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International Journal of World Research, Vol: I Issue XXXVI, December 2016, Print ISSN: 2347-937X
www.apjor.com Page 79
AN ARIMA APPROACH OF MODELLING FOREIGN CURRENCIES IN INDIA.
GIRISH.B.N* DR.NAGARAJ.H
**
*Research Scholar **Associate Professor
St.Joseph’s Research Centre. Bangalore-01 St.Joseph’s Evening College.Bangalore-01
ABSTRACT
Cointegration is a technique for testing the relationship between non-stationary time series variables. If two or more series are
themselves non-stationary, but a linear combination of them is stationary, then the series are said to be cointegration.
In the backdrop of free float currency exchange rate, this study focuses at examining the cointegration of foreign currencies in India and
to fit a suitable ARIMA model for the purpose of estimation for the chosen currency series under the study.
For this purpose Econometric techniques such as AUGMENTED DICKEY-FULLER Test(ADF Test) of stationarity, JOHANSEN’S TEST
OF C OINTEGRATION and a host of ARIMA processes were employed. ARIMA models were further probed by subjecting them to Model
Adequacy Diagnosis.
Results thus obtained showed that there is no cointegration among foreign currencies in India .
Key words: Foreign currencies, stationarity, cointegration, autoregressive process, ACF and PCF
INTRODUCTION
Foreign exchange is the conversion of one country's currency into that of another. In a free economy, a country's currency is valued
according to factors of supply and demand. In other words, a currency's value can be pegged to another country's currency, such as the
U.S. dollar, or even to a basket of currencies. A country's currency value also may be fixed by the country's government or any other
specialized agency such as RBI. However, most countries float their currencies freely against those of other countries, which keep them
in constant fluctuation and allow the countries to evaluate their currency’s true value
In Indian scenario until 1973, the Indian rupee followed a fixed exchange rate regime wherein the rupee was pegged to the pound
sterling. With the breakdown of the Bretton Woods system in the early 1970s, India switched over to a system of managed exchange
rates. During this period, the nominal exchange rate was the operating variable to achieve the intermediate target of a medium–term
equilibrium path of the real effective exchange rate. REER fell 2consistently between 1980-81 and 1992-93 from 104.48 to 57.08. In
early 1990s, India was faced with a severe balance of payment crisis due to the significant rise in oil prices, the suspension of remittances
from the Gulf region and several other exogenous developments. Amongst the several measures taken to tide over the crisis, was a
devaluation of the rupee in July 1991 to maintain the competitiveness of Indian exports
Liberalization has radically changed India’s foreign exchange sector. Since 1991, the rigid four-decade old, fixed exchange rate system
replete with severe import and foreign exchange controls and a thriving black market is being replaced with a less regulated, ―market
driven‖ arrangement. While the rupee is still far from being ―fully floating‖ (many studies indicate that the effective pegging is no less
International Journal of World Research, Vol: I Issue XXXVI, December 2016, Print ISSN: 2347-937X
www.apjor.com Page 80
marked after the reforms than before), the nature of intervention and range of independence tolerated have both undergone significant
changes. With an over-abundance of foreign exchange reserves, imports are no longer viewed with fear and skepticism. The Reserve
Bank of India and its allies now intervene occasionally in the foreign exchange markets not always to support the rupee but often to avoid
appreciation in its value
In this backdrop, this study aims at studying the behavior of few selected Foreign currencies that are expressed in terms of rupee
1) OBJECTIVES OF THE STUDY
o To verify whether any cointegration exist among Foreign Currencies in India
o To formulate a suitable VAR model of cointegration of foreign currencies in India
o To test whether the series under study are Auto regressive integrated moving averages and their respective orders
o To propose a suitable AR or ARIMA model that reflects the characteristics of the given series
2) REVIEW OF LITERATURE
The review of the past studies shows that the presence or absence of cointegration can throw meaningful insight into the working of
Foreign Currencies. Though extensive research has been carried out in the backdrop of market driven exchange rate regime, the author
could not find sufficient work on modeling of foreign exchange series. Few of the research articles reviewed for the study are listed
below:
P, Prabheesh K.; D, Malathy et al.(2007) In their article ―Demand for Foreign Exchange Reserves in India: A Cointegration Approach‖,
used cointegration and vector error correction approach, and estimated India's demand for foreign exchange reserves over the period
1983-2005. Their results establish that the ratio imports to GDP, the ratio of broad money to GDP, exchange rate flexibility and interest
rate differential determine India's long run reserves demand function. Their empirical results show that reserve accumulation in India is
highly sensitive to capital account vulnerability and less sensitive to its opportunity cost. The speed of adjustment coefficient of vector
error correction model suggests that Reserve Bank of India has to engage in more active reserve management practices.
Kanchan Datta (2014) In their paper an attempt had been taken to enquire the relationship between exchange rate and trade balance in
India, by taking 36 currency trade based effective exchange rate both nominal and real.ADF Tests and Cointegration tests were conducted
and the study showed that increase of trade balance of our country is one of the important reasons for depreciating our currency
Padhan(2011) analyzed the determinants and stability of money demand functions, as per new definitions of monetary aggregates,.
Quarterly Data from 1996Q2 to 2009Q2, for various monetary aggregates, interest rates,exchange rates, stock prices and GDP were
considered. The cointegration tests, error correction mechanism, Granger causality and CUSUM tests had been applied for empirical
analysis. The estimated results disclosed the existence long-run and short-run relationship among the variables. Unidirectional Granger
causality was found from GDP and Stock Prices to monetary, new monetary as well as liquidity aggregates. Also similar result repeated
from interest rates to money demand functions. The CUSUM and CUSUMQ tests supported the existence of stability of each money
demand functions. All the three variables, except exchange rate, affect the money demand of both types of specification
3) METHODOLOGY OF THE STUDY
The present study is focused at studying the cointegration of Foreign Currencies and their respective modeling. For this purpose three
foreign currencies are selected i…e EURO,POUND STERLING and US DOLLARS and their average monthly exchange rates as
recorded in Indian National Rupees and announced by Reserve Bank of India are considered.
The average monthly rates are collected for a period of fifteen years starting from January 2000 till Oct 2016.
Econometric Tools such as Unit root Tests for stationarity and Johansen test of cointegration were conducted to analyze the data.
Modeling was done by running various ARIMA models at differenced levels. Empirical calculations and formulations have been
obtained by using GRETL software version 1.9.92
International Journal of World Research, Vol: I Issue XXXVI, December 2016, Print ISSN: 2347-937X
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4) DATA ANALYSIS AND PRESENTATION OF RESULTS
5 a) TEST OF COINTEGRATION
Two variables are said to be cointegrated when a linear combination of the two variables is stationary implying that there is a long term
relationship existing between them. Lack of cointegration suggests that no such relationship exists.
The co-integration test represents the gesticulation of long run equilibrium relationship between two variables say yt and xt let both are
integrated at one, that is yt ~ I(1) and xt ~ I(1). Then yt and xt are said to be cointegrated if there exist a β such that yt - β xt is I (0).This is
denoted by saying yt and xt are CI (1,1).that is yt and xt are cointegrated. Different types of co-integration techniques are available for the
time series analysis. These tests include the Engle and Granger test (1987), Stock and Watson procedure (1988) and Johansen’s method
(1988).
The most popular system method is the Johansen (or Johansen and Juselius, JJ)method, based on canonical correlations (Johansen 1988;
Johansen and Juselius 1990), that provides two likelihood ratio (LR) tests. The first, trace test, tests the null hypothesis that there are at
most r (0 ≤r ≤n) cointegrating vectors, or equivalently, n–r unit roots. The second, maximum eigenvalue test, tests the null hypothesis that
there are r cointegrating vectors against the alternative of r+1 cointegrating vectors. Johansen and Juselius recommend the second test as
better. Reimers (1992) argues through a Monte Carlo study of the Johansen LR test that the test statistic be corrected for the number of
estimated parameters to obtain satisfactory size properties in small samples. The correction is by replacing T by T–np in the test statistic,
where T is the number of observations, n is the number of variables and p is the lag length of the VA R .(Pillai-2001)
Therefore as a first step it is important to check the stationarity of the given time series variables at level series and to check their order
differences
5 b )TEST OF STATIONARITY
In order to test for the existence of unit roots, and to determine the degree of differencing necessary to induce stationarity, we have
applied the Augmented Dickey –Fuller test (ADF Test)
Given an observed time series Dickey and Fuller consider three differential-form autoregressive equations to detect the
presence of a unit root:
t is the time index,
α is an intercept constant called a drift,
β is the coefficient on a time trend,
γ is the coefficient presenting process root, i.e. the focus of testing,
p is the lag order of the first-differences autoregressive process,
et is an independent identically distributed error/ residual term.
International Journal of World Research, Vol: I Issue XXXVI, December 2016, Print ISSN: 2347-937X
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The difference between the three equations concerns the presence of the deterministic elements α (a drift term) and βt (a linear time
trend). The focus of testing is whether the coefficient γ equals to zero, what means that the original process has a unit
root; hence, the null hypothesis of γ = 0 (random walk process) is tested against the alternative hypothesis γ < 0 of stationarity.
The following are the ADF Test results for the chosen variables
Table 5b.1: ADF Test at Level variables
MODEL
CURRENCIES
Test without constant
model: (1-L)y = (a-1)*y(-1) + ...
+ e
Test with constant
model: (1-L)y = b0 + (a-1)*y(-1)
+ ... + e
Test with constant and trend
model: (1-L)y = b0 + b1*t +
(a-1)*y(-1) + ... + e
EURO
tau_nc(1)0.860151
p-value 0.8955
tau_c(1) =-1.1898
p-value 0.6812
tau_ct(1) =3.02202
p-value 0.126
POUND
tau_nc(1) =0.800
p-value 0.8852
tau_c(1) =0.49974
p-value 0.889
tau_ct(1) =-1.1599
p-value 0.9173
US DOLLARS
tau_nc(1) =1.0401
p-value 0.9223
tau_c(1) = -0.1170
p-value 0.9459
tau_ct(1) = -0.5576
p-value 0.9809
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Interpretation of results
In all the variables at their level order the γ( tau) values are not significant against their table values, which is further confirmed by their
corresponding p –values which has led us to conclude that Null Hypothesis that γ=0 or H0=1 cannot be rejected
In other words the given currency series have unit root and hence are
Non-Stationary in nature
Table 5b.2 ADF Test at First Differences of Variables
MODEL
CURRENCIES
Test without constant
model: (1-L)y = (a-1)*y(-1) +
... + e
Test with constant
model: (1-L)y = b0 + (a-
1)*y(-1) + ... + e
Test with constant and trend
model: (1-L)y = b0 + b1*t +
(a-1)*y(-1) + ... + e
EURO
tau_nc(1) = -4.088
p-value
=4.482e-005
tau_c(1) = -8.1566
p-value
= 1.728e-013
: tau_ct(1) = -8.14
p-value
= 6.379e-013
POUND
tau_nc(1) = -8.66
p-value
=7.928e-016
tau_c(1) = -8.7208
p-value
=3.456e-015
: tau_ct(1) = -8.71
p-value
=6.824e-015
US DOLLARS
tau_nc(1) = -4.286
p-value
=1.938e-005
tau_c(1) = -4.4140
p-value
=0.0001
tau_ct(1) = -4.5660
p-value =0.001125
International Journal of World Research, Vol: I Issue XXXVI, December 2016, Print ISSN: 2347-937X
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Interpretation of results
In all the variables at their First order differences the γ( tau) values are significant against their table values, which is further confirmed
by their corresponding p –values that are highly significance which has led us to conclude that Null Hypothesis that γ =0 or H0=1 is
rejected and alternative hypothesis that γ<0 or H1<1 is accepted
In other worde the given currency series at their First order differences do not have unit root and hence are
Stationary in nature
Now we know that the given variables have First order differences, The next step is to apply the multivariate cointegration test of
Johansen (1988, 1991) and Johansen’s-Juselius (1990,1992), estimated through maximum likelihood estimation procedure. Two tests
statistics such as λ trace and λ maximum eigen value is used to determine the number of cointegration vector. For n variable cases if at
least one(r=1) cointegrating vector is present, it is sufficient to conclude that the variables are cointegrated. The number of cointegrating
vector is estimated through VAR model for which it is necessary to specify the number of lag length in the autoregressive process. We
have started with 1 lag and maximum of 8 is taken in the process. The lag length of 2 is chosen based on Akaike Information Criteria,
Schwarz Bayesian Criteria and log likelihood ratio tests, which is theoretically and practically justified. The following table shows the
test result obtained
TABLE 5a.1: JOHANSEN’S TEST OF COINTEGRATION RESULT
Rank Eigenvalue Trace test p-value Lmax test p-value
0 0.048454 15.729 [0.7363] 9.0891 [0.8222]
1 0.018707 6.6403 [0.6251] 3.4559 [0.9025]
2 0.017251 3.1844 [0.0743] 3.1844 [0.0743]
5a)Interpretation of results
Both the Trace test and Eigenvalue Test indicate that the null hypothesis cannot be rejected that thereare no r cointegrating vectors and
the same has been confirmed by their respective p-values .
Therefore we can conclude that the foreign currencies in India are not cointegrated.
5) MODELLING OF FOREIGN CURRENY SERIES BY USING ARIMA MODELS
The next focus of the study is to analyze the auto regressive nature of the chosen currency series by applying ARIMA models
The acronym ARIMA stands for Auto-Regressive Integrated Moving Average. Lags of the stationarized series in the forecasting
equation are called "autoregressive" terms, lags of the forecast errors are called "moving average" terms, and a time series which needs to
be differenced to be made stationary is said to be an "integrated" version of a stationary series. Random-walk and random-trend models,
autoregressive models, and exponential smoothing models are all special cases of ARIMA models.
A nonseasonal ARIMA model is classified as an "ARIMA(p,d,q)" model, where:
International Journal of World Research, Vol: I Issue XXXVI, December 2016, Print ISSN: 2347-937X
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p is the number of autoregressive terms,
d is the number of nonseasonal differences needed for stationarity, and
q is the number of lagged forecast errors in the prediction equation.
The forecasting equation is constructed as follows. First, let y denote the dth
difference of Y, which means:
If d=0: yt = Yt
If d=1: yt = Yt - Yt-1
If d=2: yt = (Yt - Yt-1) - (Yt-1 - Yt-2) = Yt - 2Yt-1 + Yt-2
In terms of y, the general forecasting equation is:
ŷt = μ + ϕ1 yt-1 +…+ ϕp yt-p - θ1et-1 -…- θqet-q
To identify the appropriate ARIMA model for Y, i…e AR(1), AR(2), …, and MA(1), MA(2), … etc..we begin by determining the order
of differencing (d) needing to stationarize the series by examining ACF and PACF functions of each currency series with the help of
correlogram .
After identifying the probable models the Akaike Information Criteria (AIC)and Schwartz Bayesian Criteria (SBC/BIC) are used to select
that ARIMA(p,d,q) model for which the AIC and BIC are minimum.
6 a) ARIMA MODEL FITTING FOR EURO CURRENCY
ACF and PACF of EURO
International Journal of World Research, Vol: I Issue XXXVI, December 2016, Print ISSN: 2347-937X
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The EURO correlogram clearly suggest that the EURO variable has unit root and PACF graph reveals that there is one significant spike
in lag 1 of the variable .
6 b) RESULT OF AR(1,1,0) FOR EURO
Model 1: ARIMA, using observations 2000:02-2015:05 (T = 184)
Estimated using Kalman filter (exact ML)
Coefficient
std. error z-value p-value
const
0.148959
0.139936 1.064 0.2871
phi_1
0.216300
0.0729031 2.967 0.0030 ***
6 c) RESULT OF MA(0,1,1) process for EURO
Model 1: ARIMA, using observations 2000:02-2015:05 (T = 184)
Estimated using Kalman filter (exact ML)
coefficient std. error z-value p-value
const
0.148305 0.131863 1.125 0.2607
theta_1
0.199560 0.0689058 2.896 0.0038 ***
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6 d) RESULT OF ARIMA(2,1,2) process for EURO
Model 2: ARIMA, using observations 2000:02-2015:05 (T = 184)
coefficient std. error z-value p-value
const
0.148848 0.114888 1.296 0.1951
phi_1
1.22012 0.0934824 13.05 6.20e-039 ***
phi_2
−0.918615 0.119093 −7.713 1.22e-014 ***
theta_1
−1.10871 0.160392 −6.912 4.76e-012 ***
theta_2
0.847351 0.157968 5.364 8.14e-08 ***
TABLE 6 d.1) MODAL ADEQUACY DIAGNOSIS OF EURO MODELS
Adequacy
Tests
Arima
process
Test of Normality Test of Auto covariance Test of ARCH Result
AR(1,1,0)
Chi-square(2) = 17.842
with p-value 0.00013
Ljung-Box Q' = 9.65022,
withp-value
= 0.5621
LM
= 15.2951
with p-value = 0.225695
Conditions fulfilled
except Normality
MA(0,1,1)
Chi-square(2) = 19.377
with p-value
0.00006
Ljung-Box Q' = 10.1308,
with p-value = 0.5187
LM = 16.6249
with p-value = 0.164259
Conditions fulfilled
except Normality
ARIMA(2,1,2)
Chi-square(2) = 11.773
with p-value 0.00278
Ljung-Box Q' = 6.66031,
with p-value = 0.5737
LM = 13.3283
with p-value = 0.345638
Conditions fulfilled
except Normality
Model Adequacy tests confirmed that the above mentioned ARIMA models are suitable for ARIMA Model fitting.
International Journal of World Research, Vol: I Issue XXXVI, December 2016, Print ISSN: 2347-937X
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Now we will select the best ARIMA model for EURO based on Akaike Information Criteria (AIC)and Schwartz Bayesian Criteria
(SBC/BIC)
Scores
Models
Akaike Information Criteria (AIC)
Schwartz Bayesian Criteria (SBC/BIC)
AR(1,1,0) 675.0920
684.7368
MA(0,1,1)
675.7801
685.4250
ARIMA(2,1,2)
676.8368
696.264
Both AIC and BIC scores are least in AR(110) Process. Therefore the best fit for EURO is AR(110) Model
6) ARIMA MODEL FITTING FOR POUND CURRENCY
ACF and PACF of POUND
The above correlogram clearly suggest that the POUND variable has unit root and PACF graph reveals that there is one significant spike
in lag 1 of the variable
International Journal of World Research, Vol: I Issue XXXVI, December 2016, Print ISSN: 2347-937X
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7 a) RESULT OF AR(1,1,0) FOR POUND
Model 3: ARIMA, using observations 2000:02-2015:05 (T = 184)
Estimated using Kalman filter (exact ML)
TABLE 7 a.1
Coefficient
std. error z-value p-value
const
0.152669
0.172733 0.8838 0.3768
phi_1
0.184718
0.0736260 2.509 0.0121 **
7 b) RESULT OF MA(0,1,1) process for POUND
Model 4: ARIMA, using observations 2000:02-2015:05 (T = 184)
Estimated using Kalman filter (exact ML)
coefficient std. error z-value p-value
const
0.152179 0.167580 0.9081 0.3638
theta_1
0.189574
0.0732506 2.588 0.0097 ***
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TABLE 7 C.1MODAL ADEQUACY DIAGNOSIS OF POUND MODELS
Adequacy
Tests
Arima
process
Test of Normality Test of Auto covariance Test of ARCH Result
AR(1,1,0)
Chi-square(2) = 25.273
with p-value 0.00000
Ljung-Box Q' = 6.57022,
with p-value = 0.8327
LM = 11.2115
with p-value = 0.510882
Conditions fulfilled
except Normality
MA(0,1,1)
Chi-square(2) = 27.250
with p-value 0.00000
Ljung-Box Q' = 6.66948,
with p-value = 0.8252
LM = 11.5929
with p-value = 0.4789
Conditions fulfilled
except Normality
Model Adequacy tests confirmed that the above mentioned ARIMA models are suitable for ARIMA Model fitting
Now we will select the best ARIMA model for POUND based on Akaike Information Criteria (AIC)and Schwartz Bayesian Criteria
(SBC/BIC)
Scores
Models
Akaike Information Criteria (AIC)
Schwartz Bayesian Criteria (SBC/BIC)
AR(1,1,0)
766.9821
776.6269
MA(0,1,1) 766.8070 776.4519
Both AIC and BIC scores are least in MA(011) Process. Therefore the best fit for POUND is MA(011) Model
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7) ARIMA MODEL FITTING FOR US DOLLAR CURRENCY
ACF AND PACF OF US DOLLARS
8a) RESULT OF AR(1,0,1) process for US DOLLAR
Model 1: ARIMA, using observations 2000:02-2015:05 (T = 184)
Estimated using Kalman filter (exact ML)
coefficient std. error z -value p-value
Const
0.112237 0.0896930 1.251 0.2108
phi_1 0.311683 0.0700155 4.452 8.52e-06 ***
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8b) MA(0,1,1) process for US DOLLAR
Model 2: ARIMA, using observations 2000:02-2015:05 (T = 184)
Estimated using Kalman filter (exact ML)
coefficient std. error z-value p-value
Const
0.111655 0.0819765 1.362 0.1732
theta_1
0.328532 0.0714742 4.597 4.30e-06 ***
8 b.1)MODAL ADEQUACY DIAGNOSIS OF US DOLLARS MODELS
Adequacy
Tests
Arima
process
Test of Normality Test of Auto
covariance Test of ARCH Result
AR(1,1,0)
Chi-square(2) =
20.4588
with p-value =
3.60937e-005
Chi-square(11) =
16.6131
p-value=0.1199
LM = 32.5813
with p-value =
0.00112479
None of the conditions
fulfilled
MA(0,1,1)
Chi-square(2) =
21.1129
with p-value = 2.6025e-
005
Chi-square(11) =
15.2435
p-value=
-0.171
LM = 28.9639
with p-value =
0.00398893
None of the conditions
fulfilled
All the models tested upto the second differences failed to fulfil model adequacy tests and no ARIMA model can be fitted for the US
DOLLARS series.
8) CONCLUSIONS
The study aimed at examining the cointegration of foreign currencies in India and attempted to construct suitable ARIMA models for the
chosen currency series.
Based on the results obtained by applying various econometric techniques, the following conclusions have been drawn.
1) There is no cointegration among Foreign Currencies in India which further signals that the Foreign Currency Markets in India
are moving towards Informational efficiency, which is required be further researched.
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2) AR(110) model and MA(011) model were best suited for forecasting EURO currency series and POUND currency series
respectively.
3) None of the ARIMA models fulfilled model adequacy tests at second differences and no model is fitted to the series.This shows
that further research is needed to understand the unique characteristic of US DOLLAR series
9) REFERENCES:
1. Kanchan Datta, ―Relationship between Currency Depreciation and Trade Balance in India- An Econometric Study.‖ Journal of
Finance and Economics, vol. 2, no. 3 (2014): 83-89. doi: 10.12691/jfe-2-3-5.
2. P, Prabheesh K, D, Malathy , R, Madhumati - : South Asian Journal of Management. Volume: 14. Issue Management.
Volume: 14. Issue: 2 Publication date: April-June 2007. Page number: 36+
3. Dickey, D. A. & Fuller, W. A. (1981). Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica,
49, 1057-1072.
4. Engel, R. F. & Granger, C. W. J. (1987). Co-integration & error-correction: representation, estimation, & testing. Econometrica,
Vol. 55, 251-276.
5. Johansen, S. & Juselius, K. (1990). Maximum likelihood estimation and inferences on cointegration—with applications to the
demand for money. Oxford Bulletin of Economics and Statistics, 52.169-210.
6. Purna Chandra Padhan: Stability of Demand for Money in India: Evidence from Monetary and Liquidity Aggregates
International Journal of Economics and Finance Vol. 3, No. 1; February 2011 www.ccsenet.org/ijef
7. N. Vijayamohanan Pillai : (2001)ELECTRICITY DEMAND ANALYSIS AND FORECASTING THE TRADITION IS
QUESTIONED Working paper 312 Centre for Development Studies
8. WEB SITES VISITED
http://www.investordictionary.com/definition
nse-india.com/content/press/feb2003c.pdf
ccsenet.org/journal/index.php/ijbm/article/download/8949/7921
ssrn.com/abstract=2551396