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Rescaled variance tests for seasonal stationarity

  • Kemal Caglar Gogebakan ORCID logo EMAIL logo
Published/Copyright: July 2, 2021
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Studies in Nonlinear Dynamics & Econometrics
From the journal Studies in Nonlinear Dynamics & Econometrics Volume 26 Issue 4

Abstract

This paper introduces rescaled variance [V/S] tests for seasonal stationarity. The V/S statistic is designed by Giraitis, L., P. Kokoszka, R. Leipus, and G. Teyssière. 2003. “Rescaled Variance and Related Tests for Long Memory in Volatility and Levels.” Journal of Econometrics 112: 265–94 to be the mean corrected versions of the KPSS statistic. In the seasonal context, Canova, F., and B. E. Hansen. 1995. “Are Seasonal Patterns Constant over Time? A Test for Seasonal Stability.” Journal of Business & Economic Statistics 13: 237–52 present the seasonal generalization of the KPSS statistic. In this regard, I aim to strengthen the work of Canova, F., and B. E. Hansen. 1995. “Are Seasonal Patterns Constant over Time? A Test for Seasonal Stability.” Journal of Business & Economic Statistics 13: 237–52 [CH] by mean correction in the seasonal framework. I obtain the asymptotic distributions of the seasonal V/S tests. The V/S tests enjoy better power performance than the CH tests while exhibiting similiar size performance. Furthermore, by data pre-filtering, I propose robustified versions of the V/S statistics to eliminate the unattended unit root problem observed in the CH tests.

Keywords: KPSS statistic; Lagrange multiplier; seasonal stationarity; unit root; V/S statistic
JEL Classification: C12; C14; C22
Corresponding author: Kemal Caglar Gogebakan, Department of Economics, Bilkent University, Ankara, Turkey, E-mail:

Present address: Kemal Caglar Gogebakan, Cancer Early Detection Advanced Research Center, Oregon Health and Science University, Portland, Oregon, U.S.A., E-mail:

Acknowledgments

I would like to thank Bruce Mizrach and two anonymous reviewers for their constructive comments. Furthermore, I am grateful to Emine Sila Ozdemir Gogebakan for reading the earlier version of this manuscript.

  1. Author contribution: The author has accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The author declare no conflicts of interest regarding this article.

Appendix A. Proofs of lemmas and theorems

First, I introduce a preliminary lemma in order to utilize in proving Theorem 1 as follows:

Lemma 1

Under the null of seasonal stationarity, the following convergence holds:

(27) 1 T F ̂ ⌊ T t ⌋ = 1 T ∑ t = 1 ⌊ T t ⌋ f t e ̂ t ⇒ ( Ω f ) 1 / 2 W s − 1 0 ( t )

where W s − 1 0 ( t ) is a vector of Brownian bridge with dimension s − 1 and Ω f is the long-run covariance matrix of f t e t defined in (4).

Proof of Lemma 1

Under the null, from the model defined in Eq. (3), I have

(28) f t e ̂ t = f t ( μ − μ ̂ ) + f t x t ′ ( β − β ̂ ) + f t f t ′ ( γ − γ ̂ ) + f t e t

Under the asssumption of OLS estimation, I have first order condition as 1 T f t ∑ t = 1 T e ̂ t = 0 . So, I have the following equation

(29) 0 = 1 T ∑ t = 1 T f t x t ′ ( β − β ̂ ) + 1 T ∑ t = 1 T f t f t ′ ( γ − γ ̂ ) + 1 T ∑ t = 1 T f t e t

Therefore, by substracting (29) from (28), I obtain

(30) f t e ̂ t = f t ( μ − μ ̂ ) + f t x t ′ − 1 T ∑ t = 1 T f t x t ′ ( β − β ̂ ) + f t f t ′ − 1 T ∑ t = 1 T f t f t ′ ( γ − γ ̂ ) + f t e t − 1 T ∑ t = 1 T f t e t

For the convergence operations, the first term in the RHS of Eq. (30) becomes

(31) 1 T ∑ t = 1 ⌊ T t ⌋ f t ( μ − μ ̂ )

There are two possibilities. If ⌊Tt⌋ in Eq. (31) is a multiple of the number of seasons s, since f t is assumed to be zero-mean process, the value of (31) becomes 0. Otherwise, as T → ∞ I have

sup 0 ≤ t ≤ 1 | 1 T ∑ t = 1 ⌊ T t ⌋ f t | → p 0

and as it is stated in Caner (1998), ( μ − μ ̂ ) is o p (1). The same method can be applied to the third term.

The second term in the RHS of Eq. (30) can be handled similarly. Following Phillips and Solo (1992), by the invariance principle of the linear processes,

1 T ∑ t = 1 ⌊ T t ⌋ f t x t ′ − 1 T ∑ t = 1 T f t x t

converges and is O p (1). And by Pötscher (1991), ( β − β ̂ ) is o p (1).

Finally, to complete the proof, by using functional central limit theorem of Billingsley (2013), the convergence of the fourth term is as follows:

1 T ∑ t = 1 ⌊ T t ⌋ f t e t − 1 T ∑ t = 1 T f t e t = 1 T ∑ t = 1 ⌊ T t ⌋ f t e t − ⌊ T t ⌋ T 1 T ∑ t = 1 ⌊ T t ⌋ f t e t ⇒ ( Ω f ) 1 / 2 ( W s − 1 ( t ) − t W s − 1 ( 1 ) )

where W s − 1 ( t ) − t W s − 1 ( 1 ) = W s − 1 0 ( t ) and Ω f is the is the long-run covariance matrix of f t e t . □

Proof of Theorem 1

I will only prove Part (i) to prove the theorem. For the other parts, similiar techniques can be used. First, consider the first term

(32) 1 T 2 ∑ t = 1 T F ̂ t ′ A 1 ( A 1 ′ Ω ̂ f A 1 ) − 1 A 1 ′ F ̂ t

By definition, I know that A1 is full rank (m − 1) × a matrix where a ≤ m − 1. Therefore A 1 ′ F ̂ t is an a × 1 vector with covariance A 1 ′ Ω ̂ f A 1 . I know that Ω ̂ f is a consistent estimator of Ω f . Therefore, with the help of Lemma 1 and continuous mapping theorem, I can conclude that

(33) 1 T A 1 ′ F ̂ ⌊ T t ⌋ = 1 T ∑ t = 1 ⌊ T t ⌋ A 1 ′ f t e ̂ t ⇒ ( A 1 ′ Ω f A 1 ) 1 / 2 W a 0 ( t )

where W a 0 ( t ) is a Brownian bridge with dimension a. Therefore by continuous mapping theorem

1 T 2 ∑ t = 1 T F ̂ t ′ A 1 ( A 1 ′ Ω ̂ f A 1 ) − 1 A 1 ′ F ̂ t ⇒ ∫ 0 1 W a 0 ( t ) ′ W a 0 ( t ) d t

The second term is the mean correction factor as follows:

1 T 3 ∑ t = 1 T F ̂ t ′ A 1 ( A 1 ′ Ω ̂ f A 1 ) − 1 A 1 ′ ∑ t = 1 T F ̂ t

With the similar arguments above, I have the following convergence result:

1 T 3 / 2 A 1 ′ ∑ t = 1 T F ̂ t ⇒ ( A 1 ′ Ω f A 1 ) 1 / 2 ∫ 0 1 W a 0 ( t )

Therefore, the second term converges to

1 T 3 ∑ t = 1 T F ̂ t ′ A 1 ( A 1 ′ Ω ̂ f A 1 ) − 1 A 1 ′ ∑ t = 1 T F ̂ t ⇒ ∫ 0 1 W a 0 ( t ) ′ d t ∫ 0 1 W a 0 ( t ) d t

The proofs for Part (ii) and (iii) can be done by setting A1 = Is−1, A = ( 0 ̃ I 2 0 ̃ ) ′ and A1 = (01)′ respectively. □

Proof of Theorem 2

Provided that Ω ̂ f is positive semidefinite, Ω ̂ f * will also be positive semidefinite. Rest of the proof is exactly as for Theorem 1, replacing Ω ̂ f and F ̂ t by Ω ̂ f * and F ̂ t * . □

B. Tables for critical values and simulation results

Table 1:

Critical values for U V / S p , by degrees of freedom (p) and significance level (ζ).

p ζ
1% 5% 10%
1 0.263 0.186 0.152
2 0.416 0.308 0.267
3 0.534 0.420 0.373
4 0.640 0.527 0.476
5 0.753 0.635 0.572
6 0.864 0.735 0.673
7 0.967 0.832 0.763
8 1.083 0.933 0.862
9 1.176 1.025 0.963
10 1.293 1.123 1.054
Table 2:

Size performance of seasonal rescaled variance tests (V/S) and Canova and Hansen (1995) (CH) tests (τ = 0).

V/S tests CH tests
q = 0 q = T1/3 q = 0 q = T1/3
b π π/2 J π π/2 J π π/2 J π π/2 J
T = 100 0.50 0.0461 0.0488 0.0406 0.0290 0.0195 0.0139 0.0448 0.0509 0.0447 0.0373 0.0325 0.0214
0.95 0.0493 0.0489 0.0415 0.0301 0.0223 0.0141 0.0489 0.0467 0.0421 0.0474 0.0291 0.0182
1.00 0.0486 0.0465 0.0411 0.0324 0.0235 0.0149 0.0496 0.0477 0.0476 0.0443 0.0321 0.0211
T = 400 0.50 0.0502 0.0502 0.0500 0.0448 0.0387 0.0315 0.0497 0.0544 0.0519 0.0438 0.0382 0.0385
0.95 0.0533 0.0507 0.0499 0.0443 0.0384 0.0328 0.0502 0.0468 0.0487 0.0443 0.0405 0.0346
1.00 0.0536 0.0528 0.0504 0.0462 0.0390 0.0353 0.0523 0.0497 0.0518 0.0438 0.0424 0.0367

  1. The results are reported under the asymptotic 5% significance level. π is the test for Nyquist frequency, π/2 is for harmonic frequency pairs and J is for joint frequencies.

Table 3:

Power performance of seasonal rescaled variance tests (V/S) and Canova and Hansen (1995) (CH) tests (τ = 0.1).

V/S tests CH tests
q = 0 q = T1/3 q = 0 q = T1/3
DGP b π π/2 J π π/2 J π π/2 J π π/2 J
T = 100 DGP1 0.50 0.5642 0.0229 0.3807 0.4533 0.0130 0.1682 0.5682 0.0316 0.4299 0.4877 0.0265 0.3169
0.95 0.5940 0.0238 0.4280 0.4551 0.0150 0.1814 0.5904 0.0304 0.4622 0.4966 0.0330 0.3341
1.00 0.5954 0.0248 0.4363 0.4553 0.0156 0.1803 0.5802 0.0295 0.4692 0.4924 0.0266 0.3270
DGP2 0.50 0.0226 0.5470 0.4498 0.0233 0.3593 0.2192 0.0287 0.5884 0.5290 0.0368 0.5087 0.4121
0.95 0.0265 0.5508 0.4584 0.0237 0.3641 0.2276 0.0363 0.5980 0.5337 0.0370 0.5006 0.4012
1.00 0.0304 0.5507 0.4625 0.0251 0.3635 0.2270 0.0343 0.5983 0.5340 0.0378 0.5171 0.4140
DGP3 0.50 0.4771 0.4378 0.6768 0.4341 0.3230 0.4492 0.5039 0.5274 0.7456 0.4917 0.4798 0.6349
0.95 0.5309 0.4641 0.7219 0.4321 0.3321 0.4584 0.5398 0.5475 0.7715 0.5009 0.4884 0.6544
1.00 0.5343 0.4575 0.7185 0.4342 0.3355 0.4611 0.5456 0.5390 0.7645 0.4708 0.4842 0.6291
T = 400 DGP1 0.50 0.9855 0.0038 0.9625 0.9501 0.0306 0.8874 0.9688 0.0075 0.9286 0.9134 0.0381 0.8364
0.95 0.9876 0.0064 0.9686 0.9530 0.0410 0.8898 0.9745 0.0110 0.9449 0.9154 0.0418 0.8388
1.00 0.9877 0.0090 0.9714 0.9498 0.0403 0.8910 0.9746 0.0106 0.9491 0.9166 0.0393 0.8424
DGP2 0.50 0.0043 0.9973 0.9954 0.0389 0.9897 0.9841 0.0078 0.9917 0.9880 0.0439 0.9775 0.9667
0.95 0.0076 0.9979 0.9960 0.0393 0.9925 0.9846 0.0110 0.9930 0.9904 0.0425 0.9796 0.9701
1.00 0.0077 0.9971 0.9951 0.0394 0.9902 0.9827 0.0118 0.9929 0.9883 0.0478 0.9758 0.9606
DGP3 0.50 0.9491 0.9844 0.9998 0.9505 0.9909 0.9995 0.9724 0.9753 0.9981 0.9170 0.9770 0.9978
0.95 0.9620 0.9821 0.9996 0.9497 0.9910 0.9991 0.9420 0.9754 0.9993 0.9115 0.9764 0.9981
1.00 0.9677 0.9840 1.000 0.9517 0.9908 0.9993 0.9432 0.9765 0.9999 0.9133 0.9796 0.9970

  1. The results are reported under the asymptotic 5% significance level. π is the test for Nyquist frequency, π/2 is for harmonic frequency pairs and J is for joint frequencies.

Table 4:

Power performance of seasonal rescaled variance tests (V/S) and Canova and Hansen (1995) (CH) tests (τ = 0.2).

V/S tests CH tests
q = 0 q = T1/3 q = 0 q = T1/3
DGP b π π/2 J π π/2 J π π/2 J π π/2 J
T = 100 DGP1 0.50 0.8415 0.0053 0.6850 0.7136 0.0087 0.3765 0.7978 0.0054 0.6664 0.6797 0.0180 0.5139
0.95 0.8726 0.0065 0.7514 0.7139 0.0084 0.3988 0.8276 0.0118 0.7166 0.6880 0.0154 0.5315
1.00 0.8688 0.0091 0.7588 0.7137 0.0089 0.4075 0.8262 0.0134 0.7191 0.6802 0.0195 0.5281
DGP2 0.50 0.0064 0.8994 0.8455 0.0161 0.7625 0.6105 0.0093 0.8754 0.8385 0.0282 0.7954 0.7183
0.95 0.0089 0.8990 0.8498 0.0173 0.7629 0.6169 0.0140 0.8799 0.8491 0.0282 0.7954 0.7183
1.00 0.0087 0.9004 0.8541 0.0169 0.7606 0.6120 0.0134 0.8879 0.8427 0.0285 0.7899 0.7249
DGP3 0.50 0.6771 0.7468 0.9507 0.6718 0.6908 0.8481 0.6647 0.7766 0.9408 0.6426 0.7517 0.8946
0.95 0.7490 0.7401 0.9667 0.6848 0.7061 0.8652 0.7250 0.7704 0.9542 0.6639 0.7708 0.8980
1.00 0.7555 0.7357 0.9691 0.6804 0.7084 0.8553 0.7349 0.7803 0.9592 0.6555 0.7688 0.9019
T = 400 DGP1 0.50 0.9983 0.0002 0.9925 0.9813 0.0134 0.9341 0.9958 0.0006 0.9820 0.9487 0.0223 0.8841
0.95 0.9994 0.0004 0.9978 0.9797 0.0333 0.9417 0.9977 0.0009 0.9902 0.9516 0.0381 0.8974
1.00 0.9994 0.0009 0.9974 0.9780 0.0362 0.9350 0.9974 0.0010 0.9891 0.9489 0.0432 0.8849
DGP2 0.50 0.0004 1.000 1.000 0.0392 0.9993 0.9976 0.0003 0.9998 0.9994 0.0409 0.9965 0.9928
0.95 0.0001 1.000 1.000 0.0422 0.9991 0.9982 0.006 0.9994 0.9991 0.0410 0.9965 0.9928
1.00 0.0004 1.000 1.000 0.0445 0.9992 0.9982 0.0010 0.9998 0.9994 0.0416 0.9957 0.9924
DGP3 0.50 0.9657 0.9988 1.000 0.9781 0.9988 1.000 0.9454 0.9956 1.000 0.9512 0.9937 0.9958
0.95 0.9827 0.9920 1.000 0.9788 0.9991 1.000 0.9694 0.9895 1.000 0.9512 0.9937 0.9998
1.00 0.9868 0.9922 1.000 0.9782 0.9984 0.9999 0.9754 0.9888 1.000 0.9538 0.9954 0.9998

  1. The results are reported under the asymptotic 5% significance level. π is the test for Nyquist frequency, π/2 is for harmonic frequency pairs and J is for joint frequencies.

Table 5:

Finite sample performance of seasonal rescaled variance tests (V/S): data prefiltering.

τ = 0.1 τ = 0.2
q = 0 q = T1/3 q = 0 q = T1/3
π π/2 π π/2 π π/2 π π/2
T = 100 DGP1 0.6023 0.0510 0.4765 0.0502 0.8723 0.0496 0.7445 0.0464
DGP2 0.0550 0.5721 0.0515 0.3843 0.0503 0.9032 0.0473 0.7704
DGP3 0.5456 0.4575 0.4425 0.3655 0.7653 0.7452 0.6804 0.7084
T = 400 DGP1 0.9862 0.0489 0.9523 0.0553 0.9994 0.0462 0.9780 0.0362
DGP2 0.0476 0.9974 0.0544 0.9913 0.0522 1.000 0.0532 0.9992
DGP3 0.9774 0.9761 0.9523 0.9904 0.9868 0.9922 0.9872 0.9984

  1. The results are reported under the asymptotic 5% significance level. π is the test for Nyquist frequency, π/2 is for harmonic frequency pairs.

Table 6:

Size performance of seasonal rescaled variance tests (V/S): additive outlier case.

Pθ2 = 0.4 Pθ2 = 0.8
π π/2 J π π/2 J
T = 100 0.0567 0.0543 0.0507 0.0721 0.0716 0.0686
T = 400 0.0624 0.0615 0.0611 0.0778 0.0769 0.0732

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Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/snde-2021-0004).

Received: 2021-01-06
Revised: 2021-06-14
Accepted: 2021-06-17
Published Online: 2021-07-02

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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  4. Forecasting transaction counts with integer-valued GARCH models
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https://doi.org/10.1515/snde-2021-0004
Keywords for this article
KPSS statistic; Lagrange multiplier; seasonal stationarity; unit root; V/S statistic

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. What does Google say about credit developments in Brazil?
  4. Forecasting transaction counts with integer-valued GARCH models
  5. Asymmetries in the monetary policy reaction function: evidence from India
  6. A mixture autoregressive model based on Gaussian and Student’s t-distributions
  7. Time-specific average estimation of dynamic panel regressions
  8. Rescaled variance tests for seasonal stationarity
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