Exact Distribution of the Mean Reversion Estimator in the Ornstein-Uhlenbeck Process * Yong Bao † Department of Economics Purdue University Aman Ullah ‡ Department of Economics University of California, Riverside Yun Wang § School of International Trade and Economics University of International Business and Economics August 31, 2014 Abstract: Econometricians have recently been interested in estimating and testing the mean reversion param- eter (κ) in linear diffusion models. It has been documented that the maximum likelihood estimator (MLE) of κ tends to over estimate the true value. Its asymptotic distribution, on the other hand, depends on how the data are sampled (under expanding, infill, or mixed domain) as well as how we spell out the initial condition. This poses a tremendous challenge to practitioners in terms of estimation and inference. In this paper, we provide new and significant results regarding the exact distribution of the MLE of κ in the Ornstein-Uhlenbeck process under different scenarios: known or unknown drift term, fixed or random start-up value, and zero or positive κ. In particular, we employ numerical integration via analytical evaluation of a joint characteristic function. Our numerical calculations demonstrate the remarkably reliable performance of our exact approach. It is found that the true distribution of the MLE can be severely skewed in finite samples and that the asymptotic distributions in general may provide misleading results. Our exact approach indicates clearly the non-mean-reverting behav- ior of the real federal fund rate. JEL Classification: C22, C46, C58 Key Words: Distribution, Mean Reversion Estimator, Ornstein-Uhlenbeck Process * We are thankful to Yacine A¨ ıt-Sahalia, Peter Phillips, and Jun Yu, for helpful comments. We also benefited from discussions with Victoria Zinde-Walsh on the subject matter. † Corresponding Author: Department of Economics, Purdue University, 403 W. State Street, West Lafayette, IN 47907, USA. E-mail: [email protected]. ‡ Department of Economics, University of California, Riverside, CA 92521, USA. E-mail: [email protected]. § School of International Trade and Economics, University of International Business and Economics, Beijing, China. E-mail: [email protected].
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Exact Distribution of the Mean Reversion Estimator in
the Ornstein-Uhlenbeck Process∗
Yong Bao†
Department of Economics
Purdue University
Aman Ullah‡
Department of Economics
University of California, Riverside
Yun Wang§
School of International Trade and Economics
University of International Business and Economics
August 31, 2014
Abstract: Econometricians have recently been interested in estimating and testing the mean reversion param-eter (κ) in linear diffusion models. It has been documented that the maximum likelihood estimator (MLE) of κtends to over estimate the true value. Its asymptotic distribution, on the other hand, depends on how the dataare sampled (under expanding, infill, or mixed domain) as well as how we spell out the initial condition. Thisposes a tremendous challenge to practitioners in terms of estimation and inference. In this paper, we providenew and significant results regarding the exact distribution of the MLE of κ in the Ornstein-Uhlenbeck processunder different scenarios: known or unknown drift term, fixed or random start-up value, and zero or positive κ.In particular, we employ numerical integration via analytical evaluation of a joint characteristic function. Ournumerical calculations demonstrate the remarkably reliable performance of our exact approach. It is found thatthe true distribution of the MLE can be severely skewed in finite samples and that the asymptotic distributionsin general may provide misleading results. Our exact approach indicates clearly the non-mean-reverting behav-ior of the real federal fund rate.
JEL Classification: C22, C46, C58
Key Words: Distribution, Mean Reversion Estimator, Ornstein-Uhlenbeck Process
∗We are thankful to Yacine Aıt-Sahalia, Peter Phillips, and Jun Yu, for helpful comments. We also benefited from discussionswith Victoria Zinde-Walsh on the subject matter.†Corresponding Author: Department of Economics, Purdue University, 403 W. State Street, West Lafayette, IN 47907, USA.
E-mail: [email protected].‡Department of Economics, University of California, Riverside, CA 92521, USA. E-mail: [email protected].§School of International Trade and Economics, University of International Business and Economics, Beijing, China. E-mail:
denotes the PDF of φ− φ and can be calculated from F ′φ
(y) = π−1∫∞0
Im (∂ϕ(u, v)/∂v|v=−uy) du. As pointed
out by Hillier (2001), the density function of a ratio of normal quadratic forms (for instance, in the case when
κ > 0 and x0 is random), can be nonanayltic at some points. Thus in this paper we focus on the CDF only.
1Since we are interested in studying the finite sample properties of κ, the initial condition x0 matters and we include it in theestimation procedure. This stands in contrast to the convention of Hurwicz (1950). For the case of known µ, if µ 6= 0, one cansimply define yi = xi − µ and work with yi.
2Note that (2.6) holds regardless of the distribution assumption.3If we discard x0 in formulating φ, then the Imhof (1961) technique is still applicable, as we can define φ in terms of quadratic
forms in the random vector (x1, x2, · · · , xn)′.
5
2.1 Joint Characteristic Function
For us to be able to use (2.7) to derive (2.6) for κ − κ, an essential task is to evaluate the joint characteristic
function of the numerator and denominator in defining φ − φ. To facilitate presentation, we first introduce
some notation. Let 0n be an n × 1 vector of zeros, In be the identity matrix of size n, ιn be an n × 1 vector
of ones, Mn = In − n−1ιnι′n, ei,n be a unit/elementary vector in the n-dimensional Euclidean space with its
i-th element being 1. Given an (n− 1)× (n− 1) matrix Cn−1, we define ACn as an n× n matrix with its lower
left block being Cn−1 and all other elements being zero, and BCn as an n× n matrix with its upper left block
being Cn−1 and all other elements being zero. When Cn−1 = In−1, we simply put An = AIn and Bn = BI
n.
For an n× n matrix Cn, we use cn,ij to denote its ij-th element, and c(ij)n to denote the ij-th element of C−1n ,
whenever it exists. Throughout, V n is an n× n matrix with its ij-th element φ|i−j|.
When κ > 0, define zi = (xi−µ)/σε, i = 0, · · · , n, and z = n−1∑ni=1 zi−1.Obviously, φ =
∑ni=1 zi−1zi/
∑ni=1 z
2i−1
for the case of known µ, and φ =∑ni=1(zi−1 − z)zi/
∑ni=1(zi−1 − z)2 for the case of unknown µ.
In the case of κ = 0, the parameter µ vanishes, and we define zi = xi/σε and ¯z = n−1∑ni=1 zi−1. In practice,
we may still proceed to estimate φ from the discrete AR(1) model without or with an intercept even when the
parameter µ is not defined. Correspondingly, φ =∑ni=1 zi−1zi/
Consequently, when κ > 0, we can write the numerator and denominator (Y1 and Y2) in formulating
φ − φ in terms of quadratic forms in either zn (when x0 is fixed) or ζn+1 (when x0 is random). Define
the following matrices/vectors (note that these matrices and vectors depend on u, v, and model parameters; we
have suppressed their arguments):
Rn = In +(φ2 + 2iuφ− 2iv
)Bn − (φ+ iu) (An +A′n),
δn = iu(In − 2φA′n)Mne1,n + 2ivA′nMne1,n + φe1,n,
Sn = In + φ2Bn − φ(An +A′n)− iu(MnAn +A′nMn) + 2i(uφ− v)A′nMnAn,
T n+1 = (1− φ2)V −1n+1 − iu(AMn+1 +AM ′
n+1) + 2i(uφ− v)BMn+1.
6
Then we can present the characteristic function ϕ(u, v) under the different scenarios as follows (see the appendix
for detailed derivation; when κ = 0, Rn, δn, and Sn are defined with φ = 1 in their expressions).
Characteristic Function of φ− φ
κ µ (intercept in AR(1) or not) x0 ϕ (u, v)
> 0 Known (no intercept) Fixed exp[z202
(1− |Rn+1|
|Rn|
)]|Rn|−1/2
Random√
1− φ2(|Rn+1| − φ2 |Rn|
)−1/2Unknown (with intercept) Fixed exp
[−φ
2z202 − i(uφ− v)z20
(1− 1
n
)+ 1
2z20δ′nS−1n δn
]|Sn|−1/2
Random√
1− φ2 |T n+1|−1/2
= 0 No intercept Fixed exp[z202
(1− |Rn+1|
|Rn|
)]|Rn|−1/2
With intercept exp[−φ
2z202 − i(u− v)z20
(1− 1
n
)+ 1
2 z20δ′nS−1n δn
]|Sn|−1/2
With ϕ (u, v) in hand, we can directly use (2.7) to derive the distribution function of φ, which in turn
can be plugged into (2.6) to obtain the distribution function of κ. Clearly, in this procedure we need to find
determinants (|Rn| , |Rn+1| , |Sn| , and |T n+1|) and inverses (S−1n and T−1n+1). When n is small, this can be
done easily numerically. But for a moderately large n, for example, T = 10, h = 1/252, n = 2520 (daily data
over 10 years), this can be quite time consuming. Instead of numerically evaluating these determinants and
inverses, we proceed to derive direct analytical expressions, presented in Appendix B, by utilizing the special
structures of these matrices.
The essential merit of our approach is computational efficiency based on analytical, instead of numerical,
determinants and inverses. We make no attempt to control computational accuracy when applying numerical
integration in (2.7); instead we rely on Matlab’s adaptive Gauss-Kronrod quadrature integral procedure quadgk
by setting numerical absolute error tolerance (1e− 12). The eigenvalue-based approach via Imhof (1961), when
applicable, might be able to manually control the integration error and truncation error, see Davies (1973, 1980),
Ansley et al. (1992), and Lu and King (2002). Our numerical exercise in the next subsection demonstrates that
our numeral integration approach in fact produces very accurate results. Also, as emphasized before, the Imhof
(1961) approach is not always applicable.
7
2.2 Numerical Results
In this section, we conduct Monte Carlo simulations to illustrate the finite sample performance of our exact
distribution in comparison with the “true” distribution and the asymptotic distribution. The data generating
process follows the OU model in (2.4), and the error term is generated from a normal distribution. The
asymptotic distribution results are available from Zhou and Yu (2010).4 Note that under the infill asymptotics,
the results are conditional on the initial x0.
We set T = 1, 2, 5, 10, h = 1/12, 1/52, 1/252, κ = 0.01, 0.1, 1, µ = 0, 0.1, σ = 0.1, x0 = µ or x0 ∼
N(µ, σ2/(2κ)). Compared with Zhou and Yu (2010), we have a more comprehensive experiment design, so as
to have a better understanding of the finite-sample distributions. For the fixed start-up case (x0 = 0), we also
consider κ = 0. As pointed out in Zhou and Yu (2010), the values of 0.01 and 0.1 for κ are empirically realistic
for interest rate data while the value of 1 is empirically realistic for volatility.
Tables 1–4 report the cumulative distributions of T (κ− κ), where the “true” distribution results come from
1,000,000 replications, and we make comparison of the exact (p), true (pedf ), and asymptotic results under the
three asymptotics (pexp, pmix, pinf ).5 In calculating the exact distribution with the analytical characteristic
function, we still need to implement numerical integration in (2.7). Appendix C discusses how to overcome
the problem of discontinuity of the square root function in the complex domain. The complete experiment
results are available from the corresponding author upon request. To save space, we report only the results for
h = 1/12, 1/252 in Tables 1–4 (each with two panels corresponding to T = 1, 10, respectively). Tables 1 and
2 report the cumulative distributions of T (κ− κ) under a fixed start-up when κ = 0.01, with x0 = µ = 0, 0.1,
respectively, and Tables 3 and 4 report the results when x0 is random.
Several striking features are present in these tables. First, the exact distribution results match to at least
the third decimal place with those obtained by 1 million simulations, in all the cases considered. This indicates
high accuracy of the exact results calculated by our numerical integration algorithm. In consistent with the
asymptotic results in Zhou and Yu (2010), there is no much difference between the results under the expanding
and mixed domains, and the infill asymptotics provide relatively better performance. Yet, the asymptotic
distribution under the infill domain may still provide poor approximation to the true distribution when the
data span is short, especially so in the left tails. While increasing data frequency does not affect much the
4Zhou and Yu (2010) did not give the expanding and infill asymptotic distribution results when κ = 0 and µ 6= 0. Thiscorresponds to the scenario, in a discrete framework, when no intercept is present in the true model, but a constant term isincluded in the regression. The expanding and infill asymptotic distribution results easily follow via the generalized delta method.
5In simulating the asymptotic (non-normal) results, 10,000 replications are used and a sample size of 5,000 is used to approximatethe integrals involving the Brownian motion by the discrete Riemann sums, with the exception that the infill asymptotic resultswhen x0 is random are calculated as averaging over 2,000 replications, where in each replication, x0 ∼ N(µ, σ2/(2κ)) and thediscrete AR(1) process is of sample size of 2,000.
8
asymptotic distributions, it does affect the true distribution, and the remarkable performance of the exact
distribution is robust to data frequency, as well as to data span and other aspects of model specification.
Second, the true distribution of κ is highly skewed to the right. Normality is a terrible approximation of the
finite-sample distribution of κ. Moreover, we can infer from these tables the exact/true median of T (κ− κ) in
all cases are substantially positive. (A direct calculation of the median is also possible, see the last paragraph
in this section to follow.) This suggests that κ can significantly over estimate κ in finite samples. This degree
of overestimation does not decrease with a higher data frequency (given a fixed data span). This is in line with
the observations made by Phillips and Yu (2005) and Tang and Chen (2009). On the other hand, increasing
data span might help alleviate this problem, though somewhat marginally.
Third, how the initial observation is spelled out affects significantly the exact distribution of κ. For example,
for the fixed start-up case, the exact distribution is less skewed to the right when x0 = 0 compared with when
x0 6= 0. Comparing the random start-up case versus the fixed start-up case with a known drift term (Table
3 versus Table 1), we see quite a difference in the exact distributions across the two cases and the former one
is less skewed; on the other hand, with a unknown drift term (Table 4 versus Table 2), there is virtually no
significant difference in the exact distributions across the two cases.6 This feature is related to the role of initial
observation in the unit-root test literature, see, for example, Muller and Elliott (2003) and Elliot and Muller
(2006). It also suggests that the conclusions in Tsui and Ali (1992, 1994) with x0 discarded should be examined
with more scrutiny.
Given the CDF function (2.7), one might be tempted to calculate the quantile function F−1κ (t), t ∈ [0, 1]
by Newton’s method of interpolation. However, this involves calculation of the PDF function, which requires
another round of numerical integration, in addition to the possible problem pointed out by Hillier (2001).
Instead, we suggest employing a very simple bisection search algorithm. Since it is relatively cheap to simulate
the asymptotic results and we have observed that the infill asymptotic results are more reliable compared with
the expanding and mixed asymptotic results, we start with the t-th empirical quantile of the simulated sample
for approximating the in-fill asymptotic results, say c0. If Fκ(c0) < t, we set c1 as the min {2t, 1}-th empirical
quantile of the simulated sample. (Typically, Fκ(c1) > t. If not, one can set c1 as the min {ct, 1}-th empirical
quantile of the simulated sample, c = 3, 4, · · · , until one finds Fκ(c1) > t.) If Fκ(c0) > t, we set c1 as the
t/2-th empirical quantile of the simulated sample. (Typically, Fκ(c1) < t. If not, one can set c1 as the ct-th
empirical quantile of the simulated sample, c = 1/3, 1/4, · · · , until one finds Fκ(c1) < t.) Given the two initial
points c0 and c1, a bisection search can then be straightforwardly applied to search numerically for F−1κ (t). This
6The effects of the initial observation x0 can be examined more carefully by looking at the characteristic functions presentedin Section 2.1. For fixed x0, the characteristic functions behave differently under zero and non-zero x0. When the drift term isunknown, we see that in the characteristic function, the exponent has terms involving α that dominate the initial value x0.
9
algorithm is in a similar spirit of the algorithm in Lu and King (2002).
3 An Empirical Example
The linear diffusion model has been used to study the short-term interest rate in the literature. Even though the
term structure literature has documented that the short-term interest rate is highly persistent, an agreement
has yet to reach among economists regarding whether there is a unit root in the time series.
Figure 1: Real Federal Fund Rate
-2
-1
0
1
2
3
4
5
90 92 94 96 98 00 02 04 06 08 10 12
Figure 1 displays the monthly real federal fund rate from January 1990 to October 2012.7 If we use the
augmented Dickey-Fuller test or Phillips-Perron test, we fail to reject the null of unit root at any conventional
level. If we use the KPSS test, on the other hand, we fail to reject the null of stationarity at any conventional
level. The mixed results are in line with the observation from Bierens (2000).8
If we are willing to use the linear diffusion model for the real federal fund rate, then based on our exact
distribution approach, we can construct straightforwardly the exact confidence intervals of the mean reversion
parameter κ.9 For comparison, we also report the confidence intervals under the infill sampling scheme.
7The real federal fund rate is calculated as the effective H-15 federal fund rate adjusted for the core PCE inflation rate. Theformer is retrieved from www.federalreserve.gov and the latter is retrieved from www.bea.gov.
8One possibility, as discussed in Bierens (2000), is that the series is nonlinear trend stationary.9The distribution of κ depends on the diffusion and drift parameters. We set them equal to the estimated values from the
sample. (Recall from Tang and Chen (2009) that estimation biases of the diffusions and drift parameters are virtually zero.) Also,we regard the first sample observation as fixed and treat the drift term as unknown.
10
Confidence Intervals of κ for Real Federal Fund Rate
In the sequel, let c0 = 1 + φ2 + 2i(uφ − v), c1 = −(φ + iu), c2 = 2i(u + v − uφ), and c3 = i(2φu − 2v − u).
Since φ > 0, the case of c0 = 2c1, corresponding to φ = −1 and v = 0, is ruled out.
Determinant of Rn
Note that Rn is a tridiagonal matrix with its main diagonal elements rn,ii = c0, i = 1, · · · n − 1, rn,ii = 1,
i = n, and sub- and super-diagonal elements c1. Expanding along the last row of Rn leads to
|Rn| = |Cn−1| − c21 |Cn−2| , (B.7)
where the determinants of Cn−1 and Cn−2 follow from (B.1).
b2 = (c0 −√c20 − 4c21)/2. This can be numerically unstable when b1 is close to b2. Using polar representation, we write b1 =
c1 (cos θ + i sin θ) , b2 = c1 (cos θ − i sin θ). Then it follows that when c0 6= ±2c1, |Cn| = [cn1 sin((n + 1)θ)]/ sin θ. We thankRaymond Kan for pointing out this result and the results (B.3)–(B.6) to follow.
14
Determinant of Sn
Note that
A′nιnι′nAn =
ιn−1ι′n−1 0n−1
0′n−1 0
, ιnι′nAn =
ιn−1ι′n−1 0n−1
ι′n−1 0
.
So
Sn = In + φ2Bn − φ(An +A′n
)− iu(MnAn +A′nMn) + 2i(uφ− v)A′nMnAn
= In + φ2Bn − φ(An +A′n
)− iu(An +A′n) + 2i(uφ− v)A′nAn
+n−1iuιnι′nAn + n−1iuA′nιnι
′n − 2n−1i(uφ− v)A′nιnι
′nAn
= Rn +i
n
[uιnι
′nAn + uA′nιnι
′n − 2(uφ− v)A′nιnι
′nAn
]= Rn +
i
n
2(u+ v − uφ)ιn−1ι′n−1 uιn−1
uι′n−1 0
≡
∆n−1 an−1
a′n−1 1
,
where
∆n−1 = Cn−1 +c2nιn−1ι
′n−1, an−1 =
iu
nιn−1 + c1en−1,n−1.
Then immediately,
|Sn| = |∆n−1|(1− a′n−1∆
−1n−1an−1
), (B.8)
where
|∆n−1| = |Cn−1|(
1 +c2nι′n−1C
−1n−1ιn−1
), (B.9)
and
∆−1n−1 = C−1n−1 −c2
n+ c2ι′n−1C−1n−1ιn−1
C−1n−1ιn−1ι′n−1C
−1n−1. (B.10)
Keep in mind that (B.8) is valid only if ∆n−1 is nonsingular; (B.9) is valid only if Cn−1 is nonsingular; (B.10)
is valid only if Cn−1 is nonsingular and n+ c2ι′n−1Cn−1ιn−1 6= 0.
From (B.1), we see that |Cn−1| 6= 0. From (B.3), we also see that n+ c2ι′n−1Cn−1ιn−1 6= 0 for any positive
integer n. Further, these two conditions ensure that |∆n−1| 6= 0 so that the expression for |Sn| given by (B.8)
is valid. With |Cn−1| from (B.1) and ι′n−1C−1n−1ιn−1 from (B.3), we can easily calculate |∆n−1| in (B.8) via
15
(B.9). Note that the remaining term
a′n−1∆−1n−1an−1 = −u
2
n2ι′n−1C
−1n−1ιn−1 + c21e
′1,n−1C
−1n−1e1,n−1
+2ic1u
ne′1,n−1C
−1n−1ιn−1
− 2ic1c2u
n(n+ c2ι′n−1C−1n−1ιn−1)
ι′n−1C−1n−1ιn−1e
′1,n−1C
−1n−1ιn−1
+c2u
2
n2(n+ c2ι′n−1C−1n−1ιn−1)
(ι′n−1C−1n−1ιn−1)2
− c21c2
n+ c2ι′n−1C−1n−1ιn−1
(e′1,n−1C−1n−1ιn−1)2, (B.11)
where ι′n−1∆−1n−1ιn−1, e
′1,n−1C
−1n−1ιn−1, and e′1,n−1C
−1n−1e1,n−1 follow directly from (B.3), (B.4), and (B.5),
respectively.
Inverse of Sn
With ∆−1n−1 given by (B.10), the inverse of Sn is straightforward:
S−1n =
∆−1n−1 0n−1
0′n−1 0
+
∆−1n−1an−1
−1
( a′n−1∆−1n−1 −1
)1− a′n−1∆
−1n−1an−1
, (B.12)
if ∆n−1 is nonsingular and 1− a′n−1∆−1n−1an−1 6= 0. We have already showed that ∆n−1 is nonsingular. From
(B.11), we can verify that 1− a′n−1∆−1n−1an−1 6= 0 for any positive integer n.
Note that we need S−1n as δ′nS−1n δn appears in the CF (A.1). Given δn = iu(In − 2φA′n)Mne1,n +
2ivA′nMne1,n + φe1,n, we partition it as follows:
δn =
iu(e1,n−1 − 1nιn−1) + 2i(φu−v)
n ιn−1 + φe1,n−1
− iun
≡
δ1:n−1,n
− iun
.
16
So
δ′nS−1n δn = δ′n
∆−1n−1 0n−1
0′n−1 0
δn
+
δ′n
∆−1n−1an−1
−1
( a′n−1∆−1n−1 −1
)δn
1− a′n−1∆−1n−1an−1
= δ′1:n−1,n∆−1n−1δ1:n−1,n
+(δ′1:n−1,n∆−1n−1an−1)2 + 2iu
n δ′1:n−1,n∆−1n−1an−1 − u2
n2
1− a′n−1∆−1n−1an−1
, (B.13)
where a′n−1∆−1n−1an−1 follows from (B.11) and
δ′1:n−1,n∆−1n−1δ1:n−1,n =c23n2ι′n−1C
−1n−1ιn−1 +
2c1c3n
e′1,n−1C−1n−1ιn−1
+c21e′1,n−1C
−1n−1e1,n−1
− c2c23
n2(n+ c2ι′n−1C−1n−1ιn−1)
(ι′n−1C
−1n−1ιn−1
)2− c21c2
n+ c2ι′n−1C−1n−1ιn−1
(e′1,n−1C−1n−1ιn−1)2
− 2c1c2c3
n(n+ c2ι′n−1C−1n−1ιn−1)
e′1,n−1C−1n−1ιn−1ι
′n−1C
−1n−1ιn−1, (B.14)
δ′1:n−1,n∆−1n−1an−1 =iuc3n2ι′n−1C
−1n−1ιn−1 − c21e′1,n−1C
−1n−1en−1,n−1
−c1c2ne′1,n−1C
−1n−1ιn−1
+c21c2
n+ c2ι′n−1C−1n−1ιn−1
(e′1,n−1C−1n−1ιn−1)2
− iuc2c3
n2(n+ c2ι′n−1C−1n−1ιn−1)
(ι′n−1C−1n−1ιn−1)2
+c1c
22
n(n+ c2ι′n−1C−1n−1ιn−1)
e′1,n−1C−1n−1ιn−1ι
′n−1C
−1n−1ιn−1. (B.15)
which can be calculated by substituting (B.3)–(B.6).
For (B.16) to be valid, ∆∗n needs to be nonsingular; for (B.17) to be valid, ∆n−1 needs to be nonsingular;
for (B.18) to be valid, ∆n−1 needs to be nonsingular and 1 + 2i(uφ− v)− 2i(uφ− v)/n− b′n−1∆−1n−1bn−1 6= 0.
We have already shown that ∆n−1 is nonsingular. Note that
b′n−1∆−1n−1bn−1 =
c23n2ι′n−1C
−1n−1ιn−1 −
2c1c3n
e′1,n−1C−1n−1ιn−1
+c21e′1,n−1C
−1n−1e1,n−1
− c2c23
n2(n+ c2ι′n−1C−1n−1ιn−1)
(ι′n−1C−1n−1ιn−1)2
− c21c2
n+ c2ι′n−1C−1n−1ιn−1
(e′1,n−1C−1n−1ιn−1)2
+2c1c2c3
n(n+ c2ι′n−1C−1n−1ιn−1)
e′1,n−1C−1n−1ιn−1ι
′n−1C
−1n−1ιn−1, (B.19)
which can be evaluated with analytical expressions from (B.3)–(B.5), and we can verify that 1 + 2i(uφ − v) −
2i(uφ− v)/n− b′n−1∆−1n−1bn−1 6= 0 for any positive integer n.
For us to use (B.16), we still need expression for a∗′n∆∗−1n a∗n. By substitution,
a∗′n∆∗−1n a∗n = a′n−1∆−1n−1an−1
+−u2 − 2inua′n−1∆
−1n−1bn−1 + n2(a′n−1∆
−1n−1bn−1)2
n2 + 2in2(uφ− v)− 2in(uφ− v)− n2b′n−1∆−1n−1bn−1
, (B.20)
where a′n−1∆−1n−1an−1 is given by (B.11), b′n−1∆
−1n−1bn−1 is given by (B.19), and
a′n−1∆−1n−1bn−1 = − iuc3
n2ι′n−1C
−1n−1ιn−1 +
c1c2ne′1,n−1C
−1n−1ιn−1
+c21e′1,n−1C
−1n−1en−1,n−1
+iuc2c3
n2(n+ c2ι′n−1C−1n−1ιn−1)
(ι′n−1C−1n−1ιn−1)2
− c21c2
n+ c2ι′n−1C−1n−1ιn−1
(e′1,n−1C−1n−1ιn−1)2
− c1c22
n(n+ c2ι′n−1C−1n−1ιn−1)
e′1,n−1C−1n−1ιn−1ι
′n−1C
−1n−1ιn−1, (B.21)
which can be evaluated with analytical expressions from (B.3)–(B.6).
19
Appendix C: Numerical Integration
Given the characteristic functions in Section 2.1, we need to implement numerical integration to calculate (2.6)
via (2.7). This can be straightforwardly implemented using Matlab’s quadgk command. One caveat to note
is that the square root function in the complex domain is not continuous. One choice is to follow Perron
(1989) to identify explicitly the discontinuous points by grid search and then integrate by parts. The search,
however, might be inefficient and time-consuming. Instead, we use the following algorithm so that the integrand
function for quadgk is always continuous. Let g (t) =√a(t) + ib(t) denote the integrand function in question
with t ∈ [l, u] . quadgk requires the integrand function to accept a vector (t1, t2, · · · , tn) and returns a vector of
output. Let θi = arg (a(ti) + ib(ti)) ∈ [−π, π] and denote ai = a (ti) , bi = b (ti) , and gi = g (ti) .
1. Start with t1 and set g1 = sqrt (a1 + ib1) . Set k = 0.
2. Beginning with t2, if ai < 0, ai−1 < 0, and bibi−1 <= 0, set k = k + 1; otherwise, k is unchanged. Set
gi =√a2i + b2i (cos (θ∗i /2) + i sin (θ∗i /2)) , where θ∗i = θi + 2kπ.
20
Tab
le1,
Pan
elA
(T=
1):
Pr(T
(κ−κ
)≤w
),F
ixed
x0,κ
=0.
01,x0
=µ
=0,σ
=0.
1
Month
lyD
ail
yw
ppedf
pexp
pmix
pinf
ppedf
pexp
pmix
pinf
-50.
0002
0.00
02
0.0
000
0.0
000
0.0
000
0.0
001
0.0
000
0.0
000
0.0
000
0.0
000
-30.
0037
0.00
36
0.0
000
0.0
000
0.0
013
0.0
013
0.0
013
0.0
000
0.0
000
0.0
014
-20.
0201
0.02
02
0.0
000
0.0
000
0.0
109
0.0
110
0.0
112
0.0
000
0.0
000
0.0
105
-1.5
0.04
870.
0485
0.0
000
0.0
000
0.0
317
0.0
323
0.0
326
0.0
000
0.0000
0.0
310
-10.
1126
0.11
29
0.0
000
0.0
000
0.0
884
0.0
892
0.0
888
0.0
000
0.0
000
0.0
876
-0.8
0.15
110.
1515
0.0
000
0.0
000
0.1
249
0.1
261
0.1
253
0.0
000
0.0000
0.1
245
-0.6
0.19
580.
1963
0.0
000
0.0
000
0.1
679
0.1
691
0.1
697
0.0
000
0.0000
0.1
688
-0.4
0.24
390.
2444
0.0
023
0.0
023
0.2
173
0.2
190
0.2
186
0.0
023
0.0023
0.2
171
-0.2
0.29
280.
2931
0.0
787
0.0
786
0.2
682
0.2
688
0.2
686
0.0
787
0.0786
0.2
682
-0.1
0.31
690.
3174
0.2
398
0.2
398
0.2
933
0.2
946
0.2
932
0.2
398
0.2398
0.2
934
-0.0
10.
3380
0.33
84
0.4
718
0.4
718
0.3
153
0.3
154
0.3
153
0.4
718
0.4
718
0.3
157
-0.0
010.
3401
0.34
05
0.4
972
0.4
972
0.3
175
0.3
180
0.3
175
0.4
972
0.4972
0.3
179
00.
3403
0.34
08
0.5
000
0.5
000
0.3
177
0.3
183
0.3
177
0.5
000
0.5
000
0.3
181
0.00
10.
3406
0.34
10
0.5
028
0.5
028
0.3
180
0.3
185
0.3
180
0.5
028
0.5
028
0.3
184
0.01
0.34
270.
3432
0.5
282
0.5
282
0.3
202
0.3
207
0.3
202
0.5
282
0.5282
0.3
203
0.1
0.36
320.
3638
0.7
602
0.7
602
0.3
411
0.3
421
0.3
414
0.7
602
0.7
602
0.3
414
0.2
0.38
530.
3856
0.9
213
0.9
214
0.3
644
0.3
651
0.3
644
0.9
213
0.9
214
0.3
649
0.4
0.42
710.
4271
0.9
977
0.9
977
0.4
081
0.4
095
0.4
088
0.9
977
0.9
977
0.4
086
0.6
0.46
600.
4660
1.0
000
1.0
000
0.4
492
0.4
512
0.4
507
1.0
000
1.0
000
0.4
516
0.8
0.50
240.
5026
1.0
000
1.0
000
0.4
889
0.4
904
0.4
899
1.0
000
1.0
000
0.4
907
10.
5366
0.53
72
1.0
000
1.0
000
0.5
255
0.5
267
0.5
263
1.0
000
1.0000
0.5
273
1.5
0.61
110.
6115
1.0
000
1.0
000
0.6
050
0.6
053
0.6
051
1.0
000
1.0
000
0.6
055
20.
6711
0.67
13
1.0
000
1.0
000
0.6
700
0.6
691
0.6
687
1.0
000
1.0000
0.6
695
30.
7592
0.75
94
1.0
000
1.0
000
0.7
628
0.7
638
0.7
634
1.0
000
1.0000
0.7
647
50.
8617
0.86
18
1.0
000
1.0
000
0.8
757
0.8
746
0.8
742
1.0
000
1.0000
0.8
755
80.
9318
0.93
18
1.0
000
1.0
000
0.9
490
0.9
485
0.9
484
1.0
000
1.0000
0.9
498
200.
9906
0.99
04
1.0
000
1.0
000
0.9
980
0.9
980
0.9
981
1.0
000
1.0
000
0.9
982
500.
9995
0.99
95
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
100
1.00
001.
0000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0000
1.0
000
21
Tab
le1,
Panel
B(T
=10):
Pr(T
(κ−κ
)≤w
),F
ixed
x0,κ
=0.
01,x0
=µ
=0,σ
=0.
1
Month
lyD
ail
yw
ppedf
pexp
pmix
pinf
ppedf
pexp
pmix
pinf
-50.
0000
0.00
00
0.1
319
0.1
318
0.0
000
0.0
000
0.0
000
0.1
318
0.1
318
0.0
000
-30.
0015
0.00
15
0.2
513
0.2
512
0.0
014
0.0
013
0.0
013
0.2
512
0.2
512
0.0
014
-20.
0124
0.01
24
0.3
274
0.3
274
0.0
114
0.0
116
0.0
115
0.3
274
0.3
274
0.0
120
-1.5
0.03
550.
0356
0.3
687
0.3
687
0.0
347
0.0
341
0.0
342
0.3
687
0.3687
0.0
348
-10.
0940
0.09
38
0.4
116
0.4
115
0.0
924
0.0
918
0.0
919
0.4
115
0.4
115
0.0
929
-0.8
0.13
100.
1308
0.4
290
0.4
290
0.1
291
0.1
288
0.1
289
0.4
290
0.4290
0.1
284
-0.6
0.17
490.
1745
0.4
467
0.4
466
0.1
734
0.1
725
0.1
729
0.4
466
0.4466
0.1
715
-0.4
0.22
260.
2221
0.4
644
0.4
644
0.2
207
0.2
194
0.2
206
0.4
644
0.4644
0.2
192
-0.2
0.27
150.
2710
0.4
822
0.4
822
0.2
703
0.2
677
0.2
698
0.4
822
0.4822
0.2
694
-0.1
0.29
570.
2952
0.4
911
0.4
911
0.2
958
0.2
928
0.2
941
0.4
911
0.4911
0.2
935
-0.0
10.
3171
0.31
66
0.4
991
0.4
991
0.3
169
0.3
152
0.3
155
0.4
991
0.4
991
0.3
153
-0.0
010.
3192
0.31
88
0.4
999
0.4
999
0.3
191
0.3
163
0.3
176
0.4
999
0.4999
0.3
175
00.
3194
0.31
90
0.5
000
0.5
000
0.3
194
0.3
166
0.3
178
0.5
000
0.5
000
0.3
177
0.00
10.
3197
0.31
93
0.5
001
0.5
001
0.3
195
0.3
168
0.3
181
0.5
001
0.5
001
0.3
179
0.01
0.32
180.
3213
0.5
009
0.5
009
0.3
216
0.3
190
0.3
201
0.5
009
0.5009
0.3
199
0.1
0.34
270.
3422
0.5
089
0.5
089
0.3
424
0.3
404
0.3
410
0.5
089
0.5
089
0.3
415
0.2
0.36
540.
3647
0.5
178
0.5
178
0.3
655
0.3
637
0.3
639
0.5
178
0.5
178
0.3
641
0.4
0.41
070.
4082
0.5
356
0.5
356
0.4
101
0.4
077
0.4
077
0.5
356
0.5
356
0.4
077
0.6
0.45
050.
4494
0.5
533
0.5
534
0.4
518
0.4
493
0.4
493
0.5
534
0.5
534
0.4
500
0.8
0.48
920.
4882
0.5
710
0.5
710
0.4
904
0.4
883
0.4
882
0.5
710
0.5
710
0.4
891
10.
5204
0.52
39
0.5
884
0.5
885
0.5
264
0.5
244
0.5
244
0.5
885
0.5885
0.5
252
1.5
0.60
300.
6017
0.6
313
0.6
313
0.6
059
0.6
027
0.6
033
0.6
313
0.6
313
0.6
044
20.
6663
0.66
53
0.6
726
0.6
726
0.6
683
0.6
664
0.6
674
0.6
726
0.6726
0.6
681
30.
7607
0.76
02
0.7
487
0.7
488
0.7
636
0.7
616
0.7
627
0.7
488
0.7488
0.7
629
50.
8720
0.87
17
0.8
681
0.8
682
0.8
735
0.8
735
0.8
740
0.8
682
0.8682
0.8
739
80.
9467
0.94
68
0.9
631
0.9
632
0.9
484
0.9
484
0.9
487
0.9
632
0.9632
0.9
489
200.
9977
0.99
77
1.0
000
1.0
000
0.9
981
0.9
982
0.9
981
1.0
000
1.0
000
0.9
982
501.
0000
1.00
00
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
100
1.00
001.
0000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0000
1.0
000
22
Tab
le2,
Pan
elA
(T=
1):
Pr(T
(κ−κ
)≤w
),F
ixed
x0,κ
=0.
01,x0
=µ
=0.
1,σ
=0.
1
Month
lyD
ail
yw
ppedf
pexp
pmix
pinf
ppedf
pexp
pmix
pinf
-50.
0001
0.00
01
0.0
000
0.0
000
0.0
000
0.0
001
0.0
000
0.0
000
0.0
000
0.0
000
-30.
0016
0.00
16
0.0
000
0.0
000
0.0
003
0.0
004
0.0
004
0.0
000
0.0
000
0.0
004
-20.
0060
0.00
60
0.0
000
0.0
000
0.0
021
0.0
020
0.0
022
0.0
000
0.0
000
0.0
021
-1.5
0.01
140.
0114
0.0
000
0.0
000
0.0
049
0.0
045
0.0
051
0.0
000
0.0000
0.0
049
-10.
0209
0.02
09
0.0
000
0.0
000
0.0
108
0.0
113
0.0
112
0.0
000
0.0
000
0.0
108
-0.8
0.02
630.
0264
0.0
000
0.0
000
0.0
146
0.0
152
0.0
151
0.0
000
0.0000
0.0
144
-0.6
0.03
290.
0330
0.0
000
0.0
000
0.0
194
0.0
188
0.0
199
0.0
000
0.0000
0.0
192
-0.4
0.04
080.
0409
0.0
023
0.0
023
0.0
258
0.0
263
0.0
262
0.0
023
0.0023
0.0
251
-0.2
0.05
010.
0501
0.0
787
0.0
787
0.0
332
0.0
339
0.0
339
0.0
787
0.0787
0.0
324
-0.1
0.05
530.
0554
0.2
398
0.2
398
0.0
373
0.0
379
0.0
383
0.2
398
0.2398
0.0
369
-0.0
10.
0603
0.06
03
0.4
718
0.4
718
0.0
417
0.0
425
0.0
425
0.4
718
0.4
718
0.0
412
-0.0
010.
0609
0.06
08
0.4
972
0.4
972
0.0
421
0.0
429
0.0
430
0.4
972
0.4972
0.0
418
00.
0609
0.06
09
0.5
000
0.5
000
0.0
422
0.0
430
0.0
430
0.5
000
0.5
000
0.0
418
0.00
10.
0610
0.06
09
0.5
028
0.5
028
0.0
423
0.0
430
0.0
431
0.5
028
0.5
028
0.0
419
0.01
0.06
150.
0615
0.5
282
0.5
282
0.0
427
0.0
435
0.0
436
0.5
282
0.5282
0.0
423
0.1
0.06
690.
0669
0.7
602
0.7
603
0.0
474
0.0
484
0.0
483
0.7
603
0.7
603
0.0
470
0.2
0.07
320.
0732
0.9
213
0.9
214
0.0
532
0.0
541
0.0
539
0.9
214
0.9
214
0.0
525
0.4
0.08
710.
0869
0.9
977
0.9
977
0.0
655
0.0
665
0.0
665
0.9
977
0.9
977
0.0
649
0.6
0.10
230.
1021
1.0
000
1.0
000
0.0
798
0.0
809
0.0
809
1.0
000
1.0
000
0.0
791
0.8
0.11
900.
1187
1.0
000
1.0
000
0.0
952
0.0
399
0.0
967
1.0
000
1.0
000
0.0
950
10.
1368
0.13
66
1.0
000
1.0
000
0.1
130
0.1
145
0.1
143
1.0
000
1.0000
0.1
125
1.5
0.18
540.
1851
1.0
000
1.0
000
0.1
620
0.1
643
0.1
641
1.0
000
1.0
000
0.1
635
20.
2380
0.23
80
1.0
000
1.0
000
0.2
187
0.2
207
0.2
206
1.0
000
1.0000
0.2
186
30.
3477
0.34
75
1.0
000
1.0
000
0.3
423
0.3
423
0.3
419
1.0
000
1.0000
0.3
410
50.
5460
0.54
56
1.0
000
1.0
000
0.5
641
0.5
634
0.5
629
1.0
000
1.0000
0.5
658
80.
7392
0.73
86
1.0
000
1.0
000
0.7
840
0.7
792
0.7
787
1.0
000
1.0000
0.7
812
200.
9562
0.95
62
1.0
000
1.0
000
0.9
879
0.9
873
0.9
872
1.0
000
1.0
000
0.9
877
500.
9977
0.99
77
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
100
1.00
001.
0000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0000
1.0
000
23
Tab
le2,
Pan
elB
(T=
10):
Pr(T
(κ−κ
)≤w
),F
ixed
x0,κ
=0.
01,x0
=µ
=0.
1,σ
=0.
1
Month
lyD
ail
yw
ppedf
pexp
pmix
pinf
ppedf
pexp
pmix
pinf
-50.
0000
0.00
00
0.1
319
0.1
318
0.0
000
0.0
000
0.0
000
0.1
318
0.1
318
0.0
000
-30.
0005
0.00
05
0.2
513
0.2
512
0.0
004
0.0
004
0.0
004
0.2
512
0.2
512
0.0
004
-20.
0025
0.00
26
0.3
274
0.3
274
0.0
023
0.0
022
0.0
022
0.3
274
0.3
274
0.0
024
-1.5
0.00
560.
0058
0.3
687
0.3
687
0.0
052
0.0
052
0.0
051
0.3
687
0.3687
0.0
054
-10.
0121
0.01
24
0.4
116
0.4
115
0.0
115
0.0
114
0.0
114
0.4
115
0.4
115
0.0
120
-0.8
0.01
600.
0165
0.4
290
0.4
290
0.0
152
0.0
153
0.0
152
0.4
290
0.4290
0.0
158
-0.6
0.02
130.
0215
0.4
467
0.4
466
0.0
199
0.0
202
0.0
203
0.4
466
0.4466
0.0
210
-0.4
0.02
760.
0279
0.4
644
0.4
644
0.0
260
0.0
257
0.0
265
0.4
644
0.4644
0.0
274
-0.2
0.03
540.
0357
0.4
822
0.4
822
0.0
333
0.0
341
0.0
341
0.4
822
0.4822
0.0
351
-0.1
0.03
980.
0401
0.4
911
0.4
911
0.0
376
0.0
384
0.0
385
0.4
911
0.4911
0.0
393
-0.0
10.
0441
0.04
45
0.4
991
0.4
991
0.0
420
0.0
425
0.0
426
0.4
991
0.4
991
0.0
436
-0.0
010.
0446
0.04
49
0.4
999
0.4
999
0.0
424
0.0
430
0.0
431
0.4
999
0.4999
0.0
442
00.
0446
0.04
50
0.5
000
0.5
000
0.0
425
0.0
430
0.0
432
0.5
000
0.5
000
0.0
443
0.00
10.
0447
0.04
50
0.5
001
0.5
001
0.0
426
0.0
431
0.0
432
0.5
001
0.5
001
0.0
444
0.01
0.04
510.
0455
0.5
009
0.5
009
0.0
430
0.0
434
0.0
437
0.5
009
0.5009
0.0
447
0.1
0.04
990.
0503
0.5
089
0.5
089
0.0
475
0.0
484
0.0
484
0.5
089
0.5
089
0.0
492
0.2
0.05
550.
0560
0.5
178
0.5
178
0.0
534
0.0
540
0.0
540
0.5
178
0.5
178
0.0
547
0.4
0.06
800.
0685
0.5
356
0.5
356
0.0
662
0.0
664
0.0
665
0.5
356
0.5
356
0.0
671
0.6
0.08
220.
0829
0.5
533
0.5
534
0.0
797
0.0
805
0.0
807
0.5
534
0.5
534
0.0
819
0.8
0.09
790.
0988
0.5
710
0.5
710
0.0
954
0.0
962
0.0
965
0.5
710
0.5
710
0.0
978
10.
1151
0.11
62
0.5
884
0.5
885
0.1
123
0.1
135
0.1
137
0.5
885
0.5885
0.1
148
1.5
0.16
400.
1651
0.6
313
0.6
313
0.1
613
0.1
627
0.1
629
0.6
313
0.6
313
0.1
632
20.
2195
0.22
07
0.6
726
0.6
726
0.2
175
0.2
186
0.2
190
0.6
726
0.6726
0.2
172
30.
3394
0.34
06
0.7
487
0.7
488
0.3
393
0.3
398
0.3
398
0.7
488
0.7488
0.3
383
50.
5585
0.55
97
0.8
681
0.8
682
0.5
624
0.5
613
0.5
613
0.8
682
0.8682
0.5
584
80.
7740
0.77
42
0.9
631
0.9
632
0.7
822
0.7
789
0.7
796
0.9
632
0.9632
0.7
781
200.
9857
0.98
56
1.0
000
1.0
000
0.9
884
0.9
880
0.9
881
1.0
000
1.0
000
0.9
885
501.
0000
1.00
00
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
100
1.00
001.
0000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0000
1.0
000
24
Tab
le3,
Pan
elA
(T=
1):
Pr(T
(κ−κ
)≤w
),R
an
domx0,κ
=0.
01,µ
=0,σ
=0.
1
Month
lyD
ail
yw
ppedf
pexp
pmix
pinf
ppedf
pexp
pmix
pinf
-50.
0000
0.00
00
0.0
000
0.0
000
0.0
099
0.0
000
0.0
000
0.0
000
0.0
000
0.0
099
-30.
0001
0.00
01
0.0
000
0.0
000
0.0
158
0.0
001
0.0
001
0.0
000
0.0
000
0.0
155
-20.
0008
0.00
09
0.0
000
0.0
000
0.0
210
0.0
005
0.0
005
0.0
000
0.0
000
0.0
203
-1.5
0.00
250.
0025
0.0
000
0.0
000
0.0
250
0.0
018
0.0
018
0.0
000
0.0000
0.0
242
-10.
0084
0.00
84
0.0
000
0.0
000
0.0
339
0.0
071
0.0
070
0.0
000
0.0
000
0.0
328
-0.8
0.01
440.
0145
0.0
000
0.0
000
0.0
423
0.0
127
0.0
127
0.0
000
0.0000
0.0
412
-0.6
0.02
640.
0265
0.0
000
0.0
000
0.0
581
0.0
241
0.0
242
0.0
000
0.0000
0.0
580
-0.4
0.05
360.
0536
0.0
023
0.0
023
0.0
919
0.0
506
0.0
508
0.0
023
0.0023
0.0
892
-0.2
0.13
430.
1340
0.0
787
0.0
787
0.1
715
0.1
302
0.1
297
0.0
787
0.0787
0.1
651
-0.1
0.24
300.
2432
0.2
398
0.2
398
0.2
658
0.2
382
0.2
378
0.2
398
0.2398
0.2
613
-0.0
10.
4202
0.42
08
0.4
718
0.4
718
0.4
232
0.4
152
0.4
145
0.4
718
0.4
718
0.4
227
-0.0
010.
4406
0.44
11
0.4
972
0.4
972
0.4
427
0.4
355
0.4
348
0.4
972
0.4972
0.4
427
00.
4428
0.44
34
0.5
000
0.5
000
0.4
449
0.4
377
0.4
370
0.5
000
0.5
000
0.4
450
0.00
10.
4451
0.44
56
0.5
028
0.5
028
0.4
471
0.4
400
0.4
393
0.5
028
0.5
028
0.4
472
0.01
0.46
530.
4658
0.5
282
0.5
282
0.4
669
0.4
602
0.4
596
0.5
282
0.5282
0.4
676
0.1
0.63
610.
6365
0.7
602
0.7
603
0.6
472
0.6
315
0.6
310
0.7
603
0.7
603
0.6
526
0.2
0.74
450.
7454
0.9
213
0.9
214
0.7
733
0.7
404
0.7
403
0.9
214
0.9
214
0.7
814
0.4
0.83
900.
8394
0.9
977
0.9
977
0.8
816
0.8
360
0.8
360
0.9
977
0.9
977
0.8
822
0.6
0.88
030.
8807
1.0
000
1.0
000
0.9
218
0.8
781
0.8
780
1.0
000
1.0
000
0.9
199
0.8
0.90
410.
9045
1.0
000
1.0
000
0.9
414
0.9
025
0.9
024
1.0
000
1.0
000
0.9
405
10.
9200
0.92
05
1.0
000
1.0
000
0.9
530
0.9
189
0.9
189
1.0
000
1.0000
0.9
526
1.5
0.94
460.
9451
1.0
000
1.0
000
0.9
684
0.9
441
0.9
443
1.0
000
1.0
000
0.9
676
20.
9590
0.95
93
1.0
000
1.0
000
0.9
760
0.9
590
0.9
591
1.0
000
1.0000
0.9
748
30.
9750
0.97
51
1.0
000
1.0
000
0.9
837
0.9
756
0.9
757
1.0
000
1.0000
0.9
826
50.
9886
0.98
86
1.0
000
1.0
000
0.9
901
0.9
897
0.9
897
1.0
000
1.0000
0.9
895
80.
9954
0.99
52
1.0
000
1.0
000
0.9
937
0.9
966
0.9
965
1.0
000
1.0000
0.9
935
200.
9995
0.99
95
1.0
000
1.0
000
0.9
975
0.9
999
0.9
999
1.0
000
1.0
000
0.9
974
501.
0000
1.00
00
1.0
000
1.0
000
0.9
990
1.0
000
1.0
000
1.0
000
1.0
000
0.9
990
100
1.00
001.
0000
1.0
000
1.0
000
0.9
995
1.0
000
1.0
000
1.0
000
1.0000
0.9
995
25
Tab
le3,
Pan
elB
(T=
10):
Pr(T
(κ−κ
)≤w
),R
an
domx0,κ
=0.
01,µ
=0,σ
=0.
1
Month
lyD
ail
yw
ppedf
pexp
pmix
pinf
ppedf
pexp
pmix
pinf
-50.
0000
0.00
00
0.1
319
0.1
318
0.0
265
0.0
000
0.0
000
0.1
318
0.1
318
0.0
260
-30.
0002
0.00
02
0.2
513
0.2
512
0.0
418
0.0
002
0.0
001
0.2
512
0.2
512
0.0
407
-20.
0019
0.00
19
0.3
274
0.3
274
0.0
548
0.0
019
0.0
017
0.3
274
0.3
274
0.0
527
-1.5
0.00
630.
0065
0.3
687
0.3
687
0.0
639
0.0
062
0.0
062
0.3
687
0.3687
0.0
611
-10.
0235
0.02
37
0.4
116
0.4
115
0.0
850
0.0
231
0.0
230
0.4
115
0.4
115
0.0
811
-0.8
0.04
110.
0410
0.4
290
0.4
290
0.1
053
0.0
405
0.0
406
0.4
290
0.4290
0.1
013
-0.6
0.07
360.
0731
0.4
467
0.4
466
0.1
432
0.0
729
0.0
729
0.4
466
0.4466
0.1
404
-0.4
0.13
420.
1338
0.4
644
0.4
644
0.2
081
0.1
334
0.1
332
0.4
644
0.4644
0.2
077
-0.2
0.23
940.
2393
0.4
822
0.4
822
0.2
931
0.2
385
0.2
385
0.4
822
0.4822
0.2
944
-0.1
0.31
020.
3084
0.4
911
0.4
911
0.3
441
0.3
073
0.3
076
0.4
911
0.4911
0.3
458
-0.0
10.
3760
0.37
40
0.4
991
0.4
991
0.3
991
0.3
730
0.3
730
0.4
991
0.4
991
0.4
016
-0.0
010.
3805
0.38
04
0.4
999
0.4
999
0.4
052
0.3
795
0.3
797
0.4
999
0.4999
0.4
077
00.
3812
0.38
12
0.5
000
0.5
000
0.4
059
0.3
803
0.3
804
0.5
000
0.5
000
0.4
084
0.00
10.
3819
0.38
19
0.5
001
0.5
001
0.4
065
0.3
810
0.3
811
0.5
001
0.5
001
0.4
091
0.01
0.38
850.
3884
0.5
009
0.5
009
0.4
127
0.3
875
0.3
878
0.5
009
0.5009
0.4
153
0.1
0.45
120.
4510
0.5
089
0.5
089
0.4
779
0.4
502
0.4
505
0.5
089
0.5
089
0.4
813
0.2
0.51
350.
5134
0.5
178
0.5
178
0.5
553
0.5
126
0.5
132
0.5
178
0.5
178
0.5
588
0.4
0.61
130.
6108
0.5
356
0.5
356
0.7
052
0.6
106
0.6
111
0.5
356
0.5
356
0.7
094
0.6
0.68
040.
6803
0.5
533
0.5
534
0.7
949
0.6
799
0.6
801
0.5
534
0.5
534
0.7
980
0.8
0.73
090.
7308
0.5
710
0.5
710
0.8
479
0.7
305
0.7
311
0.5
710
0.5
710
0.8
509
10.
7694
0.76
93
0.5
884
0.5
885
0.8
805
0.7
691
0.7
695
0.5
885
0.5885
0.8
821
1.5
0.83
490.
8349
0.6
313
0.6
313
0.9
208
0.8
348
0.8
349
0.6
313
0.6
313
0.9
206
20.
8763
0.87
59
0.6
726
0.6
726
0.9
393
0.8
763
0.8
765
0.6
726
0.6726
0.9
389
30.
9249
0.92
49
0.7
487
0.7
488
0.9
585
0.9
251
0.9
251
0.7
488
0.7488
0.9
576
50.
9677
0.96
76
0.8
681
0.8
682
0.9
744
0.9
680
0.9
680
0.8
682
0.8682
0.9
738
80.
9891
0.98
90
0.9
631
0.9
632
0.9
838
0.9
894
0.9
896
0.9
632
0.9632
0.9
834
200.
9997
0.99
97
1.0
000
1.0
000
0.9
934
0.9
998
0.9
998
1.0
000
1.0
000
0.9
933
501.
0000
1.00
00
1.0
000
1.0
000
0.9
973
1.0
000
1.0
000
1.0
000
1.0
000
0.9
973
100
1.00
001.
0000
1.0
000
1.0
000
0.9
987
1.0
000
1.0
000
1.0
000
1.0000
0.9
987
26
Tab
le4,
Pan
elA
(T=
1):
Pr(T
(κ−κ
)≤w
),R
an
domx0,κ
=0.
01,µ
=0.
1,σ
=0.
1
Month
lyD
ail
yw
ppedf
pexp
pmix
pinf
ppedf
pexp
pmix
pinf
-50.
0001
0.00
01
0.0
000
0.0
000
0.0
000
0.0
001
0.0
000
0.0
000
0.0
000
0.0
000
-30.
0016
0.00
16
0.0
000
0.0
000
0.0
003
0.0
004
0.0
004
0.0
000
0.0
000
0.0
003
-20.
0061
0.00
60
0.0
000
0.0
000
0.0
021
0.0
020
0.0
023
0.0
000
0.0
000
0.0
021
-1.5
0.01
140.
0114
0.0
000
0.0
000
0.0
049
0.0
045
0.0
053
0.0
000
0.0000
0.0
049
-10.
0210
0.02
10
0.0
000
0.0
000
0.0
109
0.0
113
0.0
115
0.0
000
0.0
000
0.0
109
-0.8
0.02
640.
0264
0.0
000
0.0
000
0.0
148
0.0
152
0.0
155
0.0
000
0.0000
0.0
147
-0.6
0.03
310.
0330
0.0
000
0.0
000
0.0
197
0.0
189
0.0
205
0.0
000
0.0000
0.0
197
-0.4
0.04
100.
0410
0.0
023
0.0
023
0.0
258
0.0
265
0.0
269
0.0
023
0.0023
0.0
258
-0.2
0.05
030.
0504
0.0
787
0.0
787
0.0
334
0.0
341
0.0
344
0.0
787
0.0787
0.0
335
-0.1
0.05
560.
0556
0.2
398
0.2
398
0.0
378
0.0
384
0.0
389
0.2
398
0.2398
0.0
379
-0.0
10.
0606
0.06
05
0.4
718
0.4
718
0.0
421
0.0
427
0.0
433
0.4
718
0.4
718
0.0
422
-0.0
010.
0611
0.06
10
0.4
972
0.4
972
0.0
426
0.0
432
0.0
437
0.4
972
0.4972
0.0
426
00.
0612
0.06
11
0.5
000
0.5
000
0.0
426
0.0
434
0.0
438
0.5
000
0.5
000
0.0
427
0.00
10.
0612
0.06
11
0.5
028
0.5
028
0.0
427
0.0
435
0.0
438
0.5
028
0.5
028
0.0
427
0.01
0.06
180.
0617
0.5
282
0.5
282
0.0
432
0.0
437
0.0
443
0.5
282
0.5282
0.0
432
0.1
0.06
720.
0671
0.7
602
0.7
603
0.0
479
0.0
487
0.0
491
0.7
603
0.7
603
0.0
479
0.2
0.07
350.
0736
0.9
213
0.9
214
0.0
535
0.0
544
0.0
548
0.9
214
0.9
214
0.0
536
0.4
0.08
740.
0873
0.9
977
0.9
977
0.0
661
0.0
668
0.0
674
0.9
977
0.9
977
0.0
662
0.6
0.10
280.
1026
1.0
000
1.0
000
0.0
805
0.0
813
0.0
818
1.0
000
1.0
000
0.0
805
0.8
0.11
940.
1194
1.0
000
1.0
000
0.0
965
0.0
974
0.0
979
1.0
000
1.0
000
0.0
966
10.
1373
0.13
74
1.0
000
1.0
000
0.1
141
0.1
150
0.1
154
1.0
000
1.0000
0.1
142
1.5
0.18
610.
1862
1.0
000
1.0
000
0.1
642
0.1
649
0.1
653
1.0
000
1.0
000
0.1
643
20.
2387
0.23
88
1.0
000
1.0
000
0.2
210
0.2
214
0.2
217
1.0
000
1.0000
0.2
211
30.
3485
0.34
85
1.0
000
1.0
000
0.3
433
0.3
430
0.3
436
1.0
000
1.0000
0.3
434
50.
5467
0.54
63
1.0
000
1.0
000
0.5
658
0.5
641
0.5
648
1.0
000
1.0000
0.5
659
80.
7397
0.73
95
1.0
000
1.0
000
0.7
825
0.7
796
0.7
800
1.0
000
1.0000
0.7
826
200.
9562
0.95
61
1.0
000
1.0
000
0.9
885
0.9
873
0.9
872
1.0
000
1.0
000
0.9
885
500.
9977
0.99
77
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
100
1.00
001.
0000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0000
1.0
000
27
Tab
le4,
Pan
elB
(T=
10):
Pr(T
(κ−κ
)≤w
),R
an
domx0,κ
=0.
01,µ
=0.
1,σ
=0.
1
Month
lyD
ail
yw
ppedf
pexp
pmix
pinf
ppedf
pexp
pmix
pinf
-50.
0000
0.00
00
0.1
319
0.1
318
0.0
000
0.0
000
0.0
000
0.1
318
0.1
318
0.0
000
-30.
0005
0.00
05
0.2
513
0.2
512
0.0
004
0.0
004
0.0
004
0.2
512
0.2
512
0.0
004
-20.
0026
0.00
26
0.3
274
0.3
274
0.0
023
0.0
023
0.0
024
0.3
274
0.3
274
0.0
023
-1.5
0.00
760.
0060
0.3
687
0.3
687
0.0
054
0.0
055
0.0
054
0.3
687
0.3687
0.0
054
-10.
0127
0.01
29
0.4
116
0.4
115
0.0
118
0.0
120
0.0
118
0.4
115
0.4
115
0.0
119
-0.8
0.01
680.
0172
0.4
290
0.4
290
0.0
159
0.0
161
0.0
158
0.4
290
0.4290
0.0
160
-0.6
0.02
230.
0226
0.4
467
0.4
466
0.0
211
0.0
213
0.0
210
0.4
466
0.4466
0.0
212
-0.4
0.02
900.
0292
0.4
644
0.4
644
0.0
276
0.0
271
0.0
275
0.4
644
0.4644
0.0
277
-0.2
0.03
710.
0373
0.4
822
0.4
822
0.0
355
0.0
358
0.0
353
0.4
822
0.4822
0.0
357
-0.1
0.04
170.
0419
0.4
911
0.4
911
0.0
401
0.0
403
0.0
398
0.4
911
0.4911
0.0
402
-0.0
10.
0462
0.04
65
0.4
991
0.4
991
0.0
445
0.0
335
0.0
443
0.4
991
0.4
991
0.0
447
-0.0
010.
0467
0.04
70
0.4
999
0.4
999
0.0
450
0.0
450
0.0
447
0.4
999
0.4999
0.0
452
00.
0467
0.04
70
0.5
000
0.5
000
0.0
450
0.0
341
0.0
448
0.5
000
0.5
000
0.0
452
0.00
10.
0468
0.04
71
0.5
001
0.5
001
0.0
451
0.0
451
0.0
448
0.5
001
0.5
001
0.0
453
0.01
0.04
730.
0475
0.5
009
0.5
009
0.0
456
0.0
455
0.0
453
0.5
009
0.5009
0.0
457
0.1
0.05
220.
0525
0.5
089
0.5
089
0.0
504
0.0
411
0.0
501
0.5
089
0.5
089
0.0
506
0.2
0.05
800.
0584
0.5
178
0.5
178
0.0
562
0.0
524
0.0
559
0.5
178
0.5
178
0.0
564
0.4
0.07
100.
0712
0.5
356
0.5
356
0.0
691
0.0
799
0.0
689
0.5
356
0.5
356
0.0
694
0.6
0.08
570.
0859
0.5
533
0.5
534
0.0
837
0.0
840
0.0
834
0.5
534
0.5
534
0.0
840
0.8
0.10
190.
1022
0.5
710
0.5
710
0.0
999
0.1
003
0.0
994
0.5
710
0.5
710
0.1
002
10.
1196
0.12
00
0.5
884
0.5
885
0.1
176
0.1
180
0.1
172
0.5
885
0.5885
0.1
180
1.5
0.16
970.
1704
0.6
313
0.6
313
0.1
679
0.1
683
0.1
674
0.6
313
0.6
313
0.1
684
20.
2260
0.22
67
0.6
726
0.6
726
0.2
248
0.2
251
0.2
244
0.6
726
0.6726
0.2
253
30.
3466
0.34
68
0.7
487
0.7
488
0.3
468
0.3
470
0.3
459
0.7
488
0.7488
0.3
473
50.
5649
0.56
44
0.8
681
0.8
682
0.5
678
0.5
677
0.5
670
0.8
682
0.8682
0.5
683
80.
7778
0.77
74
0.9
631
0.9
632
0.7
833
0.7
827
0.7
818
0.9
632
0.9632
0.7
835
200.
9860
0.98
59
1.0
000
1.0
000
0.9
885
0.9
883
0.9
882
1.0
000
1.0
000
0.9
885
501.
0000
1.00
00
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
100
1.00
001.
0000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0
000
1.0000
1.0
000
28
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