AN ARIMA APPROACH OF MODELLING FOREIGN CURRENCIES IN …apjor.com/ijrp/downloads/0901201715.pdf ·...

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



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



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.



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



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



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:



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



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



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 ***



6 d) RESULT OF ARIMA(2,1,2) process for EURO



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)


with p-value

0.00006

Ljung-Box Q' = 10.1308,


LM = 16.6249



except Normality

ARIMA(2,1,2)



Ljung-Box Q' = 6.66031,


LM = 13.3283



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 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



7 a) RESULT OF AR(1,1,0) FOR POUND



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




const

0.152179 0.167580 0.9081 0.3638

theta_1

0.189574

0.0732506 2.588 0.0097 ***



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)



Ljung-Box Q' = 6.57022,


LM = 11.2115



except Normality

MA(0,1,1)



Ljung-Box Q' = 6.66948,


LM = 11.5929



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



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



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 ***



8b) MA(0,1,1) process for US DOLLAR




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.



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

http://www.ccsenet.org/ijef

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