how efficient is the extended kalman filter at estimating ... · the cross section of yields are...
TRANSCRIPT
How Efficient is the Extended Kalman Filter
at Estimating Shadow-Rate Models?
Jens H. E. Christensen
Federal Reserve Bank of San Francisco
101 Market Street, Mailstop 1130
San Francisco, CA 94105
Preliminary and incomplete draft. Comments are welcome.
Abstract
I perform a carefully orchestrated simulation exercise to study the efficiency of estimating an
established Gaussian shadow-rate model with the extended Kalman filter. First, despite a near
unit-root property imposed on the most persistent factor, finite-sample bias remains for the esti-
mated persistence of other factors in the model. Second, parameters determined primarily from
the cross section of yields are sensitive to sample length, data quality, and sampling frequency.
Third, the accuracy of the filtered state variables improves with data quality and sampling fre-
quency, but is insensitive to sample length. Also, filtering deteriorates when yields are severely
compressed against the zero lower bound. Importantly, though, both the accuracy of the esti-
mated parameters and the filtering performance are in general close to that reported elsewhere
for affine Gaussian models estimated with the standard Kalman filter. Based on this evidence,
I recommend using the extended Kalman filter for estimation of U.S. shadow-rate models, but
question its use for estimation of Japanese shadow-rate models where interest rates have been
severely depressed for prolonged periods.
JEL Classification: C13, C58, G12, G17
Keywords: arbitrage-free Nelson-Siegel models, finite-sample bias
The views in this paper are solely the responsibility of the author and should not be interpreted as reflecting theviews of the Federal Reserve Bank of San Francisco or the Board of Governors of the Federal Reserve System.
This version: August 24, 2015.
1 Introduction
Modeling the term structure of interest rates is challenging and more so when rates are near their
zero lower bound as has been the case in the U.S. and elsewhere in recent years. To account for the
asymmetric behavior of bond yields near the zero lower bound, researchers and practitioners alike are
increasingly relying on so-called shadow-rate term structure models, where the instantaneous risk-free
rate is a truncated version of an underlying unobserved “shadow” process. Due to their nonlinear
yield function, estimation of shadow-rate models is non-standard and typically performed with the
extended Kalman filter, see Kim and Singleton (2012) and Christensen and Rudebusch (2015a) for
examples. However, little is known about the advantages and disadvantages of estimating shadow-
rate models in this way. Therefore, in this paper, I undertake a set of simulation experiments to
study the efficiency of the extended Kalman filter at estimatng Gaussian shadow-rate models.
In a recent paper, Christensen et al. (2015b, henceforth CLR) use simulations to analyze the
efficiency of the standard Kalman filter for estimation of arbitrage-free Nelson-Siegel (AFNS) models
with and without stochastic volatility. Here, I follow an approach similar to theirs and perform a
carefully orchestrated simulation study based on the established Gaussian shadow-rate AFNS model
described in Christensen and Rudebusch (2015b) and henceforth referred to as the B-CR model.1 I
then use the simulated data as input into model estimations based on the extended Kalman filter.
First, I find that, despite a near unit-root property imposed on the most persistent factor, finite-
sample bias remains for the estimated persistence of other factors in the model. This suggests that,
rather than correcting state variables individually, finite-sample bias correction should ideally be
performed on the joint dynamic system of the state variables as also argued by Bauer et al. (2012).
Second, the results show that parameters determined primarily from the cross section of yields
are sensitive to sample length, data quality, and sampling frequency.
Third, the accuracy of the filtered state variables improves with data quality and sampling fre-
quency, but is insensitive to sample length. Furthermore, a more detailed analysis of the infrequent
cases where the filtering deteriorates in quality reveals that this mainly happens when yields are
unusually low and severely compressed against the zero lower bound.
Importantly, though, both the accuracy of the estimated parameters and the filtering performance
are in general close to that reported by CLR for their simulation exercises based on the affine Gaussian
AFNS model where the standard Kalman filter is an efficient estimator equivalent to exact maximum
likelihood estimation. Based on this evidence, I conclude that, for estimation of Gaussian shadow-
rate models, the extended Kalman filter appears to be almost as efficient as the standard Kalman
filter is for affine Gaussian models. As a consequence, I recommend its use for estimation of U.S.
shadow-rate models. However, given that Japanese interest rates have been severely depressed for
1This model is also used in Christensen et al. (2015a).
1
prolonged periods since 1995, the results would question its use for Japanese shadow-rate models.
The rest of the paper is structured as follows. Section 2 describes the B-CR model, while Section
3 details the extended Kalman filter estimation used throughout the paper. Section 4 lays out the
simulation exercise and Section 5 presents the results. Section 6 concludes the paper.
2 The B-CR Model
In the shadow-rate AFNS model introduced in Christensen and Rudebusch (2015a), the shadow risk-
free rate is unconstrained and defined as the sum of level and slope just like in the original AFNS
model class, while the short rate respects a nonnegativity constraint tied to the shadow rate:
st = Lt + St, rt = max0, st. (1)
The dynamics of the state variables used for pricing under the Q-measure remain as in the original
AFNS model class:
dLt
dSt
dCt
=
0 0 0
0 λ −λ0 0 λ
θQ1
θQ2
θQ3
−
Lt
St
Ct
dt+Σ
dWL,Qt
dW S,Qt
dWC,Qt
, λ > 0, (2)
where Σ is the constant covariance (or volatility) matrix.
By implication, the yield on the shadow discount bond maintains the popular Nelson and Siegel
(1987) factor loading structure
yt(τ) = Lt +
(1− e−λτ
λτ
)St +
(1− e−λτ
λτ− e−λτ
)Ct −
A(τ)
τ, (3)
where A(τ)/τ is the same maturity-dependent yield-adjustment term as in regular AFNS models,
see Christensen et al. (2011) for details.
The corresponding instantaneous shadow forward rate is given by
ft(τ) = − ∂
∂TlnPt(τ) = Lt + e−λτSt + λτe−λτCt +Af (τ), (4)
2
where the yield-adjustment term in the instantaneous forward rate function is given by
Af (τ) = −∂A(τ)∂τ
= −1
2σ211τ
2 − 1
2(σ221 + σ222)
(1− e−λτ
λ
)2
−1
2(σ231 + σ232 + σ233)
[ 1
λ2− 2
λ2e−λτ − 2
λτe−λτ +
1
λ2e−2λτ +
2
λτe−2λτ + τ2e−2λτ
]
−σ11σ21τ1− e−λτ
λ− σ11σ31
[1λτ − 1
λτe−λτ − τ2e−λτ
]
−(σ21σ31 + σ22σ32)[ 1
λ2− 2
λ2e−λτ − 1
λτe−λτ +
1
λ2e−2λτ +
1
λτe−2λτ
].
Krippner (2013) provides a formula for the zero lower bound instantaneous forward rate, ft(τ),
that applies to any Gaussian model
ft(τ) = ft(τ)Φ
(ft(τ)ω(τ)
)+ ω(τ)
1√2π
exp(− 1
2
[ft(τ)ω(τ)
]2),
where Φ(·) is the cumulative probability function for the standard normal distribution, ft(τ) is the
shadow forward rate, and ω(τ) is related to the conditional variance, v(τ, τ + δ), appearing in the
shadow bond option price formula as follows
ω(τ)2 =1
2limδ→0
∂2v(τ, τ + δ)
∂δ2.
Within the shadow-rate AFNS model, ω(τ) takes the following form
ω(τ)2 = σ2
11τ + (σ2
21+ σ2
22)1− e−2λτ
2λ+ (σ2
31+ σ2
32+ σ2
33)[1− e−2λτ
4λ− 1
2τe−2λτ − 1
2λτ2e−2λτ
]
+2σ11σ211− e−λτ
λ+ 2σ11σ31
[− τe−λτ +
1− e−λτ
λ
]+ (σ21σ31 + σ22σ32)
[− τe−2λτ +
1− e−2λτ
2λ
].
Therefore, the zero-coupon bond yields that observe the zero lower bound, denoted yt(τ), are easily
calculated as
yt(τ) =
1
τ
∫ t+τ
t
[ft(s)Φ
(ft(s)ω(s)
)+ ω(s)
1√2π
exp(− 1
2
[ft(s)ω(s)
]2)]ds. (5)
Similar to the affine AFNS model, the shadow-rate AFNS model is completed by specifying the
price of risk using the essentially affine risk premium specification introduced by Duffee (2002), so
the real-world dynamics of the state variables can be expressed as
3
dLt
dSt
dCt
=
κP11
κP12
κP13
κP21
κP22
κP23
κP31
κP32
κP33
θP1
θP2
θP3
−
Lt
St
Ct
dt+
σ11 0 0
σ21 σ22 0
σ31 σ32 σ33
dWL,Pt
dWS,Pt
dWC,Pt
. (6)
In this unrestricted case, both KP and θP are allowed to vary freely relative to their counterparts
under the Q-measure. However, I focus on the case with the same KP and θP restrictions as in
Christensen and Rudebusch (2012, 2015b)
dLt
dSt
dCt
=
10−7 0 0
κP21 κP22 κP23
0 0 κP33
0
θP2
θP3
−
Lt
St
Ct
dt+Σ
dWL,Pt
dW S,Pt
dWC,Pt
. (7)
I label this shadow-rate model the “B-CR model.”2
3 The Extended Kalman Filter Estimation
Thanks to the nonlinear measurement equation (5), I estimate the B-CR model using the extended
Kalman filter. In the following, I first describe the standard Kalman filter that is efficient for affine
Gaussian models before I proceed to a description of the extended Kalman filter that I use in the
remainder of the paper to estimate the B-CR model.
For affine Gaussian models, in general, the conditional mean vector and the conditional covariance
matrix are
EP [XT |Ft] = (I − exp(−KP∆t))θP + exp(−KP∆t)Xt,
V P [XT |Ft] =
∫ ∆t
0e−KP sΣΣ′e−(KP )′sds,
where ∆t = T − t. I compute conditional moments of discrete observations and obtain the state
transition equation
Xt = (I − exp(−KP∆t))θP + exp(−KP∆t)Xt−1 + ξt,
where ∆t is the time between observations. In the standard Kalman filter, the measurement equation
would be affine, in which case
yt = A+BXt + εt.
2Following Kim and Singleton (2012), the prefix “B-” refers to a shadow-rate model in the spirit of Black (1995).
4
The assumed error structure is
(ξt
εt
)∼ N
[(0
0
),
(Q 0
0 H
)],
where the matrix H is assumed diagonal, while the matrix Q has the following structure:
Q =
∫ ∆t
0e−KP sΣΣ′e−(KP )′sds.
In addition, the transition and measurement errors are assumed orthogonal to the initial state.
Now, I consider Kalman filtering, which is used to evaluate the likelihood function.
Normally, the state variables are assumed to be stationary, in which case the filter is initialized
at the unconditional mean and variance of the state variables under the P -measure: X0 = θP and
Σ0 =∫∞0 e−KP sΣΣ′e−(KP )′sds, which are calculated using the analytical solutions provided in Fisher
and Gilles (1996). In case of the B-CR model, the state variables have a near unit root and the
Kalman filter algorithm is initialized in a different way as explained below.
Denote the information available at time t by Yt = (y1, y2, . . . , yt), and denote model parameters
by ψ. Consider period t−1 and suppose that the state update Xt−1 and its mean square error matrix
Σt−1 have been obtained. The prediction step is
Xt|t−1 = EP [Xt|Yt−1] = ΦX,0t (ψ) + ΦX,1
t (ψ)Xt−1,
Σt|t−1 = ΦX,1t (ψ)Σt−1Φ
X,1t (ψ)′ +Qt(ψ),
where ΦX,0t = (I−exp(−KP∆t))θP , ΦX,1
t = exp(−KP∆t), and Qt =∫∆t
0 e−KP sΣΣ′e−(KP )′sds, while
∆t is the time between observations.
In the update step at time t, Xt|t−1 is improved by using the additional information contained in
Yt. We have that
Xt = E[Xt|Yt] = Xt|t−1 +Σt|t−1B(ψ)′F−1t vt,
Σt = Σt|t−1 −Σt|t−1B(ψ)′F−1t B(ψ)Σt|t−1,
where
vt = yt −E[yt|Yt−1] = yt −A(ψ)−B(ψ)Xt|t−1,
Ft = cov(vt) = B(ψ)Σt|t−1B(ψ)′ +H(ψ),
H(ψ) = diag(σ2ε (τ1), . . . , σ2ε(τN )).
5
At this point, the Kalman filter has delivered all ingredients needed to evaluate the Gaussian log
likelihood, the prediction-error decomposition of which is
log l(y1, . . . , yT ;ψ) =
T∑
t=1
(− N
2log(2π) − 1
2log |Ft| −
1
2v′tF
−1t vt
),
where N is the number of observed yields. The likelihood is numerically maximized with respect to
ψ using the Nelder-Mead simplex algorithm. Upon convergence, standard errors are obtained from
the estimated covariance matrix,
Ω(ψ) =1
T
[ 1T
T∑
t=1
∂ log lt(ψ)
∂ψ
∂ log lt(ψ)
∂ψ
′]−1,
where ψ denotes the estimated model parameters.
This completes the description of the standard Kalman filter. However, in shadow-rate models,
zero-coupon bond yields are not affine functions of the state variables. Instead, the measurement
equation takes the general form
yt = z(Xt;ψ) + εt.
In the extended Kalman filter used in this paper, this equation is linearized through a first-order
Taylor expansion around the best guess of Xt in the prediction step of the Kalman filter algorithm.
Thus, in the notation introduced above, this best guess is denoted Xt|t−1 and the approximation is
given by
z(Xt;ψ) ≈ z(Xt|t−1;ψ) +∂z(Xt;ψ)
∂Xt
∣∣∣Xt=Xt|t−1
(Xt −Xt|t−1).
Now, by defining3
At(ψ) ≡ z(Xt|t−1;ψ)−∂z(Xt;ψ)
∂Xt
∣∣∣Xt=Xt|t−1
Xt|t−1 and Bt(ψ) ≡∂z(Xt;ψ)
∂Xt
∣∣∣Xt=Xt|t−1
,
the measurement equation can be given in an affine form as
yt = At(ψ) +Bt(ψ)Xt + εt,
and the steps in the algorithm proceeds as previously described for the standard Kalman filter.
However, due to the approximation above, the optimization of the likelihood function is referred to
as quasi maximum likelihood.
Normally, the unconditional distribution of the state variables is used to start the Kalman filter.
However, with a unit-root property imposed on the Nelson-Siegel level factor, the joint dynamics
3The derivatives involved are calculated numerically.
6
of the state variables are no longer stationary and the unconditional distribution does not exist.
Instead, I follow Duffee (1999) and derive a distribution for the starting point of the Kalman filter
based on the yields observed at the first data point in the sample.
As already noted, the yield function in shadow-rate models is nonlinear
yt = z(Xt;ψ) + εt,
which complicates the adaptation of Duffee’s (1999) approach. However, for U.S. Treasury data that
date back before the financial crisis it is the case that yields are far away from the zero lower bound.
Under those circumstances the shadow-rate AFNS model—like any shadow-rate model—collapses to
its equivalent regular model since the option to hold currency is far out of the money. Thus, the
yield function for the initial yield observation will be well approximated by4
yt = A(ψ) +B(ψ)Xt + εt, εt ∼ N(0,H),
where A(ψ) and B(ψ) are calculated from the corresponding regular AFNS model.
For the first set of observations, I rewrite this equation as
y1 = A(ψ) +B(ψ)X0 + ε0 ⇐⇒ B(ψ)X0 = y1 −A(ψ) − ε0.
Now, multiply from the left on both sides by B(ψ)′ to obtain
B(ψ)′B(ψ)X0 = B(ψ)′(y1 −A(ψ)) −B(ψ)′ε0.
Then, X0 can be isolated by using the inverse of B(ψ)′B(ψ)
X0 = (B(ψ)′B(ψ))−1B(ψ)′(y1 −A(ψ)) − (B(ψ)′B(ψ))−1B(ψ)′ε0.
Here, ε0 is normally distributed with a mean of zero and a covariance matrix equal to H. By
implication, X0 follows a normal distribution with the following properties
X0 ∼ N [(B(ψ)′B(ψ))−1B(ψ)′(y1 −A(ψ)), (B(ψ)′B(ψ))−1B(ψ)′HB(ψ)(B(ψ)′B(ψ))−1].
This is the normal distribution used to start the Kalman filter when a unit-root property is assumed.5
4Obviously, for the simulated yield samples studied later on, it will not always be the case that the initial yields arefar from the zero lower bound, but even then the committed error is likely to be relatively small and only applies tothe first observation date.
5Note that this approach generalizes to estimation of non-Gaussian affine models where nonstationarity is required.See Duffee (1999) for an example.
7
1985 1990 1995 2000 2005 2010 2015
02
46
810
12
Rat
e in
per
cent
Ten−year yield Five−year yield Two−year yield Three−month yield
Figure 1: Treasury Yields.
The figure shows three-month, two-year, five-year, and ten-year weekly U.S. Treasury zero-coupon bond yields
from January 4, 1985, to October 31, 2014.
KP KP·,1 KP
·,2 KP·,3 θP Σ
KP1,· 10−7 0 0 0 σ11 0.0069
(0.0001)KP
2,· 0.1953 0.3138 -0.4271 0.0014 σ22 0.0112
(0.1474) (0.1337) (0.0904) (0.0364) (0.0002)KP
3,· 0 0 0.4915 -0.0252 σ33 0.0257
(0.1200) (0.0087) (0.0004)
Table 1: Parameter Estimates for the B-CR Model.
The estimated parameters of the KP matrix, θP vector, and diagonal Σ matrix are shown for the B-CR model.
The estimated value of λ is 0.4700 (0.0026). The numbers in parentheses are the estimated parameter standard
deviations. The quasi maximum log likelihood value is 71,408.90.
4 The Simulation Exercise
To begin the simulation exercise, I first estimate the B-CR model on a weekly sample of U.S. Treasury
yields containing eight yields with maturities from three months to ten years covering the period
January 4, 1985, until October 31, 2014. Four of the eight yield series are shown in Figure 1.
The estimated model parameters are shown in Table 1 and identical to the estimates reported in
8
Christensen and Rudebusch (2015b).6 I use these parameters as the true parameters in the model
simulations.
Second, thanks to the unit-root property imposed on the level factor in the B-CR model, the
unconditional distribution of the state variables is not well defined. As a consequence, starting
values for each simulation cannot be obtained in the usual way. Instead, I generate the initial
conditions in each simulation by randomly drawing a state variable vector from the set of filtered
state variables implied by the B-CR model estimation using U.S. Treasury yields as described above.
This set contains a total of 1,557 such vectors. This is repeated N = 1,000 times.
Third, to simulate sample paths for the three state variables, I approximate the continuous-time
dynamics in equation (7) using the Euler approximation.7 To provide a stylized example,
dXit = κPii (θ
Pi −Xi
t)dt+ κPij(θPj −Xj
t )dt+ σiidWP,it
is approximated using
Xit = Xi
t−1 + κPii (θPi −Xi
t−1)∆t+ κPij(θPj −Xj
t−1)∆t+ σii√∆tzit, zit ∼ N(0, 1),
where I fix ∆t at a uniform value of 0.0001.
In one set of exercises, I simulate 1,000 samples with a length of ten years observed at weekly
and monthly frequency. It is important to emphasize that it is the same 1,000 sample paths that
are being simulated according to the algorithm above independent of the sampling frequency. This
is done to make the results as comparable as possible estimation by estimation. In the other set
of exercises, I simulate 1,000 samples with a length of thirty years observed at weekly and monthly
frequency. Again, it is the same 1,000 sample paths that are being simulated according to the
algorithm independent of the sampling frequency. Furthermore, it is the same 1,000 starting values
for the three state variables that are used in all exercises, again in an attempt to make the results as
comparable as possible throughout.
In the fourth step, these simulated factor paths are converted into simulated zero-coupon yields
with eight maturities, 0.25, 0.5, 1, 2, 3, 5, 7, and 10 years, using the yield function in equation (5).
Finally, Gaussian i.i.d. measurement errors are added to the bond yields. In one set of exercises,
the measurement error standard deviation is fixed uniformly at σε = 1 basis point. In the other set
of exercises, it is fixed uniformly at σε = 10 basis points. It should be noted that the simulated
measurement errors are the same independent of the value of σε. Also, it is important to note
that this may cause the simulated yields to be negative provided the measurement error shocks are
sufficiently negative. As an alternative, the simulated yields including the measurement errors could
6See Christensen and Rudebusch (2015b) for details of the data.7Thompson (2008) is an example.
9
be truncated at zero, but the effect of doing this should be small and is left for future research.
In the final step, all these simulated yield samples are used as input in estimations of the B-CR
model based on the extended Kalman filter where each estimation is started at the true parameters
shown in Table 1. Thus, we are estimating the true model, which should provide the cleanest read
possible on the efficiency of the extended Kalman filter for the estimation of Gaussian shadow-rate
models such as the B-CR model.
5 Results
In this section, I first provide a detailed description of the results from the simulated ten-year monthly
samples. Second, I summarize the results from the simulated ten-year weekly samples before I end
the section with a brief description of the results from the simulated thirty-year samples.
5.1 Analysis of Ten-Year Monthly Samples
Table 2 reports the summary statistics for the estimated parameters from the simulated ten-year
samples at monthly frequency with low and high noise in the data. First, I note that there is notable
upward bias in the absolute values of the estimated mean-reversion parameters. This suggests that
imposing high persistence for the level factor in AFNS models may address the finite-sample bias
problem for that factor, but it does not necessarily mitigate it for other factors in the model. This is
also clear from Figure 2, which shows the distribution of the estimated mean-reversion parameters
across all 1,000 estimations for the data with high noise. It is only the estimates of κP23 that tend to
fall in a fairly narrow range. Furthermore, since the parameters in KP are primarily determined from
the time-series dimension, it is worth emphasizing that these results are not sensitive to the data
quality, and they suggest that a full correction of finite-sample bias should target the joint dynamics
of all state variables as also argued by Bauer et al. (2012).
Second, it is noted that the finite-sample bias in the estimated mean-reversion parameters has a
negative effect on the accuracy of the estimated mean parameters; in particular the estimated mean
of the slope factor, θP2 , is poorly identified. This is partly a result of the fact that the level factor
with its near unit-root property affects the dynamics of the slope factor through κP21. However, even
for the curvature factor, which is not affected by any other factors, estimates of θP3 fall in a wide
range relative to the ranges of mean estimates reported by CLR from simulated ten-year monthly
samples of their Gaussian AFNS model.
Turning to the three volatility parameters in the Σ matrix, I note that they are well determined
with almost identical means and medians, both close to the true values, and the standard deviations
of their estimates are also small. Importantly, though, their accuracy is sensitive to the quality of the
data as a low value of σε decreases the dispersion of their estimated values. This result applies to all
10
Ten-year samples, σε = 1 bpParameter
True Mean Std. dev. 5% 1st quartile Median 3rd quartile 95%
κP21
0.1953 0.5599 1.1269 -1.1297 -0.0860 0.4431 1.1643 2.4673κP22
0.3138 0.8315 0.5910 0.1197 0.4312 0.6984 1.1114 1.9742κP23
-0.4271 -0.5163 0.3567 -1.0879 -0.7187 -0.5116 -0.3017 0.0514κP33
0.4915 0.9424 0.5836 0.1044 0.5481 0.8471 1.2443 2.0223
σ11 0.0069 0.0068 0.0003 0.0062 0.0066 0.0068 0.0070 0.0073σ22 0.0112 0.0112 0.0009 0.0098 0.0106 0.0112 0.0117 0.0125σ33 0.0257 0.0261 0.0021 0.0231 0.0247 0.0260 0.0273 0.0293
θP2
0.0014 -0.1901 4.9064 -0.2041 -0.0494 0.0054 0.0808 0.3603θP3
-0.0252 -0.0142 0.1676 -0.0594 -0.0348 -0.0239 -0.0110 0.0107
λ 0.4700 0.4716 0.0055 0.4644 0.4683 0.4708 0.4743 0.4816
σε 0.0010 0.0001 0.0001 0.0000 0.0001 0.0001 0.0001 0.0001
Ten-year samples, σε = 10 bpsParameter
True Mean Std. dev. 5% 1st quartile Median 3rd quartile 95%
κP21
0.1953 0.4208 1.1653 -1.1977 -0.1504 0.3836 0.9444 1.9515κP22
0.3138 0.8799 1.4997 0.1869 0.4438 0.6961 1.0558 1.9599κP23
-0.4271 -0.5105 0.4641 -1.0408 -0.6858 -0.5060 -0.3315 -0.0486κP33
0.4915 0.9709 0.6104 0.2445 0.5447 0.8447 1.2484 2.1176
σ11 0.0069 0.0067 0.0008 0.0054 0.0062 0.0067 0.0073 0.0081σ22 0.0112 0.0108 0.0014 0.0086 0.0102 0.0110 0.0116 0.0127σ33 0.0257 0.0260 0.0033 0.0209 0.0241 0.0262 0.0281 0.031
θP2
0.0014 -0.0035 0.4111 -0.1692 -0.0560 -0.0001 0.0603 0.2479θP3
-0.0252 -0.0229 0.0417 -0.0559 -0.0373 -0.0248 -0.0133 0.0064
λ 0.4700 0.4685 0.0252 0.4303 0.4549 0.4685 0.4825 0.5078
σε 0.0010 0.0010 0.0001 0.0010 0.0010 0.0010 0.0010 0.0010
Table 2: Summary Statistics of Estimated Parameters from Simulated Ten-Year Monthly
Samples of the B-CR Model.
The table reports the summary statistics of the estimation results from N = 1,000 simulated data sets of the
B-CR model, each with a length of ten years at monthly frequency and a uniform measurement error standard
deviation of σε = 1 basis point and σε = 10 basis points, respectively.
three factors, and it suggests that the values of the volatility parameters are determined to a large
extent from their impact on the cross-sectional fit of yields rather than from the time series properties
of the state variables, which are the same in the simulated data by construction and independent of
the value of σε.
Focusing on the estimates of λ, Table 2 shows that this parameter is well determined in the
estimation without any measurable bias and with a small standard deviation. Since λ only affects
the risk-neutral Q-dynamics, it is exclusively determined from the cross section of yields and therefore
sensitive to the quality of the data.
As for filtering, Table 3 reports the summary statistics for the mean absolute error of the filtering
of the three state variables in the ten-year monthly samples. The results show that the filtering of
11
0 200 400 600 800 1000
−4
−2
02
46
Estimation No.
Par
amet
er e
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ate
(a) κP
21.
0 200 400 600 800 1000
01
23
4
Estimation No.
Par
amet
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(b) κP
22.
0 200 400 600 800 1000
−3
−2
−1
01
2
Estimation No.
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(c) κP
23.
0 200 400 600 800 1000
01
23
45
Estimation No.
Par
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(d) κP
33.
Figure 2: Estimated Mean-Reversion Parameters from Simulated Ten-Year Monthly
Samples of the B-CR Model.
Illustration of the estimated mean-reversion parameters in the KP matrix from N = 1,000 simulated data
sets of the B-CR model, each with a length of ten years sampled monthly and a uniform measurement error
standard deviation of σε = 10 basis points. The true value of each parameter is indicated with a horizontal
solid grey line.
the state variables is in general only marginally less accurate than the results reported by CLR from
simulations and subsequent estimations of a regular AFNS model based on the standard Kalman
filter. However, it is also clear that there are samples in the tail of the distribution that produce
notably higher mean absolute filtered errors, a phenomenon not observed for regular AFNS models.
12
State Mean absolute fitted error, ten-year samples, σε = 1 bpvariable Mean Std. dev. 5 percentile 1st quartile Median 3rd quartile 95 percentileLt 2.31 0.75 1.68 1.81 2.01 2.61 3.75St 4.61 5.83 1.45 1.59 1.88 5.02 16.23Ct 8.92 6.26 4.56 5.01 5.76 10.85 21.70
State Mean absolute fitted error, ten-year samples, σε = 10 bpsvariable Mean Std. dev. 5 percentile 1st quartile Median 3rd quartile 95 percentileLt 13.70 2.07 11.35 12.42 13.24 14.37 17.46St 16.34 9.24 10.73 11.83 12.83 16.47 36.14Ct 42.70 12.97 32.40 35.47 38.38 44.25 68.90
Table 3: Summary Statistics of Mean Absolute Fitted Errors of the Filtered State Vari-
ables from Simulated Ten-Year Monthly Samples of the B-CR Model.
The table reports the summary statistics of the mean absolute fitted error of the three state variables from
N = 1,000 simulated data sets of the B-CR model, each with a length of ten years sampled monthly and a
uniform measurement error standard deviation of σε = 1 basis point and σε = 10 basis points, respectively.
All numbers are measured in basis points.
0 2 4 6 8 10
0.04
0.05
0.06
0.07
0.08
0.09
0.10
Fact
or v
alue
Simulated level factor Estimated level factor
(a) Lt.
0 2 4 6 8 10
−0.
04−
0.02
0.00
0.02
0.04
0.06
0.08
Fact
or v
alue
Simulated slope factor Estimated slope factor
(b) St.
0 2 4 6 8 10
−0.
050.
000.
050.
10
Fact
or v
alue
Simulated curvature factor Estimated curvature factor
(c) Ct.
Figure 3: Estimated Factor Paths with Median Error from Simulated Ten-Year Monthly
Samples of the B-CR Model.
Illustration of the estimated factor paths of the three state variables from the sample that produces the median
mean absolute filtering error for the curvature factor across N = 1,000 simulated data sets of the B-CR model,
each with a length of ten years at monthly frequency and a uniform measurement error standard deviation of
σε = 10 basis points. Also shown are the true simulated factor paths from that sample.
In the following, I analyze the cause of this difference in filtering accuracy.
5.1.1 When Does Filtering Fail in Shadow-Rate Models?
As indicated by the summary statistics in Table 3, the filtering in shadow-rate models based on the
extended Kalman filter is accurate most of the time even when the noise in the data is high. To
provide an example of this, Figure 3 shows the true and filtered state variables from the sample
13
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
Rat
e in
per
cent
Ten−year yield Five−year yield Two−year yield Three−month yield
(a) Simulated yields.
1995 2000 2005 2010 2015
01
23
45
67
89
Rat
e in
per
cent
US ten−year yield US two−year yield US one−year yield Japanese ten−year yield Japanese two−year yield Japanese one−year yield
(b) U.S. and Japanese yields.
Figure 4: Time Series of Simulated and Observed Yields.
Panel (a) shows the simulated monthly zero-coupon bond yields from the sample that produces the maximum
mean absolute filtering error for the curvature factor across N = 1,000 simulated data sets of the B-CR model,
each with a length of ten years at monthly frequency and a uniform measurement error standard deviation of
σε = 1 basis point. The yields shown have maturities: Three-month, two-year, five-year, and ten-year. Panel
(b) shows U.S. Treasury and Japanese government bond yields at daily frequency from January 1995 to July
2015. The yields shown have maturities: one-year, two-year, and ten-year.
that generates the median mean absolute filtering error for the curvature factor (38.4 basis points)
across the 1,000 simulated data sets of the B-CR model, each with a length of ten years at monthly
frequency and a uniform measurement error standard deviation of σε = 10 basis points. It is evident
that the filtering is successful in this case.
To better understand the failure of the extended Kalman filter for the samples in the tail with
much higher mean absolute filtered errors, I focus on the high-quality sample that generates the least
accurate filtering as measured by the mean absolute filtered error for the curvature factor. Despite
ideal conditions, the mean absolute filtered errors for the three factors are 7.5 basis points, 50.1 basis
points, and 54.0 basis points, respectively, for this particular sample. From Table 3 it follows that
these are large filtered errors even relative to the low-quality ten-year monthly samples. Four of the
eight yield series from this sample are shown in Figure 4(a). Clearly, this sample is characterized by
a very prolonged period of extremely low and compressed interest rates, even at longer maturities.8
In terms of practical relevance, this simulated sample of yields do share similarities with the sample
of Japanese government bond yields analyzed in Kim and Singleton (2012), extended by Christensen
8Thanks to the added Gaussian measurement errors, the simulated yields can become negative when they are nearthe zero lower bound.
14
0 2 4 6 8 10
−0.
02−
0.01
0.00
0.01
0.02
0.03
0.04
Fact
or v
alue
Simulated level factor Estimated level factor
(a) Lt.
0 2 4 6 8 10
−0.
10−
0.08
−0.
06−
0.04
−0.
020.
000.
02
Fact
or v
alue
Simulated slope factor Estimated slope factor
(b) St.
0 2 4 6 8 10
−0.
10−
0.06
−0.
020.
020.
04
Fact
or v
alue
Simulated curvature factor Estimated curvature factor
(c) Ct.
Figure 5: Estimated Factor Paths with Maximum Error from Simulated Ten-Year
Monthly Samples of the B-CR Model.
Illustration of the estimated factor paths of the three state variables from the sample that produces the maxi-
mum mean absolute filtering error for the curvature factor across N = 1,000 simulated data sets of the B-CR
model, each with a length of ten years at monthly frequency and a uniform measurement error standard
deviation of σε = 1 basis point. Also shown are the true simulated factor paths from that sample.
and Rudebusch (2015a), and shown in Figure 4(b). Since both of these studies estimate Gaussian
shadow-rate models based on the extended Kalman filter, the accuracy of their estimated state
variables could be questioned based on these findings.9 On the other hand, the U.S. Treasury yields
also shown in Figure 4(b) so far have not reached such depressingly low levels, which suggests that
those data are much less likely to pose a problem for the extended Kalman filter.
Figure 5 shows the individual true and filtered state variables for this sample. It is noteworthy
that the filtering of the level factor tends to stay on track even under such extreme circumstances.
Thus, this factor is reliably estimated as long as long-term yields are not compressed against the
zero lower bound. On the other hand, the filtering of the slope and curvature factors can be thrown
off for extended periods when short- and medium-term yields are compressed against the zero lower
bound.
Figure 6 shows the true and estimated shadow-rate paths from this particular sample. When
yields are particularly low during the years 2 to 6, the estimated value of the shadow rate can be off
by as much as 200 basis points in either direction. This lack of econometric identification is caused by
the fact that the shadow rate is deep in negative territory during this period. More importantly, these
findings reinforce the concerns raised by Christensen and Rudebusch (2015a) about the accuracy of
estimated shadow rates and their use as a measure of the stance of monetary policy when bond yields
are constrained by the zero lower bound. In particular, it is ironic that it is exactly at the times when
9To reach a more final conclusion regarding this question would require repeating the simulation exercise usingparameters estimated from their Japanese yield sample. This is left for future research.
15
0 2 4 6 8 10
−10
−8
−6
−4
−2
02
Rat
e in
per
cent
Simulated shadow rate Estimated shadow rate
Figure 6: Estimated Shadow Rate.
Illustration of the estimated shadow-rate path from the sample that produces the maximum mean absolute
filtering error for the curvature factor across N = 1,000 simulated data sets of the B-CR model, each with a
length of ten years at monthly frequency and a uniform measurement error standard deviation of σε = 1 basis
point. Also shown are the true simulated shadow-rate path from that sample.
such an analysis would be most useful that it is most unreliable, namely when yields are severely
constrained by the zero lower bound.
Furthermore, these results leave an interesting empirical question whether other nonlinear filter-
ing and estimation methods would be able to overcome the problems encountered with the extended
Kalman filter when the nonlinearities in the data are particularly severe. For Japanese data, Chris-
tensen and Rudebusch (2015a) report almost identical results using the unscented Kalman filter
relative to the standard extended Kalman filter, which in itself suggests that the unscented Kalman
filter likely would not suffice. However, the robustness of their result is unknown and is a topic left
for future research.
5.2 Analysis of Ten-Year Weekly Samples
Given that I, like Christensen and Rudebusch (2015b), estimate the B-CR model using weekly U.S.
Treasury data, the empirically relevant case is really with data at weekly frequency, which is the
focus of this section.
Table 4 reports the summary statistics of the estimated parameters based on the 1,000 simulated
16
Ten-year samples, σε = 1 bpParameter
True Mean Std. dev. 5% 1st quartile Median 3rd quartile 95%
κP21
0.1953 0.4007 0.9299 -1.0517 -0.0735 0.3606 0.9142 1.8652κP22
0.3138 0.7932 0.5185 0.1927 0.4480 0.6833 1.0223 1.7764κP23
-0.4271 -0.5098 0.2833 -0.9753 -0.6784 -0.5089 -0.3231 -0.0753κP33
0.4915 0.9347 0.5377 0.2590 0.5538 0.8382 1.2057 2.0475
σ11 0.0069 0.0068 0.0002 0.0066 0.0067 0.0069 0.0070 0.0071σ22 0.0112 0.0112 0.0004 0.0106 0.0110 0.0112 0.0114 0.0118σ33 0.0257 0.0259 0.0009 0.0246 0.0253 0.0258 0.0264 0.0272
θP2
0.0014 0.0247 0.2808 -0.1630 -0.0536 -0.0014 0.0565 0.2479θP3
-0.0252 -0.0253 0.0328 -0.0552 -0.0371 -0.0253 -0.0135 0.0023
λ 0.4700 0.4704 0.0021 0.4674 0.4691 0.4703 0.4715 0.4739
σε 0.0001 0.0001 0.0001 0.0000 0.0001 0.0001 0.0001 0.0001
Ten-year samples, σε = 10 bpsParameter
True Mean Std. dev. 5% 1st quartile Median 3rd quartile 95%
κP21
0.1953 0.4373 2.4085 -1.0587 -0.1585 0.3213 0.8306 1.8221κP22
0.3138 0.9321 3.3836 0.2038 0.4436 0.6623 1.0372 1.8160κP23
-0.4271 -0.5116 0.3328 -0.9899 -0.6820 -0.5073 -0.3360 -0.0805κP33
0.4915 0.9560 0.5687 0.2718 0.5612 0.8306 1.2316 2.0675
σ11 0.0069 0.0068 0.0005 0.0060 0.0065 0.0068 0.0072 0.0076σ22 0.0112 0.0110 0.0010 0.0096 0.0106 0.0111 0.0116 0.0122σ33 0.0257 0.0258 0.0019 0.0229 0.0246 0.0259 0.0271 0.0286
θP2
0.0014 -0.0009 0.1968 -0.1602 -0.0553 -0.0024 0.0502 0.1936θP3
-0.0252 -0.0244 0.0449 -0.0554 -0.0373 -0.0253 -0.0131 0.0037
λ 0.4700 0.4704 0.0122 0.4508 0.4633 0.4702 0.4774 0.4898
σε 0.0010 0.0010 0.0001 0.0010 0.0010 0.0010 0.0010 0.0010
Table 4: Summary Statistics of Estimated Parameters from Simulated Ten-Year Weekly
Samples of the B-CR Model.
The table reports the summary statistics of the estimation results from N = 1,000 simulated data sets of the
B-CR model, each with a length of ten years at weekly frequency and a uniform measurement error standard
deviation of σε = 1 basis point and σε = 10 basis points, respectively.
ten-year weekly samples. First, the upward finite-sample bias for κP21 is reduced fairly notably when
the sampling frequency is increased. For the other mean-reversion parameters the improvement is
more modest. However, the standard deviations of their estimates are lowered by a meaningful
amount. Second, the sizable variation in the mean parameters drops significantly when the sampling
frequency is increased, but remains unsatisfactorily large, in particular for estimates of θP2 . Third,
for the parameters determined primarily from the cross section of yields (Σ and λ), the already high
accuracy is further improved. As a result, the standard deviations of their parameter estimates are
very low when the sampling frequency is weekly. These findings are consistent with CLR, who report
similar results when they increase sampling frequency.
Turning to the filtering of the state variables, Table 9 reports the summary statistics for the mean
17
State Mean absolute fitted error, ten-year samples, σε = 1 bpvariable Mean Std. dev. 5 percentile 1st quartile Median 3rd quartile 95 percentileLt 1.99 0.44 1.67 1.75 1.83 2.04 2.83St 2.87 3.33 1.47 1.54 1.64 2.57 9.13Ct 6.68 3.72 4.60 4.78 5.10 6.91 14.43
State Mean absolute fitted error, ten-year samples, σε = 10 bpsvariable Mean Std. dev. 5 percentile 1st quartile Median 3rd quartile 95 percentileLt 10.25 1.33 8.96 9.45 9.92 10.64 12.73St 12.67 7.06 8.97 9.45 10.03 12.12 26.98Ct 32.72 9.24 26.12 27.81 29.33 33.59 52.54
Table 5: Summary Statistics of Mean Absolute Fitted Errors of the Filtered State Vari-
ables from Simulated Ten-Year Weekly Samples of the B-CR Model.
The table reports the summary statistics of the mean absolute fitted error of the three state variables from N =
1,000 simulated data sets of the preferred AFNS0 model, each with a length of ten years sampled weekly and
a uniform measurement error standard deviation of σε = 1 basis point and σε = 10 basis points, respectively.
All numbers are measured in basis points.
absolute errors of the filtered state variables from the individual samples. By comparing the results
to those from the monthly samples shown in Table 3, it is clear that the filtering of the state variables
improves when the sampling is increased from monthly to weekly frequency, and the improvement is
larger the more noisy the data is.
As for the samples that cause the extended Kalman filter problems, they are practically the same
independent of the sampling frequency. To illustrate, the sample that generated the maximum mean
absolute filtering error for the curvature factor in the ten-year monthly samples now generates the
second largest mean absolute filtering error for the curvature factor.10 The estimated shadow rate
from this weekly sample is shown in Figure 7 and exhibits the same material deviation for the years
from 2 to 6 as noted in Figure 6 at monthly sampling frequency. Thus, to move to higher frequency
data is not a remedy for the filtering problem in the extended Kalman filter under extreme conditions,
at most it can mitigate it modestly.
5.3 Analysis of Thirty-Year Monthly Samples
In this section, I analyze the results obtained for the thirty-year monthly samples simulated from the
B-CR model.
For a start, Table 6 contains the summary statistics for the 1,000 estimated parameter sets
we obtain from these thirty-year monthly samples. For the mean-reversion parameters, there is a
significant reduction in their finite-sample bias that is also noted in Figure 8, which shows their
10The sample that generates the largest mean absolute filtered error for the curvature factor in the weekly datagenerates the fifth largest mean absolute filtered error for the curvature factor in the monthly data.
18
0 2 4 6 8 10
−10
−8
−6
−4
−2
02
Rat
e in
per
cent
Simulated shadow rate Estimated shadow rate
Figure 7: Estimated Shadow Rate.
Illustration of the estimated shadow-rate path from the sample that produces the second highest mean absolute
filtering error for the curvature factor across N = 1,000 simulated data sets of the B-CR model, each with a
length of ten years at weekly frequency and a uniform measurement error standard deviation of σε = 1 basis
point. Also shown are the true simulated shadow-rate path from that sample.
individual distributions across the 1,000 samples. Furthermore, they exhibit the unusual pattern
that their bias and dispersion decline when the quality of the data declines. For the parameters
determined primarily from the cross section, i.e., λ, σ11, σ22, and σ33, we see a reduction of close to
50% in their dispersion when we triple the length of the sample. For the mean parameters, θP2 and
θP3 , we see an even sharper reduction in the dispersion. This is tied to the significant reduction in the
bias of the mean-reversion parameters. However, it is still the case that the near unit-root property
of the level factor is able to cause significant deviations in the estimates of κP21 and θP2 even in these
thirty-year samples.
Finally, Table 7 contains the summary statistics for the mean absolute errors of the filtered
state variables from the individual samples. From the table we infer that the sample length does
not matter for the accuracy of the filtering of the state variables in shadow-rate models estimated
with the extended Kalman filter, and this conclusion is robust to the quality of the data used.
Furthermore, the magnitude of the filtering problems for the extended Kalman filter in the upper
tail of the distribution is slightly exacerbated when the sample length is increased. Thus, neither
higher frequency nor longer samples will solve the filtering problem in the extended Kalman filter
19
Thirty-year samples, σε = 1 bpParameter
True Mean Std. dev. 5% 1st quartile Median 3rd quartile 95%
κP21
0.1953 0.4190 0.4240 -0.1302 0.1681 0.3510 0.6056 1.1744κP22
0.3138 0.5213 0.2478 0.1809 0.3581 0.4794 0.6434 0.9777κP23
-0.4271 -0.4935 0.2030 -0.8411 -0.6053 -0.4788 -0.3661 -0.1884κP33
0.4915 0.6484 0.2764 0.2508 0.4643 0.6211 0.8053 1.1442
σ11 0.0069 0.0068 0.0002 0.0064 0.0066 0.0068 0.0069 0.0071σ22 0.0112 0.0111 0.0006 0.0102 0.0107 0.0111 0.0115 0.0123σ33 0.0257 0.0258 0.0011 0.0240 0.0250 0.0257 0.0264 0.0275
θP2
0.0014 0.0184 0.4824 -0.0663 -0.0114 0.0157 0.0516 0.1763θP3
-0.0252 -0.0232 0.0641 -0.0409 -0.0287 -0.0216 -0.0150 -0.0033
λ 0.4700 0.4720 0.0039 0.4674 0.4695 0.4714 0.4737 0.4785
σε 0.0010 0.0001 0.0001 0.0000 0.0001 0.0001 0.0001 0.0001
Thirty-year samples, σε = 10 bpsParameter
True Mean Std. dev. 5% 1st quartile Median 3rd quartile 95%
κP21
0.1953 0.2772 0.2695 -0.1314 0.1139 0.2573 0.4154 0.7501κP22
0.3138 0.4578 0.1799 0.2328 0.3384 0.4251 0.5486 0.7821κP23
-0.4271 -0.4723 0.1357 -0.6968 -0.5525 -0.4668 -0.3875 -0.2716κP33
0.4915 0.6355 0.2586 0.3078 0.4531 0.5879 0.7667 1.115
σ11 0.0069 0.0068 0.0005 0.0061 0.0065 0.0068 0.0071 0.0076σ22 0.0112 0.0111 0.0007 0.0100 0.0107 0.0111 0.0115 0.0120σ33 0.0257 0.0258 0.0017 0.0231 0.0248 0.0258 0.0269 0.0285
θP2
0.0014 0.0043 0.0492 -0.0692 -0.0197 0.0029 0.0256 0.0796θP3
-0.0252 -0.0250 0.0100 -0.0420 -0.0312 -0.0249 -0.0181 -0.0092
λ 0.4700 0.4716 0.0139 0.4490 0.4630 0.4717 0.4800 0.4928
σε 0.0010 0.0010 0.0001 0.0010 0.0010 0.0010 0.0010 0.0010
Table 6: Summary Statistics of Estimated Parameters from Simulated Thirty-Year
Monthly Samples of the B-CR Model.
The table reports the summary statistics of the estimation results from N = 1,000 simulated data sets of
the B-CR model, each with a length of thirty years at monthly frequency and a uniform measurement error
standard deviation of σε = 1 basis point and σε = 10 basis points, respectively.
under extreme conditions.
5.4 Analysis of Thirty-Year Weekly Samples
In this section, I analyze the results obtained for the thirty-year weekly samples simulated from the
B-CR model. This simulation exercise maps directly to the U.S. Treasury yield data used in this
paper and analyzed in Christensen and Rudebusch (2015b) and therefore merits special attention.
For this final exercise, Table 8 contains the summary statistics for the 1,000 estimated parameter
sets. First, I note that the finite-sample bias for the mean-reversion parameters is smaller than in
the thirty-year monthly samples and relatively modest in absolute size. Second, it is the case that
the volatility (Σ), mean (θP ), and λ parameters are all estimated without bias in these thirty-year
20
0 200 400 600 800 1000
−1.
0−
0.5
0.0
0.5
1.0
1.5
2.0
Estimation No.
Par
amet
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stm
ate
(a) κP
21.
0 200 400 600 800 1000
0.0
0.5
1.0
1.5
2.0
Estimation No.
Par
amet
er e
stm
ate
(b) κP
22.
0 200 400 600 800 1000
−2.
0−
1.5
−1.
0−
0.5
0.0
Estimation No.
Par
amet
er e
stm
ate
(c) κP
23.
0 200 400 600 800 1000
0.0
0.5
1.0
1.5
2.0
Estimation No.
Par
amet
er e
stm
ate
(d) κP
33.
Figure 8: Estimated Mean-Reversion Parameters from Simulated Thirty-Year Monthly
Samples of the B-CR Model.
Illustration of the estimated mean-reversion parameters in the KP matrix from N = 1,000 simulated data
sets of the B-CR model, each with a length of thirty years sampled monthly and a uniform measurement error
standard deviation of σε = 10 basis points. The true value of each parameter is indicated with a horizontal
solid grey line.
weekly samples independent of the quality of the data. This is very encouraging for the corresponding
estimation results reported in Christensen and Rudebusch (2015b) as they are likely to be free of
bias. Moreover, since these findings are qualitatively on par with the results CLR report based on
forty-year weekly simulations of their Gaussian AFNS model estimated with the standard Kalman
21
State Mean absolute fitted error, thirty-year samples, σε = 1 bpvariable Mean Std. dev. 5 percentile 1st quartile Median 3rd quartile 95 percentileLt 2.48 1.19 1.75 1.89 2.16 2.62 4.12St 5.28 6.59 1.52 1.73 2.99 5.86 16.93Ct 9.76 7.31 4.80 5.38 7.48 11.09 22.38
State Mean absolute fitted error, thirty-year samples, σε = 10 bpsvariable Mean Std. dev. 5 percentile 1st quartile Median 3rd quartile 95 percentileLt 14.05 2.70 12.01 12.67 13.35 14.42 18.31St 17.17 9.35 11.29 12.19 13.61 18.35 36.34Ct 43.61 12.17 34.16 36.38 39.61 46.09 66.66
Table 7: Summary Statistics of Mean Absolute Fitted Errors of the Filtered State Vari-
ables from Simulated Thirty-Year Monthly Samples of the B-CR Model.
The table reports the summary statistics of the mean absolute fitted error of the three state variables from
N = 1,000 simulated data sets of the preferred AFNS0 model, each with a length of thirty years sampled
monthly and a uniform measurement error standard deviation of σε = 1 basis point and σε = 10 basis points,
respectively. All numbers are measured in basis points.
filter, I conclude that for the sample length and frequency analyzed here the extended Kalman filter
delivers fully satisfactory results that appear to be able to match those a fully efficient estimator
would produce.
To end the section, Table 9 contains the summary statistics for the mean absolute errors of the
filtered state variables from the individual samples. We see the same pattern as observed for the
ten-year samples, namely that increasing the sampling frequency does improve filtering accuracy and
by about the same magnitude as we observed in the switch between ten-year monthly and weekly
samples. Hence, it continues to be the case that sample length matters little for filtering accuracy
in shadow-rate models estimated with the extended Kalman filter. Also, the filtering errors in the
problematic samples in the upper tail of the distribution are of a similar magnitude as in the ten-year
samples.
6 Conclusion
In this paper, I perform a series of carefully orchestrated simulation exercises to study the efficiency
of the extended Kalman filter in estimating the established Gaussian shadow-rate model analyzed in
Christensen and Rudebusch (2015b).
Based on the results I make the following notable observations. First, parameters determined
primarily from the cross section (Σ,λ) are estimated without bias based on the extended Kalman
filter. Second, the finite-sample bias for parameters determined primarily from the times-series
dimension (KP and θP ) appears to be no more severe than in Gaussian models where the standard
Kalman filter is an efficient and consistent estimator. Third, filtering of the state variables is only
22
Thirty-year samples, σε = 1 bpParameter
True Mean Std. dev. 5% 1st quartile Median 3rd quartile 95%
κP21
0.1953 0.2882 0.2582 -0.0819 0.1284 0.2682 0.4121 0.7263κP22
0.3138 0.4521 0.1667 0.2369 0.3340 0.4223 0.5482 0.7696κP23
-0.4271 -0.4701 0.1223 -0.6777 -0.5466 -0.4637 -0.391 -0.2702κP33
0.4915 0.6365 0.2359 0.3251 0.4680 0.5962 0.7657 1.0985
σ11 0.0069 0.0069 0.0001 0.0067 0.0068 0.0069 0.0069 0.0070σ22 0.0112 0.0112 0.0002 0.0109 0.0111 0.0112 0.0114 0.0116σ33 0.0257 0.0258 0.0005 0.0251 0.0255 0.0258 0.0261 0.0265
θP2
0.0014 0.0076 0.0508 -0.0588 -0.0172 0.0053 0.0304 0.0783θP3
-0.0252 -0.0244 0.0096 -0.0406 -0.0305 -0.0245 -0.0178 -0.0092
λ 0.4700 0.4703 0.0012 0.4686 0.4696 0.4703 0.4710 0.4721
σε 0.0001 0.0001 0.0001 0.0000 0.0001 0.0001 0.0001 0.0001
Thirty-year samples, σε = 10 bpsParameter
True Mean Std. dev. 5% 1st quartile Median 3rd quartile 95%
κP21
0.1953 0.2527 0.2389 -0.0938 0.1127 0.2354 0.3823 0.6619κP22
0.3138 0.4391 0.1613 0.2379 0.3268 0.4077 0.5250 0.7353κP23
-0.4271 -0.4702 0.1191 -0.6737 -0.5505 -0.4658 -0.3909 -0.2761κP33
0.4915 0.6356 0.2448 0.3144 0.4589 0.5893 0.7591 1.1083
σ11 0.0069 0.0068 0.0003 0.0064 0.0066 0.0068 0.0070 0.0073σ22 0.0112 0.0112 0.0004 0.0105 0.0110 0.0112 0.0114 0.0118σ33 0.0257 0.0258 0.0010 0.0241 0.0252 0.0258 0.0264 0.0274
θP2
0.0014 0.0019 0.0452 -0.0625 -0.0224 0.0001 0.0237 0.0709θP3
-0.0252 -0.0249 0.0098 -0.0421 -0.0310 -0.0247 -0.018 -0.0094
λ 0.4700 0.4706 0.0068 0.4597 0.4664 0.4705 0.4750 0.4809
σε 0.0010 0.0010 0.0000 0.0010 0.0010 0.0010 0.0010 0.0010
Table 8: Summary Statistics of Estimated Parameters from Simulated Thirty-Year
Weekly Samples of the B-CR Model.
The table reports the summary statistics of the estimation results from N = 1,000 simulated data sets of
the B-CR model, each with a length of thirty years at weekly frequency and a uniform measurement error
standard deviation of σε = 1 basis point and σε = 10 basis points, respectively.
unsatisfactory when yields are unusually low and very compressed against the zero lower bound, a
phenomenon only really observed in Japanese yield data.
Based on these observations I recommend using the extended Kalman filter to estimate U.S.
shadow-rate models, but question its use for estimation of Japanese shadow-rate models where inter-
est rates have experienced the kind of severe prolonged compression near the zero lower bound that
I find can cause the extended Kalman filter significant difficulty.
A key remaining question is whether other nonlinear filtering techniques would be able to over-
come the problems encountered with the extended Kalman filter in the exercises in this paper. And if
so, what the tradeoffs are in terms of other aspects of the estimation performance and computational
challenges. These are all questions left for future research.
23
State Mean absolute fitted error, thirty-year samples, σε = 1 bpvariable Mean Std. dev. 5 percentile 1st quartile Median 3rd quartile 95 percentileLt 2.13 0.92 1.72 1.77 1.88 2.12 3.17St 3.41 4.44 1.50 1.57 1.92 3.17 10.86Ct 7.35 5.23 4.73 4.94 5.65 7.53 15.04
State Mean absolute fitted error, thirty-year samples, σε = 10 bpsvariable Mean Std. dev. 5 percentile 1st quartile Median 3rd quartile 95 percentileLt 10.58 1.95 9.29 9.67 10.04 10.78 13.72St 13.37 7.22 9.25 9.65 10.58 13.79 26.95Ct 33.93 9.94 27.28 28.51 30.23 35.35 51.92
Table 9: Summary Statistics of Mean Absolute Fitted Errors of the Filtered State Vari-
ables from Simulated Thirty-Year Weekly Samples of the B-CR Model.
The table reports the summary statistics of the mean absolute fitted error of the three state variables from
the N = 1,000 simulated data sets of the preferred AFNS0 model, each with a length of thirty years sampled
weekly and a uniform measurement error standard deviation of σε = 1 basis point and σε = 10 basis points,
respectively. All numbers are measured in basis points.
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