行政院國家科學委員會專題研究計畫 成果報告
新的一致性條件動差檢定
研究成果報告(精簡版)
計 畫 類 別 ： 個別型
計 畫 編 號 ： NSC 96-2415-H-004-008-
執 行 期 間 ： 96 年 08 月 01 日至 97 年 10 月 31 日
執 行 單 位 ： 國立政治大學經濟學系
計 畫 主 持 人 ： 林馨怡
計畫參與人員： 碩士班研究生-兼任助理人員：周東慶
處 理 方 式 ： 本計畫可公開查詢
中 華 民 國 97 年 12 月 11 日
1
Introduction
Many economic and econometric models are represented by conditional moment restrictions, for example, the rational expectation model, the market disequilibrium model, the conditional probability model, the discrete choice model and the nonlinear simultaneous equations model. The validity of those model specifications is evaluated by testing the associated moment conditions. The resulting test is the conditional moment test or M-test which has been developed by Newey (1985), Tauchen (1985), and White (1987). However, these conditional moment tests may not be consistent because they check only necessary conditions of the conditional moment restrictions and there exist alternatives that cannot be detected by these testing procedures. Therefore, our research focuses on constructing consistent conditional moment tests.
There is an abundance of literature on constructing conditional moment tests. The non-parametric tests employ a nonnon-parametric estimator to a non-parametric conditional moment function that equals zero under the null. See, for example, Eubank and Spiegelman (1990), Lewbel (1995), Hong and White (1995), Fan and Li (1996), Li and Wang (1998), Chen and Fan (1999), Zheng (1998a, 1998b, 2000), Horowitz and Spokoiny (2001), Delgado and Gonz´alez Manteiga (2001), Li, Hsiao, and Zinn (2003) and Tripathi and Kitamura (2003), among others. However, the test statistics of nonparametric tests are subjective in choos-ing smoothchoos-ing parameters and could be computationally costly. Another approach of the conditional moment test is based on infinitely many unconditional moment functions with uncountably many weighted functions indexed by continuous nuisance parameters (Stich-combe and White, 1998). Therefore, a consistent conditional moment test can be obtained by checking these orthogonality functions. Bierens (1982, 1984, 1990), de Jong (1996), Bierens and Ploberger (1997), and Bierens and Ginther (2001) choose the exponential function while Stute (1997), Stute, Thies and Zhu (1998), Koul and Stute (1999), and Stute and Zhu (2002) use the indicator function as the weight functions. The former tests are henceforth called the Bierens test while the latter tests are the the Stute test.
It is noted that both the Bierens and the Stute tests are not asymptotically pivotal in general; that is their limiting distributions depend on model characteristics and critical val-ues cannot be tabulated. For example, the auxiliary nuisance parameters in the exponential weight function of the Bierens test lead to the limiting distributions depending on the data generating process. Although Bierens and Ploberger (1997) derived case-independent upper bounds of the critical values to solve this problem, their test may be too conservative in practice. The Stute test is also not asymptotically pivotal because of the estimation effects (Durbin, 1973) and it is case dependent. Stute, Gonz´alez Manteiga and Presedo Quind¨ımil
(1998), Whang (2000, 2001, 2004), and Dominguez and Lobato (2006) try to avoid the prob-lem by using the bootstrap to approximate the limiting distributions. Stute, Thies and Zhu (1998), Koul and Stute (1999) and Stute and Zhu (2002) employe the martingale transforma-tion of Khmaladze (1981) to obtain asymptotically distributransforma-tion-free test statistics. However, due to the difficulty of having a proper definition of a multitime parameter martingales, these tests, except for Song (2007), cannot be carried out with multivariate regressors. Note that in Song (2007), the nonparametric estimation of the conditional moment function is required but a high dimensional nonparametric estimation is very complicated to compute. Kuan and Lin (2008) propose a test which centers the estimation effect out by an average of empirical processes and thus obtained an asymptotically pivotal test. Their test can be applied to multivariate regressor cases and the limiting distribution is a sup-norm of a multi-parameter Kiefer type process.
This paper proposes a consistent conditional moment test by checking an infinite set of unconditional moment conditions with indicator weight functions. The test statistic is based on the subsampling marked empirical process with sample size b instead of the whole size n such that b < n. The subsampling, investigated by Politis and Romano (1994) and Poli-tis, Romano and Wolf (1999) is a method of estimating the distribution of an estimator or test statistic by drawing subsamples from the original data. Existing studies in the litera-ture include Andrews and Guggenberger (2005), Chernozhukov and Fern´andes-Val (2005), Guggenberger and Wolf (2004), Hong and Scaillet (2006), Linton, Massoumi and Whang (2005) and Whang (2004). The subsampling method has not been used to construct the test statistics, which is done in this article. Other advantages of this paper are: (i) Instead of computing the sample average of the conditional moment function with the whole sample, the test statistic is obtained by the subsampling marked empirical process. The estimation effect disappears when the relative sample size of subsampling to that of the whole sample is zero asymptotically. Therefore, the proposed test does not suffer from the Durbin problem and is asymptotically pivotal. (ii) The present paper differs from the Stute test in that they use a martingale transformation technique which is applied to the model with a univariate regressor, and the test subsamples the marked empirical processes and can be applied to models with multivariate regressors. (iii) Unlike the tests using nonparametric smoothing methods, the proposed test therein avoids the user-chosen smoothing methodology. (iv) The test statistic can be computed using any √n-consistent estimator and different estimation methods can be applied. (v) The proposed test does not use the bootstrap, the martingale transformation, or the nonparametric method, resulting in significant simplifications in com-puting the test statistics. One disadvantage of our test is that it is powerful against local alternatives with rates b−1/2, but the proposed test is incapable of detecting local
alterna-tives at rate n−1/2. Our test shares the same disadvantage of most nonparametric tests. The Monte Carlo simulations show that the proposed test has good finite sample performances and the test is robust with respect to different values of b.
This paper is arranged as follows. Section 2 presents the conditional moment restriction and the proposed test. Section 3 shows the consistency of the test and the asymptotic behavior under different local alternatives. Section 4 shows the Monte carlo simulation results and follows the conclusions in section 5. All proofs are given in the Appendix.
2
A New Test
2.1 Conditional Moment Restrictions Consider general conditional moment restrictions
IE[m(y, x, βo) x] = 0, (1)
where IE[·|x] denotes the expectation conditional on the information set of x and m(·) is a function on data, and {y, x} is a sequence of random variables with x = (x1, · · · , xk)0 and
parameters β ∈ B with B ∈ Rk. The conditional moment restrictions can be obtained from existing models such as the parametric nonlinear regression models where m(y, x, βo) is the difference between y and g(x0, β), with g(·) a nonlinear function. To test the condition moment restrictions, the null and alternative hypotheses are as follows. The null hypothesis is the conditional moment function being equal to zero:
H0: IP IE(m(y, x, βo)
x) = 0 = 1, for some βo∈ B,
against the alternative hypothesis is, for all β ∈ B, IE(m(y, x, β) x) 6= 0 with a positive probability:
H1: IP IE(m(y, x, β)
x) = 0 < 1, for all β ∈ B, with B ∈ Rk a compact set.
As has been shown by Stinchcombe and White (1998), the conditional moment condition (1) equals infinitely many unconditional moment functions
IE[m(y, x, βo)ω(x, ξ)] = 0, ∀ξ ∈ Rk, (2)
where ω(·) is an infinite set of weights indexed by continuous parameters ξ and ω(·) may be any analytic function that is not polynomial. Therefore, testing (2) constructs a consistent conditional moment test. For example, Bierens (1982, 1984, 1990), de Jong (1996) and Bierens
and Ploberger (1997) and Bierens and Ginther (2001) use the exponential weighted function ω(x, ξ) = exp(x0ξ) for their integrated conditional moment test. Stute (1997), Stute, Thies and Zhu (1998), Koul and Stute (1999) and Stute and Zhu (2002) take the indicator function
ω(x, ξ) = 11{x≤ξ}:= 11{x1≤ξ1}· · · 11{xk≤ξk},
where 11A denotes the indicator function of even A. The present paper implements the
indi-cator function and the conditional moment restrictions (1) can be rewritten by the infinitely many unconditional moment functions as follows:
IE[m(y, x, βo)11{x≤ξ}] = 0, ∀ξ = (ξ1, · · · , ξk)0 ∈ Rk. (3)
The above moment functions allow for multivariate regressors.
2.2 Test Statistics
The specification test considered in this paper examines infinitely many unconditional mo-ment functions (3) that are equivalent to the conditional momo-ment restriction (1) and therefore it is a consistent conditional moment test. To test the moment function IE[m(y, x, βo)11{x≤ξ}]
being equal to zero, it is natural to consider the normalized sample average of the moment function: Mn(ξ; βo) := 1 √ n n X t=1 m yi, xi, βo 11{xi≤ξ},
with {yi, xi}ni=1 a sequence of random variable, and 11{xi≤ξ} = 1{xi1≤ξ1}· · · 1{xik≤ξk}. The
function m yi, xi, β 11{xi≤ξ}is considered the marked empirical process with marks given by
the moment function m. The function Mn is the average of the market empirical processes
with sample size n. Our objective is to test that Mn(ξ; βo) is close to zero or not. If
Mn(xi; βo) is close to zero, then we do not reject the null hypothesis; otherwise, we reject
the null hypothesis and conclude that the conditional moment restriction does not hold.
Since the true parameter βo is unknown, we replace βo by its consistent estimator, ˆβn, and the sample average of the marked empirical processes becomes
Mn(ξ; ˆβn) = 1 √ n n X i=1 m(yi, xi, ˆβn)11{xi≤ξ}.
Now, rewrite the process Mn:
Mn(ξ; ˆβn) = Mn(ξ; βo) + 1 √ n n X i=1 m(yi, xi, ˆβn) − m(yi, xi, βo) 11{xi≤ξ}.
If m(yi, xi, β) is once differentiable with first derivative ∇βm(yi, xi, βo), then Mn(ξ; ˆβn) = Mn(ξ; βo) + 1 √ n n X i=1 ∇βm(yi, xi, βo)( ˆβn− βo)11{xi≤ξ}+ op(1) = Mn(ξ; βo) + √ n( ˆβn− βo)1 n n X i=1 ∇βm(yi, xi, βo)11{xi≤ξ}+ op(1).
It is seen that Mn(ξ; ˆβn) and Mn(ξ; βo) are not asymptotically equivalent due to the
presence of the second term on the right hand side of the second equality. This term is the estimation effect discussed in Durbin (1973). The second term depends on a model characteristic that makes the test based on Mn(ξ; ˆβn) not asymptotically pivotal; this is
the well know Durbin’s problem. To eliminate the estimation effect, Stute, Thies and Zhu (1998), Koul and Stute (1999), and Stute and Zhu (2002) use the martingale transformation of Khmaladze (1981). However, due to the difficulty of having a proper definition of a multitime parameter martingales, these tests are used for univariate x; see also Bai (2003). Song (2007) extends the research to multiparameter processes but because a nonparametric estimation of the conditional moment restriction is required, his test is complicated to compute.
This paper considers a subsampling version of the Mnprocess. Instead of using the whole
sample of data with sample size n to compute the process, we use a subsample of data with sample size b to compute the sample average and construct the following process, for b < n:
Mb(ξ; ˆβn) := 1 √ b b X i=1 m yi, xi, ˆβn 11{xi≤ξ},
where ˆβncan be any√n-estimator associated with the model of interest by the whole sample. Mb permits the following expansion:
Mb(ξ; ˆβn) = Mb(ξ; βo) + 1 √ b b X i=1 ∇βm(yi, xi, βo)( ˆβn− βo)11{xi≤ξ}+ op(1) = Mb(ξ; βo) + r b n √ n( ˆβn− βo) " 1 b b X i=1 ∇βm(yi, xi, βo)11{xi≤ξ} # + op(1).
It is interesting to see that if b → ∞, n → ∞ and b/n → 0, and there are some regularity conditions, then the second term on the right-hand-side of the second equality of the above equality converges to zero. Thus, Mb(ξ; ˆβn) and Mb(ξ; βo) are asymptotically equivalent.
Subsampling the marked empirical process annihilates the estimation effect. Kuan and Lin (2008) considers a partial sum of the data and eliminate the estimation effect by centering a sequentially marked empirical process. Let D(Rk) be the space of the cadlag function on Rk endowed with the Skorohod topology. Here, Mb is in D(Rk). In what follows, let
⇒ denote the convergence in distribution, and −→ denote the convergence in probability.p The following assumptions are sufficient for the weak convergence of the subsampling marked empirical processes.
[A1] {yi, xi}ni=1 is ergodic and strictly stationary where xi has the continuous distribution
function F and the density function is f .
[A2] (i) IE[m(yi, xi, β)2|xi] < ∞,
(ii) IEm(yi, xi, β)4= κ < ∞,
(iii) IE[m(yi, xi, β)4||xi||1+η] < ∞, for some η > 0.
[A3] The conditional density fxi|Fi−1 is bounded and continuous, where Fi−1is the σ algebra
generated by x1, · · · , xi−1.
[A4] m(·) is once continuously differentiable in a neighborhood βo and satisfies
IE " sup β∈Bo |∇βm(yi, xi, β)| # < ∞,
where Bo denotes a neighborhood of βo.
[A5] ˆβn is a√n-consistent estimator; that is √n ˆβn− βo = Op(1).
Assumption [A1] permits data with weak dependence. Assumptions in [A2] restrict the dependence of the moment function. Given [A2] (i), the conditional variance function σ2(xi)
of m(yi, xi, β) is defined with
σ2(u) := varm(yi, xi, β)
xi = u].
For ξ = (ξ1, · · · , ξk)0 and u = (u1, · · · , uk)0, define
V (ξ) := IEσ2(xi)11{xi≤ξ} = Z ξ −∞ σ2(u)Fx(du), with Rξ −∞ := Rξ1 −∞· · · Rξk
−∞. Following Koul and Stute (1999), assumptions [A2](ii) and (iii)
together with [A3] are required to obtain the uniform tightness in the space D[−∞, ∞]. Assumption [A4] is a standard smoothness assumption. [A4] can be relaxed to non-smooth moment function when considering the stochastic equicontiunity of m. Assumption [A5] is weak and could be applied to most existing estimation method. In the following, we obtain the weak convergence of Mb.
Theorem 2.1. Under H0 and assumptions [A1]-[A5], if b → ∞, n → ∞ and b/n → 0, then
one has:
Mb(ξ; ˆβ) ⇒ B V (ξ),
The limiting distribution of Mb is a centered Gaussian process which is a multi-parameter
Brownian motion process on [0, 1]k with covariance function
V (ξ1∧ ξ2) = Z ξ1∧ξ2 −∞ σ2(u)F (du), where Rξ1∧ξ2 −∞ = Rξ11∧ξ21 −∞ · · · Rξ1k∧ξ2k
−∞ . In particular, when xi is a univariate, the process B
is the standard Brownian motion. The limit of Mb(ξ; βo) and that of Mb(ξ; ˆβn) are the
same and the Durbin problem disappears because of the convergence rate of b to infinity is slower that that of n. In addition, it is seen that V (ξ) plays an important in the test. Since V (ξ) still depends on the distribution of xi and σ2, the process Mb(ξ; ˆβn) is not
asymptotically distribution free. When σ2(xi) = σ20 (the conditional homoskedasticity case),
which is a constant, we obtain V (ξ) = σ20F (ξ). We follow Koul and Stute (1999) to consider an estimator of V (ξ): ˆ Vn(ξ) = 1 n n X i=1 m2(yi, xi, β)11{xi≤ξ}, ξ ∈ R k.
Tests for H0can be based on an appropriately scaling of Mb. Consider a consistent estimator
ˆ
σb2 for σ20 in homoskedasticity case. We have the “scale invariant”version of subsampling marked empirical processes:
˜ Mb(ξ; ˆβn) := 1 √ bσˆ −1 b b X i=1 m yi, xi, ˆβn 11{xi≤ξ}.
Theorem 2.2. Under H0 and assumptions [A1]-[A5], if b → ∞, n → ∞, b/n → 0 and
ˆ σ2
b → σ20, then one has
˜
Mb(ξ; ˆβn) ⇒ B F (ξ),
with B(·) a standard Brownian sheet.
The computational counterpart of the scaled invariant version of ˜Mb(ξ; ˆβ) is considered
as follows: ˜ Mb(xj; ˆβn) := 1 √ bσˆ −1 b b X i=1 m yi, xi, ˆβn 11{xi≤xj}, j = 1, · · · , n,
where each realization xj is used as a ξ in the indicator function. Consider two goodness-of-fit
statistics, the Kolmogorov-Smirnov and Cramer-von Mises test statistics:
KSn= sup xj∈Rk ˜Mb(xj; ˆβn) ,
CMn= 1 n n X j=1 ˜ Mb(xj; ˆβn)2.
By Theorem 2.2 and the continuous mapping theorem, for large n, one has, with ω ∈ [0, 1]k
KSn⇒ sup ξ∈Rk B F (ξ) = sup 0≤ω≤1 B(ω) , and CMn= Z ∞ −∞ ˜ Mb(ξ; ˆβn)2F (dξ) ⇒ Z ∞ −∞ B F (ξ)2 F (dξ) = Z 1 0 B(ω)2dω.
The critical values of the test statistics KSn and CMn can be found in existing literature;
see the book of Shorack and Wellner (1986). It is interesting to note that the proposed test is asymptotically pivotal and the limiting distribution of the proposed test does not depend on a data generating process. Therefore, we have the following corollary.
Corollary 2.3. Under all the assumptions in Theorem 2.2.
KSn⇒ sup 0≤ω≤1 B(ω) , CMn⇒ Z 1 0 B(ω)2dω,
with B(·) the standard Brownian sheet.
3
Power of the Tests
To investigate the power performance of the proposed test, two types of alternatives are considered. One is the general type of alternatives:
H1: IE[m(y, x, βo)|x] = µ(x) 6= 0, ∀ ξ = (ξ1, · · · , ξk) ∈ Rk,
and the other is the local alternatives:
H1L: IEm(y, x, βo) x = δ(x)
√ b ,
with δ(x) 6= 0. We then have the following theorem.
Theorem 3.1. Assume assumptions [A1]-[A5] hold. Assume also b → ∞, n → ∞ and b/n → 0. Therefore:
(i) Under the fixed alternative H1:
˜
(ii) Under the local alternatives H1L:
˜
Mb(ξ; ˆβn) ⇒ B(F (ξ)) + σ−10 IE[δ(xi)11{xi≤ξ}].
By Theorem 3.1 and the continuous mapping theorem, we additionally have the following corollary.
Corollary 3.2. Assume assumptions [A1]-[A5] hold. Assume also b → ∞, n → ∞ and b/n → 0. Therefore:
(i) Under the fixed alternative H1:
KSn→ ∞,
CMn→ ∞.
(ii) Under the local alternatives H1L:
KSn⇒ sup ξ∈Rk B(F (ξ)) + σ−10 IE[δ(xi)11{x i≤ξ}] CMn⇒ Z 1 0 B(F (ξ)) + σ−10 IE[δ(xi)11{xi≤ξ}] 2 dω.
The first part of Corollary 3.2 gives the consistency of the proposed test and the second part implies that the proposed test has nontrivial powers against local alternatives at rate b−1/2. However, there may exist local alternatives at rate n−1/2 as follows.
H2L: IEm(y, x, βo) x = δ(x)
√ n .
We have Theorem 3.3.
Theorem 3.3. Assume assumptions [A1]-[A5] hold. Assume also b → ∞, n → ∞ and b/n → 0. Under the local alternatives HL
2:
˜
Mb(ξ; ˆβn) ⇒ B(F (ξ)).
Under the local alternatives H2L, the limiting distribution of ˜Mb(ξ; ˆβn) is the same as
that under the null hypothesis; see Theorem 2.1. The proposed test is incapable of detecting local alternatives at rate n−1/2. Therefore, our test shares the same disadvantage of most nonparametric tests; see the discussion in Whang (2000) therein. The proposed test has no local power against this type of alternatives. Thus our test cannot be used in this scenario.
4
Monte Carlo Simulations
This section reports some simulation results to examine the finite sample performance of the test statistic KSn. We consider the following null data generating processes (DGPs).
(A) yi = xi1+ ei, (B) yi = xi1+ 5 + ei, (C) yi = xi1+ exp(zi) + ei, (D) yi = xi2+ xi3+ ei, (E) yi = xi2+ xi3+ 5 + ei, (F) yi = xi2+ xi3+ exp(zi) + ei.
Here xi1, xi2, xi3 and zi are independent and identically distributed (i.i.d.) N (0, 1)
distribu-tion and ei is i.i.d. N (0, σ02) with σ20 = 1, 2, 3, 4. The test statistic KSn for one regressor
is: KS1 = max j 1 √ bσˆ −1 1 b X i=1 (yi− x0i1βˆ1)11{xi1≤xj} ,
for DGPs (A), (B) and (C) and a statistic for two regressors:
KS2 = max j 1 √ bσˆ −1 2 b X i=1 (yi− x0i2βˆ2− x0i3βˆ3)11{xi2≤xj1}11{xi3≤xj2} ,
for DGPs (D), (E) and (F) where ˆβ1, ˆβ2, ˆβ3 are least square estimates, ˆσ21 = b−1
Pb
i=1(yi−
x0i1βˆ1)2 and ˆσ22= b−1
Pb
i=1(yi− x0i2βˆ2− x0i3βˆ3)2. In each simulation experiment, the number
of replications is 2000. The significance level is 0.05. We choose different values of b in this simulation. The choice of b is considered for the formula b = np with p = 0.5, 0.55, · · · , 0.95.
Table 1, given σ02 = 1, reports the rejection frequencies of the tests for different values of n and p. For DGPs (A) and (D), the rejection values are finite sample sizes of the test. In the column of DGP (A), all values are close to the significance level 0.05 except at the values of p = 0.5. However, in the column of DGP (D), the proposed test is under-sized for large p. We can see that when the number of regressors of the regression increases, or if b increases, then the sizes of the test are lower. For DGPs (B), (C), (E) and (F), the rejection rates are the finite sample powers of the proposed test. In columns of DGPs (B) and (E) that have fixed alternatives, the finite sample powers are 1 showing that the test has good power performances in different values of n and b(orp). In addition, the values on columns of
DGPs (C) and (F) present that the test performs well when the alternatives are a random variable. For DGP (C), there are good power performances of the test for large values of p. When n increases, the powers of the test are closer to 1. For DGP (F), powers are lower for n = 100, and as n and b(orp) increase, the powers increase. To sum up, the test has correct sizes for one regressor and is slightly under-sized for two regressors in the regression. When there are fixed alternatives, the power performances are very good. The powers of the test increase along with both n and b. Table 2 reports the rejection frequencies of the test for six DGPs with different σ02 and p. The sample size is 500. The finite sample performances in Table 2 are similar to those in Table 1. Moreover, we find that when the variety of error term increases, the powers of the test decrease.
5
Conclusions
This paper proposes a consistent conditional moment test based on infinitely many uncondi-tional moment conditions. The test statistic is a subsampling marked empirical processes and the Durbin problem is eliminated as the convergence rate of the subsampling size is slower than that of the whole sample size. We thus obtain an asymptotically pivotal test. The proposed test is consistent against a general type of alternatives and is powerful against local alternatives at rates b−1/2. However, our test is not powerful against n−1/2 local alternatives. In addition, the test performs well in finite sample simulations and the power performances are good with most values of b.
Table 1: Rejection frequencies of the conditional moment tests KS1 KS2 n p (A) (B) (C) (D) (E) (F) 100 0.50 0.076 1.000 0.742 0.057 1.000 0.596 0.55 0.067 1.000 0.817 0.043 1.000 0.661 0.60 0.057 1.000 0.877 0.037 1.000 0.775 0.65 0.048 1.000 0.947 0.038 1.000 0.847 0.70 0.044 1.000 0.977 0.036 1.000 0.919 0.75 0.047 1.000 0.992 0.031 1.000 0.968 0.80 0.049 1.000 0.999 0.024 1.000 0.985 0.85 0.042 1.000 1.000 0.021 1.000 0.995 0.90 0.046 1.000 0.999 0.025 1.000 0.999 0.95 0.041 1.000 1.000 0.018 1.000 0.999 200 0.50 0.063 1.000 0.863 0.046 1.000 0.783 0.55 0.045 1.000 0.937 0.038 1.000 0.867 0.60 0.051 1.000 0.978 0.040 1.000 0.950 0.65 0.053 1.000 0.992 0.031 1.000 0.981 0.70 0.039 1.000 0.997 0.035 1.000 0.990 0.75 0.054 1.000 1.000 0.023 1.000 0.998 0.80 0.048 1.000 1.000 0.028 1.000 1.000 0.85 0.043 1.000 1.000 0.023 1.000 1.000 0.90 0.042 1.000 1.000 0.025 1.000 1.000 0.95 0.049 1.000 1.000 0.022 1.000 1.000 500 0.50 0.068 1.000 0.964 0.057 1.000 0.950 0.55 0.042 1.000 0.989 0.030 1.000 0.982 0.60 0.047 1.000 0.998 0.035 1.000 0.994 0.65 0.044 1.000 0.999 0.036 1.000 0.998 0.70 0.045 1.000 1.000 0.040 1.000 0.999 0.75 0.040 1.000 1.000 0.030 1.000 1.000 0.80 0.055 1.000 1.000 0.028 1.000 1.000 0.85 0.040 1.000 1.000 0.029 1.000 1.000 0.90 0.036 1.000 1.000 0.019 1.000 1.000 0.95 0.035 1.000 1.000 0.028 1.000 1.000 Note: The significant level is 0.05. b = np. The values in the
3rd and 6th columns are the finite sample sizes and the values in the 4th, 5th, 7th and 8th columns are the finite sample powers of the proposed test.
Table 2: Rejection frequencies of the conditional moment tests KS1 KS2 σ02 p (A) (B) (C) (D) (E) (F) 2 0.50 0.058 1.000 0.906 0.037 1.000 0.862 0.55 0.053 1.000 0.972 0.041 1.000 0.955 0.60 0.043 1.000 0.995 0.036 1.000 0.984 0.65 0.044 1.000 0.999 0.037 1.000 0.998 0.70 0.049 1.000 1.000 0.039 1.000 0.999 0.75 0.044 1.000 1.000 0.029 1.000 1.000 0.80 0.052 1.000 1.000 0.030 1.000 1.000 0.85 0.040 1.000 1.000 0.033 1.000 1.000 0.90 0.040 1.000 1.000 0.022 1.000 1.000 0.95 0.036 1.000 1.000 0.023 1.000 1.000 3 0.50 0.060 1.000 0.843 0.050 1.000 0.797 0.55 0.050 1.000 0.942 0.039 1.000 0.915 0.60 0.054 1.000 0.987 0.037 1.000 0.972 0.65 0.051 1.000 0.999 0.039 1.000 0.994 0.70 0.048 1.000 1.000 0.034 1.000 0.999 0.75 0.037 1.000 1.000 0.033 1.000 1.000 0.80 0.046 1.000 1.000 0.038 1.000 1.000 0.85 0.052 1.000 1.000 0.030 1.000 1.000 0.90 0.040 1.000 1.000 0.023 1.000 1.000 0.95 0.041 1.000 1.000 0.027 1.000 1.000 4 0.50 0.048 1.000 0.776 0.042 1.000 0.703 0.55 0.053 1.000 0.897 0.042 1.000 0.845 0.60 0.044 1.000 0.969 0.037 1.000 0.949 0.65 0.042 1.000 0.992 0.031 1.000 0.988 0.70 0.048 1.000 0.999 0.033 1.000 0.999 0.75 0.046 1.000 1.000 0.035 1.000 1.000 0.80 0.053 1.000 1.000 0.038 1.000 1.000 0.85 0.047 1.000 1.000 0.027 1.000 1.000 0.90 0.043 1.000 1.000 0.024 1.000 1.000 0.95 0.042 1.000 1.000 0.029 1.000 1.000 Note: The significant level is 0.05. b = np. The values in the
3rd and 6th columns are the finite sample sizes and the values in the 4th, 5th, 7th and 8th columns are the finite sample powers of the proposed test.
Appendix
Proof of Theorem 2.1. By assumption [A4], the subsampling marked empirical process Mb permits the Taylor expansion:
1 √ b b X i=1 m yi, xi, ˆβn 11{xi≤ξ} = √1 b b X i=1 m yi, xi, βo 11{xi≤ξ}+ 1 √ b b X i=1 ∇βm yi, xi, βo(ˆβn− βo)11{xi≤ξ}+ op(1).
Because b/n → 0 and assumption [A5],
√ b( ˆβn− βo) = r b n √ n( ˆβn− βo)−→ 0.p
In addition, by assumptions [A1] and [A4], H¨older’s inequality and ergodic theorem, we have the following law of large numbers of ergodic and stationary sequence:
1 b b X i=1 ∇βm yi, xi, βo 11{xi≤ξ} p −→ IE∇βm yi, xi, βo 11{xi≤ξ} , we then obtain 1 √ b b X i=1 ∇βm yi, xi, βo(ˆβn−βo)11{xi≤ξ}= " 1 b b X i=1 ∇βm yi, xi, βo 11{xi≤ξ} # √ b( ˆβn−βo)−→ 0.p Therefore, 1 √ b b X i=1 m yi, xi, ˆβ 11{xi≤ξ}= 1 √ b b X i=1 m yi, xi, βo 11{xi≤ξ}+ op(1).
Mb(ξ; ˆβn) and Mb(ξ; βo) are asymptotically equivalent. Estimating parameter β does not
affect the limiting distribution of the statistic and the Durbin problem does not appear.
The process Mb belongs to the Shorohod space D(Rk) and the weak convergence of
Mb(ξ; βo) in the space D(Rk) to a continuous limit is implies by the tightness of Mb and
the finite dimensional convergence of Mb(ξ; βo). In the following, we first follow Bickel
and Wichura (1971), Koul and Stute (1999) and Dom´ınguez and Lobato (2004) to show the tightness of Mb and then the weak convergence of Mb(ξ; βo). Define I1 = (s1, t1] =
×k
j=1(s1j, t1j], and I2 = (s2, t2] = ×kj=1(s2j, t2j] be two subsets in Rk. Then I1 and I2 are
neighbor subsets if and only if for some j∗ ∈ {1, 2, · · · , k}, (s1
j∗, t1j∗] 6= (s2j∗, tj2∗], ×kj6=j∗(s1j, t1j] =
×k
the process Mb indexed by a parameter in Rk has an associated process indexed by the
intervals as follows: for h = 1, 2,
Mb(Ih; β) := 1 √ b b X i=1 m(yi, xi; β)11{xi∈Ih} = 1 X e1=0 · · · 1 X ek=0 (−1)k−Pj=1,··· ,kejM b(sh1+ e1(th1− sh1), · · · , shk+ ek(thk− shk); β),
which is the increment of Mb around Ih. Let m(yi, xi; β) = mi. Following Bickel and Wichura
(1971, Theorem 3 and example II), if
IE Mb(I1; β)2, Mb(I2; β)2 = 1 b2IE " b X i=1 mi11{xi∈I1} #2" b X i=1 mi11{xi∈I2} #2 .
is bounded, then for any λ > 0 and γ > 1,
IP(Mb≥ λ) ≤ λ−4µ(I1∪ I2)γ,
with some measure µ. The above result asserts that the process Mb is tight.
Let Fi denote the natural filtration. Under H0 and assumption [A1], {mi11{xi≤τ }, Fi−1}
is a strictly stationary and ergodic martingale difference sequence. When a subindex appears once in the summation, the corresponding term is zero by the law of iterated expectation and the martingale difference property. Moreover, since I1 and I2 are disjoint sets, when
a subindex appears more than twice, the corresponding term is zero. Therefore, similar to Koul and Stute (1999) and Dom´ınguez and Lobato (2004),
IE Mb(I1; β)2, Mb(I2; β)2 = 1 b2IE b X i=1 m2i11{xi∈I1} i−1 X j=1 mj11{xj∈I2} 2 + 1 b2IE b X i=1 m2i11{xi∈I2} i−1 X j=1 mj11{xj∈I1} 2 .
The first and the second terms in the above equation are similar and the only difference is the indexing set Ih; we then focus on the first term. By assumption [A2](i),
1 b2 b X i=1 IE m2i11{xi∈I1} i−1 X j=1 mj11{xj∈I2} 2 = 1 b2 b X i=1 IE σ2(xi, Fi−1)11{xi∈I1} i−1 X j=1 mj11{xj∈I2} 2 = 1 b2 b X i=1 IE Z I1
σ2(u, Fi−1)fxi|Fi−1(u)du
i−1 X j=1 mj11{xj∈I2} 2 .
By Fubini’s Theorem, the above equation equals 1 b2 b X i=1 Z I1 IE
σ2(u, Fi−1)fxi|Fi−1(u)
i−1 X j=1 mj11{xj∈I2} 2 du.
Using Cauchy-Schwarz’s inequality, we have
1 b2 b X i=1 Z I1 IE
σ2(u, Fi−1)fxi|Fi−1(u)
i−1 X j=1 mj11{xj∈I2} 2 du ≤ 1 b2 b X i=1 Z I1 n IEσ2(u, F
i−1)fxi|Fi−1(u)
2o1/2 IE i−1 X j=1 mj11{xj∈I2} 4 1/2 du.
By Burkholder’s inequality and the moment inequality yield, with some constant C,
IE i−1 X j=1 mj11{xi∈I2} 4 ≤ C IE i−1 X j=1 m2j112{xi∈I2} 2
≤ C(i − 1)2IE(m4111{x1∈I2}).
It follows that 1 b2 b X i=1 IE m2i11{xi∈I1} i−1 X j=1 mj11{xj∈I2} 2 ≤ 1 b2 b X i=1 Z I1 n IEσ2(u, F
i−1)fxi|Fi−1(u)
2o1/2 C(i − 1)2IE(m4 111{x1∈I2}) 1/2 du = 1 b2C IE(m 4 111{x1∈I2}) 1/2 b X i=1 (i − 1) Z I1 n
IEσ2(u, Fi−1)fxi|Fi−1(u)
2o1/2 du.
IE(m4111{x1∈I2}) ≤ IE(m 4
1) which is bounded by assumption [A2] (ii). In addition, from
Koul and Stute (1999), R
I1{IE[σ 2(u)f
xi|Fi−1(u)]
2}1/2du is bounded by assumptions [A2](iii)
and [A3]; see more detail discuss therein. Therefore, under H0 and assumptions [A1]–[A3],
the process Mb is tight. Note that our assumption [A2] (ii) and (iii) are similar to the
assumption (A)(a) in Koul and Stute (1999). In Dom´ınguez and Lobato (2004), they use stricter conditions (see, [A7] and [A8]) and H¨older’s inequality to obtain the boundness of IE(m4
111{x1∈I1}).
Under assumptions [A1] and [A2] (i), and by a central limit theorem for the ergodic stationary martingale difference sequence, we have for any τ ∈ Rk,
For ξ1, ξ2 ∈ Rk, Cov Mb(ξ1; βo), Mb(ξ2; βo) = 1 b b X i=1 IEm(yi, xi; βo)211{xi≤ξ1}11{xi≤ξ2} p −→ Z ξ1∧ξ2 −∞ σ2(u) F (du) = V (ξ1∧ ξ2),
where the first equality holds by the property of martingale difference sequence. Since V (ξ) is nondecreasing and nonnegative, Mb admits a asymptotically distributed as B(V (ξ)), where
B(·) is a standard Brownian sheet. 2
Proof of Theorem 2.2. In this proof, we show that using a consistent estimator ˆσb2 to replace σ02does not affect the asymptotics of the scale invariant subsampling marked empirical process. Rewrite the process ˜Mb(ξ; ˆβn) :
1 √ bσˆ −1 b b X i=1 m yi, xi, ˆβn 11{xi≤ξ} = ˆσ−1b − σ−10 1 √ b b X i=1 m yi, xi, ˆβn 11{xi≤ξ}+ 1 √ bσ −1 0 b X i=1 m yi, xi, ˆβn 11{xi≤ξ}.
Since ˆσb−1− σ0−1= op(1), and by Theorem 2.1,
b−1/2 b X i=1 m yi, xi, ˆβn 11{xi≤ξ}= Op(1), then ˜ Mb(ξ; ˆβn) = 1 √ bσ −1 0 b X i=1 m yi, xi, ˆβn 11{xi≤ξ}+ op(1). Denote ˜ Mbo(ξ, β) := √1 bσ −1 0 b X i=1 m yi, xi, β 11{xi≤ξ}.
The processes ˜Mb(ξ; ˆβn) and ˜Mbo(ξ, ˆβn) have the same limiting distribution. In addition,
sim-ilar to the proof of Theorem 2.1, replacing βo by ˆβnin ˜Mbo does not affected the asymptotics
of ˜Mbo. It follows that
˜
and it suffices to focus on the limiting behavior of ˜Mbo(ξ; βo). The tightness of Mb can be
ob-tained in Theorem 2.1 as σ2o is continuous. Since {m yi, xi, β 11{xi≤ξ}, Fi−1} is a martingale
difference sequence, we then use the central limit theorem for ergodic and stationary martin-gale difference sequence to obtain the limiting distribution, which is a Gaussian process with zero mean and for ξ1, ξ2 ∈ Rk,
Cov ˜ Mbo(ξ1, βo), ˜Mbo(ξ2, βo) = 1 bσ −2 0 b X i=1 IEm(yi, xi; βo)211{xi≤ξ1}11{xi≤ξ2} p −→ Z ξ1∧ξ2 −∞ F (du) = F (ξ1∧ ξ2).
Hence, ˜Mb(ξ; ˆβn) ⇒ B(F (ξ)), with B a Brownian sheet. 2
Proof of Theorem 3.1. M˜b(ξ; ˆβn) and ˜Mbo(ξ; βo) are asymptotically equivalent from (4).
It suffices to discuss the limit of ˜Mbo(ξ; βo) under two different types of alternatives.
For part (i), rewrite ˜Mbo(ξ; βo):
1 √ bσ −1 0 b X i=1 m yi, xi, βo 11{xi≤ξ} = √1 bσ −1 0 b X i=1 m yi, xi, βo − µ(xi) 11{xi≤ξ}+ 1 √ bσ −1 0 b X i=1 µ(xi)11{xi≤ξ}.
Under H1 and assumptions [A1]–[A5], by the previous proofs, the first part of the above
equation converges to B(F (ξ)). In addition, if IE|µ(xi)11{xi≤ξ}| < ∞, the probability limit
of b−1/2σ−10 Pb i=1µ(xi)11{xi≤ξ} will be 1 √ bσ −1 0 b X i=1 IE[µ(xi)11{xi≤ξ}] = √ bσ−10 IE[µ(xi)11{xi≤ξ}]. As b → ∞, ˜Mbo(ξ; βo) → ∞. Thus ˜ Mb(ξ; ˆβn) → ∞.
For part (ii), rewrite ˜Mo
b(ξ; βo): 1 √ bσ −1 0 b X i=1 m yi, xi, βo 11{xi≤ξ} = √1 bσ −1 0 b X i=1 m yi, xi, βo − δ(x√i) b 11{xi≤ξ}+ 1 bσ −1 0 b X i=1 δ(xi)11{xi≤ξ}.
Under H1Land assumptions [A1]–[A5], by the previous proofs, the first part of the above equa-tion converges to B(F (ξ)). If IE|δ(xi)11{xi≤ξ}| < ∞, the probability limit of b
−1σ−1 0
Pb
i=1δ(xi)11{xi≤ξ}
will be σ0−1IE[δ(xi)11{xi≤ξ}]. Therefore, under H L
1, ˜Mb(ξ; ˆβn) converges to a Brownian sheet
process plus a non-zero constant term σ0−1IE[δ(xi)11{xi≤ξ}]. 2
Proof of Theorem 3.3. Similar to the proof of Theorem 3.1, rewrite ˜Mo
b(ξ; βo): 1 √ bσ −1 0 b X i=1 m yi, xi, βo 11{xi≤ξ} = √1 bσ −1 0 b X i=1 m yi, xi, βo − δ(x√i) n 11{xi≤ξ}+ 1 √ b√nσ −1 0 b X i=1 δ(xi)11{xi≤ξ}.
The probability limit of the second term on the right-hand-side of the above equation will be
1 √ b√nσ −1 0 b X i=1 δ(xi)11{xi≤ξ}= √ b √ n " σ−10 1 b b X i=1 δ(xi)11{xi≤ξ} # p −→ 0, with b/n → 0 and b−1Pb i=1δ(xi)11{xi≤ξ} p
−→ IE[δ(xi)11{xi≤ξ}]. Therefore, ˜Mb(ξ; ˆβn)
con-verges to a Brownian sheet process under both H0and H2L. 2
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