Economics Letters 97 (2007) 133 – 137 www.elsevier.com/locate/econbase
Testing for causality in variance under nonstationarity in variance Paulo M.M. Rodrigues a,1 , Antonio Rubia b,⁎ a
b
Faculty of Economics, University of Algarve, Campus de Gambelas, 8005-139 Faro, Portugal Department of Financial Economics, University of Alicante, Campus de San Vicente, CP 03080, Spain Received 25 August 2006; received in revised form 12 December 2006; accepted 12 February 2007 Available online 14 June 2007
Abstract Van Dijk, Osborn and Sensier [van Dijk, D., Osborn, D.R., Sensier, M., 2005, Testing for causality in variance in the presence of breaks. Economics Letters 89, 193–199.] have recently shown, through Monte Carlo simulation, that causality-in-variance tests may suffer severe finite sample size distortions in the presence of neglected structural breaks. In this paper we provide the theoretical foundation to characterize such departures. The asymptotic theory allows to depict a more complete and general analysis. © 2007 Elsevier B.V. All rights reserved. Keywords: Structural changes; Volatility; Causality; Variance breaks JEL classification: C12; C22
1. Introduction In a recent paper, van Dijk, Osborn and Sensier (2005) [DOS] show, through Monte Carlo simulation, that causality-in-variance tests applied to macroeconomic series may suffer from severe finite sample size distortions in the presence of neglected structural breaks in variance. In this paper, we provide the theoretical justification for these findings as a particular case of a fairly general class of nonstationary volatility processes. Our analysis shows that the size departures are not only a small-sample effect but will also remain asymptotically because of the failure to consistently estimate cross-correlations in this ⁎ Corresponding author. Tel./fax: +34 965 90 36 21. E-mail addresses:
[email protected] (P.M.M. Rodrigues),
[email protected] (A. Rubia). 1 Tel.: +351 289 817 571; fax: +351 289 815 937. 0165-1765/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2007.02.032
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context. Hence, procedures intended to account for the potential nonstationary nature of volatility, such as the structural-break pre-testing proposed in DOS, are deserved. 2. Test for causality in variance Let ε t = (ε1t, ε2t)′ be a square-integrable bivariate time-series where ε it ≔ σ it η it , i = 1, 2, and σ it and η it are characterized in Assumption A below. Given the volatility estimates σ ˆ it, the squared 2 2 ˆ ˆ ˆ 22(0)) − 1 with (k) = γ (k)(γ (0)γ sample cross-correlation coefficient at lag k is estimated as ρ ˆ 12 12 11 PT 2 2 −1 ˆ ˆ ˆ gˆ ii ð0Þ ¼ T t¼1 ðgit −1Þ ; git ¼ eit =rit , and
gˆ 12 ðkÞ ¼
8 T X > > −1 > T ðgˆ 21t −1Þðgˆ 22;t−k −1Þ; > < > > > > : T
t¼kþ1 Tþk X −1
ðgˆ 21t −1Þðgˆ 22;t−k −1Þ;
if kz0 ð1Þ if kb0:
t¼1
Cheung and Ng (1996) Ppropose testing the null that ε2t does not cause ε1t in variance through the test statistic T CN ¼ T Kk¼1 qˆ 212 ðkÞ, which is asymptotically distributed as a χ K2 . The reverse hypothesis that ε1t does not cause ε2t in variance can be tested analogously, summing the squaredcorrelations from k = − K to − 1. In finite P samples in which K/T may be relatively large, the ⁎ ¼ T K ðT =ðT−kÞÞ2qˆ 2 ðkÞ, may provide a better control of asymptotically equivalent statistic T CN 12 k¼1 the empirical size. Hong (2001) generalizes the idea of using a weighted sum of sample squaredcorrelations and a normalized test statistic pffiffiffiffiffiffiffiffiffiffiffiffiffiffiin which all the cross-correlations may be used, PTproposes −1 ϖ2 ðk; KÞqˆ 212 ðkÞ−MðϖÞ= 2V ðϖÞ, where ϖ(k; K ) is weighting function (e.g., i.e., T H⁎ ¼ ½T k¼1 Bartlett kernel) and M(ϖ) and V(ϖ) are (known) normalizing constants such that T H⁎ is asymptotically distributed as a standard normal. 3. Asymptotic analysis The asymptotic results are discussed under the following assumption. Assumption A. i) ηit is an i.i.d. series with E(ηit) = 0, E(ηit2) = 1, and E(|ηit|2(4+δ)) b ∞ for some δ N 0 and all t. ii) σi[t/T] ≔ ωi(t/T), with ωi(s): {s ∈ (−∞, 1]} → R+ being a non-stochastic, measurable and uniformly bounded function which satisfies a Lipschitz condition except at a finite number of possible points of discontinuity. Assumption A permits a wide class of nonstationary paths in the univariate volatility processes such as, multiple structural breaks or polynomial trends, among others, with the case of homoskedastic stationarity holding for ωi(t/T) = σi. The positive innovation variance is required only to be bounded and to display a limited number of jumps. A detailed discussion of the class of variance processes allowed is given in Cavaliere (2004). In order P to save space, we focus on the effects related to the unconditional sample variance estimators ¯ r 2i ¼ T −1 Tt¼1 e2it used in the Monte Carlo simulation in DOS, and discuss only the case for testing causality in one direction (k N 0), as the reverse case is completely analogous. A more general proof is available from the authors upon request. The asymptotic behavior involved in the sample estimation of the ⁎ and T ⁎ tests are provided below. The proofs are sketched in Appendix A. cross-correlations used in the TCN H
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Lemma 1. For any 0 b k b T define κ = limT→∞ k/T. Under Assumption A, for any i = 1, 2, it follows as T → ∞ that: R1 R1 P P i) T −1 Tt¼1 evit YEðgvit Þ 0 xvi ðrÞdr and T −1 Tt¼kþ1 evit YEðgvit Þ j xvi ðrÞdr; vN1. ii) T −1
R1 2 2 2 2 t¼kþ1 e1t e2;t−k Yð1−jÞEðg1t g2;t−k Þ j
PT
x21 ðrÞx22 ðr−jÞdr.
R1 s 2 ˆ 2it ¼ e2it =r Proposition 1. Under the same conditions as inLemma 1, define g ; n ¼ ¯ ¯ 2i ¼ i;s i 0 xi ðrÞdr; r P T 2 2 of {η ˆ 1t, η ˆ 2t } T −1 t¼1 e2it ; i ¼ 1; 2, and assume that the sample cross-covariances and cross-correlations R 1 are computed for k N 0R according to(1). Given the constant terms AðjÞ :¼ ðn1;2 n2;2 Þ−1 j x21 ðrÞx22 ðr−jÞdr; 1 Di ðj; sr Þ :¼ ðni;s Þ−1 j x2i ðsr Þdr, define BðjÞ ¼ ð1−jÞð1 þ AðjÞÞ−D1 ðj; rÞ−D2 ðj; r−jÞ. Then, as T → ∞, ð2Þ gˆ 12 ðkÞYð1−jÞAðjÞg12 ðkÞ þ BðjÞ :¼ g12⁎ðkÞ and −1 −1 2 2 qˆ 212 ðkÞYg2⁎ 12 ðkÞ½C 1 ðVarðg1t Þ þ 1Þ−1 ½C 2 ðVarðg2t Þ þ 1Þ−1
ð3Þ
with Ci ¼ ni;4 =ðni;2 Þ2 . 4. Discussion Some important theoretical implications emerge from the previous analysis. The simulations carried out in DOS focus on the case of a single break. This is a particular case of the more general case analyzed here. 1) If k is kept fixed as T diverges (e.g., κ = 0), then ˆγ12(k) → γ12(k) only if at most one of the series is nonstationary. If both series are nonstationary, the cross-covariances are generally biased even in asymptotic samples, with the total effect depending on the specific characteristics of the data generation process. In any case, in finite samples in which k/T may be large, small-sample biases will remain even 1 after correction factors intended to remove (1 − κ), such as T(T − k)− , since R 1 2 applying R 1 the 2 j x1 ðrÞdrV 0 xi ðrÞdr. Intuitively, the proportion of the sample which is not considered when computing the kth order covariance represents a loss of information about the total variation of the nonstationary volatility process P which is nevertheless present in the sample variance. 2) The sample estimator T −1 Tt¼1 ðgˆ 2it −1Þ fails to converge in probability to Var(ηit2 ) if σit is nonstationary. Even though cross-covariances may be estimated consistently under some restrictions, cross-correlations are not, due to the asymptotic bias which is characterized by the nonstationarity of the fourth-order moment and the (cross-products) of the Ci ratios. 2 2 , ε2t } under AssumptionpA, 3) Finally, the central limit theory applies to the partial sums of {ε1t ffiffiffiffi and it can be shown under the null hypothesis that i) if only one series is nonstationary, then Tqˆ 12 ðkÞ converges to a non-standard normal distribution with zero mean and variance depending on a nuisance ratio givenpby pffiffiffiffibias caused by nonstationarity, and ii) if both series are ffiffiffiffi the asymptotic ⁎ and TH⁎ diverges to infinite as nonstationary, then Tqˆ 12 ðkÞ ¼ Op ð T Þ þ Op ð1Þ and hence TCN T → ∞.
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Therefore, using standard critical values leads generally to size distortions even asymptotically, where the extent of the size departure is data-dependent and is characterized by the nonstationary nature of the volatility. Acknowledgement We thank Dick Van Dijk and an anonymous referee for comments and suggestions. Financial support from POCTI/FEDER (grant ref. POCTI/ECO/49266/2002) and the SEJ2005-09372/ECON and GRJ0613 projects is gratefully acknowledged. Appendix A. Technical Appendix R1 Proof of Lemma 1. i) Define Zit,v = εitv −ωiv(t/T )E(ηitv) for any v ≤ 4 and ni;v ¼ 0 xvi ðrÞdr; fi;v ¼ sup0 V r V1 xvi ðrÞ. From Assumption A, it follows that EðZit;v jF it Þ ¼ Eðxvi ðt=TÞÞðEðevit −Eðgvit ÞÞjF it−1 Þ ¼ 0, where F it ¼ rðgij ; jVtÞ, and, hence, fZit;v ; F it g forms a martingale difference sequence (MDS). Minkowski's inequality shows that ||εitv −ωiv(t/T )E(ηitv)||2v+δ ≤ζi,v||(εitv −E(ηitv))||2v+δ, where this term is uniformly bounded because under Assumption Axvi ðd Þ is integrable on the interval [0, 1] for any finite v, and ||ηit||p b ∞ for any p ≤ 2 (4 +δ) and all t. Hence, the strong law of large numbers (SLLN) for MDS applies, and it follows a:s P P P P R ðtþ1Þ=T that T −1 Tt¼1 evit −T −1 Tt¼1 xvi ðt=TÞEðgvit ÞY 0. Also, Eðgvit ÞT −1 Tt¼1 xvi ðt=TÞ ¼ Eðgvit Þ Tt¼1 t=T P xvi ð½rT=T Þdr ¼ Eðgvit Þni;v . Thus, we conclude that T −1 Tt¼1 evit YEðgvit Þni;v . Applying a similar reasoning R1 P allows us to show that, for any 0 b κb 1 such that k ¼ ½jT ; T −1 Tt¼kþ1 evit YEðgvit Þ j xvi ðrÞdr. ii) The proof 2 2 2 2 2 2 (t/T ) =ω12(t/T )ω22(t/T ) and Yt =ε1t ε2t −ω12 (t/T )E(η1t η2t). From follows the same lines as that of i). Denote ω12 1+δ Cauchy–Schwartz's inequality E|Yt| b ∞ under Assumption A and clearly EðYt jF t−1 Þ ¼ 0. As in i) above, it follows from Minkowski's inequality and under Assumption A that fYt ; F t g is an uniformly L2-bounded P MDS and the SLLN for MDS can again be applied. Noting that T −1 Tt¼1 x212 ðt=T ÞEðg21t g22t Þ ¼ R R P P ðtþ1Þ=T 2 1 x12 ð½rT =T ÞdrYEðg21t g22t Þ 0 x212 ðrÞdr, it follows T −1 Tt¼1 e21t e22t Yðg12 ð0Þ þ 1Þ Eðg21t g22t Þ Tt¼1 t=T R1 2 2 2 2 0 x1 ðrÞx2 ðrÞdr, with γ12(0) =E(η1tη2t) −1. The R 1 along the same lines to show for any 0 b κ b 1 that P result can readily be extended T −1 Tt¼kþ1 e21t e22t−k Yð1−jÞðg12 ðkÞ þ 1Þ 0 x21 ðrÞx22 ðr−jÞdr. □ Proof of Proposition 1. i) For any 0 b k b T, define κ = limT → ∞k/T, 0 ≤ r ≤ 1, and the constant terms AðjÞ; Di ðj; sv Þði ¼ 1; 2Þ and BðjÞ as in the main text (see Proposition 1). The k-th sample cross2 2 /σ¯12 , ε2t /σ¯12 } is given by covariance based on {ε1t T X 1 gˆ 12 ðk Þ ¼ e21t e22;t−k þ 2 T r¯21 r ¯ 2 t¼kþ1
T T T−k 1 X 1 X 2 e − e2 − T T r¯21 t¼kþ1 1t T r¯22 t¼kþ1 2;t−k
! ¼ I aT þ ðI bT Þ; say:
From Lemma 1 ii) it follows that I aT Yð1−jÞAðjÞðg12 ðkÞ þ 1Þ, while Lemma 1 i) and Slutsky's theorem makes clear (for v = 2, i = 1, 2) that I bT Yð1−jÞ−D1 ðj; rÞ−D2 ðj; r−jÞ, and thus gˆ 12 ðkÞYð1−jÞ P PT 4 −1 AðjÞg12 ðkÞ þ BðjÞ. To compute the cross-correlations note that T −1 Tt¼1 ðgˆ2it −1Þ2 ¼ ¯r−4 i T t¼1 eit −1.
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P From Lemma 1 i) we know that T −1 Tt¼1 e4it YEðg4it Þni;4 , while (σ ¯i2)2 → (ξi,2)2. Therefore,noting that Var −2 R1 2 P T 2 2 2 4 −1 2 ˆ x ðrÞdr (ηit ) = E(ηit ) − 1, it is easy to see that T i t¼1 ðgit −1Þ YCi Varðgt−1 Þ þ ðC i −1Þ, with Ci ¼ 0 R1 4 x ðrÞdr. From these results, (3) follows straightforwardly. □ i 0 References Cavaliere, G., 2004. Unit root tests under time-varying variances. Econometric Reviews 23, 259–292. Cheung, Y.W., Ng, L.K., 1996. A causality in variance test and its implications to financial market prices. Journal of Econometrics 72, 33–48. Hong, Y., 2001. A test for volatility spillover with applications to exchange rates. Journal of Econometrics 103, 183–224. van Dijk, D., Osborn, D.R., Sensier, M., 2005. Testing for causality in variance in the presence of breaks. Economics Letters 89, 193–199.