GROUP 6ANURAAG MONDALAMAN NOGIA NITIN KUMARNITIN NATH SINGH

FORECASTING INDIA’S FOOD PRICE INDEX USING ARIMA MODEL

Objective of the Project

To model the monthly Commodity Food Price Index from the

year 2000 onwards using a time series analysis .

To forecast India's food price index for the next month(April)

using ARIMA model

Data Used

The data used has been obtained from www.indexmundi.com

The Commodity Food Price Index of 15 years(Monthly) has

been taken in the project.

The period taken into consideration is March 2000 to March

2015.

Commodity Food Price Index

“Everyone eats. As a result, everyone is affected to some

degree by food price changes.”

The Commodity Food Price Index is a measure of the monthly

change in international prices of a basket of food commodities.

It consists of the average of five commodity group price indices

(representing 55 quotations), weighted with the average export

shares of each of the groups for 2002-2004.

Commodity Food Price Index includes Cereal, Vegetable Oils,

Meat, Seafood, Sugar, Bananas, and Oranges Price Indices.

Data Description

Graph 1

Following is a plot of data for each month from March

2000 onwards. We can see the wide fluctuations in Food Price

Index in the year 2008 owing to the financial downturn that

was witnessed by the world economy which had an impact on

all the sectors.

3/1/

2000

8/1/

2000

1/1/

2001

6/1/

2001

11/1

/200

1

4/1/

2002

9/1/

2002

2/1/

2003

7/1/

2003

12/1

/200

3

5/1/

2004

10/1

/200

4

3/1/

2005

8/1/

2005

1/1/

2006

6/1/

2006

11/1

/200

6

4/1/

2007

9/1/

2007

2/1/

2008

7/1/

2008

12/1

/200

8

5/1/

2009

10/1

/200

9

3/1/

2010

8/1/

2010

1/1/

2011

6/1/

2011

11/1

/201

1

4/1/

2012

9/1/

2012

2/1/

2013

7/1/

2013

12/1

/201

3

5/1/

2014

10/1

/201

4

3/1/

2015

0

50

100

150

200

250

Price

Price

Graph 2

The next plot depicts the yearly mean price index from the

year 2000 onwards. It can be seen that the mean food price

Index has been rising from the year 2000 onwards.

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 20150

20

40

60

80

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140

160

180

200

Mean Price Index

Mean Price Index

Modelling the series

The Box-Jenkins Methodology for univariate time-series has been used.

To obtain the results EViews8 was used. The three primary phases in building a Box-Jenkins Time

Series Model

Phase I-Identification Data Preparation by checking for and making series

stationary Model Selection by using ACF and PACF

Phase II-Estimation and Testing Estimation of model deriving its MLE parameter estimates. Diagnosticas check ACF/PACF of residuals

Phase III-Forecasting Use models to forecast.

Phase I-Identification

Checking for Stationarity

For checking stationarity we used the Augmented

Dickey-Fuller test (ADF) on the price series.

Using the results of the test we found that the

series is non stationary

Results on following slide:

Null Hypothesis: PRICE has a unit rootExogenous: NoneLag Length: 1 (Automatic - based on SIC, maxlag=13)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic 0.075322 0.7055Test critical values: 1% level -2.577945

5% level -1.94261410% level -1.615522

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test EquationDependent Variable: D(PRICE)Method: Least SquaresDate: 04/22/15 Time: 00:45Sample (adjusted): 3 181Included observations: 179 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

PRICE(-1) 0.000171 0.002270 0.075322 0.9400D(PRICE(-1)) 0.466881 0.066794 6.989908 0.0000

R-squared 0.213725 Mean dependent var 0.333352Adjusted R-squared 0.209283 S.D. dependent var 4.562015S.E. of regression 4.056647 Akaike info criterion 5.649701Sum squared resid 2912.780 Schwarz criterion 5.685314Log likelihood -503.6482 Hannan-Quinn criter. 5.664142Durbin-Watson stat 1.979674

UNIT ROOT TEST FOR PRICE

In order to make the series stationary we differenced the original price series and generated a new series as price 1 with first difference

Next, we tested this new series for stationarity again by using Augmented Dickey-Fuller test (ADF)

By observing the results we found that the new series in stationary.

Results on following slide

Null Hypothesis: PRICE1 has a unit rootExogenous: NoneLag Length: 0 (Automatic - based on SIC, maxlag=13)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -8.027300 0.0000Test critical values: 1% level -2.577945

5% level -1.94261410% level -1.615522

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test EquationDependent Variable: D(PRICE1)Method: Least SquaresDate: 04/22/15 Time: 00:40Sample (adjusted): 3 181Included observations: 179 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

PRICE1(-1) -0.532686 0.066359 -8.027300 0.0000

R-squared 0.265751 Mean dependent var -0.034581Adjusted R-squared 0.265751 S.D. dependent var 4.720946S.E. of regression 4.045300 Akaike info criterion 5.638560Sum squared resid 2912.873 Schwarz criterion 5.656366Log likelihood -503.6511 Hannan-Quinn criter. 5.645780Durbin-Watson stat 1.980119

UNIT ROOT TEST FOR PRICE 1

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25 50 75 100 125 150 175

PRICE

-30

-20

-10

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25 50 75 100 125 150 175

PRICE1

Phase II-Estimation and Testing

Model estimation

For initial values of AR and MA to model a mean generating function, we use Correlogram ( ACF & PACF).

Further We try different combinations of AR and MA to make a suitable ARIMA model.

Date: 04/22/15 Time: 01:06Sample: 1 181Included observations: 180

Autocorrelation Partial Correlation AC PAC Q-Stat Prob

1 0.462 0.462 39.115 0.0002 0.194 -0.02... 46.061 0.0003 0.023 -0.07... 46.157 0.0004 0.064 0.109 46.908 0.0005 -0.02... -0.11... 47.065 0.0006 -0.12... -0.11... 50.020 0.0007 -0.21... -0.11... 58.883 0.0008 -0.26... -0.14... 72.063 0.0009 -0.16... 0.037 77.115 0.000

1... -0.03... 0.064 77.396 0.0001... 0.055 0.060 77.972 0.0001... -0.02... -0.09... 78.059 0.0001... -0.10... -0.13... 80.127 0.0001... -0.04... 0.020 80.458 0.0001... 0.069 0.044 81.401 0.0001... 0.031 -0.07... 81.590 0.0001... -0.00... 0.025 81.591 0.0001... -0.01... 0.017 81.648 0.0001... 0.063 0.076 82.450 0.0002... 0.061 -0.02... 83.213 0.0002... 0.028 -0.07... 83.378 0.0002... -0.04... -0.07... 83.857 0.0002... -0.01... 0.069 83.897 0.0002... -0.05... -0.06... 84.564 0.0002... -0.04... -0.01... 84.984 0.0002... -0.11... -0.11... 87.866 0.0002... -0.02... 0.119 88.018 0.0002... -0.11... -0.14... 90.986 0.0002... -0.12... -0.11... 94.593 0.0003... -0.12... -0.07... 98.224 0.0003... 0.004 0.102 98.227 0.0003... 0.072 0.088 99.372 0.0003... 0.090 0.026 101.20 0.0003... 0.153 0.051 106.43 0.0003... 0.161 0.041 112.32 0.0003... 0.155 0.008 117.80 0.000

Use ACF and PACF to identify appropriate models.

(2,1,2) ARIMA modelDependent Variable: PRICE1Method: Least SquaresDate: 04/22/15 Time: 00:58Sample (adjusted): 4 181Included observations: 178 after adjustmentsConvergence achieved after 38 iterationsMA Backcast: 2 3

Variable Coefficient Std. Error t-Statistic Prob.

C 0.323378 0.519277 0.622748 0.5343AR(1) -0.356228 0.265981 -1.339301 0.1822AR(2) 0.078652 0.199671 0.393908 0.6941MA(1) 0.849953 0.257539 3.300290 0.0012MA(2) 0.337116 0.144118 2.339173 0.0205

R-squared 0.233238 Mean dependent var 0.332135Adjusted R-squared 0.215509 S.D. dependent var 4.574855S.E. of regression 4.052016 Akaike info criterion 5.663994Sum squared resid 2840.459 Schwarz criterion 5.753370Log likelihood -499.0955 Hannan-Quinn criter. 5.700238F-statistic 13.15603 Durbin-Watson stat 2.002627Prob(F-statistic) 0.000000

Inverted AR Roots .15 -.51Inverted MA Roots -.42+.40i -.42-.40i

AR (1) and AR(2) are not significant.

(2,1,4) ARIMA modelDependent Variable: PRICE1Method: Least SquaresDate: 04/22/15 Time: 09:37Sample (adjusted): 4 181Included observations: 178 after adjustmentsConvergence achieved after 32 iterationsMA Backcast: 0 3

Variable Coefficient Std. Error t-Statistic Prob.

C 0.283847 0.617801 0.459446 0.6465AR(1) 1.158921 0.068222 16.98751 0.0000AR(2) -0.759547 0.063225 -12.01335 0.0000MA(1) -0.711078 0.091795 -7.746343 0.0000MA(2) 0.494012 0.095217 5.188289 0.0000MA(3) 0.163527 0.091151 1.794015 0.0746MA(4) 0.321004 0.084025 3.820340 0.0002

R-squared 0.295191 Mean dependent var 0.332135Adjusted R-squared 0.270460 S.D. dependent var 4.574855S.E. of regression 3.907525 Akaike info criterion 5.602217Sum squared resid 2610.956 Schwarz criterion 5.727343Log likelihood -491.5973 Hannan-Quinn criter. 5.652959F-statistic 11.93646 Durbin-Watson stat 2.027108Prob(F-statistic) 0.000000

Inverted AR Roots .58+.65i .58-.65iInverted MA Roots .66+.73i .66-.73i -.31+.48i -.31-.48i

MA(3) is not significant.

Dependent Variable: PRICE1Method: Least SquaresDate: 04/22/15 Time: 09:35Sample (adjusted): 5 181Included observations: 177 after adjustmentsConvergence achieved after 37 iterationsMA Backcast: 1 4

Variable Coefficient Std. Error t-Statistic Prob.

C 0.298671 0.588436 0.507567 0.6124AR(1) 0.907052 0.243977 3.717785 0.0003AR(2) -0.414339 0.325189 -1.274150 0.2044AR(3) -0.230172 0.217124 -1.060093 0.2906MA(1) -0.472554 0.228338 -2.069534 0.0400MA(2) 0.255119 0.234405 1.088369 0.2780MA(3) 0.283872 0.143706 1.975374 0.0499MA(4) 0.405752 0.100965 4.018761 0.0001

R-squared 0.297927 Mean dependent var 0.340960Adjusted R-squared 0.268847 S.D. dependent var 4.586313S.E. of regression 3.921640 Akaike info criterion 5.615041Sum squared resid 2599.095 Schwarz criterion 5.758596Log likelihood -488.9312 Hannan-Quinn criter. 5.673262F-statistic 10.24512 Durbin-Watson stat 2.000051Prob(F-statistic) 0.000000

Inverted AR Roots .60+.64i .60-.64i -.30Inverted MA Roots .66-.73i .66+.73i -.43-.48i -.43+.48i

(3,1,4) ARIMA model

AR(2), AR(3) & MA(2) is not significant.

(3,1,2) ARIMA model

Dependent Variable: PRICE1Method: Least SquaresDate: 04/22/15 Time: 00:56Sample (adjusted): 5 181Included observations: 177 after adjustmentsConvergence achieved after 19 iterationsMA Backcast: 3 4

Variable Coefficient Std. Error t-Statistic Prob.

C 0.333699 0.565676 0.589912 0.5560AR(1) -0.892963 0.081777 -10.91950 0.0000AR(2) -0.287029 0.102906 -2.789239 0.0059AR(3) 0.375963 0.077938 4.823885 0.0000MA(1) 1.396576 0.037482 37.26039 0.0000MA(2) 0.964087 0.036342 26.52815 0.0000

R-squared 0.246262 Mean dependent var 0.340960Adjusted R-squared 0.224223 S.D. dependent var 4.586313S.E. of regression 4.039541 Akaike info criterion 5.663450Sum squared resid 2790.360 Schwarz criterion 5.771116Log likelihood -495.2153 Hannan-Quinn criter. 5.707115F-statistic 11.17389 Durbin-Watson stat 2.003172Prob(F-statistic) 0.000000

Inverted AR Roots .43 -.66+.65i -.66-.65iInverted MA Roots -.70-.69i -.70+.69i

All the parameters are significant.

Diagnostic checking

Using (3,1,2) ARIMA model we find that all the parameters are significant, hence this is model can be used for forecasting.

We proceed to the next step i.e. diagnostic testing of the model.

Correlogram of residualsDate: 04/22/15 Time: 15:19Sample: 1 181Included observations: 161

Autocorrelation Partial Correlation AC PAC Q-Stat Prob

1 -0.49... -0.49... 40.527 0.0002 0.016 -0.30... 40.570 0.0003 -0.00... -0.21... 40.575 0.0004 -0.01... -0.17... 40.596 0.0005 -0.00... -0.15... 40.596 0.0006 -0.00... -0.13... 40.597 0.0007 0.021 -0.08... 40.669 0.0008 -0.03... -0.10... 40.836 0.0009 -0.00... -0.11... 40.837 0.000

1... 0.006 -0.11... 40.843 0.0001... 0.017 -0.08... 40.894 0.0001... 0.012 -0.03... 40.921 0.0001... -0.05... -0.10... 41.492 0.0001... -0.03... -0.21... 41.683 0.0001... 0.121 -0.07... 44.306 0.0001... -0.09... -0.14... 45.823 0.0001... 0.051 -0.11... 46.301 0.0001... -0.07... -0.23... 47.332 0.0001... 0.096 -0.15... 49.045 0.0002... 0.030 -0.02... 49.210 0.0002... -0.01... 0.064 49.240 0.0002... -0.09... -0.06... 50.976 0.0002... 0.052 -0.05... 51.491 0.0012... -0.04... -0.11... 51.836 0.0012... 0.132 0.111 55.175 0.0002... -0.19... -0.12... 62.600 0.0002... 0.169 0.002 68.160 0.0002... -0.08... 0.010 69.440 0.0002... 0.034 0.114 69.669 0.0003... -0.09... -0.08... 71.417 0.0003... 0.055 -0.11... 72.020 0.0003... 0.039 -0.05... 72.330 0.0003... -0.06... -0.02... 73.256 0.0003... 0.052 -0.04... 73.818 0.0003... 0.011 -0.03... 73.846 0.0003... 0.021 0.027 73.942 0.000

Unit root test for stationarity of residuals

Null Hypothesis: D(RESID) has a unit rootExogenous: NoneLag Length: 10 (Automatic - based on SIC, maxlag=13)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -8.636451 0.0000Test critical values: 1% level -2.579315

5% level -1.94280510% level -1.615400

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test EquationDependent Variable: D(RESID,2)Method: Least SquaresDate: 04/22/15 Time: 15:17Sample (adjusted): 20 181Included observations: 162 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

D(RESID(-1)) -8.352108 0.967076 -8.636451 0.0000D(RESID(-1),2) 6.394433 0.922708 6.930072 0.0000D(RESID(-2),2) 5.494756 0.853548 6.437550 0.0000D(RESID(-3),2) 4.688108 0.770407 6.085234 0.0000D(RESID(-4),2) 3.969352 0.681566 5.823872 0.0000D(RESID(-5),2) 3.240392 0.586127 5.528480 0.0000D(RESID(-6),2) 2.571765 0.483985 5.313730 0.0000D(RESID(-7),2) 1.932902 0.380934 5.074112 0.0000D(RESID(-8),2) 1.246360 0.277041 4.498833 0.0000D(RESID(-9),2) 0.639663 0.174266 3.670605 0.0003D(RESID(-10),2) 0.181622 0.080966 2.243192 0.0263

R-squared 0.832361 Mean dependent var 0.022037Adjusted R-squared 0.821259 S.D. dependent var 11.12582S.E. of regression 4.703752 Akaike info criterion 6.000084Sum squared resid 3340.918 Schwarz criterion 6.209736Log likelihood -475.0068 Hannan-Quinn criter. 6.085206Durbin-Watson stat 1.999566

Diagnostic Checking

From the unit root test for residuals we find that residuals are stationary

Adjusted R2 is around 82%

Phase III-Forecasting

-4.6

-5.64

-3.76

-4.2200

6-

0.67803

Change in the prices for the forecasted month -1.43478

Forecast price for april 2015= 142.52

Change in the price for the forecasted month =Coefficient + AR(i)*price(-1) + MA(i)*Resid(-1)

Using above model, the FPI values for the month of April has been predicted.

Variable Coefficient Std. Error t-Statistic Prob.

C 0.333699 0.565676 0.589912 0.556

AR(1) -0.89296 0.081777 -10.9195 0

AR(2) -0.28703 0.102906 -2.78924 0.0059

AR(3) 0.375963 0.077938 4.823885 0

MA(1) 1.396576 0.037482 37.26039 0

MA(2) 0.964087 0.036342 26.52815 0

Conclusion

The Commodity Food price index continues on following the downward trend as per the model.