Modeling and Forecasting by using Time Series ARIMA Models

Modeling and Forecasting by using Time Series ARIMA Models

Mustafa M. Ali Alfaki Research Scholar,School of Basic Science Sam Higginbottom Institute of Agriculture,

Technology and Sciences Allahabad, India

Abstract:– In this paper we introduce a brief review about Box-Jenkins models. Auto-Regressive Integrated Moving Average, Known as ARIMA models. It is a good method to forecast for stationary and non stationary time series. According to the data which obtained from the monthly sales for Naphtha product in Azzawiya Oil Refining Company Libya, then we determine an optimal model. The results of this study showed that the suitable and efficient model to represent the data of the time series according to AIC, SIC, and MSE criteria with the smallest values as well as the Box-

ljung test is the ARIMA(1,1,1), whose equation is :

Dr. (Ms.) Shalini Bhawana Masih Assistant Professor,School of Basic Science Sam Higginbottom Institute of Agriculture, Technology and Sciences

Allahabad, India

appropriate model for this time series. This model is the assessed to determine how well its the data.

Non-seasonal Box-Jenkins Models for a Stationary Series:

That is known as (ARIMA) Models, it is an Autoregressive Model AR(p) as the first part, and Moving Average Model MA(q) a second part, and the third part I(d) represents the differences required by the time-series in order to be stationary. Some models of time series may be non-

Zt

0.6010 Z

t 1

1.1713

t 1

• t

  stationary of the same kind but they would become Stationary after a lot of differences and changes, so that the

  According to the results of ARIMA (1,1,1), the amounts of the monthly sales for Naphtha product Azzawiya Oil Refining Company Libya, have been forecasted for the period from Jan: 2015 to Dec: 2020. Those values showed harmony with their counterparts in the original time series. It provided us with a future image of the reality of the monthly sales for Naphtha product in Azzawiya Oil Refining Company Libya.

  model which represents this process will vary from the original model because it will include those differences and changes made by the model by taking appropriate number of operational differences process.

  Autoregressive Models AR(p):

  Which has the general form:

  Key Words : Forecasting, Box-Jenkins, Auto-Regressive Integrated Moving Average (ARIMA), Auto-Regressive (AR), Moving Average (MA), Autocorrelation Function (ACF), Partial Autocorrelation Function (PACF).

  Zt 1 Z t1 2 Zt2 …p Zt p t

  Where :

  , t p

  (1)

  Zt = dependent variable at time t.

  INTRODUCTION

  The planning process is the most important step in the develop plan of any country, it becomes adequate if the

  statistical methods that has been applied are based on powerful scientific basis.

  The time series is a set of well-defined data items collected at successive points at uniform time intervals. Time series analysis is an important part in statistics, which analyzes data set to study the characteristics of the data and helps in predicting future values of the series based on the characteristics. Forecasting is important in fields like

  Zt 1 , Zt 2 , , Zt p = dependent variable at time lags t

  1, t 2, . . . , t p respectively.

  i : Coefficients to be estimated, as:

  i 1 , 2 ,3 , … p , 1 P 1

  t

  .

  t : Series of random errors, with mean zero and constant variance. i.e. N (0, 2 )

  Moving Average Model MA (q):

  Which has the general form:

  finance, industry, etc. The Autoregressive Integrated Moving Average (ARIMA) is based on ARMA Model. The

  t t

  1

  t1

  2

  t2

  …. q

  tq

  (2)

  difference is that ARIMA Modelconverts a non-stationary data to a stationary data before working on it. This analysis uses Box-Jenkins ARIMA modeling techniques to find an

  Where:

  Zt = dependent variable at time t.

  t : Series of random errors, with mean zero and constant

  variance. i.e.

  N (0, 2 )

  t

  .

  t 1 ,

  t 1 , … t q = Errors in previous time periods

  that are incorporated in the response Zt .

  i : Parameters of moving average model. .

  q : Degree Model.

  Mixed Models ARMA ( p , q ):

  Which has the general form:

  t 1 t1 2 t2 … p t p t 1 t1 … q tq

  Autoregressive Integrated Moving Average Model ARIMA (p,d,q):

  1 t 1 … q t q

  The general formula of Autoregressive Integrated Moving Average Models (ARIMA) That is written as :

  Figure (1): Graphical representation of the series Naphtha

  Table (1): ACF and PACF of the series Naphtha

  t 1 t 1 2 t 2 … p t p …

  (4)

  d t pd t

  Building ARIMA Model and Forecasting:

  The data used in this study consist of the monthly sales for Naphtha production in Azzawiya Oil Refining Company Libya, for the period (2008 2014).

  Stationary

  In order to apply certain techniques for identifying and ARIMA model for the data, we must determine the form of stationary of the original time series and we will draw the autocorrelation function (ACF), and the partial autocorrelation function (PACF) of data and draw confidence interval of (ACF) and (PACF) to detect the stationary or non-stationary of time series, as well as the use of the Q test, (Box- Ljung test) where:

  Through the Figure (1) and table (1) of the original series of Correlation Coefficients and figures of ACF and PACF, we note that there is non- stationary in the data of original series as there are some values outside the confidence interval, and that the significant value of the Coefficients autocorrelation function using the Q test :

  m 2 2 2

  Q m

  Where:

  n ( n 2 ) k ~

  k 1 n k

  ( m )

  (5)

  Q = 245.99 > 25, 0.05 37.652

  And to make the series stationary we make differences from First level while taking the natural logarithm of the

  k

  = the residual autocorrelation at lag k.

  data and then we can obtain the following results:

  n = the number of residuals.

  m = the number of time lags includes in the test.

  And we will test of unit root by Augmented Dickey-Fuller (ADF) test on the original data provided evidence of the presence of a unit root we difference the data to counteract the effect of this unit root. We depended on Eviews.5 program to determine tables and figures we can obtain the following results:

  Figure (2): Graphical representation of the series D(Log(Naphtha))

  Table (2): ACF and PACF of the series D (log(Naphtha))

  Through the figure (2) and table (2) of modified series of correlation coefficients and figures of ACF and PACF we note that there is stationary in the data of series and most of the values within the confidence interval, and that the Significant value of autocorrelation coefficients by using the Q test was:

  Table (3): ADF test results for series D(Log(Naphtha))

  Through the data of the table (3) we note that all calculated statistics of Dickey Fuller in all models are less than the corresponding tabular values, then there is no a unit root in the series, and the series D(Log(Naphtha)) is stationary.

  Model Identification:

  Through the table (2), we observe the presence of moving averages models MA (1), MA (3) and MA (6), though the autocorrelation function. And the presence of autoregressive models AR (1), AR (3) and AR (6), through partial autocorrelation function, and for more accuracy in reconciling the best model among Box-Jenkins models, possible models of ARIMA (p, d, q) have been applied, and calculation of each of them SC, AIC and MSE are as shown in the following table:

  Table (4): Compared to a set of valuesof AIC , SIC , MSE

  From the table (4), we find that the model ARIMA (1,1,1) is the one who gives the lower values of the previous standards, these values were as follows:

  MSE = 0.3506 , AIC=1.8629 , SC=1.9510

  So this model was relied on to be an appropriate model for this series.

  Model Diagnostics:

  Residual diagnostic tests and the over fitting process are used here to determine the goodness of fit of the ARIMA(1,1,1) model to the original time series.

  • Residual Diagnostics:

    Q = 13.586 <

    2

    25, 0.05

    37.652

    We have been used diagnostic the ACF for the residuals. Here we see that the ACF values are all within the 95%,

    We then test by Advanced Dickey Fuller and the estimation of models is as follows:

    indicating that there is no correlation amongst the residuals. This plot is used as an indicator of the independence of the residual.

    Table (5): ACF and PACF for Residual ARIMA (1,1,1)

    analyzed ( Naphtha product). The final model is of the following form:

    Table (6): Estimated model parameters of Naphtha sales model

    We obtained the model in the form:

    Zt

    0.6010 Z t1 1.1713 t 1 t

    (6)

    From the table (5) we note that the most of the coefficients fall within the confidence interval, as well as the statistical (Q):

    Forecasting

    After the identification of the model and its adequacy check, it is used to forecast the sales for Naphtha product in Azzawiya Oil Refining Company Libya in the period (January 2015 to December 2020). The forecasting results are presented in figure (4)

    5.0E+12

    Q = 10 <

    2

    25,0.05

    37.652

    4.0E+12

    Therefore Residuals represent white noise.

    Normal distribution test:

    The residuals follow the normal distribution in the following diagram:

    3.0E+12

    2.0E+12

    1.0E+12

    Series: Residuals

    Sample 2008M03 2014M12

    Observations 82

    Mean 0.043139

    Median 0.081123

    Maximum 1.366176

    Minimum -2.837595

    Std. Dev. 0.594179

    Skewness -2.329773

    Kurtosis 12.04281

    Jarque-Bera 353.5694

    Probability 0.000000

    32

    28 0.0E+00

    24

    20

    16

    12

    8

    2015 2016 2017 2018 2019 2020

    NAPHTF

    Figure (4): Forecasted Naphtha Sales

    CONCLUSION

    4

    0

    -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

    Figure (3): Testing of normal distribution of residuals series

    The distribution of residuals are almost symmetrical.

    Autocorrelation test for errors (Durbin-Watson's statistic):

    The value of D-W = 1.9924, it is lies within the confidence interval, hence it has no Auto correlation of errors. Then all the results of the residuals tests confirm the validity of the estimated model ARIMA (1,1,1) to represent the time series.

    Final Model

    Therefore conclude that the ARIMA(1,1,1) model is the best ARIMA model for the original time series being

    The aim of this analysis was to determine an appropriate ARIMA model for the sales Monthly data for Naphtha product. In particular we were interested in forecasting future for sales of company using this model. It is concluded that the ARIMA (1,1,1), where were the value of MSE, AIC and SC for the model are very small. Hence the model is appropriate to forecast the sales of Naphtha for the next six years.

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