2015

JADAVPUR UNIVERSITY

DEPARTMENT OF ECONOMICSPG I

SEMESTER II

16-Apr-15

TIME SERIES ANALYSIS OF REAL

GDP AND SHARE OF

AGRICULTURE AND ALLIED

SECTOR

ANKITA MONDAL-001400302024

SAYANTAN BAIDYA-001400302042

SOUMI BHATTACHARYA-001400302043

DEEPANWITA SAHA-001400302045

KRISHNENDU HALDER-001400302055

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CONTENTS

MODEL JUSTIFICATION………………………………………………...14

3

4

5

6

31

30

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ACKNOWLEDGEMENT

We are grateful to the faculty of Department of Economics (Jadavpur University) for

their unwavering support and cooperation. Working on this project has given us the opportunity to gather immense knowledge

regarding econometric tools and economic analysis that will surely benefit us significantly in our careers in the future. We thank our professor Dr. Arpita Dhar

immensely for setting us this task of preparing and presenting this project. We are extremely grateful and thankful to her for her tireless guidance without which it would not have been possible for us to make progress in our endeavour. We also

take this opportunity to thank our department for providing us with a functioning computer laboratory and library facilities which helped us to fulfil all our needs

regarding our project. Moreover, we are also grateful to our friends and families for their constant support and help.

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ABSTRACT

This paper is an endeavor to examine the relationship between the total GDP of INDIA & GDP

in agriculture and allied sector in India. Empirical evidence is obtained by applying the Time

Series Analysis on the annual data collected from the MINISTRY OF STATISTICS AND

PROGRAMME IMPLEMENTATION. Using Box Jenkins Approach, Unit Root test, we derive

results which show that there is no long run significant and positive relation between total

GDP of INDIA & GDP in agriculture and allied sector.

Keywords : Economics liberalization, agricultural reform, sectoral reform India ,time series,

inflation, growth,

JEL Classification : O1, O5, Q2,c22, h62, o47

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INTRODUCTION

Time series refers to a sequence of observations following each other in time, where

adjacent observations are correlated. This can be used to model, simulate, and forecast behavior for a system. Time series models are frequently used in fields such as economics, finance, biology, and engineering.

The Wolfram Language provides a full suite of time series functionality, including standard models such as MA, AR, and ARMA, as well as several extensions. Time series models can be simulated, estimated from data, and used to produce forecasts of future behavior.

The Indian agriculture sector accounts for 18 per cent of India's gross domestic product (GDP) and employs just a little less than 50 per cent of the country's workforce. This sector has made considerable progress in the last few decades with its large resources of land, water and sunshine. India is presently the world's largest producer of pulses and the second largest producer of rice and wheat.

The country is also the largest producer, consumer and exporter of spices and spice products in the world and overall in farm and agriculture outputs, it is ranked second. From canned, dairy, processed, frozen food to fisheries, meat, poultry, and food grains, the Indian agro industry has plenty of areas to choose for business.

The Department of Agriculture and Cooperation under the Ministry of Agriculture is the nodal organisation responsible for the development of the agriculture sector in India. Under it, several other bodies such as the National Dairy Development Board (NDDB) work for the development of the other allied agricultural sectors.

“Agriculture is not crop production as popular belief holds - it's the production of food and

fiber from the world's land and waters. Without agriculture it is not possible to have a city,

stock market, banks, university, church or army. Agriculture is the foundation of

civilization and any stable economy.”

By Allan Savory

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ECONOMETRIC THEORY-TIME SERIES

ANALYSIS

A time series is a sequence of data points, measured typically at successive points in

time spaced at uniform time intervals. This implies that time series data have a natural temporal

ordering which makes time series analysis distinct from other common data analysis problems, in

which there is no natural ordering of the observations. Basically, time series analysis comprises

methods for analyzing time series data in order to extract meaningful statistics and other

characteristics of the data, and time series forecasting is the use of a model to predict future

values based on previously observed values.

All data are for the period of 1954-2013, for a total of 60 observations.

IMPORTANCE OF STATIONARY STOCHASTIC PROCESS

A random or stochastic process is a collection of random variables ordered in time.

Continuous variables are denoted by Y(t) and discrete variables are denoted by Yt. The stationary

stochastic process has received a great deal of attention and scrutiny by analysts. A stochastic

process is constant when the mean and variance is constant over time and its covariance depends

on the distance/gap/lag between two time periods and not the actual time at which covariance is

calculated. Such a process is weakly stationary/ covariance stationary/ second order stationary/

wide sense stochastic process.

Definition of a stationary stochastic process:

Let Yt be a stochastic time series with these properties:

Mean: E (Yt = μ Variance: var (Yt) = E (Yt – μ 2 = σ2

Co a ia e: k = E [(Yt − μ Yt+k − μ ] Whe e k, the covariance (or auto covariance) at lag k, is the covariance between the values of Yt

and Yt+k, that is, et ee t o Y alues k pe iods apa t. If k = , e o tai 0, which is simply the

variance of Y (= σ2).

For the purpose of forecasting, we will use a stationary time series because

non stationary time series have a time varying mean or time varying variance or maybe

both. Then, we can study behaviour only for the concerned time period. Therefore,

generalization to other time periods is not a possibility.

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DIFFERENCE STATIONARY PROCESS (DSP) AND TREND STATIONARY

PROCESS (TSP):

The distinction between stationary and non stationary stochastic processes (or time

series) has a crucial bearing on whether the trend observed in the constructed time series is

deterministic or stochastic. If the trend in a time series is completely predictable and not variable,

it is called a deterministic trend, whereas if it is not predictable, it is called a stochastic trend. In

the following model of the time series Yt;

Yt = β + β t + β Yt− + ut

Where ut =a white noise error term, t = time measured chronologically.

The process will be called a Pure random walk under condition if

β1 = , β2 = , β3 = 1

Or,

Yt = Yt− + ut,

which is a non stationary process,

But Δ Yt = (Yt − Yt− ) = ut,

I.e. the first difference of the process becomes stationary. Such a process is called Difference

stationary process (DSP).

The process is called a Random walk with drift under condition:

β1 ≠ , β2 = , β3 = 1,

Or, Yt = β1 + Yt− + ut

which is a random walk with drift and is therefore non statio a . If it s itte as (Yt − Yt− = Δ Yt = β + ut, it ea s Yt ill e hi it a positi e β1 > o egati e β1 < 0) trend. Such a

trend is called a stochastic trend. This is a DSP process because the non stationarity in Yt can be

eliminated by taking first differences of the time series.

The process is called a Deterministic trend under condition:

β1 ≠ , β2 ≠ , β3 = 0

Or,

Yt = β1 + β2t + ut ,

which is called a trend stationary process (TSP). The mean of Yt is β1 + β2t, is not constant, but its

a ia e σ2 is. Ho e e the alue of ea a e esti ated afte k o i g alues of β1

a d β2.

If the mean of Yt is subtracted from Yt, the resulting series will be stationary, hence the name

trend stationary. Stationarity has been obtained after removing trend,

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THE UNIT ROOT TEST:

We need to check whether a series is trend stationary (TS) or difference stationary

(DS) process. TS and DS process have both statistical and economic justifications.

The statistical problems are:

The first problem lies in trend removing method .The process- use regression to remove trend

from the TS process and then, use differencing to remove for the DS process, but if difference for

the TS and regress for the DS process then the autocorrelation gives spurious results.

The second problem is the distribution of the test statistics are not standard distributions like

t, F or normal distributions. We need to check them case by case.

Considering the following Autoregressive model:

Yt = αYt-1 +ut --------------------------- (A)

where - ≤ α ≤ , ut is the white noise error term.

If α= , e fa e the u it oot p o le , e uatio A e o es a a do alk odel ithout drift,

making it a non stationary stochastic process. Hence we can regress Yt on its lagged value Yt-1 and

fi d out if the esti ated α is statisti all e ual to .If α≤ , i.e., if the a solute alue of α is less than 1, then it can be shown that the time series Yt is stationary. This forms the general idea of the

unit root test.

On subtracting Yt-1 from Yt, one obtains,

Yt-Yt-1 = αYt-1-Yt-1+ut

= α-1)Yt-1+ut

Δ Yt = Yt-1 +ut --------------------- (B)

Whe e, = α- , Δ is the fi st-difference operator.

In practice, instead of eliminating equation A we estimate and test the null hypothesis that

= . If = , the α= , e ha e a u it oot, ti e se ies u de o side atio is o statio a .

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Turning to equation B, we take the first difference of Yt and regress them on Yt-1 and

see if the estimated slope coefficient in this regression ( ̂) is zero or not. If it is zero, we conclude

that Yt is non stationary. If it is negative, we conclude that Yt is stationary. We cannot use the t-

test i this ega d e ause u de the ull h pothesis that = , the t alue of the esti ated coefficient of Yt-1 does not follow the t distribution even in large samples.

The other alternative shown by Dicky and Fuller is that under null hypothesis that

= , the esti ated t alue of the oeffi ie t of Yt-1 in equation B follows the Ƭ (tau) statistic.

The tau statistic or test is known as the Dickey-Fuller (DF) test, if the h pothesis that = is rejected; we can use the t-test.

The DF test is estimated under three different null hypotheses:

Yt is a a do alk: Δ Yt= Yt-1 +ut

Yt is a a do alk ith d ift: Δ Yt= β+ Yt-1 +ut

Yt is a a do alk ith d ift a ou d a sto hasti t e d: Δ Yt= β + β t+ Yt-1+ ut, where

t is the time or trend variable.

AUGMENTED DICKEY-FULLER TEST:

To check for stationarity, we go for the Unit root test for each individual series by

applying Augmented Dickey Fuller test statistic, applying Akaike Info Criterion, with trend

and intercept for knowing whether the series is trend stationary or difference stationary. It is

an augmented version of the Dickey–Fuller test for a larger and more complicated set of time

series models.

Our null and alternative hypotheses are following:

H0: unit root exists in the series.

H1: unit root does not exist.

Our objective is to reject the hypothesis so that the series can be said to be stationary.

The testing procedure for the ADF test is the same as for the Dickey–Fuller test but it is

applied to the model

� �t= + t+ �t-1+ ̂1 ��t-1+⋯+ ̂p-1 ��t-p+1+ �t

he e α is a o sta t, β the oeffi ie t o a ti e t e d a d p the lag o de of the auto eg essi e p o ess. I posi g the o st ai ts α = a d β = o espo ds to odelli g a a do alk a d usi g the o st ai t β = o espo ds to odelling a random walk with a

drift.

The unit root test is then carried out under the null hypothesis =0 against the alternative

h pothesis of < . O e a alue fo the test statisti τ= � ( ) is, computed it can be

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compared to the relevant critical value for the Dickey–Fuller Test. If the test statistic is less (this

test is non symmetrical so we do not consider an absolute value) than the (larger negative) critical

value, then the null hypothesis of =0 is rejected and no unit root is present.

CO-INTEGRATION: If two or more series are individually integrated (in the time series sense) but some linear

combination of them has a lower order of integration, then the series are said to be co

integrated. Two variables will be co integrated if they have a long-term, or equilibrium,

relationship between them.

AR, MA, ARMA, and ARIMA Modelling Of the Time Series:

If the time series is stationary, we can model it in a variety of ways:

Autoregressive (AR) Process

Moving Average (MA) Process

Auto Regressive Moving Average (ARMA) Process

Autoregressive Integrated Moving Average (ARIMA) Process

If Yt~I(1) and Xt~ I(1)

• Then Yt- βXt~) which implies ut~) -1) which implies ut~I(0)

• Then Yt and Xt are co integrated. The regression equation Yt= βXt+ut makes sense as Yt and

• Xt do not drift too much in long run.

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THE BOX –JENKINS APPROACH:

The objective of Box- Jenkins is to identify and estimate a statistical model which

can be interpreted as having generated the sample data. If this estimated model is then to be

used for forecasting we must assume that the features of this model are constant through

time, and particularly over future time periods. Therefore, it is compulsory that we have

either a stationary time series or a time series that is stationary after one or more

differencings.

This process is clearly based on three steps:

1. IDENTIFICATION

2. ESTIMATION

3. DIAGNOSTIC CHECKING

If the model is

adequate, then

go for forecasting

If the model is

not adequate,

then go to step 1

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Step-1: Identification

Here, we find out the appropriate values of p, q, and d by observing the

correlograms completely based on two important statistical tools - autocorrelation coefficient

and partial autocorrelation coefficient.

1) If the autocorrelation coefficient changes very slowly or did not die out, then

non stationary is obtained and the series must be differenced to get the stationary series. So

the stationary series is the ARMA.

2) For MA (p) we get autocorrelation coefficient to be 0 for all k>p and the partial

autocorrelation coefficient taper off by increasing the k. We obtain the cut-off point of the

autocorrelation by looking at the sample auto correlation..

3) For AR (q) the partial autocorrelation becomes 0 for all k>q and the

autocorrelation taper off by increasing k. to get the cut off pint of the partial autocorrelation

function we need to use the estimatio usi g + /√T.

4) If neither partial autocorrelation nor the autocorrelation coefficient have any

cut off points then ARMA process is identified and lag lengths of AR and MA are obtained

from the special pattern of the two functions.

Step- 2: Estimation

We estimate the parameters of the autoregressive and moving average terms included in the

model by simple least squares or by some nonlinear (in parameter) estimation methods.

Step- 3: Diagnostic Checking

Having chosen a particular ARIMA model, and having estimated its parameters, we next see

whether the chosen model fits the data reasonably well or not.

In order to check the adequacy of the model we can have two possibilities

(1) to over fit the specific model

(2) to do a residual analysis.

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A shortcut way of remembering is presented in the following chart.

Shape Indicated Model

Exponential, decaying to zero

Autoregressive model. Use the partial

autocorrelation plot to identify the order of

the autoregressive model.

Alternating positive and negative, decaying

to zero

Autoregressive model. Use the partial

autocorrelation plot to help identify the order.

One or more spikes, rest are essentially zero

Moving average model, order identified by

where plot becomes zero.

Decay, starting after a few lags

Mixed autoregressive and moving average

(ARMA) model.

All zero or close to zero

Data are essentially random.

High values at fixed intervals

Include seasonal autoregressive term.

No decay to zero

Series is not stationary.

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MODEL AND JUSTIFICATION

Our model:

Where,

Yt is total real GDP of India,

Zt is real GDP of agriculture & allied sector in India.

Agriculture Growth Rate in India GDP had been growing earlier but in the last few years it is

constantly declining. Still, the Growth Rate of Agriculture in India GDP in the share of the

country's GDP remains the biggest economic sector in the country.

India GDP means the total value of all the services and goods that are produced within the territory

of the nation within the specified time period. The country has the GDP of around US$ 1.09 trillion

in 2007 and this makes the Indian economy the twelfth biggest in the whole world.

The growth rate of India GDP is 9.4% in 2006- 2007. The agricultural sector has always been an

important contributor to the India GDP. This is due to the fact that the country is mainly based on

the agriculture sector and employs around 60% of the total workforce in India. The agricultural

sector contributed around 18.6% to India GDP in 2005.

Agriculture Growth Rate in India GDP in spite of its decline in the share of the country's GDP plays

a very important role in the all round economic and social development of the country. The Growth

Rate of the Agriculture Sector in India GDP grew after independence for the government of India

placed special emphasis on the sector in its five-year plans. Further the Green revolution took place

in India and this gave a major boost to the agricultural sector for irrigation facilities, provision of

agriculture subsidies and credits, and improved technology. This in turn helped to increase the

Agriculture Growth Rate in India GDP.

The agricultural yield increased in India after independence but in the last few years it

has decreased. This in its turn has declined the Growth Rate of the Agricultural Sector in India

GDP. The total production of food grain was 212 million tonnes in 2001- 2002 and the next year it

declined to 174.2 million tonnes. Agriculture Growth Rate in India GDP declined by 5.2% in 2002-

2003. The Growth Rate of the Agriculture Sector in India GDP grew at the rate of 1.7% each year

between 2001- 2002 and 2003- 2004. This shows that Agriculture Growth Rate in India GDP has

Yt = α +β Zt +ut

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grown very slowly in the last few years.

Agriculture Growth Rate in India GDP has slowed down for the production in this sector has

reduced over the years. The agricultural sector has had low production due to a number of factors

such as illiteracy, insufficient FINANCE, and inadequate marketing of agricultural products. Further

the reasons for the decline in Agriculture Growth Rate in India GDP are that in the sector the

average size of the farms is very small which in turn has resulted in low productivity. Also the

Growth Rate of the Agricultural Sector in India GDP has declined due to the fact that the sector

has not adopted modern technology and agricultural practices. Agriculture Growth Rate in India

GDP has also decreased due to the fact that the sector has insufficient irrigation facilities. As a

result of this the farmers are dependent on rainfall, which is however very unpredictable.

Agriculture Growth Rate in India GDP has declined over the years. The Indian government must

take steps to boost the agricultural sector for this in its turn will lead to the growth of Agriculture

Growth Rate in India GDP.

"As per latest estimates released by Central Statistics Office (CSO) the share of agricultural

products/Agriculture and Allied Sectors in Gross Domestic Product (GDP) of the country was 51.9

per cent in 1950-51, which has now come down to 13.7 per cent in 2012-13 at 2004-05 prices,"

Minister of State for Agriculture

The Indian agriculture sector accounts for 18 per cent of India's gross domestic product (GDP) and employs

just a little less than 50 per cent of the country's workforce. This sector has made considerable progress in the

last few decades with its large resources of land, water and sunshine. India is presently the world's largest

producer of pulses and the second largest producer of rice and wheat.

The country is also the largest producer, consumer and exporter of spices and spice products in the world and

overall in farm and agriculture outputs, it is ranked second. From canned, dairy, processed, frozen food to

fisheries, meat, poultry, and food grains, the Indian agro industry has plenty of areas to choose for business.

The Department of Agriculture and Cooperation under the Ministry of Agriculture is the nodal organisation

responsible for the development of the agriculture sector in India. Under it, several other bodies such as the

National Dairy Development Board (NDDB) work for the development of the other allied agricultural sectors

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ANALYSIS:-

METHODOLOGY :- We have taken the time series data of the two variables (real gdp and agriculture and allied

se to s sha e i eal gdp o e a spa of ea s 9 -2014) and have tried to identify,

estimate and cross validate the time series processes using Box- Jenkins Method.

IDENTIFICATION & ESTIMATION

OF THE STOCHASTIC PROCESS

The first step of this method is to identify the stochastic process that each of

the time series variables follow. By observing the correlogram for every series we can

infer whether the series follows AR, MA or ARMA. To get the correlogram we kept the

maximum lag length 28 and the total sample observation was 60 (included data from

1954-55 to 2013-14). The correlograms for the series are given below.

The solid vertical line in the diagram (all correlogram diagrams) represents the

zero axis; observations above the line is positive values and below the line are negative

values.

We have used the Box- Jenkins approach to throw light on the empirical analysis of

our data. According to this approach we need to follow three steps to find the long run

relationship between dependent variable and all independent variables. In our paper we have one

dependent variable i.e. growth f TOTAL REAL GDP and one independent variables i.e. GDP of

agriculture & allied sector.

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Correlogram of REAL GDP (Y)

TABLE 1. CORRELOGRAM OF REAL GDP (Y) AT LEVEL

TABLE 2 : CHECKING THE ORDER OF REAL GDP(Y):

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From the graphical representation of Y, we can identify that Y follows auto regressive process of order 1,AR(1). The partial auto-correlation function is positive and significant only for K=1 and beyond that it falls below the critical region. Moreover, the auto-correlation function is tapering as K is increasing.

We now estimate this AR(1) process. To have a stationary AR(1) process we need to

have the inverted AR root, |p|<1. But, here, we have obtained that |p| >1. This implies that for every unit change in Yt-1 there is a greater change in Yt. This

further implies that the series will continue to be non-stationary over time. Though value of the t-statistics of the AR (1) process is statistically significant, having a non-stationary series is not desirable.

So we go for differencing and see that if first differencing can remove this

nonstationarity. For that we obtain correlogram at first difference i.e. D(X).

TABLE 3: CORRELOGRAM OF REAL GDP (Y) AT 1ST

DIFFERENCE

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TABLE 4: CHECKING THE ORDER OF REAL GDP(Y) AT FIRST

DIFFERENCE:

Dependent Variable: D(Y)

Method: Least Squares

Date: 04/10/15 Time: 23:21

Sample (adjusted): 1956 2013

Included observations: 58 after adjustments Convergence achieved after 8 iterations

Variable Coefficient Std. Error t-Statistic Prob. C -111371.5 185301.8 -0.601027 0.5502

AR(1) 1.080524 0.035968 30.04138 0.0000 R-squared 0.941574 Mean dependent var 195581.1

Adjusted R-squared 0.940531 S.D. dependent var 339592.6

S.E. of regression 82813.87 Akaike info criterion 25.52045

Sum squared resid 3.84E+11 Schwarz criterion 25.59150

Log likelihood -738.0931 Hannan-Quinn criter. 25.54813

F-statistic 902.4844 Durbin-Watson stat 2.111862

Prob(F-statistic) 0.000000 Inverted AR Roots 1.08

Estimated AR process is nonstationary

First differencing of Yt gi es us ΔYt. He e e see that ∆Yt=D(Y) again follows auto regressive

process of order 1, AR(1).But through estimation we observe that the process still remains non-

stationary since the inverted AR root, |p| > 1(As indicated in the table above 1.08>1). Although

the t-statistic of the AR process is highly significant yet our estimation claims the series to be non-

stationary. Thus we go on to second differencing the series so as to remove the non-stationarity.

The correlogram of the series after second differencing is provided below

TABLE 5: CORRELOGRAM OF REAL GDP (Y) AT 2ND

DIFFERENCE

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We find that neither the partial autocorrelation function nor the auto –correlation function has a

cut-off point. So we can conclude that after second differencing the se ies Δ2Yt follows ARMA(3,3)

.The series no longer remains non-stationary as evident from the estimation of ∆ Yt= D(Y,2) given

below where the inverted AR and MA roots are less than 1 as desired. Thus we can say that the

series Y(REAL GDP) is integrated of order 2. Y~ I(2). Thus its stationary series D(Y,2) is thus said to

be integrated of order 0. D(Y,2)~ I(0).

TABLE 6 : CHECKING THE ORDER OF REAL GDP AT SECOND

DIFFERENCE:

Now we can again cross check our estimation through the augmented Dickey fuller test for the

presence of unit roots . We are testing the null hypothesis that D(Y,2) has a unit root against the

alternative hypothesis that D(Y,2) does not have one. From the analysis obtained, we can

conclude that that the augmented Dickey Fuller test statistics is statistically significant and we

reject the null hypothesis that D(Y,2) o tai s u it oot. Thus ou o lusio of Δ2Yt being a

stationary ARMA process is correct.

UNIT ROOT TEST FOR REAL GDP AT 2ND

DIFFERENCE To check for stationarity we go for the Unit Root Test for individual series by applying Augmented

Dickey Fuller test statistic, applying Akaike info criterion with trend and intercept for knowing

whether the series is trend stationary or difference stationary.

Our null hypothesis is:

H0: unit root exists in the series

H1: unit root does not exist

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Our objective is to reject the null hypothesis so that the series can be said to be stationary.

Below we have provided the table for unit root test.

TABLE 7:

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From the tabular result:

The computed value of t-statistic = -4.417592

Critical value of Augmented Dickey Fuller test statistic

at level 1% = -4.161144

at level 5% = -3.506374

at level 10% = -3.183002 Clearly, 4.417592 >4.161144, 4.417592 >3.506374 and 4.417592 >3.183002. So, we have sufficient evidence to reject the null hypothesis at 10%, 5%, and even 1%. Hence we

conclude that the unit root does not exist and the series is stationary.

From here we can imply:

D(y, 2) ~ I(0)

Y ~ I(2)

Thus real GDP follows I (2).

After the entire process of identification and estimation we move on to the third stage of the

Box-Jenkins Method namely cross validation or diagnostic checking.

CROSS VALIDATION OR DIAGNOSTIC CHECKING

By diagnostic checking we imply a methodology adopted for checking whether our model assumes

of the presence of white noise. The table for cross validation of Real GDP(Y) is given below:-

TABLE 8:- .

According to the above table the series displays a complete case of white noise and perfect fit

since the autocorrelation (pk) and partial autocorrelation (thetakk) are within bounds.

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CORRELOGRAM OF SHARE OF AGRICULTURE AND ALLIED

SECTOR TO REAL GDP (IN REAL TERMS) BASE YEAR 2004-

05.(DENOTED BY Z)

TABLE 9: CORRELOGRAM OF Z AT LEVEL

TABLE 10 CHECKING THE ORDER OF REAL GDP AT AGRICULTURE AND ALLIED

SECTOR AT LEVEL:

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From the graphical representation of Z, we can identify that Z follows auto regressive

process of order 1, AR (1). The partial auto-correlation function is positive and significant only for

K=1 and beyond that it falls below the critical region.Moreover, the auto-correlation function is

tapering as K is increasing.

We now estimate this AR (1) process. To have a stationary AR (1) process we need to

have the inverted AR root, |p|<1.But here, we have obtained that |p| >1(In the above table

inverted AR roots 1.04 >1). This implies that for every unit change in Zt-1 there is a greater change

in Zt. So the series is non- stationary.

Next we go on to first difference the series Z. The correlogram after first differencing

is provided below:-

TABLE 11: CORRELOGRAM OF Z AT 1ST

DIFFERENCING

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TABLE 12: CHECKING THE ORDER OF SHARE OF AGRICULTURE AND ALLIED

SECTOR IN REAL GDP(Z) AT FIRST DIFFERENCE:

So after first differencing we obtain that the autocorrelation and partial correlation

does not have cut off points indicating that our series D(Z) follows ARMA(1,1) and on further

estimating the process we find that the series is stationary and has inverted AR and MA roots less

than 1 as desired.

We further cross check our conclusion of D (Z) being a stationary process through the

Augmented Dickey Fuller Test for the presence of unit roots.

UNIT ROOT TEST FOR REAL GDP OF AGRICULTURE AND ALLIED

SECTOR (BASE PRICE: 2004-05) AT 1ST

DIFFERENCE

Again we check for stationarity. We go for the Unit Root Test for individual series by

applying Augmented Dickey Fuller test statistic, with Akaike info criterion with trend and intercept

for knowing whether the series is trend stationary or difference stationary.

Our null hypothesis is:

H0: unit root exists in the series

H1: unit root does not exist

Our objective is to reject the null hypothesis so that the series can be said to be

stationary. Below we have provided the table for unit root test for Z at first difference.

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TABLE 13:-

From the tabular result:

The computed value of t-statistic = -7.833550

Critical value of Augmented Dickey Fuller test statistic

at level 1% = -4.127338

at level 5% = -3.490662

at level 10% = -3.173943

Clearly, 7.833550>4.127338, 7.833550>3.490662, 7.833550>3.173943.

So, D(Z)~ I(0)

Z ~ I(1)

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Hence real GDP of agriculture and allied sector follows I(1).

We now cross validate our estimation of Z through diagnostic checking. The table for diagnostic

checking of Z is provided below :-

TABLE 14:-

According to the above table the series displays a complete case of white noise and

perfect fit since the autocorrelation (rho k) and partial autocorrelation(theta kk) are within

bounds.

Thus we conclude that by identifying and estimating the real gdp(Y) and share of

agriculture and allied sector in real gdp(Z) we find out that both the series evolve to be non-

stationary at first. But after second differencing Y and first differencing Z we succeed in making

both the process stationary wherein Y and Z follows ARMA(3,3) and ARMA(1,1) respectively. Test

for unit root supports our claim and cross validation of both the series abides by the assumption

of white noise.

DETERMINISTIC OR STOCHASTIC TREND :-

From the above observations tables and deductions we finally conclude that the initial

series Y(REAL GDP) is a difference stationary process or (DSP) since it is a non-stationary series

and has the inverted AR roots almost close to 1that is the autoregressive part of the relation is

equal to 1 and we know that he |α|= the p o ess e o es a diffe e e statio a p o ess.

Though first differencing do not give us a stationary series but further differencing gives us a

stationary series and the final D(Y,2) series follows trend stationary process(TSP) since its trend

component is significant as indicated in its unit root table(TABLE 7) and also it is stationary and has

|α| < hi h i plies that the a ia le is su je t to a constant growth trend and that the

deviations from the trend follow a stationary ARMA(3,3) process. Thus Y follows a deterministic

trend.

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Similarly Z(SHARE OF AGRICULTURE AND ALLIED SECTORS IN REAL GDP) is a

difference stationary process(DSP) since it is non-stationary and has inverted AR roots greater

than 1 or almost close to 1 for which it reduces to a difference stationary process. Further differencing of Z gives us a stationary series D(Z) which on the other hand follows Trend

Stationary Process(TSP) of ARMA(1,1) e ause he e |α| <1 and also its trend component is

significant as indicated in the unit root test for D(Z)(TABLE 12). Thus we can say that Z follows a

deterministic trend.

In generalised view if we take a variable Y and have to check whether its difference stationary we

represent it with an equation

If Y is the log of a variable then the above equation asserts that the variable grows with a constant

linear trend and its deviation from the trend follows a stationary process. Thus the series is Y is

termed TREND STATIONARY.

If the α pa a ete is so that the auto eg essi e pa t of the elatio has a u it oot, the e ha e a series that fluctuates much like that of a TSP but actually strays away from the trend. The

difference then takes the form

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According to the above graph Y4(A variable depicted in the graph) follows a TSP whereas Y5(A

variable depicted in the graph) follows a DSP.

LONG RUN RELATIONSHIP BETWEEN Y

AND Z :-

We now check whether there is any long term relationship between

REAL GDP (Y) and SHARE OF AGRICULTURE AND ALLIED SECTOR IN REAL GDP IN REAL TERMS(Z)

by testing the co-integration between the two variable.

On running the unit root test for Y and Z at level we found that both were

nonstationary. After taking 1st

difference Z became stationary so Z is I(1) but Yt is yet not

stationary after taking first difference. Finally after taking second difference Yt becomes

stationary. So Y is I(2). Therefore we cannot test cointegration between Y and Z.

The theory of co-integration as introduced by Granger (1981), uses an important

property of I(1) variables viz., there can be linear combinations of these variables that are I(0). In

case there indeed exist such linear combinations, then the variables are cointegrated.

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But since here our variables are not cointegrated thus there does not exist any long

term relationship between Y and Z as per the data of our model.( TABLE 7 FOR Y AND TABLE 12

FOR Z)

Since we cannot compute any co-integration between Y and Z thus we do not need to

find out the residual variable and compute its unit root since even if it suffices for the assumption

of white noise it is of no help in computing co-integration between Y and Z.

Y~ I(2)

Z ~ I(1)

E(Error term) satisfy the assumption of white noise so E~I(0)

Still we cannot cointegrate Y and Z.

So we can conclude that from our previous knowledge of econometric analysis

compelled us to always compute a regression between two given variables but further

knowledge on time series analysis gave us a scope to discard this conventional view of running a

regression to variables which are supposed to be correlated yet are found to be not integrated

and thus we could not find out a long run relationship of the two given variables.

So from our analysis we can state that from an apparent view REAL GDP should always have

some kind of a relationship with REAL GDP OF AGRICUYLTURE AND ALLIED SECTORS but here

from our econometric TIME-SERIES ANALYSIS and with the help of the software EVIEWS we

hereby conclude that there is no long term relationship between our two variables.

CONCLUSION:-

We see that our explanatory variables Z only needed one level of differencing to

achieve stationarity. This made the transition from non-stationarity to stationarity less

cumbersome. The proceeds of the analysis of a stationary set of data pertaining to a time

period can also be used to analyze other different sets of data pertaining to different time

periods. This would not have been possible if the data was non- stationary. Furthermore

stationarity of data also negates the possibility of spurious regression.

As per our observations, r e a l GDP of agriculture & a l l i e d s e c t o r has no

significant effect on t o t a l r e a l G D P i n I N D I A . Whatever be the scenario the

relationship between r e a l GDP of agriculture & a l l i e d s e c t o r and t o t a l r e a l G D P

i n I N D I A a controversial one in both theory and empirical findings.

There are other independent variables that directly affect TOTAL GDP. If we

incorporate those variables along with GDP of agricultural & allied sector then we may found a

positive significance in determining those effects.

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