Recursive and rolling regression-based tests of the seasonal unit root hypothesis

Journal of Econometrics 105 (2001) 309–336 www.elsevier.com/locate/econbase

Recursive and rolling regression-based tests of the seasonal unit root hypothesis Richard J. Smitha , A.M. Robert Taylorb ; ∗ a Department

of Economics, University of Bristol, Alfred Marshall Building, Bristol BS8 1TN, UK b Department of Economics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK Received 22 June 1998; revised 13 March 2001; accepted 30 April 2001

Abstract This paper is concerned with rolling and recursive regression-based implementations of tests for seasonal unit roots in a univariate time series process. These tests are based on changing subsamples of the data and thus allow one to test the conventional -xed seasonal unit root hypothesis against the alternative that the process under investigation admits a stable autoregressive root over part, if not all, of the sample at either the zero or seasonal frequencies. Asymptotic critical values are provided together with representations for the limiting distributions of these test statistics. A -nite sample size and power study of the proposed test statistics is also reported together with a discussion on the problem of lag truncation selection in the context of rolling and recursive test regressions. An application of the proposed test statistics to seasonally unadjusted U.K. consumers’ expenditure on tobacco is considered. ? 2001 Elsevier Science S.A. All rights reserved. JEL classi&cation: C12; C15; C22; C52 Keywords: Auxiliary regressions; Maximum; Minimum and di7erence of maximum and minimum seasonal unit root tests

1. Introduction Applied studies using test statistics for seasonal integration based on Hylleberg et al. (1990) (HEGY), Beaulieu and Miron (1993) and Canova ∗

Corresponding author. E-mail address: [email protected] (A.M.R. Taylor).

0304-4076/01/$ - see front matter ? 2001 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 7 6 ( 0 1 ) 0 0 0 8 3 - 5

310

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

and Hansen (1995) have indicated that seasonal unit root behaviour appears to be a relatively common feature of macroeconomic time series. See, in particular, Hylleberg et al. (1993), Canova and Hansen (1995) and Scott (1995). However, as with the non-seasonal Dickey and Fuller (1979) (DF) unit root test statistic, these approaches are predicated on the assumption that the order of integration of the process under analysis is uniquely either zero or one at the seasonal frequencies. This paper develops tests for seasonal unit roots against the possibility that the order of integration of a seasonal time series may change over time. The conventional assumption of a constant order of integration for a time series is contentious. For example, Ericsson et al. (1994) have argued that “... a series’s order of integration could di7er for di7erent time periods”, while Hendry and Mizon (1993) state that the inferences drawn from conventional unit root tests “... can only provide a rough guide since the associated tests are conditional on untested, and usually unlikely, auxiliary hypotheses concerning constancy of the parameters in the scalar representations”. To accommodate these criticisms of the conventional approach, Banerjee et al. (1992) (BLS) suggest tests for a non-seasonal unit root against the alternative that the process displays stationary autoregressive behaviour in at least some if not all of its history to date. BLS consider the minimum and maximum of a sequence of DF test statistics (together with their di7erence) computed by recursive and rolling least squares across changing sub-samples of the data and tabulate their critical values which are obtained under the unit root null hypothesis. In the non-seasonal context, BLS demonstrate that recursive and rolling regression-based DF statistics are a useful and important tool for the detection and highlighting of structural breaks and intra-sample regime shifts in the unit root properties of the time series. In practice, models with time-varying parameters might arise through a number of circumstances. Engle and Watson (1985) suggested three possible forms: unobserved causes, misspeci-ed or behavioural models. The -rst type associates the time-varying behaviour of the model parameters with an unobserved cause whereas the second recognises that model misspeci-cation may be ameliorated by allowing some of the model parameters to display time variation. The third model type stems from the Lucas (1976) critique indicating a close link between time-varying parameter models and economic policy analysis. Consequently, it is important not only to investigate the order of integration of a seasonal time series process at each of the seasonal frequencies but also to investigate the constancy or otherwise over time of the order of integration at each seasonal frequency. The former question is addressed in, inter alia, HEGY and Smith and Taylor (1998a) (ST1) for the quarterly case, while Canova and Hansen (1995) and Smith and Taylor (1999) (ST2) develop tests for arbitrary seasonal aspect. The latter issue is addressed here. Although

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

311

some authors have already recognised the need for recursive tests of seasonal unit roots (Franses, 1996, pp. 69 –70; Hall et al., 1997), as yet no formal statistical tests have been developed. Section 2 describes recursive and rolling regression-based approaches for testing the constancy over time of the order of integration at each seasonal frequency. Asymptotic representations for the proposed test-statistics are given in Section 3. Asymptotic critical values are also provided. For quarterly data, a Monte Carlo investigation into the size and power properties of these tests, relative to the full sample tests of HEGY, are presented in Section 4. An empirical application to U.K. quarterly consumers’ expenditure on tobacco is provided in Section 5. Section 6 concludes.

2. Recursive and rolling regression-based seasonal unit root tests To motivate the hypothesis structure for the recursive and rolling regressionbased test statistics discussed below, consider the time-varying autoregressive (TVAR) process {xSt+s }:

St+s (L)[xSt+s − s∗ − ∗s (St + s)] = uSt+s ; s = 1 − S; : : : ; 0;

t = 1; 2; : : : ;

(2.1)

where S denotes the seasonal aspect, for example, in the quarterly contexts of Sections 4 and 5, S = 4, and the Sth order TVAR polynomial St+s (L) ≡
1 − Si=1 i;∗St+s Li . The lag operator L operates on the process {xSt+s } in the standard manner, LSj+k xSt+s = xS(t−j)+s−k , whereas for the purely seasonal de∗ and LSj+k ∗s = ∗s−k if 1 − S 6 s − k 6 0 terministic parameters LSj+k s∗ = s−k ∗ and ∗S+s−k otherwise, k = 0; : : : ; S − 1; j = 1; 2; : : :. The data genand S+s−k eration process (DGP) (2.1) allows for the presence of di7erential seasonal intercept and time-trend terms via s∗ and ∗s , respectively, s = 1 − S; : : : ; 0. The error process {uSt+s } is assumed to follow a stationary AR(p)
process; that is, p (L)uSt+s = St+s , where the lag polynomial p (L) = 1 − pi=1 i Li is such that the roots of p (z) = 0 all lie outside the unit circle |z | = 1 and the process {St+s } is a martingale di7erence sequence (MDS) satisfying Assumption A in BLS (1992, p. 273). For notational convenience, we de-ne S ∗ = (S=2) − 1 (if S is even) and [S=2] (if S is odd) where [S=2] denotes the integer part of S=2. We assume that the investigator has available the realisation xSt+s , s = 1 − S; : : : ; 0; t = 1; : : : ; T , and the pre-sample observations xs ; s = 1 − S − p; : : : ; 0. The TVAR(S) process (2.1) may be alternatively expressed as

St+s (L)xSt+s = s; St+s + s; St+s (St + s) + uSt+s ;

(2.2)

312

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336



where s; St+s ≡ ( s∗ − Sk=1 k;∗ St+s ) + Sk=1 k k;∗ St+s ∗s−k and s; St+s ≡ ∗s −
S ∗ ∗ ∗ ∗ ∗ ∗ k=1 k; St+s s−k again with the convention s−k ≡ S+s−k and s−k ≡ S+s−k , if s − k ¡ 1 − S; k = 1; : : : ; S; s = 1 − S; : : : ; 0. Note that the seasonal deterministic terms { s; St+s }0s=1−S and { s; St+s }0s=1−S in (2.2) are also time-varying. The purpose of this paper is to describe tests for the overall seasonal unit root hypothesis H0 : St+s (L) = 1 − LS ; s = 1 − S; : : : ; 0; t = 1; : : : ; T;

(2.3)

against the alternative hypothesis H1 , that the seasonal process {xSt+s } displays stationary autoregressive behaviour in at least part of its history at, at least one of, the frequencies !k ≡ 2k=S; k = 0; : : : ; [S=2]. Under H0 , (2.2) is the seasonal unit root process with (-xed) seasonal drifts S xSt+s = s + uSt+s ; s = 1 − S; : : : ; 0; t = 1; : : : ; T;

(2.4)

where S ≡ 1 − LS and s = S ∗s ; s = 1 − S; : : : ; 0. To make the above precise, factorise the TVAR lag polynomial St+s (L) at the frequencies !k ; k = 0; : : : ; [S=2], as [S=2]

St+s (L) =



!k; St+s (L);

(2.5)

k=0

where the lag polynomial !0; St+s (L) ≡ (1− 0; St+s L) associates the time-varying parameter 0; St+s with the zero frequency !0 ≡ 0, !k; St+s (L) ≡ [1 − ( k; St+s + k; St+s i) exp(i!k )L] ×[1 − ( k; St+s − k; St+s i) exp(−i!k )L]

= [1 − 2( k; St+s cos !k − k; St+s sin !k )L + ( k;2 St+s + k;2 St+s )L2 ];

√ where i ≡ −1, corresponds to the conjugate (harmonic) seasonal frequencies (!k ; 2 − !k ), whose roots occur as the conjugate pair k; St+s ± k; St+s i with associated time-varying parameters k; St+s and k; St+s ; k = 1; : : : ; S ∗ , and !S=2; St+s (L) ≡ (1+ S=2; St+s L) corresponding to the Nyquist frequency !S=2 ≡  with the time-varying parameter S=2; St+s for S even.  Consequently, from (2.5), H0 of (2.3) may be partitioned as H0 = [S=2] k=0 H0; k , where H0; 0 : 0; S[Tr]+s = 1; s = 1 − S; : : : ; 0; r ∈ [0; 1]; H0; k : k; S[Tr]+s = 1; k; S[Tr]+s = 0; s = 1 − S; : : : ; 0; r ∈ [0; 1];

(2.6)

k = 1; : : : ; S ∗ , and H0; S=2 : S=2; S[Tr]+s = 1; s = 1 − S; : : : ; 0; r ∈ [0; 1]:

(2.7)

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

313

The hypothesis H0; 0 corresponds to a -xed unit root at the zero-frequency !0 = 0, while H0; S=2 yields a -xed unit root at the Nyquist frequency !S=2 =  if S is even. A -xed unit root at the harmonic seasonal frequencies (!k ; 2 − !k ) is obtained under H0; k ; k = 1; : : : ; S ∗ . Correspondingly, the alternative hy pothesis H1 = [S=2] k=0 H1; k , where H1; 0 : 0; S[Tr]+s = 1; r ∈ R0 ; 0; S[Tr]+s ¡ 1; r ∈ R0 ; s = 1 − S; : : : ; 0; H1; k : k;2 S[Tr]+s + k;2 S[Tr]+s = 1; r ∈ Rk ;

k;2 S[Tr]+s + k;2 S[Tr]+s ¡ 1; r ∈ Rk ; s = 1 − S; : : : ; 0;

(2.8)

k = 1; : : : ; S ∗ , and H1; S=2 : S=2; S[Tr]+s = 1; r ∈ RS=2 ;

S=2; S[Tr]+s ¡ 1; r ∈ RS=2 ; s = 1 − S; : : : ; 0:

(2.9)

In (2.8) and (2.9), {Rk }[S=2] k=0 are individually (non-empty) convex subsets of the unit interval [0; 1], for example, unions of non-overlapping sub-intervals of [0; 1]. Note that H1 neither constrains all frequencies to display a break, nor indeed does it require that breaks at di7erent frequencies must occur at the same time. Moreover, the convex subsets {Rk }[S=2] k=0 need not all be proper subsets of [0; 1]; for example, if Rk = [0; 1], then H1; k of (2.8), (2.9) corresponds to the -xed parameter alternative hypothesis considered in HEGY and ST2. Note also that the maintained hypothesis H0 ∪ H1 permits unit roots only at the frequencies !k ; k = 0; : : : ; [S=2]. In order to develop a regression-based testing procedure for H0 of (2.6) and (2.7), consider a -rst order expansion of St+s (L) in (2.5) around the seasonal unit roots exp(±i2k=S); k = 0; : : : ; [S=2], which, neglecting higher order terms, yields after writing k; St+s = 1 + k; St+s ; k = 0; S=2; k; St+s = 1 + (k; St+s =2) and k; St+s = k; St+s =2; k = 1; : : : ; S ∗ :

St+s (L) = (1 − LS ) − 0; St+s L0 (L) − S=2; St+s LS=2 (L) ∗

−L

S  k=1

k (L)(k; St+s sin !k − k; St+s (cos !k − L));

(2.10)

s = 1 − S; : : : ; 0; t = 1; : : : ; T , where 0 (L) ≡ (1 + L + · · · + LS−1 );

S=2 (L) ≡ −(1 − L + L2 − · · · − LS−1 );

∗

2

k (L) ≡ −(1 − L )

S 

[1 − 2 cos !k L + L2 ];

j=k; j=1

k = 1; : : : ; S ∗ , omitting the term −S=2; St+s LS=2 (L) if S is odd.

314

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

The seasonal unit root hypothesis H0 of (2.6) and (2.7) may then be re-expressed within the local expansion (2.10) via the rede-nitions: H0; 0 : 0; S[Tr]+s = 0; s = 1 − S; : : : ; 0; r ∈ [0; 1]; H0; k : k; S[Tr]+s = k; S[Tr]+s = 0; s = 1 − S; : : : ; 0; r ∈ [0; 1];

(2.11)

k = 1; : : : ; S ∗ , and H0; S=2 : S=2; S[Tr]+s = 0; s = 1 − S; : : : ; 0; r ∈ [0; 1]:

(2.12)

The existence of unit roots at the zero, Nyquist and harmonic seasonal frequencies therefore implies that 0; S[Tr]+s = 0; S=2; S[Tr]+s = 0 and k; S[Tr]+s = k; S[Tr]+s = 0; k = 1; : : : ; S ∗ , respectively, s = 1 − S; : : : ; 0; r ∈ [0; 1]. The alternative hypothesis H1 is equivalently stated in terms of the components H1; 0 : 0; S[Tr]+s = 0; r ∈ R0 ; 0; S[Tr]+s ¡ 0; r ∈ R0 ; s = 1 − S; : : : ; 0; H1; k : k; S[Tr]+s = 0; r ∈ Rk ; k; S[Tr]+s ¡ 0; r ∈ Rk ; s = 1 − S; : : : ; 0; (2.13)

k = 1; : : : ; S ∗ , and H1; S=2 : S=2; S[Tr]+s = 0; r ∈ RS=2 ; S=2; S[Tr]+s ¡ 0; r ∈ RS=2 :

(2.14)

In order to provide powerful tests of H0 of (2.11), (2.12) against H1 of (2.13), (2.14), we follow BLS and analyse sequences of recursive and rolling regression-based seasonal unit root tests. The recursive regression tests use [T0 ] start-up observations, where 0 ¿ 0 is the warm-up fraction, and assume that the autoregressive lag polynomial St+s (L) is time-invariant for s = 1 − S; : : : ; 0; t = 1; : : : ; [T], where  travels across the interval [0 ; 1]. The rolling regression tests assume that St+s (L) is time-invariant for s = 1 − S; : : : ; 0; t = [T]; : : : ; [T] + [T0 ], where 0 ¿ 0 is the window-fraction with [T0 ] the window-width, and allow  to travel across the interval [0; 1 − 0 ]. Consequently, within the intervals t = 1; : : : ; [T] and t = [T]; : : : ; [T]+[T0 ], respectively, we may substitute a time-invariant version of (2.10) for St+s (L) into (2.2). The sequences of auxiliary regression equations, in each case indexed by , are then given by S xSt+s = s + s (St + s) + 0 (; 0 )x0; St+s−1 + S=2 (; 0 )xS=2; St+s−1 ∗

+

S  k=1

+

p  j=1

(k (; 0 )xk; St+s−1 + k (; 0 )xk; St+s−1 ) ∗j S xSt+s−j + St+s

(2.15)

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

315

deleting the term S=2 (; 0 )xS=2; St+s−1 if S is odd, s = 1 − S; : : : ; 0. For the recursive regressions, (2.15) is estimated over the sequence t = 1; : : : ; [T],  ∈ [0 ; 1], while for the rolling regressions, (2.15) is estimated over t = [T]+ 1; : : : ; [T] + [T0 ];  ∈ [0; 1 − 0 ]. To avoid cumbersome notation, we have assumed that for each increment in t a full seasonal aspect S of data is used by the investigator. However, in practice, applied researchers will wish to increment the sequences of auxiliary recursive and rolling regressions in (2.15) both in s and t; that is, to add one observation at a time. As detailed in Remark 3.4 below, the asymptotic results of Section 3 remain unaltered in this circumstance. In the sequence of auxiliary regressions in (2.15), we have de-ned the transformations of the level process {xSt+s } corresponding to !k = 2k=S, k = 0; : : : ; [S=2]: x0; St+s ≡ 0 (L)xSt+s ; xS=2; St+s ≡ S=2 (L)xSt+s ; xk; St+s ≡ −k (L)(cos !k − L)xSt+s ; xk; St+s ≡ k (L) sin !k xSt+s ;

(2.16)

k = 1; : : : ; S ∗ , where the lag polynomials k (L); k = 0; : : : ; [S=2], are given above in (2.10). Therefore, within the above intervals, we may use the regression-based testing procedures developed in ST2 for the presence or otherwise of unit roots at the zero and seasonal frequencies !k ; k = 0; : : : ; [S=2]. This allows the practitioner to test individually for a unit versus a shifting autoregressive root at a particular frequency while remaining ambivalent concerning the presence or otherwise of a unit root at the remaining frequencies. Hence, the individual null hypotheses appropriate for the samples indicated in (2.15) above corresponding to the partition of H0 of (2.11) and (2.12) are de-ned by H0; k (; 0 ) : k (; 0 ) = 0; k = 0; S=2; H0; k (; 0 ) : k (; 0 ) = k (; 0 ) = 0; k = 1; : : : ; S ∗ , with  ∈ [0 ; 1] in the case of the recursive sequences of auxiliary regressions in (2.15), and  ∈ [0; 1 − 0 ] in the case of the rolling sequences of regressions in (2.15). Using least squares (LS) estimation of the recursive and rolling sequences of auxiliary regressions in (2.15), we therefore consider sequences of t- and F-statistics for the individual null hypotheses stated above denoted as: tk (; 0 ); k = 0; S=2, and Fk (; 0 ); k = 1; : : : ; S ∗ , where  ∈ [0 ; 1] in the case of the recursive sequence of statistics and  ∈ [0; 1 − 0 ] in the case of the rolling sequences statistics. Commensurate with the structure of the partitions of H0 and H1 given above, the t-statistics, tk (; 0 ), are one-sided against the corresponding stationary alternative H1; k (; 0 ) : k (; 0 ) ¡ 0; k = 0; S=2, while to test for a conjugate pair of unit roots at the kth harmonic frequency pair (!k ; 2 − !k ), the overall F-tests Fk (; 0 ) for H0; k (; 0 ) respectively are considered, k = 1; : : : ; S ∗ , in all cases with  ∈ [0 ; 1] and  ∈ [0; 1 − 0 ] for the recursive and rolling cases, respectively. In order to avoid the inherent multiple testing problem, particularly for larger values of S, in the same spirit as Ghysels et al. (1994)

316

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

and Taylor (1998), joint frequency recursive and rolling regression F-tests,  [S=2] F1:::[S=2] (; 0 ) and F0:::[S=2] (; 0 ), for [S=2] k=1 H0; k (; 0 ) and k=0 H0; k (; 0 ), respectively, may be opposite, where again  ∈ [0 ; 1] and  ∈ [0; 1 − 0 ] for the recursive and rolling cases, respectively. The overall null hypothesis at frequency !k considered in the recursive and rolling sequences of auxiliary regressions in (2.15), obtained as  travels over the respective intervals [0 ; 1] and [0; 1 − 0 ], is  identical to H0; k of (2.11).However, as indicated above, although H1; k = [0 ;1] H1; k (; 0 ) and H1; k = [0; 1−0 ] H1; k (; 0 ) for the recursive and rolling sequences, respectively, it should be expected that tests for H0; k (; 0 );  ∈ [0 ; 1], or H0; k (; 0 ),  ∈ [0; 1 − 0 ], that have power against, respectively, H1; k (; 0 );  ∈ [0 ; 1], or H0; k (; 0 );  ∈ [0; 1−0 ], should also be powerful against H1; k ; k = 0; : : : ; [S=2]. We follow BLS and consider certain functions of the sequences of rolling and recursive regression t- and F-statistics detailed above to test the time invariant seasonal unit root hypothesis H0 = [S=2] k=0 H0; k of (2.11), (2.12) against  the time-varying alternative hypothesis H1 = [S=2] k=0 H1; k of (2.13), (2.14). For the case of the sequence of recursive statistics from (2.15), we consider the maximal and minimal statistics from these sequences, together with their respective di7erences, viz., tkmax (0 ) = max[0 ;1] tk (; 0 );

tkmin (0 ) = min[0 ;1] tk (; 0 );

tkdi7 (0 ) = tkmax (0 ) − tkmin (0 );

k = 0; S=2;

Fkmax (0 ) = max[0 ;1] Fk (; 0 );

Fkmin (0 ) = min[0 ;1] Fk (; 0 );

Fkdi7 (0 ) = Fkmax (0 ) − Fkmin (0 );

k = 1; : : : ; S ∗ :

(2.17)

The corresponding quantities from the sequence of rolling statistics from (2.15) may be de-ned analogously with the optimisations taking place instead over the interval [0; 1 − 0 ]. In practice, of course, the optimisation of both the recursive and rolling t- and F-statistics will also be taken over s. In addition to the maximal and minimal t- and F-statistics above, we also consider maximal, minimal and di7erence statistics, de-ned in a similar manner to those in (2.17), obtained from the other sequences of recursive and rolling regression F-statistics discussed above. We denote these statistics by max min di7 (0 ); F1:::[S=2] (0 ); F1:::[S=2] (0 )); (F1:::[S=2] max min di7 (F0:::[S=2] (0 ); F0:::[S=2] (0 ); F0:::[S=2] (0 ));

(2.18)

where the optimisation is understood to have been taken over the interval [0 ; 1] in the case of the recursive sequences of statistics from (2.15), and over [0; 1 − 0 ] in the case of the rolling sequences.

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

317

In our initial presentation we allow for the possibility that, under H0 of (2.3), there may be di7erential seasonal drift; see (2.4). Other scenarios for the deterministics, which are special cases thereof, may be identi-ed within (2.1) and, thus, the recursive and rolling sequences of auxiliary regressions from (2.15). We delineate the six cases of interest for the deterministics as follows: (i) no intercept, no trend: ∗s = 0; ∗s = 0; s = 1 − S; : : : ; 0; (ii) constant intercept, no trend: ∗s = ; ∗s = 0; s = 1 − S; : : : ; 0; (iii) seasonal intercepts, no trend: ∗s = 0; s = 1 − S; : : : ; 0; (iv) constant intercept, constant trend: ∗s = ; ∗s = ; s = 1 − S; : : : ; 0; (v) seasonal intercepts, constant trend: ∗s = ; s = 1 − S; : : : ; 0; (vi) seasonal intercepts, seasonal trends: as in (2.1) and (2.15). However, in all cases, we will continue to adopt the notation outlined above to represent the corresponding sequences of t- and F-statistics, and functions thereof. 3. Asymptotic representations Firstly, in Theorem 3.1 below, the asymptotic distributions of the sequences of recursive regression t- and F-statistics, tk (; 0 ); k = 0; S=2, and Fk (; 0 ); k = 1; : : : ; S ∗ , are derived under H0 of (2.3). Theorem 3.2 then provides the limiting representations for the max; tkmax (0 ) and Fjmax (0 ); min, tkmin (0 ) and Fjmin (0 ), and di7erence, tkdi7 (0 ) and Fjdi7 (0 ); k = 0; S=2 and j = 1; : : : ; S ∗ , recursive regression t- and F-statistics of (2.17). The asymptotic distributions of the sequences of rolling regression t- and F-statistics and of the recursive and rolling regression joint frequency F-statistics follow as immediate corollaries. In deriving our results we consider {uSt+s } ∼ AR(p), as in (2.1), but we restrict p to be -nite; cf. BLS. However, in a similar manner to Zivot and Andrews (1992, p. 257), it may be conjectured that if {uSt+s } follows a stationary and invertible ARMA process with martingale di7erence innovations and, hence, admits a stationary AR(∞) representation, then, by setting p = o(T 1=3 ) in (2.15), the results given below will continue to have asymptotic validity; cf. Said and Dickey (1984). Given that the e7ective sample size at each recursion (2.15) is S[T], one should set p = o([T]1=3 ) in the augmented regression, estimated over s = 1 − S; : : : ; 0; t = 1; : : : ; [T];  ∈ [0 ; 1]; 0 ¿ 0. For rolling estimation of the augmented regression (2.15), the e7ective sample size is of constant length S[T0 ] throughout the sequence and setting p = o([T0 ]1=3 ) should suPce. In the results given below, the superscript # relates to the cases (i) – (vi) for the deterministics in the sequence of auxiliary regressions in (2.15) de-ned at the end of Section 2. Hence, for the zero frequency !0 tests: case (i): # = 0; cases (ii) and (iii): # = 1; cases (iv), (v) and (vi): # = 2. For the seasonal frequency !k tests, k = 1; : : : ; [S=2]: cases (i), (ii) and (iv): # = 0;

318

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

cases (iii) and (v): # = 1; case (vi): # = 2. In the analysis which follows, weak convergence is denoted by “ ⇒ ”. S−1 and, for We de-ne the independent standard Brownian motions {Wj (r)}j=0 0 j = 0; : : : ; S − 1, their recursive counterparts Wj (r; ) ≡ Wj (r),  1  1 Wj (s) d s (3.1) Wj (r; ) ≡ Wj (r) −  0 and Wj2 (r; )

≡

Wj1 (r; )

− 12

−3



 r− 2







0

  s− Wj1 (s; ) d s: 2

(3.2)

Theorem 3.1. Under H0 of (2:3); the recursive t- and F-statistics; tk (•; 0 ); k = 0; S=2; and Fk (•; 0 ); k = 1; : : : ; S ∗ ; estimated from the auxiliary regression (2:15); indexed by  ∈ [0 ; 1]; 0 ¿ 0; possess limiting representations: Ak (•) ≡ tk∗ (•; 0 ); k = 0; S=2; (Bk (•))1=2

tk (•; 0 ) ⇒ Fk (•; 0 ) ⇒

1 Ak; (•)2 + Ak;  (•)2 ≡ Fk∗ (•; 0 ); 2 Bk (•)

(3.3) k = 1; : : : ; S ∗ ;

(3.4)

where 

Ak () =



0

Wk# (r; ) d Wk# (r; ); Bk () =

 0



Wk# (r; )2 d r; k = 0; S=2;

and; for k = 1; : : : ; S ∗ ;   # # Ak; () = [Wk# (r; ) d Wk# (r; ) + WS−k (r; ) d WS−k (r; )]; 0



Ak;  () =

0



Bk () =

0





# # [Wk# (r; ) d WS−k (r; ) − WS−k (r; ) d Wk# (r; )];

# [Wk# (r; )2 + WS−k (r; )2 ] d r:

Proof. Follows directly from the asymptotic convergence results given in Section 4 of ST2 and those of Zivot and Andrews (1992). Corollary 3.1. For  ∈ [0 ; 1]; 0 ¿ 0; under H0 of (2:3); the sequence of recursive F-statistics; F1:::[S=2] (•; 0 ) and F0:::[S=2] (•; 0 ); estimated from the

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

319

auxiliary regression (2:15); are such that

S∗  1 ∗ ∗ 2 F k (• ;  0 ) F1:::[S=2] (•; 0 ) ⇒ t (•; 0 ) + 2 S − 1 S=2 k=1 ∗ ≡ F1:::[S=2] (•; 0 );

1 F0:::[S=2] (•; 0 ) ⇒ S

(3.5)



t0∗ (•; 0 )2

∗

+

∗ tS=2 (•; 0 )2

∗ ≡ F0:::[S=2] (•; 0 );

+2

S 



Fk∗ (•; 0 )

k=1

(3.6)

∗ (•; 0 )2 if S is odd. dropping the term tS=2

Proof. Follows immediately from the asymptotic orthogonality results in the appendix to ST2. Remark 3.1. From (3.3), for given #, the t0 (•; 0 ) and tS=2 (•; 0 ) sequences of t-statistics possess identical and independent limiting representations. If # = 2, these are identical to the limiting distribution of the recursive sequence of DF t-statistics for a regression containing an intercept and trend given in BLS (Theorem 1, pp. 273–274). Furthermore, the Fk (•; 0 ) sequences possess independent and identical limiting distributions across k = 1; : : : ; S ∗ , which are independent of those for t0 (•; 0 ) and tS=2 (•; 0 ); cf. (3.3) and (3.4). Remark 3.2. The results in Theorem 3.1 and Corollary 3.1 apply uniformly in . The marginal limiting distributions at any -xed  are those that would be obtained using conventional (-xed ) asymptotics; cf. ST2 (Section 4). Remark 3.3. Under H0 of (2.3), corresponding limiting representations for the rolling sequences of t- and F-statistics, indexed by  ∈ [0; 1 − 0 ], obtain as a consequence of Theorem 3.1. They have the same structure as those given in (3.3) – (3.6) but with the limits of integration changed from [0; ] to S−1 , [; +0 ] and the independent Brownian motion functionals {Wj# (r; ; 0 )}j=0 # S−1 replacing {Wj (r; )}j=0 , where, for j = 0; : : : ; S − 1,   0 −3 2 1 r− Wj (r; ; 0 ) ≡ Wj (r; ; 0 ) − 120 2   +0  0 × s− Wj1 (s; ; 0 ) d s; (3.7) 2 

320

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

cf. (3.2), Wj1 (r; ; 0 ) ≡ Wj (r) −

1 0



+0



Wj (s) d s;

(3.8)

cf. (3.1), and Wj0 (r; ; 0 ) ≡ Wj (r). Remark 3.4. The above results recurse or roll on the t index alone. Practitioners, however, will wish to add one extra observation to the sample at a time rather than S observations as above. Doing so, however, does not alter the asymptotic representations given in Theorem 3.1 and Corollary 3.1 for the recursive regression statistics or change Remark 3.3 for the rolling regression statistics as the additional terms are all oP (1). Remark 3.5. The power study in Section 4 indicates that it is advantageous also to consider reverse recursive (rolling) sequences of t- and F-statistics. These sequences are computed by reversing the order of the data and then proceeding exactly as above for the forward recursive (rolling) statistics. An adaptation of the limiting distribution theory provided in Leybourne (1995) for the seasonal case shows that the limiting null representations for the backward recursive sequences of t- and F-statistics are of the same form as those given in (3.3) – (3.6) except that the limits of integration are changed from [0; ] to [1 − ; 1]. The limiting representations for the backward rolling test statistic sequences are also of the same form as those in (3.3) – (3.6) except the limits of integration are changed from [;  + 0 ] to [1 −  − 0 ; 1 − ] as are those in (3.7) – (3.8); see Remark 3.3. Clearly, the backward and recursive forms of both types of statistics are asymptotically dependent. Theorem 3.2. Under H0 of (2:3); tkmax (0 ) ⇒ sup tk∗ (; 0 ) ≡ tkmax∗ (0 ); [0 ;1]

tkmin ⇒ inf tk∗ (; 0 ) ≡ tkmin∗ (0 ); [0 ;1]

tkdi7 (0 ) ⇒ tkmax∗ (0 ) − tkmin∗ (0 ); k = 0; S=2;

(3.9)

Fkmax (0 ) ⇒ sup Fk∗ (; 0 ) ≡ Fkmax∗ (0 ); [0 ;1]

Fkmin (0 ) ⇒ inf Fk∗ (; 0 ) ≡ Fkmin∗ (0 ); [0 ;1]

Fkdi7 (0 ) ⇒ Fkmax∗ (0 ) − Fkmin∗ (0 ); k = 1; : : : ; S ∗ :

(3.10)

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

321

∗ max∗ max (; 0 ) ≡ F1:::[S=2] (0 ); F1:::[S=2] (0 ) ⇒ sup F1:::[S=2] [0 ;1]

min∗ min ∗ F1:::[S=2] (0 ) ⇒ inf F1:::[S=2] (; 0 ) ≡ F1:::[S=2] (0 ); [0 ;1]

(3.11)

di7 max∗ min∗ (0 ) ⇒ F1:::[S=2] (0 ) − F1:::[S=2] (0 ); F1:::[S=2] max ∗ max∗ (0 ) ⇒ sup F0:::[S=2] () ≡ F0:::[S=2] (0 ); F0:::[S=2] [0 ;1]

∗ min∗ min F0:::[S=2] () ≡ F0:::[S=2] (0 ); (0 ) ⇒ inf F0:::[S=2] [0 ;1]

di7 (0 ) F0:::[S=2]

max∗ min∗ ⇒ F0:::[S=2] (0 ) − F0:::[S=2] (0 ):

(3.12)

Proof. Follows directly from (3.3) – (3.6) and the continuous mapping theorem via the continuity of the sup and inf functionals; see, inter alia, McCabe and Tremayne (1993, p. 186). Remark 3.6. Theorem 3.2 indicates that each of the (tkmax (0 ); tkmin (0 ); tkdi7 (0 )) statistics have identical and independent limiting distributions across k = 0, S=2. Furthermore, (Fkmax (0 ); Fkmin (0 ); Fkdi7 (0 )) have identical and independent limiting distributions across k = 1; : : : ; S ∗ and are asymptotically independent of (tkmax (0 ); tkmin (0 ); tkdi7 (0 )); k = 0; S=2. Remark 3.7. The limiting representations for the corresponding t- and Fstatistics from the sequence of rolling regression t- and F-statistics are as in (3.9) – (3.12), but with the optimisation now taken over the interval [0; 1 − 0 ]; see Remark 3.3. Remark 3.8. We have considered the maximum and minimum values from sequences of statistics de-ned only in terms of the t index. As in Remark 3.4, however, this leads to no loss of generality asymptotically. Remark 3.9. The limiting representations in (3.9) – (3.12), regardless of whether the statistics involved are taken from recursive or rolling sequences of t- and F-statistics, depend upon the parameter 0 . To conclude this section we provide simulated critical values for the limiting representations (3.9) – (3.10) given in Theorem 3.2. 1 The asymptotic critical 1 A suite of GAUSS programs to replicate both the asymptotic and -nite sample Monte Carlo experiments reported in this paper, together with a selection of -nite-sample critical values for the case of quarterly data (S = 4), is available from web site, http:== web.bham.ac.uk= R.Taylor= The GAUSS programs may also be used to develop corresponding results for other values of 0 , S and T .

322

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

Table 1 Asymptotic critical values of the tkmax (0 ), tkmin (0 ), tkdi7 (0 ), Fjmax (0 ), Fjmin (0 ) and Fjdi7 (0 ), k = 0; S=2, j = 1; : : : ; S ∗ , recursive (0 = 0:25) seasonal unit root testsa tkmin (0 ) #

tkmax (0 )

0.010

0.025

0 −3:24 1 −4:05 2 −4:64

−2:91 −3:76 −4:35

0.050 −2:64 −3:52 −4:11

0.100 −2:32 −3:25 −3:85

Fjmin (0 )

−1:09 −2:02 −2:40

0.025 −0:82 −1:71 −2:14

0.050 −0:57 −1:46 −1:92

0.100 0.900 −0:26 −1:16 −1:67

Fjmax (0 )

#

0.010

0.025

0.050

0.100

0 1 2

0.36 1.82 3.61

0.60 2.39 4.26

0.88 2.95 4.89

1.25 3.70 5.67

a The

0.010

tkdi7 (0 )

0.010 4.71 8.32 12.08

3.25 3.30 3.18

0.950

0.975

0.990

3.70 3.71 3.53

4.11 4.08 3.86

4.57 4.50 4.26

0.975

0.990

Fjdi7 (0 ) 0.025 5.58 9.45 13.32

0.050 6.42 10.45 14.53

0.100 0.900 7.55 11.80 16.13

4.42 7.13 9.64

0.950 5.24 8.11 10.82

6.04 9.06 11.95

7.07 10.29 13.41

parameter # is as de-ned for each of the statistics immediately above Theorem 3.1.

Table 2 Asymptotic critical values of the tkmax (0 ), tkmin (0 ), tkdi7 (0 ), Fjmax (0 ), Fjmin (0 ) and Fjdi7 (0 ), ˙ seasonal unit root testsa k = 0; S=2, j = 1; : : : ; S ∗ , rolling (0 = 0:33) tkmax (0 )

tkmin (0 ) #

0.010

0 −3:86 1 −4:71 2 −5:20

0.025 −3:54 −4:45 −4:94

0.050 −3:29 −4:23 −4:73

0.100 −3:00 −3:99 −4:49

−0:64 −1:19 −1:81

0.025 −0:41 −0:98 −1:61

0.050 −0:19 −0:79 −1:45

0.100

0.900

0.950

0.975

0.990

0.09

4.36 4.65 4.24

4.78 5.02 4.56

5.17 5.35 4.84

5.61 5.74 5.20

0.950

0.975

0.990

−0:56 −1:26

Fjmax (0 )

Fjmin (0 ) #

0.010

0.025

0.050

0.100

0 1 2

0.15 0.84 2.44

0.25 1.11 2.84

0.38 1.37 3.22

0.57 1.71 3.68

a The

0.010

tkdi7 (0 )

0.010 5.70 11.05 14.92

Fjdi7 (0 ) 0.025 6.63 12.19 16.19

0.050 7.50 13.30 17.37

0.100 8.61 14.73 18.93

0.900 5.61 10.71 13.54

6.53 11.83 14.79

7.37 12.93 15.98

8.50 14.36 17.48

parameter # is as de-ned for each of the statistics immediately above Theorem 3.1.

values were obtained via direct Monte Carlo simulation of the limiting functionals given in (3.9) – (3.10) in a similar fashion to that outlined in Appendix B of Zivot and Andrews (1992). The simulations were programmed using the RNDN function of GAUSS 3.1 on a Pentium 400 MHz micro-computer for samples of length 1000 with 80,000 replications for each simulation. We have followed BLS and set 0 = 0:25 for the recursive test statistics and 0 = 0:33˙ for the rolling test statistics; see the discussion in BLS, (1992, p. 277). Simulated asymptotic critical values for the recursive t- and F-statistics of (3.9) and (3.10) are provided in Table 1. Table 2 reports the corresponding

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

323

critical values for the rolling statistics. As an example of how to use these tables, the asymptotic 0.05 level critical value for the minimum recursive t0min (0:25) statistic of (2.17) from the auxiliary regression (2.15) including seasonal dummies and seasonal trends is −4:11, whatever the value of S. ˙ statistic The corresponding critical value for the minimum rolling t0min (0:33) min (0:25) is −4:73. The asymptotic 0.05 level critical values for the recursive tS=2 min ˙ statistics from this regression are −3:52 and −4:23, and rolling tS=2 (0:33) respectively. It is worth noting that the critical values for the maximum, minimum and di7erence zero-frequency recursive and rolling tests are very similar to the corresponding critical values for the sequence of DF regressions containing an intercept and trend tabulated in BLS (Table 1, p. 277) as should be expected from Theorem 3.2. Moreover, for a given value of #, the critical values for the zero and Nyquist frequencies coincide; cf. Theorem 3.2. It should be re-emphasised that these critical values are only relevant for 0 = 0:25 in the case of the recursive test statistics and 0 = 0:33˙ in the case of the rolling test statistics. Other choices of 0 would deliver di7erent critical values in each case, as is clear from Theorem 3.2. 4. Finite-sample results In this section, Monte Carlo methods are used to investigate the -nitesample properties of the test statistics proposed in Section 2 for the quarterly case (S = 4). Speci-cally, we evaluate the size properties of the tests in the presence of serially correlated innovations and the power properties of the tests against shifting autoregressive roots relative to the performance of the corresponding full-sample HEGY tests. The sequence of recursive regression test statistics is computed from the auxiliary regression (2.15) estimated over s = 1 − S; : : : ; 0; t = 1; : : : ; [T],  ∈ [0 ; 1], with 0 = 0:25 throughout. In the case of the rolling regression test-statistics, (2.15) was estimated over s = 1 − S; : : : ; 0, t = [T] + 1; : : : ; [T] + [T0 ],  ∈ [0; 1 − 0 ], with 0 = 0:33˙ throughout. In all cases, we focus on the benchmark sample size 4T = 200 (representative results for other values of T are reported where they are of interest) and consider the sequences of auxiliary regressions (2.15) containing seasonal intercepts and a constant (non-seasonal) trend. Although Section 3 gave some asymptotic guidance on selecting the lag truncation order p in (2.15), these rates for p o7er little guidance for the practical choice of lag length. In this section we have adopted the following procedure. First, the sequential rule of Ng and Perron (1995) and Beaulieu and Miron (1993, pp. 318–319), was used to determine the lag truncation order for the auxiliary regression (2.15) estimated over the full sample, s = 1 − S; : : : ; 0; t = 1; : : : ; T , with a maximal

324

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

lag order pmax = 4 and tests on the signi-cance of the lagged variables conducted at the 0.25 level, as recommended in Taylor (1997). The resultant lag order was then imposed upon each sub-sample regression considered in both the recursive and rolling sequences. Corresponding results for regressions containing no lagged dependent variables (unaugmented regressions) or a -xed number of lags are also reported in Smith and Taylor (1998b) (ST3). All unit root tests were performed at a nominal -nite-sample 0.05 level, although neither other choices of the nominal level nor the use of asymptotic critical values altered the results qualitatively. The relevant -nite sample critical values were simulated from the DGP 4 x4t+s = u4t+s , s = − 3; : : : ; 0, t = 1; : : : ; T , where {u4t+s } ∼ IN (0; 1), with the test regressions speci-ed as above. The critical values used in Section 5 were also generated in this way. All experiments were programmed using the RNDN function of GAUSS 3.1 on a Pentium 400 MHz micro-computer using 40,000 replications for each simulation. 4.1. Size properties Table 3 summarizes the -nite-sample size properties of the recursive and rolling regression test statistics when the error process displays weak parametric autocorrelation and compares them with those of the full sample HEGY tests. Table 3 reports the actual size of the test-statistics based on a nominal 0.05 level from (2.15) using the data-dependent choice of the lag truncation order p outlined above, when the true DGP for {x4t+s } is the SARIMA(1; 0; 0) × (0; 1; 1)4 model: (1 − L)4 x4t+s = (1 + )L4 )u4t+s ; s = − 3; : : : ; 0; t = − 4; : : : ; 50;

(4.1)

where {u4t+s } ∼ IN (0; 1) and u4j+s = x4j+s = 0, j 6 − 5, and the -rst 20 observations are discarded to control for the e7ects of the initial conditions. For  ∈ {0:4; 0:6} with ) = 0, the data-dependent method of lag selection appears to have e7ectively purged the -rst order autocorrelation in the error process. All tests display actual size approximately at the nominal level although the (one-sided) rolling statistic t0di7 (0 ) is somewhat over-sized. No size distortions at all were seen for larger values of T . For ) ∈ {−0:2; −0:4} with  = 0 and now setting pmax = 8 in the lag selection procedure to allow for the higher order of autoregression needed to approximate such autocorrelation, a strong similarity is observed between the size distortions for the full sample HEGY tests and their ‘maximum’ and ‘minimum’ recursive and ‘maximum’ rolling counterparts at all frequencies. When pmax = 0,  = 0 and ) = − 0:4, all of the reported test statistics have actual size lying between 0.30 and 0.40 for a nominal 0.05 level; see ST3

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

325

Table 3 Size of seasonal unit root tests (4T = 200; nominal 0:05 level), DGP (4.1): auxiliary regressions include seasonal dummies and a linear trenda Recursive statistics

Rolling statistics

t0

t0min (0 )

t0max (0 )

t0di7 (0 )

t0min (0 )

t0max (0 )

t0di7 (0 )

0.4 0.0 0.6 0.0 0.0 −0:4 0.0 −0:2

0.05 0.04 0.11 0.08

0.05 0.04 0.10 0.07

0.04 0.04 0.11 0.08

0.06 0.07 0.03 0.04

0.05 0.05 0.08 0.06

0.03 0.03 0.12 0.08

0.08 0.10 0.03 0.04



t2

t2min (0 )

t2max (0 )

t2di7 (0 )

t2min (0 )

t2max (0 )

t2di7 (0 )

0.4 0.0 0.6 0.0 0.0 −0:4 0.0 −0:2

0.05 0.05 0.11 0.07

0.04 0.04 0.11 0.07

0.04 0.04 0.11 0.07

0.05 0.04 0.03 0.04

0.04 0.04 0.08 0.06

0.04 0.04 0.13 0.08

0.04 0.04 0.03 0.04



F1

F1min (0 )

F1max (0 )

F1di7 (0 )

F1min (0 )

F1max (0 )

F1di7 (0 )

0.05 0.05 0.11 0.06

0.05 0.05 0.11 0.06

0.05 0.05 0.10 0.06

0.04 0.05 0.08 0.06

0.04 0.04 0.13 0.07

0.05 0.05 0.10 0.06

0.05 0.05 0.08 0.06



)

)

)

0.4 0.0 0.6 0.0 0.0 −0:4 0.0 −0:2 a The

third column of the table refers to the full-sample HEGY test, the fourth to sixth the recursive maximum, minimum and di7erence statistics, respectively, and the seventh to ninth columns the rolling maximum, minimum and di7erence statistics respectively. For the recursive ˙ statistics, 0 = 0:25, while for the rolling statistics 0 = 0:33.

(Tables 3–5). The adopted lag selection procedure therefore seems to be reasonably ePcacious at redressing nominal size levels in both the full-sample and sub-sample tests. Additional experiments (not reported here) showed that, also for 4T = 200, increasing pmax to 12 adequately redressed the empirical size of all of the statistics in Table 3. Moreover, as expected, for -xed pmax , i.e. one that does not increase with T , the size distortions seen when ) = 0 increase, ceteris paribus, as T increases, in both the recursive, rolling and full sample HEGY tests. 4.2. Empirical power This sub-section compares the empirical power properties of the rolling and recursive tests with the full sample tests of HEGY against shifting autoregressive root processes. Tables 4 – 6 report the (size adjusted) power of the statistics at a nominal 0.05 level from the auxiliary regression (2.15) with the lag truncation p again chosen according to our data-dependent rule, when the true DGP for {x4t+s }

326

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

Table 4 Power of zero frequency unit root tests (4T = 200; nominal 0:05 level): auxiliary regressions include seasonal dummies and a linear trenda Recursive statistics

Rolling statistics

t0

t0min (0 )

t0max (0 )

t0di7 (0 )

t0min (0 )

t0max (0 )

t0di7 (0 )

DGP (4.2) – (4.3) 1.0 0:8 F R 1.0 0.6 F R 0.8 1.0 F R 0.6 1.0 F R

0.21 0.15 0.22 0.15 0.15 0.20 0.16 0.21

0.11 0.35 0.13 0.75 0.35 0.12 0.76 0.14

0.09 0.28 0.09 0.34 0.28 0.10 0.35 0.10

0.04 0.10 0.06 0.32 0.10 0.06 0.32 0.06

0.13 0.10 0.40 0.34 0.10 0.13 0.33 0.41

0.20 0.14 0.21 0.14 0.14 0.20 0.13 0.21

0.05 0.06 0.16 0.20 0.07 0.05 0.20 0.17

DGP (4.2) – (4.4) 1.0 0.8 F R 1.0 0.6 F R 0.8 1.0 F R 0.6 1.0 F R

0.14 0.14 0.15 0.15 0.17 0.19 0.20 0.20

0.14 0.17 0.18 0.20 0.16 0.20 0.31 0.32

0.12 0.12 0.12 0.12 0.18 0.18 0.21 0.20

0.04 0.04 0.06 0.06 0.05 0.06 0.15 0.14

0.14 0.14 0.38 0.39 0.10 0.10 0.19 0.19

0.19 0.18 0.19 0.20 0.15 0.14 0.14 0.15

0.05 0.05 0.15 0.15 0.06 0.06 0.11 0.11

11

12

F=R

a The fourth column of the table refers to the full-sample HEGY test, the -fth to seventh the recursive maximum, minimum and di7erence statistics, respectively, and the eighth to tenth columns the rolling maximum, minimum and di7erence statistics, respectively. For the recursive ˙ The column headed F=R denotes statistics, 0 = 0:25, while for the rolling statistics 0 = 0:33. whether the data were reversed (R) prior to computing the test statistics, or not (F).

is the TVAR(4) model:

4t+s (L)(x4t+s − s∗ ) = u4t+s ∼ IN (0; 1); s = − 3; : : : ; 0; t = − 4; : : : ; 50 (4.2) with u4j+s = x4( j−1)+s = 0, j 6 − 5, xs−20 = s∗ , s = − 3; : : : ; 0, and the -rst 20 observations are again discarded to control for the e7ects of the initial conditions. We consider two di7erent mechanisms for the time-varying AR(4) polynomial 4t+s (L). Firstly,

4t+s (L) = (1− 11 L)(1+ 21 L)(1+ 31 L2 ); s = −3; : : : ; 0; t = −4; : : : ; 25;

4t+s (L) = (1− 12 L)(1+ 22 L)(1+ 32 L2 ); s = −3; : : : ; 0; t = 26; : : : ; 50 (4.3)

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

327

Table 5 Power of biannual frequency unit root tests (4T = 200; nominal 0:05 level): auxiliary regressions include seasonal dummies and a linear trenda Recursive statistics

Rolling statistics

t2

t2min (0 )

t2max (0 )

t2di7 (0 )

t2min (0 )

t2max (0 )

t2di7 (0 )

DGP (4.2) – (4.3) 1.0 0.8 F R 1.0 0.6 F R 0.8 1.0 F R 0.6 1.0 F R

0.24 0.13 0.23 0.12 0.13 0.25 0.12 0.24

0.13 0.50 0.13 0.87 0.50 0.12 0.88 0.13

0.11 0.24 0.11 0.22 0.25 0.11 0.24 0.10

0.06 0.25 0.04 0.56 0.24 0.07 0.54 0.04

0.22 0.14 0.58 0.42 0.14 0.22 0.43 0.59

0.24 0.13 0.23 0.10 0.12 0.24 0.10 0.24

0.08 0.09 0.19 0.27 0.09 0.09 0.28 0.18

DGP (4.2) – (4.4) 1.0 0.8 F R 1.0 0.6 F R 0.8 1.0 F R 0.6 1.0 F R

0.15 0.16 0.16 0.16 0.13 0.14 0.13 0.13

0.18 0.18 0.22 0.22 0.19 0.20 0.40 0.38

0.12 0.14 0.14 0.14 0.12 0.13 0.12 0.10

0.06 0.04 0.05 0.06 0.15 0.15 0.35 0.35

0.20 0.19 0.57 0.57 0.13 0.14 0.27 0.26

0.18 0.18 0.16 0.16 0.12 0.12 0.10 0.10

0.08 0.07 0.25 0.25 0.08 0.08 0.19 0.18

21

22

F=R

a The fourth column of the table refers to the full-sample HEGY test, the -fth to seventh the recursive maximum, minimum and di7erence statistics, respectively, and the eighth to tenth columns the rolling maximum, minimum and di7erence statistics, respectively. For the recursive ˙ The column headed F=R denotes statistics, 0 = 0:25, while for the rolling statistics 0 = 0:33. whether the data were reversed (R) prior to computing the test statistics, or not (F).

and secondly

4t+s (L)=(1 − 11 L)(1 + 21 L)(1 + 31 L2 ); s= − 3; : : : ; 0; t = − 4; : : : ; 16;

4t+s (L)=(1 − 12 L)(1 + 22 L)(1 + 32 L2 ); s= − 3; : : : ; 0; t =17; : : : ; 34;

4t+s (L)=(1 − 11 L)(1 + 21 L)(1 + 31 L2 ); s= − 3; : : : ; 0; t =35; : : : ; 50: (4.4) That is, we consider alternatives to the unit root hypothesis where, in the -rst case (4.2), (4.3), the autoregressive root at each frequency potentially varies across the -rst and second halves of the full data-span and, in the second case (4.2), (4.4), across the second vis-Sa-vis the -rst and -nal thirds of the full data-span. In each case, re-parameterising via s∗ = 0 + 2 cos[s] + 1 sin[s=2] + 1 cos[s=2], s = − 3; : : : ; 0, design (4.2) emphasises that, for

328

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

Table 6 Power of annual frequency unit root tests (4T = 200; nominal 0:05 level): auxiliary regressions include seasonal dummies and a linear trenda Recursive statistics

Rolling statistics

F1

F1min (0 )

F1max (0 )

F1di7 (0 )

F1min (0 )

F1max (0 )

F1di7 (0 )

DGP (4.2) – (4.3) 1.0 0.8 F R 1.0 0.6 F R 0.8 1.0 F R 0.6 1.0 F R

0.25 0.10 0.25 0.11 0.08 0.25 0.10 0.24

0.12 0.21 0.15 0.24 0.19 0.12 0.24 0.12

0.11 0.31 0.12 0.76 0.31 0.12 0.76 0.16

0.10 0.33 0.12 0.76 0.26 0.10 0.74 0.15

0.25 0.12 0.25 0.11 0.12 0.26 0.11 0.25

0.20 0.13 0.48 0.35 0.12 0.20 0.34 0.48

0.16 0.10 0.42 0.32 0.10 0.16 0.33 0.42

DGP (4.2) – (4.4) 1.0 0.8 F R 1.0 0.6 F R 0.8 1.0 F R 0.6 1.0 F R

0.16 0.16 0.18 0.18 0.10 0.11 0.10 0.10

0.14 0.13 0.15 0.15 0.10 0.11 0.11 0.11

0.17 0.17 0.23 0.23 0.13 0.15 0.27 0.28

0.12 0.13 0.18 0.20 0.11 0.16 0.27 0.32

0.19 0.19 0.19 0.19 0.12 0.12 0.11 0.11

0.18 0.17 0.46 0.45 0.12 0.12 0.24 0.27

0.15 0.15 0.42 0.42 0.11 0.11 0.23 0.25

2

31

2

32

F=R

a The fourth column of the table refers to the full-sample HEGY test, the -fth to seventh the recursive maximum, minimum and di7erence statistics, respectively, and the eighth to tenth columns the rolling maximum, minimum and di7erence statistics, respectively. For the recursive ˙ The column headed F=R denotes statistics, 0 = 0:25, while for the rolling statistics 0 = 0:33. whether the data were reversed (R) prior to computing the test statistics, or not (F).

a stable root at frequency !k ∈ {0; =2; }, the corresponding trigonometric seasonal intercept is preserved unless the root at frequency !k is of modulus one in which case it is annihilated. This design therefore avoids the problem of spurious sharp jumps to zero at the break dates, a problem remarked upon in BLS (p. 278). We consider the following values for the parameters of

4t+s (L) in our experiments: ij ∈ {0:6; 0:8; 1}, i = 1; 2; 3, j = 1; 2. More precisely, Table 4 reports the power of the zero frequency tests when either

11 ¡ 1 or 12 ¡ 1 and ij = 1, i = 2; 3, j = 1; 2. In Tables 4 – 6, the column headed F=R denotes whether the data were reversed (R) prior to computing the test statistics, or not (F); see Remark 3.6. The sizes of unit root tests at the biannual and annual frequency tests are not reported since in no case did these di7er signi-cantly from the nominal level. Analogously, the power of the biannual and annual frequency tests is reported in Tables 5 and 6, respectively. Allowance was also made for simultaneous movements from the

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

329

-xed unit root null at more than one frequency but hardly altered the reported results. Firstly, for design (4.3), Tables 4 – 6 indicate that the -nite-sample power of the recursive tests is strongly dependent on which half of the data is stationary at the relevant frequency. Using Table 4 as a representative example, when the process has a zero frequency unit root in the -rst half of the sample ( 11 = 1) and a stationary root of 0.6 in the second half ( 12 = 0:6), none of the recursive test statistics display nominal power in excess of 0.15. The full-sample HEGY t0 statistic has power 0.23 whereas the power of the (one-sided) rolling regression t0min (0 ) statistic is 0.40. When the data are stationary at the zero frequency in the -rst half of the data ( 11 = 0:6, 12 = 1), the power of the (one-sided) recursive t0min (0 ) statistic is 0.76 compared to 0.16 for the t0 statistic and 0.33 for the (one-sided) rolling t0min (0 ) statistic. Throughout Tables 4 – 6, the power of the full-sample and rolling HEGY tests are far less sensitive to which half of the sample data is stationary, if anything displaying a reverse pattern to the recursive tests with power somewhat higher when the -rst half of the data is non-stationary. However, in the -rst case ( 11 = 0:6, 12 = 1:0), the reverse recursive (one-sided) t0min (0 ) statistic has power of 0:75. These results are expected given that the forward recursive tests include the initial data while the reverse recursive tests include the end data. At most 66 stationary observations can be used in computing the rolling statistics, regardless of which half of the data are stationary, while, at most, all 100 stationary observations will be available to compute the recursive statistics when the data are stationary in the -rst half of the data (and similarly for the reverse recursive tests when the data are stationary in the second half of the sample). Moreover, from Tables 4 – 6, there appears to be rather less to choose between the power of the forward and reverse full sample and rolling tests, as might be expected. Therefore, in practice, computing both forward and reverse recursive statistics at each frequency would appear sensible. An overall test statistic based on some function of the two statistics such as their minimum might prove useful. Clearly, new tables of critical values would be required; a GAUSS program to calculate these can be obtained from the authors upon request. For the second design (4.4), when the middle third of the data are stationary, the rolling statistics considerably outperform both the full-sample and recursive tests. Notice that, in this case, the size of the window in the rolling tests coincides with the amount of contiguous stationary data. For example, from Table 5, when 21 = 1 and 22 = 0:6, the rolling t2min (0 ) statistic displays power of 0.57 while t2 and the recursive t2min (0 ) statistics have power of 0.16 and 0.22, respectively. When the middle third of the data is non-stationary, the power of the rolling tests declines somewhat; for 21 = 0:6 and 22 = 1, the rolling t2min (0 ) statistic has power 0.27 while t0 and the recursive t2min (0 ) statistics have power 0.13 and 0.40, respectively. Not surprisingly, for the

330

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

symmetric design (4.4), in contradistinction to the asymmetric design (4.3), there is virtually no di7erence between the forward and reverse sample tests. Furthermore, from Tables 4 – 6, certain of the maximum and minimum statistics perform better for di7erent frequencies. In Table 4, both the (onesided) recursive and rolling t0min (0 ) statistics display superior power properties; similarly, the (one-sided) recursive and rolling t2min (0 ) statistics in Table 5. Conversely, in Table 6, it is the recursive and rolling F1max (0 ) statistics that display the superior power. These results are not unexpected given the nature of the critical regions for the respective t- and F-statistics. For the zero (biannual) frequency tests, the hypothesis of a -xed zero (biannual) frequency unit root is rejected in favour of the alternative of a shifting unit-to-stationary zero (biannual) frequency root for large negative values of the resulting statistic. In both the recursive and rolling settings, the statistics tkmin (0 ), k = 0; 2, give most weight to their respective alternative and should therefore select the time-period at which the test “most rejects” the unit root. Conversely, for the harmonic frequency k = 1, because rejection occurs for “large” values of the F-statistic, the maximum value of the sequence of F-tests should be expected to display the highest power, ceteris paribus. Moreover, the powers of tj and the recursive and rolling tjmax (0 ) statistics, j = 0, 2, and F1 , and the recursive and rolling F1min (0 ) statistics under both designs are little a7ected by the degree of deviation of the relevant

ij parameters from unity. For example, in Table 5, for 22 = 1:0, t2 , and the recursive and rolling t2max (0 ) statistics show no increase in power between

21 = 0:8 and 0.6, while the recursive and rolling t2min (0 ) and t2di7 (0 ) statistics all demonstrate a clear increase in power. Similar results are seen for other values of the ij parameters. These results raise suspicions concerning the consistency of tk and the recursive and rolling tkmax (0 ), k = 0; 2, and F1 and the recursive and rolling F1min (0 ) statistics against shifting autoregressive root processes which appear to be con-rmed for larger values of T . To illustrate, for design (4.4) and 21 = 1:0, 22 = 0:8 (0.6) with 4T = 400, t2 and the rolling t2max (0 ) statistics display power of 0.18 (0.17) and 0.19 (0.17), respectively, which are little di7erent from those values in Table 5 for 4T = 200 while the rolling t2min (0 ) and t2di7 (0 ) statistics have power of 0.68 (0.98) and 0.27 (0.70), respectively. The method of selecting the lag truncation parameter p used in Sections 4.2 and 4:3 above appears ePcacious, preserving test size whilst simultaneously retaining test power. We will therefore also adopt this approach in Section 5. Alternative methods include applying the sequential rule of Ng and Perron (1995) and Beaulieu and Miron (1993) to each sub-sample regression considered although this would necessarily be time consuming. Alternatively, the Phillips–Perron-type HEGY tests of Breitung and Franses (1996) might be adopted with the spectral density estimator re-evaluated for each sub-sample regression considered.

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

331

5. An empirical application As a practical illustration of the test statistics developed in this paper, an application to real quarterly, seasonally unadjusted, U.K. consumers’ expenditure on tobacco is considered. This series is observed for the sample period 1955.1 to 1996.1 and was extracted from the U.K. O.N.S. macroeconomic database. Fig. 1 plots the logarithm of this series. Up until the mid-1970s, the series displays very slight upward trending behaviour coupled with the marked, but somewhat irregular, intra-year movements that tend to characterize seasonal unit root processes. After the mid-1970s, however, the trend in the series shows a continuous decline. Between the start of this decline and the beginning of the 1980s the seasonal pattern in the series appears consistent with that prior to the mid-1970s. However, at the start of the 1980s, the series su7ers a sharp drop and, almost simultaneously, the seasonality in the series undergoes a vast diminution with very small seasonal movements remaining afterwards. As will become clear, the full sample HEGY tests completely fail to pick this e7ect up even though a break in the seasonal pattern is quite obvious in Fig. 1. We investigate the log tobacco consumption series for the presence or otherwise of seasonal unit roots, in particular, whether or not the series admits

Fig. 1.

332

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

Table 7 Seasonal unit root tests for UK consumption dataa Statistic

a Signi-cant

t0

t2

t1

t1

F1

F1:::2

F0:::2

−1:96

−2:25

−2:90

−1:99c

6:13b

5:75b

5.38

outcomes at the 0.10 and 0.05 levels are Tagged

b

and

c,

respectively.

the 4 factor in its autoregressive representation using both full sample and sub-sample regressions. To do so we specialise (2.15) to the quarterly case, viz., p (L)4 x4t+s = s + (4t + s) + 0 x0; 4t+s−1 + 2 x2; St+s−1 +1 x1; St+s−1 + 1 x1; St+s−1 + St+s ;

(5.1)

where {x4t+s } denotes the log tobacco consumption series, and, from (2.16), x0; 4t+s ≡ (1 + L + L2 + L3 )x4t+s ; x2; 4t+s ≡ −(1 − L + L2 − L3 )x4t+s ; x1; 4t+s ≡ −L(1 − L2 )x4t+s ; x1; 4t+s ≡ −(1 − L2 )x4t+s :

(5.2)

In the context of (5.1), the stationary lag polynomial p (L) was speci-ed using the procedure outlined in Section 4, initiating the search at pmax = 8 using 0.25 level lag signi-cance tests. The -nal preferred (full sample) model included the lagged variables 4 x4s+s−k , k = 1; 5; 6; 8, which were then used in computing the full-sample HEGY tests and the sequences of rolling and recursive test regressions. Seasonal intercepts and a constant trend, Case (v) of (2.15) detailed at the end of Section 3, were included in (5.1) for both full sample and sub-sample estimation so as to render the resulting unit root tests similar with respect to the starting values of the process and to the possibility of non-seasonal drift under the null hypothesis; see ST1 for further discussion. Table 7 reports the conventional quarterly HEGY tests, t0 , t2 and F1 for the exclusion of x0; 4t+s−1 , x2; 4t+s−1 and (x1; 4t+s−1 ; x1; 4t+s−1 ) from (5.1), respectively, together with the joint frequency F-tests F1:::2 and F0:::2 for the exclusion of (x2; 4t+s−1 ; x1; 4t+s−1 ; x1; 4t+s−1 ) and (x0; 4t+s−1 ; x2; 4t+s−1 ; x1; 4t+s−1 ; x1; 4t+s−1 ) from (5.1), respectively. These were calculated from (5.1), estimated using data from 1958.1 to 1996.1, inclusive. The data from 1955.1 to 1957.4, inclusive, constitute the pre-sample data. The corresponding maximum, minimum and di7erence test statistics derived from the estimation of (5.1) by recursive, reverse recursive (see Remark 3.5) and rolling LS are reported in Table 8. The recursive statistics were obtained by estimating (5.1) recursively with the choice of 0 = 0:25 corresponding to 38 observations. For the -rst subsample (5.1) was therefore estimated using data from 1958.1 to 1967.2, inclusive, with the t- and F-statistics of the previous paragraph computed for this sub-sample, the second from 1958.1 to

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

333

Table 8 Recursive, reverse recursive and rolling regression seasonal unit root tests for UK consumption dataa Statistic t0min (0 ) t0max (0 ) t0di7 (0 ) t2min (0 ) t2max (0 ) t2di7 (0 ) F1min (0 ) F1max (0 ) F1di7 (0 ) min ( ) F1:::2 0 max ( ) F1:::2 0 di7 ( ) F1:::2 0 min ( ) F0:::2 0 max ( ) F0:::2 0 di7 ( ) F0:::2 0

Recursive −2:54

0.44 2.98 −3:29b −1:41b 1:88 3:21b 9:40b 6:20 3:27c 7:61b 4.34 3:40c 7:24 3:84

Reverse recursive −4:07b

0.03 4:10c −2:47 1:44 3:91c 0:60 9:40b 8:77c 0:41 8:72c 8:32c 1:05 13:90d 12:85d

Rolling −3:41

0.19 3.60 −6:19d −0:16 6:03d 1:07c 19:28d 18:21d 0:73 25:05d 24:32d 1:16 21:30d 20:14d

a In the case of the recursive and reverse recursive statistics  = 0:25. For the 0 ˙ Signi-cant outcomes at the 0.10, 0.05 and 0.01 levels rolling statistics 0 = 0:33. are Tagged b , c and d , respectively.

1967.3, and so on, with the -nal (116th) subsample regression coinciding with the full-sample regression above. The maximum and minimum, together with their di7erence, over the resulting sequences of 116 of each of the above tand F-statistics were computed, as in (2.17) and (2.18). The reverse recursive statistics were computed in the same way but applied to the reversed data, so that for the -rst subsample (5.1) was estimated over data corresponding to 1993.1 to 1983.4, inclusive, the second 1993.1 to 1983.3, and so on with the last (116th) recursion estimated over 1993.1 to 1955.1. Again the maximum, minimum and di7erences were taken over the sequences of t- and F-statistics. The rolling statistics were obtained, estimating (5.1) using a rolling window ˙ In the of length 50 observations, corresponding to the choice of 0 = 0:33. -rst subsample, (5.1) was therefore estimated over the sample period 1958.1 to 1970.3, inclusive, the second subsample estimated over 1958.2 to 1970.4, inclusive, and so on with the -nal (103rd) subsample, (5.1) estimated over the period 1983.3 to 1996.1, inclusive. The maximum, minimum and di7erences were again taken over the resulting sequences of t- and F-statistics. Adoption of an overall signi-cance level of 0.05 for the full sample joint frequency F0:::2 statistic implies an approximate 0.01 level for the zero and Nyquist frequency t-statistics t0 and t2 , an (approximate) 0.025 level for the harmonic frequency F1 statistic and a corresponding (approximate) 0.0375 level for the joint seasonal frequency F1:::2 statistic; (these approximations are

334

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

also germane for the test statistics reported in Table 8). Table 7 reveals that, on the basis of the reported HEGY tests, application of the 4 -lter appears to be statistically acceptable. Although some Tagged statistics are shown for certain aspects of the partition of H0 of (2.11) and (2.12), these individual outcomes are not suPciently signi-cant (using the above approximations) to overturn H0 at an overall 0.05 signi-cance level. Again adopting an overall 0.05 signi-cance level, the joint frequency maxmax (0:25), of Table 8 does not reject the -xed seaimum recursive statistic, F0:::2 sonal unit root hypothesis H0 of (2.3) whereas the corresponding minimum min (0:25), does reject H0 at the same level. However, recursive statistic, F0:::2 generally speaking, the recursive tests appear to be in broad agreement with the full-sample HEGY tests providing marginally more evidence against the -xed seasonal unit root hypothesis. The joint frequency maximum and di7erence reverse recursive statistics, max di7 (0:25) and F0:::2 (0:25), respectively, and joint frequency maximum and F0:::2 max di7 ˙ respectively, all reject ˙ and F0:::2 (0:33), (0:33) di7erence rolling statistics, F0:::2 the -xed seasonal unit root hypothesis H0 at the 0:01 level. In the case of the rolling statistics, similar patterns of rejection are also seen for the maximum max di7 ˙ and F1:::2 ˙ (0:33) (0:33) and di7erence joint seasonal frequency statistics, F1:::2 respectively. Moreover, in the case of the rolling statistics, the minimum and ˙ and t2di7 (0:33), ˙ respectively, and di7erence Nyquist frequency tests, t2min (0:33) ˙ the maximum, minimum and di7erence harmonic frequency tests F1min (0:33), max di7 ˙ and F1 (0:33), ˙ respectively, all reject the hypothesis of a -xed F1 (0:33) unit root, at the approximate signi-cance levels given above. However, no rejections are seen for the zero frequency rolling statistics. The remaining reverse recursive tests provide less (more) evidence against the -xed unit root hypothesis than do the corresponding rolling (forward recursive) statistics. To be precise, although consumers’ expenditure on tobacco is deemed to be seasonally di7erence stationary by full sample HEGY tests, the recursive and, in particular, rolling variants of the HEGY tests developed in this paper suggest that the series may in fact display some form of time-varying autoregressive behaviour. That the greatest evidence against the -xed seasonal unit root hypothesis is provided by the reverse recursive and rolling tests is consonant with the Monte Carlo evidence reported in Section 4.2, noting from Fig. 1 that the consumption series appears to change from a seasonal unit root process to a seasonally stationary process at the start of the 1980s. 6. Conclusions This paper has been concerned with recursive and rolling least squares implementations of the regression-based seasonal unit root test statistics of, inter alia, HEGY, Beaulieu and Miron (1993) and ST2. We have proposed

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

335

a number of t- and F-statistics, calculated from estimating the auxiliary regression of interest over changing subsamples of the data. In particular, we focussed on the maximum and minimum values of these sequences of tests, together with the di7erence between these extrema. Representations for the limiting distributions of these statistics under the -xed seasonal unit root null hypothesis are provided, together with simulated asymptotic critical values. We have also investigated the robustness of the -nite-sample size of these test statistics, together with the full sample tests of HEGY, to a variety of SARIMA(1; 0; 0) × (0; 1; 1)4 processes when using -nite-sample critical values appropriate for a quarterly random walk process. Additionally, we have investigated the power of these tests relative to the full-sample HEGY tests when the true process is a shifting autoregressive root process. These results show that the recursive and rolling test statistics display comparable size distortions to their full sample counterparts but display considerably better power properties against shifting autoregressive root processes. Suggestions are also made concerning the problem of lag truncation selection in the context of the rolling and recursive test statistics. In an empirical investigation into the unit root properties of (log) U.K. consumers’ expenditure on tobacco, the (reverse) recursive, and in particular, rolling tests developed in this paper provide considerably more evidence against the -xed seasonal unit root hypothesis than do conventional full sample HEGY tests. Acknowledgements The authors gratefully acknowledge -nancial support for this research provided by the Economic and Social Research Council of the United Kingdom under research grants R 00023 7334 and R 00429 334349, respectively. We are also grateful to an Associate Editor and two anonymous referees for their helpful comments and suggestions on earlier drafts of this paper. References Banerjee, A., Lumsdaine, R.L., Stock, J.H., 1992. Recursive and sequential tests of the unit-root and trend-break hypotheses: theory and international evidence. Journal of Business and Economic Statistics 10, 271–287. Beaulieu, J.J., Miron, J.A., 1993. Seasonal unit roots in aggregate U.S. data. Journal of Econometrics 55, 305–328. Breitung, J., Franses, P.H., 1996. On Phillips–Perron-type tests for seasonal unit roots. Econometric Theory 14, 200–221. Canova, F., Hansen, B.E., 1995. Are seasonal patterns constant over time? a test for seasonal stability. Journal of Business and Economic Statistics 13, 1–16. Dickey, D.A., Fuller, W.A., 1979. Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427–431.

336

R.J. Smith, A.M. Robert Taylor / Journal of Econometrics 105 (2001) 309–336

Engle, R.F., Watson, M.W., 1985. Applications of Kalman -ltering in econometrics. DP no. 1187, Harvard Institute of Economic Research. Ericsson, N.R., Hendry, D.F., Tran, H.-A., 1994. Cointegration, seasonality, encompassing, and the demand for money in the UK. In: Hargreaves, C.P. (Ed.), Nonstationary Time Series Analysis and Cointegration, Oxford University Press, Oxford (Chapter 7). Franses, P.H., 1996. Periodicity and Stochastic Trends in Economic Time Series. Cambridge University Press, Cambridge. Ghysels, E., Lee, H.S., Noh, J., 1994. Testing for unit roots in seasonal time series: some theoretical extensions and a Monte Carlo investigation. Journal of Econometrics 62, 415–442. Hall, S.G., Psaradakis, Z., Sola, M., 1997. Co-integration and changes in regime: the Japanese consumption function. Journal of Applied Econometrics 12, 151–168. Hendry, D.F., Mizon, G.E., 1993. Evaluating dynamic econometric models by encompassing the VAR. In: Phillips, P.C.B. (Ed.), Models, Methods and Applications of Econometrics: Essays in Honour of A.R. Bergstrom. Blackwell, Oxford, pp. 272–300. Hylleberg, S., Engle, R.F., Granger, C.W.J., Yoo, B.S., 1990. Seasonal integration and cointegration. Journal of Econometrics 44, 215–238. Hylleberg, S., JHrgensen, C., SHrensen, N.K., 1993. Seasonality in macroeconomic time series. Empirical Economics 18, 321–335. Leybourne, S.J., 1995. Testing for unit roots using the forward and reverse Dickey–Fuller regression. Oxford Bulletin of Economics and Statistics 57, 559–571. Lucas Jr., R.E., 1976. Econometric policy evaluation: a critique. In: Brunner, K., Melzter, A. (Eds.), Carnegie-Rochester Conferences on Public Policy, Supplementary Series to Journal of Monetary Economics, Vol. 1. North Holland, Amsterdam, pp. 19–46. McCabe, B.P.M., Tremayne, A.R., 1993. Elements of Modern Asymptotic Theory with Statistical Applications. Manchester University Press, Manchester. Ng, S., Perron, P., 1995. Unit root tests in ARMA models with data-dependent methods for the selection of the truncation lag. Journal of the American Statistical Association 90, 268–281. Said, S.E., Dickey, D.A., 1984. Test for unit roots in autoregressive-moving average models of unknown order. Biometrika 71, 599–609. Scott, A., 1995. Why is consumption so seasonal? Oxford University Applied Economics Discussion Paper, No. 172. Smith, R.J., Taylor, A.M.R., 1998a. Additional critical values and asymptotic representations for seasonal unit root tests. Journal of Econometrics 85, 269–288. Smith, R.J., Taylor, A.M.R., 1998b. Recursive and rolling regression-based tests of the seasonal unit root hypothesis. Discussion Papers in Economics, No. 98=19, University of York. Smith, R.J., Taylor, A.M.R., 1999. Regression-based seasonal unit root tests. Department of Economics Discussion Paper. Number 99-15, University of Birmingham. Taylor, A.M.R., 1997. On the practical problems of computing seasonal unit root tests. International Journal of Forecasting 13, 307–318. Taylor, A.M.R., 1998. Testing for unit roots in monthly time series. Journal of Time Series Analysis 19, 349–368. Zivot, E., Andrews, D.W.K., 1992. Further evidence on the great crash, the oil-price shock, and the unit-root hypothesis. Journal of Business and Economic Statistics 10, 251–270.