Similarity Measures between Buying and Selling Prices of Various
Exchange Rates with the inclusion of Multidimensionality and
Multicollinearity
DENNIS WEN WEI NG1,*, YUN FAH CHANG1, PREMAGOWRIE SIVANANDAN1,
WEI SHEAN NG2
1School of Accounting & Finance,
Taylor’s University,
1, Jln Taylors, 47500 Subang Jaya, Selangor,
MALAYSIA
2Department of Mathematical and Actuarial Sciences,
Universiti Tunku Abdul Rahman,
Jalan Sungai Long, Bandar Sungai Long, 43000 Kajang, Selangor,
MALAYSIA
*Corresponding Author
Abstract: - This paper introduces a new statistical model, the multidimensional measurement error model with
multicollinearity, to study the relationship between buying and selling prices of foreign exchange rates of
various currencies. Such a model is needed due to the possible rise in multidimensionality and multicollinearity
issues that may occur due to the movement of the financial markets towards stock market integration of various
countries since the occurrences of the financial crisis as well as the part result of globalization. As this
integration involves the participation of various countries, it will affect the foreign exchange rate. Hence, the
analyses are performed on seven currencies against the Malaysian Ringgit where four models’ performances
are compared. From this research, it can be concluded that the proposed model comparatively performs better
than the other models in representing the relationship of the stationary prices and performs as well as existing
models towards non-stationary prices. It can also be seen that the Japanese Yen is the currency that has the
strongest influence and closer similar trends towards other currencies while the Great British Pound showed
otherwise.
Key-Words: - measurement error model, similarity measures, multidimensionality, multicollinearity, foreign
exchange rate, currencies.
Received: July 19, 2023. Revised: February 21, 2024. Accepted: April 15, 2024. Published: May 10, 2024.
1 Introduction
Stock market integration was found to exist in the
emerging market especially in Asia since the 1997
Asian financial crisis. In the beginning, numerous
researchers’ attention had been centered on
examining the integration among five ASEAN
emerging stock markets (Malaysia, Thailand,
Indonesia, Singapore, and the Philippines). [1],
summarizes that the five ASEAN stock markets are
moving towards enhanced integration among
themselves. Based on the recent market cap, the two
largest stock markets in the world are NYSE and
NASDAQ followed by the Shanghai Stock
Exchange (SSE), EURONEXT, and the Japan Stock
Exchange (JPX). Hence, [2], focused on examining
the long-run co-movements between the Malaysian
stock market and the two largest stock markets in
the world, the United States and Japan, and was
extended by [3], by adding China, one of the largest
trading partners of Malaysia. The findings stated
that in the long run, Malaysia’s stock market is
significantly influenced by its major trading
partners’ stock markets.
As such integration will affect various countries,
we would like to analyze the relationship of an
index that could be directly impacted by such an
event, which is the fluctuations in foreign exchange
rates. Before any integration could take place, a
better understanding of the trend and performance
of the exchange rates needs to be studied on both the
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buying and selling prices. As described by [4], the
relationship between the exchange rate and stock
returns exists as postulated by the flow-oriented
theory of exchange rate behavior. Also, it is
common knowledge that foreign exchange trading is
a risky financial instrument due to its volatility,
where even in recent events, the Central Bank of
Malaysia would need to intervene in the market to
stabilize the currency, [5]. In addition, it should not
only at a one-currency basis, but also on a
combination of currencies as the rise and fall of the
currencies among various countries usually occur
concurrently due to various reasons such as foreign
direct investments, international trading, demand,
and supply in the foreign exchange market, and
common financial regulations. However, performing
such a study will normally give rise to two common
issues, which are multidimensionality and
multicollinearity.
The occurrence of multidimensionality where the
presence of multiple x and multiple y are considered
as well as multicollinearity where the variables
studied are related or influenced by one another in
finance have always been a study of great interest.
As multicollinearity can only exist when there is
multidimensionality, they normally co-exist with
one another. Much research managed to show
evidence of its presence in various financial
applications. In terms of multidimensionality, [6],
studies ten different types of commodities listed on
ASEAN’s five stock markets. On Bursa Malaysia’s
website, there is a list of thirteen commodities’
indices that are being categorized. Regarding
foreign exchange rates, Bank Negara Malaysia
(BNM) recorded four different trading times as well
as twenty-seven exchange rates with the Malaysian
Ringgit (MYR), [7]. Hence, it can be observed that
the involvement of multidimensionality is
important, especially in the application context of
this study.
On multicollinearity, multicollinearity tests were
performed in studies such as [8], that modeled the
Indonesian Composite Index and performed the
multicollinearity tests using the VIF (Variance
Inflation Factor) approach where it was proven that
multicollinearity exists among the predictor
variables. From [9], the Apple Inc. stock returns and
two tests on multicollinearity were performed,
which are the VIF and condition index. From his
study, both test results concluded that
multicollinearity exists due to a strong correlation
among the explanatory variables. Adding on, [10],
compared two classical methods in detecting
multicollinearity in a time series data for finance as
well as economic sectors where both methods
showed that multicollinearity exists. Besides that,
foreign exchange rates also seem to contain
multicollinearity between the bank forecast,
univariate time series forecast along with the
forward price for exchange rates, [11], between the
various countries, [12], as well as between the
opening, opening low, and closing prices of stock
indices, which is similar to the foreign exchange
rate, [9].
Though the presence of multicollinearity can be
observed in various financial applications, a
common assumption where the variables are
independent or are not correlated to one another is
posited in many regression models. However, the
assumption of independence could be violated in
some cases and lead to issues in constructing the
model. Firstly, the estimated coefficients of the
model can be unstable and will vary greatly from
one sample to the next. Secondly, it undermines the
statistical significance of independent variables
since it might increase the standard errors of the
estimated coefficient as stated in [13], which might
result in inaccurate statistical analyses. It is evident
in [14], where two Monte Carlo simulations were
performed, and it was shown that the presence of
multicollinearity is capable of causing theory testing
problems with different levels of impact depending
on its severity. [15], also stated that the relationship
direction between explanatory variables has the
biggest impact on the variance inflation factor
especially when the variables are prone to errors and
the measurement error influences multicollinearity
in all circumstances.
Hence, to resolve the issue of multicollinearity
needs, the multidimensional measurement error
model with multicollinearity (MMEMc) was
developed. As the name implies, the MMEMc
model can cater to multidimensional data that also
contains multicollinearity where it functions in
measuring the similarity or dissimilarity between
two sets of data. To better understand the
relationship and trend of the foreign exchange rates,
we would need to study the gap between the buying
and selling prices of the foreign exchange rates
along with combinations of various currencies
where the similarity or dissimilarity measure could
be performed. Hence, we can apply the MMEMc
model in fitting the buying (x) and selling (y) prices
of the foreign exchange data which may potentially
contain the presence of multicollinearity among the
various countries. Through this study, the results
may assist financial investors in maximizing their
profits through foreign exchange trading and be
prepared for possible stock market integration
shortly.
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Aside from the introduction in Section 1, Section
2 introduces the proposed MMEMc model along
with its closed-form equations and properties.
Section 3 presents the results and discussion of the
foreign exchange rate application. Conclusions and
comments on future research are drawn in Section 4.
2 Research Methodology
In this section, the data collection and data
massaging procedure will be listed in section 2.1. In
section 2.2, assumptions along with the closed-form
equation of the MMEMc model will be stated.
2.1 Data Collection and Massaging
The foreign exchange rate data is obtained from the
Central Bank of Malaysia (Bank Negara Malaysia)
website, [7], where all exchange rates are in terms
of the Malaysian Ringgit (MYR). From the website,
the data available for both the buying and selling
prices are at 0900 hours, which is the opening price,
1130 hours as the best price, 1200 as the midday
price, and 1700 hours as the closing price. In this
example, the selected data are with the best price,
which are at 1130 hours as it is mentioned that the
rates at 1130 are the best counter rates offered by
selected merchant banks from the website. The best
counter rates at 1130 hours are the only rates where
the buying price is generally higher than the selling
price while prices from the other trading times
showed otherwise. To obtain the highest returns
possible from investing in the foreign exchange
market, we would, hence, analyze the trading time
that offers the best price. However, only seven
currencies’ data are available, comprising of the
United States Dollar (USD), Great British Pound
(GBP), European Euro (EUR), 100 Japanese Yen
(JPY100), Australian Dollar (AUD), Canadian
Dollar (CAD) and Singapore Dollar (SGD), which
will be used in this study. These currencies are
selected because all except the SGD are the six most
popular currencies traded in the world as
documented in [16], while SGD is selected due to its
historical relationship with Malaysia and is also one
of the most traded Asia’s currencies just after
JPY100, [17].
Foreign exchange rate data is known to be a time
series data. However, the models that are applied
and compared are mostly meant for stationary data
or in other words, data that do not contain time
series properties. Hence, the Augmented Dickey-
Fuller (ADF) test by [18], will be used to identify
the data whether it is stationary data or non-
stationary data. The test statistic used in the
approximate calculation of the p-value by [19] and
[20], where a table of values can be referred to for
various sample sizes, will be tested at a 5%
significance level. If the test shows that data is non-
stationary, the differencing procedure will be
performed repetitively and tested with the ADF test
until it is converted into a stationary one.
From the data differencing procedure, the non-
stationary foreign exchange rate data is now
converted into stationary data. The conversion
changes the prices of foreign exchange rate data to
the increase or decrease in prices of foreign
exchange rate. Hence, the slope and intercept
parameters now represent the gap between the
increase or decrease of the buying and selling
prices.
2.2 The MMEMc Model
The MMEMc model is a model extended from
[21]’s multidimensional unreplicated linear
functional relationship (MULFR) model to include
multicollinearity. The proposed model contains the
following assumptions. and
are two linearly related
unobserved true values of vector variables with p
dimensions and n observations such that:
, (1)
where are the intercepts and
is the single slope of the linear function. The term
dimension refers to the number of variables that are
being studied in vector variable x and vector
variable y. Two corresponding random vector
variables and
are observed and subjected with
errors and
′ such that:
. (2)
For all and ,
both error vectors are normally distributed with
1. ,
2. ,
3. for
all (errors among dimensions),
4. ,
5. and for all
(errors among observations).
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Therefore, the error vectors follow a multivariate
normal distribution that and
where:
and
Let
. Then
are variance-covariance where
Using the MLE method and the assumption of
we have the following parameters closed-form
equations:
(3)
where
and
.
(4)
for p > 1 dimensions and
,
and
with
,
,
,
.
where the subscript, s represents that the operations
have to be the same for both the expressions while
the sign with subscript t is independent. From
Equation (4), we observe that there are four possible
solutions where due to the presence of the negative
sign, there is a possibility of obtaining a complex
number. Hence, only the real part of the solution
and the one that returns the highest log-likelihood
will be selected.
For the case of p = 1, we have the following
closed-form after the derivations.
(5)
Equation (5) has two solutions. The solution that
returns the highest log likelihood is selected.
(6)
(7)
(8)
The estimated parameters of the MMEMc model
are proven to be unbiased, consistent, and
asymptotically normal distributed and they are able
to provide explanations for the data. The estimated
parameters of and represent the size of the gap
between the two vector variables under study. The
estimated error-variance parameter of represents
the errors that occur among the daily data within the
same dimension while the error-covariance,
represents the errors that occur among the different
dimensions. The value measures the similarity or
dissimilarity between two vector variables which
assist in providing the strength of the relationship
between them. Stronger relationships allow more
accurate estimations while weaker ones will be
harder to predict.
3 Results and Discussion
The MMEMc model can assist in measuring the
relationship between the buying (x) and selling (y)
prices of the foreign exchange data which may
potentially contain the presence of multicollinearity
among the various currencies. In this study, we will
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be analyzing the scenario where the dimensions of
the vector variables are represented by the various
currencies as there is a possibility of
multicollinearity to occur between them.
Relating the parameters to this application, the
MMEMc model estimated parameters of and
represent the size of the gap between the buying and
selling prices which will help investors to estimate
the potential returns that may be gained from this
investment through the prediction of the buying and
selling prices. On the other hand, the values of
represent the errors that occur among the daily data
within the same currency while represents the
errors that occur among the different dimensions.
The estimated represents the strength of
relationship between the buying and selling prices
where large values that are close to 1 represent
strong relationship or high similarity while smaller
values close to 0 represent weak relationship or low
similarity. The results obtained will represent the
currencies and the combinations of currencies that
have the strongest or weakest relationship between
the buying and selling prices which will assist in
obtaining the highest returns assuming that there is
severe multicollinearity among the currencies.
To compare the performance of the MMEMc
model, the MULFR, linear regression, and canonical
correlation analysis models will be used. The slope
and error parameters will be compared between the
MMEMc and MULFR model while the
coefficient will be compared across all the four
models mentioned. For the linear regression, the
will be calculated by obtaining the average value of
the from each of the dimensions.
In this section, the correlation matrices of the
Malaysian currency with other countries will be
displayed in section 3.1. The results and discussions
of the estimated parameters will be explained in
section 3.2 for seven countries as an overall view of
all the combinations, section 3.3 for one country
analysis, and section 3.4 for six countries analyses.
For the intercept parameter, , the number of values
follows the number of dimensions under study. Due
to the larger number of values, only the value that
has the highest magnitude will be displayed in this
manuscript.
3.1 Correlation Among Currencies
To determine whether multicollinearity is present
among the dimensions under study, the correlation
coefficients are calculated. In [22], study, it was
mentioned that a correlation coefficient between
0.637 and 0.771 by [23], might indicate a significant
relationship among the dimensions. However, [22],
stated a general rule of thumb with a correlation
coefficient threshold of 0.8 where any value more
than 0.8 implies a serious multicollinearity problem.
Hence, we would set a benchmark of 0.63 where
any values greater than 0.63 imply that
multicollinearity exists in the study while 0.8 will be
the benchmark for a high level of multicollinearity.
From Table 1, the country that shows the highest
level of multicollinearity among the other countries
in the stationary buying price is the SGD which it
correlates to the USD, JPY100, and CAD with
0.6107, 0.6380 and 0.7085, respectively. On the
stationary selling prices, it can be seen from Table 2
that there is a low correlation among the countries.
Combining the two sets of data, the correlation
values based on Table 3 are now much lower than
0.63, and hence the presence of multicollinearity
towards the data is relatively small.
Table 1. Correlation of the Stationary Buying Prices
among Currencies
Currency
USD
GBP
EUR
JPY100
AUD
CAD
SGD
US
1.00
0.27
0.31
0.47
0.16
0.46
0.61
GBP
0.27
1.00
0.23
0.27
0.14
0.35
0.42
EUR
0.31
0.23
1.00
0.13
0.25
0.36
0.45
JPY100
0.47
0.27
0.13
1.00
0.11
0.44
0.64
AUD
0.16
0.14
0.25
0.11
1.00
0.31
0.30
CAD
0.46
0.35
0.36
0.44
0.31
1.00
0.71
SGD
0.61
0.42
0.45
0.64
0.30
0.71
1.00
Table 2. Correlation of the Stationary Selling Prices
among Currencies
Currency
USD
GBP
EUR
JPY100
AUD
CAD
SGD
US
1.00
0.35
0.22
0.13
0.26
0.24
0.10
GBP
0.35
1.00
0.25
0.06
0.32
0.20
0.05
EUR
0.22
0.25
1.00
0.07
0.24
0.12
0.04
JPY100
0.13
0.06
0.07
1.00
0.08
0.04
0.01
AUD
0.26
0.32
0.24
0.08
1.00
0.32
0.15
CAD
0.24
0.20
0.12
0.04
0.32
1.00
0.13
SGD
0.10
0.05
0.04
0.01
0.15
0.13
1.00
Table 3. Correlation of the Stationary Buying and
Selling Prices among Currencies
Currency
USD
GBP
EUR
JPY100
AUD
CAD
SGD
USD
0.31
0.11
0.09
0.03
0.05
0.04
0.04
GBP
0.09
0.29
0.08
0.00
0.08
0.05
0.02
EUR
0.11
0.14
0.03
0.03
0.08
0.05
0.01
JPY100
0.21
0.12
0.14
0.10
0.07
0.03
0.02
AUD
0.05
0.12
0.07
0.02
0.19
0.09
0.00
CAD
0.19
0.21
0.14
0.00
0.21
0.25
0.04
SGD
0.25
0.21
0.16
0.04
0.16
0.10
0.01
On non-stationary data based on Table 4 and
Table 5, except for GBP, the other currencies have
some correlation among the countries in the
exchange rate. For instance, multicollinearity is
present in the USD currency with EUR, JPY100,
CAD, and SGD in both the buying and selling prices
though the multicollinearity with the SGD can be
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seen as severe in the buying price. Multicollinearity
also exists in EUR with USD, JPY100, CAD, and
SGD where the buying price in relation to SGD
shows severe multicollinearity. JPY100, has
multicollinearity with currencies like USD, EUR
and SGD with SGD showing severe
multicollinearity in the buying price. AUD only has
multicollinearity with CAD in both the buying and
selling prices. Multicollinearity can also be seen in
CAD with USD, EUR, AUD, and SGD in both the
buying and selling prices. For SGD, it has generally
multicollinearity with most of the other currencies
except for GBP and AUD with severe
multicollinearity found in the relation with USD,
EUR, and JPY100 of the buying price. Combining
both the buying and selling prices to calculate the
correlation results in a similar observation as can be
seen in Table 6.
Table 4. Correlation of the Non-Stationary Buying
Prices among Currencies
Currency
USD
GBP
EUR
JPY100
AUD
CAD
SGD
USD
1.00
0.14
0.64
0.75
0.45
0.67
0.89
GBP
0.14
1.00
0.06
-0.30
0.24
0.23
0.03
EUR
0.64
0.06
1.00
0.70
0.51
0.70
0.84
JPY100
0.75
-0.30
0.70
1.00
0.30
0.49
0.84
AUD
0.45
0.24
0.51
0.30
1.00
0.73
0.54
CAD
0.67
0.23
0.70
0.49
0.73
1.00
0.79
SGD
0.89
0.03
0.84
0.84
0.54
0.79
1.00
Table 5. Correlation of the Non-Stationary Selling
Prices among Countries
Currency
USD
GBP
EUR
JPY100
AUD
CAD
SGD
USD
1.00
0.11
0.68
0.74
0.45
0.68
0.79
GBP
0.11
1.00
0.07
-0.27
0.21
0.20
0.02
EUR
0.68
0.07
1.00
0.68
0.53
0.72
0.74
JPY100
0.74
-0.27
0.68
1.00
0.31
0.49
0.69
AUD
0.45
0.21
0.53
0.31
1.00
0.71
0.52
CAD
0.68
0.20
0.72
0.49
0.71
1.00
0.70
SGD
0.79
0.02
0.74
0.69
0.52
0.70
1.00
Table 6. Correlation of the Non-Stationary Buying
and Selling Prices among Countries
Currency
USD
GBP
EUR
JPY100
AUD
CAD
SGD
USD
0.98
0.15
0.62
0.71
0.48
0.65
0.76
GBP
0.09
0.98
0.02
-0.29
0.21
0.17
-0.01
EUR
0.68
0.11
0.96
0.68
0.56
0.73
0.73
JPY100
0.77
-0.30
0.69
0.94
0.34
0.50
0.72
AUD
0.42
0.23
0.47
0.27
0.98
0.67
0.47
CAD
0.68
0.24
0.68
0.47
0.76
0.97
0.68
SGD
0.91
0.05
0.84
0.80
0.59
0.79
0.86
From the observations described, we would like
to compare with the results of the models to check
on their consistency with the correlation results.
3.2 Overview of the Seven Countries
From the correlation results, it is now of great
interest to observe how does the proposed MMEMc
model, its previous MULFR model, and along with
other existing models fit the foreign exchange rate
data. Looking at the results modeled based on all the
seven currencies, we have the results of parameter
estimates in Table 7 and results of in Table 8 and
Table 9 for stationary and non-stationary data,
respectively.
Focusing on the stationary data, we have the
estimated parameters of the MMEMc model with
6.5893 for , value with the largest magnitude for α
at -0.0023, 0.0522 for σ as the error variance,
0.0005 for as the error covariance and at
0.3592. For the MULFR model, we have 4.6308 for
, -0.0015 for , 0.0521 for and 0.03592 for .
Comparing these two models, it shows that the
presence of the error covariance value, though
small, causes a significant impact on the estimated
parameters especially on the slope and intercept
parameters while the impact on the error variance as
well as the is quite minimal. The meaning of the
estimated parameters represents that for every RM 1
in the increase or decrease of the buying price, the
increase or decrease in the selling price rises by
value of RM 6.5893 adds with the values where -
RM 0.0023 is the value with the largest magnitude.
For the value, it implies that the errors among the
observations within the same dimension or currency
of the increase and decrease in foreign exchange
will deviate at about 5% around the mean of zero
which is relatively small. The covariance value
implies the errors among the observations between
the different dimensions or currencies of the
increase and decrease in foreign exchange will
deviate at about 0.05% around the mean of zero
which is also relatively small.
The estimated showed a value that is greater
than 1 and very small values of which may imply
that the increase or decrease in the buying price is
relatively directly proportional to that of the selling
price. The error variance and covariance values are
also relatively small which represent those minimal
errors exist between the observations within and
among the dimensions. However, the low values of
the showed otherwise due to the values of the
observation. Hence, it is evident that the relationship
between the increase or decrease in the buying price
and the selling price is not strong in both the
models. Comparing with the other two models,
linear regression and canonical correlation analysis,
the linear regression model tends to show an
value that is way smaller that the proposed model
while the canonical correlation analysis mode
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showed a value similar to the proposed model.
Therefore, it is conclusive to say that the linear
regression model may have underestimated the
relationship strength between the variables while the
other models have somewhat similar estimation
strength.
On the non-stationary data, the values of both
the MMEMc and MULFR model are somewhat
similar which is around 1. The magnitudes are
larger than those from the stationary data, but they
are relatively small with only values between 1 and
-1. The values are also similar to those of the
stationary data at approximately 0.0426 for both the
models. The error covariance estimated is also small
at 0.0003. The small values of the estimated error
parameters also showed that minimal errors exist
between the observations within and among the
dimensions. On the values for all the models,
they are relatively large with values above 0.9 with
canonical correlation showing the largest, followed
by the proposed models and finally the linear
regression models.
Table 7. Table of MMEMc and MULFR parameter
estimates for all Seven Currencies’ (USD GBP EUR
JPY100 AUD CAD SGD) Stationary and Non-
Stationary Foreign Exchange Rate Data
Parameters
Stationary
Non-Stationary
MMEMc
MULFR
MMEMc
MULFR
6.5893
4.6308
1.0078
1.0113
-0.0023
-0.0015
-0.1133
-0.1330
0.0522
0.0521
0.0426
0.0426
0.0005
N/A
0.0003
N/A
*N/A implies Not Available
Table 8. Table of MMEMc, MULFR, Linear
Regression and Canonical Correlation Analysis
models’ estimates for Seven Countries’
Stationary Foreign Exchange Rate Data
Table 9. Table of MMEMc, MULFR, Linear
Regression and Canonical Correlation Analysis
models’ estimates for Seven Countries’ Non-
Stationary Foreign Exchange Rate Data
In summary, the proposed model, MMEMc and
its prior model, MULFR are able to perform as well
as the canonical correlation analysis model with the
linear regression model and then underestimate its
estimation based of the stationary data. However,
the presence of the error covariance parameter
which caused a slight difference in the estimated
parameters between the MMEMc and MULFR
model showed that it is important to have a
parameter that can represent multicollinearity.
Therefore, the MMEMc model may be a better
performed model in the foreign exchange rate
stationary data. On the non-stationary data, the four
models showed a strong relationship between the
buying and selling prices for all the countries
combined with the canonical correlation analysis
showing the highest values, followed by the
proposed models and finally the linear regression
model.
Nevertheless, this case only shows the overall
value for the combination of all the seven
currencies, and not much information can be
obtained from it. Reasons behind the small
values even though the values are greater than 1
could not be determined via this case. Therefore, the
relationship between the buying and selling prices is
also studied at a single currency basis which will be
analyzed in the next section.
3.3 Single Currency
Looking at the results modeled based on a single
country perspective, we have the results of in
Table 10, results of α in Table 11, results of in
Table 12, and results of in Table 13 and Table 14
for stationary and non-stationary data, respectively.
For this study, we are looking at a one-dimensional
study. Hence, the estimates and results for both the
MMEMc and MULFR are the same with the error
covariance, , to be zero which will not be
displayed.
Focusing on the stationary data, SGD seems to
have the highest value of 422.2836 followed by
JPY100 at 29.4816 and CAD at 5.5084. The rest of
the currencies have values lesser than 1. A
value of 1 is placed as a benchmark as it normally
shows a strong relationship between the two
variables as can be seen through the non-stationary
data results. On the parameter, the magnitude of
the values for all the currencies are relatively small
and are lesser than 1. The parameter values are
also small with the largest value being 0.0542. This
implies that the errors among the observations
within the same dimension of the increase and
decrease in foreign exchange will deviate at about
5% around the mean of zero which is relatively
Currencies
Non-Stationary
MMEMc
MULFR
Lin. Reg.
Can. Cor.
USD GBP
EUR JPY100
AUD CAD SGD
0.9609
0.9609
0.9097
0.9909
Currencies
Stationary
MMEMc
MULFR
Lin. Reg.
Can. Cor.
USD GBP
EUR JPY100
AUD CAD SGD
0.3577
0.3592
0.0419
0.3548
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small. The values of the MMEMc and MULFR
reflect the values of the parameter. Hence, it is
observed that the highest value goes to SGD with a
value of 0.9658 followed by JPY100 at 0.9109 and
CAD at 0.7390. The next highest value calculated is
0.1302 which is the USD . The rest of the
values are lesser than 0.1. However, the values
for the other two models, the linear regression and
canonical correlation, are very much smaller in
general than the MMEMc and MULFR models,
especially for currencies such as the JPY100, CAD,
and SGD where their tend to be larger. For
smaller values of the MMEMc and MULFR , the
values for the canonical correlation model are
slightly larger which can be seen in the currencies of
USD, GBP, EUR, and AUD.
On non-stationary data which represents the
actual price, the values of the MMEMc and
MULFR models are generally around 1 with the
highest value being 1.2786 for the SGD. The α
values have generally small magnitudes with the
largest value being 0.8906. Also, the error variance
largest value can be seen to be 0.0625 which is
relatively small as well. This implies that the errors
among the observations within the same dimension
in foreign exchange will deviate at about 6% around
the mean of zero which is relatively small. On the
, they are all generally close to one with the
smallest being 0.8876. However, compared with the
values, the values do not fully reflect them as
the value of the SGD is the largest but its
value is the smallest. This occurs because of the
buying and selling prices which affect the Sxx, Sxy
and Syy values that is present in the calculation of
the . Comparing the results with the linear
regression and canonical correlation models, the
values are very close to one another with the
MMEMc and MULFR models showing the highest
values followed by the canonical correlation model
and finally the linear regression model.
Table 10. Table of MMEMc and MULFR
estimates for a Single Country’s Stationary and
Non-Stationary Foreign Exchange Rate Data
Currencies
Stationary Data
Non-Stationary Data
MMEMc
MULFR
MMEMc
MULFR
US
0.2247
0.2247
0.9924
0.9924
GBP
0.1359
0.1359
0.9530
0.9530
EUR
0.0470
0.0470
1.0451
1.0451
JPY100
29.4816
29.4816
1.0542
1.0542
AUD
0.2974
0.2974
0.9660
0.9660
CAD
5.5084
5.5084
1.0330
1.0330
SGD
422.2837
422.2837
1.2786
1.2786
Table 11. Table of MMEMc and MULFR
estimates for a Single Country’s Stationary and
Non-Stationary Foreign Exchange Rate Data
Currencies
Stationary Data
Non-Stationary Data
MMEMc
MULFR
MMEMc
MULFR
US
0.0003
0.0003
-0.0094
-0.0094
GBP
0.0001
0.0001
0.1924
0.1924
EUR
0.0003
0.0003
-0.2732
-0.2732
JPY100
-0.0117
-0.0117
-0.2521
-0.2521
AUD
0.0001
0.0001
0.0504
0.0504
CAD
-0.0006
-0.0006
-0.1608
-0.1608
SGD
-0.1059
-0.1059
-0.8906
-0.8906
Table 12. Table of MMEMc and MULFR
estimates for a Single Country’s Stationary and
Non-Stationary Foreign Exchange Rate Data
Currencies
Stationary Data
Non-Stationary Data
MMEMc
MULFR
MMEMc
MULFR
US
0.0204
0.0204
0.0247
0.0247
GBP
0.0353
0.0353
0.0475
0.0475
EUR
0.0542
0.0542
0.0475
0.0475
JPY100
0.0371
0.0371
0.0625
0.0625
AUD
0.0276
0.0276
0.0245
0.0245
CAD
0.0192
0.0192
0.0219
0.0219
SGD
0.0156
0.0156
0.0443
0.0443
Table 13. Table of MEMMc, MULFR, Linear
Regression and Canonical Correlation estimates
for a Single Country’s Stationary Foreign Exchange
Rate Data
Currencies
Stationary Data
MMEMc
MULFR
Lin. Reg.
Can.
Cor.
USD
0.1302
0.1302
0.0973
0.3120
GBP
0.0979
0.0979
0.0836
0.2892
EUR
0.0021
0.0021
0.0011
0.0325
JPY100
0.9109
0.9109
0.0106
0.1029
AUD
0.0770
0.0770
0.0374
0.1933
CAD
0.7390
0.7390
0.0631
0.2511
SGD
0.9658
0.9658
0.0002
0.0124
Table 14. Table of MEMMc, MULFR, Linear
Regression and Canonical Correlation estimates
for a Single Country’s Non-Stationary Foreign
Exchange Rate Data
Currencies
Non-Stationary Data
MMEMc
MULFR
Lin. Reg.
Can.
Cor.
USD
0.9833
0.9833
0.9671
0.9834
GBP
0.9764
0.9764
0.9555
0.9775
EUR
0.9627
0.9627
0.9237
0.9611
JPY100
0.9460
0.9460
0.8896
0.9432
AUD
0.9774
0.9774
0.9567
0.9781
CAD
0.9704
0.9704
0.9399
0.9695
SGD
0.8876
0.8876
0.7354
0.8576
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The above observations discussed can be
explained via the graphs plotted. Stationary prices
for the seven countries can be seen in Fig. 1 with
subfigures 1a, 1b, 1c, 1d, 1e, 1f and 1g representing
USD, GBP, EUR, JPY100, AUD, CAD, and SGD.
From the graphs, those with only the blue color are
values where the increase or decrease in the selling
price is larger than the buying price while the red
color represents that the increase or decrease in the
buying price is larger than the selling price. As it is
common knowledge that the increase or decrease in
selling price must be higher than the increase or
decrease in buying price in every business, hence
the blue color lines are seen more frequently in the
graphs.
Looking closer, currencies with higher values
tend to have fewer red lines observed whereas
currencies with low values tend to have more
red lines observed. Hence, it means that the
relationship of the increase or decrease in selling
price in comparison with those of the buying price is
relatively stronger which allows a more accurate
estimation of the prices. When there are more
frequent red spikes, the increase or decrease in the
prices is much harder to estimate as the prices are
not following the usual trend, hence, lower
values are estimated. Therefore, the value of the
can assist in a more reliable prediction of the
increase or decrease in the selling price given that
the increase or decrease in the buying price is
known or vice versa.
Comparing with the linear regression and
canonical correlation model, they show relatively
lower and smaller values in all the currencies
which cannot be justified from the graphs. This
shows that the models have relatively lower
predictive power in estimating the increase or
decrease of the prices. Thus, it can be concluded
that the MMEMc and MULFR models perform
better than the linear regression and canonical
correlation models in the single currency scenario. It
can also be concluded that the currencies of
JPY100, CAD, and SGD are more predictable
currencies which can be represented by the
MMEMc and MULFR model.
(a) USD Stationary Buying and Selling
Prices at 1130
(b) GBP Stationary Buying and Selling
Prices at 1130
(c) EUR Stationary Buying and Selling
Prices at 1130
(d) JPY100 Stationary Buying and
Selling Prices at 1130
(e) AUD Stationary Buying and Selling
Prices at 1130
(f) CAD Stationary Buying and Selling
Prices at 1130
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(g) SGD Stationary Buying and Selling
Prices at 1130
Fig. 1: Comparison of 7 countries 1130 Stationary
Buying and Selling prices of Foreign Exchange Rate
On the non-stationary data, the trend of prices for
the seven currencies can be seen in Fig. 2 with
subfigures 2a, 2b, 2c, 2d, 2e, 2f, and 2g representing
USD, GBP, EUR, JPY100, AUD, CAD, and SGD.
From the graphs, two colors can be seen obviously.
The buying price is represented by the red color
while the selling price is blue. At the 1130 price, it
is known to be the best price offered by merchant
banks. Hence, it can be observed that the buying
price is generally higher than the selling price which
is quite consistent throughout the seven years
though there are certain prices that show a different
trend on certain days which can be seen from the
crossovers of the two lines. Therefore, this trend
contributes to the relatively high for all the
currencies as observed by all the four models
compared.
(a) USD Non-Stationary Buying and
Selling Prices at 1130
(b) GBP Non-Stationary Buying and
Selling Prices at 1130
(c) EUR Non-Stationary Buying and
Selling Prices at 1130
(d) JPY100 Non-Stationary Buying and
Selling Prices at 1130
(e) AUD Non-Stationary Buying and
Selling Prices at 1130
(f) CAD Non-Stationary Buying and
Selling Prices at 1130
(g) SGD Non-Stationary Buying and
Selling Prices at 1130
Fig. 2: Comparison of 7 countries 1130 Non-
Stationary Buying and Selling prices of Foreign
Exchange Rate
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However, looking deeper into the graphs, it can
be observed that there are a few large spikes seen in
the graphs which show a sudden drop or increase in
the buying or selling prices. Comparing with the
values, currencies with fewer number of spikes
regardless of the buying or selling price tend to have
higher values while currencies with more spikes
tend to have lower values. This supports the
results where the SGD graph which reflects the
highest number of spikes showing the lowest
values compared to the other currencies.
Comparing among the models, most of the
values are relatively close to one another especially
the MMEMc, MULFR and canonical correlation
models with the linear regression model showing
the lowest value in all the currencies. Therefore, it
can be concluded that the MMEMc, MULFR and
canonical correlation models perform better in
estimating the selling price given that the buying
price is known while the linear regression has
slightly lower estimation capability. Also, it can be
evident that the SGD prices are slightly more
difficult to predict compared to the other currencies.
As an overall summary, the MMEMc and
MULFR models are better performed models in
estimating the stationary prices of the increase or
decrease in selling and buying prices for each single
currency compared to the linear regression and
canonical correlation models based on their
consistency. From the models, SGD, JPY100, and
CAD are the currencies that show better consistency
in the increase or decrease in their prices. Also, the
four models compared showed relatively similar
estimation strength in predicting the non-stationary
actual selling and buying prices of each country
with the linear regression showing slightly lower
capability in the estimation. This proves that the
MMEMc and MULFR models are equivalent in
estimation strength with the canonical correlation
model in modeling single currencies. Overall, the
MMEMc and MULFR models performed better if
not equivalent in modeling single currency foreign
exchange rates.
3.4 Combinations of Six Currencies
With the relationship studied between the buying
and selling prices for both the stationary and
nonstationary data on a single currency basis, we
would now like to observe which currency has the
largest impact on the others either positively or
negatively where currencies that have similar trends
tend to have stronger relationships while dissimilar
trends will have weaker relationships. Thus, an
analysis on the combination of six currencies will be
performed where seven such combinations can be
grouped. Looking at the results modeled based on a
six-currency perspective, we have the results of in
Table 15, results of in Table 16, results of in
Table 17, results of in Table 18, and results of
in Table 19 and Table 20 for stationary and non-
stationary data, respectively.
Focusing on the stationary data, there are 6 larger
values observed that are greater than 1 with USD
GBP EUR AUD CAD SGD group being lesser than
it with a value at 0.3912 and 0.3917 for the
MMEMc and MULFR models, respectively. From
here, it can be observed that the combinations with
JPY100 result in larger values which implies that
the Japanese Yen has the strongest positive
influence on the seven currencies. Furthermore, it
can also be seen that GBP is the currency that
decreases the values of the combinations greatly
where the combination of USD EUR JPY100 AUD
CAD SGD can be seen as the highest at 19.7796 and
13.3372 for the MMEMc and MULFR models,
respectively. On the values of the α where only the
ones with the highest magnitude will be shown, they
are generally quite small with the value of the
largest magnitude seen to be -0.0077 and -0.0051 by
both the MMEMc and MULFR models respectively
for the USD EUR JPY100 AUD CAD SGD.
Similarly, on the values, the values are
generally small with the largest value seen in the
combined currencies of USD GBP EUR AUD
JPY100 at 0.0558 and 0.0557 by the MMEMc and
MULFR models, respectively. On the error
covariance estimated, the values seem to be
relatively small as well with approximately 0.0006
from the USD GBP EUR JPY100 AUD SGD and
USD GBP EUR JPY100 CAD SGD combinations
as the largest value seen. These parameters imply
that minimal errors exist between the observations
within and among the dimensions. Regarding the
values, the currencies that consist of JPY100 and
also do not consist of GBP show the highest value at
0.6160 and 0.6164 by the MMEMc and MULFR
models respectively. Whereas the combinations that
do not consist of JPY100 but consist of GBP show
the lowest value at 0.0504 for both the MULFR
and MMEMc models.
Comparing across the models, the linear
regression and canonical correlation analysis models
show relatively smaller values in all the combined
currencies with the canonical correlation analysis
model showing approximately 0.33 on average and
linear regression approximately lesser than 0.1.
However, the MMEMc and MULFR models are
seen to better represent the relationship of the
increase or decrease in the prices for the six
combined currencies where the estimates can be
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justified from the studies done at a single currency
basis. Also, it can be concluded that the combined
currencies of USD EUR JPY100 AUD CAD SGD is
the better portfolio which has more similar trends
among the currencies in their increase or decrease in
the buying and selling prices using the proposed
model which allows better estimations. From the
study of combinations of six currencies, it can be
determined that the JPY100 has the strongest
influence on the seven currencies while GBP has the
weakest.
On non-stationary data, the following
explanations apply to both the MMEMc and
MULFR models. The values are generally the
same for all the currencies combinations
approximately around 1. On the largest magnitude
of the α, it is still slightly larger than those of the
stationary data, but they are still relatively small
with values that are lesser than 1 or larger than -1.
The error variances, it is similar to the stationary
data with the largest value seen at approximately
0.0451 for the combined currencies of USD GBP
EUR JPY100 AUD, and SGD. Similarly, the error
covariance also showed small values with the
largest seen at approximately 0.0003 for most of the
combined currencies except USD GBP JPY100
AUD CAD SGD and USD EUR JPY100 AUD
CAD SGD. The estimated error parameters imply
that there are only minimal errors seen between the
observations within and among the dimensions.
Regarding , the values are generally similar to
one another where they are all larger than 0.9 for
both the MMEMc and MULFR models.
Compared with other models, the canonical
correlation model seems to show the largest values
in all the combinations followed by the MMEMc
and MULFR models which are only slightly lesser,
while the linear regression is seen to be the smallest
where some combinations show values that are
lesser than 0.9. Therefore, it can be evident that the
MMEMc and MULFR models can perform almost
as well as the canonical correlation model but better
than the linear regression model in estimating the
actual buying or selling prices for a combination of
six currencies. All the combinations of the
currencies show similar strong relationships
between the buying and selling prices due to their
similar trends which can be seen in Figure 2.
Table 15. Table of MMEMc and MULFR
estimates for Six Countries’ Stationary and Non-
Stationary Foreign Exchange Rate Data
Currencies
Stationary Data
Non-Stationary Data
MMEMc
MULFR
MMEMc
MULFR
USD GBP EUR
JPY100 AUD
CAD
2.6678
1.9421
0.9980
1.0016
USD GBP EUR
JPY100 AUD
SGD
6.3976
4.5530
1.0070
1.0102
USD GBP EUR
JPY100 CAD
SGD
7.6990
5.3661
1.0117
1.0157
USD GBP EUR
AUD CAD SGD
0.3912
0.3917
0.9960
1.0002
USD GBP
JPY100 AUD
CAD SGD
7.5983
6.1621
1.0027
1.0040
USD EUR
JPY100 AUD
CAD SGD
19.7796
13.3372
1.0418
1.0396
GBP EUR
JPY100 AUD
CAD SGD
7.6201
5.7663
1.0107
1.0139
Table 16. Table of MMEMc and MULFR largest
estimates for Six Countries’ Stationary and Non-
Stationary Foreign Exchange Rate Data
Currencies
Stationary Data
Non-Stationary Data
MMEMc
MULFR
MMEMc
MULFR
USD GBP EUR
JPY100 AUD
CAD
-0.0007
-0.0004
-0.0588
-0.0786
USD GBP EUR
JPY100 AUD
SGD
-0.0022
-0.0015
-0.1089
-0.1267
USD GBP EUR
JPY100 CAD
SGD
-0.0027
-0.0018
-0.1348
-0.1576
USD GBP EUR
AUD CAD SGD
0.0003
0.0003
-0.0473
-0.0709
USD GBP
JPY100 AUD
CAD SGD
-0.0027
-0.0021
-0.0850
-0.0923
USD EUR
JPY100 AUD
CAD SGD
-0.0077
-0.0051
-0.2574
-0.2472
GBP EUR
JPY100 AUD
CAD SGD
-0.0027
-0.0020
-0.1292
-0.1471
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Table 17. Table of MMEMc and MULFR
estimates for Six Countries’ Stationary and Non-
Stationary Foreign Exchange Rate Data
Currencies
Stationary Data
Non-Stationary Data
MMEMc
MULFR
MMEMc
MULFR
USD GBP EUR
JPY100 AUD
CAD
0.0550
0.0549
0.0416
0.0416
USD GBP EUR
JPY100 AUD
SGD
0.0558
0.0557
0.0451
0.0451
USD GBP EUR
JPY100 CAD
SGD
0.0543
0.0542
0.0448
0.0448
USD GBP EUR
AUD CAD SGD
0.0473
0.0473
0.0381
0.0381
USD GBP
JPY100 AUD
CAD SGD
0.0472
0.0472
0.0416
0.0416
USD EUR
JPY100 AUD
CAD SGD
0.0425
0.0425
0.0412
0.0412
GBP EUR
JPY100 AUD
CAD SGD
0.0540
0.0540
0.0449
0.0449
Table 18. Table of MMEMc ω estimates for Six
Countries’ Stationary and Non-Stationary Foreign
Exchange Rate Data
Currencies
Stationary Data
MMEMc
Non-Stationary
Data MMEMc
USD GBP EUR
JPY100 AUD
CAD
0.0005
0.0003
USD GBP EUR
JPY100 AUD
SGD
0.0006
0.0003
USD GBP EUR
JPY100 CAD
SGD
0.0006
0.0003
USD GBP EUR
AUD CAD SGD
0.0002
0.0003
USD GBP
JPY100 AUD
CAD SGD
0.0004
0.0002
USD EUR
JPY100 AUD
CAD SGD
0.0004
0.0002
GBP EUR
JPY100 AUD
CAD SGD
0.0005
0.0003
Table 19. Table of MMEMc, MULFR, Linear
Regression and Canonical Correlation estimates
for Six Countries’ Stationary Foreign Exchange
Rate Data
Currencies
Stationary Data
MMEMc
MULFR
Lin.
Reg.
Can.
Cor.
USD GBP EUR
JPY100 AUD
CAD
0.1936
0.1970
0.0488
0.3544
USD GBP EUR
JPY100 AUD
SGD
0.3390
0.3403
0.0384
0.3312
USD GBP EUR
JPY100 CAD
SGD
0.3872
0.3884
0.0426
0.3397
USD GBP EUR
AUD CAD SGD
0.0504
0.0504
0.0471
0.3510
USD GBP
JPY100 AUD
CAD SGD
0.4992
0.4997
0.0487
0.3543
USD EUR
JPY100 AUD
CAD SGD
0.6160
0.6164
0.0349
0.3313
GBP EUR
JPY100 AUD
CAD SGD
0.3993
0.4000
0.0326
0.3495
Table 20. Table of MMEMc, MULFR, Linear
Regression, and Canonical Correlation estimates
for Six Countries’ Non-Stationary Foreign
Exchange Rate Data
Currencies
Non-Stationary Data
MMEMc
MULFR
Lin.
Reg.
Can.
Cor.
USD GBP EUR
JPY100 AUD
CAD
0.9662
0.9662
0.9387
0.9903
USD GBP EUR
JPY100 AUD
SGD
0.9604
0.9604
0.9047
0.9906
USD GBP EUR
JPY100 CAD
SGD
0.9596
0.9596
0.9019
0.9899
USD GBP EUR
AUD CAD SGD
0.9655
0.9655
0.9130
0.9903
USD GBP
JPY100 AUD
CAD SGD
0.9607
0.9607
0.9073
0.9906
USD EUR
JPY100 AUD
CAD SGD
0.9556
0.9556
0.9021
0.9896
GBP EUR
JPY100 AUD
CAD SGD
0.9581
0.9581
0.9001
0.9860
In summary, the MMEMc and MULFR models
are better performed models in estimating the
stationary prices of the increase or decrease in
selling prices and the increase or decrease in buying
prices for six combined currencies compared with
WSEAS TRANSACTIONS on BUSINESS and ECONOMICS
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Dennis Wen Wei Ng, Yun Fah Chang,
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Volume 21, 2024
the linear regression and canonical correlation
models based on their consistency. From the better
performed models, USD EUR JPY100 AUD CAD
SGD combination is the portfolio that shows similar
trends in the increase or decrease in their prices
which may allow higher returns with some risks
being hedged. For combinations that have lower ,
the currencies’ increase or decrease in prices do not
move towards the same trend which shows weak
relationship between the buying and selling prices
with the weakest seen in the USD GBP EUR AUD
CAD SGD combination. Also, it can be analyzed
that the JPY100 has the strongest influence among
the seven currencies while GBP has the weakest
which affected the overall values.
On the non-stationary prices, the four models
compared showed relatively similar strong
estimation strength in predicting the actual selling
and buying prices of each six currencies
combinations where the values are all similar with
only the linear regression showing slightly lower
capability in the estimation. This proves that the
MMEMc and MULFR models are equivalent in
estimation strength to the canonical correlation
analysis model. Also, as the error covariance
estimated is relatively small, the impact on the
values is minimal hence, not many differences can
be seen. Overall, the MMEMc and MULFR models
performed better if not equivalent to the canonical
correlation analysis model in modelling combined
three currencies foreign exchange rates.
4 Conclusion
From the discussions, we started with the study of
correlation among the seven countries followed by
an overview of the estimation results of all the seven
countries. Then, we analyzed the results for each of
the currencies individually to understand their
relationship between the buying and selling prices.
Then, a study on combinations of six currencies was
also performed to analyze the currency that has the
strongest positive or negative influence towards the
other currencies which assists in identifying the
number of possible portfolios that have better
predictability for investment. From the analyses, it
was determined that the MMEMc and MULFR
models have similar values as the error
covariance values are quite small. Though the
presence of ω, is small, it also results in some
differences in the estimated , as well as
parameters. Nevertheless, among the four models
compared, the MMEMc and MULFR models are
able to perform better than the linear regression and
canonical correlation analysis models in fitting the
stationary foreign exchange rate data where the
prices with similar trends are able to be captured by
the two models regardless of the combinations.
From the single stationary currency analyses
from the better performed models, it was mentioned
that SGD has the best relationship, based on the
values among the seven compared currencies, the
relationship between the increase or decrease of the
selling and the buying prices followed by JPY100
and CAD where they are the currencies with the
better relationships. On the other currencies, the
worst off is EUR followed by AUD, GBP, and
USD. However, as the analysis continues with a
combination of six currencies, JPY100 seems to be
the currency with the strongest influence in raising
the values while GBP has the strongest influence
in decreasing them. Hence, it is evident to conclude
that combinations that contain the currency of
JPY100 and do not contain GBP are safer portfolios
as they allow more accurate estimation of the
increase or decrease in the prices.
On the non-stationary prices, all four models
tend to show similar strength in determining the
relationship between the buying and selling prices.
All of them show strong relationships regardless of
the number of combinations and currencies with
canonical correlation analysis tend to show the
largest values, followed closely by the MMEMc and
MULFR models and finally linear regression model.
The three models with the higher values are
quite close to one another and are relatively
consistent in all the cases where they are above 0.9.
However, the linear regression model has values
slightly lower than them with some cases being less
than 0.9. Therefore, it can be said that the linear
regression model tends to underestimate in
calculating values while the other three models
have similar performances for the non-stationary
prices.
In conclusion, this study showed that
multicollinearity does exist, though not severe,
among the various currencies studied and it is
important to include its presence in a particular
model in which the MMEMc model is able to. From
the model, the JPY100 is concluded as the currency
with the strongest positive influence on all the seven
currencies studied while GBP is the worst. On
nonstationary prices, the MMEMc and MULFR
models can perform as well as the canonical
correlation model in the study of their relationship.
All the currencies and their combinations showed a
similar strong relationship between the buying and
selling prices.
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Dennis Wen Wei Ng, Yun Fah Chang,
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Nevertheless, further analyses can be extended as
future work from this point. Predictive analytics
studies can be conducted to test the accuracy of the
predicted prices with the actual ones based on the
estimated parameters. If the accuracy can be proven
to be strong, financial analysts will be able to utilize
this model in making more accurate predictions
which will help in maximizing investors' profits
through foreign exchange rate trading. Of course,
this model could also be applied to other financial
instruments or even another field of studies that uses
primary data from surveys or experiments
conducted are being used for analysis and the study
of relationships is needed to better understand the
data.
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Contribution of Individual Authors to the
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Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
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DOI: 10.37394/23207.2024.21.94
Dennis Wen Wei Ng, Yun Fah Chang,
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