【35 頁】

Cross-Sectional Effects of Common and Heterogeneous Regressors on Asymptotic Properties of Panel Autoregressive Unit Root Tests

Katsuto Tanaka
Faculty of Economics, Gakushuin University, Tokyo, Japan

The present paper deals with nonstationary panel autoregressive （AR） models, and examines cross-sectional effects of regressors on the asymptotic properties of panel unit root tests for the AR（1） coefficient. We consider various types of common and heterogeneous regressors and compute limiting local powers of tests as T →∞ for each N, where T and N are the time and cross section dimensions, respectively. Dealing with tests based on the ordinary least squares estimator （OLSE） and the generalized LSE（GLSE）, we examine how common and heterogeneous regressors affect the tests as N becomes large. It is shown that the existence of common regressors does not affect the tests asymptotically as N →∞. This means that the power of the tests remains the same even if the model contains common regressors. We further derive the limiting power envelopes of the most powerful invariant （MPI） tests, which yields the conclusion that the GLSE-based tests are asymptotically efficient, unlike the time series case.

Keywords Asymptotically efficient test, Common regressor, Cross-sectional effect, Heterogeneous regressor, Moment generating function, Numerical integration, Panel unit root tests.

Address correspondence to Katsuto Tanaka:

Faculty of Economics
Gakushuin University
Mejiro, Toshima-ku, Tokyo 171-8588
Japan
e-mail:

1. INTRODUCTION

Nonstationary panel AR models were extensively discussed in Moon and Perron （2008） and Moon, Perron, and Phillips（2007）, where the former deals with the case of heterogeneous intercepts, whereas the latter discusses the case of heterogeneous trends. In these papers, the limiting local powers of various panel AR unit root tests are computed as T and N jointly tend to ∞ under the local alternative that shrinks to the null at the rate of 1/（TN^κ）, where T is the time series dimension and N is the cross section 【36 頁】 dimension with 0 ＜ κ ＜ 1.
　　Unlike the above works, the present paper examines the effect of the cross section dimension N on the unit root tests as T →∞. This may be useful when T is bigger than N and it is desirable to see the intermediate situation rather than the final situation as both T and N go to ∞. We consider four types of regressors:（1） a common intercept and trend,（2） heterogeneous intercepts and a common trend,（3） a common intercept and heterogeneous trends, （4） heterogeneous intercepts and trends. For these models we conduct panel unit root tests based on the OLSE and GLSE of the AR coefficient, and some other tests based on these residuals. To see the cross-sectional effect, we compute limiting local powers of these unit root tests as T → ∞ for each intermediate N under the AR coefficient close to unity in the order of 1/T. It is theoretically and graphically shown that, as N becomes large, the existence of common regressors does not affect the asymptotic properties of these tests, although that of heterogeneous regressors does affect. This fact was also partly observed in panel AR models discussed in Breitung （2000） and Moon et al.（2007）. We give more detailed analysis of this fact for each intermediate value of N. We also derive the limiting powers of these tests and envelopes of the most powerful invariant （MPI） tests as N →∞, utilizing the joint moment generating functions （m.g.f.s） associated with the test statistics obtained in Nabeya and Tanaka（1990）, Tanaka（1996, Chap. 7）, and Tanaka（2017, Chap. 10）.
　　The outline of the paper is as follows. In Section 2 we present panel AR models to be dealt with in this paper. In Section 3 we compute limiting local powers of various unit root tests. In Section 3.1 we deal with OLSE-based tests, followed by GLSE-based tests in Section 3.2. The limiting power envelopes are derived in Section 3.3, and it is found that the GLSE-based tests are asymptotically efficient, unlike the time series case. The effect of temporal or cross-sectional dependence of the error term on the tests is discussed in Section 3.4. Section 4 concludes the paper. Proofs of theorems are provided in the Appendix.

2. PANEL AR MODELS

The panel AR models to be discussed in this paper are the following types:

where i refers to cross section, whereas t refers to time series. The process｛η_it ｝ is defined for all models by

【37 頁】

where it is assumed that｛η_it ｝ starts from η_i0 ＝0 for each i, and is driven by｛ε_it ｝. We initially assume｛ε_it ｝〜 i.i.d.（0, σ² ） for simplicity of presentation. The case of temporal or cross-sectional dependence will be discussed in Section 3.4.
Model A is the most restricted model with common intercept and trend. Model B has heterogeneous intercepts, whereas Model C has heterogeneous trends. Model D is the most unrestricted model with heterogeneous intercepts and trends. Note that these four models coincide with each other when N＝1.
　　For the above models we consider the panel AR unit root test

where we assume that, under H₁, ρ_i takes the following form:

with c ＞ 0 and 0 ＜ κ ＜ 1. This is a simple extension of the time series unit root test. A more general alternative allows the true value of ρ_i to be different among cross sections. Moon and Perron（2008） and Moon, Perron, and Phillips （2007） assume such an alternative, but we maintain （7） to simplify subsequent discussions.
　　Under the above setting we shall explore asymptotic properties of various unit root tests. For this purpose we define the Ornstein-Uhlenbeck（O-U） process by

where r ∈ ［0, 1］ and ｛W_i （r）｝ is the standard Brownian motion independent of ｛W_k（r）｝（i≠k） so that Y₁ （r）, . . . , Y_N （r） are i.i.d. for any r ∈［0, 1］.

3. LIMITING POWERS AND POWER ENVELOPES

We first compute the limiting local powers of various unit root tests for Models A through D. In Section 3.1 we deal with OLSE-based tests, followed by GLSE-based tests in Section 3.2. The limiting power envelopes of the MPI tests are derived in Section 3.3. The effect of temporal or cross-sectional dependence of the error term is discussed in Section 3.4.

3.1. OLSE-Based Tests

The present test was earlier considered in Moon et al. （2007）, and Moon and Perron （2008）. The limiting local power was also computed in these works as both T and N go to ∞ under a more general setting. Here we examine the cross-sectional effect of regressors as T →∞ for each N.

【38 頁】

Let _it^（M ）be the OLS residual obtained from Model M （M＝A, B, C, D）. Then we compute the estimator ^（M ） of ρ_i ＝ρ under H₀ by

where

　The following theorem describes the asymptotic distribution of ^（M ） as T → ∞ for each N, the proof of which is given in the Appendix.

Theorem 1. As T →∞ with N fixed under ρ_i ＝1−c_N / T , the asymptotic distribution of ^（M ） in Model M （M＝A, B, C, D） follows

where

【39 頁】

　Some remarks follow.

（a） When N＝1, that is, in the time series case, the distribution of Q_N^（M ） in （12） reduces to Q₁^（D ）＝U₁^（D ）/V₁^（D ） for all M. Note also that U₁^（A ）/V₁^（A ） corresponds to the popular near-unit root distribution associated with the time series model x_t＝ρx_t−1＋ε_t with ρ＝1−c/T .

（b） As N becomes large, it holds that

where the distribution of this last quantity is obtained from Model M without common regressors, which means that the effect of common regressors fades away as N becomes large.

（c） We can deal with some other variations of the above models, for which we can also consider the statistics T （^（・）−1）. For example, we can show that, as T →∞ with N fixed under ρ_i ＝1− c_N /T ,

Thus we also conclude that, for these models, the existence of common regressors does not affect the asymptotic behavior of the OLSE-based tests as N →∞, which was also described in（b）.

（d） U_i ^（C ） and V_i ^（C ） behave differently from the other quantities, which may be because U_i ^（C ）/V_i ^（C ） results from the restricted regression without intercept y_it ＝β_i t＋η_it . It can also be shown that U_i ^（M ） and V_i ^（M ） are uncorrelated under ρ＝1 for M＝B, D, but are correlated for M＝A, C. In fact, it holds that Cov（U_i ^（A ）, V_i ^（A ））＝1/3 and Cov（U_i ^（C ）, V_i ^（C ））＝1/175 when c_N＝0. These can be computed easily from the joint moment generating function（m.g.f.） described below.

To compute the distribution of Q_N^（M ） in （12） for each N, we use the joint m.g.f.m^（M ）（x, y ） of U_i ^（M ） and V_i ^（M ） defined by

【40 頁】

where, by putting μ＝ , we have［Tanaka（2017, Chap. 10）］

Then the distribution of Q_N^（M ） can be computed by using Imhof’s formula［Imhof （1961）］

Numerical computation like Simpson’s formula can be used to compute （14） by taking care of the computation of square roots of complex-valued quantities［Tanaka（1996）］.
　　Figure 1 draws the probability densities of Q_N^（B ） and Q_N^（D ） for various values of N under H₀（c_N＝0）to examine the cross-sectional effect of N. As Theorem 2 below indicates, these distributions converge to −3 and −15/2, respectively, as N becomes large. Note that Q₁^（B ）＝Q₁^（D ）. The distributions Q_N^（B ） for N ＞1 are shifted from Q₁^（D ）, whereas Q_N^（D ） for N ＞ 1 are just the convolution since P （Q_N^（D ）≤ z ）＝P（（zV_i ^（D ）−U_i ^（D ））≥0）, as is seen from（14）. The general feature of Q_N^（A ） and Q_N^（C ） are the same as Q_N^（B ）, although those densities are not presented here.

　　We next compute limiting powers of the tests based on Q_N^（M ） as N → ∞ under ρ＝1−c_N with c_N＝ 【41 頁】 c/N^κ. We need to find the limiting distribution of normalized Q_N^（M ） by suitably choosing κ. For this purpose, let us put

The joint m.g.f. m^（M ）（x, y ） of U_i ^（M ） and V_i ^（M ） shown above can be used to compute these moments using the Taylor expansion, as is shown in the Appendix. We have, by the week law of large numbers（WLLN）and the central limit theorem（CLT）,

【42 頁】

where

Then it is recognized that, for the asymptotic distribution of normalized Q_N^（M ） to be nondegenerate, c_N＝O（1/） when a₁b₀−a₀b₁≠0, and c_N＝O（1/N ^1/4） when a₁b₀−a₀b₁＝0 and a₂b₀−a₀b₂≠0. It is shown in the Appendix that c_N＝O（1/） for Q_N^（A ） and Q_N^（B ） , whereas c_N＝O（1/N ^1/4） for Q_N^（C ） and Q_N^（D ）, and we have the the following theorem.

Theorem 2. The limiting powers of the tests based on Q_N^（M ）（M＝A, B, C, D） under ρ＝1− c /（N^κT ） at the 100γ% level are given as follows:

where Φ（・） is the distribution function of N（0, 1）, and z_γ is the 100γ% point of N（0, 1）, whereas κ＝1/2 for Models A and B, and κ＝1/4 for Models C and D.

　　It follows that the OLSE-based unit root tests in Models A and B have nontrivial powers in a N ^−1/2T ⁻¹ neighborhood of unity, whereas the powers for Models C and D are nontrivial in a N ^−1/4T ⁻¹ neighborhood of unity. It is also seen that the limiting power decreases as the model complexity increases.
Figure 2 shows powers of the Q_N^（B ） - and Q_N^（D ） -tests against c＝c_N N^κ∈［0, 20］ at the 5% level for N＝1, 10, 100, ∞, where the powers for N ＜ ∞ are obtained from（14） by putting z at the 5% point of the null distribution of Q_N^（M ）, whereas those for N＝∞ are obtained from Theorem 2. It seems that the powers for N＝100 are still not well approximated by the limiting powers. This is particularly true of Model D. The powers of the Q_N^（B ） -test are higher than those of the Q_N^（D ） -test, which is also evident from Theorem 2. This means that the existence of heterogeneous trends decreases the power.

　　In the next subsection we consider the GLSE-based tests, which will be shown to be better than the

【43 頁】

OLSE-based tests.

3.2.GLSE-Based Tests

Let us express Model M （M＝A, B, C, D） as y ＝X ^（M ）γ^（M ）＋η, where X ^（M ） and γ^（M ） are the regression matrix and parameter vector in Model M, respectively, whereas

【44 頁】

Then we define the GLS residual by ^（M ）＝y − X ^{（M ）（M ）}, where

Here ⊗ is the Kronecker product and C is the T × T lower triangular matrix with （s, t ）-th element being 1 for s ≥ t and 0 otherwise. The GLSE ^（M ） of ρ can be computed following（9） with ^（M ） replaced by ^（M ）.
　　The following theorem describes the asymptotic distribution of ^（M ） as T → ∞ for each N, the proof of which is given in the Appendix.

Theorem 3. As T →∞ with N fixed under ρ＝1− c_N /T , the asymptotic distribution of ^（M ） for Model M （M ＝ A, B, C, D） follows

where

　　It is noticed that the distributional structure of the GLSE-based statistics R_N^（M ） remains the same as that of the OLSE-based statistics Q_N^（M ）. It is also seen that R_N^（A ） coincides with R_N^（B ）. The same is true of R_N^（C ） and R_N^（D ）, and these properties are also shared in the time series case［Tanaka（1996）］. The densities of R_N^（A ）（ ＝ R_N^（B ）） under H₀ are drawn at the top of Figure 3 for N＝1, 10, 30, whereas those of R_N^（C ）（＝R_N^（D ）） at the bottom for N＝1, 10, 50. The former densities are seen to be shifted from the latter as N becomes large. Both R_N^（A ） and R_N^（B ） converge to 0, whereas both R_N^（C ） and R_N^（D ） converge to −3, as is described in Theorem 4 below.

　　We next consider limiting powers of the tests based on R_N^（M ） as N → ∞, which is described in the following theorem, the proof of which is given in the Appendix.

Theorem 4. The limiting powers of the tests based on R_N^（M ）（M＝A, B, C, D） as N →∞ under ρ＝1−c /（N^κT ） at the 100γ% level are given as follows:

【45 頁】

where κ＝1/2 for Models A and B, and κ＝1/4 for Models C and D.

　　It follows from Theorem 4 that R_N^（M ） converges to 0 for M ＝ A, B, whereas it converges to −3 for M＝C, D, as was mentioned before. It is also noticed from Theorems 4 and 2 that the GLSE-based tests are better than the OLSE-based tests in Models B, C, D, although those are the same in Model A. The top of Figure 4 shows powers of R_N^（A ）（＝R_N^（B ））-tests at the 5% level for N＝1, 10, 50, ∞, whereas the bottom of Figure 4 those of the R_N^（C ）（＝ R_N^（D ））-tests. It is seen that, for Models A and B, the powers for N＝50 are reasonably well approximated by the limiting powers, whereas, for Models C and D, the aprroximation is still not good enough for N＝50. It is seen that the former powers are higher than the latter, as is anticipated from Theorem 4. This means that the existence of heterogeneous trends decreases the power, as in the OLSE-based tests.

【46 頁】

3.3. Limiting Power Envelopes

In previous subsections, dealing with Models A through D, we considered panel unit root tests based on OLSE and GLSE, for which the limiting local powers were computed and power comparisons were made among those tests, examining the cross-sectional effects. In this subsection we derive the power envelopes, from which the performance of these tests can be evaluated. The idea was earlier developed in the time series context by Elliott et al. （1996）, and was extended to the nonstationary panel data by Moon et al. （2007）. Here we derive the power envelopes for Models A through D, paying attention to the cross-sectional effects.
　　Let us consider the testing problem

where θ_N ＝θ/N^κ with θ being a known positive constant. We assume that the true value of ρ under H₁ is 【47 頁】 given by ρ（c）＝1−c_N /T with c_N＝c/N^κ. Assuming｛ε_it ｝〜 NID（0, σ²）, the Neyman-Pearson lemma tells us that the test which rejects H₀ for small values of

is MPI, where _it^（M ）（0） and _it^（M ）（1） are the GLS residuals obtained from Model M under H₀ and H₁, respectively. The residual _it^（M ）（0） is the same as the GLS residual dealt with in the last subsection, that is, ^（M ）（0）＝（M）, whereas ^（M ）（1）＝y − X ^（M ）γ^（M ）（1）, where

with Ω（θ）＝I_N ⊗ C（ρ（θ））C'（ρ（θ））. Here C（ρ（θ）） is the T × T lower triangular matrix with（s, T ）-th element being ρ^|s-t|_（θ） for s ≥ t and 0 otherwise. The test based on S_NT^（M ）（θ） with fixed θ is called the point optimal invariant（POI） test［King（1987）］.
　　The following theorem gives the weak convergence of S_NT^（M ）（θ） as T →∞ for each N, the proof of which is given in the Appendix.

Theorem 5. As T →∞ under ρ＝1− c_N /T for each N, the MPI test statisticS_NT^（M ）（θ） in（18） follows

where

with δ_N ＝1＋θ_N ＋θ²_N .

　　It is seen that the expression for S_N^（M ）（θ） in（19） is of a similar nature to Q_N^（M ） in（12） and R_N^（M） in （16）. It is also noticed that the distribution of S_N^（M ）（θ） depends on θ that is the value under H₁. Thus the MPI test based on S_N^（M ）（θ） is not uniformly best, but we can modify S_N^（M ）（θ） so that the distribution of the modified statistic does not depend on θ as N →∞. Then we can compute the limiting power of the test based on a modified statistic which yields the limiting power envelope of all the invariant tests for Model M. The following theorem gives such statistics and the power envelopes.

【48 頁】

Theorem 6. The limiting powers of the tests based on the MPI statistics S_N^（M ）（θ） in（19） at the 100γ% level as N →∞ under θ_N ＝θ/N^κ and c_N＝c/N^κ are given by

where κ＝1/2 for Models A and B, and κ＝1/4 for Models C and D.

　　The limiting powers of the modified tests give the power envelope of all the invariant tests. Comparing Theorem 6 with Theorem 4 it is seen that the power functions of the GLSE-based tests coincide with the power envelopes. Thus the GLSE-based tests are asymptotically efficient, unlike the time series case. This is a merit of panel tests as N →∞.
　　There are some other tests that are asymptotically efficient. Here we take up two such tests. Define

The test that rejects H₀ for K_NT^（M ） small is locally best invariant（LBI）, although the test is inapplicable to Models C and D because K_NT^（M ） ≡ 0 for M ＝ C, D, whereas the test that rejects H₀ for L_NT^（M ） small is LBI and unbiased（LBIU） for M＝C,D ［Tanaka（2017, Chap. 10）］. We have, as T →∞ for each N,

Since it can be shown that, for M＝A, B,

we have （K_N^（M ）−1） ⇒ N（−c, 2） by putting c_N＝c /, which implies that the K_N^（M ）-tests for M＝A, B are asymptotically efficient. It is evident that the L_N^（M ）-tests are asymptotically efficient for M＝C, D.

【49 頁】

Note that the LBI and LBIU tests in the time series case（N＝1） are asymptotically inefficient［Tanaka （1996, Chap. 9）］. We also note in passing that, if the GLS residual in （23） is replaced by the OLS residual, the resulting statistic is essentially the Durbin-Watson statistic and the corresponding test is asymptotically inefficient.

3.4. Effect of Temporal or Cross-Sectional Dependence

Here we consider the situation where there exists temporal or cross-sectional dependence of the error term ｛ε_it ｝ in （5） and examine the effect of such dependence on the test statistics obtained in previous subsections.
　　Let us first consider temporal dependence. For this purpose we assume

where φ_i （L）＝1＋φ_i 1L＋φ_i 2L ₂＋・ ・ ・ with L being the lag-operator. The distributional properties of the statistics T （^（M ）−1） in （12） and T （^（M ）−1） in （16） are affected by this relaxation. In fact, it can be shown that, as T →∞ for each N,

where U_i ^（M ） and V_i ^（M ） are defined in （12）, and W_i ^（M ） and X_i ^（M ） are defined in（16）, whereas λ_i  is the ratio of the short-run to long-run variances of ｛ε_it ｝ given by（1）. The above statistics depend on the short-run and long-run variances of the error term that characterize temporal dependence.
　　We next consider cross-sectional dependence, for which we assume that

　It then follows that

【50 頁】

Here _i^（M ）, _i^（M ）, _i^（M ）, and _i^（M ） replace U_i ^（M ）, V_i ^（M ）, W_i ^（M ）, and X_i ^（M ）, respectively, with Y_i ^（r） replaced by _i^（r）, where

and ｛_i^（r）｝ is the standard Brownian motion with

The test statistics depend on the covariances σ_ik of the error term that characterize cross-sectional dependence.
　　It is recognized from the above observations that, to use the asymptotic results obtained in previous subsections, we need to modify the statistics to make them independent of nuisance parameters. This remains to be done.

4. CONCLUDING REMARKS

Under a simple setting, we have presented a unified approach to deriving the limiting local powers of panel AR unit root tests, paying attention to the cross-sectional effect of N. For this purpose it is necessary to compute moments up to the second order of the limiting statistic in the time series direction. We found it easier to use its m.g.f., unlike in the literature. It happened that the tests that were not powerful in the time series case become more powerful in the panel case. It was also found that the existence of a common intercept and/or a common trend does not affect the asymptotic behavior of the tests. This holds for not only the tests based on OLS and GLS residuals, but also power envelopes.
　　The present approach can be applied to unit root tests for other types of panel models such as panel moving average models or panel error components models. Some simple extensions are found in Tanaka （2017, Chap. 10）. For these models the panel LBI or LBIU tests can be used and the corresponding statistics have a distributional structure similar to the panel AR unit root tests discussed in this paper. Details are reported in Tanaka（2018）.

5. APPENDIX: PROOFS OF THEOREMS

Proof of Thereom 1: We first deal with Model D. Given the OLSEs _i  and _i of α_i  and β_i , respectively, the OLS residual is

【51 頁】

The continuous mapping theorem（CMT） yields, as T →∞ with N fixed,

Thus the relation（12） is proved for Model D by the CMT.
　　We next deal with Model B, for which the OLSEs and of α and β are given by

where I_N is the identity matrix of order N, ⊗ is the Kronecker product, and

【52 頁】

We then have

where η_t＝（η_1t , . . . , η_Nt ）'. It follows that T （^（B ）−1）＝U_it^（B ）/V_it^（B ）, where it holds that, as T →∞ with N fixed,

Let us put Y （r）＝（Y₁（r）, . . . , Y_N （r））'. Following Nabeya （2000）, consider Z （r）＝HY （r）, where H  is the N ×N orthogonal matrix with the first row being i'_N / so that ｛Z（r）｝｛Y（r）｝ and Y'（r）i_N＝Y'（r）H'Hi_N ＝ Z₁（r）. Then it holds that

【53 頁】

which shows that（12） holds for M＝B by the CMT. We can prove（12） for M＝A and C similarly.

Proof of Theorem 2: Let us consider Model B. The joint m.g.f of U_i ^（B ） and V_i ^（B ） is given in the text, which can be used to compute first and second moments of these quantities as c_N → 0 by employing computerized algebra and the Taylor expansion. Here we show how to compute E（U_i ^（B ）） and Var（U_i ^（B ））. It follows from（13） that

where

with g（0）＝e^cN , a₁＝（sinhc_N）/c_N and a₂＝coshc_N. Then we have

Here it holds that

Then, using the expansions for e^-cN , e^-2cN , a₁, and a₂ as c_N → 0,

【54 頁】

we can obtain moments of U_i ^（B ）. We now have

It follows from the WLLN that Q_N^（B ） → −3 in probability. Thus we consider

The m.g.f. of U_i ^（B ）＋3V_i ^（B ） can be obtained from m^（B ）（x, y ） by putting y＝3x, which yields

Putting c_N＝c/ , we now have

which proves the second relation in the theorem. We note in passing that

so that it holds that

are uncorrelated.
　　For the other models the first two moments of U_i ^（M ） and V_i ^（M ） as c_N → 0 can be obtained from their joint m.g.f.s given in Tanaka（2017, Chap. 10）. More specifically we have

【55 頁】

These moments yield the limiting powers for M＝A, C, D shown in the theorem, which establishes Theorem 2. We note in passing that, when c_N＝0,

so that U_i ^（C ） and V_i ^（C ） are correlated, whereas U_i ^（D ） and V_i ^（D ） are uncorrelated.

Proof of Theorem 3: We first deal with Model D. Defining the T ×T  lower triangular matrix C with C （s, t ）＝1 for s ≥ t and C （s, t ）＝0 for s ＜ t, the GLSEs of α_i  and β_i  are given by

so that we have

Consider T （−1） ＝U_it ^（D ）/V_it ^（D ）. It holds that, as T →∞ with N  fixed,

【56 頁】

which proves（16） for M＝D.
　　We next consider Model B. The GLSEs of α =（α₁, . . . , α_N ）' and β are given by

which yields

Then, applying the orthogonal transformation Z（r）＝HY（r） used in the proof of Theorem 1, it holds that, as T →∞ with N  fixed,

Noting that｛Z （r）｝｛Y （r）｝, the relation（16） is proved for M＝B. We can prove（16） for M＝A and C similarly, which establishes Theorem 3.

【57 頁】

Proof of Theorem 4: It follows from Theorem 3 that, as N  becomes large,

where W_i ^（M ） and X_i ^（M ） are defined in （16）. The limiting distribution of R_N^（M ） for M＝A, B is the same as that of Q_N^（A ） in （12）, which proves the first relation. For Models C and D, W_i ^（M ）＝−1/2, whereas the m.g.f. of X_i ^（M ） is given in Tanaka（2017, Chap. 10） and

which yields R_N^（M ） →−3 in probability. Then we consider

where

Putting c_N＝c/N ^1/4, we now have

which proves the second relation, and Theorem 4 has been established.

Proof of Theorem 5: The relation（19） for Model D was proved in Tanaka（2017, Chap. 10）. Let us consider Model B, for which it holds that

where

It holds that

【58 頁】

which gives

Thus it holds that, as T →∞ with N  fixed,

where

Applying the orthogonal transformation Z（r）＝HY（r）, we obtain

where

【59 頁】

Noting that ｛Z （r）｝｛Y （r）｝ and （_it^（B ）（0）−_i,t−1^（B ）（0））²/T → σ² in probability, we prove （19） for M＝B after some manipulations. We can prove （19） for M＝A and C similarly, which establishes Theorem 5.

Proof of Theorem 6: For Models A and B we have S_N^（A ）（θ） ＝ Z_i ^（A ）（θ）/N＋o_p（1）, where Z_i ^（A ）（θ）is defined in （19） and the m.g.f. of （Z_i ^（A ）（θ）＋θ_N ）/θ²_N  is given in Tanaka（2017, Chap. 10） as

where μ ＝ Differentiation of m^（A ）（x） gives us

It follows from the CLT that, as N →∞ under θ_N = θ/N ^1/2 and c_N = c/N ^1/2,

which yields（20）.
　　For Models C and D we have S_N^（C ）（θ）＝Z_i ^（C ）（θ）/N＋o_p（1）, where Z_i ^（C ）（θ） is defined in（19） and the m.g.f. of （Z_i ^（C ）（θ）＋θ_N ）/θ²_N  is given in Tanaka（2017, Chap. 10） as

where μ ＝ . Differentiation of m^（C ）（x） gives us

It follows from the CLT that, as N →∞ under θ_N =θ/N ^1/4 and c_N =c/N ^1/4,

【60 頁】

which proves（21）, and Theorem 6 has been established.

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