Forecasting the US Dollar / Euro Exchange rate
Using ARMA Models
LIUWEI (9906360)
ABSTRACT... 3
1. INTRODUCTION ... 4
2. DATA ANALYSIS... 5
2.1 Stationary estimation ... 5
2.2 Dickey-Fuller Test... 6
3. MODEL CREATION... 8
3.1 AR Model ... 8
3.2 MA Model... 10
3.3 ARMA Models... 11
4. COMPARING THE MODELS ... 12
5. FORECASTING WITH THE BEST MODEL ... 12
6. REFERENCE... 15
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.
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 1537. 20 observations in the first month of 2005 are reserved for forecasting.
0.8 0.9 1.0 1.1 1.2 1.3 1.4
250 500 750 1000 1250 1500
EXCHANGE_RATE
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)
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 0.193817 0.9722 Test critical values: 1% level -3.434404
5% level -2.863217
10% level -2.567711
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation Dependent Variable: D(EXCHANGE_RATE) Method: Least Squares
Sample(adjusted): 2 1536
Included observations: 1535 after adjusting endpoints
Variable Coefficient Std. Error t-Statistic Prob.
EXCHANGE_RATE(-1) 0.000257 0.001324 0.193817 0.8463 C -0.000146 0.001381 -0.105866 0.9157 R-squared 0.000025 Mean dependent var 0.000119 Adjusted R-squared -0.000628 S.D. dependent var 0.006875 S.E. of regression 0.006877 Akaike info criterion -7.119954 Sum squared resid 0.072501 Schwarz criterion -7.113002 Log likelihood 5466.565 F-statistic 0.037565 Durbin-Watson stat 2.032730 Prob(F-statistic) 0.846345
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.
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 -39.78823 0.0000 Test critical values: 1% level -2.566470
5% level -1.941030
10% level -1.616560
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation Dependent Variable: D(EXCHANGE_RATE,2) Method: Least Squares
Sample(adjusted): 3 1536
Included observations: 1534 after adjusting endpoints
Variable Coefficient Std. Error t-Statistic Prob.
D(EXCHANGE_RAT E(-1))
-1.015946 0.025534 -39.78823 0.0000 R-squared 0.508039 Mean dependent var 4.17E-06 Adjusted R-squared 0.508039 S.D. dependent var 0.009804 S.E. of regression 0.006876 Akaike info criterion -7.120838 Sum squared resid 0.072484 Schwarz criterion -7.117360 Log likelihood 5462.683 Durbin-Watson stat 2.000908
We see that there is no unit root in the differenced data. The differenced exchange rate appears to be stationary.
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:
1 t;
t
t X e
X =ρ − +
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.
And we have that:
-.03 -.02 -.01 .00 .01 .02 .03 .04
250 500 750 1000 1250 1500
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): 3 1536
Included observations: 1534 after adjusting endpoints Convergence achieved after 3 iterations
Variable Coefficient Std. Error t-Statistic Prob.
AR(1) -0.015946 0.025534 -0.624518 0.5324 R-squared -0.000063 Mean dependent var 0.000122 Adjusted R-squared -0.000063 S.D. dependent var 0.006876 S.E. of regression 0.006876 Akaike info criterion -7.120838 Sum squared resid 0.072484 Schwarz criterion -7.117360 Log likelihood 5462.683 Durbin-Watson stat 2.000908 Inverted AR Roots -.02
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:
1 1
0 + − −
= t t
t a a
y φ θ .
For the model estimate, we have that:
Dependent Variable: EX_1 Method: Least Squares Sample(adjusted): 2 1536
Included observations: 1535 after adjusting endpoints Convergence achieved after 6 iterations
Backcast: 1
Variable Coefficient Std. Error t-Statistic Prob.
MA(1) -0.017267 0.025529 -0.676378 0.4989 R-squared -0.000026 Mean dependent var 0.000119 Adjusted R-squared -0.000026 S.D. dependent var 0.006875 S.E. of regression 0.006875 Akaike info criterion -7.121207 Sum squared resid 0.072505 Schwarz criterion -7.117731 Log likelihood 5466.526 Durbin-Watson stat 1.998331 Inverted MA Roots .02
3.3 ARMA Models
Also, the mixed ARMA model can be applied to first differences. For example, the ARMA(1,1) model is:
1
1 −
− + +
= t t t
t X
X φ ε θε
The estimation output for the ARMA(1,1) specification is:
Dependent Variable: EX_1 Method: Least Squares Sample(adjusted): 3 1536
Included observations: 1534 after adjusting endpoints Convergence achieved after 17 iterations
Backcast: 2
Variable Coefficient Std. Error t-Statistic Prob.
AR(1) 0.519828 0.409129 1.270572 0.2041 MA(1) -0.543137 0.402405 -1.349726 0.1773 R-squared 0.001437 Mean dependent var 0.000122 Adjusted R-squared 0.000786 S.D. dependent var 0.006876 S.E. of regression 0.006873 Akaike info criterion -7.121036 Sum squared resid 0.072376 Schwarz criterion -7.114079 Log likelihood 5463.834 Durbin-Watson stat 1.987838 Inverted AR Roots .52
Inverted MA Roots .54
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 -7.117360; for MA(1) it is -7.117731; and for ARMA(1,1) it is -7.114079.
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.
Now, we have:
Dependent Variable: EX_1 Method: Least Squares Sample(adjusted): 800 1536
Included observations: 737 after adjusting endpoints Convergence achieved after 4 iterations
Backcast: 799
Variable Coefficient Std. Error t-Statistic Prob.
MA(1) -0.029713 0.036846 -0.806411 0.4203 R-squared -0.008515 Mean dependent var 0.000669 Adjusted R-squared -0.008515 S.D. dependent var 0.006903 S.E. of regression 0.006932 Akaike info criterion -7.104007 Sum squared resid 0.035366 Schwarz criterion -7.097762 Log likelihood 2618.826 Durbin-Watson stat 1.999597 Inverted MA Roots .03
Forecasting results of the exchange rate for the data from 1537 to 1557 are as follows:
-.00006 -.00005 -.00004 -.00003 -.00002 -.00001 .00000
1540 1545 1550 1555
EX_1F
A graph that includes the observations from #800 to #1536 as well as the forecasts is given below.
-.03 -.02 -.01 .00 .01 .02 .03
1000 1250 1500
EX_1F
The graph confirms that changes in the exchange rate are predicted as zero, except for the first step.
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.
