Tests of AR(2) unit roots and cointegration with the ... - Semantic Scholar

Tests of AR(2) unit roots and cointegration with the GPH statistic Jonas Andersson∗and Johan Lyhagen†

Abstract Time domain tests of unit roots and cointegration are often made after a model search procedure has been made in order to reduce erroneus inference caused by model misspecification. An alternative to this is to use a semi parametric method like the GPH (Geweke & Porter-Hudak, 1983) test where the properties of the spectral density close to the zero frequency are exploited, see Cheung & Lai (1993) who use this to test for cointegration in a bivariate system. In this paper we further investigate the size and power properties of this test. The results provide evidence that the test has not only good power properties against AR(1) and ARFIMA(0,d,0) alternatives, which was shown by Cheung and Lai (1993) but also against AR(2) alternatives. We show that the power of the test is not as good when applied to a raw time series as when applied to residuals from a cointegrating regression. The power can, in the AR(1) case, be significantly improved by using a larger bandwith than is usually done for the GPH estimator without imposing any size distortions. For the AR(2) alternative, however, severe size distortions appear when this is done.

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Introduction

Tests for unit roots are usually made in order to investigate whether the effect of a shock in a time series variable persists or eventually vanishes. One of the most common of those tests, the Dickey-Fuller test (Dickey & Fuller, 1979), is based on a regression of the first difference on the first lag of the level. What effectively is tested for is a unit root in the AR(1) polynomial. Since the purpose is to test if the effect of a shock persists an AR(1) alternative is not necessarily the only interesting alternative hypothesis. Aware of this, Dickey and Fuller (1981) proposed a modification of the test, the augmented Dickey-Fuller (ADF) test, where they included lags of the first difference in the regression. However, it ∗ Department of Information Science, Division of Statistics, Uppsala University, Box 513, SE-751 20 Uppsala. † Department of Economic Statistics, Stockholm School of Economics, Box 6501, SE-113 83 Stockholm, Sweden.

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has later been shown that the test has low power against fractional alternatives (see e.g. Sowell, 1990). Since it is the long run properties of the time series we are interested in it seems reasonable to consider the spectral density close to the zero frequency. Cheung and Lai (1993) used this approach in testing for cointegration in a bivariate system, by basing a test statistic on the semi-parametric GPH estimator. They show that the test in fact is more powerful against both ARFIMA(0,d,0) and, more surprisingly, AR(1) alternatives. In this paper we investigate the properties of this test further. Especially, we study which bandwith to use in order to preserve the attractable size and power properties of the test and also evaluate these properties under AR(2) alternatives. In Section 2 the GPH estimator and its properties is reviewed. In Section 3 pure unit root tests are to be studied while tests of cointegration, based on the GPH estimator, are to be considered in Section 4. A conclusion closes the paper.

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The GPH test

We want to test whether the underlying process generating the time series {xt } has a unit root or not. If this is the case the spectral density of the first difference Gt = (1 − L) xt : fG (ω) = |1 − exp (−iω)|2 fx (ω) ,

(1)

where fx (ω) and fG (ω) are the spectral densities of xt and Gt , respectively, should be flat in a neighborhood of the zero frequency. This property is exploited by Cheung and Lai (1993) who used the semi parametric GPH estimator, originally suggested by Geweke and Porter-Hudak (1983), to test for a unit root in {xt }. The GPH estimator is based on the representation fG (ω) = |1 − exp (−iω)|−2(d−1) fu (ω) −2(d−1)

= (2 sin (ω/2))

fu (ω) ,

(2)

where d is the long memory parameter and fu (ω) is the spectral density of an ARMA process (or a slowly varying function). The estimator is obtained by taking the logaritm of fG (ω), adding and subtracting the log-periodogram of {Gt } , ln I (ω), and doing the same with ln (fu (0)) , i.e. ¡ ¢ ln I (ω j ) = ln (fu (0)) − (d − 1) ln 4 sin2 (ω j /2) + ln (fu (ω j ) /fu (0)) + ln (I (ω j ) /fG (ω j )) (3) 2πj ωj = , j = 1, 2, ...m and T = sample size T ¡ ¢ This is a regression of ln I (ω j ) on ln 4 sin2 (ω j /2) with Gumbel error-term. If fu (ω) is flat, i.e. when the ARMA-part of the process is just a white noise, the 2

term ln (fu (ω j ) /fu (0)) is zero for all ω j . If the ARMA-part of the process is not white noise it holds that ω j → 0 implies ln (fu (ω j ) /fu (0)) → 0. This means that as the sample size grows this term becomes negligible if the frequencies used is a function of the sample size but growing slower. Estimates of d are affected by this term and therefore, to avoid bias because of its impact, only frequencies near zero are used in the regression. The number of frequencies, m, must satisfy some conditions, see Geweke and Porter-Hudak (1983), which are satisfied by the function m = T α , 0 < α < 1. They argue that using α between 0.5 and 0.6 gives confidence intervals with accurate coverage probability. If, additionaly, the first frequencies are excluded, Robinson (1995), has derived the asymptotic distribution of the estimator. When the estimator shall constitute a basis for a test of cointegration, though, this asymptotic distribution is of no use since the error-term is not observable, and we have to use the residuals instead. Thus, the distribution has to be simulated. In order to evaluate the GPH test we will use the popular ADF test as a benchmark. The ADF test is based on the assumption that {xt } is an AR(p)process. In our study we focus on the case when p = 2, (see e.g. Oke and Lyhagen (1999) for the effect of pretesting for p): xt = φxt−1 + γxt−2 + εt .

(4)

where {εt } is a white noise process. The test statistic is then calculated as the t-statistic of xt−1 in the regression ∆xt = (1 − φ − γ)xt−1 − γ∆xt−1 + εt .

(5)

We will call this the ADF 1 test with the natural extension ADF k. Since p is rarely known we will also study the case when we, incorrectly, specifies an AR(5) model and thus uses an ADF 4 test.

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Application to AR(2) unit roots

Cheung and Lai (1993) investigated the alternative hypotheses AR(1) and ARFIMA(0,d,0). Under the null hypothesis these processes are identical and implies a white noise in the first difference. The attractive size properties will thus be preserved for these processes even if the number of ordinates in the GPH test is increased to m = T 0.9 , which was pointed out in Andersson and Lyhagen (1997). The reason for doing this should be to increase the power of the test. However, Table 1 shows that the size properties are not preserved when the data generating process is an AR(2) with a unit root. Figure 1 and Figure 2 show the differences between the logarithm of the spectral densities of the first difference of an AR(1) and different AR(2) processes, all with a unit root. The slope of the logarithm of the spectral density increases faster for the AR(2) process than for the AR(1) process as the frequencies increase. This explaines why the size is distorted in the AR(2) case. In fact, the slope is zero for all frequencies for the AR(1) process. All lines have 3

been subtracted by the value of the first frequency in order to start at the same level. In Table 1, the empirical size of the test for different AR(2) unit root processes are reported. A problem noted there but not indicated in the Figure 1 and Figure 2 are the small size obtained when φ > 1. In this case, the slope is negative. Since the size distortions were severe for larger α:s, we use α = 0.6 when we evaluate the power of the test. The results, presented in Table 2 and Table 3, shows that the power of the unit root GPH test is lower than the ADF test in all cases, a result not expected in the light of the study by Cheung and Lai (1993) who showed that the GPH test had higher power than the ADF test when applied to residuals from a cointegrating regression.

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Application to tests for cointegration

Consider the bivariate system yt + xt yt + 2xt

= u1t = u2t .

(6)

where {u1t } is a random walk and the process {u2t } is an AR(2) process. If {u2t } is a mean reverting process, {xt } and {yt } are cointegrated. In Table 4, the critical values for the ADF and GPH statistics has been obtained by simulating data from model (6) with {u2t } generated as a random walk. The number of replicates was 50000. In order to investigate how the size properties is influenced when an AR(2) process with an unit root is considered, a simulation study is performed. The results are presented in Table 5. As seen, and maybe not surprisingly with the previous section in mind, the critical values in Table 4 can be used as long as α is kept down to values not smaller than 0.6. A notable difference compared to the unit root test in the previous section is that the power of the GPH test, relative to the ADF test, is significantly higher. However, the power improvement, by using more of the ordinates (Andersson & Lyhagen, 1997), is not applicable for the AR(2) alternative since the size then is heavily distorted for α larger than 0.6. The distortions, though, is not as severe as for the unit root test.

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Conclusions

In this paper properties of the semi-parametric GPH test for unit roots/ cointegration is investigated. Contrary to previous studies, AR(2) processes are also considered. It is shown that remaining short memory greately influence the size of the test. This is due to that the slope of the spectral density of the differenced series is different from the one of white noise, i.e. under the (asymptotic) condition which the critical values are derived. The lesson to be

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learned is that an investigation of a too simplistic process, in this case an AR(1), greately reduces the generalisability of the results.

References Andersson, J. and Lyhagen, J. (1997) A note on the power of the GPH test of cointegration. Research Report 97:7, Department of Statistics, Uppsala University. Cheung, Y-W. and Lai, K. (1993) A fractional cointegration analysis of purchasing power parity. Journal of Business & Economic Statistics 11, 103-112. Dickey, D.A. and Fuller, W.A. (1979) Distribution of the estimator for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427-431. Dickey, D.A. and Fuller, W.A. (1981) Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, 1057-1072. Geweke, J. and Porter-Hudak, S. (1983) The estimation and application of long memory time series models. Journal of Time Series Analysis 4, 221-238. Oke, T. and Lyhagen, J. (1999) Small-sample properties of some tests for unit root with data-based choice of the degree of augmentation. Computational Statistics and Data Analysis 30, 457-469. Robinson, P.M. (1995) Log-periodogram regression for time series with long-range dependence. The Annals of Statistics 23, 1048-1072 Sowell, F.B. (1990) The fractional unit root distribution. Econometrica 58, 495-505.

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0.04

φ =0 φ =0.25 0.02

φ =0.5 φ =0.75 φ =1

0 1

2

3

4

5

6

7

8

9

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Figure 1: The logarithm of the spectral density of the first difference of AR(2) unit root processes with different parameter combinations. The x-axis represents the first 10 Fourier frequencies when T = 100. 0.04

φ=0 φ=0.25 0.02

φ=0.5 φ=0.75 φ=1

0 0

5

10

15

20

25

30

35

Figure 2: The logarithm of the spectral density of the first difference of AR(2) unit root processes with different parameter combinations. The x-axis represents the first 32 Fourier frequencies when T = 1000.

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φ 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45

GPH 0.5 0.0585 0.0626 0.0606 0.0646 0.0639 0.0599 0.0566 0.0592 0.0571 0.0547 0.0575 0.0524 0.0481 0.0380 0.0320

GPH 0.6 0.0736 0.0726 0.0768 0.0750 0.0805 0.0721 0.0685 0.0678 0.0663 0.0584 0.0558 0.0426 0.0372 0.0267 0.0170

GPH 0.7 0.1573 0.1515 0.1536 0.1541 0.1419 0.1328 0.1247 0.1014 0.0839 0.0688 0.0453 0.0278 0.0144 0.0076 0.0026

GPH 0.8 0.7110 0.7102 0.6813 0.6313 0.5740 0.4759 0.3617 0.2571 0.1629 0.0866 0.0367 0.0142 0.0044 0.0015 0.0002

Table 1: Size of the GPH test for a unit root in AR(2) processes, represented by combinations of φ and γ such that φ + γ = 1. The samplesize is 100 and the number of replicates 10000.

φ 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45

φ + γ = 0.95 GPH 0.6 0.0939 0.0899 0.0966 0.0961 0.0955 0.0986 0.1049 0.1029 0.0998 0.0994 0.0935 0.0836 0.0794 0.0633 0.0503

ADF 1 0.1758 0.1865 0.1945 0.2136 0.2290 0.2489 0.2668 0.2861 0.3177 0.3672 0.4130 0.4622 0.5441 0.6375 0.7397

ADF 4 0.1567 0.1591 0.1725 0.1878 0.2013 0.2150 0.2296 0.2469 0.2728 0.3095 0.3430 0.3859 0.4358 0.5196 0.6036

φ + γ = 0.9 GPH 0.6 0.1271 0.1296 0.1422 0.1508 0.1537 0.1590 0.1780 0.1811 0.1916 0.2035 0.2084 0.2081 0.2270 0.2091 0.1956

ADF 1 0.3824 0.4113 0.4439 0.4784 0.5128 0.5673 0.6105 0.6558 0.7196 0.7803 0.8431 0.9003 0.9478 0.9733 0.9923

ADF 4 0.3211 0.3373 0.3533 0.3944 0.4169 0.4585 0.4913 0.5333 0.5837 0.6360 0.6886 0.7522 0.8183 0.8807 0.9349

Table 2: Power of the GPH test for a unit root in AR(2) processes for different values of φ + γ. The samplesize is 100 and the number of replicates 10000.

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φ 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45

φ + γ = 0.8 GPH 0.6 0.2566 0.2864 0.3175 0.3472 0.3701 0.4023 0.4441 0.4810 0.5296 0.5672 0.6123 0.6719 0.7179 0.7410 0.7626

ADF 1 0.8423 0.8691 0.9014 0.9228 0.9459 0.9631 0.9778 0.9869 0.9937 0.9972 0.9990 0.9998 1.0000 1.0000 1.0000

ADF 4 0.6797 0.7084 0.7426 0.7731 0.8028 0.8369 0.8696 0.9014 0.9283 0.9447 0.9635 0.9816 0.9888 0.9955 0.9986

φ + γ = 0.7 GPH 0.6 0.4369 0.4825 0.5310 0.5764 0.6227 0.6593 0.7153 0.7628 0.8148 0.8574 0.8906 0.9240 0.9462 0.9565 0.9614

ADF 1 0.9871 0.9927 0.9941 0.9970 0.9982 0.9990 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

ADF 4 0.8881 0.9022 0.9188 0.9372 0.9511 0.9622 0.9760 0.9833 0.9894 0.9932 0.9967 0.9992 0.9993 0.9998 1.0000

Table 3: Power of the GPH test for a unit root in AR(2) processes for different values of φ + γ. The samplesize is 100 and the number of replicates 10000.

Level 1% 5% 10%

ADF 1 -3.9788 -3.3916 -3.0782

ADF 4 -3.8280 -3.2489 -2.9524

GPH 0.5 -2.9368 -1.9632 -1.5146

GPH 0.6 -2.8655 -1.9685 -1.5354

GPH 0.7 -2.8264 -1.9365 -1.5027

GPH 0.8 -2.8300 -1.9341 -1.5062

Table 4: Critical values for the GPH and ADF test of no cointegration. The sample size is 100.

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φ 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45

GPH 0.5 0.0480 0.0518 0.0527 0.0562 0.0590 0.0567 0.0515 0.0528 0.0509 0.0493 0.0478 0.0492 0.0455 0.0406 0.0404

GPH 0.6 0.0538 0.0621 0.0643 0.0658 0.0654 0.0631 0.0631 0.0576 0.0525 0.0524 0.0495 0.0430 0.0384 0.0330 0.0310

GPH 0.7 0.0968 0.1135 0.1112 0.1137 0.1091 0.0985 0.0948 0.0787 0.0662 0.0558 0.0450 0.0363 0.0277 0.0217 0.0205

GPH 0.8 0.4911 0.5106 0.4921 0.4488 0.3967 0.3226 0.2429 0.1700 0.1090 0.0662 0.0401 0.0254 0.0187 0.0171 0.0143

Table 5: Size of the GPH test for cointegration for different parameter combinations such that φ + γ = 1. The samplesize is 100 and the number of replicates 10000.

φ 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45

φ + γ = 0.95 GPH 0.6 0.0696 0.0797 0.0742 0.0867 0.0823 0.0787 0.0857 0.0842 0.0800 0.0829 0.0785 0.0709 0.0649 0.0560 0.0463

ADF 1 0.0369 0.0431 0.0516 0.0590 0.0593 0.0647 0.0729 0.0752 0.0748 0.0850 0.0893 0.0986 0.1003 0.1028 0.1069

ADF 4 0.0593 0.0598 0.0617 0.0654 0.0655 0.0739 0.0739 0.0731 0.0752 0.0794 0.0854 0.0949 0.0930 0.1077 0.1114

φ + γ = 0.9 GPH 0.6 0.0918 0.1107 0.1086 0.1241 0.1252 0.1275 0.1405 0.1427 0.1476 0.1630 0.1583 0.1604 0.1570 0.1477 0.1309

ADF 1 0.0642 0.0759 0.0865 0.1022 0.1085 0.1217 0.1373 0.1523 0.1623 0.1920 0.2163 0.2589 0.3116 0.3614 0.4242

ADF 4 0.0808 0.0858 0.0937 0.1007 0.1085 0.1185 0.1260 0.1323 0.1365 0.1559 0.1760 0.1982 0.2196 0.2606 0.3052

Table 6: Power of the GPH test for cointegration. The samplesize is 100 and the number of replicates 10000.

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φ 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45

φ + γ = 0.8 GPH 0.6 0.1856 0.2157 0.2362 0.2635 0.2851 0.3157 0.3529 0.3859 0.4198 0.4613 0.4952 0.5372 0.5727 0.5940 0.5794

ADF 1 0.1695 0.1919 0.2072 0.2251 0.2547 0.2768 0.3102 0.3390 0.3772 0.4165 0.4698 0.5335 0.5923 0.6549 0.7101

ADF 4 0.1694 0.1918 0.2072 0.2250 0.2546 0.2767 0.3101 0.3389 0.3771 0.4164 0.4697 0.5334 0.5922 0.6548 0.7100

φ + γ = 0.7 GPH 0.6 0.3290 0.3824 0.4221 0.4661 0.5022 0.5591 0.6224 0.6634 0.7132 0.7664 0.8132 0.8473 0.8782 0.9004 0.8904

ADF 1 0.3125 0.3454 0.3836 0.4151 0.4541 0.4986 0.5488 0.5940 0.6409 0.6857 0.7298 0.7896 0.8324 0.8523 0.8473

ADF 4 0.3124 0.3453 0.3835 0.4150 0.4540 0.4985 0.5487 0.5939 0.6408 0.6856 0.7297 0.7895 0.8323 0.8522 0.8472

Table 7: Power of the GPH test for cointegration. The samplesize is 100 and the number of replicates 10000.

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