TESTING FOR LONG MEMORY IN THE LKR/USD EXCHANGE RATE: EVIDENCE FROM SRI LANKA

The question of whether exchange rate markets are efficient or not, is directly related to whether or not long memory is present in the exchange rate changes. Therefore, this paper explores the nature of the data generating processes of foreign exchange rate LKR against the US Dollar (USD), (LKR/USD) by examining the long memory properties of the LKR/USD return series based on econophysics models. In this study, autocorrelation function and spectral density function are used as visual test to inspect long memory of exchange rate returns. Further, parametric-ARFIMA model, Semi-parametric test proposed by Geweke and Porter-Hudak, Local Whittle estimator and non-parametric (R/S) test are employed as inferential tests to examine the long memory properties of the LKR/USD using daily data for the period from 2005-01-03 to 2016-12-30. Kernel density of LKR/USD return series show peak and fat tail postures. Visual inspection and inferential results reveal strong evidence of long memory property in the daily LKR/USD exchange rate return. It indicates that pricing by the market participants is not efficient. The results of this study have policy implications for traders and investors in designing and implementing trading strategies. It can also be helpful in predicting expected future return. Thus, the results of this study should be useful to regulators, practitioners and investors.


Introduction
In a globalized world, today, exchange rate plays a prominent role in international trade. The behavior of foreign exchange rates is of great importance to international investors. The nature of exchange rate dynamics is important for traders and investors in foreign currency markets. Understanding of long range dependence (long memory) of exchange rate return dynamics can be helpful in estimating expected return hence in designing, and implementing trading strategies. Hence, the study of the long memory property of exchange rate return is important for market participants. Long memory of exchange rate is a topic that has not received its due attention from academics in Sri Lanka.
The aim of this study is to examine long memory in the LKR/USD exchange rate in two steps. First, we employ traditional unit root tests. Second, we apply fractionally integrated ARMA models that are more flexible long memory models and selected some long memory tests.
This article is organized as follows: Section 2 briefly reviews the literature. Section 3 defines thelong memory model. Section 4 describes the data and methodology of the study. Section 5 discusses the empirical results obtained from various econometric techniques. The final section presents the conclusions.

Literature Review
There exist many empirical studies that test for the presence of long memory in the financial and economic time series. For example, Soofi, Wang and Zhang (2006), Kumar and Maheswaran (2015), Alptekin (2006). Granger and Joyeux (1980), Hosking (1981) proposed the autoregressive fractionally integrated moving average (ARFIMA) model to study long memory processes. Mandelbrot (1972) used the R/S analysis to study long memory. Peters (1991) found evidence of long memory for using R/S approach to study the long memory of daily exchange rate data of US dollar, Japanese yen, GBPound, Euro, and Singapore dollar. Cheung (1993) first studied time series properties of five major nominal exchange rates series using ARFIMA model that provides a direct and convenient frame work to study both short and long memory behavior. Cheung found the statistical evidence for long memory using various estimation techniques (R/S, LO R/S, GPH,). Corazza and Malliaris (2002) found evidence that exchange rate return follows a fractional Brownian motion (long memory) by using Hurst exponent method.
De Boef and Granato (1997) reviewed that the data are long memory processes but do not have unit roots, especially in the range 0<d<1. Even though it does not contain a unit root, it does have long memory, whereby shocks to the series persist for at least 12 months.
However, there are no studies inves -tigating the memory properties of Sri Lankan exchange rates using fractional integration technique. So, this study will be a new attempt to study about LKR/USD exchange rate dynamics based on recent econometric time series analysis in Sri Lanka. A comprehensive understanding of time series and statistical properties of LKR/USD exchange rate in Sri Lanka might provide useful implications for the direction of future research and effective exchange rate and monetary, and trade policies. Therefore, this study would contribute significantly to the existing knowledge.

Long-Memory Time Series Model
A covariance stationary time series is said to exhibit long memory if it satisfies the following condition ) (k ρ is the autocorrelation at lag k. This equation states that the sum of the absolute autocorrelation is infinite and non-summable (McLeod &Hipel, 1978). If the limit value is finite then the process possesses short memory.

data and variables
Data used in this study are daily exchange rate series of LKR/USD. The sample covers from 3 rd January, 2005 to 30 th December 2016. The variable used in this study is LKR/USD which is collected from the website of Central Bank of Sri Lanka. Total number of observations is 2896. The daily changes of the exchange rate is measured by the return series which is defined as below t r represents logged returns at day t., Pt represents closing exchange prices at day t. The Hodrick-Prescott (HP) filter is used to extract the trend component of the series.

Analytical methods
This study uses various econometric techniques to get robust results as each estimation techniques may have limits. Exploratory data analysis (EDA) and parametric, semi-parametric and nonparametric estimation procedures are used to identify and estimate memory parameter of LKR/USD exchange rate dynamics. These techniques can uncover the underlying structure of dynamic behavior of the LKR/USD exchange rates.

Exploratory data analysis
EDA; line graphs, autocorrelation function, kernel regression line, confidence ellipse , spectral density function are used to find novel and useful information that might otherwise remain unknown.

Unit root tests
The Augmented Dickey-Fuller (ADF) test , the Phillips and Perron (PP) test and the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test are implemented to understand the stationarity of the series (test for unit root). These tests have low power against fractional integration alternatives.

The Rescaled Range Statistic
Among various methods that measure long memory is the best known method,the "range over standard deviation statistic" or (R/S) to detect long range dependence of a series. It is defined as: is the range of the first n values, and S(n) is their standard deviation. The R/S statistic is the range of partial sums of deviations of a time series from its mean, rescaled by its standard deviation. This statistic was introduced by Mandelbrot and Wallis (1969) and Hurst (1951). R/S statistic provides the Hurst exponent ( ) , the series indicates persistent behavior/long memory, referred as fractional Brownian motion..
, then the series is called anti-persistent (mean-reverting).

Geweke& Porter-Hudak, "d"
Geweke& Porter-Hudak, (1983), (GPH), proposed a more robust estimation method based on spectral analysis,a seminonparametric procedure, to test for long memory in terms of fractionally integrated process. According to the GPH method, given the periodogram ) ( j I ω of variable y one can estimate fractional differencing parameter "d" by using the model of a log periodogramregression: The null hypothesis of the GPH is 0 : "there is no long memory" Granger and Joyeux (1980) and Hosking (1981) introduced the ARFIMA , parametric approach to test the long memory property in the LKR/USD return.

Autoregressive fractional integrated moving average (ARFIMA) model
The ARFIMA(p,d,q) process can be written as : ae lag polynomials of finite orders. . The parameter d is allowed to assume any real value. Long-memory processes are stationary processes whose autocorrelation functions decay more slowly than short-memory processes. The memory property of a process depends on the value of d. when , the process indicates a stationary process with a long memory. Hence, the existence of long memory can be determined by testing for the statistical significance of the sample differencing parameter d. Hosking (1981) showed that the autocorrelation, (.) ρ , of an ARFIMA processes is proportional of ]. It implies that the autocorrelations of the ARFIMA processes decay hyperbolically to zero as ∞ → k

Basic Features of LKR/USD exchange rate Dynamics in Sri Lanka
The basic features of LKR/USD series are examined by visual inspection using various graphs. Visual inspection indicates that the series seems to be non-stationary. Exchange rate changes defined as return  Return series seem to have 0 mean. But, when we extract the trend using Hodrick-Prescott (HP) filter, the trend of return is not linear and having little upward movement ( Figure 2).

Empirical behavior of Autocorrelation function structure of LKR/USD, Return i). Lag Plot -Scatter plot with confidence ellipse
The lagged scatter plot with confidence ellipse, nearest neighbour fit show the dependence nature of the LKR/USD dynamics in level as well as in the rate of   changes. Figure 3 exhibits two lag scatter plots with 95 confidence ellipse one for LKR/USD and the other one for HP-trend Return series. These figures suggest that LKR/USD and HP filtered trend return series are positively auto-correlated. Hence, the series is having long range dependence. Figure 5 shows the ACF of return decays slowly as a function of time lag. However, it seems that it does not explicitly exhibit long memoryat the beginning and not have a smooth behavior. Sample autocorrelations of return of LKR/ USD are statistically significant at a 5 percent level for long lags. In order to overcome this muddle, we employ HP filter and test for long memory.  Figure 6 shows the ACF of HP filtered return series that shows strong evidence of long memory. ACF decays very slowly with lags.

Figure 6 Autocorrelation function for HP filtered trend series of return of LKR/USD
These sample ACFs show that the impact of a shock t ε on exchange rate series does not diminish over time. The past shocks continue to play a significant role in determining the future exchange rate path.
The p value for (Ljung-Box test statistic) the joint significance of the correlation coefficients, indicates that they are significantly different from zero even for large lags. This implies the autocorrelation exist for longer time.

Visual Inspection of Spectral Density Function
In addition, spectral methods are useful to uncover key characteristics of economic time series for model building. Granger (1966) describes how the spectral shape of an economic variable concentrates spectral mass at low frequencies, declining smoothly as frequencies increases. The series y t displays long memory if its spectral density, f Y increases without limit as angular frequency tends to zero.   In order to validate the results of visual inspection, the inferential analysis is further performed using non-parametric, s e m i -p a r a m e t r i c a n d p a r a m e t r i c approaches.

Unit root test Results
The results of standard unit root tests ADF, PP, and KPSS show the LLKR/USD are non-stationary, I(1) and return series (first difference) is stationary, I(0). All these tests only count the integer order of integration of the series,I(1) or I(0). It has not identified the fractional integration.

Probability values are in ( ). Critical values are [ ]
It is interesting to note that LKR/USD return series does not have significant long memory estimates. However, HP filtered trend of return series shows the evidence of long memory.

Semi-Parametric GPH Estimate " d "
We compute the fractional integration parameter using algorithms built in GRETL. There are two methods we use here to estimate fractional parameter "d" (i) proposed by Geweke and Porter-Hudak method (d GPH, ) (ii) Local Whittle estimator (d LW ): both of these methods estimate "d" using a frequency domain maximum likelihood estimation. The results are given in Table 6. However, fractional integration can be analyzed by R/S, GPH, ARFIMA model, FIGARCH model.

The Rescaled Range Statistic -Hurst Estimator (H)
The results of the rescaled range analysis ( ).
In the case of return series, It has not shown any evidence of long memory due to noise. Hence, the noise is separated from the trend. HP filtered trend series werederived from return series to estimate memory parameter. R/S estimate for HP filtered series is 0.87 with probability 0.000.

ARFIMA Model
ARFIMA model is used to estimate long memory parameter using maximum likelihood estimation method. Table 7 shows Long memory parameter estimates for the selected ARFIMA models for the return series. In return-LKR/USDcases, estimates of long memory parameter dand AR parameters which are highly statistically significant at the 5 percent level. LM parameter estimates lie in the interval (0, 0.5), implying that the series exhibit long memory. Long memory indicates that shocks to LKR/USD may persist over a long period of time. These evidences imply predictability of future LKR/USD based on historical prices.
The impulse response function (IRF) (Campbell andMankiw, 1987, Watson 1986)of estimated ARFIMA model is one measure of the LM in exchange rate changes. The IRF measures the effect of a unit shock on the k-period -ahead exchange rate change. The estimated IRF shows (Figure 7) a positive shock to LKR/ USD return and has an immediate positive impact on LKR/USD with the impacts disappearing at a very hyperbolic rate after first step, for longer time lag period. The magnitude of the response is positive for longer periods. In this study, we estimated fractional difference parameter (LM) for exchange rate changes of LKR/USD in Sri Lanka using fractional integration methods. A battery of nonparametric, semiparametric and parametric tests were used to investigate long memory for LKR/USD return series. All these tests show that return series have long memory and they are fractionally integrated. Our results provide support for the hypothesis of longrange dependence in Sri Lankan exchange rates relative to the US dollar. Further, our results are evident against unit root but for long memory. It is noted that the fractional difference parameters are significantly different from 1 as well as from 0. The persistent behavior of LKR/USD leads to market inefficiency. Further, the findings of this study have policy implications for traders and foreign investors in designing and implementing trading strategies. It can also be helpful in predicting expected future return and volatility. Thus, the results of this study should be useful to regulators, practitioners and investors. Central bank interventions are thought to play an important role in explaining the dynamics of exchange rates. However, Central bank interventions and long memory in exchange rate dynamics are not considered in the study. Therefore, future empirical work in this area should attempt to investigate the Central bank interventions and long memory in exchange rate dynamics.