Exploring the empirical nexus among Indian GDP, Imports and Exports

Exploring the empirical nexus among
Indian GDP, Imports and Exports
Using VAR Model approach
M.Sc. STATISTICS Project
Author:
K.Harsha Vardhan,
Enrollment No.: 09ILMB05,
I.M.Sc. Maths 10th Semester,
School of Mathematics,
University of Hyderabad.
Supervisors:
Prof. Bandi Kamaiah, Prof. Shoba Naresh Sharma,
School of Economics, School of Mathematics,
University of Hyderabad. University of Hyderabad.
May 5, 2014 May 5, 2014

DECLARATION
This is to declare that the subject matter included in this project is the outcome of
the project work carried out during the months of August 2013 - April 2014 by me
under the guidance and supervision of Prof. Bandi Kamaiah, School of Economics,
Prof. Shoba Naresh Sharma, School of Mathematics, University of Hyderabad.
Date: 05/05/2014 [K.Harsha Vardhan]
Hyderabad

CERTIFICATE
This is to certify that the project work presented in this thesis entitled “Exploring
the empirical nexus among Indian GDP, Imports and Exports using VAR
approach”, submitted to University of Hyderabad for the partial fulfillment of the
degree of Integrated Master of Science in Mathematics, is a work carried out by
K.Harsha Vardhan under our supervision at School of Mathematics, University of
Hyderabad, Hyderabad. This project work has not been submitted to this or any
other university partially or fully for the award of any degree or diploma.
Prof. Bandi Kamaiah
(Project Supervisor)
Dean
School of Mathematics
Prof. Shoba Naresh Sharma
(Project Supervisor)

ACKNOWLEDGEMENT
Foremost, I would like to express my sincere gratitude to my advisor Prof. Bandi
Kamaiah and Prof.Shoba Naresh Sharma for their continuous support of my
project and research. Their guidance helped me in all the time of research and
writing of this report. Your advice on both research as well as on my career have
been priceless.
I am also very thankful to Zikrullah Khan, Ph.D. scholar and Rajendra Narayan,
Ph.D. Scholar for their valuable advice and guiding me throughout the work as
well as report. Finally I would like to thank all my friends for supporting me to
finish the project.

Contents
1 INTRODUCTION................................................................................................... 6
2 GROSS DOMESTIC PRODUCT............................................................................... 6
2.1 GDP of India.................................................................................................. 7
3 PROPOSITION...................................................................................................... 8
4 METHEDOLOGY.............................................................................................. 9
4.1 Univariate Time series model........................................................................ 9
4.2 Multivariate time series model ................................................................... 10
4.3 VAR (p) model ........................................................................................... 10
4.4 Test for stationarity .................................................................................... 11
4.5 Least square method .................................................................................. 12
4.5 Forecasting ................................................................................................. 14
4.6 Impulse response function.......................................................................... 14
5 DATA................................................................................................................. 16
5.1 Source of the data....................................................................................... 16
5.2 Description of the data ............................................................................... 16
6 Empirical resul .................................................................................................. 28
7 CONCLUSION .................................................................................................... 34
8 REFERENCES...................................................................................................... 35

1 INTRODUCTION
A time series is a sequence of data points, measured typically at successive
points in time spaced at uniform time intervals. If the data has only one variable
then we can directly use univariate time series model. But in general if we take a
data of time series there can be many variables which will be affecting dependent
variable, so multivariate time series model is used. In multivariate time series
model vector autoregression model (VAR), vector moving average model (VAR)
are used to predict the future values of dependent variable.
A VAR model describes the evolution of a set of k variables
(called endogenous variables) over the same sample period (t = 1, ..., T) as
a linear function of only their past values.
2 GROSS DOMESTIC PRODUCT
Gross domestic product (GDP) is the market value of all officially recognized final
goods and services produced within a country in a year, or other given period of
time. GDP per capita is often considered an indicator of a country's standard of
living.

Here we take GDP in Expenditure approach – all expenditure incurred by
individuals during 1 year. GDP is the sum of consumption, investment,
government spending and net exports.
How GDP is measured according to exports and imports of goods and services and
we take other variables unchanged.
Y =C+I+G+(X-M), where
Y = Gross domestic product
C = consumption
I = investment
G = government spending
X = exports
M = imports
2.1 GDP of India
The Economy of India is the 8th-largest in the world by nominal GDP and
the third-largest by purchasing power parity. India is the 16th-largest exporter and
the 8th-largest importer of goods and services in the world. The GDP value of
India represents 2.97 percent of the world economy. GDP in India averaged

485.65 USD Billion from 1970 until 2012, reaching an all-time high of 1872.90
USD Billion in 2011 and a record low of 63.50 USD Billion in 1970.
3 PROPOSITION
How GDP is influenced by imports and exports of goods and services, not only
GDP but also how exports and imports are affected mutually. The three variables
are GDP, Exports and Imports.
To build a model for three variables, we use the method of vector auto regression.
And for this we need to know briefly about the vector autoregression time series
model (VAR).
It is an econometric model used to capture the linear interdependencies among
multiple time series. Vector autoregressive model is simply a multivariate time
series model. VAR models generalize the univariate autoregression (AR)
models by allowing for more than one evolving variable.

4 METHEDOLOGY
4.1 Univariate Time series model
In univariate time series model the past history of a single variable is used to model
the behavior of that same series.in general, a variable at time t Yt is modeled as a
function of past values of that variable plus current and past random error terms.
The difference between univariate and multivariate time series models, therefore,
is that in a multivariate model Yt is modeled as a function of current and past
values of other variables and their random error terms as well.
Yt = aYt-1+Ut |a| < 1
E(Ut) = 0
E(Ut
2
) = 2
E (YtUs) = 0 t<s
The AR (1) model expresses the current value of Y as the sum of fraction of the
previous values of Y and a random white noise disturbance terms. The condition
for U to be white noise process are given by
E (Ut) = 0
E (Ut
2
) = 2
E (YtUs) = 0 t<s
Because Yt=aYt-1+Ut |a| < 1 does not include a constant term, Y can be
viewed as a variable with a zero mean or as the deviation of a variable from its
non-zero mean.[1]

4.2 Multivariate time series model
In a multivariate time series model the interaction between several variables is
used to forecast each individual variable. Thus, in a multivariate time series model
the forecast at time t of a variable Y is a function of past values of itself and
current and past values of all other variables in the system. A VAR model
describes the evolution of a set of k variables (called endogenous variables) over
the same sample period (t = 1... T) as a linear function of only their past evolution.
The variables are collected in a k × 1 vector Yt, which has as the ith
element Yi,t the
time t observation of variable Yi. For example, if the ith
variable is GDP, then Yi,t is
the value of GDP at t.
4.3 VAR (p) model
VAR representation
Yt = α+ Ф1 Yt-1+ Ф2 Yt-2+………+ Фp Yt-p+£t
where α is a k × 1 vector of constants (intercept), Φi is a k × k matrix (for every i =
1, ..., p) and £t is a k × 1 vector of error terms The i-periods back observation Yt-i is
called the ith
lag of y. Vector white noise
E(£t) =0
E(£t, £t
`
) = ∑
E(£t, £s
`
) = 0

£ of different time periods should not be correlated.
All of these are independent of time t.
VAR(p) representation in lag operators
Ф(L)Yt = α+£t
VMA representation Yt=Ф(L)-1
α+ Ф(L)-1
£t
Estimation and uses of VAR representation
VAR(1) Yt=α+ ФYt-1+£t
For K=2
y1t = α1 + ф11 y1t-1 + ф12 y1t-1 + £1t ----(A)
y2t = α2 + ф21 y1t-1 + ф22 y1t-1 + £2t ---(B)
(A)and (B) can each be estimated by ordinary least square method (if they are
stationary)
4.4 Test for stationarity
The time series data is said to be stationary if its mean, variance at different lags
are time independent. If the data is non-stationary then we have to make the
data as stationary by first difference of the observations. We use Augmented

Dickey-Fuller (ADF) test to check whether the given time series data is
stationary or not.
Null hypothesis: unit root
Alternate hypothesis: not a unit root
ADF Test statistic:
DF =
̂
̂
By the ADF test we can say that the data over a time period is stationary or not.
4.5 Least square method
The method of least square is used to estimate β0 and β1, so we will estimate β0 and
β1 so that the sum of squares of the difference between the observations yi and the
straight line is minimum.
yi = β0 + β1xi + εi i=1,2,………..,n
Sample regression model, written in terms of the n pairs of data (yi,xi)
(i=1,2,………..,n). Thus the least squares criterion is
S(β0, β1) = ∑ yi-β0-β1xi)2
The least square estimators of β0 and β1, say β0 and β1, must satisfy
= -2∑ yi-β0-β1xi) =0

= -2∑ yi-β0-β1xi)xi =0
Simplifying these two equations yields
σβ0 + β1∑ xi) = ∑ yi)
Β0∑ xi) + β1∑ xi)2
=∑ yixi)
These equations are called least square normal equations. The solution to the
normal equation is
�̂0 = y – �̂ 1x
An important preliminary step in model building and impulse response analysis is
the selection of the VAR lag order. For finding the lag we use some commonly
used lag-order selection criteria to choose the lag order, such as AIC, HQ, SC and
FPE.
Using Akaike Information Criterion to choose lag order.
AIC = -2 ( ) +
We take p that delivers smallest AIC value.
We estimated the parameters using OLS method and by AIC we estimated lag.
�1 =

4.5 Forecasting
VAR(p) model
t = ̂+̂1 t-1+ ̂2 t-2 + _ _ _ _ + ̂p t-p
So we need data t =1, 2,_ _ _ T to estimate the parameters
E(YT+1|IT) = E(YT+1|YT,YT-1,_ _ _)
T+1 = ̂+̂1 T+ ̂2 T-1 + _ _ _ _ + ̂p T-p+1
E(YT+2|IT) = ̂+̂1 YT+1+ ̂2 T + _ _ _ _ + ̂p T-p+2
But we don’t know the value of YT+1 so we use T+1
σow T+2 =̂ +̂1 T+1 + ̂2 T + _ _ _ _ + ̂p T-p+2
4.6 Impulse response function
Impulse response function is a shock to the VAR system. Impulse responses
identify the responsiveness of the dependent variables (endogenous variable) in a
VAR when a shock is put to the error term such as £1t and £2t at the equation below.
Unit shock is applied to each variable and we see its effects on the VAR system.
For example if we take money and consumption then the model is
Money = B1 + B2*consumptiont-I + B3*moneyt-I + £1t

Consumption=B4 + B5*moneyt-I + B6*consumptiont-I + £2t
A change in £1t will bring a change in money. It will change consumption and also
money during the next period. So we give a shock to the innovation or residual,
that is on £1t and £2tof the above VAR model to see how it affects the whole VAR
model. But for calculating impulse responses, the ordering of the elements is
important.
For the VAR model with one lag we write:
yt = β0 +β1yt-1 +vt
If we substitute in for yt-1 we get:
yt = β0 + β1(β0 +β1yt-1+vt-1) + vt
= β0 + β1β0 + β1
2
yt-2 + β1vt-1 + vt
Doing this substitution over and over, we get
yt = β0
*
+ β1
k
yt-k + ∑ β1
j
vt-j where ( β0
*
= (I-β1)-1
β0)
Now yt is a function of a weighted sum of the intervening values of the error vector
vt-j. How do elements of yit respond to past shocks on the jth
element (vjt) the
answer is obtained by taking the derivative of yt with respect to vt-j above.

When we plot [β1
k
]i,j as a function of k, we see how future values of variable i are
impacted by a one unit change in variable j, k periods in the past. This is called the
impulse response function of variable i to a change in variable j.
This is the primary method used to understand the implied dynamics of a VAR
model. It answers the basic question of how a change in one variable affects the
system in the future. So powers of the matrix β1 determine how a change in one
variable today affects the future values.
5 DATA
5.1 Source of the data
The data has been taken from the reserve bank of India about the country’s GDP
gross domestic product, Imports and Exports. The gross domestic product (GDP) at
market price is taken from macroeconomics aggregate. The imports data has been
taken from imports of principals commodities. The exports data has been taken
from exports of principal commodities. This is a quarterly data taken from 1996 to
2009.
5.2 Description of the data
There are 54 observations in this data. The values in the table are given in millions
of rupees. The green line in the graph is the GDP, in the initial period (1996) the
GDP of India was 3884.64 million rupees and increased to 9065.69million rupees
in 2012.
From the graph below we can see the seasonality effect there is a sudden
decrease in starting of the year periods 1,5,9,13…..

Here period 1 is 1st
quarter in 1996
Period 15 is 3rd
quarter in 1999
Till the period 30 the value of imports and exports are almost similar.
Figure 1 Graph between period(time) vs. value(million rupees)
Because there is seasonality effect first we must deseasonalize the data. By using
moving average process. After deseasonalising the data, the graph is given below.
Here most of the seasonal effect is removed

Figure 2 Graph between period (time) vs. value(million rupees) after
deseasonalising
As the values of the GDP, exports and imports are increasing at higher rate so we
take log of the observations. For the log values of the observations are not
stationary so we take the first difference of the log observations.
0.00
2000.00
4000.00
6000.00
8000.00
10000.00
12000.00
0 10 20 30 40 50 60
imports
GDP
exports
Period(time)
Value(millionrupees)

ADF test for log observations data
ADF
t-statistic
p-value 1%
t-statistic
5%
t-statistic
10%
t-statistic
Log(exports) -1.3750 0.5876 -3.5600 -2.9172 -2.5966
Log(imports) -1.1536 0.6877 -3.5600 -2.9862 -2.5966
Log(GDP) 0.9850 0.9958 -3.5713 -2.9224 -2.5992
Now ADF test for log observations for first difference
ADF
t-statistic
p-value 1%
t-statistic
5%
t-statistic
10%
t-statistic
DEX -10.9318 0.0001 -3.5626 -2.9187 -2.5972
DIM -6.7163 0.0000 -3.5626 -2.9187 -2.5972
DGD -3.0837 0.0344 -3.5713 -2.9224 -2.5992
Null hypothesis: unit root
Alt hypothesis: no unit root
Here the p-values in table one are greater than 0.05 so we do not reject null
hypothesis. But in second table the values are less than 0.05 so we reject null
hypothesis rather we accept alternate hypothesis.
We can say that the values in the second table are stationary.

Vector Autoregression Estimates
Standard errors in ( ) & t-statistics in [ ]
DEX DGD DIM
DEX(-1) -0.24764 0.046402 0.149475
-0.14236 -0.02998 -0.12771
[-1.73957] [ 1.54775] [ 1.17044]
DEX(-2) -0.28438 0.092871 0.107426
-0.12201 -0.0257 -0.10946
[-2.33078] [ 3.61426] [ 0.98145]
DGD(-1) -0.86132 -0.40804 1.18672
-0.72579 -0.15285 -0.6511
[-1.18675] [-2.66955] [ 1.82263]
DGD(-2) -0.18309 0.247695 0.973316
-0.71951 -0.15153 -0.64547
[-0.25447] [ 1.63464] [ 1.50791]
DIM(-1) 0.381693 -0.0661 0.015283
-0.16911 -0.03561 -0.15171
[ 2.25709] [-1.85598] [ 0.10074]
DIM(-2) 0.151114 -0.02301 -0.01481

-0.17207 -0.03624 -0.15437
[ 0.87819] [-0.63485] [-0.09593]
C -0.04319 -0.01803 0.015421
-0.02728 -0.00575 -0.02447
[-1.58301] [-3.13831] [ 0.63008]
R-squared 0.28529 0.430934 0.095303
Adj. R-squared 0.18783 0.353334 -0.028065
Sum sq. resids 0.388237 0.017219 0.312449
S.E. equation 0.093934 0.019782 0.084268
F-statistic 2.927244 5.553285 0.77251
Log likelihood 52.02222 131.4699 57.56018
Akaike AIC -1.765577 -4.881171 -1.982752
Schwarz SC -1.500425 -4.616019 -1.7176
Mean dependent -0.026399 -0.016505 -0.027005
S.D. dependent 0.104231 0.0246 0.08311
Determinant resid covariance (dof adj.) 2.11E-08
Determinant resid covariance 1.35E-08
Log likelihood 244.9096
Akaike information criterion -8.780768
Schwarz criterion -7.985310

From the above table we can say that
If DEX is dependent variable then DEX(-1), DEX(-2), DIM(-1),
DIM(2), DGD(-1), DGD(-2), C are the independent variables.
If DGD is dependent variable then DEX(-1), DEX(-2), DIM(-1),
DIM(-2), DGD(-1), DGD(-2), C are the independent variables.
If DIM is dependent variable then DEX(-1), DEX(-2), DIM(-1),
DIM(-2), DGD(-1), DGD(-2), C are the independent variables.
t-statistic =
From this table we cannot explain the influence of independent variable on the
dependent variable, so we again do the estimation by ordinary least square method.
If we do estimation by ordinary least square method then we get the p-values
by which we can explain about the influence of independent variables on the
dependent variable.

Estimation Method: Least Squares
Coefficient Std. Error t-Statistic Prob.
C(1)
-0.247639 0.142357 -1.739565 0.0843
C(2)
-0.284383 0.122012 -2.330782 0.0213
C(3)
-0.861324 0.725786 -1.186747 0.2375
C(4)
-0.183094 0.719511 -0.25447 0.7995
C(5)
0.381693 0.169109 2.257086 0.0256
C(6)
0.151114 0.172074 0.878189 0.3814
C(7)
-0.043187 0.027281 -1.583014 0.1158
C(8)
0.046402 0.02998 1.547749 0.1241
C(9)
0.092871 0.025696 3.614262 0.0004
C(10)
-0.408041 0.15285 -2.669546 0.0085
C(11)
0.247695 0.151529 1.63464 0.1045
C(12)
-0.066099 0.035614 -1.855979 0.0657
C(13)
-0.023006 0.036239 -0.634849 0.5266
C(14)
-0.018031 0.005745 -3.138315 0.0021

C(15)
0.149475 0.127708 1.170436 0.2439
C(16)
0.107426 0.109457 0.981449 0.3282
C(17)
1.18672 0.651103 1.82263 0.0706
C(18)
0.973316 0.645473 1.50791 0.134
C(19)
0.015283 0.151707 0.100739 0.9199
C(20)
-0.014809 0.154368 -0.09593 0.9237
C(21)
0.015421 0.024474 0.630081 0.5297
Determinant residual covariance 1.35E-08
Equation: DEX = C(1)*DEX(-1) + C(2)*DEX(-2) + C(3)*DGD(-1) +
C(4)*DGD(-2) + C(5)*DIM(-1) + C(6)*DIM(-2) + C(7)
Observations: 51
R-squared 0.285290 Mean dependent var -0.026399
Adjusted R-squared 0.187830 S.D. dependent var 0.104231
S.E. of regression 0.093934 Sum squared resid 0.388237
Durbin-Watson stat 2.462939

Equation: DGD = C(8)*DEX(-1) + C(9)*DEX(-2) + C(10)*DGD(-1) +
C(11)*DGD(-2) + C(12)*DIM(-1) + C(13)*DIM(-2) + C(14)
Observations: 51
Adjusted R-squared 0.353334 S.D. dependent var 0.024600
Equation: DIM = C(15)*DEX(-1) + C(16)*DEX(-2) + C(17)*DGD(-1) +
C(18)*DGD(-2) + C(19)*DIM(-1) + C(20)*DIM(-2) + C(21)
Observations: 51
Adjusted R-squared -0.028065 S.D. dependent var 0.083110

After doing the estimation method through least square we get all the coefficients
of the model
C(1) – C(7) are the coefficients of model DEX
C(8) – C(14) are the coefficients of model GDP
C(15) – C(21) are the coefficients of model DIM
So we got the p values from the ordinary least square method.
If the p value is greater than 5% then we accept the null hypothesis. If p-value is
less than 5% we reject null hypothesis and accept alternate hypothesis.
H0 : there is no influence on the dependent variable
H1 : there is influence on the dependent variable
For example:
If we want to see the influence of DEX(-2) on the DGD for this equation
Equation: DGD = C(8)*DEX(-1) + C(9)*DEX(-2) + C(10)*DGD(-1) +
C(11)*DGD(-2) + C(12)*DIM(-1) + C(13)*DIM(-2) + C(14)
The coefficient of DEX(-2) is c(9).the p-value for C(9) is 0.0004 < 0.05 so we
reject null hypothesis and accept alternate hypothesis, that means there is
significant influence of DEX(-2) on DGD.

In the similar way we can find the combined influence of the independent variables
on the dependent variable. For this we use wald test.
Example:
If we want to see the combined influence of DIM(-1), DIM(-2) on the DEX then
H0 : C(5)=C(6)=0 which means there is no influence
H1 : there is influence on the dependent variable (DEX)
Wald Test:
Test Statistic Value Df Probability
Chi-square 6.0727 2 0.0480
Null Hypothesis: C(5)=C(6)=0
Here the p-value is 0.048 so we reject null hypothesis, that means there is
combined influence of DIM(-1) and DIM(-2) on the DEX.

6 EMPIRICAL RESULTS
Impulse response function of VAR is to analysis dynamic effects of the system
when the model received the impulse. It is a shock given to the variable. The shock
can be of different forms 1)War in the country 2)the owner of the company died
3)sudden fall in the stock market etc..
In our VAR model, we have three variables. We can work the response between
these variables in order to display the response function clearer we plot the graphs
given below.
Table1:Response of DEX
Period DEX DGD DIM
1 0.093934 0 0
2 -0.01748 -0.00957 0.031484
3 -0.02454 0.017695 0.009845
4 0.011775 0.000737 -0.0128
5 0.009468 -0.00056 -0.00111
6 -0.0048 -0.00324 0.004796
7 -0.00272 0.003535 0.000495
8 0.001906 -0.00117 -0.00199
9 0.001007 0.000541 0.000187
10 -0.00085 -0.00092 0.000579

This is the matrix when there is a one unit change in the variable DEX.
The values in the 10th
period in the table are so close to zero, that means there
won’t be any affect in the future due to the change in one variable(DEX).
After 10 periods the variables are not affected by any other shocks.
Figure 3 Graph of table1 Response of DEX on DEX, DGD, and DIM
Blue line is the impulse response of DEX to DEX, when the impulse is DEX, the
DEX value in the first period is so high and there is sudden decrease in the second
period and became zero as the periods progressed.
Red line is the impulse response of DGD to DEX, there is no significant effect on
DGD as the periods increased it got flattered towards the zero line.

Green line is the impulse response of DIM to DEX, there is no significant change
in DIM the values just fluctuated on the zero line, as the periods progressed it
almost became zero.
Table2:Response of DGD
Period DEX DGD DIM
1 -0.00634 0.018738 0
2 0.006892 -0.00878 -0.00545
3 0.003079 0.005898 0.001705
4 -0.00309 -0.0056 0.001502
5 -0.00025 0.005284 9.84E-05
6 0.000863 -0.00356 -0.00112
7 0.000328 0.002482 0.000536
8 -0.00055 -0.00208 1.59E-05
9 9.26E-05 0.00175 8.70E-05
10 6.58E-05 -0.00131 -0.00023
This is the matrix when there is a unit change in variable DGD
won’t be any affect in the future due to the change in one variable(DGD).

Figure 3 Graph of table2 Response of DGD on DGD, DIM, DEX
Impulse response of DEX to DGD, the values started from negative and increased
in the very next period and as periods increased the values almost became equal to
zero.
Impulse response of DGD to DGD, it started from a high value in the starting and
fluctuated around zero line and in the tenth period also it didn’t become zero.
Impulse response of DIM to DGD, in the starting it just fluctuated around the zero
line and in the last it almost reached zero.

Table3:Response of DIM
Period DEX DGD DIM
1 0.000829 0.017223 0.082485
2 0.006526 0.0225 0.001261
3 0.00957 0.006472 -0.00297
4 0.004865 -0.00017 0.001506
5 -0.00162 0.001013 0.002652
6 -0.00072 0.000838 5.56E-05
7 0.001095 0.000369 -0.00067
8 0.000335 -0.00035 0.000125
9 -0.00035 0.000146 0.000308
10 -7.77E-05 1.90E-05 -6.45E-05
This is the matrix when there is a unit change in variable DIM
won’t be any affect in the future due to the change in one variable(DIM).

Figure 4 Graph of table3 Response of DIM on DIM, DGD, and DEX
When the impulse response of DEX to DIM, at starting only the values are close to
zero line and values are just around zero line till tenth period.
When the impulse response of DGD to DIM, it started from positive value and
after 4 periods it came close to zero and from there it has no significant change in
the values of DGD till the tenth period.
When the impulse response of DIM to DIM, at the start it has high positive value
and just after 1 period there is a sudden fall and after that it doesn’t have any
significant changes, at last it settled at zero line.

7 CONCLUSION
VAR is used to find the linear interdependencies among multiple time series, here
we have three variables GDP, imports and exports which are also interdependent,
we forecasted Y variable using its own past and present values and also past values
of other variables. Y variable is GDP and other variables are imports and exports.
We forecasted the GDP using least square method as this method gives us the p-
values which are important to analyze the influence of independent variables on
the dependent variable.
Next if we come to the impulse response function, which is the shock to the error
term in the equation resulting to the change in either exports or imports in turn
affecting the final GDP. This impulse response functions are explained by using
the graph, in which we tried to give shock to every variable and observing how
other variables have been affected by this process, the fluctuations in the graphs
shows that by giving the shock to a particular variable changes other variables too.
In the first graph we can see that the impulse response is DEX and when the shock
is applied the change in the imports (DIM) and GDP (DGD) shows us the
interdependency of the variables. In the second graph impulse response is DGD
and when shock is given then there is a change in imports (DIM) and exports
(DEX). In the similar way when shock is given to DIM then there is a change in
exports (DEX) and GDP (DGD).

8 REFERENCES
[1] Introduction to time series by Brockwell and Davis
[2] VAR a user guide by Craig S.Hakkio and CharlesS.Morris
[3]Faculty.chicagobooth.edu_jeffrey.russell_teaching_timeseries_handouts_notes3
[4] Vector auto regression by James H.Stock and Mark W.Watson
[5] Hamilton, James D. 1994 Time Series Analysis. Princeton University Press:
Princeton.
[6] http://www.youtube.com/watch?v=J6BTw2Ff95A (VAR estimation and uses)
by Ralf Becker
[7] http://www.statistics.du.se/essays/D10_Xinzhou_lucao.pdf
[8] Abdulnasser Hatemi-J(2004). Multivariate tests for autocorrelation in the stable
and unstable VAR models. Economic Modelling 21,p 85-115

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