forecasting foreign exchange rates4

1.

Forecasting foreign exchange rates:

Random Walk Hypothesis, linearity and data

frequency

Christopher Bellgard Dr Peter Goldschmidt

[email protected] [email protected]

Department of Information Management and Marketing

The University of Western Australia

Perth, Western Australia

Abstract

This research paper discusses aspects of foreign exchange rate forecasting in terms of the

Random Walk Hypothesis (RWH). The following forecasting techniques were evaluated for

profitability: Random Walks, Exponential Smoothing, AutoRegressive Integrated Moving

Average and Artificial Neural Networks. Using recent Australian-US dollar data, the

research examined the implications, of data frequency and linearity, on the RWH.

The research intention is, consequently, to advance foreign exchange rate research,

particularly of the Australian-US dollar. The importance of such research is substantiated by

the phenomenal size of the foreign exchange rate market; the Bank of International

Settlements (1993) estimated that it exceeded US$1 trillion per day.

The paper discusses how the RWH was initially discounted through statistical insignificance.

However, statistical measures of accuracy do not always have a direct bearing on

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profitability (Lee, 1996). Consequently, all model types including random walks were

ultimately compared through a trading simulation. Being based on a position-taking rule, the

trading simulation was inherently non-linear.

In direct conflict with the RWH, the trading simulation was able to surpass a 30% annualised

return, within sample. This was achieved by capturing and capitalising upon intrinsic non-

linearities. That is, foreign exchange rates exhibit complex (non-linear) patterns that may be

profitably exploited.

This paper adds to an increasing body of evidence against the RWH. Evidence, to date,

includes the following:

FX differences are leptokurtic (Hsieh, 1988; Contingency Analysis, 1997).

ANNs, including hybrid ANNs (Tan, 1995), can yield a higher profit than linear

models (Refenes, 1992; Lee and Jhee, 1994; Zhang, 1994).

Technical analysis is widely used by many investors (Hawley et al., 1993; Mehta,

1995).

Keywords

Forecasting; Random Walk Hypothesis; linearity; artificial neural networks; foreign

exchange.

Introduction

This research paper considers the Random Walk Hypothesis (RWH) through the effect of

linearity and forecasts of different data frequencies. Linear and non-linear models were

3.

evaluated on various linear and non-linear criteria. In accordance with the aim, models were

ultimately evaluated through a position-taking trading simulation, which, incidentally,

introduced further non-linearity. The RWH is addressed through the consideration of non-

linear models.

Background

This paper is derived from Bellgard (1998). In this research, one null hypothesis was: Foreign

exchange rate data exhibits complex (non-linear) patterns that may be profitably exploited.

The research question was: Are artificial neural networks superior to simple linear modelling

techniques in forecasting foreign exchange rates, at different data frequencies? Accordingly,

the objective was to investigate various modelling techniques used for forecasting foreign

exchange (FX) rates. For example, Random Walks (RWs), Exponential Smoothing,

AutoRegressive Integrated Moving Average (ARIMA) and Artificial Neural Networks

(ANNs). The aim was to identify the most profitable model if any at each frequency. It

was expected that linearity and non-linearity would influence such identification. This would,

then, have a significant impact on the RWH.

Data set

For reasons of reliability and interest, the data set comprised half-hourly observations of the

United States Dollar and Australian Dollar (USD/AUD) FX rate during 1996. Forecasting of

this particular FX rate has, incidentally, received much less research attention than the two

most traded currencies (USD and Deutschmark) recent exceptions include Manzur (1993)

and Tan (1995).

4.

From the half-hourly data set, hourly and daily observations were extracted. The first

difference was then taken for each frequency; earlier studies have found that the direction of

the forecast is more important than the actual forecast in determining the profitability of a

model (Tsoi et al., 1993a; Tsoi et al., 1993b; Sinha and Tan, 1994).

Linearity

Linearity is the property of having one dimension (Oxford Dictionary, 1992). The term is also

used to express the concept that the model possesses the properties of additivity and

homogeneity (Hair et al., 1998) meaning that a change in one variable causes a proportional

change in another variable. It follows that non-linearity is the property of having more than

one dimension. Profitably trading on such intrinsic data properties is directly relevant to the

RWH.

Random walks

A RW is a time series whose period to period changes, or first differences, are stationary

(Newbold and Bos, 1994). A time series is stationary if its mean and variance are both

constant across time and its autocorrelation1 depends only on the lag. Hence, for a stationary

time series, mean, variance and autocorrelation are all independent of time (Everett, 1997).

Additionally, the proportional changes of RWs, in a short period of time, are normally

distributed (Hull, 1995). That is, the direction and size of changes are independently and

randomly chosen from the normal distribution. The best prediction is no change; this is

known as the naive forecast.

1 Correlation between a time series and the same series with a given lag.

5.

The Random Walk Hypothesis

The Efficient Markets Hypothesis (EMH)2 states that, in an efficient market, asset prices fully

reflect all available information about the asset; investors cannot consistently earn abnormal

returns (Peirson et al., 1995). The weak form considers only past information (i.e. historical

prices). It implies that prices follow a RW in which successive price changes have zero

correlation (Trippi and Lee, 1996). This subset of the EMH is known as the RWH (Peirson et

al., 1995). Furthermore, the EMH does not support technical analysis.

Due to market efficiency, FX rates are widely viewed to be best explained as RWs (Diebold

and Nason, 1990). In addition, research by Meese and Rogoff (1983) and Hogan (1986)

supports the superiority, of RWs, over certain estimated models, like ARIMA. The ARIMA

modelling approach (Box and Jenkins, 1970), however, is more general because it includes

RWs.

EMH and ANNs

The weak EMH is mitigated by bounded rationality arguments (Simon, 1955; Simon, 1982).

Such arguments hold that efficiency is restricted by human information processing which is

inherently limited. New processing technology (such as ANNs) provides profit opportunities

for the holder of that technology, effectively providing a form of insider information.

However, the EMH implies that increased availability, of the technology, will rapidly erode

its advantages until they disappear. Meanwhile, in view of the relative novelty of neural

networks, and the implications of bounded rationality, it is at least conceivable that previously

2 Fama (1970).

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undetected regularities exist in historical asset price data, and that such regularities may yet

persist. (White, 1993).

Preliminary evidence against RWH

The study of kurtosis provides preliminary evidence against the RWH for FX rates. Kurtosis

is a measure of the fatness of a probability distributions tails. It is measured relative to a

normal distribution with the same mean and standard deviation. A leptokurtic distributions

tails are fatter and a platykurtic distributions tails are thinner than those of a

corresponding normal distribution (Contingency Analysis, 1997). This is exemplified in

Figure 1, below.

Figure 1: Normal, Leptokurtic and Platykurtic Distributions

FX rate changes tend to be leptokurtic (Hsieh, 1988; Contingency Analysis, 1997). This

means that dramatic market moves occur with greater frequency than is predicted by the

normal distribution. That is, large changes (those at the tails) have a greater probability than

they would have under a normal distribution. The existence of leptokurtic FX rates provides

evidence against the RWH.

Normal

Leptokurtic

Platykurtic

7.

Normality (stationarity), of first differences, can be assessed quantifiably with statistical tests.

Statistical test for non-stationarity

The adjusted Box-Pierce Q-Statistic indicates the overall adequacy of the RW model

(Manzur, 1993; Newbold and Bos, 1994). It is derived as follows.

Let: n = number of observations (less lags)

M = number of autocorrelations

= min (n/2, 3 n)

K = number of parameters in model

r(k) = correlation at lag k

then: -

+= =

M

k knkr

nnQ1

2 )()2(

and: Q ~ c2M-K

Hence, Q has a chi-squared (c2) distribution with M-K degrees of freedom. The observed

value of Q is significant (and the data non-stationary) if it is less than the critical (tabulated)

chi-square value. At a 5% significance level, an insignificant observed value of Q indicates

(with at least 95% certainty) that the data does not follow a RW.

Corresponding to the aim, first differences were used as the basis of prediction. The predicted

direction of price movements was more relevant, to the position-taking nature of the trading

simulation, than the actual price prediction. For completeness, however, the Q-test was

applied to the prices and price changes of the data sets. The Q-test results are shown below in

Table 1.

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Data FrequencyHalf-Hourly Hourly Daily

Critical Value 443.4 320.0 74.5Prices

Observed Value 1173.1 555.9 79.6

Critical Value 443.4 320.0 74.5Price changes

Observed Value 6346.0 3025.1 220.2

Table 1: Box-Pierce Q-Statistic Results

The observed values of the price changes are substantially higher than those of the prices.

That is, price changes are more stationary than prices. More importantly, in all cases,

observed chi-square values exceed critical values, especially at higher frequencies. Hence,

1996 USD/AUD FX prices (and price changes) fail the Box-Pierce Q-Test. 1996 USD/AUD

FX prices did not follow a RW.

This finding is replicated by the normality test macro in SPSS (refer to Figures 2, 3 and 4).

The Kolmogorov-Smirnov Test is similar to the Box-Pierce Q-Test; the sig value is the

probability that price changes follow a normal distribution. A sig value of less than 5% means

that price changes are not normally distributed, and that prices do not follow a RW.

Incidentally, this is true for all three frequencies.

Visually, the histogram of normally distributed input should be very close to the normal

curve. In all cases, the histogram is substantially taller. This is consistent with the leptokurtic

FX finding of Hsieh (1988) and Contingency Analysis (1997).

9.

Half-Hourly USD/AUD FX Price Changes (UA30D)

Case Processing Summary

17567 100.0% 0 .0% 17567 100.0%UA30DN Percent N Percent N Percent

Valid Missing Total

Cases

Tests of Normality

.164 17567 .000UA30DStatistic df Sig.

Kolmogorov-Smirnova

Lilliefors Significance Correctiona.

UA30D

.0117.0102

.0086.0070

.0055.0039

.0023.0008

-.0008

-.0023

-.0039

-.0055

-.0070

-.0086

-.0102

-.0117

Histogram

Fre

quen

cy

10000

8000

6000

4000

2000

0

Std. Dev = .00

Mean = .0000

N = 17567.00

Normal Q-Q Plot of UA30D

Observed Value

.02.010.00-.01-.02

Exp

ecte

d N

orm

al

4

2

0

-2

-4

Detrended Normal Q-Q Plot of UA30D

Observed Value

.02.010.00-.01-.02

Dev

from

Nor

mal

20

10

0

-10

-20

Figure 2: Output for Normality Test of USD/AUD half-hourly price changes

10.

Hourly USD/AUD FX Price Changes (UA60D)



Valid Missing Total

Cases

Tests of Normality


Kolmogorov-Smirnova


UA60D

.0109.0084

.0059.0034

.0009-.0016

-.0041

-.0066

-.0091

-.0116

Histogram

Fre

quen

cy

5000

4000

3000

2000

1000

0

Std. Dev = .00

Mean = .0000

N = 8783.00


Observed Value

.02.010.00-.01-.02

Exp

ecte

d N

orm

al

4

2

0

-2

-4


Observed Value

.02.010.00-.01-.02

Dev

from

Nor

mal

20

10

0

-10

-20

Figure 3: Output for Normality Test of USD/AUD hourly price changes

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Daily USD/AUD FX Price Changes (UA366D)



Valid Missing Total

Cases

Tests of Normality


Kolmogorov-Smirnova


UA366D

.0112.0088

.0062.0037

.0012-.0013

-.0038

-.0063

-.0088

-.0112

-.0137

-.0162

-.0187

-.0213

Histogram

Fre

quen

cy

140

120

100

80

60

40

20

0

Std. Dev = .00

Mean = .0001

N = 365.00


Observed Value

.02.010.00-.01-.02-.03

Exp

ecte

d N

orm

al

3

2

1

0

-1

-2

-3


Observed Value

.02.010.00-.01-.02-.03

Dev

from

Nor

mal

2

1

0

-1

-2

-3

-4

-5

Figure 4: Output for Normality Test of USD/AUD daily price changes

12.

Normally distributed data should closely follow the straight line of the Normal Plot. It should

not exhibit a pattern in the Detrended Normal Plot. Clearly, the histogram and plots are

consistent with the Kolmogorov-Smirnov Normality Test in asserting that price changes are

not normally distributed. Furthermore, the S-shape Normal Plots suggest that high-frequency

FX prices are further away from a RW than low-frequency FX prices (the daily datas Normal

Plot is straighter).

Based on the above evidence, it is not surprising that RW models were statistically

insignificant at each frequency. However, given the non-linearity of the trading simulation,

RWs were not discarded from the comparison. The logical rationale for this was that

statistical insignificance may not have a bearing on profitability, when considering non-

linearities. This is because statistical measures of accuracy do not always have a direct

bearing on profitability (Lee, 1996).

Comparing models

At each frequency, six models were compared. These comprised:

three ARIMA models3 (statistically significant)

one Recurrent Neural Network (RNN)

one RW (statistically insignificant)

one Exponential Smoothing model (statistically insignificant).

Models were initially compared using theoretical (linear) measures such as root mean squared

error (RMSE), mean absolute error (MAE) and information coefficient (Tr). Tr is also known

3 Diagnosed with an ARIMA modelling algorithm explained in Bellgard and Goldschmidt (1999a).

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as Theils Coefficient of Inequality (1966). It measures performance relative to the naive

forecast of a RW (Refenes, 1995).

All ARIMA models ranked first. This reflects the statistical significance of the ARIMA

models and the linearity of the measures. RNNs were ranked close to RWs and exponential

smoothing. The linear measures did not account for the non-linearity of the RNN models.

The next type of comparison introduced a degree of non-linearity. It measured percentage of

correctly predicted direction changes. Individual results are not important, except to report

that:

There was a big difference in model rankings between frequencies.

RWs were always ranked last.

RNN rankings varied substantially.

Considering all of the preceding measures, the findings so far suggest that:

RWs could not better ARIMA or RNNs.

Linear and non-linear measures are not necessarily consistent.

This provided enough justification to continue with the trading simulation.

Trading simulation

To make it more realistic, the trading simulation incorporated filters and transaction costs

(both measured in FX points). Filters expanded the range of available models. A market

maker with low transaction costs requires 60% correct trades to run a profitable FX desk

(Grabbe, 1996). Hence, a model was deemed acceptable if it returned a net positive gain,

14.

subject to having winning trades at least 60% of the time. At each frequency, the three most

acceptable models were collected in Table 2, below.

Model FilterTotal

AnnualisedGain

%Win

1st ARIMA(2,0,2) 0.0010 15.3% 100.0

ARIMA(2,0,0) 0.0010 15.3% 100.0

ARIMA(0,0,2) 0.0010 15.3% 100.0

2nd RNN 0.0050 14.8% 75.0Hal

f-H

ourl

y

3rd ARIMA(2,0,0) 0.0005 1.9% 60.0

1st Random Walk 0.0050 4.9% 60.0

2nd (none)

Hou

rly

3rd (none)

1st RNN 0.0000 43.9% 62.5

2nd RNN 0.0005 38.5% 60.0

Dai

ly

3rd ARIMA(0,0,1) 0.0000 30.5% 100.0

Table 2: Three most acceptable models by frequency

The three daily models are also the three most acceptable models overall. For hourly data,

RW with 0.0005 filter was the only acceptable model.

Conclusion

This study conflicts with the RWH and, hence, with the weak form of the EMH. This

apparent discovery adds to the increasing body of evidence against the RWH. The pertinent

view, in economic literature, that exchange rates follow a random walk, has been dismissed

by recent empirical work. There is now strong evidence that exchange rate returns are not

independent of past changes. (Tenti, 1996). This body of evidence includes the following:

15.

FX differences are leptokurtic (Hsieh, 1988; Contingency Analysis, 1997).

ANNs, including hybrid ANNs (Tan, 1995), can yield a higher profit than linear

models (Refenes, 1992; Lee and Jhee, 1994; Zhang, 1994).

Technical analysis is widely used by many investors (Hawley et al., 1993; Mehta,

1995).

16.

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17.

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The data set was derived from HFDF96 which was purchased from Olsen and Associates,Switzerland. SPSS for Windows is a trademark of SPSS Inc.

forecasting foreign exchange rates4

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