Bachelorarbeit

EUR/CHF Exchange rate returns and Macroeconomic data A Cointegrated vector autoregressive (VAR) approach

am Departement für Quantitative Wirtschaftsforschung der Universität Fribourg

vorgelegt von stud. rer. pol. Johannes MEIXNER aus Frankfurt

Leitung: Dr. J.-F. Emmenegger Frühlingssemester 2012 Abgabetermin: April 2012 Studentennummer: 07-203-177

Adresse des Verfassers Fritzlarer Str. 14 D-60487 Frankfurt [email protected]

EUR/CHF Exchange rate returns and Macroeconomic data

Contents Table of Contents

i

List of Figures

ii

List of Tables

iii

Introduction

1

1. Theoretical models 1.1. Univariate time series models . . . . . . . . . . . . . . . . . . . 1.1.1. Stationary processes . . . . . . . . . . . . . . . . . . . 1.1.2. Non-stationary processes . . . . . . . . . . . . . . . . . 1.2. Multivariate time series models . . . . . . . . . . . . . . . . . . 1.2.1. VAR(p) models . . . . . . . . . . . . . . . . . . . . . . . 1.2.2. Cointegration of economic time series . . . . . . . . . . 1.3. Hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1. Unit root tests . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2. Normality tests . . . . . . . . . . . . . . . . . . . . . . . 1.3.3. Cointegration tests - Johansen-Juselius approach . . . 1.3.4. Portmanteau tests for serial autocorrelation in residuals 1.3.5. Lagrange Multiplier tests for nth-order autocorrelation .

. . . . . . . . . . . .

2 2 2 8 9 9 10 12 12 13 13 15 16

. . . . . . . .

17 17 18 18 19 20 20 22 23

3. Results 3.1. Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Additional literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 28

Conclusion

31

A. Appendix A.1. Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2. Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32 32 37

Bibliography

41

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2. Empirical Model 2.1. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Explaining the EUR/CHF exchange rate returns . . . . . . . . . . . 2.3. Cointegration models . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. EUR/CHF and the trade balance . . . . . . . . . . . . . . . 2.3.2. EUR/CHF, trade balance and Swiss and EMU GDP . . . . 2.3.3. EUR/CHF, trade balance and Swiss and EMU broad money 2.3.4. EUR/CHF, trade balance and Swiss and EMU interest rates 2.3.5. EUR/CHF, Swiss and EMU GDP and real interest rates . .

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i

EUR/CHF Exchange rate returns and Macroeconomic data

List of Figures 1.1. AR(2) process with φ1 = 0.9 and φ2 = −0.15 . . . . . . . . . . . . . . . . . . .

5

2.1. Distribution of the EUR/CHF exchange rate returns in percent and fitted normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. EUR/CHF exchange rate x1 and fitted time series xˆ1,1 from model 1 . . . . . 2.3. EUR/CHF exchange rate and fitted time series xˆ1,3 from model 3 . . . . . . . 2.4. EUR/CHF exchange rate and fitted time series xˆ1,5 from model 5 . . . . . . . 2.5. Virtual 10 year real interest rates . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. EUR/CHF exchange rate and fitted time series xˆ1,7 and xˆ1,8 from model 6 . . 2.7. Virtual 1 year real interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8. EUR/CHF exchange rate and fitted time series from model 7 . . . . . . . . . 2.9. EUR/CHF exchange rate and fitted time series xˆ∗1,10 from model 7 . . . . . .

18 20 21 22 23 24 25 25 26

A.1. A.2. A.3. A.4.

Daily EUR/CHF exchange rates . . . . . . . . . . . . . . . . . . . . . . . . . Daily EUR/CHF exchange rate returns . . . . . . . . . . . . . . . . . . . . . Monthly trade balance of the ECU and Switzerland in million euros. . . . . . GDP of the European Currency Union (Nominal, in current prices) in million euros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5. Annualized EMU inflation rate . . . . . . . . . . . . . . . . . . . . . . . . . . A.6. ECU Monetary aggregate M3 (broad money) in billion Euro . . . . . . . . . . A.7. One and 10 year interest rates on German Bunds . . . . . . . . . . . . . . . A.8. Swiss GDP (Nominal, in current prices) in million Swiss Francs. . . . . . . . A.9. Swiss inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.10.Swiss Monetary aggregate M3 (broad money) in billion Swiss Francs . . . . . A.11.One and 10 year interest rates on Swiss federal obligations . . . . . . . . . .

32 32 33 33 34 34 35 35 36 36 37

ii

EUR/CHF Exchange rate returns and Macroeconomic data

List of Tables 1.1. Autocorrelation function and Partial Autocorrelation function of an AR(2) process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

5

2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7.

I(1)-ANALYSIS of model 1 without deterministic components I(1)-ANALYSIS of model 2 without deterministic components I(1)-ANALYSIS of model 3 without deterministic components I(1)-ANALYSIS of model 4 without deterministic components I(1)-ANALYSIS of model 5 without deterministic components I(1)-ANALYSIS of model 6 without deterministic components I(1)-ANALYSIS of model 7 without deterministic components

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. . . . . . .

. . . . . . .

19 20 21 22 23 24 26

A.1. A.2. A.3. A.4. A.5.

Dickey-Fuller test results for the log time series . . . . . . . . . . . . Dickey-Fuller test results for the first difference of log time series . . Basic statistics of the daily EUR/CHF exchange rate returns returns Basic statistics of the EMU GDP growth rate . . . . . . . . . . . . . Basic statistics of the Swiss GDP growth rate . . . . . . . . . . . . .

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37 38 38 38 38

iii

EUR/CHF Exchange rate returns and Macroeconomic data

Introduction Due to its heavy appreciation against foreign currencies over the last years, the nominal Swiss Franc exchange rate has come into focus of researchers, policy makers, and public media. A great variety of theories have been advanced, such as attributing the appreciation of the Swiss Franc to a "‘Safe haven characteristics"’ or an increasing risk aversion. This analysis tries to identify a long run relationship between the EUR/CHF exchange rate and a set of macroeconomic variables comprising the trade balance between Switzerland and the European Monetary Union (EMU), the respective Gross Domestic Products, interest rates, inflation rates and monetary aggregates. Foreign trade is directly influenced by the exchange rate. An appreciation of the Swiss Franc against the Euro increases the price of exported goods and services and consequently weakens Switzerlands competitive position, leading to fewer exports in the future. The opposite is true for imports. As foreign products become cheaper with an appreciating Swiss Franc, imports are expected to rise as the relative price of foreign products declines and consumers substitute domestic products. The relation between the exchange rate and interest rates is also straightforward, as future expected exchange rates can be calculated, given the nominal exchange rate and interest rates, due to the uncovered interest rate parity. The estimates are obtained by setting up a Cointegrated Vector Autoregressive model in the Regression Analysis in Time Series (RATS) 8.0 software package provided by Estima, Inc., Evanston, and their Cointegration Analysis in Time Series (CATS) 2.0 additional module. Results show that the hypothesis of normally distributed EUR/CHF exchange rate returns can be rejected. Cointegration is found for different models, with good fits for the model incorporating Swiss and EMU gross domestic product and real interest rates, and to a lesser degree broad monetary aggregates M3 of the EMU and Switzerland as well as trade balances between Switzerland and the EMU. However, due to serial autocorrelation the results need to be treated with caution. The outline of this analysis is as follows. In the first chapter, a theoretical introduction into time series analysis is given. Box-Jenkins methods are presented, vector autoregressive models are introduced and the notion of cointegration is explained. Test statistics for hypothesis tests are defined, enabling the treatment of data in the subsequent chapters. In the course of the second chapter, data sources are listed and different models explaining the EUR/CHF exchange rate are evaluated. The third chapter contains a discussion of results and additional literature.

1

EUR/CHF Exchange rate returns and Macroeconomic data

1. Theoretical models 1.1. Univariate time series models Modern time series analysis began to emerge after the seminal work of Box and Jenkins [1976]. Its methods were extended by auto-regressive (AR) processes, moving-average (MA) processes, and the combination of these two, auto-regressive moving-average processes (ARMA). Besides, an important contribution of their work is the sparse usage of variables, or parsimonious use of variables. In this section, fundamental properties and implications of the models outlined by Box and Jenkins are presented. First, the concept of stationarity is explained. Then follows a description of models which require the assumption of stationarity of the underlying process. Second, the assumption of stationarity is relaxed and concepts dealing with integrated processes are introduced. 1.1.1. Stationary processes The following definitions are taken from Lütkepohl [2004]. A stochastic process yt is called stationary if it has time-invariant first and second moments. In other words, yt is stationary if 1. E[yt ] = µ = constant for all t ∈ T and 2. E[(yt −µy )(yt−h −µy )] = γh = constant for all t ∈ T and all integers h such that t−h ∈ T . The first condition leads to yt fluctuating around its expected value µ, implying the absence of any trend. The second condition ensures that the covariance between two members of the process yt and yt−h does not depend on t but on the distance h of the members of the process. Autoregressive processes An important model within the class of univariate, stationary time series models is the autoregressive (AR) process of order p. A time series yt is regressed on its p previous values, or yt = φ1 yt−1 + φ2 yt−2 + ... + φp yt−p + t , (1.1)

2

1. Theoretical models where t is a mean-zero white noise process with constant variance E[2t ] = σ2 for any t ∈ T . Using a lag operator Lp yt = yt−p , the process can compactly be written as yt = φ1 Lyt + φ2 L2 yt + ... + φp Lp yt + t (1 − φ1 L − φ2 L2 − ... − φp Lp )yt = t φ(L)yt = t . Consider φ(L) as a polynomial φ(z), z ∈ C. The process is stable if the roots of the polynomial φ(z) lie outside the unit circle in the complex plane C. In practice, the most important processes are of order one and two. The AR(1) process yt = φyt−1 + t is stationary for |φ| < 1 with E[yt ] = 0 for all t, σ2 1 − φ2 φh σ2 γh := Cov[yt , yt−h ] = , h = 0, 1, . . . , t − 1 1 − φ2 Var[yt ] =

ρh := Corr[yt , yt−h ] = φh .

Moments of autoregressive processes of higher order p, AR(p), are derived as follows. In order to obtain the autocovariances, one multiplies yt = φ1 yt−1 + φ2 yt−2 + ... + φp yt−p + t with yt−h so that yt yt−h = φ1 yt−1 yt−h + φ2 yt−2 yt−h + ... + φp yt−p yt−h + t yt−h . After taking expectations, this yields γh = φ1 γh−1 + φ2 γh−2 + ... + φp γh−p . The error term t vanishes, as the shock in t is uncorrelated with yt−h for h > 0. Dividing γh by γ0 , one gets the autocorrelation function (ACF) ρh with ρh =

γh = φ1 ρh−1 + φ2 ρh−2 + ... + φp ρh−p , h > 0. γ0

By substituting h = 1, 2, . . . , p, a set of linear equations for φ1 , φ2 , . . . , φp in terms of autocorrelations ρ1 , ρ2 , . . . , ρp can be obtained. These equations can be solved by equivalently

3

1. Theoretical models writing 

ρ=

      

ρ1 ρ2 .. . ρp





      

      

=

1 ρ1 .. .

ρ1 1 .. .

ρ2 ρ1 .. .

ρp−1 ρp−2 ρp−3

· · · ρp−1 · · · ρp−2 .. .. . . ··· 1

       

φ1 φ2 .. . φp

       

= P φ,

(1.2)

with coefficients vector φ = (φ1 , φ2 , . . . , φp )0 . Denote by φpj , the jth coefficient in an autoregressive representation of order p, so that φpp is the last coefficient. Then, setting φ = (φp1 , φp2 , . . . , φpp )0 and inserting into (1.2) yields the Yule-Walker equations (see Box and Jenkins [1976, p.65f]). As the theoretical autocorrelations ρi , i = 1, 2, . . . , p are unknown for a given time series, they can be replaced by the estimated autocorrelations ri , i = 1, 2, . . . , p. Then, the coefficients vector φ = (φ1 , φ2 , . . . , φp )0 can be obtained with φ = P −1 r. Considering the coeffients in terms of the lag h, the quantity φhh , h = 1, 2, . . . , p is called partial autocorrelation function (PACF). Its value is nonzero for h ≤ p and zero otherwise. The variance σy2 is equal to γ0 and is γ0 = σy2 =

σ2 . 1 − φ1 ρ1 − φ2 ρ2 − ... − φp ρp

If the roots of the characteristic polynomial are real, the autocorrelation function exponentially damps out to zero. But the roots need not be real. In the AR(2) model, if φ21 − 4φ2 < 0, the roots of the characteristic polynomial are complex. Then, the autocorrelation function consists of dampened sine waves.1 Any autoregressive process can equivalently be written in terms of the errors yt =

∞ X

θj t−j ,

j=0

which is the moving-average representation of an autoregressive process, where again t is mean-zero white noise with constant variance E[2t ] = σ2 for any t ∈ T . The coefficients θj P need to be computed recursively with θj = ji=1 θj−i φi for j > 0 and θ0 = 1. By means of example, consider an AR(2) process with φ1 = 0.9, φ2 = −0.15 and errors t i.i.d. with σ 2 = 1, or equivalently yt = φ1 yt−1 + φ2 yt−2 + t = 0.9yt−1 − 0.15yt−2 + t . The characteristic polynomial equals φ(L) = 1 − 0.9L + 0.15L2 and its roots are L1 = 3 − √ √ 2.33333 = 1.47247 and L2 = 3 + 2.33333 = 4.52752. As both roots lie outside of the unit circle in the complex plane C, the process is stationary and has an infinite moving 1

Box and Jenkins [1976, p.60] show that in this case, ρk = 1+D 2 cos 2πf0 = √φ1 and tan F = 1−D 2 tan 2πf0 . 2

D k sin 2πf0 k+F sin F

with D =

√

−φ2 ,

−φ2

4

1. Theoretical models average representation. The autocorrelation function and partial autocorrelation function for the first five lags are shown in table 1.1. As can easily be seen, the autocorrelation function exponentially damps out, whereas the partial autocorrelation function has a cutoff at lag two, with higher lags close to zero. AR(2) process 5.0

2.5

0.0

-2.5

y

-5.0 200

400

600

800

1000

Simulation

Figure 1.1.: AR(2) process with φ1 = 0.9 and φ2 = −0.15

Table 1.1.: Autocorrelation function and Partial Autocorrelation function of an AR(2) process Lag 1 2 3 4 5 ACF 0.77228 0.51890 0.33096 0.20478 0.14025 PACF 0.77228 -0.19207 0.00408 -0.00439 0.04075

Moving average processes Another important but intuitive feature discussed in Box and Jenkins [1976] is the moving average process. A time series yt is the weighted sum of previous random errors t , or equivalently yt = t + θ1 t−1 + θ2 t−2 + ... + θq t−q . (1.3) Again using the lag operator Lyt = yt−1 , the process can equivalently be written yt = (1 + θ1 L + θ2 L2 + ... + θq Lq )t = θ(L)t . Consider the polynomial θ(z), z ∈ C corresponding to θ(L). It is stationary for any θi with i = 1, ..., q. If a moving average processes can be rewritten as an infinite sum of past values plus present error term t , or equivalently if it has an infinitive autoregressive representation, so

5

1. Theoretical models that yt = ∞ i=1 πi yt−i + t for some πi , the process is said to be invertible. This can be verified by evaluating the roots of the polynomial θ(z). A moving average process of order q, or MA(q), is invertible if the roots z ∈ C lie outside the unit circle in the complex plane. A MA(1) process yt = t + θ1 t−1 is invertible for |θ| < 1 and has the following moments: P

E[yt ] = 0, Var[yt ] = (1 + θ12 )σ2 ,   θσ 2 

γk = Cov[yt , yt−h ] =  ρk = Cov[yt , yt−h ] =

0    −θ12 1+θ1

 0

if k = 1 if k ≥ 2, if k = 1 if k ≥ 2.

More generally, the variance of a MA(q) process is given by γ0 = (1 + θ12 + θ22 + ... + θq2 )σ2 and

γk =

  (−θk

+ θ1 θk−1 + θ2 θk−2 + ... + θq−k θq)σ2

if k = 1, ..., q if k > q.

 0

This leads to the autocorrelation function (ACF) ρk with

ρk =

   −θk +θ1 θk−1 2+θ2 θk−22 +θq−k θq

if k = 1, ..., q

 0

if k > q.

1+θ1 +...+θq

(1.4)

Autocorrelations ρk of moving average processes are zero upon exceeding the order of the process q. In contrast to that, partial autocorrelations φhh calculated by using the Yule-Walker equations (1.2) are decaying but nonzero for any k. Autoregressive moving average processes The two models discussed beforehand can be combined into a auto-regressive movingaverage process of order (p, q) defined by yt = φ1 yt−1 + φ2 yt−2 + . . . + φp yt−p + t + θ1 t−1 + θ2 t−2 + . . . + θq t−q .

(1.5)

yt is a combination of its lagged values and the actual and lagged error terms and, using the lag operator L, can equivalently be written as (1 − φ1 L − φ2 L2 − . . . − φp Lp )yt = (1 + θ1 L + θ2 L2 + . . . + θq Lq )t φ(L)yt = θ(L)t .

6

1. Theoretical models Consider the polynomials φ(z) and θ(z), z ∈ C of the AR(p) polynomial φ(L) and the MA(q) polynomial θ(L), respectively. The process is stationary if the roots of the polynomial φ(z) lie outside the unit circle in the complex plane C and invertible if the roots of the polynomial θ(z) lie outside the unit circle in the complex plane C. An invertible stationary ARMA(p, q) process has an infinite moving average representation with yt = ψ(L)t =

∞ X

ψi t−j ,

i=0

where ψ(L) = φ−1 (L)θ(L), and an infinite autoregressive representaton with π(L)yt = yt −

∞ X

πj yt−j = t ,

i=1

where π(B) = θ−1 (L)φ(L). As φ(L)ψ(L) = θ(L), the weights ψi can be determined by ψi = φ1 ψj−1 + φ2 ψj−2 + ... + φp ψj−p − θj , j > 0, with ψ0 = 1, ψj = 0 for j < 0, and θj = 0 for j > p. The variance and autocovariances can then be computed according to the equation γk = φ1 γk−1 + ... + φp γk−p + σ2 (θk ψ0 + θk−1 ψ1 + ... + θq ψq−k ), so that γ0 = φ1 γ1 + ... + φp γp + σ2 (1 + θ1 ψ1 + θ2 ψ2 + ... + θq ψq ). It can be seen that the autocorrelations, for k ≥ q + 1, are dominated by the autoregressive part. If k ≥ q + 1, the term σ2 (θk ψ0 + θk−1 ψ1 + ... + θq ψq−k ) contains only zeros, so that γk = φ1 γk−1 + φ2 γk−2 + ... + φp γk−p

k ≥q+1

ρk = φ1 ρk−1 + φ2 ρk−2 + ... + φp ρk−p

k ≥ q + 1.

and

The identification of the orders (p, q) of an ARMA process is done via its autocorrelation function and its partial autocorrelation function. The autocorrelation function of an autoregressive progress tails off, whereas its partial autocorrelation function is zero upon exceeding lag p. The opposite can be said about moving average processes. Its autocorrelation function is zero upon exceeding lag q, while its partial autocorrelation function tails off. If both ACF and PACF tail off, a mixed process is suggested. As Box and Jenkins [1976, p.186]

7

1. Theoretical models show, the ACF of a mixed process, consisting of a pth-order autoregressive and a qth-order moving average component, is a mixture of exponentials and damped sine waves after the first q − p lags. Conversely, the PACF is a mixture of exponentials and damped sine waves after the first p − q lags. 1.1.2. Non-stationary processes Empirical time series need not be stationary. Nelson and Plosser [1982] show that the null hypothesis of logged U.S. gross national product (GNP) containing a unit root cannot be rejected. This means that the process generating the data can be presented as a random walk. In other words, yt = yt−1 + t . (1.6) The exhibition of a unit root has some important economic implications. The moments of this process are not independant of t, as can be seen below: E[yt ] =

t X

(t−i ) = 0,

i=0

Var[yt ] = tσ2 . However, the assumptions of the OLS methods require underlying time series to be stationary. Granger and Newbold [1974] show that the presence of unit roots in time series favours the identification of spurious relationships and invalidates the usual significance tests on coefficients. A time series yt is called integrated of order d if d roots of its characteristic equation lie on the unit circle and after differencing d times, the resulting time series ∆d yt is again stationary. ARIMA(p, d, q) processes Box and Jenkins [1976] first developed the class of nonstationary processes exhibiting both autoregressive and moving average characteristics. The autoregressive integrated moving average model is given by φ(L)(1 − L)d yt = θ(L)t . (1.7) In order to identify an ARIMA(p, d, q) process, it has to be differenced d times in order to achieve stationarity. The identification of the autoregressive and moving average components is similar to the identification of an ARMA model shown above.

8

1. Theoretical models

1.2. Multivariate time series models 1.2.1. VAR(p) models The univariate AR(p) model has been extended to multivariate applications by Lütkepohl [1991]. Consider a (K × 1)-dimensional vector yt = (y1t , y2t , . . . , yKt )0 of random variables yit , i = 1, 2, . . . , K. A vector autoregressive process of order p, VAR(p), is then described by yt = α + βt + A1 yt−1 + A2 yt−2 + ... + Ap yt−p + ut .

(1.8)

yt = (y1t , y2t , ..., yKt )0 is a (K × 1) vector of K random variables, the Ai are (K × K) constant coefficient matrices. α = (α1 , α2 , . . . , αK )0 is a (K × 1) vector of intercept terms, β = (β1 , β2 , . . . , βK )0 is a vector of linear trend coefficients, and ut = (u1t , u2t , ..., uKt )0 is a K-dimensional white noise process having E(ut ) = 0, E(ut u0t ) = Σu and E(ut u0s ) = 0 for t 6= s. The first two moments of a VAR(1) process yt = α + A1 yt−1 + ut are E(yt ) := (IK − A1 )−1 α Γy (h) := E(yt − α)(yt−h − α) =

∞ X

0

i Ah+i 1 ΣA1 .

i=0

As both moments do not depend on t, the process satisfies the condition of covariance stationarity. Lütkepohl [2005, p.15] demonstrates that a K-dimensional VAR(p) model can be transformed into a Kp-dimensional VAR(1) model for any order p: Yt = µ + AYt−1 + Ut where

 

Yt :=

      

yt yt−1 .. . yt−p+1

(Kp × 1)





   ,   

      

µ :=

α 0 .. . 0

    ,   

A :=

         

(Kp × 1)



  A1 A2 · · · Ap−1 Ap  u  t  IK 0 · · · 0 0       0    0 IK 0 0  , Ut :=  .  .  ..  .. .. ..   ..   . . . .   0 0 0 · · · IK 0 (Kp × Kp) (Kp × 1)

Yt is stable if |IKp − Az| = 6 0 for |z| ≤ 1.

9

1. Theoretical models 1.2.2. Cointegration of economic time series In empirical data, a relationship can often be found between two (or more) time series. Let yt = (y1t , y2t , ..., yKt )0 be a vector of K variables of interest and let the long-run equilibrium relation be defined by cyt = c1 y1t + c2 y2t + ... + cK yKt = 0, where c = (c1 , c2 , ..., cK ) is a vector of linear coefficients. In the short-run, variables are allowed to deviate from the relation so that cyt = zt , with z ∼ i.i.d(0, σz2 ), is a stationary process. If all time series in yt are I(d) and there exists a linear combination cyt which is I(d − b), the variables are cointegrated of order (d, b), briefly yt ∼ CI(d, b). Error correction models The notion of cointegration has first been advanced in error correction models. If two time series yt and xt are CI(1, 1), then ∆yt , ∆xt and yt − βxt are stationary representations. Changes in yt are explained by changes in ∆xt and by the deviation from the long-run equilibrium yt−1 − βxt−1 . ∆yt = α + θ∆xt + γ(yt−1 − βxt−1 ) + t .

(1.9)

xt may also be a vector of K variables being I(d). In that case, ∆yt = α +

K X j=1

θj ∆xjt + γ(yt−1 −

K X

βj xjt−1 ) + t .

(1.10)

j=1

The Johansen-Juselius approach to Cointegration Consider a (K × 1)-dimensional vector Xt of K variables Xt = (X1t , X2t , . . . , XKt )0 . Set up a VAR(p) model according to (1.8). Following Johansen and Juselius [1990], the VAR(p) can be written as Xt = Π1 Xt−1 + Π2 Xt−2 + ... + Πp Xt−p + t . (1.11) Πi , i = 1, 2, . . . , p are (K ×K)-dimensional coefficient matrices and t is a (K ×1)-dimensional P vector of independant identically distributed errors with variance Ω. Define Π = ( pi=1 Πi ) − I P and Γi = − pj=i+1 Πj . Consider a VAR(2). By differencing, it can be seen that, ∆Xt = Xt − Xt−1 = (Π1 − I)Xt−1 + Π2 Xt−2 + t = (Π1 + Π2 − I)Xt−1 − Π2 (Xt−1 − Xt−2 ) + t ∆Xt = ΠXt−1 + Γ1 ∆Xt−1 + t

(1.12) (1.13) (1.14)

it is possible to establish an error correction form within the general VAR process. It is to be noted that not all variables in Xt need to be integrated of the same order, as there can be multiple cointegrating relationships. Following Johansen and Juselius [1990] and

10

1. Theoretical models Johansen [1995], Π will be singular, as its rank (Π) = r < K. It can be decomposed into two (K ×r)-dimensional matrices α and β with β equal to a matrix constisting of r K-dimensional cointegration vectors. α is then a matrix of coefficients needed to write the rows of Π as linear combinations of the rows in β 0 so that Π = αβ 0 . Considering an additive linear trend in the model and removing the restriction on p, the equation (1.14) can be rewritten as ∆Xt = µ + δt + ΠXt−1 +

p−1 X

(1.15)

Γj ∆Xt−j + t

j=1

= µ + δt + αβ 0 Xt−1 +

p−1 X

(1.16)

Γj ∆Xt−j + t .

j=1

The cointegration relation between the two or more variables in the vector Xt can then be found by evaluating the product of a cointegration vector βi0 , i = 1, . . . , r and vector Xt−1 . Each row vector in βi defines a cointegration relation between the variables in Xt . Contrary to error correction models, it is not necessary for all variables Xit to be integrated of the same order. Rather, the test relies on the notion of stationarity in βi Xt . According to Johansen [1995, p.39], any vector v in RK can be decomposed into a vector in v1 ∈ sp(β⊥ ), where β⊥ is the vector orthogonal to β so that β⊥ β = 0, and a vector in v2 ∈ sp(α), where sp(z) is the linear vector space spanned by z (spanned), so that 0 0 v = v1 + v2 = β⊥ (α⊥ β⊥ )−1 α⊥ v + α(β 0 α)−1 β 0 v.

This helps in decomposing µ and δ:

µ = αµ1 + α⊥ µ2 , δ = αδ1 + α⊥ δ2 , so that 0 0 µ µ2 = (α⊥ α⊥ )−1 α⊥

is a vector of linear trend slopes in the data,

0 0 δ δ2 = (α⊥ α⊥ )−1 α⊥

is a vector of quadratic trend coefficients in the data,

µ1 = (α0 α)−1 α0 µ

is a vector of intercepts in the cointegration relations,

δ1 = (α0 α)−1 α0 δ

is a vector of linear trend coefficients in the cointegration relations.

Using this property in (1.16) yields 

∆Xt = α⊥ µ2 + α⊥ δ2 t + α[β

0

 , µ1 , δ1 ]   

Xt−1 1 t

  K−1 X  + Γj ∆Xt−j 

+ t .

j=1

11

1. Theoretical models With the knowledge of the underlying data generating process, this can be used to select the right cointegrated vector autoregressive model and to estimate the cointegration vectors accordingly. It is possible to calculate the coefficient of determination Ri2 for each variable ∆Xit in a cointegrated vector autoregressive model set up according to (1.16) [Estima, 2006, p.180]. ˆ ii Ω ˆ ii equal to the correlation matrix of the residuals to ∆Xit and with Ω Then, Ri2 = 1 − var(∆X t )i i σ ˆi =

q

ˆ ii and R2 = 1 − Ω i

Error Sum of Squares , Total Sum of Squares

which is analogous to a linear regression model.

1.3. Hypothesis testing In this section, statistics necessary to conduct hypothesis testing in the following chapters are introduced and the testing procedures in the RATS 8.0 software package is described. First, tests statistics for unit roots in univariate data are introduced. Then, a test for the normality of the distribution is shown. Third, the Johansen-Juselius tests for cointegration are displayed. Finally, portmanteau tests for serial autocorrelation in residuals are defined. 1.3.1. Unit root tests Dickey-Fuller test In the autoregressive model defined by yt = ρyt−1 + t

(1.17)

with uncorrelated and i.i.d. errors t , the null hypothesis for a unit root in the time series is H0 : ρ = 1. Dickey and Fuller [1979, p.428] show that the test statistics is τˆ =

ρˆ − 1 2 σ t=2 yt−1 Pn

where σ2

Pn

=

− ρˆyt−1 )2 n−2

t=2 (yt

and leads to the rejection of the null hypothesis if τˆ is lower than the critical value. It is to be noted that this statistics is not asymptotically distributed according to a t-distribution, but converges to a distribution first tabulated by Dickey and Fuller [1979]. Performing the Dickey-Fuller test in the software package RATS 8.0 is done by the procedure @DFUNIT [Estima, 2010b, p.101], allowing to specify several options, the time series and starting as well as end points. The standard Dickey-Fuller test is done with @DFUNIT(ttest,lag=0,DET=NONE) ts start end, displaying the test statistics and critical values for α = 0.1, 0.05, 0.01 for a time series ts in the interval from start to end.

12

1. Theoretical models Augmented Dickey-Fuller test The test regression 1.17 has been extended by Dickey and Said [1984] to yt = ρyt−1 + β1 ∆yt−1 + ... + βp−1 ∆yt−p+1 + t

(1.18)

in order to take into account autocorrelation of the error terms. The test statistics is identical to τˆ shown above. Augmented Dickey-Fuller tests can be calculated in RATS by the procedure @DFUNIT mentioned above, changing the option lag to a value greater than 0 to incorporate the terms ∆yt−i , i = 1, . . . , p. 1.3.2. Normality tests Examination of the normality of a time series can be done by calculating the Jarque-Bera test statistics (K − 3)2 n ) (1.19) JB = (S 2 + 6 4 with S= K=

1 n

Pn

1 n

Pn

i=1 (x σ3 i=1 (x σ4

− x¯)3 − x¯)4

, .

Under the null hypothesis, the underlying distribution of the process is a Normal distribution and JB ∼ χ22 . In RATS, the command statistics displays basic descriptive statistics of a time series as well as skewness, excess kurtosis and the Jarque-Bera statistics [Estima, 2010a, p.445]. 1.3.3. Cointegration tests - Johansen-Juselius approach Trace statistics In order to test a cointegration hypothesis on a matrix Π, the null hypothesis H0 : rank(Π) = r < K has to be verified. The corresponding alternative hypothesis is that the matrix Π has full rank K, H1 : rank(Π) = K. The statistics for testing the null hypothesis against the first of alternative hypothesis refers to the trace statistics λtrace = −T

n X

ˆ i ). ln(1 − λ

i=r+1

13

1. Theoretical models Maximum eigenvalue statistics For testing the null hypothesis of H0 : rank(Π) = r against the alternative hypothesis H1 : rank(Π) = r + 1 the λmax statistics is used: ˆ r+1 ). λmax = −T ln(1 − λ As the small sample properties of the trace test were found to be different from its asymptotic properties, Johansen [2002] introduced a Bartlett correction factor for the likelihood ratio test improving its finite sample properties.2 In the software package RATS, the additional component CATS 2 has to be started to be able to test hypotheses on the cointegration rank. The software does not provide a method to estimate the λmax statistics. The λtrace statistics and the Bartlett-corrected λ∗trace statistics are accessible from the menu entry I(1) -> Test Rank Statistics. Then, the test statistics as well as the 95%-fractiles and the corresponding p-values for both statistics are displayed. Testing restrictions on β There are two ways for testing hypotheses on the cointegration vectors β [Juselius, 2006, p.173]. One way is to specify si free parameters, the other way is to place mi restrictions on each vector βi . First, a constrained cointegration vector βic in terms of free parameters ϕi is set up, so that β c = (β1c , β2c , . . . , βrc ) = (H1 ϕ1 , H2 ϕ2 , . . . , Hr ϕr ). (1.20) ϕi is a (si × p1 )-dimensional coefficients matrix, Hi is a (p1 × si ) design matrix, p1 is the ˜ t−1 in the VAR model, and i = 1, . . . , r. dimension of X Alternatively, (p1 × mi )-dimensional restriction matrices Ri are specified for mi = p1 − si restrictions on βi so that R10 β1 = 0 .. . Rr0 βr = 0. It is to be noted that Ri = H⊥,i , or equivalently Ri0 Hi = 0. In order to test the hypothesis on coefficient vectors βi in the case of identical restrictions on all vectors βi , a reducted cointegrating relationship has to be set up, with ˜ t−1 + ∆Xt = γ0 + γ1 t + αϕ0 H 0 X

K−1 X

Γj ∆Xt−j + t .

(1.21)

j=1

2

This correction factor is derived by finding the expectation of the likelihood ratio test statistics and correcting it in order to have the same mean as the limit distribution.

14

1. Theoretical models The likelihood ratio test statistics is calculated by −2lnΛ = −T

r X

ˆ i )] ˆ c ) − ln(1 − λ [ln(1 − λ i

i=1

ˆ i and is asymptotically distributed as χ2 (v) with v = rm defor the r biggest eigenvalues λ grees of freedom for m restrictions on the r cointegration vectors. 1.3.4. Portmanteau tests for serial autocorrelation in residuals In order to test the null hypothesis of no autocorrelation in the residuals of an ARIMA time series, a class of tests called portmanteau tests have been established. In the univariate time series model, these tests have first been developed by Box and Pierce [1970]. Box-Pierce test The null hypothesis of a Box-Pierce test is that no serial autocorrelation ρt−h between time intervals t and t − h, h ∈ 1, 2, . . . , T is statistically different from zero. The corresponding test statistics is QBP = T

h X

ρ2i ,

(1.22)

i=1

where ρi is the autocorrelation at lag i and h is the number of lags being tested. The test statistics is asymptotically distributed with QBP ∼ χ2h and the null hypothesis is rejected when QBP > χ21−α,h . Ljung-Box test Ljung and Box [1980] have proposed a modified version of the Box-Pierce statistics with improved finite-sample properties, QLB = T (T + 2)

h X

ρ2i . i=1 T − i

(1.23)

QLB has the same limiting distribution as QBP , so that QLB ∼ χ2h . Multivariate portmanteau tests - Hosking Qs The class of portmanteau tests has been extended to the multivariate framework by Hosking [1980]. The test statistics is then Qs = T 2

h X

(T − i)−1 tr(Cˆi0 Cˆ0−1 Cˆi Cˆ0−1 ),

(1.24)

i=1

15

1. Theoretical models where Cˆi = 1/T Tt=i+1 uˆt uˆ0t−i . The statistics is distributed as Qs = χ2K 2 (h−p) , where h is the number of lags to which autocorrelation is examined, K the dimension of the VAR model under examination and p the lags within the VAR model. The test statistics is available under the CATS menu Misc -> Residual Analysis. The test statistics can be seen in the Tests for Autocorrelation section as Ljung-Box statistics and is estimated for the auto- and crosscorrelations for the first T /4 lags [Estima, 2006, p.50]. P

1.3.5. Lagrange Multiplier tests for nth-order autocorrelation The Lagrange Multiplier test developed by Godfrey [1988] tests for nth-order serial autocorrelation. Its test statistics is asymptotically distributed as LMn ∼ χ2p2 where p is the number of lags within the VAR model. If LMn > χ21−α,p2 the null hypothesis of no serial correlation up to lag n can be rejected. The test statistics is available under the CATS menu Misc -> Residual Analysis. The test statistics can be seen in the Tests for Autocorrelation section as LM(n) statistics and is estimated for the autocorrelations up to n lags with n supplied by the user.[Estima, 2006, p.50].

16

EUR/CHF Exchange rate returns and Macroeconomic data

2. Empirical Model 2.1. Data Data for EUR/CHF exchange rates is provided by the European Central Bank (ECB) (see figure A.1), on a daily basis from 01.01.1999 to 30.06.2011. In the following models, the series is named x1 . Basic statistics on the differenced log series can be seen in table A.3. Additional information on the EUR/CHF exchange rate is given in section 2.2. Data on trade balances between the European Monetary Union (EMU) and Switzerland can be obtained from the ECB (see figure A.3). A negative sign indicates a trade deficit for Switzerland and a trade surplus for the EMU. Data is called x2 in the following models. The Gross Domestic Product (GDP) time series for the EMU are calculated by the ECB as seasonally adjusted quarterly series (see figure A.4). Basic statistics on GDP growth rates are presented in table A.4. The time series is called x3a . The inflation rate in the EMU is taken from the ECB HICP statistics (see figure A.5) and is called x4a . The ECB publishes the EMU monetary aggregate M3 (broad money) (see figure A.6). The time series is saved as x5a . EMU Interest rates are obtained from the German Bundesbank. German 1 and 10 year bonds are set as reference for European interest rates (see figure A.7) and are called x6a and x7a , respectively. Swiss GDP is published by the Swiss National Bank (SNB) on a seasonally adjusted quarterly basis (see figure A.8). Descriptive statistics are shown in table A.5. The time series will be referenced as x3b . Swiss inflation rates relevant to monetary decisions are provided by the SNB on a monthly basis (see figure A.9), as is the Swiss monetary aggregate M3 (see A.10). The time series are called x4b and x5b , respectively. Swiss 1 and 10 year bonds are set as reference for Swiss interest rates (see figure A.11). The data is obtained from the SNB and will be referenced as time series x6b and x7b , respectively. All time series exhibit a unit root in the log series, whereas the first differences are stationary.

17

2. Empirical Model

2.2. Explaining the EUR/CHF exchange rate returns 225 200

Mean-9.83022e-005 Standard Deviation 0.00336 Skewness -0.10758 Excess Kurtosis 7.61048 Fitted normal distribution: N(-9.83022e-005, 1.12788e-005)

175 150 125 100 75 50 25 0 -0.03

-0.01

0.01

0.03

Figure 2.1.: Distribution of the EUR/CHF exchange rate returns in percent and fitted normal distribution Basic statistics on the daily EUR/CHF returns is shown in table A.3. The hypothesis of a zero mean cannot be rejected, but the Jarque-Bera statistics indicates that the distribution of daily returns cannot be assumed to be normal, as it exhibits excess kurtosis (Ku = 7.610482). This is consistent with estimates for other currencies (see e.g. Mandelbrot and Hudson [2004]). Over the period from 04.01.1999. to 30.06.2011, the standard deviation is equal to σ = 0.003358, but the two most extreme values in the return series are a −2.4548% loss on 24.10.2008 (which is equal to a 7.3σ-event) and a 3.02053166% gain on 12.03.2009 (which is equal to a 9.0σ-event). According to standard normal distribution tables, this should only ocurr in 0.0000000000014077628% and 0.00000000000034361403% of the cases. Consequently, the probability of such an event ocurring in the 3162 observation period is nearly zero and its occurrence heavily contrasts the normal distribution assumption.

2.3. Cointegration models In the following section, seven models will be evaluated. In the first model, it is examined whether the EUR/CHF exchange rate is cointegrated with the trade balance of Switzerland and the EMU. The second model then additionally includes the their respective gross domestic products. In the third model, GDP is replaced with the broad monetary aggregates M3 in order to investigate whether fluctutations in nominal money and the trade balance can account for changes in the exchange rate. The fourth model replaces M3 with 10-year interest rates to check on the uncovered interest rate parity. In the fifth model, the time horizon

18

2. Empirical Model of interest rates is shortened to one year. The sixth and seventh model estimate the effect of 10-year and 1-year real interest rates. 2.3.1. EUR/CHF and the trade balance The first hypothesis is that the EUR/CHF exchange rate (see figure A.1) is cointegrated with the foreign trade between Switzerland and the European Monetary Union. A model is set up for the time series x1,t and x2,t . As shown in tables A.1 and A.2, both time series are integrated of first order, or I(1), so that a combination of both time series could be cointegrated. The economic intuition is that with an increasing EUR/CHF exchange rate, the trade balance improves for Switzerland - leading to less imports and higher exports. According to equation 1.16, a system is set up with one lag, leading to





∆x1,t   = ∆x2,t













h i x α ∆x1,t−1   1  β1 β2  1,t−1  + Γ1  + t . α2 x2,t−1 ∆x2,t−1

In order to establish a model fitting the data, the economic properties of both time series need to be considered. Evaluating the exchange rate between the EMU and Switzerland, the assumption of a deterministic trend in the time series is highly improbable, especially when the time series in figure A.1 is considered, as it is fluctuating without any obvious trend. For the trade balance (seen in figure A.3), the interdependance between Switzerland and its neighbouring states has grown steadily and absolute trade balances have risen, accordingly. But the hypothesis that the mean of the differences of the log series is zero cannot be rejected, so the changes in the log series might be generated by a random walk without any deterministic trend component. As this is true for all examined variables, the focus in this chapter is on models without deterministic trend. In the following, all cointegration vectors βi∗ will be normalized on the x1 time series. Table 2.1.: I(1)-ANALYSIS of model 1 without deterministic components p-r r Eig.Value Trace Trace* Frac95 P-Value P-Value* 2 0 0.284 50.742 50.483 12.282 0.000 0.000 1 1 0.016 2.389 2.386 4.071 0.143 0.143 rank(Π) = 1 indicates 1 cointegration relation with xˆ1,1 = 2.052 × x2 . Tests for residual autocorrelation: Hosking Qs test (36 lags, 140 degrees of freedom) p-value 0.000 Godfrey LM test (1 lag, 4 degrees of freedom) p-value 0.094 Godfrey LM test (2 lags, 4 degrees of freedom) p-value 0.480 The first model is calculated over monthly intervals from January 1999 to June 2011. Evaluating the corresponding model, the rank test statistics (shown in table 2.1) indicates

19

2. Empirical Model 2.2

x1 x1,1

2.0

1.8

1.6

1.4

1.2

1.0 1999

2001

2003

2005

2007

2009

2011

Figure 2.2.: EUR/CHF exchange rate x1 and fitted time series xˆ1,1 from model 1 This graphic shows the realized EUR/CHF exchange rate x1 and the fitted series xˆ1,1 . 0 They result from a reorganization of the vector β1∗ = (1.000, −2.052)0 to a new vector 0 β˜∗ 1 = (0, 2.052)0 so that β˜∗ 1 Xt leads to an estimate of the original EUR/CHF exchange rate. one cointegrating vector with β1∗ = (1.000, −2.052)0 . A fitted time series corresponding to an artificial EUR/CHF exchange rate is shown in figure 2.2. 2.3.2. EUR/CHF, trade balance and Swiss and EMU GDP Setting up a new model with additional variables log Swiss GDP and log EMU GDP, so that Xt = (x1,t , x2,t , x3a,t , x3b,t )0 , and allowing for one lag leads to the counterintuitive result of matrix Π having full rank, as is shown in table 2.2, suggesting that at least one component of ΠXt−1 is I(1) and therefore contradicting the assumptions made previously. Consequently, this model is rejected. Table 2.2.: I(1)-ANALYSIS of model 2 without deterministic components p-r r Eig.Value Trace Trace* Frac95 P-Value P-Value* 4 0 0.681 106.403 102.054 40.095 0.000 0.000 3 1 0.472 50.400 48.981 24.214 0.000 0.000 2 2 0.259 19.066 18.764 12.282 0.003 0.003 1 3 0.085 4.357 4.340 4.071 0.042 0.042 rank(Π) = 4 indicates that Π has full rank.

2.3.3. EUR/CHF, trade balance and Swiss and EMU broad money Replacing EMU and Swiss GDP with the respective monetary aggregates M3 so that Xt = (x1,t , x2,t , x5a,t , x5b,t )0 leads to a rank test statistics (table 2.3) indicating two cointegrating ∗ ∗ relations with β3,1 = (1.000, −0.382, 0.374, −0.924)0 and β3,2 = (1.000, −1.307, −0.110, 0.162)0 .

20

2. Empirical Model ∗ The fitted time series for β3,2 corresponding to artificial EUR/CHF exchange rates is shown below. The coefficient of determination of the fitted model for ∆x1,t is R2 = 0.030. 1.70 1.65 1.60 1.55 1.50 1.45 1.40 1.35 1.30 x1 x1,3

1.25 1999

2001

2003

2005

2007

2009

2011

Figure 2.3.: EUR/CHF exchange rate and fitted time series xˆ1,3 from model 3 This graphic shows the realized EUR/CHF exchange rate x1 and the fitted series xˆ1,3 . It ∗0 results from a reorganization of the vector β3,2 = (1.000, −1.307, −0.110, 0.162)0 to a new 0 vector β˜∗ 3,2 = (0, 1.307, 0.110, −0.162)0 so that β˜∗ 3,2 Xt leads to an estimate of the original EUR/CHF exchange rate.

Table 2.3.: I(1)-ANALYSIS of model 3 without deterministic components p-r r Eig.Value Trace Trace* Frac95 P-Value P-Value* 4 0 0.767 121.750 116.773 40.095 0.000 0.000 3 1 0.536 50.402 48.983 24.214 0.000 0.000 2 2 0.216 12.785 12.583 12.282 0.041 0.044 1 3 0.017 0.857 0.854 4.071 0.411 0.412 rank(Π) = 2 indicates 2 cointegration relations with xˆ1,2 = 0.382 × x2 + 0.924 × x5a − 0.374 × x5b and xˆ1,3 = 1.307 × x2 − 0.162 × x5a + 0.110 × x5b . Tests for residual autocorrelation: Hosking Qs test (12 lags, 184 degrees of freedom) p-value 0.000 Godfrey LM test (1 lag, 16 degrees of freedom) p-value 0.043 Godfrey LM test (2 lags, 16 degrees of freedom) p-value 0.500

21

2. Empirical Model 2.3.4. EUR/CHF, trade balance and Swiss and EMU interest rates Replacing the monetary aggregates M3 with 10-year interest rates, a fourth model is fitted for Xt = (x1 , x2 , x7a , x7b )0 . Rank test statistics (shown in table 2.4) reject the existence of a cointegrating relationship (r = 0). Table 2.4.: I(1)-ANALYSIS of model 4 without deterministic components p-r r Eig.Value Trace Trace* Frac95 P-Value P-Value* 4 0 0.384 32.455 31.128 40.095 0.244 0.303 3 1 0.100 8.705 8.460 24.214 0.917 0.927 2 2 0.058 3.560 3.504 12.282 0.763 0.771 1 3 0.012 0.611 0.608 4.071 0.499 0.500 rank(Π) = 0 indicates no cointegration relation in the data. 10 year interest rates are replaced by their one-year equivalent in model 5 so that Xt = ∗ = (x1,t , x2,t , x6a,t , x6b,t ). Rank test statistics endorse two cointegrating vectors, leading to β5,1 ∗ (1.000, 9.936, 139.885, −162.529)0 and β5,2 = (1.000, 1.119, −9.656, 6.362)0 . The coefficient of determination of the fitted model for ∆x1,t is R2 = 0.200. 1.7

x1 x1,5

1.6

1.5

1.4

1.3

1.2 1999

2001

2003

2005

2007

2009

2011

Figure 2.4.: EUR/CHF exchange rate and fitted time series xˆ1,5 from model 5 This graphic shows the realized EUR/CHF exchange rate x1 and the fitted series xˆ1,5 . This ∗0 results from a reorganization of the vector β5,2 = (1.000, 1.119, −9.656, 6.362)0 to a new vector 0 β˜∗ 5,2 = (0, −1.119, 9.656, −6.362) so that β˜∗ 5,2 Xt leads to an estimate of the original EUR/CHF exchange rate.

22

2. Empirical Model Table 2.5.: I(1)-ANALYSIS of model 5 without deterministic components p-r r Eig.Value Trace Trace* Frac95 P-Value P-Value* 4 0 0.476 59.388 56.961 40.095 0.000 0.000 3 1 0.401 27.714 26.934 24.214 0.017 0.021 2 2 0.030 2.632 2.590 12.282 0.878 0.882 1 3 0.023 1.144 1.139 4.071 0.333 0.334 rank(Π) = 2 indicates 2 cointegration relations with xˆ1,4 = −9.936 × x2 − 139.885 × x5a + 162.529 × x6b and xˆ1,5 = −1.119 × x2 + 9.656 × x5a − 6.362 × x6b . Tests for residual autocorrelation: Hosking Qs test (12 lags, 184 degrees of freedom) p-value 0.000 Godfrey LM test (1 lag, 16 degrees of freedom) p-value 0.004 Godfrey LM test (2 lags, 16 degrees of freedom) p-value 0.555 2.3.5. EUR/CHF, Swiss and EMU GDP and real interest rates 0.045 0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.000 1999

2001

2003

2005

2007

2009

2011

10 year Swiss interest rate - Swiss inflation 10 year German interest rate - ECU inflation

Figure 2.5.: Virtual 10 year real interest rates Substracting annual inflation rates from 10 year interest rates, virtual real interest rates for the European Currency Union and Switzerland (x7a,t − x4a,t , x7b,t − x4b,t ) can be evaluated. Model 6 is estimated for Xt = (x1,t , x3a,t , x3b,t , x7a,t − x4a,t , x7b,t − x4b,t )0 in order to take into account the exchange rate, Swiss and EMU gross domestic product as well as the virtual real interest rates. A cointegrated VAR system is set up for Xt and three cointegrating vectors are identified (see table 2.6). Corresponding cointegration vectors are ∗ ∗ β6,1 = (1.000, 1.832, −2.121, −260.238, 688.062)0 , β6,2 = 1.000, −2.003, 2.466, −5.733, −2.423)0 ∗ and β6,3 = (1.000, 1.226, −1.552, −3.564, −0.727)0 . The coefficient of determination of the fitted model for ∆x1,t is R2 = 0.381.

23

2. Empirical Model

1.8

1.7

1.6

1.5

1.4

1.3 x1 x1,7 x1,8

1.2 1999

2001

2003

2005

2007

2009

2011

Figure 2.6.: EUR/CHF exchange rate and fitted time series xˆ1,7 and xˆ1,8 from model 6 This graphic shows the realized EUR/CHF exchange rate x1 and the fitted series xˆ1,7 and ∗0 , βi,3 , βi,4 , βi,5 )0 to a new xˆ1,8 . They result from a reorganization of the vector β6,i = (1, ββi,2 i,1 βi,1 βi,1 βi,1 0 vector β˜∗ = (0, − βi,2 , − βi,3 , − βi,4 , − βi,5 )0 so that β˜∗ 6,i Xt leads to an estimate of the original 6,i

βi,1

βi,1

βi,1

βi,1

EUR/CHF exchange rate.

Table 2.6.: I(1)-ANALYSIS of model 6 without deterministic components p-r r Eig.Value Trace Trace* Frac95 P-Value P-Value* 5 0 0.735 129.451 121.764 59.961 0.000 0.000 4 1 0.602 69.628 66.500 40.095 0.000 0.000 3 2 0.337 28.219 27.347 24.214 0.014 0.019 2 3 0.176 9.744 9.575 12.282 0.131 0.139 1 4 0.023 1.054 1.050 4.071 0.355 0.356 rank(Π) = 3 indicates 3 cointegration relations with xˆ1,6 = −1.832 × x3a + 2.121 × x3b + 260.238 × (x7a − x4a ) − 688.062 × (x7b − x4b ), xˆ1,7 = 2.003 × x3a − 2.466 × x3b + 5.733 × (x7a − x4a ) + 2.423 × (x7b − x4b ) and xˆ1,8 = −1.226 × x3a + 1.552 × x3b + 3.564 × (x7a − x4a ) + 0.727 × (x7b − x4b ). Tests for residual autocorrelation: Hosking Qs test (11 lags, 260 degrees of freedom) p-value 0.000 Godfrey LM test (1 lag, 25 degrees of freedom) p-value 0.000 Godfrey LM test (2 lags, 25 degrees of freedom) p-value 0.316

24

2. Empirical Model 0.030 0.025 0.020 0.015 0.010 0.005 0.000 -0.005 -0.010 -0.015 1999

2001

2003

2005

2007

2009

2011

1 year Swiss interest rate - Swiss inflation 1 year German interest rate - ECU inflation

Figure 2.7.: Virtual 1 year real interest rates A seventh model is fitted for the EUR/CHF exchange rate, Swiss and EMU GDP and oneyear real interest rates, estimated by substracting annualized inflation rates from one year bond interest rates. This leads to the vector Xt = (x1,t , x3a,t , x3b,t , x6a,t − x4a,t , x6b,t − x4b,t )0 . Accepting the hypothesis of two cointegrating vectors (see table 2.7) leads to estimates ∗ = (1.000, −2.870, 3.493, 16.265, −36.501)0 and β7,1 ∗ β7,2 = (1.000, −1.879, 2.301, −6.940, 2.736)0 . The coefficient of determination of the fitted model for ∆x1,t is R2 = 0.384. 4.0

x1 x1,9 x1,10

3.5

3.0

2.5

2.0

1.5

1.0 1999

2001

2003

2005

2007

2009

2011

Figure 2.8.: EUR/CHF exchange rate and fitted time series from model 7 This figure shows the the realized EUR/CHF exchange rate x1 and the fitted series xˆ1,9 ∗0 , βi,3 , βi,4 , βi,5 )0 , i = 1, 2 and xˆ1,10 . They result from a reorganization of the vector β7,i = (1, ββi,2 i,1 βi,1 βi,1 βi,1 0 to a new vector β˜∗ = (0, − βi,2 , − βi,3 , − βi,4 , − βi,5 )0 so that β˜∗ Xt leads to an estimate of the 7,i

βi,1

βi,1

βi,1

βi,1

7,i

25

2. Empirical Model original EUR/CHF exchange rate. As the time series xˆ1,9 matches the original time series x1 only very poorly, figure 2.9 shows the series xˆ1,10 in order to better asess its fit. Table 2.7.: I(1)-ANALYSIS of model 7 without deterministic components p-r r Eig.Value Trace Trace* Frac95 P-Value P-Value* 5 0 0.799 134.681 126.683 59.961 0.000 0.000 4 1 0.649 62.536 59.726 40.095 0.000 0.000 3 2 0.179 15.415 14.938 24.214 0.429 0.466 2 3 0.086 6.550 6.436 12.282 0.373 0.386 1 4 0.055 2.525 2.514 4.071 0.131 0.132 rank(Π) = 2 indicates 2 cointegration relations with xˆ1,9 = 2.870 × x3a − 3.493 × x3b − 16.265 × (x6a − x4a ) + 36.501 × (x6b − x4b ) and xˆ1,10 = 1.879 × x3a − 2.301 × x3b + 6.940 × (x6a − x4a ) − 2.736 × (x6b − x4b ). Tests for residual autocorrelation: Hosking Qs test (11 lags, 265 degrees of freedom) p-value 0.000 Godfrey LM test (1 lag, Y degrees of freedom) p-value 0.001 Godfrey LM test (2 lags, Y degrees of freedom) p-value 0.467

1.7

x1 x1,10

1.6

1.5

1.4

1.3

1.2 1999

2001

2003

2005

2007

2009

2011

Figure 2.9.: EUR/CHF exchange rate and fitted time series xˆ∗1,10 from model 7

26

EUR/CHF Exchange rate returns and Macroeconomic data

3. Results 3.1. Discussion of results Evidence of cointegration was found in models 1, 3, 5, 6 and 7. The interpretation of the results from the first model is straightforward. An increase in the trade balance is accompanied by a 2.052-fold increase in the EUR/CHF exchange rate. Comparing the fitted time series xˆ1,1 to the original EUR/CHF exchange rate return time series x1 in figure 2.2, it can be seen that the predictive power of the model is rather low and fluctuations are exaggerated. The economic interpretation that the EUR/CHF exchange rate and the trade balance are cointegrated is reasonable. With an appreciation of the Swiss Franc, and a resulting lower exchange rate, imports from the EMU to Switzerland become cheaper and exports to the EMU lose attractiveness, as their price in EUR terms rises. Therefore, with the decline of the EUR/CHF exchange rate, an increase in the trade deficit can be expected. This is supported from the data and the predictions made in the first model. In the second model, including the EUR/CHF exchange rate, trade balance and Swiss and EMU GDP, rank test statistics suggests that the matrix Π has full rank, which indicates the presence of a non-stationary component being at least I(1). This clearly conflicts with equation 1.16, where all parts of the equation are stationary by assumption, and rejects the assumption of an underlying cointegration of the variables. Economically, this means that no clear inference can be drawn on the influence of Swiss and European GDP on the exchange rate. The results from the third model, having replaced GDP by nominal money M3, suggest a ∗ ∗ and β3,2 . Therefore, the coefficients cannot cointegration plane spanned by the vectors β3,1 be easily be interpreted, as there is no unique result within the plane. However, the fitted ∗ time series xˆ1,3 for β3,2 seems to accurately fit the data (see figure 2.3). Including the broad money supply aggregates M3 for Switzerland and the European Currency union improves the precision in the cointegration relationship established in the first model. Fluctuations in nominal money supply seem to be a driving force behind the EUR/CHF exchange rate. The fourth model indicated no cointegration between the EUR/CHF exchange rate, trade balance, and 10-year Swiss and German bond interest rates, as rank test statistics show that rank(Π) = 0. Including the long-term borrowing cost of public financing does not improve the explanative power of the fitted exchange rate. In the fifth model 10-year bond interest rates are replaced by their one-year counterpart. Rank test statistics indicate that rank(Π) = 2, hence suggesting two cointegration relation∗ ∗ ∗ ships β5,1 and β5,2 . The graphical fit for xˆ1,5 , derived from β5,2 , is shown in figure 2.4. The deviations of the fitted exchange rate from the original series are rather considerable, especially when compared to the fitted series from model three. Besides, it wrongly predicts

27

3. Results a strong appreciation of the Euro in 2010, where in reality the Euro has fallen to record lows. However, inclusion of short-term public financing cost improves the fit of the estimated exchange rate compared to the first model. The sixth model establishes a relationship between EUR/CHF exchange rate, Swiss and EMU gross domestic product and 10-year virtual real interest rates. Rank test statistics indicates rank(Π) = 3, therefore establishing three cointegration relationships with cointegration ∗ ∗ ∗ . The fit of xˆ1,7 seems to be low, as it predicts a huge decline to and β6,3 , β6,2 vectors β6,1 1.3 CHF/EUR in 2008 and increases over the period 2001-2003, where the exchange rate declined. The fit of xˆ1,8 is very poor, as it counters the EUR/CHF exchange rate’s movements from 2003 to 2003. Overall, deviations from the original series seem to be higher when compared to model 5. Yet the results seems consistent with findings from model 4 that long-term real borrowing costs do not improve the precision of the predicted exchange rate, and is consistent with the results from model 2 that the predictive power of Swiss and EMU gross domestic product on the exchange rate is poor. In the seventh model, shortening the virtual real interest rates to a one-year time horizon, ∗ ∗ rank test statistics show the presence of two cointegration relationships β7,1 and β7,2 , as rank(Π) = 2. The fitted series xˆ1,9 and xˆ1,10 are shown in figure 2.8. As can easily be seen, the fit of xˆ1,9 is very poor compared to the fit of xˆ1,10 shown in figure 2.9. The fit of ∗ is good, accurately predicting the decline starting in 2007 together with a series xˆ1,10 for β7,2 slight increase in 2009 and further decline until 2011. Overall, the fitted series xˆ1,10 provides the best fit from all evaluated models. It can be seen that short-term real borrowing costs, together with GDP, form a close relationship with the EUR/CHF exchange rate. It is to be kept in mind that the null hypothesis of no serial correlation in the Hosking test statistics is rejected for the residuals in the first, third, fifth, sixth and seventh model. Godfrey LM test statistics do not reject the null hypothesis at the second lag in the first, third, fifth, sixth and seventh model and does not reject the null hypothesis at the first lag at a 5% level in the first model. This leads to the conclusion that the results need to be treated with caution, as the predictive power of the fitted models is heavily questioned. Residual autocorrelation could reflect omitted variables or model misspecification, which leads to the conclusion that the identified driving forces of the exchange rate might not reflect the true ecoomic conditions.

3.2. Additional literature Komlos and Flandreau [2006] use ARIMA methods to evaluate the efficiency of futures markets of the Austro-Hungarian Florin/German Reichsmark exchange rate over the time period of 1876-1914 in predicting the effective exchange rate after a one-month period. They predict an estimate of the exchange rate based on past realizations and compare it to the predicted forward rate. They find that the exchange rate is best predicted by an ARIMA(2, 1, 0) model

28

3. Results over the period 1876-1896 and an AR(1) model over the period 1896-1914, for which the Austro-Hungarian Bank adopted a "‘shadow"’ gold standard, stabilizing its exteriour value. Besides, in the second period, the forward rate improved in predicting the future spot rate, reducing the forecast error by half. Tambi [2005] evaluates exchange rates of the Indian Rupiah against different currencies, including US-Dollar, British Pound, Euro and Yen over the time period 1992-2002. He uses ARIMA models for the different exchange rates. With these models, he predicts estimates for an end-of-series subsample, leading to a high goodness-of-fit. He then concludes that the notion of a unit root in Indian Rupiah exchange rates can be rejected in favour of specifying seperate ARIMA models for each currency. VAR models are generally used in order to assess spillover and pass-through effects. Stulz [2007] investigates the pass-through of exchange rate shocks to import prices, consumer prices for imported goods and total consumer prices over the period 1976 to 2004. They find that import prices quickly react to exchange rate shocks, indicating a pass-through of 0.35% to 0.37% of import prices for a 1% rise in the exchange rate. The pass through of exchange rate shocks to the CPI for imported goods is 0.27 after three months, increasing to 0.34 over two years. The reaction of total consumer price index to shocks to the exchange rate amounts to 0.09 after three months and increases to 0.18 after two years. He also finds that there is a structural change in the pass-through relations in 1993, resulting in lower pass-through in the later period. He argues that in the later sub-period, the Swiss National Bank has more strongly committed to keep inflation at low levels, thus making it more difficult for firms to pass on higher import costs. Cointegration analysis has been used in economics in a variety of fields. Johansen and Juselius [1990] were the first to analyse money demand, focusing on Danish and Finnish money demand structures. They confirm a single cointegration relationship for Danish data and multiple cointegration relationships for Finnish data. Besides, they argue that the interpretation of coefficients in cointegration vectors βi is not unambiguous if rank(Π) > 1. Monticelli and Strauss-Kahn [1992] assess money demand of the European Monetary System and its members. They argue that spillovers between countries, such as the use of another EMS currency to carry out domestic transactions and weakened sovereign authority over monetary decisions as a consequence of free capital mobility, foster the stability of a money demand on a supranational level. Hansen and Kim [1995] examine on German money demand and assess the possibility of structural changes in the underlying money demand function. They find that demand for M1 has two structural breaks. The first structural break is found in 1973, when the Bundesbank announced a regime shift from targeting the liquidity ratio of commercial banks to central bank money consolidated in M1 (see Willms [1983]). The second structural break occurs in 1990 due to the German reunification and the resulting German Monetary Union. Karfakis and Phipps [1996] investigate the cointegration of the US-Dollar / AustralianDollar exchange rate with terms of trade, relative price levels and interest rate differentials.

29

3. Results They find that the USD/AUD exchange rate is mainly influenced by the first two variables, whereas short-term as well as long-term interest rate differentials as a proxy for the uncovered interest rate parity are insignificant at conventional significance levels. Paresh Kumar Narayan and Zheng [2010] inquire a possible cointegration between gold and oil spot and future prices. They argue that with rising oil prices, the customer price index increases and due to secondary-round effects, this leads to an increase in the gold price. Cointegration is found for spot markets and for future markets up to 10 months.

30

EUR/CHF Exchange rate returns and Macroeconomic data

Conclusion In this work, fundamental notions of time series processes have been introduced. Theoretical notions of stationary autoregressive and moving-average processes have been discussed, as well as their non-stationary counterpart, the autoregressive integrated moving average process. The autoregressive process has then been generalized to a K-dimensional vector autoregressive (VAR) process of order p. The concept of cointegration has been introduced and the Johansen-Juselius approach to cointegration has been discussed. Test statistics have been introduced, laying the foundation of subsequent chapters. It was found that the distribution of EUR/CHF exchange rate returns is not normal, as it exhibits strong excess kurtosis and several outliers with very low probability. Cointegration models have examined the relationship between the exchange rate and macroeconomic data, with the main interest of identifying forces driving the exchange rate. No cointegration was found for exchange rate, gross domestic product and long-term interest rates. Evidence suggests that, within limitations, the EUR/CHF exchange rate reacts to changes in real interest rates of Switzerland and Germany, and, to a lesser degree, broad monetary aggregates M3 of the European Monetary Union and Switzerland as well as trade balances between Switzerland and the EMU. However, the evidence on cointegration of the EUR/CHF exchange rate with macroeconomic variables is rather weak. Besides, additional literature making use of Box-Jenkins techniques, VAR models and cointegration have been discussed. It was seen that these models have been applied to a wide range of analyses and that cointegration is found for various economic relationships.

31

EUR/CHF Exchange rate returns and Macroeconomic data

A. Appendix A.1. Graphics 1.7

1.6

1.5

1.4

1.3

1.2

1.1 1999

2001

2003

2005

2007

2009

2011

2009

2011

Figure A.1.: Daily EUR/CHF exchange rates

0.04

0.03

0.02

0.01

0.00

-0.01

-0.02

-0.03 1999

2001

2003

2005

2007

Figure A.2.: Daily EUR/CHF exchange rate returns

32

A. Appendix

-500 -750 -1000 -1250 -1500 -1750 -2000 -2250 -2500 -2750 1999

2001

2003

2005

2007

2009

2011

Figure A.3.: Monthly trade balance of the ECU and Switzerland in million euros.

2400000 2300000 2200000 2100000 2000000 1900000 1800000 1700000 1600000 1500000 1999

2001

2003

2005

2007

2009

2011

Figure A.4.: GDP of the European Currency Union (Nominal, in current prices) in million euros.

33

A. Appendix

0.04

0.03

0.02

0.01

0.00

-0.01 1999

2001

2003

2005

2007

2009

2011

2009

2011

Figure A.5.: Annualized EMU inflation rate

10000

9000

8000

7000

6000

5000

4000 1999

2001

2003

2005

2007

Figure A.6.: ECU Monetary aggregate M3 (broad money) in billion Euro

34

A. Appendix

6

5

4

3

2

1

0 1999

2001

2003

2005

1yr BUND

2007

2009

2011

10yr BUND

Figure A.7.: One and 10 year interest rates on German Bunds

150000

140000

130000

120000

110000

100000

90000 1999

2001

2003

2005

2007

2009

2011

Figure A.8.: Swiss GDP (Nominal, in current prices) in million Swiss Francs.

35

A. Appendix

2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 1999

2001

2003

2005

2007

2009

2011

2007

2009

2011

Figure A.9.: Swiss inflation

750

700

650

600

550

500

450 1999

2001

2003

2005

Figure A.10.: Swiss Monetary aggregate M3 (broad money) in billion Swiss Francs

36

A. Appendix 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 1999

2001

2003 1yr EIDG

2005

2007

2009

2011

10yr EIDG

Figure A.11.: One and 10 year interest rates on Swiss federal obligations

A.2. Tables Time series lnEURCHF lnBalance lnECUBIP ECUHVPI lnECUM3 lnCHBIP lnCHM3 CHINFL

1%(**) -2.56642 -2.57931 -2.60902 -2.57931 -2.5794 -2.60902 -2.57931 -2.57931

5%(*) -1.93943 -1.94195 -1.94726 -1.94195 -1.9420 -1.94726 -1.94195 -1.94195

10% -1.61566 -1.61681 -1.61922 -1.61681 -1.6168 -1.61922 -1.61681 -1.61681

T-statistics -1.54165 -1.53213 7.79992 -0.23775 10.8311 6.66383 5.01963 -0.63118

Table A.1.: Dickey-Fuller test results for the log time series

37

A. Appendix

Time series gEURCHF gBalance gECUBIP gECUHVPI gECUM3 gCHBIP gCHM3

1%(**) -2.5664 -2.5794 -2.60998 -2.57941 -2.57931 -2.60998 -2.57931

5%(*) -1.9394 -1.9420 -1.94742 -1.94197 -1.94195 -1.94742 -1.94195

10% -1.6157 -1.6168 -1.61929 -1.61681 -1.61681 -1.61929 -1.61681

T-statistics -55.8555** -19.1268** -2.00456* -9.66262** -6.27830** -2.23014* -9.20909**

Table A.2.: Dickey-Fuller test results for the first difference of log time series

Statistics on Series GEURCHF Observations 3162 Sample Mean -0.000098 Standard Error 0.003358 t-Statistic (Mean=0) -1.645933 Skewness -0.107576 Kurtosis (excess) 7.610482 Jarque-Bera 7636.983946

Skipped/Missing Variance SE of Sample Mean Signif Level (Mean=0) Signif Level (Sk=0) Signif Level (Ku=0) Signif Level (JB=0)

96 0.000011 0.000060 0.099877 0.013572 0.000000 0.000000

Table A.3.: Basic statistics of the daily EUR/CHF exchange rate returns returns

Statistics on Series GECUBIP Observations 49 Sample Mean 0.008162 Standard Error 0.007277 t-Statistic (Mean=0) 7.851306 Skewness -2.614220 Kurtosis (excess) 9.647536 Jarque-Bera 245.840184

Variance SE of Sample Mean Signif Level (Mean=0) Signif Level (Sk=0) Signif Level (Ku=0) Signif Level (JB=0)

0.000053 0.001040 0.000000 0.000000 0.000000 0.000000

Table A.4.: Basic statistics of the EMU GDP growth rate

Statistics on Series GCHBIP Observations 49 Sample Mean 0.007187 Standard Error 0.007543 t-Statistic (Mean=0) 6.669650 Skewness -0.762222 Kurtosis (excess) 1.124101 Jarque-Bera 7.324547

Variance SE of Sample Mean Signif Level (Mean=0) Signif Level (Sk=0) Signif Level (Ku=0) Signif Level (JB=0)

0.000057 0.001078 0.000000 0.034735 0.135342 0.025674

Table A.5.: Basic statistics of the Swiss GDP growth rate

38

EUR/CHF Exchange rate returns and Macroeconomic data

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