Forecasting the US Dollar / Euro Exchange rate Using ARMA Models

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1 Forecasting the US Dollar / Euro Exchange rate Using ARMA Models LIUWEI ( ) - 1 -

2 ABSTRACT INTRODUCTION DATA ANALYSIS Stationary estimation Dickey-Fuller Test MODEL CREATION AR Model MA Model ARMA Models COMPARING THE MODELS FORECASTING WITH THE BEST MODEL REFERENCE

3 Abstract The aim of this paper is to forecast the exchange rate of US Dollar / Euro in the month of February 2005, using different methods, such as AR, MA, and ARIMA. Comparing the models by the Schwarz criterion, the best model is selected. The Euro/ Us Dollar exchange rate data is selected from the Austrian National Bank statistical data base

4 1. Introduction We could see here the data of the exchange rate of US Dollar / Euro from the beginning 1999 till end of 2004, total 1537 observations. The data points for the year 1999 are 1 to 259, for 2000 they are 260 to 514, for 2001 they are 515 to 769, for 2002 they are 770 to 1023, for 2003 they are 1024 to 1278, for 2004 they are 1279 to observations in the first month of 2005 are reserved for forecasting EXCHANGE_RATE - 4 -

5 The curve of the dollar/euro rate shown above tells the history of the Euro. At the introduction of the Euro the rate was almost 1.16 US dollars, and the rate went down for two years. In 2001 the exchange rate reached a minimum and after that year euro went up. In 2004 the Euro increased to almost 1.4 US dollars. I use linear Time Series Analysis as the tool for the empirical analysis, based on ARIMA models. 2. Data Analysis 2.1 Stationary estimation Stationary modeling is based on the assumption that the process is in a particular state of statistical equilibrium. A stochastic process is said to be strictly stationary if its properties are unaffected by a change of time origin. (Box and Jenkins) - 5 -

6 2.2 Dickey-Fuller Test A very simple way of determining whether the data is stationary or not, is a Unit Root Test. In the E-Views Software we use the Augmented Dickey Fuller Test. The following is the E-Views output of the test: Null Hypothesis: EXCHANGE_RATE has a unit root Exogenous: Constant Lag Length: 0 (Automatic based on SIC, MAXLAG=23) t-statistic Prob.* Augmented Dickey-Fuller test statistic Test critical values: 1% level % level % level *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(EXCHANGE_RATE) Method: Least Squares Sample(adjusted): Included observations: 1535 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. EXCHANGE_RATE(-1) C R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) In this version without added trend, the unit-root test confirms that there is a unit root in the data generating process (DGP). Thus, it appears that the DGP is not stationary

7 Next, we take first differences of the data and check whether this transformation implies stationarity. Now, the variable under investigation is called D(Exchange_Rate). Now we have that: Null Hypothesis: D(EXCHANGE_RATE) has a unit root Exogenous: None Lag Length: 0 (Automatic based on SIC, MAXLAG=12) t-statistic Prob.* Augmented Dickey-Fuller test statistic Test critical values: 1% level % level % level *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(EXCHANGE_RATE,2) Method: Least Squares Sample(adjusted): Included observations: 1534 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. D(EXCHANGE_RAT E(-1)) R-squared Mean dependent var 4.17E-06 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat We see that there is no unit root in the differenced data. The differenced exchange rate appears to be stationary

8 3. Model Creation 3.1 AR Model Autoregressive models are often well suited for modeling economic time series. An autoregressive model of first order can be written as: X ρ X + e ; t = t 1 t In short, the value of today depends on the value of yesterday. I would like to say the behavior of the exchange market should be under some influence from the price increase of yesterday or the day before yesterday. I think we should use the AR(1) or AR(2) to estimate the first difference of the data

9 And we have that: EX_1 The graph shows the first difference of the exchange rate data that we found to be stationary. I first use the AR(1) model, that is: Dependent Variable: EX_1 Method: Least Squares Sample(adjusted): Included observations: 1534 after adjusting endpoints Convergence achieved after 3 iterations Variable Coefficient Std. Error t-statistic Prob. AR(1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat Inverted AR Roots

10 3.2 MA Model The moving-average model can also be a good model for economic data. For example, the MA model of first order is: y t = 0 + at θ1at 1 φ. For the model estimate, we have that: Dependent Variable: EX_1 Method: Least Squares Sample(adjusted): Included observations: 1535 after adjusting endpoints Convergence achieved after 6 iterations Backcast: 1 Variable Coefficient Std. Error t-statistic Prob. MA(1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat Inverted MA Roots

11 3.3 ARMA Models Also, the mixed ARMA model can be applied to first differences. For example, the ARMA(1,1) model is: X t = φ X t 1 + εt + θεt 1 The estimation output for the ARMA(1,1) specification is: Dependent Variable: EX_1 Method: Least Squares Sample(adjusted): Included observations: 1534 after adjusting endpoints Convergence achieved after 17 iterations Backcast: 2 Variable Coefficient Std. Error t-statistic Prob. AR(1) MA(1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat Inverted AR Roots.52 Inverted MA Roots

12 4. Comparing the Models To choose the best model here, we should know the criterion first. Here I would like to choose the Schwarz criterion. The smaller the value of the Schwarz criterion, the better is the estimation. We collect here the values for all three models: the Schwarz criterion for AR(1) is ; for MA(1) it is ; and for ARMA(1,1) it is So here the best Model is MA(1). 5. Forecasting with the Best Model For the complete sample, MA(1) is the best. However, I would like to focus on the data after observation #800. Why I would like to choose the date after 800. First, from the graph, we can see the strong rising trend in the Euro/ Us Dollar exchange rate. Second, at the beginning of the year 2002, some political and monetary political issues has been changed. So the data after 800 is good for us to estimate the changes in the exchange rate

13 Now, we have: Dependent Variable: EX_1 Method: Least Squares Sample(adjusted): Included observations: 737 after adjusting endpoints Convergence achieved after 4 iterations Backcast: 799 Variable Coefficient Std. Error t-statistic Prob. MA(1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat Inverted MA Roots.03 Forecasting results of the exchange rate for the data from 1537 to 1557 are as follows: EX_1F

14 A graph that includes the observations from #800 to #1536 as well as the forecasts is given below EX_1F The graph confirms that changes in the exchange rate are predicted as zero, except for the first step

15 6. Reference Fuller (1976): Introduction to Statistical Time Series; John Wiley. Box/ Jenkins (1976): Time Series Analysis, Forecasting and Control; Prentice-Hall Hall, Cuthbertson, Taylor (1992): Applied Econometric Techniques; Phillip Allan Mills (1990): Time series techniques for economists; Cambridge. Granger/ Newbold (1986): Forecasting economic time series; Academic Press Montgomery, Johnson, Gardiner (1990): Forecasting and Time series Analysis; McGraw-Hill Brockwell/ Davis (1991): Time Series: Theory and Methods; Springer Granger (1989): Forecasting in Business and Economics; Academic Press

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