LM TEST FOR THE CONSTANCY OF REGRESSION COEFFICIENT WITH MOVING AVERAGE INNOVATION

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1 Sankhyā : he Indian Journal of Statistics 22, Volume 64, Series B, Pt.2, pp LM ES FOR HE CONSANCY OF REGRESSION COEFFICIEN WIH MOVING AVERAGE INNOVAION By MEIHUI GUO and C.C. SHEN National Sun Yat-Sen University, aiwan SUMMARY. Assume that the time series can be decomposed into the sum of a deterministic trend, a random walk and a MA() innovation. We are interested in testing the constancy of regression coefficient against the random walk alternative. In this paper, we derive the exact Lagrange multiplier test for the above hypotheses and proposed an asymptotically equivalent test statistic. Furthermore, the null and alternative limiting distributions of the test statistic are derived. Finally, we compare the finite sample size and power of the proposed test statistic with that of Kwiatkowski et al. (992). An example of testing the global warming hypothesis is also demonstrated.. Introduction In this paper, we consider the following model: { yt = α t + z tβ + ν t α t = α t + µ t, () where (i) {y t } is a sequence of scalar observations, whereas z t = (, t), β = (β, β ) ; (ii) ν t = ε t + θε t is an MA() process with unknown parameter θ, {ε t } and {µ t } are independent of each other and are i.i.d., respectively, with E(ε t ) =, E(ε 2 t ) = σε 2 >, E(µ t ) =, E(µ 2 t ) = σµ 2 ; (iii) {α t } starts with the initial value α =. he model () belongs to a class of state space models developed in control engineering, which has also been used frequently in the financial economics literature, see for example Fama and Gibbons (982), Brown, Kleidon and Marsh (983) and rzcinka (982). Paper received May 998; revised April 22. AMS (99) subject classification. Primary 62H5, 62P2; secondary 62M. Key words and phrases. Lagrange multiplier test, moving average.

2 lm test for the constancy of regression coefficient 25 When σ 2 µ >, one component α t of the coefficient vector (α t, β) varies over time, the model () is thus also referred to as varying coefficient regression model, see for example, Nicholls and Pagan (985). Whether σ 2 µ is zero or not is a critical issue for the applications of the model, thus we consider the following hypothesis testing problem: H : r = σ2 µ σ 2 ε = against H a : r >. (2) It is obvious from (2) that, the process {y t } is trend stationary under the null hypothesis and is differencing stationary under the alternative. Most of the tests in the literature are derived for i.i.d. innovation {ν t } (which is corresponding to θ = in model ()). See for example, Brown, Durbin and Evans (975) Garbade (977), also the exact test of LaMotte and McWhorter (978) and Shively (988), the locally best invariant (LBI) tests of Nyblom and Mäkeläinen (983), Nabeya and anaka (988) and Leybourne and Mc- Cabe (989), and the Lagrange multiplier (LM) test of anaka (983) and Kwiatkowski et al. (KPSS) (992). he null asymptotic distributions of the above tests are then derived for normal i.i.d. or α mixing innovation {ν t }. o accomplish test for correlated {ν t }, KPSS first proposed a modified LM test for strong mixing {ν t }. In this studies, assuming a MA() innovation {ν t }, we proposed an asymptotical LBI test which in general attains better sizes and higher power than KPSS s modified test. In the following, we explain briefly KPSS s procedure of extending the LM test derived from i.i.d. {ν t } to non i.i.d. situation. he LM test statistic for i.i.d. innovation {ν t } is ˆη = t= St 2 ˆσ ε 2, (3) where S t = t i= e i, t =, 2,...,, is the partial sum process of the residuals, and e i s are the residuals from the least squares regression, i =, 2,..., and ˆσ ε 2 = i= e 2 i is the estimate of the error variance from the regression. o allow for temporal dependence, KPSS assume that {ν t } in model () satisfy the (strong mixing) regularity conditions of Phillips and Perron (988, p.336). Define the long-run variance as σ 2 = lim E ( i= ν i ) 2. hey adopt the following estimate of σ 2, s 2 (l) = ˆγ() + 2 l s= ( s ) ˆγ(s) (4) l +

3 26 meihui guo and c.c. shen where ˆγ(s), s =, 2,..., l is the sample autocovariance function of the residuals {e i }. KPSS, proceeding in the spirit of Phillips (987) and Phillips and Perron (988), proposed the following modified LM test for level-stationary and trend-stationary which are denoted by ˆη and ˆη, respectively: ˆη (or ˆη ) = 2 t= S 2 t s 2 (l), (5) where S t = t i= (y i ȳ) for ˆη and S t = t i= (y i ˆβ ˆβ i) for ˆη, ȳ is the sample mean and ˆβ, ˆβ are the least square estimators of β and β. hat is, the denumerator ˆσ 2 ε of (3) was modified to s 2 (l) by a nonparametric approach. It has been shown that, under H where V (r) = W (r) rw (r), and D ˆη V (r) 2 dr, (6) D ˆη V 2 (r) 2 dr, (7) where V 2 (r) = W (r) + (2r 3r 2 )W () + ( 6r + 6r 2 ) W (s)ds, and W (r) is a Wiener process (Brownian motion). hroughout, we use the symbol D to represent weak convergence of the associated probability measures. Although, KPSS s test is an easy handy test for correlated {ν t }, yet their simulation results show the test has serious size distortion problem and difficulty of selecting l while computing s 2 (l) for AR() errors. In this studies, we adopt a parametric approach to resolve these problems. In particular, our approach will avoid the problem of choosing l for s 2 (l) and achieve better finite sample sizes than KPSS especially when the MA coefficient θ is negative, also attains higher power than KPSS when both tests achieve the correct size. In this paper, we will derive the exact LM test then proposed an asymptotically equivalent test statistic for the hypothesis testing problem (2) when {ν t } is an MA() process. We will also derive the null and alternative asymptotic distributions of our proposed test statistic and compare its finite sample power with KPSS s by simulation. Finally, note that the model () can also be expressed as the following form: ( B)y t = β + µ t + ( + θb)( B)ε t = β + ε t + aε t + bε t 2 for some white noise process {ε t } with variance σ 2 ε = θ b σ2 ε, where a and b

4 lm test for the constancy of regression coefficient 27 satisfy the following equations: + a 2 + b 2 b ( a + ) b = [2(θ2 θ + ) + r] θ (θ )2 = θ (8) with r=σµ/σ 2 ε. 2 In other words, the reduced form of model () is an ARIMA(,,2). Since by (8), a + b + = is implied by the hypothesis r = σµ/σ 2 ε 2 =, our testing problem can also be applied to testing the moving average unit root problem of an ARIMA(,,2) process, which has received much attention recently (see, for example, anaka (99), Saikkonen and Luukkonen (993) and say (993)). However, there are still different aspects between these two approaches. For example, there is a clearly defined distinction between the random walk and linear component, and no infinite past (α = ) in the state space model (), but not in the ARIMA model. Furthermore, the parameter mapping (see (8)) is not necessary oneto-one for these two models. Saikkonen and Luukkonen (993, p.598) also mentioned that the nonstationarity in these two models are generated in a different way. he article is organized as follows. In section 2, we derive the proposed test statistic and its null and alternative asymptotic distributions. he consistency of the proposed test is also proved. In section 3, simulation results are given to compare the finite sample sizes and power of our test with KPSS test. An example of testing the global warming hypothesis is also considered. Section 4 is the conclusion. All the proofs are given in Appendix. 2. LM est First, we give a brief introduction to the LM test procedure. Assume L(ψ) is a likelihood function, and let ψ denote the ML estimator of ψ under the null hypothesis H. he LM (Lagrange multiplier) test statistic has { } { } the form, LM = log L( ψ)ψ I ( ψ ψ) log L( ψ), where I( ) is ψ the Fisher information matrix. he null hypothesis H is rejected for large values of LM. If H is true, ψ will be close to the unrestricted MLE, ˆψ, since ψ log L( ˆψ) =, ψ log L( ψ) will also be close to zero. Furthermore, since ψ log L( ψ) = ψ log L( ˆψ) + ( ψ ˆψ) 2 log L( ˆψ) ψ ψ

5 28 meihui guo and c.c. shen thus LM = ( ψ ˆψ) = ( ψ 2 ˆψ) 2 ψ ψ log L( ˆψ), log L( ˆψ)I ( ψ) log L( ˆψ)( ψ ˆψ). ψ ψ ψ ψ 2 Under H, for large sample, under some smoothness condition, I( ψ) has the same probability limit as 2 ψ ψ log L( ˆψ), the limiting distribution of LM is thus the same as that of the likelihood ratio (LR) test statistic, ( ψ 2 ˆψ) log L( ˆψ)( ψ ˆψ). hat is, the Lagrange multiplier procedure leads to a test with the same asymptotic distribution as the LR test ψ ψ statistic under the null hypothesis. One distinguishing feature of the LM test is that it only entails the estimation of the restricted model under the null hypothesis. Furthermore, the result of King and Hillier (985) can be applied to show that the LM test is equivalent to a LBI test for the one-sided hypotheses test (2). Now for the model (), under i.i.d. and normality assumptions of {ε t }, the log-likelihood function L(σµ, 2 φ) is given by L(σ 2 µ, φ) = c 2 log Σ 2 (y Zβ) Σ (y Zβ) where c is a constant, φ = (β, θ, σε) 2, β=(β, β ), y=(y, y 2,..., y ), Z = (, t), = (,, ), t = (, 2,, ), Σ = σεd 2 + σµv 2, V = (min(s, t)) and + θ 2 θ θ + θ 2 θ D = θ + θ 2.. (9) θ θ + θ 2 Since the hypothesis test (2) is a one sided test, the LM test statistic can be written as (anaka, 983) L(, φ) σµ LM = 2 (â ĉ ˆB ĉ), () /2

6 lm test for the constancy of regression coefficient 29 where φ is the MLE of φ under H, ( 2 L(, â = E φ) ) ( (σµ) 2 2, ˆB 2 L(, φ) ) ( = E, ĉ 2 L(, = E φ) ) φ φ σµ φ 2. () In the above equations, and φ are plugged in after taking derivative or expectation. heorem. Assume that {y t } satisfy the model () and {ε t } are i.i.d. N(, σε) 2 random variables. hen the LM test statistic of the hypothesis test (2), defined by () is LM = 2 τ [ tr( D V ) + σ 2 ε (y Z β) D V D (y Z β)], (2) where β, σ ε 2 are the constrained MLE s of β and σε 2 under H, respectively. D is the matrix D defined by (9) with θ replaced by θ (the constrained MLE of θ under H ) and τ = 2(â ĉ ˆB ĉ) σ 2 ɛ. Remark. It was proved (heorem 3.2 Guo and Shen, 997) that 2 tr( D V ) converges to and τ converges to 2(+θ) 2 2 6(+θ) 2 in probability, respectively. In the proof of heorem 2, we show that (+θ)2 (y 2 σɛ 2 Z β) D V D (y Z β) converges in distribution to a functional of Brownian motion. In accordance with this, we propose to use LM = ( + θ) 2 2 σ ε 2 (y Z β) D V D (y Z β) (3) as the test statistic for trend-stationary. Similarly, when β = in the model (), that is { yt = α t + β + ε t + θε t (4) α t = α t + µ t, we use LM = ( + θ) 2 2 σ ε 2 (y β ) D V D (y β ) (5) as the test statistic for level-stationary. It is well known that the matrix V can be factorized as LL, where L is a lower triangular matrix of ones. Since premultiplication of a vector by L forms a backwards cumulative sum, the

7 22 meihui guo and c.c. shen numerators of LM and LM can be viewed as a sum of squares of backwards weighted partial sum of the residuals (computed from m.l.e). Remark 2. Since D = (D, α) (d () ij ), d () ij = ( D α ) if i = j; θ if i j = ; otherwise, = (D D ) + (αα ), where D = and α = (,,...,, θ). herefore, D = (D D ) M, where M defn = (D D ) (αα )(D D ) + α (D D ) α It is easy to derive that D = (d ( ) ij ), where and d ( ) ij = m st = θ2 2 ( θ 2 ) θ2 +2 = (m st ). if i = j; if i > j; ( θ) k if j i = k, k =,...,, s t ( θ) 2i s ( θ) 2j t. i= j= In the proof of the following theorems, we derive the convergence order of M multiplied by related quantities (e.q. t Mt = o( 3 ) ). After comparing with the convergence order of (D D ), we found that M is negligible compared with (D D ) while deriving the asymptotic distributions of LM and LM. heorem 2. Assume that {y t } satisfy the model (4). Let β and θ be the true values of β and θ, respectively and let β be the constrained MLE of β under the null hypothesis. hen under H, we have (i) β β = t= ν t + o p ( /2 ), where ν t = ε t + θ ε t, (ii) LM D V (r)2 dr, where V (r) is defined as in (6). Furthermore, under H a we have LM D ( a W (s)ds) 2 da (W (s)) 2, ds

8 lm test for the constancy of regression coefficient 22 where W (s) = W (s) W (t)dt. heorem 3. Assume that {y t } satisfy the model (). Let β, β and θ be the true values of β, β and θ, respectively and let β and β be the constrained MLE s under H of β and β, respectively. hen under H, we have (i) β β = 2 ν t = ε t + θ ε t t= ν t t= ( t)ν t + o p ( /2 ), where (ii) β β = 6 2 t= ν t 2 3 t= ( t)ν t + o p ( 3/2 ), (iii) LM D V 2(r) 2 dr, where V 2 (r) is defined as in (7). Furthermore, under H a we have where LM D ( a W (s)ds) 2 da (W (s)) 2, ds W (s) = W (s) + (6s 4) W (t)dt + ( 2s + 6) tw (t)dt. Remark 3. By heorem 2 and heorem 3, we deduce that both LM and LM are of O p ( ) under H a, thus the tests are consistent. However, for KPSS s test statistic, ˆη = O p ( l ), where l is defined in (4). o achieve consistency of s 2 (l), it s necessary that l as. hus, we expect LM (or LM ) will have better sizes and higher power than ˆη (or ˆη ) in finite sample case. 3. Size and Power in Finite Samples In this section, we study the size and power of LM and LM in finite samples. he results are based on simulations, using the normal random number generator of IMSL library. Note that the values of (β, β ) in the model () and the value of β in model (4) will not affect the finite sample power functions of LM and LM. In our simulation, we set β =. for model (4), see able, and set β =. and β =. for model (), see able 2. he sample sizes in our simulation studies are 3, 5,, and 2 respectively with, replication. he results are presented in able

9 222 meihui guo and c.c. shen and able 2. First, we consider the size of the tests. Since under H, that is r = σ2 µ =, LM σε 2 and LM have the same null limiting distribution as ˆη and ˆη their asymptotic upper tail critical values can be found in p.66 able, KPSS(992). o compute the sizes of ˆη and ˆη, we have to decide the value of l for s 2 (l) defined in (4). Schwert (989) and KPSS suggest to choose l =, l 4 = [4(/) /4 ] and l 2 = [2(/) /4 ]. Since the innovations {ν t } in this paper are assumed to be a MA() process, ˆσ 2 = ( + θ) 2 σ ε 2 and ˆσ 2 2 = ˆγ() + 2ˆγ(), (where ˆγ(s) is the sample autocovariance function of the regression residuals) are the maximum likelihood estimator (m.l.e) and the moment estimator of the long run variance σ 2 = lim E( i= ν i ) 2, respectively. Compared with ˆσ 2 2, obviously s2 (l ) = ˆγ() will underestimate σ 2 when θ > and overestimate σ 2 when θ <. We also observe, in the simulation study, that ˆη and ˆη with either l 4 or l 2 have worse sizes and smaller power than LM and LM. herefore we will not present the results of ˆη and ˆη in the tables. Instead, we consider the following two statistics ˆη, (ˆη, ) and ˆη,2 (ˆη,2 ) for level and trend stationary models, respectively: ˆη, (ˆη, ) = t= S 2 t ˆσ 2 and ˆη,2 (ˆη,2 ) = t= S 2 t ˆσ 2 2 and perform simulation for these two statistics. Base on the results of able and able 2, we compare the sizes and the power of LM (LM ), ˆη, (ˆη, ), and ˆη,2 (ˆη,2 ) for θ < and θ >, respectively. (I) When θ <, we have (i) all size of LM and LM are less than or equal to.5; (ii) most of the size of ˆη, (ˆη, ) and ˆη,2 (ˆη,2 ) are excessively over.5. Except when θ is close to zero and is large, ˆη, (ˆη, ) and ˆη,2 (ˆη,2 ) attain size close to.5 (i.e θ =.2, ), yet their power are smaller than LM (LM ), especially when r = and =. (II) When θ >, we observe that (i) the power of LM (LM ) dominate the other statistics, however the size of LM (LM ) is in excess of.5 when is small and θ is close to (i.e θ =.8, 2 for LM and θ =.5, 5, θ =.8, 2 for LM ) (ii) the size of ˆη, (ˆη, ) and ˆη,2 (ˆη,2 ) are all close to.5. speaking, ˆη, (ˆη, ) has higher power than ˆη,2 (ˆη,2 ). Generally

10 lm test for the constancy of regression coefficient 223 able. Size and Power of LM, ˆη, and ˆη,2 5% level θ r = σ2 µ 4 2 σɛ 2 (size) (size) 4 2 = 3 = 5 LM ˆη, ˆη, LM ˆη, ˆη, LM ˆη, ˆη, LM ˆη, ˆη, LM ˆη, ˆη, LM ˆη, ˆη, = = 2 LM ˆη, ˆη, LM ˆη, ˆη, LM ˆη, ˆη, LM ˆη, ˆη, LM ˆη, ˆη, LM ˆη, ˆη,

11 224 meihui guo and c.c. shen able 2. Size and Power of LM, ˆη, and ˆη,2 5% level θ r = σ2 µ 4 2 σɛ 2 (size) (size) 4 2 = 3 = 5 LM ˆη, ˆη, LM ˆη, ˆη, LM ˆη, ˆη, LM ˆη, ˆη, LM ˆη, ˆη, LM ˆη, ˆη, = = 2 LM ˆη, ˆη, LM ˆη, ˆη, LM ˆη, ˆη, LM ˆη, ˆη, LM ˆη, ˆη, LM ˆη, ˆη,

12 lm test for the constancy of regression coefficient 225 Example. We study 42 yearly deviations of global temperature, ending in 997. he data set is given in the Appendix. For testing the global warming hypothesis, one may want to test β = as well to see whether the temperature is really increasing linearly or is just some positive excursion of a random walk. he time plot and sample ACF of the original series {y t } and the differenced series {y t y t } are given in Figure. he ACF of the differenced series shows that the reduced form of the data has an ARIMA(,,2) structure. he estimated parameters are ˆβ =.457, ˆβ =.42 ( with p-value = ), and the test statistic LM =.435 which is greater than the 95% critical value of LM (=.46, see KPSS p.66, able ). herefore, we conclude that there are both trend and random walk components in this temperature data. Figure. (a) the original series (b) the differenced series

13 226 meihui guo and c.c. shen 4. Conclusion From heorem 2 and heorem 3, we know that LM (LM ), ˆη (ˆη ), ˆη, (ˆη, ) and ˆη,2 (ˆη,2 ) have the same null and alternative limiting distributions. he simulation results of = 2 also sustain these results. However, their finite sample distributions are different and depends heavily on the value of the MA() parameter θ. Deduce from the summary of the simulation results, LM (LM ) and ˆη, (ˆη, ) are two competitive statistics. When the MA() coefficient θ <, the residuals will tend to be negatively correlated. In this case, the size of LM (LM ) are less than or equal to.5, nevertheless most of the size of ˆη, (ˆη, ) are excessive large. In the situation when both tests have about the correct size and is moderate, the power of LM (LM ) is much larger than ˆη, (ˆη, ) especially when r =. his is in accordance with the result that we mention early in Introduction that the derived exact LM test is equivalent to a LBI test. When the MA() coefficient θ >, the residuals will tend to be positively correlated. In this case the power function of LM (LM ) dominate ˆη, (ˆη, ). However, LM (LM ) has excessive size when θ is close to while is not large enough. Proof of heorem. First, the numerator of () equals 5. Appendix L(, φ) σ 2 µ = 2 σ 2 ε tr( D V ) + 2 σ 4 (y Z β) D V D (y Z β). ε Next, we derive the denominator of (), (â ĉ ˆB ĉ) 2. It can be shown that â = 2 σ 4 ε tr(g ) 2 ; (6) ( and ĉ = ˆB =, ( σ 2 ε(z D Z) H 2 σ 4 ε tr(g ), ) 2 σ 2 ε tr(g G 2 ) ; (7) ), (8) where G = D V, G 2 = D D, (9) where D is the first derivative of D w.r.t. θ at θ. ( 2 σ 4 H = ε tr(g 2 ) 2 2 σ εtr(g 2 2 ) tr(g 2 ) 2 [tr(g 2 )] 2 2 σ εtr(g 2 2 ) 2 ).

14 lm test for the constancy of regression coefficient 227 herefore, (â ĉ ˆB ĉ) 2 = τ = { tr(g 2 ) 2 σ ε 2 τ, where [ tr(g 2 2) [tr(g 2 )] 2 [tr(g )] 2 tr(g 2 2) 2tr(G G 2 )tr(g )tr(g 2 ) + [tr(g G 2 )] 2]} 2. (2) Hence LM = 2 τ [ tr( D V ) + σ 2 ε (y Z β) D V D (y Z β) ]. Proof of heorem 2. (i) Since β L(, φ) = ( σ 2 ε D) (y β ) =, β = ( D ) D y. By Remark 2, D = ( D D ) M. hus, β β = ( D D ) ν Mν ( D, D ) M where ν = (ν, ν 2,..., ν ). By direct computation we have M = G ( ( θ) + ( θ 2 )( + θ) ) 2 = O(), Mν = G ( θ 2 ) ( ) j θ j+ ( θ 2j )ν j = o p ( ), where G = Finally, since θ 2 +2 θ 2. ( D D ) = j= 2 t ( θ) j = ( + θ) 2 + O(), t= j= and D D ν = = ( ) ( θ) t+ t+ t= + θ ( θ) j ν t+j j= ( + θ) ν 2 k + o p ( ), k=

15 228 meihui guo and c.c. shen we have β β = ν t + o p ( /2 ). t= (ii) Note that y β = ν + (β β ), thus we can write (y β ) D V D (y β ) = (A ) + (B ) + (C ), where (A ) = ν D V D ν ( ) 4 2 t = + θ ν j+ + o p ( 2 ); t= j= (B ) = (β β ) D V D (β β ) ( ) 4 3 ( ) 2 = + θ ν t + o p ( 2 ); 3 t= (C ) = 2ν D V D (β β ) ( ) 4 t = 2 + θ t ( ν j+ t= j= t= ν t ) + o p ( 2 ). he above equalities are obtained by plugging in D = ( D D ) M and direct computation of the orders of M multiplied by related quantities. By Hamilton (994) p.56, we have ( ) 2 (y β ) D V D (y β ) D 2 σε 2 [W (r) rw ()] 2 dr. +θ Finally, since the constrained MLE s σ 2 ε and θ are consistent estimators of σ 2 ε and θ ( e.g. Lemma 3.2. of Guo, 989), LM D [W (r) rw ()] 2 dr. (iii) Let e t = t i= µ i + ν t, then y t = β + e t under H a. Similarly as in (i), we have hus β β = t= e t + o p ( 2 ) = t µ i + o p ( 2 ), t= i= (y β ) D V D (y β )

16 lm test for the constancy of regression coefficient 229 = = ( ) 4 2 t (e j+ + β β ) +o p ( 4 ) + θ t= j= ( ) 4 t j+ + θ µ i 2 t µ i +o p ( 2 ) +o p ( 4 ), t= j= i= t=i= and (+ θ) 4 4 (y β ) D V D (y β ) ( a ( ) 2 D σµ 2 W (r) W (s)ds dr) da. As LM is defined as in (5), it only remains to prove the weak convergence of σ 2 ε under H a. Since we have σ 2 ε L(, φ)= 2 ( σ 2 ε +σε 4 (y Z β ) D (y Z β )=, σ ɛ 2 = (y β ) D (y β ) = 2 t ( θ) j t+j (µ i + β β ) t= j= i= = ( ) 2 ( t + θ µ j t ) 2 µ i + o p ( 2 ). t= j= t= i= Finally, by Hamilton (994), ( + θ) 2 σ2 ɛ D σµ 2 (W (r) W (s)ds) 2 ds. Proof of heorem 3. ( ) β (i) Since = (Z D Z) β Z D y and by Lemma 2 we have D = ( D D ) M, thus ( ) ( ) β β = β β [ ] Z ( D D ) Z Z MZ [ ] Z ( D D ) ν Z Mν.

17 23 meihui guo and c.c. shen By direct computation, we have ( θ ) i, ( θ ) i,, D Z = i= i= ( θ ) i (2) i, ( θ ) i (i + ),, i= i= and D ν = ( θ ) j ν j, ( θ ) j ν j+,, ν. (22) j= j= hus and β β = ( + θ ) 2 t ( θ ) j ν t+j t= j= 6 t 2 t ( θ ) j ν t+j + o p ( /2 ) t= j= = 2 ν t 6 t= 2 ( t)ν t + o p ( /2 ), t= β β = ( + θ ) 6 t 2 ( θ ) j ν t+j t= j= + 2 t 3 t ( θ ) j ν t+j + o p ( 3/2 ) t= j= = 6 2 ν t + 2 t= 3 ( t)ν t + o p ( 2/3 ). t= (ii) Note that (y Z β) D V D (y Z β) = (A2 ) + (B 2 ) + (C 2 ), where (A 2 ) is the same as (A ) in the Proof of heorem 2(i), (B 2 ) = (β β) Z D V D Z(β β) ( ) ( 4 3 = + θ 3 (β β ) (β β ) (β β )(β β ) ) + o p ( 2 ),

18 lm test for the constancy of regression coefficient 23 (C 2 ) = ν D V D Z(β β) ( ) 4 = + θ (β β t ) t t= j= ( ) +(β β t ) t t2 2 t= By Hamilton (994) p.56, we have ν j+ ν j+ j= + o p ( 2 ). 2 (y Z β) D V D (y Z β) ( ) D 2 [( ) { 2 σε 2 W 2 (r)dr + + θ 5 W 2 () + 6 ( ) 2 W (r)dr } 5 5 W () W (r)dr ( +2 W () (2r 3r 2 )W (r)dr )] + W (r)dr ( 6r + 6r 2 )W (r)dr = ( + θ ) 2 σ 2 ε + ( 6r + 6r 2 ) ( W (r) + (2r 3r 2 )W () W (s)ds) 2 dr. Finally, since under H both σ 2 ε and θ are consistent, the results are thus obtained. (iii) he result can be obtained similarly as in the proof of heorem 2(iii). Data set of Example

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