行政院國家科學委員會專題研究計畫 成果報告
對不相容假說的穩健性排序檢定
研究成果報告(精簡版)
計 畫 類 別 ： 個別型
計 畫 編 號 ： NSC 95-2415-H-004-009-
執 行 期 間 ： 95 年 08 月 01 日至 96 年 07 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：周恒章、葉憶婷
處 理 方 式 ： 本計畫可公開查詢
中 華 民 國 96 年 09 月 20 日
1
Introduction
Different economic or econometric theories usually suggest non-nested models in theoretical and empirical researches. Tests for non-nested hypotheses, henceforth the non-nested tests, are impor-tant because researchers are able to choose the true model from non-nested models by the tests. The pioneering works of Cox (1961, 1962), Atkinson (1970) and Pesaran and Deaton (1978) become available to comparing non-nested models. Several papers, such as Davidson and MacK-innon (1981), Fisher and McAleer (1981), Gourieroux, Monfort and Trognon (1983), Mizon and Richard (1986) and Vuong (1989), discuss the theoretical methods for non-nested tests. Many papers apply the non-nested tests in empirical applications; complete surveys can be found in Gourieroux and Monfort (1994) and McAleer (1995). It is often the case that empirical fact dis-plays non-Gaussian behavior, such as models with heavy tail or influential outliers. Most of the existing testing procedures are designed for model with Gaussian (normal) distribution and are not robust with respect to misspecification of error distribution.
Aguirre-Torres and Gallant (1983) and Hall (1985) have suggested non-nested tests that in-corporate M-estimators and base on classical testing procedure. Although M-estimator is a robust estimator in general, their tests using classical procedure are lack of robustness for model where the error distribution is assumed non-Gaussian. To the best of our knowledge, only Victoria-Feser (1997) constructs a robust non-nested test. She considers a Lagrange multiplier version of the Cox test and extends the optimal bounded influence parametric tests of Heritier and Ronchetti (1994) for testing non-nested hypotheses. Her test limits the influence of small contamination in the data and is robust to model deviations. In order to derive her test statistic, one must specify an explicit density function under the null hypothesis to obtain the log-likelihood function and maximum likelihood (ML) estimators of the model. The test is thus restrictive and strong when applying in practice. In addition, similar to the Cox test, the test statistic involves a very difficult integration problem such as the Cox test and are not easy to compute for applied theorists.
In this paper, we propose a robust testing procedure for the non-nested hypotheses. Sev-eral features are as follows. First, the proposed test extends the rank score test of Gutenbrunner
et al. (1993); this class of rank test plays an important role especially when the empirical
phe-nomenons are non-Gaussian. Second, the rank test statistic is based on the regression rankscore process that is computed from parametric linear programming method of quantile regression. Specifications of explicit density functions are not required. Also, we do not need to estimate ML estimators and only root-n consistent estimators of non-nested models can be used in the pro-posed test. Different from the test of Victorian-Feser (1997), it is easy to compute our statistics. In addition, the propsed test is easy to implement by existing software. Unlike the non-nested test in general use, the simulation or bootstrapping methods are not required. Third, we show that under
very weak assumptions, the proposed test statistic has asymptoticallyχ2distribution and the test
is asymptotic distribution-free. Fourth, the proposed test can be extended to test one model against several competing models. The choice of multiple model selection becomes available. Fifth, lo-cal powers of our robust rank score test are derived. Finally, Monte Carlo simulations results are provided and show that the proposed test has good finite sample performances. Comparing with the J test, the rank score tests is robust when the error term is non-Gaussian. Moreover, unlike the existing non-nested literature, our test is robust to the relative number of regressors in the two hypotheses.
This paper is organized as follows. In section 2, the rank score tests for non-nested hypotheses are proposed. We consider both single and multiple alternatives. Local alternatives of our test are discussed in section 3. Some Monte Carlo simulation results are presented in section 4. Section 5 is our conclusion of this paper.
2
Rank Score Tests
2.1 Motivations and Setup
Suppose that we want to choose between two linear models as follows:
H0: y = Xβββ + e0,
H1: y = Zγγγ + e1,
where the dependent variable y is an n × 1 matrix, explanatory variables X and Z are n × p and
n ×q matrices, and e0and e1are error terms, respectively. X and Z are two matrices which contain
different variables and the models of H0and H1are non-nested. To test non-nested hypotheses H0
and H1, we consider the following artificial nesting model:
y =(1 − λ)Xβββ + λZγγγ + e, (1)
where λ = 0 means that the null hypothesis is correct and λ = 1 means that the alternative hypothesis is correct. Under this artificial nesting model, we are able to reconsider the non-nested hypotheses as
H0:λ = 0,
H1:λ = 1.
Davidson and MacKinnon (1981) estimatedγγγ in (1) by its ML estimator ˜γγγ and then estimated
λ from model
y = Xβββ∗+λZ ˜γγγ + e, (2)
with error density e independently and identically distributed (i.i.d.) as normal distributions. Davidson and MacKinnon introduced the J test that uses the classical t statistic for ˆλ to test the non-nested hypotheses λ = 0. The J test is based on the classical testing procedure and is not robust to the misspecification of error density. For example, if the error distributions of the non-nested models are non-Gaussian, the J test may lead to incorrect inference.
To overcome the non-robustness problem discussed above, Victoria-Feser (1997) proposes a robust test using the optimal bounded influence parametric tests of Heritier and Ronchetti (1994). The test is to limit the influence of small contamination in the data. Victoria-Feser (1997) considers a Lagrange multiplier version of the Cox test and bounds the level influence function of the test. As one can see that her test bounds the effect of the outlier and is a first paper for robust test for non-nested hypotheses. It is however, in the context of ML method, complete density functions of the models should be specified. In addition, one needs to compute the ML estimator in her test. This makes her test statistic very complicated to compute (see p.722-723 for the computation of her test statistic). Her test is thus restrictive and not operational in practice.
2.2 A Robust Test
In this article, a robust testing procedure for non-nested hypotheses is proposed. The proposed test is based on the rank score test of Gutenbrunner et al. (1993), that tests the parametric hypothesis for quantile regression. For testing non-nested models in H0 and H1, we now apply the rank
score test to test λ = 0 in the artificial nesting model (2). When λ = 0, the restricted model becomes y = Xβββ + e, and a regression rankscore process ˆa(t), introduced by Gutenbrunner and Jureˇckov´a (1992) is obtained by solving from
ˆ
a(t) = arg max y0a X0a = (1 − t)X01n, a ∈ [0, 1]n
, (3)
with 1n an n × 1 vector of ones. It is noted that problem (3) is the dual problem of the objective
function of quantile regression in linear programming. ˆa(t) can be obtained easily from the exist-ing software since the quantile regression has been available in the standard toolbox of researcher’s desk.
with bounded variation. Integratingϕ(t) with respect to ˆai(t) from zero to one constructs ˆ bi = − Z 1 0 ϕ(t)d ˆai(t). (4)
Denote ˆb with i th element ˆbi to be a “score” vector and contains ˆbi, i = 1, · · · , n. ϕ(t) is also
called a score generating function that generates score ˆbi. The underlying idea of the rank score
test is to check whether scores ˆb are sufficiently close to zero. Intuitively, ˆb can be interpreted as functions of ranks of residualss from restricted regression model y = Xβββ∗ +e. Of course,
different score-generating functionsϕ(·) lead to different ˆb. Three commonly used score functions are Wilcoxon scores, normal scores and sign-median scores. We compare the power performances of our test by different score generating functions in Section 4.
Extending the test of Gutenbrunner et al. (1993), we propose a rank score test for Davidson and Mackinnon’s artificial nesting model (1) and define
S ˆβββ, ˆγγγ = √1
n Z ˆγγγ − ˜Z
0ˆ
b,
where ˆγγγ can be any consistent estimator of the restricted model, and ˜Z is the linear projection of
Z ˆγγγ on X: ˜ Z = X(X0X)−1X0Z ˆγγγ . It follows that S ˆβββ, ˆγγγ = √1 n MXZ ˆγγγ 0ˆ b,
where MX =I − X(X0X)−1X0. The rank score test for non-nested hypotheses is defined as:
R := S ˆβββ, ˆγγγ 0Vˆ−1S ˆβββ, ˆγγγ /A2(ϕ), where ˆV = n−1(MXZ ˆγγγ )0(MXZ ˆγγγ ), and A2(ϕ) = Z 1 0 ϕ(t) − Z 1 0 ϕ(t)dt 2 dt, (5)
for some score function ϕ(·). The proposed test statistic is only composed of data X and Z, an estimator ˆγγγ , and ˆb, and is easy to computed. In our test, we do not need to specify the complete density function and the estimating of ML estimator is not required. The proposed test is thus easy to implement. In the following, we show that the limiting distribution of the proposed test is chi-square distribution with one degree of freedom.
Let X = [X ˜Z] be an n × ( p + 1) matrix and {xi, i = 1, · · · , n} the i-th vector of X. Denote
ei, i = 1, · · · , n to be the ith element of error vector e and the conditional distribution functions of
error term ei conditional on information setF are denoted as Fei|F(·), i = 1, · · · , n. In addition,
Theorem 2.1. If (i) Fei|F(·), i = 1, · · · , n are i.i.d. and absolutely continuous with continuous
densities fei|F(·) uniformly bounded away from 0 and ∞. (ii) (a) x1 = 1n, with 1n an n × 1
vector of ones, (b) n−1X0X → D, a positive definite matrix, (c) n−1Pni =1||xi||4 = O(1), (d)
maxi =1,··· ,n||xi|| = O(n1/4/ log n). (iii) ˆγγγ is a consistent estimator of γγγβββ. (iv) ˆV → V :=
IE0[(MXZγγγβββ)0(MXZγγγβββ)]/n, a positive definite matrix. Under the null hypothesis,
R ⇒χ12.
Proof. Under the assumptions (i) and (ii), by the same arguments in Theorems 4.1 and 5.1 of
Gutenbrunner et al. (1993), we have
S ˆβββ, ˆγγγ = √1 n MXZ ˆγγγ 0ˆ b = √1 n MXZ ˆγγγ 0 b + op(1), where b = − Z 1 0 ϕ(t)d(τ − 1{yi−x0iβββτ<0}). In addition, rewrite 1 √ n MXZ ˆγγγ 0 b = √1 n MXZγγγβββ 0 b +√1 nMXZ ˆγγγ − γγγβββ 0 b = √1 n MXZγγγβββ 0 b + ˆγγγ − γγγβββ√1 n MXZ 0 b = √1 n MXZγγγβββ 0 b + op(1),
where the last equality holds because ˆγγγ − γγγβββ =op(1) and n−1/2 M XZ
0
b = Op(1). By central
limit theorem and under the null hypothesis, 1
√
n MXZ ˆγγγ
0
b ⇒ N 0, VA2(ϕ).
Therefore, under assumption (iii), one has
R ⇒χ12.
2.3 Multiple Alternatives
The testing procedure introduced in the aforementioned can be extended to the choice of multiple alternatives. Suppose that there are k different non-nested alternatives as follows:
H11: y = Z1γγγ1+e1, · · ·
where Z1, · · · , Zkare n × q1, · · · , n ×qkmatrices,γγγ1, · · · , γγγkare associated parameters,
respec-tively, and e1, · · · , ek represent error terms. To test H0against multiple alternatives H11, · · · , H1k,
we combine these non-nested hypotheses into an artificial nesting model as Davidson and McK-innon (1981) and McAleer (1983):
y = Xβββ∗+Wλλλ + e,
where W = Z1γγγˆ1, · · · , Zkγγγˆk, ˆγγγ1, ˆγγγ2, · · · , ˆγγγk are consistent estimators of γγγ1, γγγ2, · · · , γγγk,
respectively, andλλλ = (λ1, · · · , λk) contains k elements, e = (e1, · · · , en) is an error vector in this
model. We thus can test the multiple non-nested hypotheses by testing whether all elements ofλλλ significantly differ from zero or not: H0:λλλ = 0.
For the multiple non-nested alternative case, the test statistic is defined as
Rk := S0kVˆ −1 k Sk/A2(ϕ), where Sk = 1 √ n(MXW) 0ˆ b,
with ˆbi the i th element of ˆb, ˆVk =n−1(MXW)0(MXW), and ˆbi and A2(ϕ) are the same as defined
in (4) and (5). We have the following theorem.
Corollary 2.2. If (i) Fei|F, i = 1, · · · , n are i.i.d. and absolutely continuous with continuous
densities fei uniformly bounded away from 0 and ∞. (ii) (a) x
∗
1 = 1n, with 1n an n × 1
vec-tor of ones, (b) n−1
X∗0X∗ → D∗, a positive definite matrix, (c) n−1Pni =1||x ∗
i||4 = O(1), (d)
maxi =1,··· ,n||x∗i|| = O(n1/4/ log n), and (iii) ˆVk → Vk∗ := IE0[(MXZγγγβββ)0(MXZγγγβββ)]/n, Under
the null hypothesis, Rk ⇒χk2,
a chi-square distribution with k degree of freedom.
3
Local Powers of the Test
The local powers of our test are considered in this section. Pesaran (1982) and Ericsson (1983) have compared the local powers of non-nested tests. Consider a local alternative as
H1n :λn =
λ0
√ n.
Similar to (2), the resulting artificial nesting model is
y = Xβββ∗+λnZ ˆγγγ + e,
with e =(e1, · · · , en). As sample size n increases, the model converges to the null model.
Denote IE1nas the expectation under H1n. Under H1n,
IE1n1{yi−x0iβββτ<0}|F = IE1n1{ei<−√λ0nz0iγγγ }ˆ |F = Fei|F −√λ0 nziγγγˆ ,
and taking Taylor expansion at zero,
Fei|F −√λ0 nziγγγˆ =τ − λ 0 √ nz 0 iγγγˆ fei|F(0) + op(1). It follows that IE1n[bi|F ] = IE1n − Z 1 0 ϕ(t)dτ − 1{yi−x0iβββτ<0} F = Z 1 0 ϕ(t)dhτ − IE1n 1{yi−x0iβββτ<0} Fi = λ 0 √ nz 0 iγγγˆ Z 1 0 ϕ(t)d fei|F(0)
where the second equality holds by assuming the expectation and integration are interchangeable. If the conditional distribution are i.i.d., fe1|F(0) = fe|F(0),
IE1n[b|F ] = λ 0 √ nZ ˆγγγ Z 1 0 ϕ(t)d fe|F(0).
From the proof of Theorem 2.1, it is known that S ˆβββ, ˆγγγ = n−1/2 MXZγγγβββ
0 b + op(1). It follows that IE1n h S ˆβββ, ˆγγγ i= IE1n 1 √ n MXZγγγβββ 0 IE1n[b|F ] = λ 0 n IE MXZγγγβββ 0 Z ˆγγγ Z 1 0 ϕ(t)d fe|F(0) = λ 0 n ω(ϕ, f )IE1n h MXZγγγβββ 0 Zγγγβββi , with R01ϕ(t)d fe|F(0) = R 1
0 ϕ(t)d fe|F(0) = ω(ϕ, f ), where the third equality holds by ˆγγγ is a
consistent estimator ofγγγβββ.
Therefore, under H1n, S ˆβββ, ˆγγγ is asymptotically normal distributed with mean
λ 0 n ω(ϕ, f )IE1n h MXZγγγβββ 0 Zγγγβββ i
and variance VA2(ϕ). Under H
1n,R is noncentralχq2distribution with non-centrality parameter
λ 0 n 2ω2(ϕ, f ) A2(ϕ) IE1n h MXZγγγβββ 0 Zγγγβββi 0 V−1IE1n h MXZγγγβββ 0 Zγγγβββi .
4
Monte Carlo Simulations
We use Monte carlo simulations to investigate the finite sample performances and robustness of the proposed tests. The data generating process (DGP) is specified as follows. Given a weight
ω ∈ [0, 1],
y =(1 − ω)Xβββ + ωZγγγ + e, (6)
where X, Z are n × p, n × q random matrices i.i.d. from N(0, 1) except that x01=1n, andβββ and γγγ are p × 1, q × 1 vector of ones. The replication of each simulation is 3000 and we compute the
rejection probabilities to check the finite sample performances. Whenω = 0, the nesting model (6) becomes the null model of H0; the resulting rejection probabilities are the finite sample sizes of
our test. Whenω = 1, the nesting model becomes the alternative model of H1and we can obtain
the finite sample power. The nominal level is 5% in this section. Four scenarios are considered: the error terms e are i.i.d. from standard normal distribution, normal distribution with mean 0 and variance 4, the t distribution with 2 degree of freedom, and the standard cauchy distribution.
To examine the finite sample size, we first consider that the error term is i.i.d. drawn from
N(0, 1) and ω = 0. Different score generating functions, sign, Wilcoxon and normal, are
con-sidered in the simulation. Table 1 reports the rejection probabilities of the test with sample sizes
n = 50 and n = 300. In Table 1, the finite sample size is over-sized which is common in
non-nested J test; see Godfrey and Pesaran (1983) and Gourieroux and Monfort (1994). The fi-nite sample size is more accurate when q is small and is greater than the nominal level when q is large relative p. For example, when q = 2, finite sample sizes from p = 2 to p = 6 are 8.3%, 6.6%, 6.2%, 6.8%, 5.6% which are close to the nominal size. Moreover, when p = 4, finite sample sizes from q = 2 to q = 6 are 6.2%, 7.9%, 9.4%, 11.4%, 12.7% which become greater when q is large relative to p. For different score generating functions, the finite sample performances of our test are similar but the finite sample size is smaller with sign score generating function. In addition, the finite sample size is greater when the sample size n = 50 than the finite sample size when n = 300. This result comes from the asymptotic effect.
To examine the finite sample power, we show finite sample power functions of our test in Figures 1–2 with three different score generating functions. The error term is i.i.d. drawn from standard normal distribution. The sample sizes is 300. Figure 1 is power functions for non-nested
Table 1: Finite Sample Sizes, % n=50 n=300 sign p q=2 q=3 q=4 q=5 q=6 q=2 q=3 q=4 q=5 q=6 2 7.0 10.7 12.5 16.1 18.8 8.3 10.7 13.1 14.0 17.9 3 7.2 9.0 10.9 12.5 16.0 6.6 7.7 10.4 12.4 14.6 4 6.3 9.3 10.3 11.0 13.8 6.2 7.9 9.4 11.4 12.7 5 7.3 8.1 9.2 9.7 12.8 6.8 6.8 9.5 11.6 10.8 6 6.4 8.0 8.1 9.2 10.4 5.6 7.9 7.4 8.9 9.9 Wilcoxon p q=2 q=3 q=4 q=5 q=6 q=2 q=3 q=4 q=5 q=6 2 8.6 13.3 17.6 21.2 25.8 8.6 12.4 17.1 19.1 26.2 3 7.8 11.2 13.3 16.3 20.3 7.8 9.6 14.1 15.9 19.8 4 7.7 11.6 13.0 14.8 18.1 7.8 8.5 10.9 14.0 16.5 5 8.3 9.6 11.3 12.5 15.8 7.2 9.1 11.8 13.5 14.8 6 6.5 9.2 9.6 12.6 13.7 7.3 9.3 9.4 10.5 13.0 normal p q=2 q=3 q=4 q=5 q=6 q=2 q=3 q=4 q=5 q=6 2 8.5 13.7 17.5 21.3 27.1 8.7 12.2 17.5 20.3 27.5 3 7.1 10.9 13.4 15.7 20.9 8.2 10.3 13.8 17.3 20.4 4 7.1 11.2 12.5 14.2 17.9 7.7 9.1 12.0 14.0 17.3 5 7.5 8.8 10.7 11.9 15.4 6.9 9.1 11.5 14.1 15.0 6 5.7 8.8 9.1 12.7 12.0 7.3 8.8 9.8 10.8 13.9
models with p = q = 3. The horizontal axis isω and the vertical axis is the rejection probability. Whenω deviates from zero, the rejection probabilities are finite sample powers. From Figure 1, we can see that our test has good power performances in all three score generating functions. When ω is about ±0.25, the finite sample power approximates 1. The tests with Wilcoxon and normal score generating functions have almost the same power function. The test with sign score generating function has the most accurate finite sample sizes than the test with Wilcoxon and normal score generating functions. The finite sample power of the test with Wilcoxon and normal score generating functions is greater than the one of the test with sign score function. Figure 2 plots the power functions of our test for p = 5, q = 2 with Wilcoxon, normal and sign score generating functions. Our test has finite sample sizes which are close to the nominal size in all three score generating functions. Whenω = ±0.2, the finite sample power approximates 1 and thus the finite sample performances of the proposed test are very good. Similar to the case of
p = q = 3, finite sample powers of the test with Wilcoxon and normal score generating functions
are greater than the ones with sign functions. This result is intuitive since the error term is drawn from the normal distribution.
To examine the robustness of our test, we consider several non-standard scenarios and compare the power performances of our test and the J test. First, Figure 3 plots finite sample power functions of our test and the J test with the error term i.i.d. from N(0, 1). Figures 4–5 plot finite sample power functions of our test and the J test when the error term is i.i.d. drawn from N(0, 4). Both the tests are over-sized but our test has more accurate finite sample size than the one of the J test. Therefore, the over-sized problem is less serious in our test. In addition, powers of the J test are greater than powers of our test. Like Godfrey and Pesaran (1983), we adjust our test and the J test to see whether high power comes form high size. We find that our test has slightly high powers whenω 6= 0. This shows that high power performances of the J test come from high size. Finally, we can see that our test and the J test both perform better when the error term is from N(0, 1) than when the error term is from N(0, 4).
Figures 6, 7, 8 plot finite sample power functions of our test and the J test when the error term is i.i.d. drawn from the t2distribution for non-nested models with p = 2, q = 7 and p = q = 3
and p = 6, q = 3, respectively. In Figures 6 and 7, sizes of our test are 8.2% and 8.6% which are close to the nominal size. Sizes of the J test are 17.9% and 20.8% which are much greater than the nominal size. The power of our test approximates 1 when ω = ±0.3 and the power of the J test approximates 1 whenω are around 0.6 to ±0.8. This shows that our test has better power performances in cases when p = q = 3 and p = 6, q = 3. The power of our test approximates 1 more rapid than the power of the J test. In addition, in Figure 8, the p = 2, q = 7 case, the J test performs poorly when the number of regressors in the alternative hypothesis is large relative to the one in the null hypothesis. Our test is less sensitive to the relative number of regressors in
the two hypotheses. In addition, the J test is very sensitive to the relative number of regressors of the two non-nested hypotheses, while the rank score test is not. Therefore, our test is more robust than the J test.
Figures 9, 10, 11 plot finite sample power functions of our test and the J test when the error term is i.i.d. drawn from the standard Cauchy distribution for non-nested models with p = q = 3,
p = 6, q = 3,and p = 2, q = 7. In Figures 9 and 10, sizes of our test are 6.4% and 5.6% and sizes
of the J test are 28.6% and 29.0%. Our test has correct finite sample size while the finite sample size of the J test are badly distorted when the errors are not assumed to be normal. The powers of our test increase largely when ω deviates from 0 and our test has good power performances under Cauchy distribution. The powers of the J test increase very slowly whenω deviates from 0. Whenω = ±1, the powers of our test is double of the powers of the J test. In Figure 11, our test also has better power than the J test but the performance of the J test is poor. To sum up, when the DGP is non-Gaussian distribution, the small sample simulations show that our test is robust in two ways. First, our test is robust to different data generating process, especially the non-standard distribution. Second, our test is robust with respect to the number of the regressors of non-nested models.
5
Conclusion
Robust testing procedures of non-nested tests are desired in theoretical and empirical researches. Unlike the optimal bounded influence parametric test considered by Victoria-Feser (1997), we have suggested another robust test for non-nested tests by extending the regression rank score test of Gutenbrunner et al. (1993). We introduce a test statisticR for two non-nested hypotheses and
the statisticRk for multiple non-nested alternatives. The limiting distributions of our test
statis-tics in this paper are χ2distributions and our test is asymptotically distribution free. Moreover,
limiting distributions of the proposed test under the local alternative are non-centralχ2 distribu-tions. Monte Carlo simulations show that our test has good finite sample performances against non-nested alternatives. Our test is robust for testing non-nested hypotheses under non-Gaussian error terms and is robust to the relative number of regressors in the two hypotheses.
References
Aguirre-Torres, V. and R. Gallant (1983). The null and non-null asymptotic distribution of the Cox test for multivariate nonlinear regression: alternatives and a new distribution-free Cox test, Journal of Econometrics, 21, 5–33.
Atkinson, A. (1970). A method for discriminating between models, Journal of the Royal Statistical
Society, Series B, 32, 323–353.
Cox, D. (1961). Tests of separate families of hypotheses, in Fourth Berkeley Symposium on
Math-ematical Statistical Association, 1, 105–123.
(1962). Further results on tests of separate families of hypotheses, Journal of the Royal
Statistical Society: series B, 24, 406–424.
Davidson, R. and J. MacKinnon (1981). Several tests for model specification in the presence of alternative hypotheses, Econometrica, 49, 781–793.
Ericsson, N. R. (1983). Asymptotic properties of instrumental variables statistics for testing non-nested hypotheses, Review of Economic Studies, 50, 287–304.
Fisher, G. and M. McAleer (1981). Alternative procedures and associated tests of significance for non-nested hypotheses, Journal of Econometrics, 16, 103–119.
Godfrey, L. and M. Pesaran, (1983). Tests of non-nested regression models small sample adjust-ments and Monte Carlo evidence, Journal of Econometrics, 21, 133–154.
Gouri´eroux, C. and A. Monfort (1994). Testing non-nested hypotheses, in: R. Engle and D. Mcfadden, eds., Handbook of Econometrics, Vol. 4, Amsterdam: North–Holland.
Gouri´eroux, C., A. Monfort and A. Trognon (1983). Testing nested or non-nested hypotheses,
Journal of Econometrics, 21, 83–115.
Gutenbrunner, C. and J. Jureˇckov´a (1992). Regression rank scores and regression quantiles,
An-nals of Statistics, 20, 305–330.
Gutenbrunner, C., J. Jureˇckov´a, R. Koenker and S. Portnoy (1993). Tests of linear hypotheses based on regression rank scores, Nonparametric Statistics, 2, 307–331.
Hall, A. (1985). A simplified method of calculating the distribution free Cox test, Economic
Letters, 18, 149–151.
Heritier, S. and E. Rochetti (1994). Robust bounded-influence tests in general parametric models,
Journal of the American Statistical Association, 89, 897–904.
Koenker, R. and G. Bassett (1978). Regression quantile, Econometrica, 46, 33–50.
McAleer, M. (1983). Exact tests of a model against non-nested alternatives, Biometrika, 70, 285– 288.
(1995). The significance of testing empirical non-nested models, Journal of Econometrics,
67, 149–171.
non-nested hypotheses, Econometrica, 54, 657–678.
Pesaran, M. (1982). Comparison of local power of alternative tests of non-nested regression mod-els, Econometrica, 50, 1287–1305.
Pesaran, M. H. and A. S. Deaton (1978). Testing non-nested nonlinear regression models,
Econo-metrica, 46, 677–694.
Victoria-Feser, M.-P. (1997). A robust test for non-nested hypotheses, Journal of the Royal
Statis-tical Society, Series B, 59, 715–727.
Vuong, Q. (1989). Likelihood ratio tests for model selection and non-nested hypotheses,
wilcoxon normal
sign
Figure 1: Finite sample powers ofR with p = 3 and q = 3.
wilcoxon normal
sign
R
J
Figure 3: Finite sample powers under N(0, 1) distribution and p = 3 and q = 3.
J R
J R
Figure 5: Finite sample powers under N(0, 4) distribution and p = 6 and q = 3.
R J
R J
Figure 7: Finite sample powers under t2distribution and p = 6 and q = 3.
J R
R
J
Figure 9: Finite sample powers under Cauchy distribution and p = 3 and q = 3.
R
J
J
R
