Forecasting the US Dollar / Euro
Exchange rate
Using ARMA Models
LIUWEI (9906360)
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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
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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.
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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.
1.4
1.3
1.2
1.1
1.0
0.9
0.8
250
500
750
1000
EXCHANGE_RATE
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1250
1500
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)
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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
Augmented Dickey-Fuller test statistic
0.193817
Test critical values:
1% level
-3.434404
5% level
-2.863217
10% level
-2.567711
*MacKinnon (1996) one-sided p-values.
Prob.*
0.9722
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
EXCHANGE_RATE(-1)
0.000257 0.001324
0.193817
C
-0.000146 0.001381 -0.105866
R-squared
0.000025 Mean dependent var
Adjusted R-squared
-0.000628 S.D. dependent var
S.E. of regression
0.006877 Akaike info criterion
Sum squared resid
0.072501 Schwarz criterion
Log likelihood
5466.565 F-statistic
Durbin-Watson stat
2.032730 Prob(F-statistic)
Prob.
0.8463
0.9157
0.000119
0.006875
-7.119954
-7.113002
0.037565
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.
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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
Augmented Dickey-Fuller test statistic
-39.78823
Test critical values:
1% level
-2.566470
5% level
-1.941030
10% level
-1.616560
*MacKinnon (1996) one-sided p-values.
Prob.*
0.0000
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
D(EXCHANGE_RAT -1.015946
0.025534 -39.78823
E(-1))
R-squared
0.508039 Mean dependent var
Adjusted R-squared
0.508039 S.D. dependent var
S.E. of regression
0.006876 Akaike info criterion
Sum squared resid
0.072484 Schwarz criterion
Log likelihood
5462.683 Durbin-Watson stat
Prob.
0.0000
4.17E-06
0.009804
-7.120838
-7.117360
2.000908
We see that there is no unit root in the differenced data. The differenced
exchange rate appears to be stationary.
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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 t = ρX t −1 + et ;
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.
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And we have that:
.04
.03
.02
.01
.00
-.01
-.02
-.03
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
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Inverted AR Roots
-0.000063
-0.000063
0.006876
0.072484
5462.683
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Durbin-Watson stat
-.02
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0.000122
0.006876
-7.120838
-7.117360
2.000908
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 = φ 0 + at − θ 1 a t −1 .
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
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
-0.000026
-0.000026
0.006875
0.072505
5466.526
Inverted MA Roots
.02
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Durbin-Watson stat
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0.000119
0.006875
-7.121207
-7.117731
1.998331
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 = φX t −1 + ε t + θε t −1
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)
MA(1)
0.519828
-0.543137
0.409129
0.402405
1.270572
-1.349726
0.2041
0.1773
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Inverted AR Roots
Inverted MA Roots
0.001437
0.000786
0.006873
0.072376
5463.834
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Durbin-Watson stat
.52
.54
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0.000122
0.006876
-7.121036
-7.114079
1.987838
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.
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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
MA(1)
-0.029713
0.036846 -0.806411
R-squared
-0.008515 Mean dependent var
Adjusted R-squared
-0.008515 S.D. dependent var
S.E. of regression
0.006932 Akaike info criterion
Sum squared resid
0.035366 Schwarz criterion
Log likelihood
2618.826 Durbin-Watson stat
Inverted MA Roots
.03
Prob.
0.4203
0.000669
0.006903
-7.104007
-7.097762
1.999597
Forecasting results of the exchange rate for the data from 1537 to 1557 are
as follows:
.00000
-.00001
-.00002
-.00003
-.00004
-.00005
-.00006
1540
1545
1550
EX_1F
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1555
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.
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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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