [6] Bannor, R.K. and Sharma, M. (2015). Spatial price transmission of groundnut markets of Rajasthan. Indian Journal of Economics and Development, 11(4), [7] Bannor, R.K. (2015). Marketing of kinnow in Rajasthan. Unpublished PhD thesis submitted to Post Graduate Studies of SK Rajasthan Agricultural University. [8] Bhardwaj S. P. (2011). Singificance of Market Information System (MIS) in Agricultural Development. Indian Journal of Agricultural Marketing. 25( 3 ): [9] Brockwell, P., and Davis, R. (2002). Introduction to Time Series and Forecasting. Springer International Edition. [10] Burark, S.S., Pant,D.C., Sharma H. and Bheel, S. (2013). Price forecast of Coriander- A case study of the Kota Market of Rajasthan. Indian Journal of Agricultural Marketing. 27( 3):72. [11] Harvey, A.C. and Todd, E.C. (1983). Forecasting econometric time series with structuralbox-jenkins models (with discussion). Journal of Business and Economic Statistics.1(4): Retrieved from [12] Hosking, J. R. M. (1981). Fractional differencing. Biometrika,68: [13] Juselius K (2006). The Co-integrated VAR Model: Methodology and Applications.Oxford University Press. [14] Kwasi, B. R., and Sharma, A. (2015). Comparing the Forecasting Power of Multivariate VAR and Univariate ARIMA Models: A Case of Groundnut Prices in Bikaner District of Rajasthan. Asian Journal of Research in Business Economics and Management, 5(3), [15] Kwasi, B. R. (2015). Long run and short run causality of rice consumption by urbanization and income growth in Ghana. ACADEMICIA: An International Multidisciplinary Research Journal, 5(2), [16] Kwasi, B.R., and Kobina B.J. (2014). Forcasting of Cassava Prices In The Central Region of Ghana Using Arima Model. Intercontinental Journal of Marketing Research Review. 2(8). [17] Kwasi, B. R., and Kobina, B. J. (2014). Cassava markets integration analysis in the central region of Ghana. Indian Journal of Economics and Development, 10(4), [18] Mafimisebi T.E., Agunbiade B.O. and Mafimisebi O.E. (2014). Price Variability, Co-integration and Exogeniety in the Market for Locally Produced Rice: A Case Study of Southwest Zone of Nigeria. Retrieved from [19] Mishra, P., Sahu, P. K., Dhekale, B. S., and Vishwajith, K. P. (2015). Modeling and forecasting of wheat in India and their yield sustainability. Indian Journal of Economics and Development, 11(3), [20] Paul, R. K. (2014). Forecasting Wholesale Price of Pigeon Pea Using Long Memory Time-Series Models. Agricultural Economics Research Review, 27(2), [21] Stock J.H. and Watson, M.W. (2011). Introduction to Econometrics (3rd Edition). Boston, MA: Pearson. Open Access Journals Blue Ocean Research Journals 13