Hybrid time-series models for forecasting agricultural commodity prices

Abstract

Agricultural price forecasting has become a promising area of research in recent times. ARIMA model has been most widely used technique during the last few decades for this purpose. When the assumption of homoscedastic error variance is violated then ARCH/GARCH models are applied in order to capture the changes in the conditional variance of the time-series data. The ANN approach can also be applied in the field of forecasting of real time-series data successfully as an alternative to the traditional forecasting models. Real-world time-series data are rarely pure linear or nonlinear in nature, sometimes contain both the pattern together. In this situation a hybrid approach of combining the forecasts from a linear time-series model (ARIMA) and from a nonlinear time-series model (GARCH, ANN) has the better forecasting performance. The hybrid methodology namely ARIMA-GARCH and ARIMA-ANN have been applied for modelling and forecasting of wholesale potato price in Agra market of India. A comparative assessment has also been made in terms of Mean absolute percentage error (MAPE) and Root mean square error (RMSE) among the hybrid and their individual counterpart as far as forecasting is concerned. It is observed that ARIMA-ANN hybrid model outperforms the other combinations and individual counterpart for the data under consideration. R software package has been used for the data analysis.

Keywords

ARIMA ANN GARCH hybrid model potato price

1. Introduction

Time-series forecasting is an important statistical analysis technique used as a basis for manual and automatic planning in many application domains (Gooijer & Hyndman, 2006). Time series forecasting is an important area of forecasting in which past observations of the same variable are collected and analyzed to develop a model describing the underlying relationship. Forecasting plays a crucial role in business, industry, government and institutional planning. Sometimes we have little knowledge about the underlying data generating process. In this situation modeling approach becomes useful. Much effort has been devoted over last few decades to develop and improve several time-series forecasting models. There are several linear time-series models available in literature. One of the important and widely used technique for analysis of univariate time-series data is Box Jenkins’ Autoregressive integrated moving average (ARIMA) methodology (Box et al., 2007). The ARIMA model is so popular due to its statistical properties. ARIMA is a flexible class of models including pure autoregressive (AR) models, pure moving average (MA) models, combined AR and MA (ARMA) models. In addition to ARIMA, various exponential models can also be used to forecast a linear time-series process. But one of the major limitations of these models is the pre-assumed linear form of the models. This assumption of linearity limits the application of ARIMA model to real time-series data. Some of the applications of this model can be found in Paul and Das (2010, 2013), Paul et al. (2013, 2014), Paul (2015).

Linear models are not able to describe any changes in the conditional variances present in the real data. To tackle this situation, Engle (1982) defined the Autoregressive conditional heteroscedastic (ARCH) models in which significant presence of autocorrelation in the squared residual series is considered. But the ARCH models give satisfactory forecast only with large number of parameters which has necessitated the emergence of more parsimonious version that is Generalized ARCH (GARCH) models (Bollerslev, 1986). In GARCH models the unconditional autocorrelation function has slow decay rate.

Unlike the traditional model-based methods, artificial neural network (ANN) is a data-driven, self-adaptive, nonlinear, nonparametric method of forecasting. Many nonlinear processes that have unknown functional relationship can be modeled by the ANN models. There are many empirical evidences that nonlinear models perform well for long term forecasting whereas the linear models are suitable for short range forecasting. So there is a need of combining the linear and nonlinear models in order to get more accurate forecast. Sometimes it becomes too difficult in practice to decide whether a time-series process is generated using a linear or nonlinear model or whether a particular forecasting method is appropriate than the other in getting out-of-sample forecasts. In this situation generally a number of models are tried and the best model is selected based on some information criteria viz. Akaike’s information criteria (AIC) or Bayesian information criteria (BIC) or Hannan and Quinn (HQ) criteria. However, there is no guarantee that the final selected model will give best forecast if there are some influential factors like sample variation, structural change present in the data. As solution to this problem different forecasting methods can be combined to get the final forecast. Real time-series data are rarely pure linear or nonlinear in nature. They often contain both linear and nonlinear components in the structure which make it necessary to combine linear and nonlinear models to capture the existing pattern in the dataset more accurately. It is almost universally agreed that no single forecasting method will be the best choice in every situation. Most of the real-world problems are complex in nature and any single model is not able to capture several patterns uniformly. Therefore, combining of different models is important to increase the chance of capturing different patterns and improve the forecasting performance. Paul (2015) has combined ARIMAX model with GARCH and Wavelet technique and showed the improvement in forecasting accuracy as compared to individual counterpart. Paul and Sinha (2016) have compared the performance of ARIMAX and NARX model for forecasting crop yield. In the present investigation ARIMA-GARCH, ARIMA-ANN models along with the individual models like ARIMA and GARCH have been applied to the real data set.

2. Time-series forecasting models

There are several approaches of time-series modeling. Some of the traditional linear time-series models are moving average, exponential smoothing and ARIMA. To overcome the deficiencies of linear time-series models and to capture certain nonlinear pattern in the time-series data several nonlinear time-series models are available in the literatures. The most commonly used nonlinear time-series models are the bilinear model, the Threshold autoregressive model (TAR), the Exponential autoregressive (ExpAR) and the ARCH model. In the present investigation an attempt has been made to apply ARIMA, GARCH and its family of models, ANN model and some hybrid models. A brief description of the models is given below.

2.1 The ARIMA model

In an ARIMA model, it is assumed that the future value of a variable is a linear function of past values of the variable itself and random errors also. It is a linear univariate time-series model which expresses a time-series process, say, $\left\{y_{t}\right\},t=1,2,\ldots,n$ in terms of three sets of parameters as

$\varphi(L)y_{t}=\left(1-L\right)^{-d}\theta(L)e_{t}$ (1)

where $y_{t}$ and $e_{t}$ are the actual value and random error at the time $t$ , respectively; $\varphi(L)$ and $\theta(L)$ are the polynomial of lag operator $L$ of order $p$ and $q$ respectively with root outside the unit circle; random errors, $e_{t}$ are assumed to be independent and identically distributed with mean zero and variance $\sigma^{2}$ . In general an ARIMA model is denoted as ARIMA ( $p$ , $d$ , $q$ ) where $p$ , $d$ and $q$ represents the order of autoregression, integration (differencing) and moving average respectively. ARIMA is a general class of models which can represent both stationary and nonstationary processes by allowing $d$ value as 0 and 1 or 2 respectively.

2.2 The ARCH model

Linear Gaussian models are not able to describe non constant conditional error variance which is present in many real time-series data. To handle such a situation, Engle (1982) has introduced the ARCH models in which significant presence of autocorrelation of squared residuals is considered. The ARCH ( $q$ ) model for the series $\{\epsilon_{t}\}$ , $t=1,2,\ldots,n$ is defined by specifying the conditional distribution of $\{\epsilon_{t}\}$ given the information available up to $t-1$ . The process $\{\epsilon_{t}\}$ is ARCH ( $q$ ) if the conditional distribution of $\{\epsilon_{t}\}$ given the available $\psi_{t-1}$ information is

$\epsilon_{t}|\psi_{t-1}\sim N(0,h_{t})\text{ and }\epsilon_{t}=\sqrt{h_{t}}% \varepsilon_{t},$

where $\{\varepsilon_{t}\}$ is a white noise process that means $\{\varepsilon_{t}\}$ is a sequence of independent and identically distributed (i.i.d) random variables with mean zero and variance 1, i.e., $\varepsilon_{t}\sim iid\left(0,1\right)$ and the conditional variance $h_{t}$ is defined as

$h_{t}=a_{0}+\sum\nolimits_{i=1}^{q}a_{i}\epsilon_{t-i}^{2},∼{}a_{0}>0,a_{i}% \geqslant 0∼{}\forall\mathrm{∼{}and∼{}}\sum\nolimits_{i=1}^{q}a_{i}<1$ (2)

But ARCH model has drawback that, when the order of ARCH model is very large, estimation of a large number of parameters is required which is really a cumbersome process. Also, ARCH model is not parsimonious model.

2.3 The GARCH model

To overcome these difficulties of ARCH model, Bollerslev (1986) proposed GARCH model in which conditional variance is also a linear function of its own lags.

A GARCH ( $p$ , $q$ ) process has the following form

$h_{t}=a_{0}+\sum\nolimits_{i=1}^{q}a_{i}\epsilon_{t-i}^{2}+\sum\nolimits_{j=1}% ^{p}{a_{j}h_{t-j}}=a_{0}+a(L)\epsilon_{t}^{2}+b(L)h_{t}$ (3)

where $a(L)$ and $b(L)$ are the finite polynomial in the lag operator $L$ of order $p$ and $q$ respectively. The conditional variance defined by Eq. (3) has the property that the autocorrelation function of the squared residuals, $\epsilon_{t}^{2}$ , if exists, decay slowly. The most popular model is GARCH (1,1). A sufficient condition for the conditional variance to be positive is:

$a_{0}>0,a_{i}\geqslant 0,i=1,2,\ldots,q,b_{j}\geqslant 0,j=1,2,\ldots,p$

The first step of a GARCH process is to check for conditional heteroscedasticity of the squared residual series $\epsilon_{t}^{2}$ which is known as the ARCH test. There are two tests available in the literature for ARCH test. The first one is Ljung-Box test where the null hypothesis is that the first $m$ lags of autocorrelation functions of the $\epsilon_{t}^{2}$ series are zero. The second test for conditional heteroscedasticity is the Lagrange multiplier (LM) test. After detecting the ARCH effect the parameters of the model is estimated using the Gaussian maximum likelihood estimation (GMLE) method. In the last step the most suitable model is selected on the basis of minimum AIC or SBC value and lowest RMSE. Applications of this model in agriculture can be found in Paul et al. (2009, 2014) and Ghosh et al. (2011).

2.4 The ANN approach to time-series modeling

The working principle of ANN is based on the human brain by making the right connections. Like the structure of neuron, ANN comprises of several layers namely: input layer that receives external information; one or more hidden layer that performs mathematical operations on the data and an output layer that produces the results. All the layers are connected through an acyclic arc (Khashei & Bijari, 2010).

ANNs are more flexible computing system for modeling a wide variety of nonlinear problems. There are two Artificial neural network topologies – feed-forward and feedback. In feed-forward topology the flow of information is unidirectional and there is no feedback path where as in feedback topology feedback paths are there. These two topologies are demonstrated in Fig. 1.

Figure 1.

Artificial Neural Network topologies – feed-forward (left) and feedback (right).

The application of neural network structure for solving a particular time-series problem involves determination of number of layers and total number of nodes in the structure which is done on experimentation basis. It is established that single hidden layer with sufficient number of nodes at the hidden layer and adequate data for initialization. In neural network determination of number of input nodes which are lagged observations of same variable plays an important role in model building. Determination of output nodes is relatively easy. It is suggested that model with small number of nodes at hidden layer results in improved out-of-sample forecasting performance.

Single hidden layer feed-forward network is most widely used for modeling and forecasting of time-series data. The model is organized by a structure of three layers of processing units connected by acyclic arc. The relationship between one output ( $y_{t}$ ) and the $p$ inputs ( $y_{t-1},y_{t-2},\ldots,y_{t-p}$ ) is represented as

$y_{t}=\alpha_{0}+\sum\limits_{j=1}^{q}{\alpha_{j}g\left(\beta_{0j}+\sum\limits% _{i=1}^{p}\beta_{ij}y_{t-i}\right)}+\varepsilon_{t},$ (4)

where $\alpha_{j}$ ( $j=0,1,2,\ldots,q$ ) is the weight attached to the connection from the $j^{\rm th}$ hidden node to the output node and $\beta_{ij}$ ( $i=0,1,2,\ldots,p;j=0,1,2,\ldots,q$ ) represents the weight attached to the connection between $i^{\rm th}$ input node and $j^{\rm th}$ node of hidden layer; $p$ is the number of input nodes (trapped delay) and $q$ is the number of nodes at hidden layer and $g$ is the activation function which is usually nonlinear sigmoid function. The logistic function is often used as hidden layer transfer function, that is,

$g(x)=\frac{1}{1+\exp(-x)}$ (5)

Hence the ANN model performs a nonlinear functional mapping from the past observations ( $y_{t-1},y_{t-2},\ldots,y_{t-p}$ ) to the future value $y_{t}$ , i.e.,

$y_{t}=f\left(y_{t-1},y_{t-2},\ldots,y_{t-p},\bm{w}\right)+\varepsilon_{t},$ (6)

where $\bm{w}$ is the vector of all parameters and $f$ is a function determined by the network structure and connection weights. The Eq. (6) indicates that there is one node at output layer and is used for one-step ahead forecasting.

There are some similarities between ARIMA and ANN models. Both of them include a variety of models with different orders. Data transformation is sometimes needed to obtain best forecasts. A relatively large sample is necessary to fit a suitable model.

3. Tests for nonlinearity: Brock-Dechert-Scheinkman (BDS) test

The presence of nonlinearity pattern in a time-series data can be tested using BDS test. After making the data stationary by differencing, suitable linear model (e.g. ARMA ( $p$ , $q$ ), exponential smoothing etc.) is fitted to capture the linear component from the series. On the extracted residuals of the fitted linear model, the BDS test is used to test the null hypothesis that the residuals are independent and identically distributed (i.i.d.) against the alternative hypothesis that there exists hidden nonlinearity, hidden nonstationarity or other type of structure in the residuals. The detail computational procedure can be found in Brock et al. (1996).

4. The hybrid methodology

Zhang (2001) proposed a hybrid approach that decomposes a time-series process into its linear and nonlinear component. The hybrid model considers the time-series $y_{t}$ as a combination of both linear and nonlinear components. That is,

$y_{t}=L_{t}+N_{t}$ (7)

where $L_{t}$ and $N_{t}$ represent the linear and nonlinear component present in the given data respectively. These two components are to be estimated from the data. This hybrid method of combining forecasting has following steps:

1) 1)

First, a linear time-series model, say, ARIMA is fitted to the data.

At the next step residuals are obtained from the fitted linear model. The residuals will contain only the nonlinear components. Let $e_{t}$ denotes the residual at the time $t$ from the linear model, then

$e_{t}=y_{t}-\hat{L}_{t}$ (8)

where $\hat{L}_{t}$ is the forecast value for the time $t$ from the estimated linear model.

Diagnosis of residuals is done to check if there is still linear correlation structures left in the residuals. The residuals are tested for nonlinearity by using BDS test.

Once the residuals confirm the nonlinearity, then the residuals are modeled using a nonlinear model, say, ARCH. And also obtain the forecast values, $\hat{N}_{t}$ for the residual series.

Finally the forecasted linear and nonlinear components are combined to obtain the aggregated forecast values as

$\hat{y}_{t}=\hat{L}_{t}+\hat{N}_{t}$ (9)

The hybrid approaches can be graphically represented by Figs 2 and 3.

Figure 2.

Schematic representation of ARIMA-GARCH hybrid methodology.

Table 1

Descriptive statistics of potato prices in Agra market

Statistics	Agra
Observations	149
Mean (Rs/quintal)	652.33
Minimum	175
Median	536.00
Maximum	2107.00
SD	367.86
CV	56.39
Skewness	1.52
Kurtosis	5.51

Note: SD: standard deviation; CV: coefficient of variation.

Table 2

Seasonal factors for potato prices in the Agra market

Months	Agra
January	0.68
February	0.65
March	0.76
April	0.83
May	0.97
June	1.08
July	1.21
August	1.20
September	1.22
October	1.29
November	1.32
December	0.79

Figure 3.

Schematic representation of ARIMA-ANN hybrid methodology.

5. Illustration

For the present study potato price data belonging to Agra market in India for the period January, 2005 to May, 2017, collected from National Horticulture Research and Development Foundation (NHRDF) (http://nhrdf.org/en-us/) are used. The data series is divided into two parts: training set and testing set (holdout set). The training data set (January, 2005 to May, 2016) is used for parameter estimation and the last 12 observations i.e. from June, 2016 to May, 2017 considered as testing set is used for validation purpose and also for obtaining out-of-sample forecast.

5.1 Descriptive statistics and seasonal indices:

The descriptive statistics of potato price for Agra market are reported in Table 1. A perusal of the Table 1 indicates that average potato price in Agra market is 652. Since the CV is more than 50% it can be concluded that the variability in price of Agra market is slightly in higher sight. The series under consideration is positively skewed and leptokurtic. Original data is seasonally adjusted to eliminate the influence of seasonality in price. Table 2 shows the seasonal index values. Relatively higher values of seasonal indices are found from June to November. Being a rabi crop, the planting time of potato is 15 ${}^{\rm th}$ September–15 ${}^{\rm th}$ October and it gets ready for harvesting at the end of November. Fresh arrival starts to reach the market by the end of November onwards.

The first and foremost step in time-series analysis is to plot the data and visualize the presence of several time-series components. Figure 4a and b show the time-series plot of average monthly price of potato for original series and monthly potato price for seasonally adjusted series from January, 2005 to May, 2017 in Agra market. A perusal of this figure indicates that the price attains its higher values during the period June, 2014 to December, every year. Though the highest price has been observed in October, 2014. The time-plot of original price data also indicates that some seasonal pattern is present in the dataset and it is required some kind of seasonal adjustment.

Table 3
ADF and PP test for stationarity

Markets	Original series				1 ${}^{\rm st}$ differenced series
	ADF test		PP test		ADF test		PP test
	Test statistic	$p$ -value	Test statistic	$p$ -value	Test statistic	$p$ -value	Test statistic	$p$ -value
Agra	$-$ 2.72	0.07	$-$ 2.68	0.09	$-$ 10.35	$<$ 0.001	$-$ 10.35	$<$ 0.001

Figure 4.

Monthly price of Potato from January 2005 to May 2017 for (a) Original Series (b) Seasonally Adjusted Series in Agra Market.

5.2 Test for stationarity

Phillips-Perron (PP) and Augmented Dicky-Fuler (ADF) tests have been applied to see the presence of non-seasonal unit root in the seasonally adjusted series it was found that the null hypothesis of unit root test is not rejected at 5% level of significance indicating seasonally adjusted series are non-stationary in nature and the results are given in the Table 3. Non rejection of the null hypothesis of unit root for both the tests at 5% level of significance indicates that differencing is required to make the seasonally adjusted series stationary for the market. Rejection of null hypothesis of stationarity test for 1 ${}^{\rm st}$ differenced series reveals that no more differencing is required.

5.3 Fitting of forecasting models

5.3.1 Fitting of ARIMA model

After confirming the stationarity of the price series after one differencing, suitable ARMA model was selected based on minimum AIC and BIC criterion and observing the significance of autocorrelation and partial autocorrelations functions. Accordingly, ARIMA(1,1,0) mode is selected for seasonally adjusted price series of potato in Agra market. The parameter estimates of fitted ARIMA model are furnished in Table 4 along with their significance level ( $p$ -value).

Table 4
Parameter estimates of the ARIMA (1,1,0) of Agra market

Model	Parameters	Estimate	$p$ -value
ARIMA (1,1,0)	C	5.58	0.47
	AR (1)	$-$ 0.26	$<$ 0.01

5.3.2 Testing for ARCH effects

The presence of autocorrelation in the squared residuals of best fitted ARIMA model was investigated and reported in Table 5. It was found that the squared residuals are autocorrelated at least up to 12 lags indicating possible presence of ARCH effect. To test the presence conditional heteroscedasticity, ARCH-LM test is performed and it is found that the ARCH effect is significant up to 5 lags.

Table 5
Test for ARCH effects for seasonally adjusted series

Order	Agra
	Q-statistic	$p$ -value	LM-statistic	$p$ -value
1	9.43	$<$ 0.01	9.24	$<$ 0.01
2	9.58	0.01	10.61	0.01
3	9.65	0.02	11.12	0.01
4	9.67	0.05	11.34	0.02
5	9.87	0.08	11.66	0.04
6	9.89	0.13	11.79	0.07
7	9.89	0.19	11.81	0.11
8	10.06	0.26	11.90	0.16
9	10.09	0.34	12.18	0.20
10	10.50	0.40	12.88	0.23
11	10.92	0.45	15.00	0.18
12	11.26	0.51	15.00	0.24

Table 6

BDS test for nonlinearity of residuals

Series	Dimension (m)	Epsilon $\left(\varepsilon\right)$		Statistic	$p$ -value
Agra	2	eps (1)	56.71	3.91	$<$ 0.01
		eps (2)	113.41	3.57	$<$ 0.01
		eps (3)	170.12	3.05	$<$ 0.01
		eps (4)	226.82	2.01	0.04
	3	eps (1)	56.71	6.73	$<$ 0.01
		eps (2)	113.41	4.47	$<$ 0.01
		eps (3)	170.12	3.26	$<$ 0.01
		eps (4)	226.82	2.33	0.02

5.3.3 Testing for nonlinearity

BDS test has been employed to test the presence of any remaining structure in the residuals obtained from the fitted ARIMA model for the market under consideration. The results of the test are given in Table 6 indicating the possible presence of nonlinear patter in the residuals of ARIMA model.

5.3.4 Fitting of GARCH model

Accordingly, to capture the nonlinearity and heteroscedasticity in conditional variance, GARCH model is applied for modelling and forecasting the price series. The parameter estimates of best fitted ARIMA and GARCH model are furnished in Table 7 along their significance level.

Table 7
Parameter estimates of the ARIMA (1,1,0)-GARCH (1,1) model for Agra market

Model	Parameters	Estimate	$p$ -value
ARIMA (1,1,0)-GARCH (1,1)	Mean equation
	C	5.29	0.43
	AR (1)	$-$ 0.06	0.62
	Variance equation
	C	330.19	0.19
	ARCH(1)	0.43	$<$ 0.01
	GARCH(1)	0.68	$<$ 0.01

5.3.5 Fitting of hybrid models

Once it is confirmed that the residuals of the fitted ARIMA model contains nonlinear part and also the significant ARCH effect is present, the hybrid models namely ARIMA-ANN and ARIMA-GARCH model as discussed in Section 4 were employed to investigate the improvement in forecast accuracy as compared to the individual ARIMA and GARCH models.

6. Evaluation of forecasting performances

The prediction abilities of the ARIMA and GARCH models and the hybrid models i.e. ARIMA-ANN and ARIMA-GARCH are compared with respect to mean absolute percentage error (MAPE) and root mean squared error (RMSE) for last twelve observations (i.e. for last twelve months). The formula for computing MAPE and RMSE are given below

$\displaystyle\text{MAPE}=\frac{1}{n}\sum\limits_{t=1}^{n}\left|y_{t}-\hat{y_{t% }}\right|\times 100$ (10) $\displaystyle\text{RMSE}=\sqrt{\frac{1}{n}\sum\limits_{t=1}^{n}\left(y_{t}-% \hat{y_{t}}\right)^{2}}$ (11)

where $y_{t}$ is the actual price at time $t$ , $\hat{y_{t}}$ is the predicted price at time $t$ , and $n$ is the sample size for the hold out data. In the present investigation $n$ is 12. The values of MAPE and RMSE of different models are reported in the Table 8.

Table 8

Comparison of prediction performance of different models

Series	ARIMA	GARCH	ARIMA-GARCH	ARIMA-ANN
Comparison based on MAPE
Agra	16.16	16.54	16.00	11.31
Comparison based on RMSE
Agra	172.71	189.47	169.44	123.90

7. Conclusions

The accuracy of a statistical model is the fundamental feature to select that particular model and to take many important decisions. Box-Jenkins’s ARIMA methodology is most popular method of forecasting of a linear time-series process. In many of the practical situations, the assumptions of linearity and homoscedastic error variance which are two most crucial assumptions of ARIMA model are violated. In such cases, nonlinear time series models are called for. GARCH family of models is the most widely used nonlinear time series models in literature. The hybrid methodology that decomposes a time-series into its linear and nonlinear part and then model each part separately before they are combined for getting final forecast is described in detail here. The above methodology has been applied in forecasting the wholesale price of potato in Agra market. The residuals obtained from the fitted ARIMA model was tested using BDS test which reveals that nonlinearity pattern exists in the residual series. The comparison of forecast performance among the ARIMA, GARCH, ARIMA-GARCH and ARIMA-ANN hybrid models has been carried out. It is seen that the hybrid models perform better than the individual counterpart i.e. ARIMA and GARCH models with respect to minimum MAPE and RMSE value. The residuals from finally fitted hybrid model are examined and it is found that the residuals are independent and normally distributed ensuring the adequacy of model selected.

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Hybrid time-series models for forecasting agricultural commodity prices

Abstract

Keywords

1. Introduction

2. Time-series forecasting models

2.1 The ARIMA model

4. The hybrid methodology

5.1 Descriptive statistics and seasonal indices:

Table 3 ADF and PP test for stationarity

5.3 Fitting of forecasting models

5.3.1 Fitting of ARIMA model

Table 4 Parameter estimates of the ARIMA (1,1,0) of Agra market

Table 5 Test for ARCH effects for seasonally adjusted series

5.3.4 Fitting of GARCH model

Table 7 Parameter estimates of the ARIMA (1,1,0)-GARCH (1,1) model for Agra market

6. Evaluation of forecasting performances

References

Table 3
ADF and PP test for stationarity

Table 4
Parameter estimates of the ARIMA (1,1,0) of Agra market

Table 5
Test for ARCH effects for seasonally adjusted series

Table 7
Parameter estimates of the ARIMA (1,1,0)-GARCH (1,1) model for Agra market