Self-Normalized Dickey-Fuller Tests for a Unit Root

Transcription

1 Self-Normalized Dickey-Fuller Tests for a Unit Root Gaowen Wang Department of Finance and Banking, Takming College Wei-Lin Mao Department of Economics, National Cheng-Chi University Keywords: domain of attraction, normal law, linear processes, long memory. JEL classification: C2, C3, C22. Tel.: GWWang@takming.edu.tw. Research supported by the National Science Council of the R.O.C. Grant NSC H Corresponding author. Address: No. 64, Sec. 2, Chih-nan Rd., Wenshan, Taipei 623, Taiwan, R.O.C. Tel.: Fax: maotai@nccu.edu.tw.

2 Abstract In this paper, we establish corresponding functionals of self-normalized partial sums for the Dickey-Fuller unit root tests. Under very weak conditions, we derive the asymptotic distributions of the tests with errors being iid random variables, short-memory linear processes, and long-memory linear processes. Under iid environment, the finite second moment condition used in previous research is weakened. For the short-memory linear processes with absolutely summable coefficients, we impose no other conditions on the coefficients. As to the long-memory linear processes, when the coefficients are assumed to be regularly varying with index in (, /2), the only extra condition on the coefficients is automatically satisfied.

3 Introduction Let {u k } k be a sequence of nondegenerate random variables. Set S n = n k= u k and Vn 2 = n k= u2 k. The asymptotic behavior of the self-normalized sums, {S n /V n } n, has been extensively studied in probability theory (Logan et al., 973; Egorov, 996; Giné et al., 997; Kruglov and Petrovskaya, 200; Csörgő et al., 2003; Chistyakov and Götze, 2004; Hall and Wang, 2004). It was proved that, as n, {S n /V n } converges weakly to a Gaussian distribution if and only if u k are independent and identically distributed (iid) random variables with E(u k ) = 0 and in the domain of attraction of the normal law, denoted by u k D N (Egorov, 996; Giné et al., 997). Particularly, S n and V n display synchronous behavior, implying that no moment conditions are needed for the limit distributions of the self-normalized sums (Chistyakov and Götze, 2004). In this paper, we employ the self-normalized sums to investigate asymptotic behavior of the Dickey-Fuller (DF) tests for a unit root (Dickey and Fuller, 979, 98). Testing unit root hypothesis is an important issue in econometrics and has been extensively studied in past years; see, for examples, Stock (994) and Phillips and Xiao (998) for recent surveys. The goal of this paper is threefold. The first one is to establish corresponding functionals of the self-normalized partial sums for the DF tests. The second goal is to demonstrate that the resulting functionals provide a convenient way to derive asymptotics for the tests by allowing a much wider class of

4 error processes. However, much research has been done in this direction. In this paper we restrict our attention to errors being assumed to be iid (a benchmark case), short-range dependent, and long-range dependent, respectively. Note that unit root tests in the presence of long-range dependence are recognized to be of increasing importance in econometrics. There has recently been a surging interest in such models. Investigations by Lubian (999), Wang et al. (2002, 2003), and Wu and Min (2004) have been at the forefront of research in this field. Nevertheless, from theoretical point of view, it is important to know what assumptions can be made for the models under consideration such that the asymptotics of the DF tests still remain asymptotically valid. The third major goal of this paper is to apply the resulting self-normalized DF test statistics to address this issue. Under iid environment, it is well-known that a finite second moment assumption on the underlying errors u k is often required for deriving asymptotics for the DF tests (White, 958; Dickey and Fuller, 979, 98; Phillips, 987; Phillips and Perron, 988; Chan and Wei, 988). In contrast, in this paper the required conditions are that u k D N and E(u k ) = 0. Clearly, our moment conditions are weaker since E(u 2 k ) = is allowable. It represents a significant improvement of the previously known results. On the other hand, the focus in this paper will be on linear processes which provide an excellent vehicle for modeling both short-range and longrange dependence. The process u k = j=0 θ jε k j, θ j R, where {ε i } i Z is a 2

5 sequence of iid(0,) innovations, is said to be short-memory if j=0 θ j < and long-memory if j=0 θ j = but j=0 θ2 j < (see Hall, 992). In the context of short-memory linear processes, the coefficients θ j are typically restricted to be one-summable (i.e. j=0 j θ j < ) or ½-summable (i.e. j=0 j/2 θ j < ) for validating the functional central limit theorem (FCLT) (Said and Dickey, 984; Phillips, 987; Stock, 994; Phillips and Xiao, 998; Chang and Park, 2002; Wang et al., 2003). For more results, see references cited in these papers and Phillips and Solo (992) for a demonstration of the asymptotics for linear processes. Recently, the studies by Wang et al. (2002) and by Wu and Min (2004) have obtained the invariance principles for short-memory linear processes under much weaker conditions than the ½-summability. However, our results for the asymptotic distributions of the DF tests only require that both j=0 θ j 0 and j=0 θ j <. Since we impose less restrictive assumptions on the coefficients θ j, it can be viewed as an improvement of the results in the latter two papers. As to the long-memory linear processes, the long-run variance of u k does not exist, so that the FCLT may not apply unless the norming constant has a faster rate than n and the innovations ε i have finite fourth moment, as is done in Lubian (999) who employed the invariance principle of Davydov (970) to derive asymptotics for the DF tests. As in Wang et al. (2002), in this paper only finite second moments for ε i are required, which abolishes the restriction of E(ε 4 i ) <, used in previous works (see, Wang et 3

6 al. (2003), Wu and Min (2004) and references therein). On the other hand, the condition j=n+ θ j = o( j=0 θ j) that we impose in this paper is sufficiently weak. In particular, when the coefficients have the form θ j = l(j)/j β for j, where β (/2, ) and l is a slowly varying function, we show that the imposed condition is automatically satisfied (see Remark 2 in Section 4). Therefore, our results are new and represent a significant improvement of the previously known results. The paper is organized as follows. The self-normalized DF test statistics are given in Section 2. These self-normalized statistics are applied in Section 3 to iid errors and in Section 4 to errors being, respectively, short-memory and long-memory linear processes with iid innovations. Section 5 concludes the paper. 2 Self-Normalized Unit Root Test Statistics In this section, we first review some known facts about the DF tests for a unit root and then present their self-normalized versions. Suppose {y t } t is generated according to y t = φy t + u t with φ =. () Given y 0 = 0 and n observations y,..., y n, to test φ = against φ <, two appropriate test statistics for the null are as follows: ( ˆρ n := n( ˆφ n ) = n 2 t= 4 y 2 t ) ( n ) y t u t, (2) t=

7 ˆτ n := ( ˆφ n ) ( n t= y2 t ˆσ 2 u ) /2, (3) where ˆφ n = ( n t= y2 t ) n t= y t y t is the least squares estimator (LSE) of φ in () by regressing y t on y t, and ˆσ 2 u = (n ) n t= (y t ˆφ n y t ) 2. Alternatively, by regressing y t on y t with a constant term c, the analog of the two statistics in (2) and (3) can be expressed, respectively, as [ ] [ ] ˆρ µn := n( ˆφ µn ) = n 2 (y t ȳ ) 2 (y t ȳ )u t (4), n t= t= [ n ˆτ µn := ( ˆφ t= µn ) (y t ȳ ) 2 ] /2, (5) ˆσ 2 µu where ˆσ 2 µu = (n 2) n t= (y t ĉ n ˆφ µn y t ) 2, ȳ = n n t= y t, and ˆφ µn and ĉ n are the LSEs of φ and c, respectively. These four test statistics have been well-studied and used as a standard for detecting the presence of a unit root in (). In the case that the error sequence {u t } is assumed to be iid with zero mean and finite positive variance, a number of authors (White, 958; Dickey and Fuller, 979, 98; Phillips, 987; Phillips and Perron, 988; Chan and Wei, 988) have reported the asymptotic distributions of the unit root statistics, which are given by: (see Chapter 7 of Hamilton (994) for full details) ˆρ n W ρ := W 2 () 2 0 W 2 (r)dr, (6) ˆτ n W τ := W 2 () 2 [ ] /2, (7) 0 W 2 (r)dr [ 2 W 2 () ] W () 0 ˆρ µn W ρµ := W (r)dr [ ] 0 W 2 2, (8) (r)dr 0 W (r)dr 5

8 ˆτ µn W τµ := [ 2 W 2 () ] W () 0 { W (r)dr [ ] }, (9) 0 W 2 2 /2 (r)dr 0 W (r)dr where {W (r) : 0 r } stands for a standard Brownian motion, and the symbol signifies the weak convergence of the space D[0, ] endowed with the Skorohod topology. Note that Dickey and Fuller (979, 98) in fact derived alternative but equivalent representations for the asymptotic distributions above (see Corollary 3..3 of Chan and Wei (988)). We now proceed to elaborate on the self-normalized DF tests. Again, let S n = u k, Vn 2 = k= u 2 k. (0) The quotient S n /V n is of the so-called self-normalized sum. We set S n /V n = 0 if V n = 0. Under () with y 0 = 0, we have the representations y t = S t = t k= u k and S 0 = 0. It then follows that k= n yt 2 = St 2, t= t= ȳ = n y t = n S t. () n t= t= Note further that y 2 t = (y t + u t ) 2 = y 2 t + 2y t u t + u 2 t, (2) implying that ( y t u t = yn 2 2 t= t= u 2 t ) = 2 ( S 2 n Vn 2 ). (3) Since, as n, n/(n d) tends to for finite d, it is easy to show that ˆσ 2 u = n u 2 t + o p () = n V n 2 + o p () and ˆσ µu 2 = n V n 2 + o p (), (4) t= 6

9 where the notation o p () denotes a term converging in probability to zero. Similarly, denote by O p () a sequence of random variables that is bounded in probability. Using (0) (4), the four test statistics in (2) (5) can be expressed in terms of functionals of the self-normalized partial sum processes {S t /V n, t n}, respectively, as follows: ˆρ n = ˆτ n = ˆρ µn = = ˆτ µn = = 2 (S2 n Vn 2 ) 2 n = [(S n/v n ) 2 ] n t= S2 n t n t= (S t/v n ), (5) 2 2 (S2 n Vn 2 ) n = 2[(S n/v n ) 2 ] t= S2 t [ n V n 2 + o p (), (6) + o p ()] n t= (S t/v n ) 2 n 2 (S2 n Vn 2 ) n S n [ n t= S2 t n n t= S t ( n t= S t ) 2 ] 2 [(S n/v n ) 2 ] n (S n/v n ) n t= S t/v n [ n ( n t= (S t/v n ) 2 n ) ] 2, (7) n t= S t/v n 2 (S2 n Vn 2 ) n S n n t= S t [ n ( n ) ] 2 [ t= S t n V n 2 + o p () ] t= S2 t n n 2 [(S n/v n ) 2 ] n (S n/v n ) n t= S t/v n [ n ( n t= (S t/v n ) 2 n ) ] + o p(). (8) 2 n t= S t/v n Clearly, if the sequence {S t /V n, t n} has asymptotic distribution, so do the DF test statistics ˆρ n, ˆτ n, ˆρ µn and ˆτ µn. Recently, it was proved that the sequence {S t /V n, t n} converges weakly to a standard Brownian motion {W (r), 0 r } if and only if u k are in the domain of attraction of the normal law and E(u k ) = 0 (Logan et al., 973; Egorov, 996; Giné et al., 997; Kruglov and Petrovskaya, 200; Csörgő et al., 2003). It should 7

10 be noticed that {S t /V n, t n} does not depend on the variance of u k such that E(u 2 k ) = is allowable if the distribution function of u k belongs to the domain of attraction of the normal law. In the following section we shall show that the asymptotic behavior of the DF unit root tests can be derived without the requirement of finite second moment on the iid errors, as commonly assumed in the literature (Dickey and Fuller, 979, 98; Phillips, 987; Chan and Wei, 988; Fuller, 996). 3 DF Tests with Iid Errors Let {u k } in () be iid symmetric random variables with zero mean and a common distribution function from the domain of attraction of a normal law, i.e., u k D N. To begin, we first introduce some notation and then discuss some results for self-normalized sums related to the central limit theorem (CLT) and the invariance principle. Denote by L(Y ) the probability distribution of the random variable Y, by (A) the indicator function of a set A, and by N (0, ) the standard normal distribution. Recent results show that self-normalized sums satisfy the CLT and the invariance principle provided that u k D N and E(u k ) = 0. Giné et al. (997) proved that lim L(S x 2 P( u k > x) n/v n ) = N (0, ) if and only if lim n x E(u 2 = 0. (9) k ( u k x)) The latter condition is equivalent to saying that u k are in the D N (Gnedenko 8

11 and Kolmogorov, 968, p.72; Araujo and Giné, 980, Theorem 6.7). This self-normalized CLT was conjectured by Logan et al. (973). Note that the normal law is the limit for a wide class of distributions, including every distribution with E(u 2 k ( u k x)) being a slowly varying function. This is the case whenever a second moment exists, but E(u 2 k ( u k x)) may also vary slowly when no variance exists (Feller, 966). Thus, the self-normalized CLT above holds under weaker moment conditions than those in the classical limit theorems. In addition, Csörgő et al. (2003, Theorem ) further proved that S [nr] V n W (r), 0 r, (20) where [x] denotes the largest integer less than or equal to x. Csörgő et al s (2003) theorem is a self-normalized version of the FCLT generalizing the theorem of Giné et al. (997). We particularly note that the identically distributed assumption on the sequence {u k } is not necessary for deriving the results shown in (9) (20), see Egorov (996) and Kruglov and Petrovskaya (200). Based on (9) and (20) together with the continuous mapping theorem (Billingsley, 968, p.3) and the fact that W () has a N (0, ) distribution, it is readily seen that if u k are iid symmetric errors and if their distribution functions belong to the D N, then the asymptotic distributions of the DF test statistics in (5) (8) are given by (ˆρ n, ˆτ n, ˆρ µn, ˆτ µn ) (W ρ, W τ, W ρµ, W τµ ). (2) 9

12 Result (2) is identical to those of the corresponding statistics reported in (6) (9), but to derive these asymptotic distributions the only requirement we use is that u k D N. This weakens the finite second moment condition assumed by Dickey and Fuller (979, 98) and by Fuller (996). To illustrate that the self-normalized approach is flexible enough to be applied in a wide class of error processes, in the following section {u k } is assumed to be short-memory and long-memory linear processes, respectively. 4 DF Tests with Long- and Short-Memory Errors Let u k be generated by the linear process u k = θ(l)ε k, θ(l) = θ j L j, (22) where L is the lag operator, {ε k } k Z is an iid random sequence with zero j=0 mean and unit variance, and {θ j } j 0 is a sequence of real numbers. In the context of linear processes, it is well-known that stringent restrictions typically have to be imposed on the rate of decay of θ j. As mentioned in Introduction, {u k } is said to be stationary with short-memory if j=0 θ j < and with long-memory if j=0 θ j = but j=0 θ2 j < (see Hall, 992). Recall S t = t k= u k, t n, and V 2 n = n k= u2 k. Let S εt = t k= ε k, t n, and V 2 εn = n k= ε2 k. Then from Section 3, S ε[nr]/v εn W (r), 0 r. In this section, we establish two results, one for the 0

13 long-memory case (Theorem below), the other for the short-memory case (Theorem 2 below). Theorem. Let {y t } and {u t } be generated according to () and (22), respectively, and let ε k, k Z, be iid random variables with E(ε 0 ) = 0 and E(ε 2 0 ) =. Let θ() = but j=0 θ2 j Then we have j=n+ <. Suppose that, as n, θ j = o(θ()). (23) λ ln S [nr] V n W (r), (24) where 0 r and λ 2 ln := V 2 n /[θ()v εn ] 2 0 in probability, and (a) ˆρ n 2 W 2 () ; 0 W 2 (r)dr (b) ˆτ n in probability; (c) ˆρ µn 2 W 2 () W () 0 W (r)dr 0 W 2 (r)dr [ ; 0 W (r)dr]2 (d) ˆτ µn in probability. Proof of Theorem. To prove (24), we star by writing S [nr] V n = S [nr]/[θ()v εn ] V n /[θ()v εn ]. (25) From this, we have λ ln S [nr] /V n = S [nr] /[θ()v εn ]. Put S j,ε[nr] = [nr] k= ε k j and V 2 j,εn = n k= ε2 k j. In particular, let S ε[nr] = S 0,ε[nr] and V 2 εn = V 2 0,εn. Noting S [nr] = θ()s ε[nr] + n j=0 θ j(s j,ε[nr] S ε[nr] ) + j=n+ θ j(s j,ε[nr]

14 S ε[nr] ) and Vj,εn 2 /V εn 2 p as n, by using (23) and the fact that for any j 0, S j,ε[nr] S ε[nr] = S j,ε[nr] V j,εn S ε[nr] V εn V j,εn V εn V εn = S j,ε[nr] V j,εn S ε[nr] V εn + o p () = O p () (26) is stochastically bounded self-normalized partial sums (Giné et al., 997; Chistyakov and Götze, 2004), we have that S [nr] θ()v εn = S ε[nr] V εn + = S ε[nr] V εn + j=0 j=0 θ j (S j,ε[nr] S ε[nr] ) θ()v εn + j=n+ θ j (S j,ε[nr] S ε[nr] ) θ()v εn θ j (S j,ε[nr] S ε[nr] ) θ()v εn + o p (). (27) Note that for 0 j < [nr], S j,ε[nr] and S ε[nr] are overlapped. Put Θ j = j k= θ n+ k, j n. Note that as n, Θ n and Θ n /θ(). After a straightforward calculation, the second term of the right hand side of (27) can be written as θ j (S j,ε[nr] S ε[nr] ) j=0 θ()v εn = Θ n θ() j= Θ j ε j n Θ n V εn Θ j ε j (n [nr]). (28) Θ n By Kolmogorov s Three Series Theorem, it is easy to show that as n, n j= ε j n/v εn and n j= ε j (n [nr])/v εn converge almost surely. This fact, together with Θ n, implies that j= Θ j Θ n ε j n V εn 0 a.s. and j= j= V εn Θ j Θ n ε j (n [nr]) V εn 0 a.s. (29) by Kronecker s lemma (see Petrov, 995, p.209), where notation a.s. stands for almost surely. Then we have S [nr] θ()v εn = S ε[nr] V εn + o p () W (r), 0 r, (30) 2

15 proving (24). To deal with the term λ ln = V n /[θ()v εn ] in (25) we note that V 2 n = [θ(l)ε k ] 2 = k= j=0 θ 2 j V 2 j,εn + 2 j=0 i=j+ θ j θ i ε k j ε k i =: A + A 2. (3) k= Noting θ 2 () = j=0 θ2 j + 2 j=0 i=j+ θ jθ i, by assumption this equality implies that 2 j=0 i=j+ θ j θ i /θ 2 (). (32) Note that, as n, V 2 εn/n p and n n k= ε k jε k i p 0 by the law of large numbers (LLN). Therefore, by (32) we have A 2 θ 2 ()V 2 εn Similar to (33), it is readily seen that = 2 j=0 i=j+ θ jθ n i k= ε k jε k i θ 2 () Vεn 2 p 0. (33) A θ 2 ()V 2 εn = j=0 θ2 j θ 2 () V 2 j,εn V 2 εn p 0, (34) since Vj,εn 2 /V εn 2 p and j=0 θ2 j /θ2 () 0 by assumption. So we conclude from (33) and (34) that λ 2 ln = V 2 n /[θ()v εn ] 2 p 0. The results of parts (a) (d) follow directly from (5) (8) and (24). The proof of Theorem is complete. Remark. If θ j are of the form l(j)/j β, j, β (/2, ), for some slowly varying function l, then θ j are regularly varying with index β and {u k } 3

16 has long-range dependence. By Karamata s theorem (see Theorem.5.3 and Proposition.5. in Bingham et al. (987)), sup {θ j } θ n+ 0 as n, (35) j n+ where a n b n denotes lim n a n /b n =. In this case the condition in (23) is automatically satisfied since (35) implies that j=n+ θ j 0 as n. Remark 2. Confining attention to ε k being iid random variables, Theorem asserts that even though the coefficients θ j decay slowly, the asymptotic distribution is the standard Brownian motion (as in Wang et al. (2002)) rather than the fractional Brownian motions (cf., Lubian, 999; Wang et al., 2003; Wu and Min, 2004). Under long-range dependence, since the only condition given in the theorem automatically holds (see Remark above), we believe that the theorem imposes the weakest conditions (or no condition) on the coefficients θ j than previous ones in the literature, for example in Wang et al. (2002). Specifically, as done in the latter paper, Theorem also abolishes the condition E(ε 4 k ) <, which represents a further improvement of the result of the latter paper in the respect that the θ j are allowed to decay sufficiently slowly. Theorem 2. Let {y t } and {u t } be generated according to () and (22), respectively, and let ε k, k Z, be iid random variables with E(ε 0 ) = 0 and E(ε 2 0 ) =. Let j=0 θ j < and θ() 0. Then we have S [nr] V n λ s W (r), 0 r, (36) 4

17 where λ 2 s := j=0 θ2 j /θ2 () <, and (a) ˆρ n 2 W 2 () λ 2 s ; 0 W 2 (r)dr (b) ˆτ n W 2 () λ 2 s 2λ s [ ; 0 W 2 (r)dr] /2 (c) ˆρ µn 2[W 2 () λ 2 s] W () 0 W (r)dr 0 W 2 (r)dr [ ; 0 W (r)dr]2 (d) ˆτ µn λ s 2[W 2 () λ 2 s] W () 0 { W (r)dr 0 W 2 (r)dr [ 0 W (r)dr]2} /2. Proof of Theorem 2. The proof of Theorem 2 is similar to the one of Theorem. First, since by assumption j=0 θ j <, it is necessary to have θ j 0 as j. It implies that as n, j=n+ θ j 0. By this fact and by (26), similar to (27) we have S [nr] θ()v εn = S ε[nr] V εn + j=0 θ j (S j,ε[nr] S ε[nr] ) θ()v εn + o p (). (37) Put Θ j = n k=j θ k, j n. Then the second term of the right hand side of (37) can be written as j=0 θ j (S j,ε[nr] S ε[nr] ) θ()v εn = j= Θ j ε j θ()v εn j= Θ j ε [nr]+ j θ()v εn =: M M 2. (38) Again, since the coefficients θ j are absolutely summable, it is sufficient to show that Θ j /θ() 0 as j. Noting that V 2 εn/n p as n, it thus follows from the Toeplitz lemma (Stout, 974, p.20) that E(M 2 ) = n n j= [ Θ j /θ()] 2 0. Then the weak LLN (Petrov, 995, p.34) implies that M p E(M ) = 0. By the same method used above, we can also show that M 2 p 0, and thus details are omitted. Now, we conclude from (37) and (38) that S [nr] /[θ()v εn ] = S ε[nr] /V εn + o p () W (r), 0 r. 5

18 Next, define λ 2 sn = V 2 n /[θ()v εn ] 2. Similar to (3), (33) and (34), we have that λ 2 sn p j=0 θ2 j /θ2 () =: λ 2 s <. Then (36) follows immediately from this and the result of the previous paragraph. Finally, the results of parts (a) (d) follows directly from (5) (8) and (36). The proof of Theorem 2 is complete. Remark 3. For short-memory linear processes with iid innovations, Theorem 2 holds under no condition on the coefficients θ j except that both j=0 θ j 0 and j=0 θ j <. Again, the required moment conditions are E(ε 2 k ) < rather than E(ε4 k ) < (cf. Wu and Min, 2004). It is easily seen that the theorem improves the previous similar results given by Phillips and Solo (992), Phillips and Xiao (998), Chang and Park (2002), Wang et al. (2002), and Wu and Min (2004). 5 Conclusion In contrast to the classical FCLT, the self-normalized version of the theorem possesses a major advantage that moment conditions are not the main prerequisite for convergence of the distribution of statistics based on the self-normalized partial sums (Giné et al., 997; Csörgő et al., 2003; Hall and Wang, 2004; Chistyakov and Götze, 2004). The paper demonstrates that the self-normalized limit theorems are particularly useful in characterizing the asymptotic distributions of the DF tests for a unit root. Thus, it seems 6

19 reasonable to expect that it would be helpful in deriving asymptotic distributions for many statistics arising from the study of nonstationary time series. 7

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