Japanese yen-US dollar exchange rate

Introduction

This research paper looks into the Univariate Time Series analysis of the Japanese Yen-United States Dollar Exchange Rate Data (daily basis) during a period of 2000-2009 with 2477 observations. I will first test if the data is stationary, and also empirically test the random-walk hypothesis, and test for the presence of residuals.

The Japanese yen is the third most-traded currency in the forex market after the US dollar and the Euro. It is also widely used as a reserve currency after the US dollar, the euro, and pound sterling. In February 2007, The Economist estimated that the yen is 15% undervalued against the dollar.

Barbara Rossi(2005) in her research paper discusses that for some countries the Random Walk Hypothesis does not hold and that exchange rates are random walks. This raises the possibility that economic models that were previously rejected not because the fundamentals were completely unrelated to exchange rate fluctuations, but because the relationship was unstable over time and, thus, difficult to capture by Granger Causality tests or by forecast comparisons. Christopher J.Neely (2006) discusses in his paper, that to understand the effects of of foreign exchange intervention, it is important to understand the use of high-frequency data and event studies. Yuko Hashimoti,et al, (2008) in their research showed that using tick-by-tick data of the dollar-yen exchange rates, in the actual transaction platform, does have some predictable information on the direction of the next price movement, thus refuting random walk hypothesis. Meese and Rogoff(1983), found that in a simple random walk model, in which the forecasted value is the most recent realization, it outperforms various forecasting models, including those using economic fundamentals as predictors.

Economic Theory

Random Walk Hypothesis

The random walk hypothesis is built on the financial theory that financial market prices develop according to a random walk, and therefore their prices cannot be predicted. Economists believe that these prices are completely random, because of the efficiency of the market.

A time series is said to be stationary if it has a constant mean, a constant variance and a covariance which depends only on the time between lagged observations - the dependence between successive observations does not change over time.

Given a time series as Yt = a+ b Yt-1 + et

It can be rewritten as (1 - bL)Yt = a + et L is the lag operator

For a series to be stationarity, b must not be exceed unity in absolute value. Hence stationarity requires that -1<b<1.

However, many of the variables studied in financial markets are non-stationary. One such type of time series are called random walks, because they slowly wander upwards or downwards, but with no real pattern. The other non-stationary series, which shows a definite trend either upwards or downwards is called random walk with a drift.

Random Walk without Drift.

Suppose ut is a white noise error term with mean 0 and variance ?2. Then the series Yt is said to be a random walk if

Yt = Yt-1 + ut

In the random walk model, the value of Y at time t is equal to its value at time (t - 1) plus a random shock.

Random Walk with Drift. It can be represented by:

Yt = δ + Yt-1 + ut

where δ is known as the drift parameter.

The Random Walk Model (RWM) is a specific case of a general class of stochastic processes known as integrated processes. The RWM without drift is non-stationary, but its first difference, is stationary. Therefore, we call the RWM without drift integrated of order 1, denoted as I(1).

If a time series Yt is stationary to begin with (i.e., it does not require any differencing), it is said to be integrated of order zero, denoted by

Yt ~I(0).

Random walks exhibit the Markov and Martingale properties.

Analysis

I first downloaded the data, needed for the analysis, and saved it as a .csv file. I then set my directory, and called the data into my directory. I gave the command to give me the column names. I input the acf and pacf commands.

I then defined my data parameters to plot my graph.

It is clear from the graph that a random walk exists. We have to test to see if it is a pure random walk, or a random walk with drift.

I then plotted the acf and pacf. Acf is autocorrelation function and pacf is partial autocorrelation function. In acf the autocorrelation coefficients starts at a very high value, and declines very slowly with the lags. Thus it seems this dataset is non-stationary.

The pacf correlogram also suggests the data is unstationary.

Next, I want to find out if the data has a unit root,(i,e) whether it is stationary. I used the augmented Dickey-Fuller Test (ADF).

To select the lag lengths, either I have to minimize the AIC, or select a large number of lags and delete insignificant lags sequentially. I first performed the test with 10 lags.

Null hypothesis: Unit root is present(implying non-stationarity)

The tau-statistic of 0.1321 is greater than the Critical values at levels of 1%, 5% and 10%, so we do not reject the null hypothesis and a unit root is present

The tau-statistic of -2.3922 is greater than the Critical values at levels of 1%, 5% and 10%.

The phi3 value of 2.8657 is less than the Critical values at levels of 1%, 5% and 10%.

Both the results of the tau-statistic, and phi3 value conclude, that we cannot reject the null hypothesis, and unit root is present.

The phi2 value of 1.9463 is less than the Critical values at levels of 1%, 5% and 10%, this shows that the series behaves like a pure random walk, without a linear trend or drift.

The tau-statistic of -2.3862 is greater than the Critical values at levels of 1%, 5% and 10%, so we cannot reject the null hypothesis and a unit root is present.

I next performed the test with 3 lags.

Null hypothesis: Unit root is present(implying non-stationarity)

The tau-statistic of 0.1542 is greater than the Critical values at levels of 1%, 5% and 10%, so we do not reject the null hypothesis and a unit root is present

The tau-statistic of -2.3187 is greater than the Critical values at levels of 1%, 5% and 10%.

The phi3 value of 2.6942 is less than the Critical values at levels of 1%, 5% and 10%.

Both the results of the tau-statistic, and phi3 value conclude, that we do not reject the null hypothesis, and unit root is present.

The phi2 value of 1.8359 is less than the Critical values at levels of 1%, 5% and 10%, this shows that the series behaves like a pure random walk, without a linear trend or drift.

The tau-statistic of -2.3128 is greater than the Critical values at levels of 1%, 5% and 10%, so we do not reject the null hypothesis and a unit root is present

Therefore, both the results at lag 10, and lag 3 are similar, and confirm a unit root is present, (i.e) the data is not stationary. To make the data stationary, we will take the first difference of the time series (I1), and see if it becomes stationary. This will confirm, that only 1 unit root is present.

If it still does not become stationary, then more than 1 unit root is present. If it is not stationary at I(1), I will have to differentiate it by two times (i.e) I(2).

I now test the differentiated time series with the ADF Test to check for non-stationarity with 3 lags.

Null hypothesis: Unit root is present(implying non-stationarity)

The tau-statistic of -28.721 is lower than the Critical values at levels of 1%, 5% and 10%, so we reject the null hypothesis and a unit root is not present.

The tau-statistic of -28.715 is lower than the Critical values at levels of 1%, 5% and 10%.

The phi3 value of 412.2747 is greater than the Critical values at levels of 1%, 5% and 10%.

Both the results of the tau-statistic, and phi3 value conclude, that we reject the null hypothesis, and unit root is not present.

The phi2 value of 274.8503 is greater than the Critical values at levels of 1%, 5% and 10%, this shows that the time series does not behave like a pure random walk, and has a linear trend or drift.

The tau-statistic of -28.7191 is lower than the Critical values at levels of 1%, 5% and 10%, so we reject the null hypothesis and a unit root is not present.

So from the ADF test on the differentiated time series I(!), it can be concluded that the data is stationary, and it has only 1 unit root.

Another unit root test which can be performed is the Phillips-Perron Test, which I have not carried out here.

I have plotted the graph of the differentiated stationary time-series.

Now , I can go ahead and estimate random walk models, to confirm whether the time series has pure random walk or random walk with drift. Please note that, the ADF test confirmed a random walk with drift or trend.

For this, first I will generate the lags, and combine the data.

Then I estimate the random walk model(without the intercept).

Null hypothesis: Random Walk is present

Computed p-value 0.07724

Critical value(1%) 0.01

Critical value(5%) 0.05

Critical value(10%) 0.10

The computed p-value is greater than the critical value at 1% and 5% significance levels, higher p- values compared to their critical values are insignificant, so we do not reject the null hypothesis at 1% and 5% significance levels(pure random walk is present). The computed p- value is smaller than the critical value at the 10% significance level, so this lower p-value is more significant than its critical value, so we reject the null hypothesis at 10% significance level (pure random walk is not present).

Now I estimate the random walk model with drift

Null hypothesis: Random Walk with drift is present

Computed p-value 0.07686

Critical value(1%) 0.01

Critical value(5%) 0.05

Critical value(10%) 0.10

The computed p-value is greater than the critical value at 1% and 5% significance levels, higher p- values compared to their critical values are insignificant, so we do not reject the null hypothesis at 1% and 5% significance levels(random walk with drift is present). The computed p- value is smaller than the critical value at the 10% significance level, so this lower p-value is more significant than its critical value, so we reject the null hypothesis at 10% significance level (random walk with drift is not present).

Next I perform, the BDS Test, to confirm if the time-series is iid (independent identically distributed), or the residuals are iid.

Null: Residuals are iid

Alternative: Alternative non-linear structure to be modelled

BDS Test Residuals for Random Walk Model

Since p-values are higher than the critical values corresponding to 1%, 5% and 10% significance levels, we cannot reject the null of the presence of iid residuals. So the linear random walk model is suitable to fit the univariate time-series data.

Null: Residuals are iid

Alternative: Alternative non-linear structure to be modelled

BDS Test Residuals for Random Walk Model with Drift

Since p-values are higher than the critical values corresponding to 1%, 5% and 10% significance levels, we cannot reject the null of the presence of iid residuals. So the linear random walk model is suitable to fit the univariate time-series data.

Conclusion:

From the empirical analysis of the time-series data of Japanese Yen-US Dollar Exchange Rates, my conclusions are:

  1. After the initial data-analysis, the time-series data was found to be stationary.
  2. After the data was differenced one time, to the order (I1), it became stationary.
  3. The acf and pacf, showed a graphical interpretation of non-stationarity.
  4. The ADF test carried out, confirmed the presence of a unit root, on the raw data, and the existence of 1 unit root on the differenced data. It also confirmed the presence of a random walk with drift or trend.
  5. In the random walk model (without intercept), The computed p-value is greater than the critical value at 1% and 5% significance levels, higher p- values compared to their critical values are insignificant, so we do not reject the null hypothesis at 1% and 5% significance levels(pure random walk is present). The computed p- value is smaller than the critical value at the 10% significance level, so this lower p-value is more significant than its critical value, so we reject the null hypothesis at 10% significance level (pure random walk is not present).
  6. In the random walk model(with drift), the computed p-value is greater than the critical value at 1% and 5% significance levels, higher p- values compared to their critical values are insignificant, so we do not reject the null hypothesis at 1% and 5% significance levels(random walk with drift is present). The computed p- value is smaller than the critical value at the 10% significance level, so this lower p-value is more significant than its critical value, so we reject the null hypothesis at 10% significance level (random walk with drift is not present).
  7. In the BDS test for residuals in the random walk model(without intercept) and the random walk model with drift, both show that since p-values are higher than the critical values corresponding to 1%, 5% and 10% significance levels, we cannot reject the null of the presence of iid residuals. So both the linear models are suitable to fit the univariate time-series data.

References:

  • Cowpertwait, P.S.P, Metcalfe, A.V. (2009), Introductory Time Series with R, Springer Publications
  • Gujarati,D.N, (2003),Basic Econometrics, 4th Edition, MacGraw-Hill
  • Kirchgässner, G., Jürgen,Wolters (2007),Introduction to Modern Time Series Analysis, Springer Publications
  • Macdonald,J, Braun, J(2007), Data Analysis and Graphics Using R: An Example-Based Approach, 2nd Edition, Cambridge University Press
  • Tsay, R.S (2005),Analysis of Financial Time Series, 2nd Edition, John Wiley & Sons Inc.
  • Watsham, T.J, Parramore K. (1997), Quantitative Methods in Finance, 1st Edition, Gray Publishing
  • An Analysis of Recent Studies of the Effect of Foreign Exchange Intervention
  • (Christopher J. Neely)(December 2005), International Business & Economics Research Journal - December 2006 Volume 5, Number 12
  • Meese, Richard A. and Rogoff, Kenneth. "Empirical Exchange Rate Models of the Seventies: Do They Fit Out of Sample?" Journal of International Economics, February 1983, 14(1), pp. 3-24.
  • Sharma,A (2009), Quantitative Methods in Finance, Lecture 3, 4,and 5, Course Notes, University of Bradford, School of Management

Websites:

  • Triennial Central Bank Survey(December 2007), Foreign exchange and derivatives market activity in 2007 http://www.bis.org/publ/rpfxf07t.pdf [Accessed 14/12/2009]
  • The cheap yen is dangerous (February 2007) http://www.economist.com/finance/displaystory.cfm?story_id=8679006 [Accessed 14/12/2009]
  • On the Japanese Yen-US Dollar Exchange Rate: A Structural Econometric Model Based on Real Interest Differentials (April 1997) http://ideas.repec.org/p/cpr/ceprdp/1639.html [Accessed 14/12/2009]
  • Intraday Yen/Dollar Exchange Rate Movements:News or Noise? ( March 1991) http://www.nber.org/papers/w2703Intraday Yen/Dollar Exchange Rate Movements: News or Noise? [Accessed 14/12/2009]
  • Why testing for random walk (Unit root test of stationary)? www.hkbu.edu.hk/~billhung/econ3670/lecture/3670note02.doc [Accessed 15/12/2009]
  • Are Exchange Rates Really Random Walks? Some Evidence Robust to Parameter Instability,Barbara Rossi(2005) http://econ.duke.edu/~brossi/emptvp.pdf [15/12/2009]

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