rmb exchange rate forecasting in the context of the financial crisis

Systems Engineering — Theory & PracticeVolume 29, Issue 12, December 2009Online English edition of the Chinese language journal

Cite this article as: SETP, 2009, 29(12): 53–64

RMB Exchange Rate Forecasting in the Context of the FinancialCrisisSUN Bo1, XIE Chi∗1,21.College of Business Management, Hunan University, Changsha 410082, China

2.Center of Finance and Investment Management, Hunan University, Changsha 410082, China

Abstract: This article offers an effective solution of forecasting the RMB exchange rate volatility during the financial crisis period.

Based on the test of nonlinearity structure in the exchange rate system via the method of surrogate data, the optimal lag periods for

each specific exchange rate series were computed by autocorrelation criterion (AC) approach, and then, the structure of multilayer

perceptrons (MLP) and recurrent neural networks (RNN) were applied to build the homogeneous artificial neural network (ANN) model.

The comparison of the forecast results of ANNs with different parameters shows that, according to the specific exchange rate series,

the forecast performance of ANN models with different freedom of degrees has obvious differences in different forecast periods. The

RNN model, which contains layer feedback process, has showed great ability to explain and forecast the RMB exchange rates volatility

behavior. The optimal forecasting model for each RMB exchange rate volatility series has been found and explained.

Key words: optimal lag periods; method of surrogate data; ANNs; RNN model

1 Introduction

The American subprime lending problem revealed pre-

liminary in the spring of 2006 and gradually caused multi-

concern from both the financial theory and the practice cir-

cle. In March 2007, the problem promoted as a financial

crisis which first broke out in the USA and spread quickly

to the whole world. Since 2008, the intensified sub-prime

crisis has coursed many subprime mortgage institutions met

with bankruptcy, large quantity of investment fund forced to

close, and the global stock market turbulence. On the 14th

September, the collapse of Lehman marked the outbreak of

the global financial tsunami. Affected by the crisis, Amer-

ica, Europe, Japan, and other major economies have suffered

obvious economic recession. As China’s important trading

partners, the main consumption market continuous downturn

and the hereby caused international market turbulence, have

not only severely impact China’s export trade, but also dis-

rupt the original step of RMB appreciation, which caused

frequent fluctuations on China’s foreign exchange market

and much more uncertainty in the RMB exchange rate be-

havior.

Under the condition of the economic globalization and

the floating rate, the uncertainty change of exchange rate

may easily cause currency crisis to an economic entity or

even a financial crisis with greater damage. In addition,

via abnormal fluctuations in the relative currency of those

closely related entities, a domestic financial crisis can spread

to the world at a high speed. In the theoretical study, the ex-

change rate volatility is usually regarded as one of the most

important indicators for the prediction of currency and fi-

nancial crisis. Girton and Roper[1] first proposed the index

of exchange market pressure (EMP) and took exchange rate

volatility as a sensitive indicator of currency crisis. By sum-

marizing several previous crises, Krugman et al.[2] suggested

that some macro- and micro-economic variables may make

sense in the crisis warning function, and they emphasized the

important role of exchange rate abnormal volatility for a cur-

rency crisis early warning. McKinnon[3] also supported that

exchange rate volatility has good effectiveness and sensitiv-

ity in response to the world economic fluctuation and global

financial risk. Since then, with the theoretical basis, a large

number of researches focus on the currency crisis and finan-

cial crisis prewarning issue from different aspects, in which

the exchange rate volatility has been paid more and more

attention to [4–8].

On the other hand, exchange rate forecasting is always

the focus of attention to theoreticians. Since recent evidence

has clearly demonstrated the existence of complex nonlin-

earities in exchange rates, more and more nonparametric

and nonlinear methods have been applied to solve the prob-

lem. The artificial neural network (ANN) is one of the most

widely concerned models in exchange rate prediction area.

Early studies showed that ANN model has higher predic-

tion capability[9−12]. However, other researchers suggested

that ANNs often suffer from over-fitting, which will affect

the forecasting results[13]. To overcome this, a group of re-

searches focused on several critical factors in ANN design

process, including the input variables, lag periods, network

structure, and others, which have significant impact on ANN

prediction efficiency[14−17]. Though the existing domestic

Received date: April 30, 2008

∗ Corresponding author: Tel: +86-731-8882-3890; E-mail: [email protected]

Foundation item: Supported by the Grants from the National Social Science Foundation of China (07AJL005) and the National Higher Education Fund for

Young Teachers Award(Education 2002[123])

Copyright c©2009, Systems Engineering Society of China. Published by Elsevier BV. All rights reserved.

SUN Bo et al./Systems Engineering — Theory & Practice, 2009, 29(12): 53–64

researches have provided some basis for ANN[18−20] ap-

plication in the RMB exchange rate forecasting, the above

design factors are still seldom mentioned. Both the homo-

geneous network and the recurrent neural networks (RNN),

which are supposed to be more capable of effective learning,

have rare been modeled for the RMB exchange rate forecast-

ing.

In consideration of the above, some more capable ANN

models are built based on the RMB exchange rate data char-

acteristics during the financial crisis period. ANN models

with different design factors, freedoms, and forecasting pe-

riods were compared and discussed to offer an optimal solu-

tion for the RMB exchange rate prediction. The conclusion

will not only help to measure the crisis level but also play a

part in crisis early warning. Meanwhile, it will benefit the

solution of the RMB exchange rate issue in a more active

manner.

The rest of this is organized as follows. Section 2 in-

vestigates the ANN forecasting model and the key factors

design. The experiment design is introduced in section 3. In

section 4, the specifications of empirical data are presented.

Finally, experimental findings and conclusions are discussed

in section 5.

2 ANN model for exchange rate forecasting

The complex nonlinear characteristics within the ex-

change rate system called for a nonlinear model to study the

underlying law.ANN technique, which has the capacity of

function approximating and self-adapting, seems to be more

capable for establishing nonlinear models. Recently, many

related studies have proved that ANN can offer better solu-

tion than other forecasting methods and improve the reliabil-

ity of prediction results.

2.1 Dynamic ANN modelTo forecast the exchange rate series, there are two main

broad types according to the input selection: the heteroge-

neous network and homogeneous network. Since the hetero-

geneous network may need more consideration of variable

selection and restrict to the data availability[21], the homoge-

neous network, which utilizes data directly from exchange

rate time series being forecast, seems to have better perfor-

mance in the short-term forecast[16]. Therefore, the homo-

geneous network is hereby adopted.

(1) Layer feedback network(RNN2)

In the building process of ANN model, the first impor-

tant is to choose the network architecture. Multilayer per-

ceptrons (MLP) is perhaps the most popular network in use.

However, connections in MLP are allowed to project only

forward, which limits its ability in describing the dynamic

exchange rate system. Tenti[15] suggested that the predic-

tive ability of ANN can be significantly improved by choos-

ing appropriate network structure. He hereby proposed three

types of RNN with different local feedback, among which

layer feedback network (RNN2) seems to be better in re-

flecting exchange rate dynamic system and offer accurate

prediction[15]. A typical three-layer RNN2 structure with an

input layer, a hidden layer, an output layer, and a recurrent

layer is shown in Figure 1. In the architectures, the hidden

layer is fed back into itself through an extra recurrent layer,

which then feeds forward to activate the output layer.

Figure 1. Typical three-layer RNN2 structure

Symbolically, the output of the jth neuron in the hidden

layer can be expressed as follows:

uj,t = ϕ(n∑i=0

wijxi,t +m∑j=0

δijuj,t−1) (1)

And the final output is

yt = G(m∑j=1

vjϕ(n∑i=0

wijxi,t+m∑j=0

δijuj,t−1)) = f(Xt, Ut−1, θ)

(2)

where ϕ and G are activation functions. xi,t is the ith input,

and Xt represents the ANN input variable at time t. uj,t is

the output of the jth hidden layer units at time t, and δj is

the weight vectors of the output of the jth hidden layer units

at time t-1. i = 0, 1, · · · , n and j = 1, 2, · · · ,m. θ is the

vector of parameters containing all w′s, v′s, and δ′s.

(2) Autocorrelation criterion approach

In the homogeneous ANN models, the input vari-

ables are determined by the lag structure of the exchange

rate data, which will significantly affect ANN’s forecasting

accuracy[17]. In order to optimize the input parameter for

ANN models, the autocorrelation criterion (AC) approach

proposed by Huang[17] is adopted to seek the optimal lag

structures.

Assuming that the time series is yt, first the auto-

correlation coefficient must be estimated as the functions:

rk =

∑Tt=k+1(yt − y)(yt−k − yt−k)/(T −K)∑T

t=1(yt − y)2/T(3)

where y is the sample mean of y, estimated from:

yt−k =T∑

t=k+1

yt−k/(T − k) (4)

In Eq. (3), rk describes the autocorrelation of yt and

yt−k. The closer neighboring data observations are to one

another, the higher degree of correlation they have.

Next suppose that a(1), a(2), · · · , a(m) are the deter-

mined lag periods of yt series. If we want to add a new lag

period a(m + 1), we wish yt−a(m+1) has as high degree of

correlation to yt as possible while as low degree of corre-

lation to yt−a(1), yt−a(2), · · · , yt−a(m) as possible. First set

the upper limit of lag period N and denote a(1) = 1 and


m = 1. Then, select the parameter k by calculating the

function:

a(m+ 1) = argkmax{(|rk|)/(m∑i=1

|rk−a(i)|)} (5)

where k = a(m) + 1, a(m) + 2, · · · , n.

Next, let m = m + 1. If a(m) is less than N , the cal-

culation will continue by Eq.(5). Otherwise, the process is

stopped, and we finally get the a(m) series represented opti-

mal lag periods of yt series.

2.2 Nonlinearity detection of surrogate data methodTesting for nonlinearity is the first step to determine

whether the nonlinear ANN model is valid in exchange

rate forecasting[22]. Theiler[23] proposed the surrogate data

method, which can effectively identify the nonlinear struc-

ture in time series. The method of surrogate data involves a

careful statement of the null hypothesis, which characterizes

a candidate linear process, the generation of surrogate data

sets that are similar to the original time series but consistent

with the null hypothesis, and the computation of a discrim-

inating statistic for the original and for each surrogate data

set. The rejection of the null hypothesis will demonstrate the

existence of nonlinearity in the series. The most typical null

hypothesis test is to detect whether the data are generated

from a Gaussian linear stochastic process[23]. Based on the

null hypothesis, the surrogate data are generated by phase

randomized Fourier-transform method[24].

First step is to make discrete Fourier transform on the

observation series x(n). Suppose that the series of N val-

ues taken at regular intervals of time t = t0, t1, · · · , tN−1 =0,Δt, · · · , (N − 1)Δt, and then:

X(f) = F{x(t)} =N−1∑n=0

x(tn)e2πifnΔt (6)

Second, a phase-randomized Fourier transform X(f) is

made by rotating the phase ϕ at each frequency f by an in-

dependent random variable ψ, which is chosen uniformly in

the range [0, 2π], and we obtain

X(f) = A(f)ei[ϕ(f)+ψ(f)] (7)

where A(f) stands for the amplitude.

Third, the surrogate time series x(f) is given by the in-

verse Fourier transform:

x(t) = F−1{X(f)eiϕ(f)} (8)

To test whether the hypothesis is true, the difference

between the surrogate sets and the original data should

be test based on the constructed statistics. According to

Theiler[23], there are large flexibilities in the discriminat-

ing statistics. However, related researches have proved that

BDS test performed well in detecting nonlinearity ignored

by other statistics[25]. In this article, the surrogate data and

BDS test are hereby combined to detect the nonlinear com-

position in the RMB exchange rate series. The formula of

BDS statistic is

W (N,m, ε) =

√N [C(N,m, ε)− C(N, 1, ε)]

σ(N,m, ε)(9)

wherem is the embedding dimension,C(N,m, ε) is the cor-

relation integral, and σ(N,m, ε) is an estimate of the asymp-

totic standard deviation of C(N,m, ε)− C(N, 1, ε)m.

Finally, we use the Sigma statistic to test against the

null hypothesis, that is

S =|q0 − qsurr|σsurr

(10)

where q0 is the BDS statistic for original data and qsurr is

the mean value of BDS statistics for the surrogate sets. And

σsurr stands for the standard deviation of the surrogate val-

ues. The null of linearity is rejected if S is superior to 1.96at the 95% level of confidence.

3 Experiment design and research data

The empirical study aims to provide an effective

method of nonlinear prediction for four kinds of exchange

rate volatility including the RMB against US dollar, euro,

Japanese yen, and Hong Kong dollar during the period of

financial crisis. First, the surrogate data method is put into

test the nonlinearity in the four RMB exchange rate volatil-

ity series. Then, based on the assumption proposed by Kuan

and Liu[10] that the forecasting capability of network models

relied on diversified series by empirical results, two types

of ANN models including both feed-forward and recurrent

networks will be modeled with different lag structure. And

the forecasting performance of each model for different peri-

ods will be compared in the in-sample fit and out-of-sample

forecast. At last, DM test is adopted to prove the predictive

accuracy of the optimal model.

3.1 Design specification of experimentThere are mainly four aspects for ANN structure de-

sign: First, the adopted ANNs include both feed-forward

MLP and recurrent RNN2 model with three layers. Theoret-

ically, it has been shown that three-layer ANN can uniformly

approximate any continuous function to any desired degree

of accuracy[26]. Second, the inputs for each homogeneous

ANN model will depend on the optimal lag structure of the

four exchange rate volatility. Third, in order to control the

complexity of the ANN models and prevent from the over-

fitting problem, neurons in the hidden layer will be set equal

to the input layer[16,27]. Fourth, the number of the output

neuron is set to “1” in the one-step-ahead forecasting pro-

cess.

Specifically, the empirical study is organized as fol-

lows: (1) The surrogate method is adopted to test the non-

linear structure in the four RMB exchange rate volatility se-

ries. (2) The optimal lag periods are sought to determine the

input nodes and hidden nodes of ANNs for each volatility

series by using the AC approach. (3) Two type ANNs and

the simple random walk model (SRW) are trained and built

for each exchange rate data sets. The in-sample fit capability

for models with different parameters is compared according

to the indicators including SSE, RMSE, and MAE(see Table

1). The smaller of the indicators value, the better of the pre-

diction performance. (4) The trained well models are used to

perform the out-of-sample forecasting. Compared with the

results of random walk model, the optimal ANN models


Table 1. Criteria definition

In-sampleSSE RMSE MAE∑N

t=1(yt − yt)

2(

1N

∑N

t=1(yt − yt)

2) 1

2 1N

∑N

t=1|yt − yt|

Out-of-sampleNMSE DA

1σ2N

∑N

t=1(yt − yt)

2 1N

∑ai, ai =

{1 if (yi+1 − yi)(yi+1 − yi) > 00 otherwise

are selected based on the NMSE value and the direction in-

dicator DA(see Table 1). Lower NMSE value and higher DA

will indicate better prediction capability. (5) The forecast-

ing ability of the optimal models chosen in step (4) will be

further tested by DM test.

3.2 Data descriptionIn the experiments, the RMB exchange rate time

series is consist of 733 daily closed prices of RMB

against US dollar(USD), Euro(EUR), Japanese Yen(JPY),

and Hongkong dollar(HKD), covering the period from

January 1st, 2006 to November 30th, 2008. The raw

data were obtained from Pacific Exchange Rate Ser-

vice(http://fx.sauder.ubc.ca/data.html). On the one hand,

this article aims to study the volatility behavior of the four

major currencies on China’s foreign exchange market dur-

ing the financial crisis. On the other hand, the sample pe-

riod covers three years, which is supposed to be sufficient

to obtain the suitable prediction accuracy via ANN model

according to Walczak[28].

The natural logarithmic first difference is computed to

represent the exchange rate volatility, that is, V = ln pt+1 −ln pt, which is supposed to better reflect the exchange rate

volatility dynamic nature[29]. The four volatility series are

marked as VUSD,VEUR,VJPY, and VHKD, respectively,

and the related description results are shown in Figure 2 and

Table 2. The four volatility series each contains 732 points

of data. The first 712 points are collected as the training set,

and the last 20 points are divided into four parts (within five

trading days each) to measure the forecasting performance

of ANNs at different time levels.

The results in Figure 2 and Table 2 have shown that,

with frequent fluctuation during the study periods, the four

exchange rate volatility behavior demonstrates visibly clus-

tering effects, which have indicated that the financial crisis

has spread to the foreign exchange market.

Among them, series of VEUR and VJPY have more

intensive fluctuations than the other two. Meanwhile, less

volatility contained in series VUSD and VHKD, which may

have something to do with China’s USD-pegging regime and

Hong Kong’s linked exchange rate system. In Table 2, the

evidence also shows that all the four series rejected the null

hypothesis of a normal distribution, and each distribution has

-.03

-.02

-.01

.00

.01

.02

.03

100 200 300 400 500 600 700

VUSD

-.03

-.02

-.01

.00

.01

.02

.03

.04

.05

100 200 300 400 500 600 700

VJPY

-.03

-.02

-.01

.00

.01

.02

.03

100 200 300 400 500 600 700

VHKD

-

-

-

.03

.02

.01

.00

.01

.02

.03

.04

100 200 300 400 500 600 700

VEUR

Figure 2. Four types of RMB exchange rate volatility time series

a tail to varying degrees, which means that larger change

may appear with higher probability.

4 Empirical results and analysis

4.1 Nonlinearity detection in exchange rate seriesAccording to the method of surrogate data, first we de-

fine the null hypothesis: the four kinds of RMB exchange

rates volatility series are generated from Gaussian linear

stochastic process.

Based on the null hypothesis, 39 groups of surrogate

data for each series are generated by the phase randomized

Fourier transform. Theiler and Prichard[24] proposed that the

smallest size B of surrogate data sets to guarantee the valid-

ity of null hypothesis is given by the probability of rejecting

the null, that is, Bmin = 2/(1−α)− 1. So we take B = 39surrogates to guarantee the hypothesis validity at the 95%

level of confidence. When to compute the BDS statistics,

the embedding dimensionm is assigned values from 2 to 10,

and ε is considered to be one times the standard deviation of

the time series σ according to the recommendations of Brock

et al.[30]. Thus, the BDS statistics results of the original data

and surrogate sets for each volatility series are gathered in

Figure 3 and Table 3. Note that ′′O′′and ′′×′′ separately rep-

resents the BDS statistic for the original data and surrogate

data.

Table 2. Description results for the four exchange rate volatility series

Mean Max Min Std.dev. Skewness Kurtosis J-B value P value

VUSD –0.000229 0.003703 –0.004613 0.001024 –0.181792 4.625316 84.60228 0.0000

VEUR –0.000150 0.030288 –0.022945 0.005702 0.042533 6.499228 373.6810 0.0000

VJPY 0.000041 0.044080 –0.027606 0.006934 0.684590 7.538939 685.5369 0.0000

VHKD –0.000228 0.004061 –0.005156 0.001080 –0.190350 4.345252 59.61635 0.0000


2 3 4 5 6 7 8 9 10-5

0

5

10

15

m

RMB/USD

2 3 4 5 6 7 8 9 10 m-5

0

5

10RMB/EUR

2 3 4 5 6 7 8 9 10 m-5

0

5

10

15

RMB/JPY

2 3 4 5 6 7 8 9 10 m-

5

0

5

10

15

20

25

RMB/HKD

Figure 3. BDS statistics for the RMB exchange rate volatility

original data and each surrogate sets.

From Figure 3, it can be seen that there are obvious dif-

ference in the BDS statistics between each RMB exchange

rate original data and its corresponding surrogate sets. In

addition, the differences increased with the embedding di-

mension m up to 10. Meanwhile, the results in Table 3 show

that the value of Sigma statistic S is superior to 1.96 at the

95% level of confidence, which is to say that the null of lin-

earity should be rejected. Thus, the existence of nonlinear-

ity is confirmed within the four adopted RMB exchange rate

volatility series, though with different degrees. Hereby it is

valid to apply the nonlinear ANN models to forecast the ex-

change rate.

4.2 Optimal lag periods selection for each series

As mentioned above, the AC approach is adopted to es-

timate the optimal lag periods for each RMB exchange rate

volatility series. Since many previous researches have sug-

gested that the input nodes of ANN model is from 1 up to

10, the upper limit of lag period is hereby set as 10. Via

Eviews5.0, the autocorrelation coefficients of the four series

VUSD,VEUR,VJPY, and VHKD are computed, gathered in

Table 4.

As it is shown in Table 4, there is no significant dif-

ference between the autocorrelation coefficients of the four

series. Next, the values were put into the AC procedure, and

the optimal lag structure for the four RMB exchange rate

volatility series is gathered in Table 5.

Table 3. Results of the BDS statistics and the Sigma test

m SUSD SEUR SJPY SHKD2 6.6214 0.633794 4.327662 10.01096

3 6.6606 2.227657 6.830795 9.065451

4 6.8744 2.679086 7.186518 10.29402

5 8.1945 3.307522 7.33122 11.57157

6 9.0953 3.694304 7.916709 13.74359

7 9.8495 4.113068 8.563857 16.1475

8 11.5062 4.855061 9.226222 20.22819

9 12.6024 5.606046 10.24258 23.96047

10 13.9851 6.579798 10.94325 28.74137

Table 4. Autocorrelation coefficient of the each series

VUSD VEUR VJPY VHKDr1 –0.04853 –0.02572 –0.08172 –0.07465

r2 –0.03646 0.06414 0.070313 –0.04035

r3 –0.00088 0.05316 0.00861 0.04774

r4 0.02338 0.08297 –0.02721 0.03785

r5 0.01922 –0.03140 –0.04847 0.01010

r6 –0.05481 –0.01340 –0.00734 –0.03237

r7 0.01409 0.03995 –0.00192 0.00724

r8 –0.00026 –0.04015 0.04605 0.02186

r9 0.04643 0.02351 0.02829 0.07720

r10 0.08168 –0.03612 0.02332 0.05708

According to Table 5, the long-term relationship between

data may exist in the series, which have reflected the fea-

ture of volatility clustering in the four RMB exchange rate

behaviors. Accordingly, the input and hidden layer units of

ANN models are set in accordance with each related optimal

lag period. That is, for VUSD, the optimal lag structure is

1, 9, 10; for VEUR, it is 1, 7, 8, 10; for VJPY, it is 1, 8,

10, and for VHKD, it is 1, 9, 10. The notations for various

models are as follows: MLP(n) or RNN2(n) indicate MLP

or RNN2 model, respectively, with n lagged dependent vari-

ables as input and also n hidden layer units.

4.3 In-sample fit test of ANN modelsAccording to the above methods, related functions of

the Neural Networks Tools in software Matlab 7.5 are ap-

plied to study the RMB exchange rate volatility behavior

and hereby offer both in-sample fit and out-of-sample fore-

cast. In the training process, all neural network models that

use the symmetric sigmoid logistic function and linear func-

tion, respectively, as the activation and transfer function, are

trained for 100 epochs with back-propagation algorithm over

the training set. In order to simplify the model structure, the

feedback parameter of lag period for RNN2 model is set to

be 1. Since the in-sample conclusions for the four volatility

series are basically the same, the results for series VUSD are

listed and analyzed here as an example. The criteria results

of ANNs and the benchmark random walk model(SRW) for

VUSD are gathered in Table 6 ( and the results for other three

series can be seen in the appendix).

First it is obvious from Table 6 that the values of SSE,

RMSE, and MAE for each ANNs are much less than those

for the SRW model with different degrees of freedom. And

the indicators for RNN2 models are always better than MLPs

with the same lag periods. Meanwhile, the three crite-

ria values of networks are declining when the parameters

of lag periods are increasing, i.e, the related indicators for

RNN2 models decreased from 0.74740, 1.0, and 0.75361 to

0.44821, 0.79905, and 0.61521 with the freedom degree pa-

rameters growing up to 10. Similar results can be obtained

from the other three series, and RNN2(10) models seems to

be the best among ANNs and SRW models in the in-sample

fit test for the series including VUSD, VEUR, VJPY,

Table 5. Optimal lag structure for each volatility series

VUSD VEUR VJPY VHKDa(1) 1 1 1 1

a(2) 9 7 8 9

a(3) 10 8 10 10

a(4) — 10 — —


Table 6. In-sample fit test on VUSD series

Models SSE×1000 RMSE×1000 MAE×1000 Training set

MLP(1) 0.74740 1.0 0.75361

1 RNN2(1) 0.74740 1.0 0.75361 712

SRW 1.6 1.5 1.1

MLP(9) 0.50058 0.84384 0.65239

9 RNN2(9) 0.49335 0.83773 0.64628 712

SRW 1.6 1.5 1.1

MLP(10) 0.48676 0.82877 0.63025

10 RNN2(10) 0.44821 0.79905 0.61521 712

SRW 1.6 1.5 1.1

and VHKD. According to the empirical results, there are

three main conclusions obtained as follows: First, the non-

linear non-parameter methods perform better than the tra-

ditional linear model, and networks with dynamic structure

may have stronger ability in exchange rate behavior descrip-

tion. Second, for the adopted four RMB exchange rate his-

torical data, the lag structure of longer periods may contain

more useful predictive information. Third, with more hid-

den neurons, the learning ability of ANN models may be

enhanced for the nonlinear and complex relationship among

the exchange rate data.

4.4 Out-of-sample forecastingIn this step, the trained well networks are applied to

perform the out-of-sample forecasting for the latest 1 month

samples of the four volatility series in periods including 1

week, two weeks, three weeks, and four weeks. The pre-

dict performance compared by criteria NMSE and DA are

listed from Table 7 to Table 10. The values showed in the

four tables all have five significant figures reserved. The val-

ues in boldface and underlined represent the computed min-

imal ones. It should be noted that the DA values for SRW

model are marked as ′′−′′, for the DA indicator will be al-

ways equals to 0 according to the model’s predictive princi-

ple, that is, to forecast tomorrow in accordance with today.

According to the NMSE values in Table 7, there are sig-

nificant differences among the adopted models with various

degrees of freedom in the four prediction periods. However,

for almost all the models, the NMSE values reached the max-

imum for the first week and the minimum for the two weeks

period. The results indicate that the neural network mod-

els are more suitable to forecast the VUSD behavior in two

weeks period.

Compared with the results in Table 6, it can be found

that a good in-sample fit network is a poor guide for the out-

of-sample prediction. The optimal model in the in-sample

fit process RNN2(10) with related values of NMSE 2.0868,

1.8376, 1.6676, and 1.4413, performs even worse than the

SRW models with related values 1.9553, 1.2863, 1.7035, and

1.7638, which may suffer from the over-fitting problem. In

comparison, there is no significant trend within the direction

statistics of DA. However, ANNs with lower degrees per-

form better than those with higher degrees.

Generally, RNN2(1) can be identified as the optimal

model with the best performance according to the indicators

among the adopted models. This result not only confirms the

capability of the dynamic networks in capturing and fore-

casting the exchange rate volatility complex behavior com-

pared to the feed-forward networks and random walk model,

but also reflects the existence of the correlation within the

VUSD series data. Figure 4 also shows the prediction re-

sults of RNN2(1) model on the VUSD series, where ′′O′′

and ′′∗′′, respectively, represent the real and predicted val-

ues(similarly hereinafter).

As it is shown in Table 8, NMSE values of each ANN

model reach minimum in the period of two weeks. To some

degree, the over-fitting problems also appeared in models

with higher degrees, such as 8 and 10. Besides, with the free-

dom degrees growing, values of NMSE got a trend to first

decrease then increase, which reach the minimum with 7 lag

periods. Accordingly, RNN2(7) appears to be the best model

among all the feed-forward and recurrent neural networks

with the lowest value of NMSE and best of DA. In contrast

to the VUSD series, the results imply that there may be more

complexity in the VEUR behavior, which have shown the

properties of long memory. Intuitively, the forecasting

Table 7. Out-of-sample forecasting performance on VUSD series

Models First week Two weeks Three weeks Four weeks

NMSE DA NMSE DA NMSE DA NMSE DA

MLP(1) 1.1127 100% 1.0042 88.89% 1.0534 85.71% 1.0329 78.95%

1 RNN2(1) 1.1127 100% 1.0041 88.89% 1.0533 85.71% 1.0328 78.95%SRW 1.9553 — 1.2863 — 1.7035 — 1.7638 —

MLP(9) 1.6095 75% 1.3316 77.78% 1.2889 78.57% 1.1542 78.95%

9 RNN2(9) 1.9065 100% 1.1758 77.78% 1.3562 78.57% 1.3178 73.68%

SRW 1.9553 — 1.2863 — 1.7035 — 1.7638 —

MLP(10) 2.8554 75% 2.1968 77.78% 1.9547 78.57% 1.6654 78.95%

10 RNN2(10) 2.0868 75% 1.8376 77.78% 1.6676 78.57% 1.4413 73.68%

SRW 1.9553 — 1.2863 — 1.7035 — 1.7638 —


5 10 15 20-8

-6

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2

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8

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ty-3×10

Observations

ObservationsObservations

Observations

Figure 4. Out-of-sample prediction results for RNN2(1) model

on VUSD series.

results of RNN2(7) model on VEUR series are shown in Fig-

ure 5, which has proved that RNN2(7) performed better in

the periods of first week and two weeks.

In Table 9, the NMSE values for models on VJPYshowed an initial increase, followed by a decrease with the

periods changing from first week to four weeks, which is

quite different from series VUSD and VEUR. Generally,

the NMSE values indicated that all the adopted models per-

formed best in the first weeks, which reflects that it con-

tains much more uncertainty in the VJPY series. That is,

it is only suitable to make forecasting on VJPY at short hori-

zon. In addition, with the freedom increasing from 1 to 8 and

10, the NMSE values of ANNs are much larger than that of

SRW model, and DA values significantly reduced, even ap-

peared to be less than 50%. It not only proves that MLPs and

RNN2s have suffered the over-fitting problem with higher

degrees of freedom, but also reflects that there exists strong

correlation within the VJPY series data. That’s the reason

why the first-order neual network models can offer better

solution for the VJPY series prediction. As Figure 6 shows,

RNN2(1) is the best model for the RMB against Japanese

yen exchange rate series volatility in each predictive period.

Observations

Vo

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Vo

lati

lity

Observations

Vo

latl

ilit

yV

ola

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Observations

5 10 15 20

-0.2

-0.1

0

0.1

0.2

5 10 15

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0

0.1

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Observations

1 2 3 4 5 6 7 8 9 10-0.25

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-0.15

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0

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on VEUR series.

According to the NMSE values in Table 10, there is no

significant trend for each ANN model on the VHKD series

in the four different prediction periods. Compared with other

three series, the difference between the periods of first week

and four weeks is much less obvious, which have implied

that the fluctuations of the RMB against Hong Kong dollar

exchange rate express a more moderate manner. Specifically,

RNN2(9) is the optimal model that can offer prediction with

highest accuracy among the adopted models, which indicates

that the dynamic networks with nine inputs and hidden nodes

can provide better description for the VHKD complex non-

linear behavior. The forecasting results can also be seen in

Figure 7.

Generally, combining the results of in-sample or out-

of-sample testing, it can be obtained that the optimal fore-

casting models for the four kinds of RMB exchange rate

volatility VUSD, VEUR, VJPY, and VHKD are, respectively,

RNN2(1), RNN2(7), RNN2(1), and RNN2(9). The results

not only proved the conclusion proposed by Kuan and Liu[10]

that the forecasting capability of network models relied

Table 8. Out-of-sample forecasting performance on VEUR series



MLP(1) 1.0652 75% 1.0459 66.67% 1.0586 64.29% 1.0064 73.68%

1 RNN2(1) 1.0076 75% 1.0026 66.67% 1.0090 71.43% 1.0023 78.95%

SRW 2.1994 — 1.9167 — 2.0102 — 1.9232 —

MLP(7) 0.9448 100% 1.0402 71.43% 1.0290 73.68% 1.0046 78.95%

7 RNN2(7) 0.3996 100% 0.7259 88.89% 0.9400 85.71% 1.0122 84.21%

SRW 2.1994 — 1.9167 — 2.0102 — 1.9232 —

MLP(8) 1.6647 100% 2.3497 77.78% 2.5989 78.57% 2.2286 73.68%

8 RNN2(8) 1.7737 75% 1.6309 88.89% 2.5492 85.71% 1.8874 89.47%

SRW 2.1994 — 1.9167 — 2.0102 — 1.9232 —

MLP(10) 1.8084 75% 1.7063 77.78% 1.9908 78.57% 1.9437 73.68%

10 RNN2(10) 2.5946 100% 2.4757 77.78% 3.0087 78.57% 2.1015 78.95%

SRW 2.1994 — 1.9167 — 2.0102 — 1.9232 —


Table 9. Out-of-sample forecasting performance on VJPY series



MLP(1) 1.0327 75% 1.0713 66.67% 1.0779 71.43% 1.0494 73.68%

1 RNN2(1) 0.8957 75% 0.9833 66.67% 1.0307 71.43% 0.9897 73.68%SRW 1.9888 — 1.5987 — 1.7272 — 1.8677 —

MLP(8) 4.6910 75% 6.2925 33.33% 4.6656 50% 4.4033 57.89%

8 RNN2(8) 6.1446 100% 8.8142 55.56% 7.5617 64.29% 7.4309 52.63%

SRW 1.9888 — 1.5987 — 1.7272 — 1.8677 —

MLP(10) 4.6684 50% 5.5594 55.56% 5.4592 64.29% 4.9229 52.63%

10 RNN2(10) 2.1126 50% 2.3947 33.33% 2.1962 42.86% 2.1205 52.63%

SRW 1.9888 — 1.5987 — 1.7272 — 1.8677 —

Observations

Vola

tili

tyV

ola

tili

ty

1 2 3 4 5 6 7 8 9 10

Observations

5 10 15 20

-0.1

-0.05

0

0.05

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0.15

Observations5 10 15

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0

0.05

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Observations

Vola

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on VJPY series.

on different exchange rate series, but also indicated that the

nonlinear dynamic networks performed better than the feed-

forward networks and the traditional linear simple random

walk model in the prediction of the RMB exchange rate

volatility. In addition, it can also be confirmed that ANN

models perform quite differently in different forecasting pe-

riods.

4.5 Significant test for the forecasting modelsIn order to demonstrate the forecasting capability of the

four optimal models, further investigation is implemented

via the DM test proposed by Diebold and Mariano[31], to

Observations

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Vo

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Vola

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5 10 15 20-0.015

-0.01

-0.005

0

0.005

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0.015

Observations

5 10 15-0.015

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-0.005

0

0.005

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0.015

Observations

1 2 3 4 5 6 7 8 9 10-0.015

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0

0.005

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0.015

1 2 3 4 5-0.015

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0

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0.015


on VHKD series.

see whether there are significant differences between the op-

timal models and others with the same freedom. Accord-

ingly, a set of hypotheses for the four optimal RNN2 models

are stated. Specifically, for the VUSD series out-of-sample

forecasting, the optimal RNN2(1) model is assumed to have

no obvious superior than MLP(1) and SRW. Similarly, the

RNN2(7) model, RNN2(1) model, and the RNN2(9) model,

which performed best in the out-of-sample forecast, respec-

tively, for the VEUR, VJPY, and VHKD series, are assumed

to have no significant difference compared with other mod-

els.

Table 10. Out-of-sample forecasting performance on VHKD series



MLP(1) 1.1168 75% 1.0614 77.78% 1.1011 78.57% 1.1022 78.95%

1 RNN2(1) 1.1168 75% 1.0614 77.78% 1.1011 78.57% 1.1022 78.95%

SRW 1.8802 — 1.2989 — 1.6737 — 1.5746 —

MLP(9) 1.2728 75% 1.3864 77.78% 1.2465 78.57% 1.3922 78.95%

9 RNN2(9) 0.6993 100% 1.1504 88.89% 1.0321 85.71% 0.9963 84.21%SRW 1.8802 — 1.2989 — 1.6737 — 1.5746 —

MLP(10) 2.9463 75% 2.1453 77.78% 1.7123 78.57% 1.6529 78.95%

10 RNN2(10) 1.8506 100% 1.2077 88.89% 1.4423 85.71% 1.2713 84.21%

SRW 1.8802 — 1.2989 — 1.6737 — 1.5746 —


Table 11. Results of the DM statistics

Series Compared models DM(5) DM(10) DM(15) DM(20) Significance level of 0.5%

VUSD RNN2(1) and MLP(1) –7.2814 –10.9064 –13.3471 –15.4184 –2.58

RNN2(1)and SRW –7.8207 –10.8045 –13.2903 –15.3437 –2.58

VEUR RNN2(7) and MLP(7) –7.8207 –10.9064 –13.2903 –15.3064 –2.58

RNN2(7)and SRW –8.0904 -1-1.0084 –13.2903 –15.3437 –2.58

VJPY RNN2(1) and MLP(1) –7.5510 –10.7026 –13.1199 –15.3437 –2.58

RNN2(1)and SRW –7.2814 –10.6007 –13.1767 –15.3437 –2.58

VHKD RNN2(9) and MLP(9) –7.8207 –10.7026 –13.3471 –15.3811 –2.58

RNN2(9)and SRW –7.5510 –10.6007 –13.1767 –15.3437 –2.58

Thus, the DM test is used to measure the above hy-

potheses. The statistics for the compared models in the four

predict periods are computed, gathered in Table 11.

The results in Table 11 show that the values of DM

statistic for each group are ranked in the rejection region.

Since the null hypotheses should be rejected, the four se-

lected models including RNN2(1), RNN2(7), RNN2(1), and

RNN2(9) are the optimal models, respectively, for the series

of VUSD , VEUR, VJPY, and VHKD at the significance level

of 0.5%.

5 Concluding remarks

Taking the four main RMB exchange rates as the ob-

jects, this article aims to provide an effective method to fore-

cast the volatility series during the financial crisis period.

Based on the normal test and the nonlinearity test via sur-

rogate data method for each exchange rate volatility series,

the nonlinear, nonparametric models of ANNs were adopted

to make prediction on the four specific series. Different

ANN models including the MLP and RNN2 were applied

and compared from aspects of the in-sample fit, the out-of-

sample forecasting, and the DM test. The main conclusions

are collected as follows: (1) The visibly clustering effects

in the four exchange rate volatility behavior demonstrated

that the financial crisis has spread to the foreign exchange

market. Besides, the fluctuations are more intensive in the

series of VEUR and VJPY, which means that China’s export

trade has to face greater pressure. (2) The existence of non-

linearity in the four adopted RMB exchange rate volatility

series is confirmed; thus, it is valid to apply the nonlinear

ANN models to make forecasting. (3) The in-sample fit re-

sults indicated that the dynamic ANN models with higher

degree of freedom may have stronger ability in the exchange

rate volatility behavior description. (4) Based on the out-

of-sample forecasting comparison, we obtain the following

conclusion: first, there is no necessary connection between

the in-sample fit and the out-of-sample forecasting when us-

ing ANN models; second, the nonlinear dynamic networks

have stronger capability than the feed-forward networks, as

well as the traditional linear model in exchange rate forecast-

ing; third, ANN models perform quite differently in different

forecasting periods. (5) Via DM test, the four selected mod-

els including RNN2(1), RNN2(7), RNN2(1), and RNN2(9)

are further confirmed to be the optimal models, respectively,

for the series of VUSD, VEUR, VJPY, and VHKD. Thus,

this article has accomplished to offer a solution in the RMB

exchange rate volatility behavior description and forecasting

by combing the diverse techniques.

In addition, it is of great value to further construct

China’s financial crisis early warning system based on the

methods analyzed here, in order to improve financial risk

management and to maintain the economy stability and se-

curity.

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rmb exchange rate forecasting in the context of the financial crisis

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