A Appendix: Technical Details

Throughout the appendix, we use ∆ to denote a finite positive constant, and denote the spectral norm of a matrix A= (aij)q×p as∥A∥2 =√

relying on a finite dimension vectorθ, wherepandqare also finite, PnFt(θ) := _n¹ ∑n

t=1Ft(θ). For the first and second derivatives, ˙ℓt(β) := ^∂ℓ_∂β^t(β), ℓ¨t(β) := ^∂_∂β∂β^2ℓt(β)′ , A(θ) :=˙ ^∂A(θ)_∂θ′ ,B˙(θ) := ^∂B(θ)_∂θ′ and ˙ht(θ) := ^∂h_∂θ^t(θ)′ . We suppress the expressionE and W in ¯θn(E, W) and ht(θ;E, W), and write them as ¯θn and ht(θ) whenever doing so would not cause confusion.

We firstly present the assumptions needed in our analysis. Assumptions used for the approx-imate MLE of A¨ıt-Sahalia (1999, 2002) are presented as well, as we have used an approach that can lead to results for both the combined estimation using either the full likelihood scores or the approximated likelihood scores. Discussions to the assumptions including comparison with the conditions in the extant literatures are given in the supplementary material.

Assumption 1. (i) θ = (β^′, λ^′)^′ = (θ1,· · · , θq+d)^′ ∈ Θ which is a compact set in R^q+d, where

Assumption 2. (i) The short rate r(t) follows the time homogeneous diffusion processes (2.1) and (2.3) under measuresQ0andQ1. Assumption 1 in A¨ıt-Sahalia and Mykland (2004) is satisfied under the measure Q₀, and (2.4) holds under the measureQ₁. (ii) The pricing functions in (2.6) and (2.7) are three times differentiable with respect toθ. (iii) For fixedτ1,· · · , τM and fixedM, the

∫ _∂2

Assumption 4. The J-term expansion to the log of transition density ℓt(β) in (2.2) is ℓ^(J)_t (β) = −log√

2πδ+A¹(rt|rt−1, δ;β) +A²(rt|rt−1, δ;β) +A^(J)3 (rt|rt−1, δ;β), (A.1) where A^(J)3 (x|x0, δ;β) = log{

∑J

j=0cj(γ(x;β)|γ(x0;β);β)^δ_j!^j}

for J ≥ 1, and the expressions of the functions A1,A2, γ and c_j can be found in A¨ıt-Sahalia (1999). Let

as a approximate forht(θ;E, W) in (4.1) to establish our proposed estimator. Assumptions (A.3), (A.6) and (A.7) in Chang and Chen (2011) hold and there exist finite positive constants νk for k = 0,1,2,3, and M2 such that ν0 > 3, ν2 > ν1 > 3, ν3 > 1 and for any i1,· · · , i3 ∈ {1,· · · , q},

Assumption 5. (i) The measurement error {u0t} in (3.2) is a martingale difference array with respect to the filtration{G^t}, whereG^tis theσ-algebra generated by{(rl+1, u^′_0l)^′}l≤t. (ii) The short rate and measurement error process {(rt, u^′_0t)^′} is stationary and satisfies (3.1). (iii) We assume {(rt, u^′_0t)^′} is ρ-mixing with the ρ-mixing coefficient

The q^†×d matrix Zλ(θ0) :=Z(θ0)(0d×q, Id)^′ satisfies rank{Zλ(θ0)}=d.

In the following we present the lemmas as well as the proofs of the propositions and theorems by using these lemmas. The proofs of the lemmas and some corollaries are left in the supplementary material.

Lemma A.1. Under Assumptions 1, 2, 4 and 5, there exist positive constants M01, M02 < ∞ and ∆1, such that for any J, where J can be infinity, l = 1,2, δ ∈ (0,∆1], d ≤ K ≤ q+d, and

Lemma A.2. For every i, j, there exists a constant M31 such that E{

sup_θ∈Θ Lemma A.3. Under Assumptions 1, 2 and 4,

Pnℓ˙t(β)−E{

Proof of Proposition 4.1: According to the stationary, (A.4) and Lemma A.1, by Theorem 16.3.8 in Athreya and Lahiri (2006), the long-run covariance matrix limn→∞nVar{Pⁿht(θ0)} =

= 0 by Assumption 5. Then the proposition

can be proved together with Assumption 3. □

Lemma A.4. Under Assumptions 3, 5 and 6,Ω(δ;E, W)defined in (5.3) and Q⁰(δ;E, W)defined

. Taking the Taylor expansion at θ0 on the first oder condition to the minimization,{

Pnh˙_t(θ₀)}′

by Proposition 4.1, the stationarity, (A.4) and Theorem 16.3.8 in Athreya and Lahiri (2006). □ Proof of Theorem 5.2: Note thatψ(θ₀)E1 andψ(θ₀)E2are both of full rank. From (A.5), we have Acknowledgements: We thank the Editor, the AE and two referees for helpful comments and suggestions which have improved the presentation of the paper. The research was partially sup-ported by Natural Science Foundation of China grants 11131002 and 71371016.

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Table 1: J-tests for the Vasicek and CIR models based on the Federal fund rates and bond prices.

GMM: E¹ = (e2, e3, e4) (Moment Conditions I) 1−3 1−5 1−10 1−15 6−15 Vasicek J-statistic 10.2720 30.8105 11.4163 30.6171 32.3894

p-value 0.0059 0.0000 0.0033 0.0000 0.0000 CIR J-statistic 0.8903 1.4591 9.0660 26.4127 28.3110

p-value 0.6407 0.4821 0.0107 0.0000 0.0000 GMM: E² = (e1, e2, e3) (Moment Conditions II)

1−3 1−5 1−10 1−15 6−15 Vasicek J-statistic 10.3928 10.0876 6.8914 7.9270 7.9914 p-value 0.0055 0.0064 0.0319 0.0190 0.0184 CIR J-statistic 0.7671 1.4826 11.4109 48.9833 50.4626

p-value 0.6814 0.4765 0.0033 0.0000 0.0000

Table 2: Combined estimates based on the monthly Federal fund rates and bond prices for the CIR model (W = ˆV_n⁻¹(E¹)). Figures inside the parentheses are the standard errors of the estimates above.

MLE 1−3 1−5 1−10 1−15 6−15

κ 0.1875 0.1705 0.1635 0.2454 0.1504 0.1836 (0.0337) (0.0239) (0.0243) (0.0331) (0.0295) (0.0295) α 0.0751 0.0645 0.0696 0.0574 0.0857 0.0726

(0.0133) (0.0090) (0.0103) (0.0077) (0.0167) (0.0117) σ 0.0641 0.0544 0.0548 0.0647 0.0604 0.0607

(0.0007) (0.0004) (0.0004) (0.0007) (0.0005) (0.0005)

λ - 0.6589 0.4772 1.2344 0.0608 0.5713

- (0.4390) (0.4417) (0.5127) (0.4867) (0.4866)

b - 0.1347 0.1374 0.1655 0.1467 0.1489

- (0.0058) (0.0054) (0.0046) (0.0043) (0.0043)

a - 0.0817 0.0829 0.0852 0.0878 0.0895

- (0.0015) (0.0010) (0.0009) (0.0010) (0.0010)

Table 3: Estimated correlation matrix of the measurement errors for 1-5 years maturity under the CIR model.

τ_i 2 3 4 5

1 0.950 0.879 0.811 0.749

2 0.983 0.949 0.911

3 0.991 0.971

4 0.994

500 1000 1500 2000

0.050.150.250.35

W=I

_M

n

SD of κ

MLE M=5, MC I M=5, MC II M=15, MC I M=15, MC II

500 1000 1500 2000

0.050.150.250.35

W=V ^

n

−1

n

SD of κ

MLE M=5, MC I M=5, MC II M=15, MC I M=15, MC II

500 1000 1500 2000

0.0040.0080.012

W=I

M

n

SD of α

MLE M=5, MC I M=5, MC II M=15, MC I M=15, MC II

500 1000 1500 2000

0.0040.0080.012

W=V ^

n

−1

n

SD of α

MLE M=5, MC I M=5, MC II M=15, MC I M=15, MC II

Figure 1: Simulated standard deviation (SD) of the MLEs ( ˜βn), and the combined estimators (ˆθn) with two moment selection matricesE¹ (MC I) andE² (MC II) forκandα in the CIR model with (κ, α, σ, λ) = (0.892,0.09,√

0.033,0.1)); M denotes the number of bonds.

500 1000 1500 2000

0.0020.0040.0060.008

W=I

_M

n

SD of σ

MLE M=5, MC I M=5, MC II M=15, MC I M=15, MC II

500 1000 1500 2000

0.0020.0040.0060.008

W=V ^

n

−1

n

SD of σ

MLE M=5, MC I M=5, MC II M=15, MC I M=15, MC II

500 1000 1500 2000

0.20.30.40.50.60.7

W=I

M

n

SD of λ

M=5, MC I M=5, MC II M=15, MC I M=15, MC II

500 1000 1500 2000

0.20.30.40.50.60.7

W=V ^

n

−1

n

SD of λ

M=5, MC I M=5, MC II M=15, MC I M=15, MC II

Figure 2: Simulated standard deviation (SD) of the MLEs ( ˜βn), and the combined estimators (ˆθn) with two moment selection matricesE¹ (MC I) andE² (MC II) forσ andλ in the CIR model with (κ, α, σ, λ) = (0.892,0.09,√

0.033,0.1)); M denotes the number of bonds.

500 1000 1500 2000

0.000.100.20

W=I

_M

n

BIAS of κ

MLE M=5, MC I M=5, MC II M=15, MC I M=15, MC II

500 1000 1500 2000

0.000.100.20

W=V ^

n

−1

n

BIAS of κ

MLE M=5, MC I M=5, MC II M=15, MC I M=15, MC II

500 1000 1500 2000

0.000.100.200.30

W=I

M

n

BIAS of λ

M=5, MC I M=5, MC II M=15, MC I M=15, MC II

500 1000 1500 2000

0.000.100.200.30

W=V ^

n

−1

n

BIAS of λ

M=5, MC I M=5, MC II M=15, MC I M=15, MC II

Figure 3: Averaged absolute bias (BIAS) of the MLEs ( ˜βn), and the combined estimators (ˆθn) with two moment selection matricesE¹ (MC I) andE² (MC II) forκ andλ in the CIR model with (κ, α, σ, λ) = (0.892,0.09,√

0.033,0.1)); M denotes the number of bonds.

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