Oct 25, 2002 - Theme: Short rates contain information about future currency ... We combine interest rate and foreign exchange rate forecasting in an inter-.
Forecasting Interest Rates and Exchange Rates under Multi-Currency Quadratic Models
Markus Leippold Swiss Banking Institute, University of Zurich Liuren Wu Graduate School of Business, Fordham University
October 25th, 2002
CRIF, Fordham University
Overview
• Introduction • Multi-Currency Quadratic m + n Models • Property Analysis • Data and Estimation • Estimation Results • Summary
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Introduction Two Strands of Literature: 1. Interest Rate Forecasting • Theme: Current term structure contains information on future short rates. • A simple regression: n−1 − rt = an + bn (ftn − rt) + ent+1; ft+1 rt+n − rt = an + bn (ftn − rt) + ent+1; ¢ 1 ¡ n yt+1 − ytn+1 = an + bn ytn+1 − rt + ent+1. n
• Based on different expectation hypotheses (U-EH, L-EH, RTM-EH, YTMEH). • Literature includes e.g.: Backus, Foresi, Mozumdar, and Wu (2001), Bekaert, Hodrick, and Marshall (1997), Fama (1984b), Roll (1970).
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2. Foreign Exchange Rate Forecasting • Theme: Short rates contain information about future currency movements. • A simple regression (UIP), based on martingale hypothesis st+n − st = αn + βn (f (s)nt − st) + et+n = αn + βn (rt − rt∗) + et+n. • Literature includes: Backus, Foresi, and Telmer (2001), Cheung (1993), Engle (1996) Fama (1984a). • Regression has been studied with improved econometric techniques (e.g. Wu and Zhang (1997)). • Are deviations from EH merely a coincidence of a few empirical artifacts (Lothian and Wu (2002))? • Failing of EH tests may be due to existence of time varying risk premium. • Fama (1984a): risk premium on currency must be negatively correlated with expected depreciation and must have greater variance.
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Our Contribution • We combine interest rate and foreign exchange rate forecasting in an internally consistent way. • We propose an extension of quadratic term structure models to a multicountry setup (MCQM(m+n)). • We investigate 1. whether (whole) term structures contain information on exchange rates. 2. whether term structures of different countries share common factors, and may improve upon forecasting interest rates. 3. whether forecasting based on model outperforms forecasting based on EH.
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Executive Summary 1. Theory • In the MCQM(m+n), exchange rates can have independent movements, i.e. dynamics are not solely determined by interest rate differential. • The MCQM(m+n) can generate more volatile exchange rate dynamics than implied by differences in term structure risk premia. 2. PC Analysis • Co-movement of interest rates between – U.S. and Japan is very small. – U.S. and Germany is larger. • Exchange rate and term structure movements are – almost independent for U.S. and Japan. – ...and almost independent for U.S. and Germany.
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3. Estimation Results • MCQM(6+1) model yields superb fitting for both term structures and currencies. • Prediction of interest rates for both U.S. and Japan is excellent. • Prediction of exchange rate between U.S. dollar and Japanese yen is difficult. • Prediction of exchange rate between U.S. dollar and German mark/Euro is ... provided soon!
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Multi-Currency Quadratic m + n Models Basic Notation • The Economy: N countries with each MN time series of term structure data. • US investor is the domestic investor. • Complete markets and no-arbitrage assumption imply: · ¸ MT P (·, τ ) = Et , Mt where P (·, τ ) : default free zero-coupon bond with expiration time T = t + τ. Mt : state-price deflator. Et [·] : expectation operator under physical probability measure P.
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Assumptions Assumption 1 ((m+n) Vector of State Variables). Let Zt ≡ [Xt, Yt]> ∈ Rm+n some vector Markov process that describes the state of the economy, with Xt ∈ Rm, Yt ∈ Rn, and hXti, Ytj i = 0 for all i = 1, 2, · · · , m and j = 1, 2, · · · , n. Assumption 2 (Orthogonal Decomposition). The state-price deflator is multiplicative decomposable, Mt = ξ(Xt)ζ(Yt), where ξ(·), ζ(·) ∈ C 2(R) such that hξ, ζi = 0. The process ζt is an exponential martingale under P. Remarks: 1. Under Assumptions 1 and 2, ζ(Yt) does not enter the pricing of zero-coupon bonds: · · ¸ · ¸ ¸ ζt+τ ξt+τ ξt+τ Et = Et = P (Xt, τ ). (1) P (Z, τ ) = Et ξt ζt ξt 2. We label ξ as the term structure state-price density. 9
Assumption 3 (State Vector Dynamics). Xt ∈ Rm and Yt ∈ Rn follow dYt = −κy Ytdt + dWty ,
dXt = −κxXtdt + dWtx,
(2)
where Wtx ∈ Rm, Wty ∈ Rn are adapted standard Brownian motions with hW x, W y i = 0 and κx ∈ Rm×m, κy ∈ Rn×n. Further, let V ≡ E(XX >).
No-Arbitrage Condition The term structure state-price density, ξ, must satisfy (Duffie (1992)) , dξt = −r(Xt)dt − γξ (Xt)>dWtx. ξt where
r(Xt) : domestic instantaneous interest rate. γξ (Xt) ∈ Rm : domestic term structure risk premium.
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(3)
Multi-Currency Quadratic m + n Models Assumption 4 (MCQM(m+n)). Domestic interest rate and term structure risk premium are given by r(Xt) = Xt>Ar Xt + b> r Xt + cr , γξ (Xt) = Aξ Xt + bξ ,
(4)
where Ar , Aξ ∈ Rm×m, br , bξ ∈ Rm, cr ∈n R. The same functional forms o hold for the N −1 foreign countries, with Ajr , Ajξ , bjr , bjξ , cjr , j = 2, . . . , N . The independent martingale component of the pricing kernel is given by dζt = −γζ (Yt)dWty , ζt
(5)
where γζ (Yt) the currency risk premium and Y the currency risk factor. Moreover, γζ (Yt) = Aζ Yt + bζ , (6) where Aζ ∈ Rn×n, bζ ∈ n Rn. Theo same functional forms hold for the N − 1 foreign countries, with Ajζ , bjζ , j = 2, . . . , N .
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Interest Rates and Foreign Exchange Rates Proposition 1 (Bond Prices). Under the quadratic class, prices of zerocoupon bonds are exponential-quadratic functions of the Markov process Xt, h i > > P (Xt, τ ) = exp −Xt A (τ ) Xt − b (τ ) Xt − c (τ ) , (7) where the coefficients A(τ ) ∈ Rm×m, b(τ ) ∈ Rm, and c(τ ) ∈ R are determined by the following ordinary differential equations (see Leippold and Wu (2002a)). Proposition 2 (FX Rate Dynamics). Under Assumptions 1 - 3, exchange rate dynamics have a quadratic drift and an affine diffusion term in Zt, dStj Stj
= =
¡ ¡
j
>
r(Xt) − r (Xt) + γ(Zt) Zt>AjµZt
+
bj> µ Zt
+
cjµ
¢
¡
¢¢ ¡ ¢> j γ(Zt) − γ (Zt) dt + γ(Zt) − γ (Zt) dWt j
dt +
¡
Ajσ Zt
+
¢ j > bσ dWt, >
where γ(Zt) ≡ [γξ (Xt), γζ (Yt)]>, and Wt ≡ [Wtx, Wty ] , both of which have dimension m + n.
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(8)
Property Analysis 1. Co-movement of Interest Rates Across Countries Denote by yth the h-period yield. Then, yth = Xt>AhXt + b> h Xt + ch , where Ah = A(h∆)/(h∆),
bh = b(h∆)/(h∆),
ch = c(h∆)/(h∆).
with ∆ denoting the length (in years) of the observation interval. Cross-correlation between interest rates across countries: ¡ ¢ ∗ ∗ 2tr Ah1 V Ah2 V + b> h1 V bh2 h1 h2 ∗ Corr(yt , yt ) = q . ∗ ∗> ∗ 2 (2tr[(Ah1 V )2] + b> h1 V bh1 )(2tr[(Ah2 V ) ] + bh2 V bh2 ) Remarks: 1. A one-factor MCQM can generate negative correlation. 2. A CIR model has difficulties in reproducing negative correlation.
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2. Forecasting Interest Rates Time Series View A simple test for Expectation Hypothesis: ¡ h ¢ h−1 ft+1 − rt = ah + bh ft − rt + eht+1,
(9)
where fth is the forward rate at time t valid between t + h∆ and t + (h + 1)∆. Remarks: 1. Estimation of (9) can give regression slopes bh < 1. 2. Estimation of (9) is a purely econometric view. 3. Additional information gained by linking time series and cross-sectional characteristics in an internally consistent way, i.e. with an arbitrage-free model? Ã Requires a model with enough flexibility!!!
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Model View In our model, forward rate fth is a quadratic form. The forward regression slope bh is £¡ > ¢ ¤ 2tr Φ Ah−1Φ − Ar V (Ah − Ar ) V + (bh − br )> V (Φbh−1 − br ) bh = , 2 > 2tr ((Ah − Ar ) V ) + (bh − br ) V (bh − br ) with {Ah, bh} the quadratic coefficients for the forward rate fth, and Φ = exp(−κ∆) the weekly autocorrelation matrix for Xt. Remarks: 1. A one-factor quadratic model can generate bh < 1. 2. Compared to affine models, quadratic models offer much more flexibility to generate bh < 1 (see Leippold and Wu (2002b)). 3. Quadratic models exploit no-arbitrage information and may improve forecasts compared to the purely statistical view.
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3. Forecasting Exchange Rates Time Series View Forward premium regression: ¡ h ¢ st+h∆ − st = αh + βh fs,t − st , with h fs,t : log of h∆-period forward exchange rate at time t.
• The forward premium regression can be written as ¡ h ¢ h h Cov pt + qt , qt ¡ ¢. βh = (∆h)V ar pht + qth h pht = fs,t − Et(st+h∆) : forward risk premium, h qt = Et(st+h∆) − st : expected deprecation rate.
• Fama (1984a)’s necessary and sufficient conditions that generate βh < 0: ¡ ¢ 1. Cov pht, qth < 0, ¡ h¢ ¡ h¢ 2. V ar pt > V ar qt . 16
Model View In the MCQM model, for h=1: h i ˆξ 2tr Aˆξ V (Ar − A∗r ) V + (br − b∗r )> V b , β1 = 1 + 2tr ((Ar − A∗r ) V )2 + (br − b∗r )> V (br − b∗r ) where
¢ 1¡ > ∗ Aˆξ = Aξ Aξ − A∗> A ξ ξ ; 2
¡ ¢ ∗> ∗ ˆ ξ = 1 A> b − A b b ξ ξ ξ . 2 ξ
Remarks: 1. The slope of forward premium regression only depends upon the term structure risk premium, but not upon the currency risk premium. Nevertheless, the n currency risk factors may play an important role in the FX dynamics. 2. An MCQM(1+n) can generate negative slope coefficients. 3. Gaussian models with proportional or constant price of risk cannot account for the “anomaly”. After all, accounting for the “EH-anomalies” based on simple regressions is only a minimum requirement for model design. 17
Data and Estimation Data Description • Eight years for weekly data of LIBOR/swap rates and exchange rates for U.S. Dollar and Japanese Yen (the two most actively traded currencies in the swap market). • Weekly (Wednesday) closing mid quotes from April 6th, 1994 to April 17th, 2002 (420 observations), provided by Lehman Brothers. • LIBOR with maturities 1, 2, 3, 6 and 12 months. • US dollar swap rates 2, 3, 5, 7, 10, 15, and 30 years. Japanese Yen swap rates 2, 5, 7, 10, 20, and 30 years. • Exchange rate is represented in U.S. dollar prices per unit of Japanese yen. • In total: 24 time series of interest rates and exchange rates. • Estimation based on data before 2000 (300 observations). Data after 2000 (120 observations) for out-of-sample tests. 18
Table 1: Summary Statistics of Interest Rates and Exchange Rate Levels Maturity
Mean
Std. Dev.
Skewness
Differences Kurtosis
Auto
Mean
Std. Dev.
Skewness
Kurtosis
Auto
22.78 25.01 17.97 12.43 4.03 0.99 0.84 0.72 0.67 0.65 0.60 0.54
0.081 0.089 0.122 0.112 0.051 0.055 0.030 -0.012 -0.028 -0.039 -0.053 -0.019
A. LIBOR and Swap Rates on US Dollars 1m 2m 3m 6m 1y 2y 3y 5y 7y 10 y 15 y 30 y
5.26 5.30 5.34 5.43 5.64 5.91 6.10 6.33 6.48 6.63 6.80 6.89
1.11 1.12 1.14 1.15 1.15 1.01 0.91 0.81 0.76 0.73 0.70 0.70
-1.73 -1.75 -1.74 -1.66 -1.34 -0.80 -0.45 -0.10 0.05 0.13 0.24 0.39
2.82 2.84 2.81 2.59 1.89 0.81 0.23 -0.33 -0.52 -0.61 -0.61 -0.49
0.981 0.983 0.984 0.986 0.987 0.986 0.985 0.983 0.982 0.981 0.981 0.984
-0.45 -0.47 -0.48 -0.52 -0.51 -0.45 -0.42 -0.40 -0.37 -0.37 -0.37 -0.37
0.83 0.70 0.68 0.70 0.87 0.96 0.97 0.97 0.95 0.94 0.90 0.80
-0.60 -0.75 -1.43 -1.61 -0.68 0.13 0.16 0.19 0.21 0.23 0.18 0.26
B. LIBOR and Swap Rates on Japanese Yen 1m 2m 3m 6m 1y 2y 5y 7y 10 y 20 y 30 y
0.64 0.66 0.67 0.70 0.77 1.02 1.83 2.23 2.60 3.18 3.43
0.68 0.68 0.68 0.69 0.75 0.89 1.11 1.11 1.02 0.99 1.16
1.61 1.63 1.64 1.67 1.67 1.51 1.02 0.85 0.77 1.02 1.25
1.43 1.49 1.52 1.67 1.72 1.32 0.23 -0.13 -0.28 0.41 0.90
0.984 0.988 0.989 0.989 0.990 0.989 0.990 0.991 0.990 0.990 0.990
-0.54 -0.53 -0.53 -0.55 -0.57 -0.67 -0.83 -0.79 -0.75 -0.61 -0.58
0.61 0.41 0.37 0.36 0.42 0.61 0.69 0.67 0.64 0.79 1.03
-0.91 -1.98 -2.07 -1.60 -0.46 0.03 0.36 0.36 0.35 1.98 4.67
26.66 15.98 14.37 10.16 7.94 5.69 2.80 1.38 1.39 16.97 74.56
-0.155 0.026 0.045 0.155 0.135 0.052 0.017 0.022 0.034 -0.049 -0.032
11.64
0.81
4.53
0.048
C. Dollar Price of Yen —
0.89
0.10
0.63
0.51
0.988
19
-5.35
Factor Analysis Table 2: PCA for US and Japanese Interest Rates and Exchange Rate
Factor
1 2 3 4 5 6 7 8 9 10
Panel A: PCA on Levels
Panel B: PCA on Differences
USD
JPY
UJR
UJF
USD
JPY
UJR
UJF
81.11 18.06 0.61 0.11 0.04 0.03 0.02 0.01 0.01 0.00
83.59 11.48 3.78 0.59 0.32 0.13 0.04 0.03 0.02 0.01
63.58 27.44 5.53 2.11 0.56 0.40 0.12 0.09 0.06 0.03
63.57 27.44 5.52 2.11 0.56 0.41 0.12 0.09 0.06 0.03
75.04 16.73 3.57 1.75 1.28 0.85 0.31 0.28 0.11 0.03
62.44 17.55 11.00 4.05 2.04 1.12 0.69 0.43 0.34 0.20
51.05 19.93 11.40 5.68 3.93 2.11 1.41 1.07 0.88 0.66
51.03 19.92 11.40 5.68 3.93 2.11 1.41 1.07 0.88 0.66
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Principal Components I US Short End
US Long End
0.1
0.1
0
0.05
−0.1
0
−0.2 −0.05
−0.3
−0.1
−0.4
−0.15 50
100
150
200
250
300
350
400
50
Japanese Short End
100
150
200
250
300
350
400
350
400
Japanese Long End
5
0.4 0.2
0
0 −0.2
−5
−0.4 −0.6
−10 50
100
150
200
250
300
350
400
50
100
150
200
250
300
US dollar/Japanese yen 0.04 0.02 0 −0.02 −0.04 −0.06 50
100
150
200
250
300
350
400
Figure 1: Reconstruction of original time series using the first four PC. The plots show the relative deviation from the original time series.
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Principal Components II US Short End
US Long End 0.02
0.02 0
0.01
−0.02 0
−0.04 −0.06
−0.01
−0.08 100 200 300 Japanese Short End
400
0.5
100 200 300 Japanese Long End
400
100
400
0.02
0 −0.5
0
−1 −1.5 100
200
−0.02 400 US dollar/Japanese yen
300
200
300
0.02 0 −0.02 50
100
150
200
250
300
350
400
Figure 2: Reconstruction of original time series using the first six to nine PCs. The plots show the relative deviation from the original time series.
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Co-movement of short and long rates 1 month LIBOR
3 month LIBOR
0
−2
US JP 100
200
300
−4
400
timeLIBOR 1 month
rates
rates
−2
US DEM 100
200
300
−4
400
rates
rates 100
200
time
300
400
−4
200
300
400
100
200
0
−2
US DEM 300
−4
400
US DEM 100
200
300
400
time 7 year swap 2
0
−2
JP DEM
100
2
2
0
US JP time 7 year swap
timeLIBOR 3 month
2
−4
−4
400
0
timeLIBOR 1 month
−2
300
2
0
−4
200
timeLIBOR 3 month
2
−2
100
0
−2
US JP
rates
−4
0
rates
−2
2
rates
2
rates
rates
2
7 year Swap
−2
JP DEM 100
200
time
0
300
400
−4
JP DEM 100
200
time
Figure 3: Co-movement of US, Japanese and German interest rates.
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300
400
Table 3: Variance Decomposition on US and Japanese Interest Rates and Exchange Rate Factors
1
2
3
4
5
6
17
13.43 21.23 34.69 51.63 67.68 92.60 94.96 94.66 93.38 89.54 86.36 77.15
0.00 0.02 0.15 0.00 0.07 0.83 0.89 1.16 1.30 1.45 1.50 1.24
64.05 59.66 50.64 37.67 17.14 0.08 0.35 2.29 3.92 5.83 7.81 9.78
0.00 1.72 2.32 2.31 1.46 0.03 0.00 0.04 0.04 0.03 0.09 0.10
4.65 0.06 0.61 2.70 4.49 0.20 0.04 0.05 0.21 0.44 0.51 1.01
13.07 0.49 0.00 1.32 3.98 2.41 1.20 0.05 0.11 1.02 1.58 3.68
0.00 0.00 0.03 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.03
73.80
0.82
17.02
0.50
1.14
2.33
0.01
0.64 0.61 1.25 1.28 3.15 5.49 7.13 6.51 6.33 3.25 2.21
4.52 11.31 13.05 21.40 31.62 53.29 72.47 76.41 73.27 81.07 68.97
6.35 4.34 2.56 1.43 0.55 0.05 0.10 0.17 0.25 0.53 0.75
35.75 41.89 47.23 47.82 35.75 21.24 6.51 1.52 0.53 7.51 15.49
35.24 17.32 8.03 1.34 0.00 4.10 9.32 9.50 8.65 1.38 9.51
0.30 6.48 9.17 5.87 2.66 0.01 1.78 1.94 1.78 0.00 0.86
0.00 0.17 0.52 0.00 0.01 0.03 0.02 0.01 0.02 0.00 0.00
JP aggregated
3.80
58.63
1.16
16.48
9.58
1.61
0.03
FX
0.10
0.32
0.43
0.14
1.31
1.59
78.42
51.03
19.92
11.40
5.68
3.93
2.11
0.09
US
1m 2m 3m 6m 12m 2y 3y 5y 7y 10y 15y 30y
US aggregated JP
Total
1m 2m 3m 6m 12m 2y 5y 7y 10y 20y 30y
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Table 4: Correlations of the Mimicking PCA Portfolios with US and Japanese Interest Rates and Exchange Rate Factors US
1m 2m 3m 6m 12m 2y 3y 5y 7y 10y 15y 30y
US average JP
1m 2m 3m 6m 12m 2y 5y 7y 10y 20y 30y
JP average FX Total average
1
2
3
4
5
6
17
0.37 0.46 0.59 0.72 0.82 0.96 0.97 0.97 0.97 0.95 0.93 0.88
-0.01 -0.02 -0.04 -0.01 -0.03 -0.09 -0.09 -0.11 -0.11 -0.12 -0.12 -0.11
-0.80 -0.77 -0.71 -0.61 -0.41 -0.03 0.06 0.15 0.20 0.24 0.28 0.31
0.00 0.13 0.15 0.15 0.12 0.02 -0.00 -0.02 -0.02 -0.02 -0.03 -0.03
-0.22 -0.02 0.08 0.16 0.21 0.04 0.02 -0.02 -0.05 -0.07 -0.07 -0.10
-0.36 -0.07 0.00 0.12 0.20 0.16 0.11 0.02 -0.03 -0.10 -0.13 -0.19
0.00 -0.00 0.02 0.00 0.00 -0.00 -0.00 0.01 0.01 0.00 0.00 -0.02
0.80
-0.07
-0.17
0.04
-0.00
-0.02
0.00
0.08 0.08 0.11 0.11 0.18 0.23 0.27 0.26 0.25 0.18 0.15
0.21 0.34 0.36 0.46 0.56 0.73 0.85 0.87 0.86 0.90 0.83
-0.25 -0.21 -0.16 -0.12 -0.07 0.02 0.03 0.04 0.05 0.07 0.09
-0.60 -0.65 -0.69 -0.69 -0.60 -0.46 -0.26 -0.12 -0.07 0.27 0.39
-0.59 -0.42 -0.28 -0.12 0.01 0.20 0.31 0.31 0.29 -0.12 -0.31
0.05 0.25 0.30 0.24 0.16 -0.01 -0.13 -0.14 -0.13 0.01 0.09
-0.00 0.04 -0.07 0.00 0.01 0.02 -0.01 -0.01 0.01 0.00 -0.00
0.17
0.63
-0.05
-0.32
-0.07
0.06
-0.00
-0.03
-0.06
0.07
0.04
-0.11
-0.13
0.89
0.48
0.25
-0.11
-0.12
-0.04
0.01
0.04
25
Conclusion from PC Analysis • Upward sloping mean yield curve, higher kurtosis at the short end. • US: higher volatility and negative skewness at the short end; Japan: higher volatility at the long end, positive skewness at the short end. • PCA for single countries: 3 factors explain more than 90% variation. • For PCA based on UJF (differences), we need at least 5 factors to explain more than 90% variation. • No substantial change for PCA by adding exchange rate. • More detailed PCA analysis reveals: – Term structure movements do not have much in common. – A “common factor” is the curvature factor. – Predominant part of FX movements is not related to term structure factors.
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Estimation Methodology • We use QML with (Scaled) Unscented Kalman Filter (UKF). • Duffee and Stanton (2001) exemplify the superiority of extended Kalman filter (EKF) estimation over EMM (with SNP auxiliary model). • UKF is a relatively new filtering technique more powerful than EKF. • Builds on the principle that it is easier to approximate probability distribution than to approximate nonlinear function. • Two main advantages over EKF: 1. More accurate results. 2. More efficient implementation.
27
Estimation Results Table 5: Parameter Estimation (Preliminary) Data used are weekly from April 6th, 1994 to April 17th, 2002 (420 observations for each time series).
m
1
2
3
4
κx
0.1136
1.9683
0.2094
0.1464
A. Common Dynamics 5 6 0.5717
0.0852
n
1
κy
0.0010
B. Interest Rates U.S. m
1
Ar 0.0000 br 0.0103 cr 0.0551 κx + Aξ 0.0048 bξ −0.1456
Japan
2
3
4
5
6
0.0090 0.0152
0.0000 0.0177
0.0004 0.0002
0.0002 0.0077
0.0000 0.0000
0.3629 −0.1482
1.8092 −0.0039
0.0183 0.3590 0.0019 −0.0002
0.8505 −0.0030
1
2
0.0001 0.0000 0.0016 0.0049 0.0068 0.0558 0.0001 0.0005 −0.0024
3
4
5
6
0.0007 0.0000
0.0000 0.0131
0.0166 0.0071
0.0000 0.0117
0.0809 0.4685 0.0133 −0.6629
0.7656 0.2018
1.9343 −0.0004
C. Foreign Exchange Rates U.S. n Aζ bζ
Japan
1
1
−0.0137 0.3553
0.0012 0.0001
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Table 6: Estimation Results (Preliminary) Data used are weekly from April 6th, 1994 to April 17th, 2002 (420 observations for each time series). Forecasting Error
Fitting Error
MCQM
(3+1)
(4+1)
(5+1)
(6+1)
(3+1)
(4+1)
(5+1)
(6+1)
US
1m 2m 3m 6m 12m 2y 3y 5y 7y 10y 15y 30y
57.39 61.28 64.50 79.31 89.97 92.66 94.47 96.00 96.48 96.56 96.21 93.58
−27.41 −16.52 1.50 49.73 89.20 95.81 96.04 96.44 96.67 96.73 96.49 95.61
−18.29 −13.41 −4.97 27.82 72.35 89.13 91.91 94.10 95.09 95.76 95.82 92.48
87.49 90.65 89.73 91.44 94.37 95.31 95.83 96.32 96.57 96.77 96.81 96.92
71.41 75.55 77.68 89.36 96.60 97.69 98.95 99.86 100.00 99.77 99.16 95.65
−28.66 −17.37 1.37 50.11 92.09 99.96 99.95 99.97 100.00 99.85 99.42 97.58
30.69 35.87 40.14 61.42 90.59 99.11 99.65 99.80 99.89 99.82 99.27 94.70
95.22 98.71 97.88 97.76 99.05 99.92 99.99 99.96 99.97 99.99 99.86 99.07
JP
1m 2m 3m 6m 12m 2y 5y 7y 10y 20y 30y
97.16 97.60 97.41 96.41 94.70 95.14 96.83 96.54 95.23 93.58 81.73
91.43 92.29 92.50 92.30 92.50 93.76 97.10 97.84 97.27 91.83 73.16
95.67 96.28 96.19 95.82 94.84 95.41 97.21 96.84 95.81 92.77 80.91
96.39 97.40 97.85 98.32 98.71 98.30 98.75 99.00 97.92 95.70 84.31
98.53 99.57 99.80 99.15 97.44 96.68 98.11 98.05 97.32 95.27 82.63
98.68 99.57 99.81 99.29 98.40 98.02 99.71 99.88 99.42 93.55 74.22
97.64 98.74 98.65 98.08 96.90 97.19 99.21 99.06 98.14 95.47 83.24
98.77 99.55 99.74 99.72 99.82 99.56 99.82 99.96 98.86 96.94 85.50
−244.44
−39.02
−76.97
−8.33
99.82
100.00
99.93
98.21
−9994
−10994
−11214
−14225
FX Maximum Likelihood
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Forward Rate Regression 6
Forward Rate Regression Slopes
Forward Rate Regression Slopes
2
0
−2
−4
−6
−8
−10 0
5
10
15
20
25
30
5 4 3 2 1 0 −1 0
5
10
Maturity, Years
15
20
25
30
Maturity, Years
Figure 4: Forward Rate Regressions The solid lines are the regression slope estimates of the following regression n ft+1 − rt = an + bn (ftn − rt ) + et+1
where ftn denotes time-t forward rates between time t + n and t + n + ∆ and r is the short rate. The dashed dotted lines are the 95 percent confidence interval constructed based on the standard deviation estimates of the regression slopes. The forward rates are extracted from the data based on the (6 + 1) model.
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Forward Rate Regression - Simulation
5
0
Forward Rate Regression Slopes
Forward Rate Regression Slopes
2
−2 −4 −6 −8 −10 −12 −14 0
5
10
15
20
25
30
Maturity, Years
0 −5 −10 −15 −20 −25 −30 0
5
10
15
20
25
30
Maturity, Years
Figure 5: Forward Rate Regressions Lines are the median (solid line) and 5 percent and 95 percent percentiles of the forward rate regression based on 1,000 simulated paths of forward rates. Each path has an weekly frequency and 300 observations. The paths are simulated based on the MCQM (6 + 1) model with parameter estimates in Table 5.
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Forecasting Exchange Rate 1.3
Dollar Price of 100 Yen
1.2
1.1
1
0.9
0.8
0.7 Jan95
Jan96
Jan97
Jan98
Jan99
Figure 6: Fitting and Forecasting of Exchange Rate Fitting (solid line) and forecasting (dashed line) of observed foreign exchange rates (circles). EH regression yields ³ ´ ∆st+1 = 0.2032 − 4.3884 yt − ytj + e, R2 = 0.0020, (0.2664)
(5.6917)
where yt and ytj denotes the one-week spot rates on US dollar and Japanese yen, respectively. Monte Carlo simulation based on the model parameter estimates, mean slope estimated at 8.16, but with standard deviation 22.48!
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Summary • The MCQM(m+n) enables us to integrate Gaussian state variables, affine market price of risk, and rich nonlinear dynamics for interest rates and exchange rates into a consistent framework. • The (m+n) factor structure helps to explain the, empirically observed, independent movements in term structures and exchange rates within an arbitrage-free setup. • The MCQM(m+n) can readily explain documented evidence on EH based forecasting relations. • PCA shows little evidence of co-movements in term structure. A predominant independent factor of exchange rate movements exists. • MCQM(6+1) fits interest rates and exchange rates. Forecasting of interest rates is excellent. Forecasting of exchange rates is difficult. • Future research: analyze and integrate different currencies and term structures. 33
References Backus, D., S. Foresi, A. Mozumdar, and L. Wu, 2001, “Predictable Changes in Yields and Forward Rates,” Journal of Financial Economics, 59(3), 281–311. Backus, D., S. Foresi, and C. Telmer, 2001, “Affine Term Structure Models and the Forward Premium Anomaly,” Journal of Finance, 56(1), 279–304. Bekaert, G., R. Hodrick, and D. Marshall, 1997, “On Biases In Tests of the Expectation Hypothesis of the Term Structure of Interest Rates,” Journal of Financial Economics, 44(3), 309–348. Cheung, Y., 1993, “Exchange Rate Risk Premiums,” Journal of International Money and Finance, 12, 182–194. Duffee, G., and R. Stanton, 2001, “Estimation of Dynamic Term Structure Models,” manuscript, UC Berkley. Duffie, D., 1992, Dynamic Asset Pricing Theory. Princeton University Press, Princeton, New Jersey, second edn. Engle, C., 1996, “The Forward Discount Anomaly and the Risk Premium: Survey of Recent Evidence,” Journal of Empirical Finance, 3, 123–192. Fama, E., 1984a, “Forward and Spot Exchange Rates,” Journal of Monetary Economics, 14, 319–338. Fama, E. F., 1984b, “The Information in the Term Structure,” Journal of Financial Economics, 13, 509–528. Leippold, M., and L. Wu, 2002a, “Asset Pricing under the Quadratic Class,” Journal of Financial and Quantitative Analysis, 37(2), 271–295. , 2002b, “Design and Estimation of Quadratic Term Structure Models,” European Finance Review, forthcoming. Lothian, J. R., and L. Wu, 2002, “Uncovered Interest Rate Parity Over Past Two Centuries,” Working paper, Graduate School of Business Administration, Fordham University. Roll, R., 1970, The Behavior of Interest Rates. Basic Books, New York. Wu, Y., and H. Zhang, 1997, “Forward Premiums as Unbiased Predictors of Future Currency Depreciation,” Journal of International Money and Finance, 16(4), 609–623.
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