Testing fractional integration with monthly data

Economic Modelling 16 Ž1999. 613]629

Testing fractional integration with monthly data Luis A. Gil-AlanaU Centre for Economic Forecasting, London Business School, Sussex Place, Regent’s Park, London NW1 4SA, UK

Accepted 29 April 1999

Abstract Seasonal roots can help to explain the seasonal fluctuations in macroeconomic time series. In this paper we concentrate on monthly data and look at different versions of Robinson’s Ž1994. tests for testing unit roots and other fractionally integrated hypotheses when the root is located at zero andror at the seasonal frequencies. A Monte Carlo experiment is carried out to check the power of these tests against different fractional alternatives, and an empirical application, using Spanish monthly data for the consumer price index, is also carried out in the article. Q 1999 Elsevier Science B.V. All rights reserved. JEL classifications: C12; C15; C22 Keywords: Seasonal unit roots; Fractional integration; Monte Carlo simulations

1. Introduction Many macroeconomic time series contain seasonal components. Despite its long history, however, there is little consensus on how seasonality should be treated in empirical research on aggregate fluctuations. While it is common practice to model a seasonal component as having a deterministic Že.g. with seasonal dummies., or a stationary stochastic Že.g. with a seasonal ARMA process. components, there may be cases where it is more U

Tel.: q44-171-262-5050, ext. 3374; fax: q44-171-724-7875. E-mail address: [email protected] ŽL.A. Gil-Alana.

0264-9993r99r$ - see front matter Q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 2 6 4 - 9 9 9 3 Ž 9 9 . 0 0 0 1 7 - 6

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appropriate to allow the model of the seasonal component to drift substantially over time. This possibility is implicit in the practice of seasonal differencing Žsee, e.g. Box and Jenkins 1970., whereby a process observed s times per year would be transformed to its s-period difference, on the assumption that the process contains an integrated seasonal component. Let yt be the time series we observe. We can consider the model Ž 1 y Ls . yt s u t ,

Ž1.

where Ls is the seasonal lag operator Ži.e. Ls yt s ytys ., and u t is a stationary I Ž0. process, which might include stationary ARMA components. The lag polynomial in Eq. Ž1. can be factorized as Ž 1 y L .Ž 1 q L q L2 q . . . qLsy1 . s Ž 1 y L . S Ž L . . That is, the seasonal difference operator can be broken down into the product of the first difference operator and the moving-average seasonal filter SŽ L., containing further roots of modulus unity. In this article we concentrate on monthly data. Thus, s s 12 and Ž1 y L12 . can be decomposed into Ž 1 y L .Ž 1 q L q L2 q . . . qL11 . s Ž 1 y L .Ž 1 q L .Ž 1 q L2 .Ž 1 q L q L2 . = Ž 1 y L q L2 .Ž 1 q '3 L q L2 .Ž 1 y '3 L q L2 . indicating the presence of the unit roots 0; y1; "i; y 12 Ž 1 " '3 i . ;

1 2

Ž 1 " '3 i . ; y 12 Ž '3 " i . ;

1 2

Ž '3 " i . .

Ž2.

The first of these roots occurs at the so-called long-run Žor zero. frequency, and the remaining seasonal roots correspond to 6, 3, 9, 8, 4, 2, 10, 7, 5, 1 and 11 cycles per year, which frequencies are 0, p , "pr2, "2pr3, "pr3, "5pr6 and "pr6, respectively. In the case of Eq. Ž1. we suppose that all these roots have the same integration order, one, at all frequencies. However, we can also consider the model allowing fractional roots, for instance, Ž 1 y Ls . d yt s u t ,

Ž3.

where d can be any real number, and s s 12 if monthly data. These kind of models were proposed by Porter-Hudak Ž1990., and yt is stationary and invertible if d g Žy1r2, 1r2.. If d ) 0, the process is called long-memory Žas opposed to short-memory when d s 0., and the autocorrelations take far longer to decay to zero than the exponential rate associated to the ARMA process for instance. ŽIn fact, they decay so slowly as to be non-summable.. They are also called strong dependent processes, because of the strong dependence between observations widely separated in time. Clearly, the seasonal unit root model in Eq. Ž1. belongs to this class of processes with d in Eq. Ž3. equal to one. We can also extend the model

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in Eq. Ž3. to allow different integration orders at each of the frequencies, for instance, d

Ž 1 y L6 . 1 Ž 1 q L6 .

d2

yt s u t ,

or more generally, d

d

d

Ž 1 y L . d1 Ž 1 q L . d 2 Ž 1 q L2 . 3 Ž 1 q L q L2 . 4 Ž 1 y L q L2 . 5 Ž 1 q '3 L q L2 .

d6

d7

= Ž 1 y '3 L q L2 . yt s u t , for real values d1 ,..., d 7 , and the process will display long memory at a given frequency if the order of integration at that frequency is greater than zero. In this article, I concentrate on a testing procedure suggested by Robinson Ž1994., for testing unit roots and other fractionally integrated hypotheses, which is fairly general in the sense that it allows us to test seasonal roots at any frequency on the interval w0, p x under both the null and the alternative hypotheses. Section 2 describes this testing procedure but first briefly reviews the literature on testing seasonal unit roots in monthly data. Section 3 contains a Monte Carlo experiment to check the power of different versions of Robinson’s Ž1994. tests against fractional alternatives. In Section 4, the tests of Robinson Ž1994. are applied to Spanish monthly data of prices, and finally, Section 5 contains some concluding remarks.

2. Testing seasonal unit roots with monthly data Hasza and Fuller Ž1982. consider as their null hypothesis the model: Ž 1 y L .Ž 1 y L12 . yt s « t . They suggest estimating the equations yt s a 1 yty1 q a 2 yty12 q a 3 yty13 q « t yt s f 1 yty1 q f 2 Ž yty12 y f 1 yty13 . q « t , and then testing the restrictions w a 1 , a 2 , a 3 x s w1, 1, y1x or w f 1 , f 2 x s w1, 1x by calculating a standard F-statistic and using the proper tabulated distribution. This testing procedure has two main drawbacks. First, the test imposes two unit roots at frequency zero under the null. Second, it is unclear how the performance of the test changes when only some of the seasonal frequencies possess a unit root. Furthermore, the test imposes a particular form for the alternative creating, in the event of misspecification, residual correlation, and thus requiring many lags of the dependent variable to whiten the errors. Therefore, applications of the test are likely to suffer from low power and residual autocorrelation in the errors.

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Dickey et al. Ž1984. bypass the assumption of two unit roots by considering the null model Ž 1 y L12 . yt s « t . They suggest estimating the equation yt s r yty12 q bt q « t , and testing whether r s 1, where bt is either a constant or seasonal dummies. They calculate the distribution of various statistics when the regression equation includes a constant and seasonal dummies. As with the Hasza]Fuller test, this test suffers from the problems of interpreting rejections, low power and residual correlation. Beaulieu and Miron Ž1993. propose a testing procedure that avoids these problems by including a regressor corresponding to each of the potential unit roots. This procedure is a generalization to monthly data of the Hylleberg et al. Ž1990. procedure for quarterly data. Let yt be the series of interest, generated by a general autoregression of form

r Ž L . yt s « t , where « t is a white noise process. The testing procedure consists essentially of linearizing the polynomial r Ž L. around the zero frequency unit root plus the eleven unit roots given in Eq. Ž2.. Then we have the equation 13

y 13 ,t s

Ý p k yk ,ty1 q « t ,

Ž4.

ks1

where y 1,t s Ž 1 q L q L2 q L3 q L4 q L5 q L6 q L7 q L8 q L9 q L10 q L11 . yt y 2,t s y Ž 1 y L q L2 y L3 q L4 y L5 q L6 y L7 q L8 y L9 q L10 y L11 . yt y 3,t s y Ž L y L3 q L5 y L7 q L9 y L11 . yt y4,t s y Ž 1 y L2 q L4 y L6 q L8 y L10 . yt y5,t s y 12 Ž 1 q L y 2 L2 q L3 q L4 y 2 L5 q L6 q L7 y 2 L8 q L9 q L10 y 2 L11 . yt

y6,t s

'3 2

Ž 1 y L q L3 y L4 q L6 y L7 q L9 y L10 . yt

L.A. Gil-Alana r Economic Modelling 16 (1999) 613]629

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y 7,t s 12 Ž 1 y L y 2 L2 y L3 q L4 q 2 L5 q L6 y L7 y 2 L8 y L9 q L10 q 2 L11 . yt

y 8,t s y

'3 2

Ž 1 q L y L3 y L4 q L6 q L7 y L9 y L10 . yt

y 9,t s y 12 Ž '3 y L q L3 y '3 L4 q 2 L5 y '3 L6 q L7 y L9 q '3 L10 y 2 L11 . yt y 10 ,t s 12 Ž 1 y '3 L q 2 L2 y '3 L3 q L4 y L6 q '3 L7 y 2 L8 q '3 L9 y L10 . yt y 11 ,t s 12 Ž '3 q L y L3 y '3 L4 y 2 L5 y '3 L6 y L7 q L9 q '3 L10 q 2 L11 . yt y 12 ,t s y 12 Ž 1 q '3 L q 2 L2 q '3 L3 q L4 y L6 y '3 L7 y 2 L8 y '3 L9 y L10 . yt y 13 ,t s Ž 1 y L12 . yt . In order to test hypotheses about various unit roots, one estimates Eq. Ž4. by ordinary least squares and then compares the OLS test statistics to the appropriate finite sample distributions based on Monte Carlo results. For frequencies 0 and p , one simply examines the relevant t-statistics for p k s 0 against the alternative p k - 0, with k s 1 and 2, respectively. For the other frequencies, one tests p k s 0, where k is even, with a two-sided test. The even coefficient is zero if the series contains a unit root at that frequency. It is not zero otherwise for the seasonal frequencies other than pr2. For pr2, the coefficient is not zero if no root exists at that frequency. Under the alternative, the even coefficient may be positive or negative. If one fails to reject p k s 0, then one tests p ky1 s 0 vs. the alternative that p ky 1 - 0. The test is one-sided because the sensible alternative is that the series contains a root outside the unit circle. Under stationarity the true coefficient is less than zero. The model can be extended to allow for a constant, seasonal dummies andror a time trend. Thus, the model in Eq. Ž4. becomes 13

y 13 ,t s

12

Ý p k yk ,ty1 q m 0 q m1 t q Ý m k Sk ,t q « t , ks1

Ž5.

ks2

and the equation is again estimated by OLS though the asymptotic distribution and finite sample distribution change. All tests presented so far particularized the case of a unit root at some or all seasonal frequencies. Robinson’s Ž1994. tests described below are more general in the sense that they allow us to test any integer or fractional root of any order, and therefore do not concentrate merely on the unit root situation. Similarly to Beaulieu and Miron Ž1993., wand thus to Hylleberg et al. Ž1990.x, they improve tests of Dickey et al. Ž1984. allowing for roots at all seasonal frequencies, but unlike these tests, the model will allow us to test different amplitudes and different

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frequencies not only under the alternative but also under the null hypothesis. We now describe the testing procedure. We observe Ž yt , z t ., t s 1,2,..., n4 where yt s b 9z t q x t ,

t s 1,2, . . . ,

Ž6.

r Ž L;u . x t s u t ,

t s 1,2, . . .

Ž7.

x t s 0,

t F 0,

Ž8.

where b is a Ž k = 1. vector of unknown parameters and z t is a Ž k = 1. vector of deterministic variables that might include an intercept, a time trend andror seasonal dummies; r Ž L;u . is a prescribed function of L and the unknown Ž p = 1. parameter vector u , that depends on the model tested; and u t is an I Ž0. process with parametric spectral density f Ž l ;t . s

s2 2p

g Ž l ;t . , yp - l - p ,

where the positive scalar s 2 and the Ž q = 1. vector t are unknown but g is of known form. In general we want to test the null hypothesis H0 :

u s 0,

Ž9.

and given the functional form of r Ž L;u . we can test different hypotheses under the null. For example, if we want to test the presence of a unit root at the zero frequency, we take

r Ž L;u . s Ž 1 y L .

dq u

,

Ž 10.

with d s 1, and allowing d in Ž10. to be any real value, we can also test for fractional roots at the same zero frequency. If we believe that yt contains an extra root at frequency p , the alternative will be

r Ž L;u . s Ž 1 y L2 .

dq u

,

Ž 11 .

and similarly, adding further roots, we can consider alternatives of form

r Ž L;u . s Ž 1 y L4 .

dq u

,

Ž 12 .

or more generally,

r Ž L;u . s Ž 1 y L12 .

dq u

,

amongst other possibilities covered by Robinson’s Ž1994. tests.

Ž 13 .

L.A. Gil-Alana r Economic Modelling 16 (1999) 613]629

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Under Eq. Ž9., the residuals in Eqs. Ž6. ] Ž8. are u ˜t s r Ž L . yt y b˜9wt ,

t s 1,2, . . . , n

b˜ s

r Ž L . s r Ž L;0 . ;

ž

n

Ý wt w9t Ý wt r Ž L . yt ,

/

ts1

wt s r Ž L . z t .

ts1

Unless g is a completely known function Že.g. g ' 1, as when u t is white noise. we have to estimate the nuisance parameter vector t , for example by

ˆt s arg min t g T s 2 Žt .

Ž 14 .

where T is a suitable subset of R q and

s 2 Žt . s

2p

ny1

g Ž l j ,t .

Ý

n

y1

I Ž lj . ,

js1

where y1 I Ž l j . s Ž 2p n .

2

n

Ý u˜t e it l

,

j

lj s

2p j

ts1

n

.

The test statistic, derived from the Lagrange multiplier ŽLM. principle is n

Rˆ s

sˆ 4

a9 ˆ Aˆy1 aˆ s ˆr 9rˆ,

Ž 15 .

where

ˆr s

n1r2

Aˆ s

sˆ 2 n

2

Aˆy1r2 a, ˆ

)

sˆ 2 s s 2 Ž tˆ . ,

Ý c Ž lj . c Ž lj . j

c Ž l j . s Re

9

y

a ˆs

y2p

)

­ ­u

n

9

log r Ž e i l j ;0 . ,

/

«ˆ Ž l j . s

y1

I Ž lj . ,

j

)

Ý c Ž l j . «ˆ Ž l j . Ý «ˆ Ž l j . «ˆ Ž l j . j

ž

)

Ý c Ž l j . g Ž l j ;tˆ .

ž

­ ­t

j

9

/

y1 )

Ý «ˆ Ž l j . c Ž l j .

9

,

j

log g Ž l j ;tˆ . ,

and ÝUj is a sum over l j such that yp - l j - p . l j f Ž r l y l1 , r l q l1 ., l s 1,2,...,s, such that r l , l s 1,2,...,s - ` are the distinct poles of r Ž L.. Note that Rˆ is a function of the hypothesized differenced series which has short memory under Eq. Ž9. and thus, we must specify the frequencies and integration orders of any seasonal roots.

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Robinson Ž1994. established under regularity conditions that Rˆ ª d x p2

n ª `,

as

Ž 16.

and also the Pitman efficiency property of LM in standard problems. If p s 1, an approximate one-sided 100 a % level test of Eq. Ž9. against alternatives Ha :

u)0

Ž 17 .

rejects H 0 if ˆ r ) za , where the probability that a standard normal variate exceeds za is a , and conversely, a test of Eq. Ž9. against alternatives Ha :

u-0

Ž 18 .

rejects H 0 if ˆ r - yza . A test against the two-sided alternative u / 0, for any p, rejects if Rˆ exceeds the upper critical value of the x p2 distribution. We can compare Robinson’s Ž1994. tests with those in Beaulieu and Miron Ž1993.. Extending Eq. Ž5. to allow augmentations of the dependent variable to render the errors white noise, and deterministic paths, the auxiliary regression in Beaulieu and Miron Ž1993. is 13

f Ž L . y 13 ,t s

Ý p k yk ,ty1 q ht q « t ,

Ž 19 .

ks1

where f Ž L. is a stationary lag polynomial and ht is a deterministic process that might include an intercept, a time trend andror seasonal dummies. If we cannot reject the null hypothesis p 1 s 0 against the alternative p 1 - 0 in Eq. Ž19., the process will have a unit root at zero frequency whether or not other Žseasonal. roots are present in the model. In Robinson’s Ž1994. tests, taking Eqs. Ž7. and Ž10., with d s 1, the null wEq. Ž9.x implies a single unit root at the same zero frequency. However, instead of Eq. Ž10. we could have Eq. Ž11. or alternatively Eq. Ž12. or Eq. Ž13.. If again d s 1, under Eq. Ž9., x t displays two unit roots at frequencies zero and p in Eq. Ž11.; the same two roots at zero and p , and two complex ones corresponding to frequencies pr2 and 3pr2 in Eq. Ž12.; and all the roots appearing in Eq. Ž2. under Eq. Ž13.. Similarly, the non-rejection of the null p 2 s 0 in Eq. Ž19. will imply a unit root at frequency p independently of other possible roots, and this can be consistent with Eqs. Ž6. ] Ž8. jointly with Eq. Ž11., Ž12. or Ž13., Žagain with d s 1., among other possibilities covered by Robinson’s Ž1994. tests. Furthermore, testing sequentially the different null hypotheses in Eq. Ž19., if we cannot reject p k s 0 for k s 1,2,...12, the overall null hypothesized model in Beaulieu and Miron Ž1993. is:

f Ž L .Ž 1 y L12 . yt s ht q « t ,

t s 1,2, . . .

Ž 20 .

L.A. Gil-Alana r Economic Modelling 16 (1999) 613]629

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and we can compare it with the set-up in Robinson Ž1994., using Eqs. Ž6. ] Ž8. and Ž13. with

f Ž L. ut s «t ,

t s 1,2, . . .

Ž 21 .

which, with d s 1, under the null Ž9. becomes

f Ž L .Ž 1 y L12 . yt s f Ž L . b 9 Ž 1 y L12 . z t q « t ,

t s 1,2, . . .

Ž 22.

Clearly, if we do not include explanatory variables in Eqs. Ž6. and Ž19., Ži.e. ht s z t s 0., Eq. Ž22. becomes Eq. Ž20., and including regressors, the difference between the two models will be due purely to deterministic components. Finally, we should remark that the null x p2 limit distribution of Robinson’s Ž1994. tests is constant across specifications of r Ž L;u . and z t and thus does not require case by case evaluation of a nonstandard distribution, unlike the other tests described above.

3. Finite sample performance This section examines the finite-sample behaviour of versions of Robinson’s Ž1994. tests by means of Monte Carlo simulations. In Robinson Ž1994. a finite-sample experiment was also performed. In that paper, he looked at the rejection frequencies when the true model was a random walk, wi.e. Ž1 y L. yt s « t x, and the alternatives were either fractional wi.e. Ž1 y L.1q u yt s « t x, or autoregressive wi.e. 1 y Ž1 q u L. yt s « t x, for different values of u . Thus, under the null hypothesis Eq. Ž9., the process contains a unit root at the zero frequency, and under the first type of alternatives, yt still contains a single root at the same zero frequency. In this section we want to investigate the power of Robinson’s Ž1994. tests when the alternatives have roots at different frequencies from those observed in the true model, and also the cases when the alternatives contain less roots than those in the true model. In Tables 1]4 we look at the rejection frequencies of Robinson’s Ž1994. tests when the null model consists of Eqs. Ž6. ] Ž8. with b s 0 a priori, Ži.e. yt s x t . and r Ž L. s r Ž L;0. s Ž1 y Ls . with s s 12, 4, 2 and 1, respectively. The alternatives will be in all cases fractional, with r Ž L;u . s Ž1 y Ls .1q u for s s 12, 4, 2 and 1, and u s y1, y0.75, y0.25, 0, 0.25, 0.50, 0.75 and 1. Thus, the rejection frequencies corresponding to u s 0 when the same s is taken under the null and the alternative hypotheses will indicate the sizes of the tests. In these cases, we calculate both, the one and the two-sided test statistics. We generate Gaussian series generated by the routines GASDEV and RAN3 of Press et al. Ž1986., with 10 000 replications of each case. The sample sizes are n s 120, 240 and 360 observations and in all cases the nominal size is 5%. In Table 1 the true model is given by Ž1 y L12 . x t s u t ; u t white noise. That is, we assume that x t has 12 unit roots as in Eq. Ž2.. When the alternatives are of the

622

Table 1 Rejection frequencies of ˆ r and Rˆ in Eq. Ž15. against fractional alternatives wtrue model: Ž1 y L12 . x t s u t ; u t is white noisexa

T s 120 Ž1 y L12 .1q u Ž1 y L12 .1q u Ž1 y L4 .1q u Ž1 y L2 .1q u

Ž1 y L.1q u T s 240 Ž1 y L12 .1q u Ž1 y L12 .1q u Ž1 y L4 .1q u Ž1 y L2 .1q u

Ž1 y L.1q u T s 360 Ž1 y L12 .1q u Ž1 y L12 .1q u Ž1 y L4 .1q u Ž1 y L2 .1q u Ž1 y L.1q u a

ru

y1

y0.75

y0.50

y0.25

0.00

ˆr Rˆ Rˆ Rˆ Rˆ

0.970

0.961

0.912

0.601

0.035

0.000

0.000

0.008

0.029

0.965

0.951

0.881

0.497

0.015

0.000

0.000

0.000

0.001

0.888

0.996

1.000

1.000

1.000

1.000

1.000

1.000

1.000

0.561

0.813

0.906

0.952

0.979

0.992

0.997

0.999

0.999

0.425

0.800

0.950

0.991

0.998

0.999

1.000

1.000

1.000

ˆr Rˆ Rˆ Rˆ Rˆ

0.990

0.990

0.983

0.841

0.110

0.849

0.996

0.999

0.989

0.988

0.978

0.779

0.020

0.114

0.425

0.905

0.992

0.837

0.998

1.000

1.000

1.000

1.000

1.000

1.000

1.000

0.638

0.899

0.954

0.979

0.992

0.997

0.999

0.999

1.000

0.558

0.887

0.979

0.997

0.999

1.000

1.000

1.000

1.000

ˆr Rˆ Rˆ Rˆ Rˆ

0.994

0.995

0.995

0.949

0.729

0.999

1.000

1.000

0.994

0.995

0.994

0.916

0.028

0.706

0.996

1.000

1.000

0.807

0.998

1.000

1.000

1.000

1.000

1.000

1.000

1.000

0.678

0.926

0.971

0.987

0.995

0.998

0.998

1.000

1.000

0.613

0.920

0.985

0.998

0.999

1.000

1.000

1.000

1.000

0.00

0.000

0.035

0.0003

0.038

0.0012

10000 replications were used for each case. Sizes are in bold and the nominal size used was 5%.

0.25

0.50

0.75

1.00

L.A. Gil-Alana r Economic Modelling 16 (1999) 613]629

r Ž L;u .

Table 2 Rejection frequencies of ˆ r and Rˆ in Eq. Ž15. against fractional alternatives wtrue model: Ž1 y L4 . x t s u t ; u t is white noisexa

T s 120 Ž1 y L4 .1q u Ž1 y L4 .1q u

Ž1 y L12 .1q u Ž1 y L2 .1q u Ž1 y L.1q u T s 240 Ž1 y L4 .1q u

Ž1 y L4 .1q u Ž1 y L12 .1q u Ž1 y L2 .1q u Ž1 y L.1q u T s 360 Ž1 y L4 .1q u Ž1 y L4 .1q u

Ž1 y L12 .1q u Ž1 y L2 .1q u Ž1 y L.1q u a

ru

y1

y0.75

y0.50

y0.25

0.00

ˆr Rˆ Rˆ Rˆ Rˆ

0.311

0.367

0.383

0.154

0.001

0.604

0.566

0.447

0.153

0.178

0.768

0.676

0.489

0.227

0.665

0.917

0.998

0.647

0.968

ˆr Rˆ Rˆ Rˆ Rˆ

0.569 0.750

ˆr Rˆ Rˆ Rˆ Rˆ

0.00

0.25

0.50

0.75

1.00

0.298

0.881

0.998

1.000

1.000

0.763

0.989

1.000

1.000

0.052

0.013

0.014

0.024

0.038

1.000

1.000

1.000

1.000

1.000

1.000

1.000

1.000

1.000

1.000

1.000

1.000

1.000

0.679

0.807

0.627

0.985

1.000

1.000

1.000

0.781

0.814

0.554

0.101

0.958

1.000

1.000

1.000

0.821

0.730

0.554

0.349

0.279

0.519

0.828

0.964

0.993

0.727

0.947

0.999

1.000

1.000

1.000

1.000

1.000

1.000

0.735

0.987

1.000

1.000

1.000

1.000

1.000

1.000

1.000

0.684

0.790

0.923

0.887

0.998

1.000

1.000

1.000

0.831

0.861

0.927

0.847

0.081

0.993

1.000

1.000

1.000

0.780

0.992

1.000

1.000

1.000

1.000

1.000

1.000

1.000

0.756

0.958

0.999

1.000

1.000

1.000

1.000

1.000

1.000

0.780

0.992

1.000

1.000

1.000

1.000

1.000

1.000

1.000

0.004

0.189

0.007

0.151

L.A. Gil-Alana r Economic Modelling 16 (1999) 613]629

r Ž L;u .

10000 replications were used for each case. Sizes are in bold and the nominal size used was 5%.

623

624

L.A. Gil-Alana r Economic Modelling 16 (1999) 613]629

form Ž1 y L12 .1q u x t s u t , we look at the rejection frequencies for both the oneand the two-sided tests wi.e. ˆ r and Rˆ in Eq. Ž15.x. In both cases we observe that the sizes are too small compared with the nominal one Ž5%., though they increase as we increase the number of observations. Looking at the rejection frequencies, we observe that they are large for u ) 0, but very small for u - 0 especially when n s 120, indicating the usefulness of the tests when the sample size is small. However, increasing the sample size to 240 or 360 observations, these frequencies also increase strongly. When the alternatives are of form Ž1 y Ls .1q u x t s u t , with s s 4, 2 and 1 Žthat is, including four, two and one unit roots., we see that the rejection frequencies are also high even when the sample size is not very large Že.g. n s 120., especially for u ) 0. In Table 2 the true model is given by: Ž1 y L4 . x t s u t ; u t white noise. Looking at the one-sided test we see that the size is too small when u - 0 but too large for u ) 0 though they tend to improve as n increases. Similarly, the size of the two-sided test is too large Ž17.8%. when n is small Ž120 observations., but improves considerably as we increase the sample size to 240 or 360 observations. Looking at the rejection frequencies, the test statistics seem to perform quite well in practically all cases. The only exceptional case seems to occur when the alternatives are Ž1 y L12 .1q u x t s u t ; u ) 0 and n s 120. In this case, the frequencies are very low even for u s 1. However, taking n s 240 or 360, we observe that these frequencies are very high in all cases, suggesting the efficiency of the tests, not only under local alternatives but also under different functional forms of r Ž L;u .. Table 3 takes the true model as Ž1 y L2 . x t s u t ; u t white noise. That is, x t contains only two unit roots: one at the zero frequency and the other at the frequency p . The size of the one-sided test is too small for u - 0 but too large for u ) 0 but, as in the previous table, they improve as n increases. The size of the two-sided test is too small when n s 120 but increasing the sample size, it tends to approximate to the nominal size. In this table we observe that all rejection frequencies are quite high. As in Table 2, the poorest results are obtained when the alternative includes 12 roots and u ) 0 with n s 120, but increasing the sample size the tests improve considerably. Finally, in Table 4, we look at the rejection frequencies of Robinson’s Ž1994. tests when the true model is a pure random walk, i.e. Ž1 y L. x t s u t , and u t is white noise. As in the previous tables, the sizes of the one-sided tests are biased; they are too small for u - 0 but too large for u ) 0. Also the size of the two-sided test is too large but in all the cases, they improve as n increases. The test statistics seem to perform very well when the alternative is Ž1 y L.1q u for any value of u even when the sample size is small. For the remaining specifications, the tests behave better when u - 0. As before, the worst results are obtained when the alternatives are of the form Ž1 y L12 .1q u , that is, including the 12 roots in Eq. Ž2.. In this case, even when the sample size is relatively large, the power of the test is very low for positive values of u . That means that if the true model is a pure random walk, and we perform the test with r Ž L;u . s Ž1 y L12 .1q u and u ) 0, it is very likely that the null hypothesis will not be rejected. This should be kept in mind when performing the tests in the empirical application in the following section.

Table 3 Rejection frequencies of ˆ r and Rˆ in Eq. Ž15. against fractional alternatives wtrue model: Ž1 y L2 . x t s u t ; u t is white noisexa

T s 120 Ž1 y L2 .1q u Ž1 y L2 .1q u

Ž1 y L12 .1q u Ž1 y L4 .1q u Ž1 y L.1q u T s 240 Ž1 y L2 .1q u

Ž1 y L2 .1q u Ž1 y L12 .1q u Ž1 y L4 .1q u Ž1 y L.1q u T s 360 Ž1 y L2 .1q u Ž1 y L2 .1q u

Ž1 y L12 .1q u Ž1 y L4 .1q u Ž1 y L.1q u a

ru

y1

y0.75

y0.50

y0.25

0.00

ˆr Rˆ Rˆ Rˆ Rˆ

0.682

0.743

0.792

0.471

0.012

0.791

0.803

0.781

0.398

0.041

0.647

0.623

0.568

0.484

0.806

0.720

0.541

0.790

0.873

ˆr Rˆ Rˆ Rˆ Rˆ

0.792 0.872

ˆr Rˆ Rˆ Rˆ Rˆ

0.00

0.25

0.50

0.75

1.00

0.090

0.783

0.997

1.000

1.000

0.595

0.983

1.000

1.000

0.399

0.354

0.374

0.413

0.452

0.400

0.630

0.935

0.995

0.999

1.000

0.995

1.000

1.000

1.000

1.000

1.000

1.000

0.864

0.942

0.865

0.970

1.000

1.000

1.000

0.903

0.940

0.814

0.045

0.917

1.000

1.000

1.000

0.774

0.725

0.682

0.713

0.826

0.940

0.985

0.996

0.999

0.884

0.853

0.714

0.344

0.663

0.995

1.000

1.000

1.000

0.856

0.919

0.999

1.000

1.000

1.000

1.000

1.000

1.000

0.837

0.911

0.977

0.973

0.997

1.000

1.000

1.000

0.901

0.936

0.977

0.958

0.049

0.987

1.000

1.000

1.000

0.839

0.769

0.698

0.760

0.933

0.996

1.000

1.000

1.000

0.912

0.890

0.836

0.421

0.711

0.999

1.000

1.000

1.000

0.895

0.940

0.999

1.000

1.000

1.000

1.000

1.000

1.000

0.015

0.069

0.025

0.058

L.A. Gil-Alana r Economic Modelling 16 (1999) 613]629

r Ž L;u .

10000 replications were used for each case. Sizes are in bold and the nominal size used was 5%.

625

626

Table 4 Rejection frequencies of ˆ r and Rˆ in Eq. Ž15. against fractional alternatives wtrue model: Ž1 y L. x t s u t ; u t is white noisexa

T s 120 Ž1 y L.1q u Ž1 y L.1q u

Ž1 y L12 .1q u Ž1 y L4 .1q u Ž1 y L2 .1q u T s 240 Ž1 y L.1q u

Ž1 y L.1q u Ž1 y L12 .1q u Ž1 y L4 .1q u Ž1 y L2 .1q u T s 360 Ž1 y L.1q u Ž1 y L.1q u

Ž1 y L12 .1q u Ž1 y L4 .1q u Ž1 y L2 .1q u a

ru

y1

y0.75

y0.50

y0.25

0.00

ˆr Rˆ Rˆ Rˆ Rˆ

1.000

1.000

0.999

0.864

0.032

1.000

1.000

0.999

0.819

0.057

0.120

0.031

0.005

0.009

0.998

0.979

0.858

1.000

1.000

ˆr Rˆ Rˆ Rˆ Rˆ

1.000 1.000

ˆr Rˆ Rˆ Rˆ Rˆ

0.00

0.25

0.50

0.75

1.00

0.091

0.938

1.000

1.000

1.000

0.857

0.999

1.000

1.000

0.000

0.000

0.000

0.000

0.000

0.478

0.074

0.001

0.000

0.000

0.000

0.999

0.951

0.407

0.023

0.320

0.836

0.986

1.000

1.000

0.991

0.999

1.000

1.000

1.000

1.000

1.000

0.987

0.055

0.994

1.000

1.000

1.000

0.986

0.925

0.747

0.414

0.109

0.006

0.002

0.000

0.000

1.000

1.000

0.999

0.954

0.420

0.003

0.004

0.102

0.516

1.000

1.000

1.000

0.999

0.711

0.024

0.685

0.996

1.000

1.000

1.000

1.000

0.999

1.000

1.000

1.000

1.000

1.000

1.000

1.000

0.999

0.052

0.999

1.000

1.000

1.000

0.999

0.997

0.967

0.794

0.328

0.022

0.001

0.001

0.001

1.000

1.000

1.000

0.998

0.840

0.036

0.318

0.943

0.999

1.000

1.000

1.000

1.000

0.875

0.022

0.895

1.000

1.000

0.032

0.079

0.036

0.073

10000 replications were used for each case. Sizes are in bold and the nominal size used was 5%.

L.A. Gil-Alana r Economic Modelling 16 (1999) 613]629

r Ž L;u .

L.A. Gil-Alana r Economic Modelling 16 (1999) 613]629

627

4. Empirical application In this section Robinson’s Ž1994. tests are applied to Spanish monthly data of the consumer price index. This series starts in January 1970 and ends in December 1997. Denoting the series yt , we employ throughout the model Eqs. Ž6. ] Ž8. with r Ž L;u . s Ž1 y Ls . dq u , and z t s Ž1,t .9, t G 1, z t s Ž0,0.9 otherwise, so under the null hypothesis wEq. Ž9.x, the model becomes yt s b 1 q b 2 t q x t , Ž 1 y Ls . d x t s u t ,

t s 1,2, . . .

Ž 23 .

t s 1,2, . . .

Ž 24.

treating separately the cases b 1 s b 2 s 0 a priori, b 1 unknown and b 2 s 0 a priori, and Ž b 1 , b 2 . unknown. We model the I Ž0. disturbances u t as white noise. Thus, when d s s s 1, for example, the differences Ž1 y L. yt behave for t ) 1, like a random walk when b 2 s 0, and a random walk with drift when b 2 / 0. However, we report test statistics not merely for the null when d s 1, but for d s 0, Ž0.25., 2, thus including also tests for stationarity Ž d s 0.50. and for I Ž2. Ž d s 2., as well as other possibilities. The test statistic reported in Table 5 is the one-sided one given by ˆ r in Eq. Ž15. when s s 1. That is, we concentrate on the case when the root is located at the so-called long run or zero frequency. In view of the rules explained in Section 2, significantly positive values of ˆ r are consistent with Eq. Ž17., implying that the order of integration should be higher than d. Conversely, significantly negative values of ˆ r are consistent with Eq. Ž18., indicating that the integration order should be smaller than the one suggested by d. A notable feature of the second column in Table 5, in which b 1 s b 2 s 0 a priori, is the fact that we reject the null hypothesis in all cases except when d s 1, indicating the presence of a unit root at the zero frequency. We observe in these Table 5 Testing for a unit root at the zero frequency: ˆ r in Eq. Ž15. with r Ž L;u . as in Eq. Ž10. Žseries: C.P.I.. d

With no regressors

With an intercept

With a time trend

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

55.98 55.01 53.19 29.02 0.42a y4.20 y5.92 y6.92 y7.58

55.98 50.25 45.27 43.25 7.82 y3.81 y5.93 y6.88 y7.50

51.35 46.73 45.06 33.03 7.61 y3.15 y5.90 y6.88 y7.50

a

Non-rejection values of the null hypothesis at the 95% significance level.

L.A. Gil-Alana r Economic Modelling 16 (1999) 613]629

628

results that there is a monotonic decrease in ˆ r as d increases. Such monotonicity is a characteristic of any reasonable statistic, given correct specification and adequate sample size because, for example, if we reject the null wEq. Ž9.x against the alternative wEq. Ž17.x for d s 0.75, an even more significant result in this direction should be expected when d s 0.50 or d s 0.25. Columns 3 and 4 in Table 5 give results with, respectively, b 2 s 0 a priori Žno time trend in the undifferenced series., and both b 1 and b 2 unrestricted. In every case in both columns, ˆ r is monotonic and moreover, the null hypothesis is rejected for all values of d, suggesting that the inclusion of an intercept andror a time trend in the model might be inappropriate. Table 6 gives results of ˆ r in Eq. Ž15. with r Ž L;u . s Ž1 y L2 . dq u Žin column 2.; 4 dq u Ž1 y L . Žin column 3.; and Ž1 y L12 . dq u Žin column 4., that is, we perform the tests for testing the presence of two roots Žat frequencies 0 and p . in column 2; four roots at frequencies 0, p , pr2 and 3pr2, in column 3; and the 12 roots appearing in Eq. Ž2. in column 4, for the same values of d as in Table 5. We observe that monotonicity is always achieved across this table. Starting with the case of two roots we see that the unit root null hypothesis Ž d s 1. is rejected, and the only non-rejection case occurs at d s 1.25. This hypothesis is also non-rejected in columns 3 and 4, when four and 12 roots are considered, and in the latter case, the null hypothesis is not rejected for d s 1.50, 1.75 and 2 as well. These non-rejections, however, might be related to the low power noted in Section 3 when testing with r Ž L;u . s Ž1 y L12 .1q u in the presence of a single unit root located at the zero frequency.

5. Conclusions We have presented different versions of Robinson’s Ž1994. tests for testing unit roots and other fractionally integrated hypotheses with monthly data. The tests of Robinson Ž1994. were compared with a number of leading seasonal unit root tests

Table 6 Testing for unit roots at zero and at the seasonal frequencies: ˆ r n Ž15. with r Ž L;u . as in Eqs. Ž11. ] Ž13. d

r Ž L;u . s Ž1 y L2 .dq u

r Ž L;u . s Ž1 y L4 .dq u

r Ž L;u . s Ž1 y L12 .dq u

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

49.71 41.73 40.52 34.44 4.70 y1.10a y2.95 y3.96 y4.60

37.33 37.19 38.27 30.61 6.40 y0.26a y2.05 y3.02 y3.65

10.61 10.53 10.46 10.29 5.24 0.49a y0.59a y1.11a y1.43a

a

Non-rejection values of the null hypothesis at the 95% significance level.

L.A. Gil-Alana r Economic Modelling 16 (1999) 613]629

629

and we showed that these tests can be viewed as particular cases of Robinson’s Ž1994. tests with d s 1. A Monte Carlo experiment was also conducted in order to check the power of the tests when the fractional alternatives included roots at different frequencies from those observed in the true model. The results suggest that the tests perform reasonably well when the number of observations is sufficienttly large. The worst results are obtained when testing 12 roots if the true model has a single root at the zero frequency. Finally, an empirical application was also carried out at the end of the article studying the behaviour of the consumer price index in Spain with monthly data. The results indicate the presence of a single unit root at the zero frequency. The article can be extended in several directions. The Monte Carlo simulations can be extended to study the power of the tests when the disturbances are weakly parametrically autocorrelated, and this can also be done in the empirical application in Section 4. However, as a preliminary step, in order to check the integration order of the series, the results in the paper are quite conclusive about the presence of a unit root at the zero frequency. Additionally, the tests of Robinson Ž1994. can also be performed in a different fashion, testing different orders of integration for each one of the seasonal frequencies.

References Beaulieu, J.J., Miron, J.A., 1993. Seasonal unit roots in aggregate U.S. data. J. Economet. 55, 305]328. Box, G.E.P., Jenkins, G.M., 1970. Time Series Analysis: Forecasting and Control. Holden-Day, San Francisco, CA. Dickey, D.A., Hasza, D.P., Fuller, W.A., 1984. Testing for unit roots in seasonal time series. J. Am. Stat. Assoc. 79, 355]367. Hasza, D.P., Fuller, W.A., 1982. Testing for nonstationary parameter specifications in seasonal time series models. Ann. Stat. 10, 1209]1216. Hylleberg, S., Engle, R.F., Granger, C.W.J., Yoo, B.S., 1990. Seasonal integration and cointegration. J. Economet. 44, 215]238. Porter-Hudak, S., 1990. An application of the seasonal fractionally differenced model to the monetary aggregate. J. Am. Stat. Assoc. 85, 338]344. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T., 1986. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge. Robinson, P.M., 1994. Efficient tests of nonstationary hypotheses. J. Am. Stat. Assoc. 89, 1420]1437.