Short-term Forecasting Ability of Hybrid Models for BRIC Currencies

Abstract

This article proposes a new framework to improve short-term forecasting accuracy of exchange rates of BRIC nations, that is, Brazil (USD/BRL), Russia (USD/RUB), India (USD/INR) and China (USD/CNY). The study employs three methodologies for a 42-day forecast: hybrid models based on least square support vector machine, residual hybrid model and automatic hybrid model forecasting using R software. The results show that the proposed residual hybrid model framework, including autoregressive integrated moving average-artificial neural network (ARIMA–ANN)-TBATS, outperformed other models with Brazil and China return series reflecting the best accuracy in ANN model and India and Russia demonstrating the best accuracy in trigonometric seasonal, box-cox transformation, ARIMA residuals, trend and seasonality (TBATS) model. Further, the results indicate that Brazil and China return series follow a non-linear pattern, while India and Russia follow a non-linear complex seasonal pattern. The highest level of forecast accuracy has been observed in China followed by Brazil, India and Russia.

Keywords

Exchange rate return hybrid model artificial neural network exponential smoothing automatic forecasting least square support vector machine

Introduction

Exchange rates play a fundamental role in nearly all aspects of international financial management and are imperative for many economic agents as it can generate incentives during international trade. However, all these economic agents face exchange rate risks. The expensive consequences of these risks could be allayed by dependable exchange rate forecasts, but it is also a challenging task. Thus, a precise exchange rate forecasting is crucial, particularly in financial markets, where a poor forecast can cause huge losses. A seminal work by Meese and Rogoff (1983) established that the econometric specifications based on macroeconomic fundamentals are incapable of forecasting better than a simple random walk forecast. Zhang (1994) had ascertained that the exchange rate series are normally non-linear, and a standard linear model may not produce desired results in short-term currency trading. Kilian and Taylor (2001) proposed a non-linear econometric model and reported that their model has good predictability over the 2–3-year horizon, but they report that predicting exchange rates in the short term remains a challenge. It can be inferred from these studies that predicting exchange rates remains a tough proposition, and this could be because of the complex role of exchange rates in financial markets.

The challenge becomes tougher when predicting exchange rates in BRIC (Brazil, Russia, India and China) economies, which are more dynamic in nature. The BRIC economies hold a combined GDP (PPP) of US$20 trillion (World Bank, 2019) and are among the fastest growing emerging markets. An analysis of the behaviour of exchange rates of these countries is fundamental for different decision-makers in international financial management.

This study proposes to validate an improvement in short-term forecasts of the exchange rate returns when using hybrid models as opposed to applying only econometric or artificial intelligence models in a rolling window frame. The study has employed the following methods: autoregressive integrated moving average (ARIMA) which, in particular, explores the linear behaviour of the time series; the error, trend and seasonality (ETS) exponential smoothening, which uses three components, namely trend, error and seasonality; artificial neural network (ANN), which considers specifically the non-linear behavior of the time series; and TBATS, a state space model, which allows for Box-Cox transformation and ARMA error that can identify the complex seasonal data in the time series.

In essence, this study is an attempt to construct apt short-term predictive models for exchange rate return series of BRIC nations. The findings of the study can provide direction for short-term hedging or cash management decisions and for evaluating foreign borrowing or investment opportunities. The article is divided into the following sections: the second section provides an overview of the literature on the topic. The third section presents the details of various forecasting models used in the study. The fourth section shows results from data analysis, and the fifth section concludes the article.

Review of Literature

Academic literature started observing impediments of forecasting exchange rates after Meese and Rogoff (1983) verified that most linear models are incapable to go beyond the random walk estimate for exchange rates. This study provided a platform for academic studies to focus on the viability of non-linear models to predict exchange rates. For instance, Frankel and Froot (1990) showed that volume and volatility along with behavioural bias are as important as economic models in predicting exchange rates. A different perspective was offered by Zhang and Yu (1998) who reported that neural networks provide better forecasts than linear models for shorter horizons in case of British Pound and US dollar currency pair. In another study, Evans and Lyons (2005) challenge the findings of Meese and Rogoff (1983) and show that micro-based models have a better predictive ability than macro-based or random walk model. Bissoondeeal et al. (2008) have suggested that neural network models outperform random walk models and traditional time series models. The findings of Bissoondeeal et al. (2008) are in contrast with an earlier study conducted by Newbold et al. (1998) that found that exchange rates follow a random walk pattern. More recently, researchers have suggested the use of scientific models to predict exchange rates. For example, Galeshchuk and Mukherjee (2017) have used deep learning for change direction forecasting of exchange rates by using convolution neural networks (CNNs) and found that the model can also effectively identify directional changes in exchange rates. It can be observed that literature review provides findings both in favour and against the use of random walk, traditional or sophisticated models for predicting exchange rates.

Some recent studies have proposed non-linear models with econometric design to predict exchange rates. Khashei et al. (2009) demonstrated progress in prognostic results when using ANNs set in with autoregressive models, concluding that the hybrid model attained superior results than an ARIMA model. Dhamija and Bhalla (2011) conducted an evaluation of diverse ANN architectures for exchange rate forecasting, ruling that ANNs can be successfully used in this task. Khashei and Bijari (2011) introduced a hybrid model that employs ARIMA to identify the linear structure in data and then employs ANN to establish a model that examines the underlying data, creating the process to envisage its future performance. Many similar studies (e.g., Ince & Trafalis, 2006; Lin et al., 2012) present model comparison in exchange rate forecasting, which is a proposal emerging since the research of Bates and Granger (1969) and Clemen (1989).

Contemporary studies have also proposed model composition specifically related to currency forecasting. Bildirici et al. (2010) have created an association of ANNs in mix with a Threshold Autoregressive Vector Error Correction (TARVEC) model to find progress in the modelling of non-linear co-integration for monthly returns of real exchange rates. Galeshchuk (2016) has evaluated the ANN’s forecast accuracy in three currencies (EUR/USD, GBP/USD and JPY/USD) in daily, monthly and quarterly terms, while Yao and Tan (2000) have illustrated the superiority of using ANN, particularly in capturing the non-linear element of the exchange rate. Rivas et al. (2017) have used ANN collectively with genetic algorithm (GA) on currency exchange and reported that the new hybrid model increases the forecasting accuracy accomplished by traditional hybrid models and of the techniques used alone.

The study of literature supports the use of hybrid model proposition of Rivas et al. (2017) not only for currency forecasting but also for financial time series. Taskaya-Temizel and Casey (2005) are of the opinion that hybrid models can be standardized or mixed, that is, composed of linear and non-linear models. Fatima and Hussain (2008) have evaluated ARIMA, autoregressive conditional heteroscedasticity (ARCH)/generalized autoregressive conditional heteroscedasticity (GARCH) and ANN models, individually and collectively, and found that the hybrid models have the best performance. Li et al. (2008) applied auto-regressive generalized regression neural networks (AR-GRNN) model for financial time series forecasting and concluded that the collective model was more effective. Pradeepkumar and Ravi (2017) applied a particle swam optimization (PSO) regression with an ANN to predict volatility from financial time series and outperform other models. These studies provide evidence in support of the use of hybrid models for financial time series such as exchange rates, and this article attempts to evaluate the use of hybrid models to predict financial time series of currency data from BRICS (Brazil, Russia, India, China and South Africa) economies. An important and differentiating aspect of this article from earlier studies is the choice of data from BRIC nations mainly due to dynamic nature and prospects of emerging markets in BRIC nations.

A report published by World Bank (2019) shows that the BRICS group accounts for a 20 per cent of the global growth. O’Neill (2011) states that the contribution of BRICS nations to global GDP growth over the past two decades is important to our global financial system and difficult to ignore. Many studies have estimated the impact of trade on exchange rate volatility in developed markets (Bailey et al., 1987; Esquivel & Larrain, 2002; Koray & Lastrapes, 1989) and in developing or emerging markets (Arize et al., 2003; Chue & Cook, 2008; Maradiga et al., 2012; Rao & Padhi, 2018). However, there are hardly any studies that presented hybrid or sophisticated models to forecast the movement of BRIC currencies. Sui and Sun (2016) performed a study on the dynamic relationship between stock returns, currency, interest rates and returns of US equity index S&P 500. Their study using vector error correction model (VECM) and vector auto correction (VAR) model found that movement in foreign exchange rates has a spillover effect on stock prices in BRICS economies, and the spillover effect amplified around financial crisis period of 2007–2009. Kocaarslan et al. (2017) have performed a study using dynamic conditional correlation to derive time-varying relationships and estimate the impact of volatility expectations of currency along with oil and gold on US stock markets and BRICS nations. Their study finds that interdependence in financial and non-financial markets is driven by risk perception in stocks, oil and gold. Rao and Padhi (2018) developed a model to predict currency crisis in BRICS economies—they prepared an ‘Exchange Market Pressure Index’ and a panel probit model to predict currency crises. Their study finds that currency crisis can be predicted by integrating various macroeconomic variables like current account balance, exports, imports, real interest rate, etc. Another study conducted by Saji (2018) explores the possibility of having a currency union among BRICS nations and suggests that strong policy interaction in the region can pave way for strong union currency among BRICS members.

From literature review, it can be established that there have been academic studies that have explored the relationship between currency and other macroeconomic variables such as exports, policy interaction and stock markets in BRIC nations. There is hardly any evidence which focuses on the use of hybrid models to predict currency movement based on historical currency data. To address this research gap, this study intends to present hybrid forecasting models for BRIC currencies with the objective of adding to existing literature.

Research Methodology

Objectives of the Study

The objective of this study is to investigate whether hybrid models can accurately predict exchange rates of BRIC countries in the short run based on historical data from exchange rates. The findings from this research may purposefully contribute to academic literature on the validity of hybrid models for predicting BRIC exchange rates in the short term. The nature of models used in the study are discussed next.

Autoregressive Integrated Moving Average Model

In the case of a conventional ARIMA model, non-stationary data are transformed into a stationary time series data by using set differencing of the data points. It is represented as

[1 - \sum_{i = 1}^{p} υ_{i} R^{i}] {(1 - R)}^{q} y_{t} = [1 + \sum_{j = 1}^{r} υ_{j} R^{j}] ε_{t}

(1)

In Equation (1), p, q and r symbolize the autoregressive, integrated and moving average unit, respectively, of the process.

Error–Trend–Seasonality Model

This model uses exponential smoothening methods for the components of level, trend and seasonality at time t and is given by:

A_{t} = γ U_{t} + (1 - γ) R_{t}

(2)

C_{t} = λ R_{t} + (p - λ) C_{t - 1}

(3)

D_{t} = δ D_{t} + (1 - v) d_{t - 1}

(4)

Eividually represent the magnitude, incline and seasonal component and γ, ʋ, λ and ϕ are constants.

Artificial Neural Network Model

This method of forecasting can be applied when there is a large sample. The association between the results (A_t) and the variables used ( $A_{t - 1}$ , $A_{t - 2}$ , …, A_t − p) can be given by:

A_{t} = λ_{0} + \sum_{k = 1}^{j} λ_{k} h (α_{0 k} + \sum_{m = 1}^{l} α_{m k} A_{t - i}) + ε_{t}

(5)

where $λ_{0} (k = 0, 1, 2, \dots, l; k = 1, 2, \dots, j)$ are the model strictures, l is the numeral of input modes and j is the numeral of invisible nodes.

Trigonometric Seasonal, Box-Cox Transformation, ARIMA Residuals, Trend and Seasonality

This method uses a state space model that is a comprehensive foundation of exponential smoothening method, allowing for automatic Box-Cox transformation and ARMA errors. It is given by:

A_{t}^{k} = k^{'} B_{t - 1} + ϵ_{t}

(6)

B_{t} = L B + h \in_{t}

(7)

In the above equation, $k^{'}$ is the row vector, h is the column vector, L is the grid and $B_{t}$ is the unobserved state vector at time t.

Method 1: Hybrid Methodology Forecasting Least Square Support Vector Machine

Under this method, a hybrid methodology has been put forward for forecasting models with least square support vector machine model to the methods (ARIMA, ETS, ANN and TBATS) with the objective of allocating to each of the models the accurate weight in the concluding estimate. To this end, the study has followed the following process:

Initially, ARIMA, ETS, ANN and TBATS methods were fit to the exchange rate series data represented by Z_t, individually for obtaining the forecast. Hence, $Z_{t}^{A}$ will represent the projected data obtained from an ARIMA method, $Z_{t}^{E}$ is the projected data found from an ETS method, $Z_{t}^{T}$ is the projected data from using a TBATS method and $Z_{t}^{N}$ is the projected data obtained from an ANN method.

The study then proceeds to use the fitted values from the above output to calculate approximately the weights for each method established on least square support vector machine model. At the end, the projected output from the regression when using tee methods, viz. hybrid model 1 combining ARIMA, ANN, TBATS and hybrid model 2 combining ARIMA, ANN and ETS is:

Z_{t} = α_{0} + α_{1} Z_{t}^{A} + α_{2} Z_{t}^{N} + α_{3} Z_{t}^{T} + ε

(8)

Z_{t} = α_{0} + α_{1} Z_{t}^{A} + α_{2} Z_{t}^{N} + α_{3} Z_{t}^{E} + ε

(9)

with the limitation that $\sum_{i = 0}^{3} α_{i} = 1$ .

The projected output from the regression when using two models ANN and TBATS is:

Z_{t} = α_{0} + α_{1} Z_{t}^{N} + α_{2} Z_{t}^{T} + ε

(10)

with the limitation that $\sum_{i = 0}^{2} α_{i} = 1$ .

Finally, an equation is obtained using the output from the least square support vector machine model. The forecasted time series for the next 42 days from each procedure (represented by $Z_{t}^{(Method, FT)}$ , where method = {ARIMA, TBATS, ANN} and {IMA, TBATS, ETS}, (FT is the forecast) provide as data for the regression equation, which is given by:

Z_{t}^{(FT)} = α_{0} + α_{1} Z_{t}^{(A, FT)} + α_{2} Z_{t}^{(T, FT)} + α_{3} Z_{t}^{(N, FT)} + ε

(11)

Z_{t}^{(FT)} = α_{0} + α_{1} Z_{t}^{(A, FT)} + α_{2} X_{t}^{(T, FT)} + α_{3} X_{t}^{(E, FT)} + ε

(12)

The forecasted time series for models combining ANN and TBATS will be:

Z_{t}^{(FT)} = α_{0} + α_{1} Z_{t}^{(N, FT)} + α_{2} Z_{t}^{(T, FT)} + ε

(13)

Method 2: Residual Hybrid Forecast Model

The hybrid methodology as per method 1 used two and three forecasting models, thus resulting in hybrid 1, hybrid 2 and hybrid 3 models. Further, the study employed the fitted values from the most accurate hybrid model as ‘actual’ observations from Method 1 and applied ARIMA–ANN–TBATS in method 2. Subsequently, after the assessment of the models (ARIMA, ANN and TBATS), the concluding estimate for the next 42 days is found.

Method 3: Automatic Hybrid Forecast Model Using R Package (Version 8.10)

To evaluate the forecasting methods, another procedure was chosen using forecasts created from ARIMA, ETS and theta models; ANN in time series and trigonometric seasonal; box-cox transformation; ARIMA residuals; and trend and seasonality. This methodology has been applied to the exchange rate series of the four countries and a roll of 11 models is determined.

Accuracy Measurement

There are various measures of verifying the accuracy of the proposed forecast models, and the measure used in the study is the statistical errors analysis. The mean absolute error (MAE) measures the mean error value between the observed and adjusted series as:

MAE = \frac{1}{n} \sum_{i = 1}^{n} | v_{a} - v_{o} |

(14)

where $v_{a}$ symbolizes the individual value of the adjusted exchange rate series, $v_{o}$ symbolizes the individual value of the observed exchange rate series and n is the series order.

Data, Data analysis and Estimation Results

Data

The exchange rate data used in this study corresponded to the USD/BRL, USD/RUB, USD/INR and USD/CNY parities in daily terms for the period from 3 January 2005 to 28 December 2018 (3,548 days in the entire period). Exchange rate series are obtained from the International Monetary Fund database and represent the amount of US dollars per unit of local currency. For modelling, the series are used in terms of natural logarithm at time t., defined as Y_t., which represents the series Brazil USD/BRL, Russia USD/RUB, India USD/INR and China USD/CNY exchange rates, respectively. The exchange rate variation is defined as the first difference of the series and is given by:

Δ Y_{t} = Y_{t} - Y_{t - 1}

(15)

The graphs shown in Figures 1 –4 clearly indicate that the trend of the four exchange rates was decreasing until around mid-2008, due to the consequences of sub-prime csis, causing a rise in the exchange rates.

Figure 1.

Source: Prepared by author.

Figure 2.

Source: Prepared by author.

Figure 3.

Source: Prepared by author.

Figure 4.

Source: Prepared by author.

Table 1.

Descriptive Statistics

Particulars	Mean	Standard Deviation	Skewness	Jarque–Bera	Kurtosis
Brazil	0.015	0.969	0.826	14614.2	14.32
Russia	0.029	0.926	1.38	431728	33.27
India	0.0144	0.464	0.225	1020	6.39
China	−0.005	0.141	0.256	7912	34.38

Source: IMF database, Author’s calculations.

Table 2.

Stationary Test for Different Price Series

Country	Particulars	PP Test	p-Value
Brazil	Level	−3570.3	0.01
Brazil	Differenced	−5.91	0.77
Russia	Level	−3472.7	0.01
Russia	Differenced	−4.827	0.840
India	Level	−3745.9	0.01
India	Differenced	−9.104	0.602
China	Level	−3317.5	0.01
China	Differenced	−3.5034	0.99

Source: Author’s calculations.

Descriptive Statistics

Descriptive statistics as per Table 1 indicate that the volatility is high in exchange rates of Brazil and Russia as measured by the standard deviation. China’s exchange rate return series is the least volatile. On an average, Russian rouble gave a better return than the exchange series of other nations during the period under study. The kurtosis is greater than 3 in the case of BRIC, implying that the return series is fat tailed. This is further confirmed by the Jarque-Bera test (JB) statistics, which is significant at the 1 per cent level. The hypothesis of normality is thus rejected.

Stationary Tests

The study used Philips–Peron (PP) unit root test on the data for the determination of stationary. Table 2 shows the absence of unit root in the series tested using PP tests at level.

L-JUNG Box Tests and Testing for ARCH Effects

After obtaining the residuals $ε_{t}$ from the ARMA process, the ARCH LM test was applied to examine the presence of ARCH effect in the residuals. This has been done by regressing the squared residuals on a constant and five lags. The indices were tested by using Box and Jenkins (1976) methodology and Ljung-Box (1978) to determine the number of autoregressive (AR), moving average (MA) and ARMA terms. The output in Table 3 shows that there are ARCH effects in the return series for all the countries. Thus, the variance of the errors is non-constant for the period under the study. The series being stationery, the study proceeds with the determination of the best fitting mean eation by applying ARMA to choose the best process in modelling the conditional mean. Table 3 depicts appropriate ARMA models, fitted by using Box-Jenkins methodology, for all the indices considered. Fitted ARMA models generally follow moving average pattern and autoregressive pattern in case of Brazil, Russia and China, proving that they have a longer moving average pattern leading to persistent shocks in the market over a period of time, but India’s return series follows a pure AR pattern.

Results for the Hybrid Forecast Models 1–3

The study has fitted independently ARIMA, ANN, ETS and TBATS models for the return on the exchange rate series for the four countries. The resultant models and attributes are provided in Table 4. The study merged ARIMA, TBATS, ETS and ANN in one multiple linear regression (three models and two models, respectively), and the results found are provided as follows.

Table 3.

Parameter Estimates of ARMA Model and ARCH–LM Tests

Series	Parameter	Estimate	Arch–LM Test Value
Brazil	AR (1), MA (1) and MA (2)	AIC = 9854.84	19.106 (0.0001)
Brazil		LL = −4923.4
Russia	AR (1), AR (2), MA (1) and MA (2)	AIC = 9521.22	9.1575 (0.0001)
Russia		LL = −4755.6
India	AR (1) and AR (2)	AIC = 5602.88	33.815 (0.0001)
India		LL = −2798.44
China	AR (2) and MA (2)	AIC = −3692.01	63.257 (0.0001)
China		LL=1851

Source: Author’s calculations.

Table 4.

The Parameters of the ARIMA, ANN, ETS and TBATS

Model	Brazil	Russia	India	China
ARIMA	Arima (0, 1, 1)(0, 0, 0)	Arima (0, 1, 1)(0, 0, 0)	Arima (0, 1, 1)(0, 0, 0)	Arima (0, 1, 1)(0, 0, 0)
Co-efficient	0.0153	0.0296	0.120	−0.0054
ANN	NNAR (1, 1)	NNAR (1, 1)	NNAR (1, 1)	NNAR (1, 1)
ETS	(A, N, N)α = 1 $e^{- 04}$	(A, N, N)α = 1 $e^{- 04}$	(A, N, N)α = 1 $e^{- 04}$	(A, Ad, N)α = 0.0189β = 1 $e^{- 04}$ phi = 0.8025
TBATS	Bats (1,{0,0},0.902,−)α = −0.01925β = 0.001830	Bats (1,{2,3},−,−}α = −0.000815AR co-efficient−0.0116−0.4983MA co-efficient0.1206950.55932	Bats (1,{0,0},0,−,−)α = −0.0009003	Bats (1,{0,0},0.8,−)α = 0.8666β = 07

Source: Author’s calculations.

Analysis of Hybrid Model 1 Using Three Methods (ARIMA–ANN–TBATS)

Brazil is

- 2.2390 + 0.12259 Z^{A} + 0.16948 Z^{N} - 0.13416 Z^{T}

(16)

Russia is

- 20.53140 + 0.32431 Z^{A} + 1.11677 Z^{N} - 0.15320 Z^{T}

(17)

India is

30.3326 + 0.5547 Z^{A} - 0.4904 Z^{N} + 0.4383 Z^{T}

(18)

China is,

30.4035 - 4.7851 Z^{A} + 2.2747 Z^{N} - 1.575 Z^{T}

(19)

Analysis of Hybrid Model 2 Using Three Methods (ARIMA–ANN–ETS)

Brazil is

- 10.56 + 0.08034 Z^{A} - 0.4023 Z^{N} + 4.122 Z^{E}

(20)

Russia is

- 17.53140 + 0.31641 Z^{A} + 1.067 Z^{N} - 0.1420 Z^{E}

(21)

India is

24.2496 + 0.3120 Z^{A} + 0.1534 Z^{N} + 0.1630 Z^{E}

(22)

China is

27.31 - 4.65851 Z^{A} + 2.165 Z^{N} - 1.575 Z^{E}

(23)

Analysis of Hybrid Model 3 Using Two Methods (ANN–TBATS)

Brazil is

25.6806 + 0.1348 Z^{N} - 5.58847 Z^{T}

(24)

Russia is

18.6899 - 0.2154 Z^{N} + 0.5845 Z^{T}

(25)

India is

19.44813 + 0.20383 Z^{N} + 0.48063 Z^{T}

(26)

China is

24.2129 - 2.4013 Z^{N} - 0.2902 Z^{T}

(27)

The study employed MAE for checking forecast accuracy for the exchange rate series. The MAE for the above-mentioned models is presented in Table 5. On scrutiny, it has been found that the hybrid model 1 integrating ARIMA–ANN–TBATS has a lesser value error of MAE for BRIC as compared to other hybrid models—Brazil (MAE = 0.0154), Russia (MAE = 0.1522), India (MAE = 0.0469) and China (MAE = 0.0097).

Hybrid model 2 integrating ARIMA–ANN–ETS and hybrid model 3 integrating ANN–TBATS have a higher level of standard error in comparison to hybrid model 1. Hence, hybrid model 2 and hybrid model 3 are not processed further in residual hybrid model. The forecasted values from the three hybrid models of exchange rate data series for the BRIC nations are shown in Figures 5 –8. From the figures, it is found that the model combining ARIMA–ANN–TBATS presents the best estimate to the real time series data than other models. The exchange rate time series are influenced by various factors, and the forecast for hybrid model 1 is adequate and within confidence levels. This proves that the exchange rate time series for all the four countries exhibit patterns following a non-linear trend and have multiple seasonality behavior that changes over time.

Table 5.

Accuracy of Forecast Using the LSSVM Method

Model	MAE Values
Model	Brazil	Russia	India	China
Hybrid 1 (ARIMA, ANN and TBATS)	0.0154	0.1522	0.0469	0.0097
Hybrid 2 (ARIMA, ANN and ETS)	0.0170	0.1610	0.0608	0.0121
Hybrid 3 (ANN and TBATS)	0.0196	0.1609	0.0647	0.0136

Source: Author’s calculations.

Figure 5.

Source: Prepared by author.

Figure 6.

Source: Prepared by author.

Figure 7.

Source: Prepared by author.

Figure 8.

Source: Prepared by author.

Results for Residual Hybrid Model Forecast (Hybrid Model 1 Including ARIMA–ANN–TBATS)

The accuracy test (Table 5) for the former approach demonstrated that hybrid model 1 combining ARIMA–ANN–TBATS achieved superior results than hybrid model 2nd hybrid model 3. Therefore, in a second procedure, a residual hybrid model has been used for forecasting by using the fitted values from the hybrid model 1, which integrates ARIMA–ANN–TBATS. The output described in Table 6 shows that for Brazil and China, ANN model gives the higher value of accuracy measures (Brazil MAE = 0.0133, China MAE = 0.001465 attributes being NNAR(1,1), $\in^{2}$ . = 0.0003092 and $\in^{2}$ = 0.0004821, respectively) than for Russia and India, TBATS model provides the best accuracy measure (Russia MAE = 0.02458 and India MAE = 0.01916 attributes being BATS(1,{0,0},1,–) α = −0.0201 for both the countries and β = 0.031191 and −0.09939, respectively). From the MAE values between hybrid model 1 and residual hybrid model 1, it is evident that the residual hybrid model has the best results to be used as a prognostic method. The results also established that Brazil and China return series follow a non-linear pattern, and India and Russia follow non-linear and complex seasonal pattern as shown in Figures 9 –12.

Table 6.

Results for Residual Hybrid Model Forecast (hybrid model 1 including ARIMA, ANN and TBATS)

Model	MAE Values
Model	Brazil	Russia	India	China
ARIMA	0.0149	0.02707	0.04619	0.00159
ANN	0.0133	0.02697	0.02480	0.001465
TBATS	0.0151	0.02458	0.01916	0.00177

Source: Author’s calculations.

Figure 9.

Source: Prepared by author.

Figure 10.

Source: Prepared by author.

Results for the Automatic Hybrid Forecast Model

In the final procedure, the analysis has been conducted using automatic hybrid forecast programme in R statistical software. Table 7 shows the results of the statistical errors among the likely blend possible. In the case of Brazil (MAE = 0.5370), Russia (MAE = 0.4432) and China (MAE = 0.0659), the hybrid model ANN–TBATS showed the lowest error value. In the case of India, the lowest error value (MAE = 0.2667) results were shown for hybrid model ARIMA–ETS–ANN. The empirical results of the accuracy measures indicate that the hybrid model of ANN–TBATS performs better for Brazil, Russia and China than other models—it has the lower value of MAE. The ANN method identifies the non-linear behaviour of the data, while TBATS identifies the seasonality behaviour, which changes over time. The next combination where the value of error was least was ARIMA–ANN–TBATS for Brazil (55.75%), India (26.76%), China(6.89%) and the stand-alone ANN model (44.89%) for Russia. Since the variation in values is diminutive, the two models can be used as a prognostic model as shown in Figures 13 –16.

Figure 11.

Source: Prepared by author.

Figure 12.

Source: Prepared by author.

Table 7.

Forecast Accuracy of Automatic Forecast Model Using R (V 8.11)

Model	MAE Values
Model	Brazil	Russia	India	China
ARIMA	0.6099	0.5257	0.1269	0.0977
ETS	0.6094	0.5254	0.1269	0.0790
ANN	0.6079	0.4489	0.1280	0.0709
TBATS	0.6118	0.5285	0.1272	0.0793
ARIMA–ANN–ETS	0.5605	0.4690	0.1266	0.0698
ARIMA–ETS–TBATS	0.6098	0.5284	0.1278	0.0787
ARIMA–ANN–TBATS	0.5575	0.4722	0.1267	0.0689
ARIMA–ANN–Thetam–TBATS–ETS	0.5720	0.4821	0.1268	0.0711
ANN–ETS–TBATS	0.5581	0.5264	0.1269	0.0700
ANN–TBATS	0.5370	0.4432	0.1268	0.0659
ARIMA–TBATS	0.6111	0.5284	0.1269	0.0788

Source: Author’s calculations.

Figure 13.

Source: Prepared by author.

Figure 14.

Source: Prepared by author.

Figure 15.

Source: Prepared by author.

Figure 16.

Source: Prepared by author.

The automatic hybrid models in R have provided a higher level of standard errors in accuracy (MAE) in comparison to the residual hybrid model. Thus, the residual hybrid model has been accepted as the method that provides the best approximation to the actual exchange rate series.

Conclusion

This article analysed exchange rate time series of emerging nations BRIC based on daily data from 2005 to 2018 and used hybrid models to obtain an accurate prediction for the upcoming 42 days. To this end, three methodologies using hybrid models were used, namely LSSVM method; residual hybrid model with ARIMA, ANN and TBATS; and finally, the automatic hybrid model proposed in software R.

The results indicated that when LSSVM method was employed, hybrid model 1 integrating ARIMA–ANN–TBATS yielded a better forecast than the stand-alone models and other hybrid models by means of a 42-day rolling forecast. Accuracy measure MAE shows that the hybrid model integrating ARIMA–ANN–TBATS outperformed other methods of forecast that were considered, and the results for Brazil (MAE = 0.0154), Russia (MAE = 0.1522), India (MAE = 0.0469) and China (MAE = 0.0097) provide the lower value between the proposed models.

Next, the fitted values from hybrid model 1 were applied in building residual hybrid forecasting models, and the results found were more precise and had the best attributes to be used as the prophetic model. The results obtained proved that the proposed residual hybrid model framework outperformed other models used in the study with Brazil (MAE = 0.0133)and China (MAE = 0.001465) return series reflecting the best accuracy in ANN model and India (MAE = 0.01916) and Russia (MAE = 0.02458) demonstrating the best accuracy in TBATS model. This is more vital, as exploration for the best forecasting model is seldom incomplete without a post-processing methodology like the residual hybrid model employed in this study. The results also show that Brazil and China return series follow a non-linear pattern, while India and Russia follow non-linear complex seasonal pattern. A limitation of this study is that it remains unknown whether the residual hybrid model developed in this study can be applied to forecasting currencies of other emerging economies in South America and Africa. Another limitation of this study is that the model addresses short-term forecasts of exchange rate series, and further studies could be performed to test whether such models are applicable for exchange rate forecasts in the long run. Policymakers may consider the findings from this study to manage exchange rates specifically in case of managed currency regimes. The formidable task was to demonstrate that not all the methodologies proffered on forecasting are efficient because they are influenced by the class of the time series. Exchange rate series are impacted by various unbalanced factors, and it is crucial to work on many methods and blend them to achieve the best accuracy forecast model. The residual hybrid model proposed in this study was significantly more effectual compared to the models proposed earlier in the literature.

Footnotes

Acknowledgement

The authors are grateful to the anonymous referees and Dr Arindam Banik for their extremely useful suggestions to improve the quality of the article. Usual disclaimers apply.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

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