^{1}

^{1}

^{1}

The use of historical data is important in making the predictions, for instance in the exchange rate. However, in the construction of a model, extreme data or dirtiness of data is inevitable. In this study, AR model is used with the exchange rate historical data (January 2007 until December 2007) for USD/MYR and is divided into 1-, 3- and 6-horizontal months respectively. Since the presence of extreme data will affect the accuracy of the results obtained in a prediction. Therefore, to obtain a more accurate prediction results, the bootstrap approach was implemented by hybrid with AR model coins as the Bootstrap Autoregressive model (BAR). The effectiveness of the proposed model is investigated by comparing the existing and the proposed model through the statistical performance methods which are RMSE, MAE and MAD. The comparison involves 1%, 5% and 10% for each horizontal month. The results showed that the BAR model performed better than the AR model in terms of sensitivity to extreme data, the accuracy of forecasting models, efficiency and predictability of the model prediction. In conclusion, bootstrap method can alleviate the sensitivity of the model to the extreme data, thereby improving the accuracy of forecasting model which also have high prediction efficiency and that can increase the predictability of the model.

One of an important aspect in the financial system is the exchange rate. The exchange rate is a rate at which an order currency converted at each other, in addition also referred to as rates of exchange or exchange rates or foreign exchange rate currency arrangement. [

The Bretton-Woods system of monetary management was introduced in July 1944. However, the system failed in 1971. Since then, much attention has been given to study and forecasts on monetary management. For instance, forecasting the monetary policy in order to maintain the exchange rate. Generally, the exchange rates have a relationship with business in the financial markets. Therefore, it is important for every businessman, corporate members, academicians and anyone related to be aware of forecasting exchange rates.

However, there are constraints that can reduce the accuracy of the prediction in the next set of data that can affect the accuracy of the prediction. Outliers in data are one of the constraints that often cause a high rate of inaccuracy in forecasting. In statistics, outliers are also known as extreme data which is an observation of far deviated from the other data. According to [

The presence of outliers in the data indeed gives a significant impact on currency exchange rate. From the problems encountered, it is visibly important to establish an approach that can reduce and subsequently solve the problem. The presence of outliers will lead to a high sensitivity to the forecasting models. The consequence of this high sensitivity is that it produces less accurate results, thus providing wrong conclusions. The accuracy of forecasting is important to identify the efficiency of forecasting models used. An efficient forecasting model will in turn increase the predictability of the model. Thus, bootstrap approach will be hybridized to the AR model in this study of exchange rate currency.

Bootstrap approach which is used in this study gives significant impact on exchange rates currency. The method eventually could reduce the effect of outliers as well as provide a more accurate estimation by bootstrap procedure of sampling with replacements. The hybrid between an AR model with bootstrap approach will form a model known as BAR model. BAR models that are less sensitive to outliers allows high predictability of exchange rates currencies.

The high predictability also enables higher forecast accuracy of future exchange rates. According to [

From the proposed model, five objectives which are divided to twofold were highlighted where the first two objectives are tested for accuracy and sensitivity of the model. While the last three are more for the determination of the efficiency, forecasting predictability as well as the appropriate forecasting model. The five objectives of the study are firstly to test the accuracy of the forecasting of USD/MYR exchange rates, currency in 1-, 3- and 6-horizontal months. Secondly, is to test the sensitivity of the model with 1%, 5% and 10% outliers respectively in 1-, 3- and 6-horizontal months. The next objective is to determine the efficiency of the constructed forecasting models based on the statistical forecasting values obtained at 1-, 3- and 6-horizontal months. The fourth objective is to determine the forecasting predictability for USD/MYR exchange rates, currency in 1-, 3- and 6-horizontal months. While, the last objective is to determine the most appropriate forecasting model for USD/MYR exchange rates, currency based on forecasting accuracy, sensitivity to outliers, forecasting efficiency and forecasting predictability in 1-, 3- and 6-horizontal months.

Previous studies discussed more on Regression model with robust approach [

Bootstrap method is a computer-based method for assessing the accuracy of the statistics [

Several years later, [

As mentioned before, there are still only a few studies on the bootstrap approach in AR model. Furthermore, [

In this study the US Dollar (USD) to Malaysian Ringgit (MYR), USD/MYR exchange rate currencies are considered. The exchange rate data used in this study is taken from Bank Negara Malaysia starting from January 2007 until December 2007 which can be referred to in

Motivated by this issue, the historical exchange rate of USD/MYR is divided into 1-, 3- and 6-horizontal months which involves 227, 186 and 126 data respectively. The selected sample size is motivated by the availability of data during the study. However, this study is limited to determining the appropriate model for the exchange rates currencies of USD/MYR. To be precise, this study only involves the determination of an

appropriate forecasting model for USD/MYR exchange rate currencies and does not involve data forecasting.

AR model is about the time series as a linear function of the historical sample data. This model is quite similar to the multiple regression model where the variable of the AR model is regressed with the previous

by neglecting

for

AR model has been widely used in forecasting and predicting of the actual process, based on the analysis discrete and continuous time series. A linear AR model for order k contain adequate statements for the current observation as a combination of previous samples, which was disrupted by some noisy model:

In general, the AR (1) process is given by:

If a process does not depend on the future, such as AR (1) with

Statistics is a science of learning through experience, or in other words, through a learning curve. The statistical theory initiated one of the basic issues that are to express the accuracy of a data summary which is part of a process known as statistical inference. Bootstrapping is the latest development techniques to create certain kind of statistical inference. It is a computer-based method to determine the accuracy measure to statistical estimates. Bootstrap was introduced in 1979 as a computer-based method to estimate the standard error of

The Bootstrap method is a technique of sampling with replacement. Bootstrap method provides an estimate for any type of statistics such as standard errors, confidence intervals, distribution, etc. Generally, the bootstrap method is being implemented by taking B new samples with replacement from the observed data. Furthermore, the statistics which are to be tested for each new data set is then calculated to obtain bootstrap distribution. The main idea of the bootstrap method is to repeat the sampling of the original data to create, replicate data sets where the features of interest can be assessed. Samples can be taken directly (nonparametric bootstrap) or through the suitability model (parametric bootstrap).

Least-squares estimation method is commonly used to estimate the parameters of a linear model. In some particular assumptions, the least-squares estimator is the best unbiased linear. One of the significant assumptions of the linear model is that the error terms are normally distributed. However, the performance of the least-squares estimator is lower when the error is assumed to be normally distributed. Thus, the parameters of the AR model are estimated by the least squares method without an assumption on the error distribution [

One of the ways to deal with this problem is to apply the bootstrap technique, which does not depend on the assumption of normality. The Bootstrap method [

The Bootstrap method is sampling with replacement from a sample that is to take samples at random from the original sample. Bootstrap sampling depends on the sample itself as just as many sources owned. The principle of bootstrap equality states that the bootstrap estimator for sub sampling (bootstrap method) is equal to the estimated sample. For example, to evaluate the accuracy of N sample data (statistical sample), sampling of N bootstrap is used and hence the statistics from each bootstrap is calculated. The values of bootstrap statistics are used to evaluate the statistics’ accuracy of the original statistical samples.

The determination of performance for the implementation of bootstrap method is also based on comparisons of the statistic forecasting that is RMSE, MAE and MAD. Comparison of the statistical forecasting for the constructed model is carried out to determine the sensitivity of the model. The sensitivity of the model is determined based on the comparison of different outliers. The comparison is between the AR model and BAR model with 1%, 5% and 10% outliers respectively.

The smallest value of statistical forecast with the increase of outliers indicates a lower sensitivity of the model to outliers. While the larger value of statistical forecast with the increase of outliers indicates a higher sensitivity of the model to outliers. The achievement, determination for the implementation of the AR model with bootstrap approach, BAR will be discussed further in the next discussion on results and findings.

There are several bootstrap methods for time series. These include the periodogram bootstrap, block bootstrap and the residual bootstrap [

In this study, the development of AR models using the bootstrap approach hybrid is proposed. For this purpose, the bootstrap algorithm was developed with the following steps:

Step 1:

Consider

Step 2:

Further, sampling with replacement (bootstrap) was performed on the value of residue,

Step 3:

From the value of residual bootstrap,

Step 4:

Based on new data obtained before, get an average for the

Step 5:

From the dependent variable

Step 6:

Step 1 is repeated to estimate the parameters, which is

Step 7:

Based on the information obtained in the previous steps, obtain the bootstrap estimates,

Therefore, it can be concluded that the general AR model with hybridize bootstrap approach (BAR) Model can be written as follows:

Step 8:

For purposes of comparison in terms of efficiency, sensitivity, accuracy and predictability of the model, get the Root Mean Square Error (RMSE), Mean Absolute Error (MAE) and Median of Absolute Deviation (MAD). The evaluation and comparison of predictions are based on three statistical forecasting [

1) Root mean Square Error (RMSE),

2) Mean Absolute Error (MAE),

3) Median of Absolute Deviation (MAD),

where

In this section, the discussion concerning the effectiveness of the proposed model was performed. The least-squares estimator is widely used, especially in parameter estimation of a model. However, the obvious weakness is that it has a high sensitivity to outliers. Nevertheless, methods to reduce the sensitivity of least-squares estimator to outliers have been identified. The intended method involves the sampling with replacement technique or better known as the bootstrap method [

This section will discuss more about the application of BAR exchange rates, currency of the US Dollar to Malaysian Ringgit, USD/MYR. The forecast model estimation was obtained using the data exchange rate sample data shown in the previous

Based on Tables 1-3, the respective sample size of data are 227, 186 and 126. For the robust method, the sample data is considered to cause contamination by 1%, 5% and 10% for all three horizontals. The bootstrap approach took place after the contaminated

RMSE | MAE | MAD | |
---|---|---|---|

AR^{a} | 0.295102000 | 0.294506300 | 0.295715600 |

BAR (1%)^{b} | 0.034463560 | 0.034398430 | 0.034533960 |

BAR (5%) | 0.018752940 | 0.017589420 | 0.016694980 |

BAR (10%) | 0.016840210 | 0.016516860 | 0.018040160 |

^{a}AR refer to autoregressive model. ^{b}BAR refer to bootstrap autoregressive model for different percentages. The bold values are refer to most minimum estimation value of RMSE, MAE and MAD.

RMSE | MAE | MAD | |
---|---|---|---|

AR^{a} | 0.293880800 | 0.293142900 | 0.294708300 |

BAR (1%)^{b} | 0.047553280 | 0.047304910 | 0.047365930 |

BAR (5%) | 0.015749730 | 0.009573013 | 0.007507964 |

BAR (10%) | 0.011392750 | 0.008073250 | 0.007787416 |

^{a, b}The abbreviation can be referred to

RMSE | MAE | MAD | |
---|---|---|---|

AR^{a} | 0.302655100 | 0.301573800 | 0.303919900 |

BAR (1%)^{b} | 0.067600590 | 0.067309690 | 0.067981270 |

BAR (5%) | 0.022338350 | 0.012197780 | 0.008830690 |

BAR (10%) | 0.035483420 | 0.034912520 | 0.039091910 |

^{a, b}The abbreviation can be referred to

samples data were used in estimating the AR model. The complete procedure of bootstrap can be referred to in section 3.1 where the algorithm of sampling of the residual model of AR with the replacement obtained. Through all procedures, three hybrid models were obtained, namely BAR (1%), BAR (3%) and BAR (6%). Without disregard to the standard AR model, the evaluation of the forecast is made using the equations of RMSE, MAE and MAD in order to find the best model of estimation which literally determines the highest accurate model.

Each table shows the performance results of forecasting models for a US Dollar to Malaysian Ringgit, USD/MYR. Tables 1-3 indicate the results for 1-, 3- and 6-horizontal months for forecasting. In each panel, row (2) to (4) shows the measure of forecasting accuracy, RMSE, MAE and MAD. Forecasting accuracy measure which reported in

For 1-horizontal month in

The results obtained indicate the diversity of achievement for each model used are in horizontal month’s forecast. However, there are similarities that can be seen in terms of the best model, in order to measure the forecast accuracy for all these three horizontal months. The AR model with bootstrap approach hybrid at 10%, BAR (10%), and at 5%, BAR (5%), outliers are found tend to give smallest value of these three statistical forecasts which makes bootstrap models with 5% and 10% outliers is the best for measuring the forecast accuracy for these three horizontal months.

Nevertheless, there are still differences between the values of the result of statistical forecast for BAR (10%) model and BAR (5%) model in each 1-, 3- and 6-horizontal months. The value of statistical forecast for BAR (5%) is greater compared to the value of statistic forecast for BAR (10%) which explains that the exchange rates, currency of USD/MYR forecast is less accurate for 6-horizontal months. The forecasting reached its maximum accuracy at 3-horizontal months due to the smallest value of forecast accuracy given compared to 1-horizontal months. Thus, forecasting exchange rates, currency of the US Dollar to Malaysian Ringgit is more accurate when tested at 3-hori- zontal months by using BAR (10%) model.

One of the criteria in the selection of forecasting models is the ability of the proposed model to provide an accurate prediction despite the presence of outliers. The forecasting ability can be measured by identifying the model’s sensitivity to outliers that can be identified based on the values of forecast statistics, RMSE, MAE and MAD which indicated with various outliers [

There are three levels of outliers involved in this study, which are 1%, 5% and 10% outliers. Smaller values of forecast statistics with an increase of outliers revealed that the model has a high sensitivity to outliers, thereby enabling this model to be the most appropriate forecasting medium for a tested data.

According to

For 1- and 3-horizontal months, the AR model with bootstrap approach hybrid at 10%, BAR (10%), outliers revealed the smallest value of statistical forecast for RMSE and MAE, followed by the AR model with bootstrap hybrid at 5%, BAR (5%), outliers then followed by the AR model with bootstrap approach hybrid at 1%, BAR (1%) outliers which gives greatest value among all three models with outliers. Similarly to the 6-horizontal months data which also gives the smallest value of forecast statistic; RMSE, MAE and MAD, even on the AR model with bootstrap approach hybrid at 5%, BAR (5%) outliers, but still it produces the smaller value compared to model at 1% outliers, BAR (1%).

Overall, the AR model with the bootstrap approach hybrid, BAR, which has a greater outliers is more sensitive to the presence of extreme data that could further improve the forecasting accuracy, especially for those that is very risky to have extreme data.

A developmental model is best when it fulfills the estimation’s criteria which is known as Best, Linear, Unbiased, Efficient, or in short, BLUE. In addition, the sensibility of a model which is closely related to the sensitivity of the model to outliers need to be considered. Therefore, the bootstrap approach hybrid on the AR model is developed. The bootstrap model has a high sensitivity to outliers. This can be proved by the smaller values of statistic forecast even with the increase in the percentage of contaminated data.

Therefore, it can be concluded that the AR model with the bootstrap approach is more efficient than the AR model itself. Thus, the developed bootstrap model can be applied as a forecasting model for exchange rate data.

Research has shown that the exchange rates’ predictability is interrelated with the forecasting accuracy which can be measured by the model used in [

Predictability of exchange rates can be determined by comparing the AR model and BAR model through the values of statistics forecast; RMSE, MAE and MAD. From the previous discussion, it clearly shows the AR model with the bootstrap approach hybrid, BAR (1%), BAR (5%) and BAR (10%) has a higher forecast accuracy than the AR model.

For the exchange rates of Malaysian Ringgit to US Dollar (USD/MYR), the predictability is based on the comparison shown in Tables 1-3. The values of forecast statistics vary according to horizontal months. However, the values of forecast statistic shown by the three proposed models are smaller compared to the values of forecast statistic when using an AR model. This can be seen in the results for the 3-horizontal months in

Through comparison with the values of the forecast statistics; RMSE, MAE and MAD obtained for the AR model, each with 0.293880800, 0.293142900 and 0.294708300 respectively, it clearly shows that the forecast values of the proposed model are smaller. This proved that the predictability of the exchange rates is better by using the BAR model because it can result in more accurate forecast statistics in which it can acquire even the smallest of errors, even with the increase of outliers.

In this study, the AR model was used. However, this model is sensitive to outliers. So, in order to alleviate this problem, the BAR model was proposed. The bootstrap approach was hybridized to the AR model, and produced a new model known as the BAR model. The results revealed that the proposed model has improved the forecasting accuracy, reduced the sensitivity of the AR model to outliers, and increasing the efficiency of the forecasting model. It can be proven by the comparison of the BAR model and AR model at 1%, 5% and 10% outliers through the smaller values of RMSE, MAE and MAD. From the results obtained, the proposed model has improved the AR model. This condition, thus also improved the level of predictability of the AR model on the tested exchange rate currencies.

The authors would like to acknowledge the financial support from the Graduate School of Universiti Malaysia Terengganu (UMT) through the Graduate Financial Scheme (GFS), and also the School of Informatics and Applied Mathematic (SIAM), as well as the Research Management Centre (RMC) of University Malaysia Terengganu for their continuous and .diligent support.

Lola, M.S., David, A. and Zainuddin, N.H. (2016) Bootstrap Approaches to Autoregressive Model on Exchange Rates Currency. Open Journal of Statistics, 6, 1010-1024. http://dx.doi.org/10.4236/ojs.2016.66081