SHU-PING SHI. School of Economics, The Australian National University SUMMARY

Transcription

1 ESING FOR PERIODICALLY COLLAPSING BUBBLES: AN GENERALIZED SUP ADF ES SHU-PING SHI School of Economics, he Australian National University SUMMARY Identifying exlosive bubbles under the influence of their eriodically collasing roerty has long been a concern in bubble testing literature. In this aer, we argue that the su Augmented Dickey-Fuller (ADF) test (Phillis, Wu and Yu, 2009), which imlements a right-tail ADF test and a su test on a forward exanding samle sequence, is sensitive to the samle starting oint when there are more than one bubble collasing eisodes within the samle range. o surmount this itfall we roose an alternative method named the generalized su ADF test, which amlifies the samle sequence by varying the samle starting oint within its feasible range. his test imroves the ower of the bubble testing method significantly. We then aly both tests to the Hong Kong stock market from Octobe980 to Aril he generalized su ADF tests find evidence of exlosive behavior in the Hang Seng Index, whereas the su ADF tests suggest the oosite. Key words: rational bubble, eriodically collasing, su ADF test, generalized su ADF test JEL classification: C22 address: shuing.shi@anu.edu.au (SHU-PING SHI). I am very grateful to Jun Yu and Peter C.B. Phillis for advice in the early stage of this research at Singaore Management University. I thank Heather Anderson, Farshid Vahid and om Smith for many valuable discussions and suggestions. 2 January 200

2 INRODUCION he literature on the identification of rational bubbles from market fundamentals stems from the Lucas asset ricing model. Most econometric tests of bubbles, namely, the cointegration based test (Diba and Grossman, 988), West s two-ste test (West, 987), the variance bounds test (Shiller, 98, LeRoy and Porter, 98) and the intrinsic bubbles test (Froot and Obsterfeld, 99), begin with the following equation (for an overview of econometric tests of bubbles, see Gurkaynak (2008)): P t = ( ) i E t (D t+i) + B t () i=0 + r f where P t is the after-dividend rice of the asset (i.e stock rice), and D t is the ayoff received from the asset (i.e. dividend) and r f is the risk-free interest rate. B t defines the bubble comonent, which has an exlosive roerty E t (B t+ ) = ( + r f ) B t. (2) his equation imlies that bubbles cannot o and restart (Diba and Grossman, 998). Provided that negative asset rices are imossible (irole, 982; Wu, 997), a multilicative form between B t and ε t is more reasonable than an additive form. hat is, B t+ = ( + r f ) B t ε t+, where E (ε t ) =. herefore, if B t equals zero at time t, it will stay at zero for all future eriods. However, Evans (99) argues that it is ossible that bubbles collase to a non-zero value and continue to grow at some exlosive rate deending on the bubble size. 2 Furthermore, Evans (99) shows via simulation that the conventional cointegration based test, which relies on a right-tail unit root test (with an exlosive alternative hyothesis), is incaable of detecting exlosive bubbles under the influence of the eriodically collasing roerty. 3 2 Blanchard (979) also notes the eriodically collasing roerty of bubbles. 3 he failure of the conventional cointegration based test is further studied in Charemza and Deadman (995) with the setting of bubbles with stochastic exlosive roots. 2

3 his argument has led to a number of aers which roose bubble testing methods that have some ower in detecting eriodically collasing bubbles. One of the revalent methods is the su ADF test (or the forward recursive ADF test) ut forward by Phillis, Wu and Yu (2009, PWY hereafter). hey roose to imlement the unit root test reeatedly on a forward exanding samle sequence and make inference based on the su value of the corresonding ADF statistic sequence. hey show that, comared to the conventional stationarity test, the su ADF test imroves the ower significantly in the resence of eriodically collasing bubbles. In this aer, we argue that the testing value of the su ADF test relies greatly on the starting oints of samles. Namely, if the starting oint of a samle is selected so that the samle includes more than one bubble collasing eisodes, the test may fail to reveal the existence of bubbles. o overcome this itfall of the su ADF test, we roose an alternative method named the generalized su ADF test. he generalized su ADF test is also based on the idea of reeatedly imlementing the ADF test; however, it extends the samle sequence to a broader range. Instead of fixing the starting oints of the samles (namely, on the first observation of the total samle), the generalized su ADF test extends the samle sequence by changing the starting oint of each samle over a feasible range, and suerimosing exanding samle sequences onto each starting oint. Consistent with the su ADF test, the samle sequence is designed (i) to cature the exlosive hase within the total samle and (ii) to ensure that there are sufficient observations to achieve estimation efficiency. herefore, the generalized su ADF test, which covers more samles, is exected to outerform the su ADF test in finding the most exlosive hase with the total samle, given an identical smallest samle size. he asymtotic distribution of the generalized su ADF statistic is then comared to that of the su ADF test. he imrovement of the generalized su ADF test over the su ADF test is demonstrated by erforming both tests on a simulated asset rice series. Furthermore, 3

4 based on the Lucas asset ricing model and the Evans bubble model, we calculate the owers of these two methods and find a significant gain in the generalized su ADF test. We then aly the su ADF test and the generalized su ADF test to the Hong Kong stock market from Octobe980 to Aril he outline of this chater is as follows. Section 2 discusses the rationale of the conventional cointegration based bubble test. he su ADF test and the generalized su ADF test, along with the asymtotic distribution of the su ADF statistic and the generalized su ADF statistic, are described in Section 3. Section 4 exlores the sensitivity of the su ADF test to the starting oint of the estimation samle via exerimenting on simulated asset rices. We then imlement the generalized su ADF test on the same simulated data series to show the advantage of the test. Power comarison is conducted in Section 5. An alication of these tests to the Hong Kong Stock index is resented in Section 6. Section 7 concludes the aer. 2 HE CONVENIONAL COINEGRAION BASED ES Based on the exlosive roerty of bubbles, Diba and Grossman (988) recommend the strategy of using a stationarity test for the logarithmic asset rices and observable market fundamentals, such as the logarithmic dividends. he conventional stationarity test is based on the standard Augmented-Dickey-Fuller test or Phillis- Perron test (Phillis and Perron, 998), but has an exlosive alternative hyothesis. Consider the model y t = α + βy t + k ψ i y t i + ε t (3) i= where y t is the logarithmic asset rice or the logarithmic dividend, ε t N (0, σ 2 ) and k is the number of lags. he significance test (Ng and Perron, 200) is used to determine the lag order. he null hyothesis is β = 0, which imlies that y t is a unit root rocess ( y t is stationary). he alternative hyothesis is β > 0, meaning that y t is exlosive ( y t is non-stationary). 4

5 When there are no bubbles in the market, equation () imlies that ( ρ) f t ρe d d t = κ + e d ρ j E t [ d t+j ], (see Aendix A.) (4) j= where t = log(p t ), d t = log(d t ), ρ = ( + r f ) and κ = (ρ ) ( ) + ρe d ( d ), where and d are the resective samle means of t and d t. his equation manifests the rationale of the cointegration based bubble tests. If the first order difference of the logarithmic dividend d t is stationary, t and d t should be cointegrated with vector [( ρ), ρe d ] in normal market states. Due to the ossible resence of unobservable market fundamentals such as intangible caital (Li, 2005), we cannot use evidence of nonstationarity in the first order difference of the asset rices t to conclude that there are bubbles. However, the reverse inference can be established. Namely, if no evidence of nonstationarity is found, the ossibility of bubbles can be ruled out. 3 HE SUP ADF ES AND HE GENERALIZED SUP ADF ES Suose the regression samle starts from the r th fraction of the total samle and ends at fraction r 2, where r 2 = + r w and r w is the fraction of the samle size in the regression. he number of observations in the regression is w = [ r w ], where [.] signifies the integer art of its argument and is the total number of observations. he su ADF test roosed by PWY imlements the ADF test reeatedly on a forward exanding samle sequence. he starting oint of the samle sequence is fixed at 0, so the ending oint of each samle r 2 is equal to r w. he samle window r w exands from r 0 to, where r 0 is the smallest samle window (selected to ensure estimation efficiency) and is the largest samle window (total samle size). he su ADF statistic is defined as su rw [r0,] ADF rw, and it is denoted by SADF. Under the null hyothesis that the true rocess is a random walk without drift, the 5

6 asymtotic distribution of the su ADF statistic is [ SADF L r rw w 0 W dw su r ] 2 w rw 0 W dr.w (r w ) r w [r 0,] rw /2 { rw rw 0 W 2 dr [ r w 0 W (r) dr] 2} /2, where W is the standard Wiener rocess (see aendix B and C. for the roof). Comared with the su ADF test, the generalized su ADF test extends the samle sequence to include more samles. Besides exanding the samle window r w, the generalized su ADF test allows the samle starting oint to vary within its feasible range, which is from 0 to r w. he regression starts from the first observation when = 0, and when = r w, the regression samle covers the last observation. he resective ADF statistic is denoted by ADF rw. We define the generalized su ADF statistic to be the largest ADF statistic over the feasible ranges of r w, and we denote this statistic by GSADF. hat is, GSADF = su r w [r 0,] { su [0, r w] ADF rw and the corresonding window size of the GSADF is referred to as the otimum window size r w, namely r w = arg su r w [r 0,] { su ADFr rw [0, r w] Under the null hyothesis that the true rocess is a random walk without drift, the asymtotic distribution of the generalized su ADF statistic is (see aendix C.) su r w [r 0,] su [0, r w] r 2 = +r w r w [ r2 W dw r ] 2 w r2 W (r) dr. [W (r 2 ) W ( )]. rw /2 } } { r r2 w W 2 dr [ r2 W dr ] 2 } /2., and It is well known that the Wiener rocess has indeendent increments with distribution W (r 2 ) W ( ) N (0, r w ). We can then infer that the generalized su ADF test nests the su ADF test. 6

7 Suose the true rocess is a random walk with drift, then both the su ADF statistic and the generalized su ADF converge to the standard normal distribution. hus, SADF and GSADF test statistics can be comared to the usual t tables to erform an asymtotically valid test (see aendix C.2 for the roof). In ractice, r 0 is inversely related to the total number of observations. If is small, r 0 needs to be large enough to achieve estimation efficiency. If is large, r 0 can be set to be a smaller number so that we will not miss any oortunity to cature the most exlosive hase. o obtain the asymtotic critical values of the ADF statistic distributions under the null hyothesis that the true rocess is a random walk, we resort to simulation. One of the key stes is to simulate the standard Wiener rocess. Since the Wiener rocess is continuous and stochastic, we can only generate a ath samled with a finite number of oints. Suose that t, t 2,, t N are equally saced within a finite interval. At each oint, we generate a Gaussian random variable with mean 0 and variance /N. he value of W (r) is the sum of the first r increments. he asymtotic critical values under the null hyothesis that the true rocess is a random walk without drift are dislayed in able. he simulated asymtotic able. he asymtotic critical values of the ADF tests with constant (true rocess is a random walk without drift) ADF SADF GSADF Stationary Alternative H : β < 0 % % % Exlosive Alternative H : β > 0 0% % % Note: he number of discrete oints for aroximating the Wiener rocess and the integral are 5,000 and 2,000 resectively. he smallest samle fraction r 0 for the su ADF statistic and the generalized su ADF statistic is 0.. 7

8 critical values for the ADF test are consistent with those in Fuller (996, able 0.A.2). he right-tail critical values of the generalized su ADF test are larger than those of the su ADF test. 4 SIMULAION SUDY In this section, we demonstrate how the the su ADF test and the generalized su ADF test work when we testing a samle eriod that contains more than one bubble collases. 4. Generating the test samle We first simulate an asset rice series based on the Lucas asset ricing model and the Evans s bubble model. he simulated asset rices consist of a market fundamental comonent P f t, which combines a random walk dividend rocess and the Lucas asset ricing equation 4 to obtain (see Aendix A.2) D t = µ + D t + ε Dt, ε Dt N ( ) 0, σd 2 (5) P f t = µρ ( ρ) 2 + ρ ρ D t (6) and a bubble comonent roosed in Evans (99) such that B t+ = ρ B t ε B,t+, if B t < b (7) B t+ = [ ζ + (πρ) θ t+ (B t ρζ) ] ε B,t+, if B t b. (8) 4 An alternative data generating rocess, which assumes that the logarithmic dividend is a random walk with drift, is as follows: ln D t = µ + ln D t + ε t, ε t N ( 0, σd 2 ) P f t = ρ ex ( µ + ) 2 σ2 d ρ ex ( µ + )D t. 2 σ2 d 8

9 his has the roerty that E t (B t+ ) = ( + r f ) B t. µ is the drift of the dividend rocess, σd 2 is the variance of the dividend, ρ = + r f > and ε B,t = ex (y t τ 2 /2) with y t NID (0, τ 2 ). ζ is the remaining size after the bubble collase. θ t follows a Bernoulli rocess which takes the value with robability π and 0 with robability π. hat is, θ t takes the value if the bubbles survive at eriod t; otherwise, it takes 0. Equation (7) states that a bubble grows exlosively at rate ρ when its size is less than b. If the size is greater than b (equation (8)), the bubble grows at a faster rate ((πρ) > ρ ) but with π robability of collasing. Fig.. Simulated data with samle size=400 We set the arameters in the data generating rocess as in Evans (99): B 0 = 0.5, α =, π = 0.85, ζ = 0.5, ρ = 0.952, τ = 0.05, µ = , D 0 =.3, and σ 2 D = Stock rices are then calculated as the sum of the market fundamental comonent and the bubble comonent: P t = P f t + 20B t as in Evans (99). he samle size is set to 400. Figure deicts one realization of the data generating rocess. As we can observe from this grah, there are four obvious sires within the samle. hose sires are either results of bubbles collasing or bubble-like volatilities in asset rices. 4.2 Generating the aroriate critical values Since the data generating rocess involves a constant term, we simulate critical values for these tests under the null hyothesis that the true rocess is a random walk with drift. Critical values dislayed in able 2 are obtained from 5, 000 Monte 9

10 Carlo simulations with 400 observations. he maximum lag order of the significant test is set to 2 for all tests in this aer. he smallest window size r 0 considered is 0., which contains 40 observations. As we can see from the table, the 5% right-tail critical value of the ADF test, the su ADF test and the generalized su ADF test are.26, 2.85 and 4.75 resectively. able 2. Critical values of the ADF tests with constant (true rocess is a random walk with drift) ADF SADF GSADF Stationary Alternative H : κ < 0 % % % Exlosive Alternative H : κ > 0 0% % % Note: critical values of both tests are obtained from 5,000 Monte Carlo simulations with samle size 400. he maximum lag order is set to 2. he smallest samle has 40 observations (r 0 = 0.). 4.3 Performing the su ADF test and the generalized su ADF test In this subsection, we first imlement the su ADF test on the whole samle range. o illustrate the instability of the su ADF test, we reeat the test on a sub-samle which contains fewer sires. Furthermore, to show the advantage of the generalized su ADF test, we conduct the test on the same simulated data series (with the whole samle rang). he smallest window size considered in the su ADF test for the whole samle is 0., which contains 40 observations. he ADF statistic sequence of the su ADF test is dislayed in Figure 2 and the eak of the sequence is the defined su ADF statistic. he su ADF statistic of the simulated data series is.20, which is smaller than the 5% critical value of the su ADF statistic herefore, we conclude that there are no bubbles in this samle. 0

11 Fig. 2. he su ADF test (a) Samle: to 400 with r 0 = 0. (b) Samle: 20 to 400 with r 0 = 0.2 Suose the su ADF test starts from the 20 th observation, which is right before the two largest sires. he smallest regression window also contains 40 observations (r 0 = 0.2). he resective ADF statistic sequence is dislayed in Figure 2. he su ADF statistic obtained from this samle is 6.47 and it is greater than hus, we confirm the existence of bubbles. As we can see, the su ADF test fails to reveal the existence of bubbles when the whole samle is utilized, whereas by re-selecting the starting oint of the samle to exclude sires before the largest one, it manages to confirm the existence of bubbles. Both exeriments above can be viewed as secial cases of the generalized su ADF test, where the samle starting oints are fixed. In the first exeriment, 5 From 200 observations, the 5% critical value obtained from Monte Carlo simulation with 5,000 relications is 2.68 (r 0 = 0.2).

12 the samle starting oint of the generalized su ADF test is set to 0. he samle starting oint of the second exeriment is fixed at he conflicting results we obtained from these two exeriments also demonstrate the imortance of a varying starting oint in the generalized su ADF test. We then aly the generalized su ADF test to the simulated asset rices. he otimum window size is 65 uon considering the sequence of window size from 40 to Figure 3 illustrates the ADF statistic sequence with the otimum window size. he generalized su ADF statistic of the simulated data is 7.35, which is greater than the 5% critical value It imlies that the generalized su ADF test find evidence of bubble existence. Comared to the su ADF test, the generalized su ADF identifies bubbles without mining the samle starting oint, which is an obvious imrovement. 7 Fig. 3. he generalized su ADF test Notice that both the su ADF test and the generalized su ADF test are tests of the exlosivity of the largest sire within the samle range. herefore, only the eak of their ADF statistic sequences can be comared to the resective 5% critical value. o exlore the significance of the second highest ADF statistics in the sequences, 6 o imrove the comutation seed, the maximum window size is set 0.3 (=20/400) instead of. he maximum window size normally can be set according to the feature of resective data series. 7 We observe similar henomenon from the alternative data generating rocess where the logarithmic dividend is a random walk with drift. Parameters in the alternative data generating rocess are set as in Hall et al. (999): B 0 = 0.5, α =, π = 0.85, ζ = 0.5, ρ = 0.952, τ = 0.05, µ = 0.03, D 0 = 0.26, σ 2 D = 0.06, and P t = P f t + 250B t. 2

13 we need critical values for the second largest ADF statistics or we have to exclude the largest sire from the samle range. he same argument alies to other sires in the samle range. herefore, we cannot conclude that the rest of the sires in the simulated data series are not exlosive simly based on the su ADF test or the generalized su ADF test. 5 POWER AND SIZE COMPARISON his section comares the owers and sizes of the su ADF test and the generalized su ADF test. he basic idea of ower comarison is to reeatedly generate data series with exlosive and eriodically collasing roerties, and comare the roortion of simulations for which the method draws the right conclusion that there exists bubbles. For the size comarison, we need to reeatedly generate a data series without bubbles, and comare the roortions of the simulations for which the method draws the wrong conclusion that there exists bubbles. he data generating rocess of the ower comarison is the same as the simulation in section 4, which is the summation of a market fundamental comonent and a bubble comonent. he data generating rocess of the size comarison is based on equation (5) and equation (6), which is the market fundamental comonent. he critical values of the su ADF test and the generalized su ADF test are dislayed in able 2. he number of iterations for ower and size calculations are,000. he exlosive alternative is tested at the 5% significance level. he samle size equals 400. he smallest samles of both tests are set the same as these in calculating resective critical values, which are both set to have 40 observations. able 3. Powers and Sizes of the ADF tests (obs.=400) ADF SADF GSADF Power Size (5%) Note: the number of iterations for ower and size calculation equals 000. he smallest samle has 40 observations (r 0 = 0.). 3

14 able 3 deicts the calculated owers and sizes of these methods. As shown in Evans (99), the conventional ADF test erforms oorly in the resence of eriodically collasing bubbles. We confirm the ower imrovement of the su ADF test as in PWY. he generalized su ADF test which is roosed to overcome the itfall of the su ADF test increases the testing ower from 0.67 to Furthermore, there is no significant difference in the sizes of these three different methods at the 5% level. hus, we conclude that the generalized su ADF test erforms better in revealing the existence of exlosive behavior. 8 able 4. Powers and Sizes of the ADF tests (obs.=200) ADF SADF GSADF Power Size (5%) Note: he 5% critical values are obtained from 5,000 times Monte Carlo simulations with samle size 200, which are.09, 2.82 and 4.30 for the ADF test, su ADF test and the generalized su ADF test resectively. he number of iterations for ower and size calculation equals 000. he smallest samle has 40 observations (r 0 = 0.2). Similar ower and size atterns are observed in able 4, where the samle size is 200. he owers of the su ADF test and the generalized su ADF test decrease with the samle size; nevertheless, the ower of the generalized su ADF test remains higher than the su ADF test. he size difference between the su ADF test and the generalized su ADF test again is not significant. 6 APPLICAION: HONG KONG SOCK MARKE o highlight the differences between the su ADF test and the generalized su ADF test, we investigate the resence of bubbles in Hong Kong stock market. Many aers have studied the evidence of bubble existence in Hong Kong stock market (e.g., Sornette and Johansen (200), Zhou and Sornette (2003) and Cajueiro and abak 8 Powers of the ADF test, the su ADF test and the generalized su ADF test with samle size 400 under the alternative data generating rocess, where we assume that the logarithmic dividend is a random walk, are 0.42, 0.94 and resectively. 4

15 (2006), among others). Although the samle eriods and the samle frequencies examined by these aers are different, most of them find evidences of of bubble existence. Our data is samled monthly over the eriod from Octobe980 to Aril 2009, constituting 343 observations. he data comrises the Hang Seng Index (HSI) and the consumtion rice index. he Hang Seng Index is downloaded from Datastream International. he consumtion rice index (October 2004-Setember 2005 =00) is obtained from the Hong Kong Monetary Authority. he consumtion rice index is used to convert the stock rices into real series. Fig. 4. Real Hang Seng Index (normalized to 00 at the beginning of data series) Figure 4 illustrates the behavior of the real Hang Seng Index (normalized to 00 at the beginning of the data series) during the data eriod. As we can see from the grah, the real Hang Seng Index fluctuates throughout the samle range. It is extremely volatile in the nine years sanning from 994 to 2002 due to the 997 Asian financial crisis, and a considerable increase that occurred over the eriod from Aril 2003 to November he eak of this increase was 6.97 times bigger than the starting oint of the series. he real Hang Seng Index, then droed quickly so that by March 2009 it was only 2.65 times that of the starting oint. his last change is obviously related to the subrime crisis. We aly the su ADF test and the generalized su ADF test to the logarithmic real HSI. able 5 resents critical values for these two tests and these were obtained from 5,000 times Monte Carlo simulation under the null hyothesis that the true 5

16 able 5. he su ADF test and the generalized su ADF test of the logarithmic real Hang Seng Index SADF GSADF Log real HSI Stationary Alternative H : β < 0 % % % Exlosive Alternative H : β > 0 0% % % Note: he otimum window of the logarithmic real Hang Seng Index has 37 observations. Critical values of both tests are obtained from 5,000 Monte Carlo simulations with samle size 343 under the null hyothesis that the true rocess is a random walk without drift. he smallest window is set to have 34 observations. rocess is a random walk without drift. In both erforming the ADF regressions and calculating the critical values, the smallest samle considered has 34 observations. From able 5, the su ADF statistic of the logarithmic real Hang Seng Index is.25, which is smaller than the 0% right-tail critical value.55 and greater than the 0% critical value for the stationary alternative of the su ADF test Based on the su ADF test, we conclude that the logarithmic real Hang Seng Index has a unit root. he generalized su ADF statistic of the logarithmic real Hang Seng index is 5.02, which is greater than the resective 0% critical value of the exlosive alternative, his suggests that the Hang Seng Index is exlosive based on the generalized su ADF test, which contradicts the result from the su ADF test. 7 CONCLUSION he su ADF test, also referred to as the forward recursive ADF test, imlements the ADF test reeatedly on a sequence of forward exanding samles. he generalized su ADF test can be viewed as a rolling window ADF test with a double-su 6

17 otimum window selection criteria 9. hat is, we select an otimum window size using the double-su criteria and imlements the ADF test reeatedly on a sequence of samles, which moves the otimum window frame gradually toward the end of the samle. By exerimenting on simulated asset rices, we show the itfall associated with the su ADF test inability to find bubbles when there are many sires in the samle range. In contrast to the su ADF test, the generalized su ADF test is able to surmount this roblem and we show that it significantly imroves the ower to finding bubbles. We aly both the su ADF test and generalized su ADF test to the Hang Seng Index from Octobe980 to Aril his series contained many sires before the rices soared in 2006 and subsequently crashed in 2007 (the well-known subrime crisis). his is similar to the simulated scenario that highlighted the itfall of the su ADF test and the test results are consistent with our exectation. he generalized su ADF suggests that there is exlosive behavior in the Hang Seng Index, whereas the su ADF test does not. REFERENCES [] O. J. Blanchard. Seculative bubbles, crashes and rational exectations. Economics Letters, 3: , Nov 979. [2] D. O. Cajueiro and B. M. abak. esting for rational bubbles in banking indices. Physica A, 366: , [3] J. Y. Cambell and R. J. Shiller. he dividend-rice ratio and exectations of future dividends and discount factors. he Review of Financial Studies, (3):95 228, 989. [4] W. W. Charemza and D. F. Deadman. Seculative bubbles with stochastic 9 First, we calculate the su value of the ADF statistic over the feasible ranges of the window starting oints for a fixed window size. hen, we calculate the su value of the su ADF statistic over the feasible range of window sizes. 7

18 exlosive roots: he failure of unit root testing. Journal of Emirical Finance, 2:53 63, 995. [5] B.. Diba and H. I. Grossman. Exlosive rational bubbles in stock rices? he American Economic Review, 78(3): , Jun 988. [6] D. A. Dickey and W. A. Fuller. Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74(366):427 43, June 979. [7] G. W. Evans. Pitfalls in testing for exlosive bubbles in asset rices. he American Economic Review, 8(4): , Se. 99. [8] K. A. Froot and M. Obstfeld. Intrinsic bubbles: he case of stock rices. American Economic Review., 8:89 24, 99. [9] R. S. Gurkaynak. Econometric tests of asset rice bubbles: aking stock. Journal of Economic Surveys, 22():66 86, [0] J. D. Hamilton. A new aroach to the economic analysis of nonstationary time series and the business cycle. Econometrica, 57(2): , March 989. [] J. D. Hamilton. ime Series Analysis. Princeton University Press, edition, 994. [2] S. F. LeRoy and R. D. Porter. he resent-value relation: ests based on imlied variance bounds. Econometrica, 49: , 98. [3] N. Li. Intangible caital and stock rices. Singaore National University, Working Paer, [4] S. Ng and P. Perron. Lag length selection and the construction of unit root tests with good size and ower. Econometrica, 69(6):59 554, 200. [5] P. C. B. Phillis and P. Perron. esting for a unit root in time series regression. Biometrika, 75(2): , 988. [6] P. C. B. Phillis, Y. Wu, and J. Yu. Exlosive behavior in the 990s Nasdaq: When did exuberance escalate asset values? International Economic Review, forthcoming, [7] R. J. Shiller. Do stock rices move too much to be justified by subsequent changes in dividends? American Economic Review., 7:42 436, 98. 8

19 [8] D. Sornette and A. Johansen. Significance of log-eriodic recursors to financial crashes. Quantitative Finance, :452 47, 200. [9] J. irole. On the ossibility of seculation under rational exectations. Econometrica, 50(5):63 8, Se 982. [20] K. D. West. A secification test for seculative bubbles. Quarterly Journal of Economics., 02: , 987. [2] Y. Wu. Rational bubbles in the stock market: Accounting for the U.S. stockrice volatility. Economic lnquiry, XXXV:309 39, 997. [22] W. Zhou and D. Sornette. Evidence of worldwide stock market log-eriodic anti-bubble since mid Physica A, 330: , [23] E. Zivot and D. W. K. Andrews. Further evidence on the great crash, the oil-rice shock, and the unit-root hyothesis. Journal of Business & Economic Statistics, 0(3):25 270, Jul

20 A A SIMPLE ASSE PRICING MODEL Consider a simle asset ricing model where risk-neutral investors choose between consumtion and holding a risky asset. Suose there exists a risk-free interest rate r f, the eriod-to-eriod arbitrage condition for the asset is 0 P t = ρe t (P t+ + D t+ ) (A.) where ρ = ( + r f ), P t is the after-dividend rice of the asset (i.e stock rice), and D t is the ayoff received from the asset (i.e. dividend). Iterating equation (A.) forward, we can obtain P t = j= ρ j E t D t+j + lim j ρ j E t (P t+j ). (A.2) he first comonent of equation (A.2) is defined as the market fundamental of the asset rices P f t and the second comonent is defined as the bubble comonent B t. hose are, P f t = ρ j E t D t+j j= B t = lim j ρ j E t P t+j (A.3) (A.4) he conditional exectation of ρb t+ is E t (ρb t+ ) = lim j ρ j+ E t [E t+ P t++j ] herefore, we have E t (B t+ ) = ( + r f ) B t. = lim j ρ j+ E t P t++j = lim k ρ k E t P t+k = B t 0 By imosing these restrictions, we illustrate a simlified version of the cointegration relationshi between stock rices and dividends. For a general descrition see Cambell and Shiller (989). 20

21 A. HE COINEGRAION RELAIONSHIP Define t = ln P t and d t = ln D t, equation (A.) can be rewritten as [ ] e t = ρe t e t+ + e d t+ Alying the aylor series exansion at the samle mean and d, t κ + ρe t [ t+ + e d d t+ ] where κ = (ρ ) ( ) + ρe d ( d ). By iterating forward, we can get: t κ ( + ρ + ρ ) + Since ρ <, so we can get j= ρ j E t ( e d d t+j ) + lim j ρ j E t ( t+j ) t κ ρ + j= ρ j E t ( e d d t+j ) + lim j ρ j E t ( t+j ) If there is no bubble in the market b t = 0; t = f t. hen, t = f t κ ρ + ( ) ρ j E t e d d t+j j= Multilying both sides by ( ρ), ( ρ) f ( ) t = κ + ( ρ) ρ j E t e d d t+j j= ( ) = κ + ρ j E t e d ( ) d t+j ρ j E t e d d t+j j= j=2 ( ) = κ + ρe t e d [ d t+ + ρ j E t e d (d t+j d t+j ) ] j=2 = κ + ρe d [ ] d t + ρe t e d [ ] d t+ + ρ j E t e d d t+j j=2 he last equation can be rewritten as ( ρ) f t ρe d d t = κ + e d ρ j E t [ d t+j ]. j= 2

22 A.2 DAA GENERAING PROCESS: HE MARKE COMPONEN Suose dividend D t follows a unit root rocess with drift D t = µ + D t + ε t, ε t N ( 0, σ 2 D). Iterating backwards, the dividend rocess can be rewritten as j D t+j = jµ + D t + ε t+i. i= (A.5) Combining equation (A.3) and equation (A.5), the market fundamental comonent of asset rice can be written as P f t = ρ j E t D t+j = µ jρ j + j= j= j= ρ j D t = µρ ( ρ) 2 + ρ ρ D t. Suose the logarithmic dividend follows a unit root rocess with drift ln D t = µ + ln D t + ε t, ε t N ( 0, σ 2 d). Iterating backwards, the dividend rocess can be rewritten as ( D t+j = ex jµ + ) j ε i= t+i D t (A.6) Combining equation (A.3) and equation (A.6), the market fundamental comonent of asset rice can be written as P f t = ρ j E t D t+j = j= j= ( = ρ j ex jµ + j= 2 jσ2 d ) = ρ ex ( µ + 2 σ2 d ρ ex ( )D µ + t 2 σ2 d ( ρ j E t [ex jµ + ) ] j ε i= t+i D t ) D t 22

23 B PROPOSIIONS AND PROOFS Proosition Let u t = ψ (L) ε t = Σ j=0ψε t j, where Σ j=0j. ψ j < and {ε t } is an i.i.d sequence with mean zero, variance σ 2 and finite fourth moment. Define y t = t u s with y 0 = 0, r 2 = + r w and γ j E (u t u t j ) = σ 2 s=0 ψ s ψ s+j for j = 0,, 2. hen, we can calculate that (a) (b) /2 (c) 3/2 (d) 2 (e) [ y t ε L t σ 2 r2 ψ () W (r) dw (r) ] 2 r w ε t L σ [W (r 2 ) W ( )] y t yt 2 (f) 3/2 L r2 ψ () σ W (r) dr L σ 2 [ψ ()] 2 r2 u t j 0 where W is the standard Wiener rocess. y t u t j 0, j = 0,, [W (r)] 2 dr Claim y t = ψ () t ε s + η t η 0, where η t = j=0 α j ε t j, η 0 = j=0 α j ε j and α j = (ψ j+ + ψ j+2 + ), which is absolutely summable. Please refer to Hamilton (994, ch ) for the roof. Claim 2 [ r2 ] ε2 t r w.σ 2. Since [ rw] r w as N, so by the law of large numbers, ε 2 t = [ r w]. [ r w ] ε 2 t r w σ 2. Claim 3 X (.) L σw (.), where X (r) is the samle mean of the first r th fraction of observations {ε t } t=. 23

24 Define a random walk rocess Z t = t ε s, then X (r) = [ r] ε s = Z [ r]. For any given realization, X (r) is a ste function in r, with 0 for 0 r < X (r) = Z fo r < 2. By the definition of X (r), X (r) = [ r] ε s = Z for r = [ r] [ r] [ r] ε s. By the central limit theorem, [ r] [ r] ε L s N (0, σ 2 ). Since [ r] r when, so X (r) L N (0, rσ 2 ). By the functional central limit theorem, we have X (.) σ where W is the standard Wiener rocess. L W (.) or X (.) L σw (.) Claim 4 Z2 [ ] and Z [ r2 ] = ε s. L σ 2 [W ( )] 2 and Z2 L σ 2 [W (r 2 )] 2, where Z [ r ] = [ ] ε s Define S (r) = [ X (r) ] 2, which can be written as 0 for 0 r < S (r) = Z 2 fo r < 2 (B.) Z 2 for r = 24

25 By Claim 3 and the continuous maing theorem, we have Z2 [ ] = S ( ) L σ 2 [W ( )] 2 Z2 = S (r 2 ) L σ 2 [W (r 2 )] 2. Claim 5 t ε s.ε L t σ [ 2 r2 W (r) dw (r) r 2 w]. he definition of Z t imlies that Z t = Z t + ε t. Summing Z t ε t from [ ] to and dividing by, Z t ε t = t=[ 2 Z2 2 Z2 [ ] 2 ] ε 2 t. herefore, combining Claim 4 and Claim 2, we can get Z t ε L t { 2 σ2 [W (r 2 )] 2 [W ( )] 2 } [ r2 r w = σ 2 Claim 6 (η t η 0 ) ε t 0 as. W (r) dw (r) 2 r w ]. Assumtions in the roosition ensure that {(η t η 0 ) ε t } t= is a martingale difference sequence with finite variance, so we comlete the roof. Claim 7 / ( η [ r] η 0 ) 0 as. Assumtions in the roosition ensure that {η t η 0 } t= is a martingale difference sequences with finite variance, so we comlete the roof. (a) From Claim, we have y t ε t = = ψ () ( ) t ψ () ε s + η t η 0 ε t t ε s ε t + (η t η 0 ) ε t. 25

26 herefore, by Claim 5 and Claim 6, we have [ y t ε L t σ 2 r2 ψ () W (r) dw (r) ] 2 r w. (b) From Claim 3, /2 ε t = [X (r 2 ) X ( )] L σ [W (r 2 ) W ( )]. (c) Define M (r) = / [ r] u s. From the definition y t = t u s, we can write M (r) as 0 for 0 r < M (r) = y fo r < 2. y for r = It follows that M (r) = / y [ r] = / [ r] ψ () ε s + η [ r] η 0 (from Claim ) = ψ () ( / From Claim 3 and Claim 7, we can get ) [ r] ε s + / ( η [ r] η 0 ) M (r) L ψ () σw (r). Integrating M (r) from to r 2, r2 = 3/2 s=[ ] ( y[ r ] M (r) dr =. y L r2 s ψ () σ W (r) dr. + + y ) 26

27 (d) Define N (r) = [ M (r) ] 2. We can write N (r) as 0 for 0 r < N (r) = y 2 fo r < 2 y 2 for r = Integrating r from to r 2, r2 N (r) dr = ( y 2 [ r ] ) + + y2 = 2 y s. 2 s=[ ] By the continuous maing theorem, 2 y2 t L σ 2 [ψ ()] 2 r2 [W (r)] 2 dr. (e) Since [ rw] r w as N, so by the law of large numbers, u t j = [ r w] [ r w ] u t j rw.e (u t ) = 0. Claim 8 y t u t L 2 ψ ()2 σ 2 [W (r 2 )] 2 2 (r 2 ) γ 0. By definition of y t = t u s, we have y t u t = 2 y[ 2 r 2 ] 2 u 2 s [ ] u 2 s. From art (d), y 2 = [ /2 ( u + + u [ r2 ])] 2 L σ 2 [W (r 2 )] 2. Further,more, [ ] u 2 s = [ r 2] u 2 s = [ ] [ ] u 2 s [ ] u 2 s r 2.γ 0.γ 0. 27

28 herefore, y t u t L 2 σ2 [W (r 2 )] 2 2 (r 2 ) γ 0. Claim 9 For j =, 2,, y t u L t j 2 σ2 [W (r 2 )] 2 j 2 (r 2 ) γ 0 + r w γ s. We observe that y t = y t j + j u t s, which imlies that +j y t u t j = +j y t j u t j + j u t s u t j +j (B.2) From Claim 8, we have +j y t j u t j L 2 σ2 [W (r 2 )] 2 2 (r 2 ) γ 0. Since +j j u t s u t j r w j γ s, equation B.2 converges to +j y t u L t j 2 σ2 [W (r 2 )] 2 j 2 (r 2 ) γ 0 + r w γ s. Combining with the fact that [ ]+j y t u t j 0, y t u L t j 2 σ2 [W (r 2 )] 2 j 2 (r 2 ) γ 0 + r w γ s. (f) From Claim 8 and Claim 9, we have 3/2 y t u t j = /2. y t u t j 0, for j = 0,,. 28

29 Proosition 2 Let u t = ψ (L) ε t = Σ j=0ψ j ε t j, where Σ j=0j. ψ j < and {ε t } is an i.i.d sequence with mean zero, variance σ 2 and finite fourth moment. Define y t = αt + t u s with y 0 = 0, r 2 = + r w and γ j E (u t t t j ) = σ 2 s=0 ψ s ψ s+j for j = 0,, 2. hen, we can calculate that (a) 3/2 (b) /2 (c) 2 (d) 3 (e) y t ε t 3/2 ε t L σ [W (r 2 ) W ( )] y t yt 2 (f) 2 α 2 r w ( + r 2 ) α2 ( r r) 3 u t j 0 where W is the standard Wiener rocess. y t u t j 0,, j = 0,, α (t ) ε t Claim 0 ξ t = Σ t u s = ψ () t ε s + η t η 0, where η t = j=0 α j ε t j, η 0 = j=0 α j ε j and α j = (ψ j+ + ψ j+2 + ), which is absolutely summable. he roof is the same as Claim. (a) From Claim 0, we have ( ) 3/2 y t ε t = 3/2 t α (t ) + ψ () ε s + η t η 0 ε t = 3/2 α (t ) ε t + ψ () 3/2 + 3/2 (η t η 0 ) ε t. t ε s ε t herefore, by Claim 5 and Claim 6, we have 3/2 y t ε t 3/2 α (t ) ε t. 29

30 (b) he roof is the same as (b) in Proosition. (c) Define M (r) = / [ r] u s. Let ξ t = t u s, we can write M (r) as 0 for 0 r < M (r) = ξ fo r < 2. ξ for r = It follows that M (r) = / ξ [ r] = / [ r] ψ () ε s + η [ r] η 0 (from Claim 0) = ψ () ( / ) [ r] From Claim 3 and Claim 7, we can get ε s + / ( η [ r] η 0 ) M (r) L ψ () σw (r). Integrating M (r) from to r 2, Since y t = αt + ξ t, r2 M (r) dr =. = 3/2 2 s=[ ] ξ t y s = 2 = α 2 ( ξ[ r ] + + ξ L r2 ψ () σ W (r) dr. s=[ ] αt + 2 [ r w ] ( + [ ]) 2 α 2 r w ( + r 2 ) ξ t t + 2 ) s=[ ] ξ t 30

31 (d) Define N (r) = [ M (r) ] 2. We can write N (r) as 0 for 0 r < N (r) = ξ 2 fo r < 2 ξ 2 for r = Integrating r from to r 2, r2 N (r) dr = ( ξ 2 ) [ r ] + + ξ2 = 2 ξt 2. By the continuous maing theorem, hen, 3 2 ξt 2 yt 2 = α 2 3 L σ 2 [ψ ()] 2 r2 t [W (r)] 2 dr. ξt α 3 α 2 ( tξ t r r) 3 (e) he roof is the same as (e) in Proosition. Claim 3/2 y t u t L 0. By definition of y t = α + y t + u t, we have y t u t = 2 [ y 2 t y 2 t α 2 2 αy t 2u t α u 2 t ] = 2 y2 2 y2 [ ] α 2 α 2 [ r ] α u s u s u 2 s 2 [ ] y t u 2 s. 3

32 From art (d), we know ξ 2 = [ /2 ( u + + u [ r2 ])] L σ 2 [W (r 2 )] 2. So, 2 y 2 = 2 { α α + ξ 2 } α 2 r 2 2. Similarly, we can get 2 y 2 [ ] α 2 r 2. From art (c), 2 y t α r 2 w ( + r 2 ). Also, [ ] u 2 s = [ r 2] u 2 s = [ ] [ ] u 2 s [ ] u 2 s r 2.γ 0.γ 0 herefore, 2 y t u t α 2 ( + r 2 ) (r 2 r w ) = 0 Claim 2 For j =, 2,, 2 y t u t j L 0. We observe that y t = α (j + ) + y t j + j u t s, which imlies that [ r 2 2 ] y t u t j = 2 j α (j + ) + y t j + u t s u t j +j +j From Claim, we have = 2 +j + 2 j u t s u t j +j 2 +j α (j + ) u t j + 2 y t j u t j L 0. +j y t j u t j Since +j j u t s u t j r w j γ s, equation (B.3) converges to 2 +j y t u t j L 0 (B.3) 32

33 Combining with the fact that 2 [ ]+j y t u t j 0, 2 y t u t j L 0. (f) From Claim, and Claim 2, for j = 0,,, we have 2 y t u t j L 0. C HE ASYMPOIC DISRIBUION OF HE GENERALIZED SUP ADF SAISIC Consider the ADF model y t = k= φ k y t k + α + βy t + ε t where ε t N (0, σ 2 ). Suose the samle starts from the fraction of the total samle and ends at fraction r 2, where [0, r w ], r 2 = + r w, r w [r 0, ] is the window size fraction and r 0 is the smallest fraction considered (0 < r 0 < ). he deviation of the OLS estimate ˆθ from the true value θ is given by ˆθ θ = X t X t X t ε t (C.) where X t = [u t u t 2... u t k y t ], θ = [ψ ψ 2... ψ k α β] and [.] signifies the integer art of its argument. C. RUE PROCESS IS A RANDOM WALK WIHOU DRIF Assume the intimal value y 0 = 0. Under the null hyothesis that α = β = 0, we have y t = t u s, where u t = ( φ L φ 2 L 2 φ L ) ε t = ψ (L) ε t. From (e) and (f) of Proosition, we know that the robability limit of X tx t is a block diagonal. herefore, we only need to obtain the last 2 2 comonents of [ r2 ] X tx t and the last 2 comonent of X tε t to calculate the ADF 33

34 statistics, which are Σ Σy t Σy t Σyt 2 and Σε t Σy t ε t resectively, where Σ denoting summation over t = [ ], [ ]+,,. Based on roosition 2, the scaling matrix should be Υ = diag (, ). Pre-multilying equation (C.) by Υ, results in Υ α = β β Υ X t X t Υ ( 2) ( 2) Υ X t ε t ( 2) [ Consider the matrix Υ [ r2 ] X ] tx t ( 2) ( 2) Υ, 0 Σ Σy t 0 Σ = 0 Σy t Σyt 2 0 3/2 Σy t L r w ψ () σ r 2 ψ () σ r 2 W (r) dr σ 2 [ψ ()] 2 r 2 W (r) dr. [W (r)] 2 dr 3/2 Σy t 2 Σyt 2 [ and the matrix Υ [ r2 ] X tε t ]( 2), 0 Σε t /2 Σε t L = 0 Σy t ε t Σy t ε t σ 2 ψ () [ r2 σ [W (r 2 ) W ( )] W (r) dw (r) r ] 2 w Under the null hyothesis that β = 0, ˆα r L w A C = ˆβ A B D A 2 r w B B A A C r w D 34

35 where r2 A = ψ () σ W (r) dr B = σ 2 [ψ ()] 2 r2 [W (r)] 2 dr C = σ [W (r 2 ) W ( )] D = σ 2 ψ () [ r2 W (r) dw (r) 2 r w herefore, the generalization of the Dickey-Fuller test when lagged changes of y are included in the regression is ˆβ.ψ () L r [ r2 w W dw r 2 w r w r2 ]. ] r2 W dr. [W (r 2 ) W ( )] W 2 dr [ r2 W dr ] 2 o calculate the t-statistic of ˆβ, we need to find the standard error of ˆβ. Since the variance of ˆθ is var (ˆθ) = var = σ 2 X t X t X t X t X t ε t herefore, variance of ˆβ is σ 2 multily the last element of [ [ r2 ] X tx t]. We know that ˆα Σ Σy var = σ 2 t ˆβ Σy t Σyt 2 so, the variances of ˆβ can be calculated as follows: ˆα var = var ˆβ Υ ˆα ˆβ 35

36 0 Σ Σy = σ 2 t 0 0 Σy t Σyt 2 0 Σ 3/2 Σy = σ 2 t r L σ 2 w A 3/2 Σy t 2 Σyt 2 A B Hence, the t-statistic of ˆβ is ˆβ se ( ˆβ) = ˆβ se ( ˆβ ) L AC r ( wd A 2 r w B. rw σ 2 r w B A 2 = r [ r2 w W dw r 2 w /2 w ) /2 ] r2 W dr. [W (r 2 ) W ( )] { r r2 w W 2 dr [ r2 W dr ] 2 } /2 he asymtotic distribution of the t-statistic of the generalized su ADF statistic is su r w [r 0,] su [0, r w] r 2 = +r w r w [ r2 W dw r ] 2 w r2 W dr. [W (r 2 ) W ( )]. rw /2 { r r2 w W 2 dr [ r2 W dr ] 2 } /2 In the su ADF test, the starting oint is fixed at 0, so r 2 = + r w = r w. herefore, the asymtotic distribution of the su ADF statistic is su r w [r 0,] [ r rw w 0 W dw r 2 w { rw rw 0 [W (r)] 2 dr [ r w /2 w 0 W (r) dr] 2} /2. ] rw 0 W dr.w (r w ) C.2 RUE PROCESS IS A RANDOM WALK WIH DRIF Assume the initial value y 0 = 0. Under the null hyothesis that β = 0, we have y t = α + u t = αt + t u s, where u t = ( φ L φ 2 L 2 φ L ) ε t = ψ (L) ε t and α = α ( φ φ 2 φ ) = ψ () α. From (e) and (f) of Proosition, we know that the robability limit of X tx t 36

37 is a block diagonal. herefore, we only need to obtain the last 2 2 comonents of [ r2 ] X tx t [ r2 ] and the last 2 comonent of X tε t to calculate the ADF statistics, which are Σ Σy t Σε t and Σy t Σyt 2 Σy t ε t resectively, where Σ denoting summation over t = [ ], [ ] +,,. Based on roosition, the scaling matrix should be Υ = diag ( /2, 3 2 ). Premultilying equation (C.) by Υ, results in Υ α α = β β Υ X t X t ( 2) ( 2) Υ [ Consider the matrix Υ [ r2 ] X ] tx t ( 2) ( 2) Υ, Υ X t ε t /2 0 Σ Σy t /2 0 Σ 2 Σy t = 0 3/2 Σy t Σyt 2 0 3/2 2 Σy t 3 Σyt 2 L r w αr 2 w ( + r 2 ) 3 α2 (r2 3 r) 3 [ and the matrix Υ [ r2 ] X tε t ]( 2), ( 2) αr 2 w ( + r 2 ) = V (C.2) /2 0 Σε t /2 Σε t /2 Σε t = 0 3/2 Σy t ε t 3/2 Σy t ε t 3/2 Σ α (t ) ε t 37

38 he variance-covariance matrix of the last two elements, /2 Σε [ ] t E /2 Σε t 3/2 Σ α (t ) ε t = σ 2 W, 3/2 Σ α (t ) ε t where herefore, we have α 2 (r2 2 r 2 ) W =. α 2 (r2 2 r) 2 3 α2 (r2 3 r) 3 /2 Σε t L h 2 N ( 0, σ 2 W ) 3/2 Σy t ε t (C.3) Combining equation C.2 and equation C.3, it follows that /2 (ˆα α) L V h 2 N ( 0, σ 2 V W V ). 3/2 ˆβ Let R = + r 2, R 2 = r 2 2 r 2 = r w R, R 3 = r 3 2 r 3 ; V W V = r w αr 2 wr α 2 wr 3 α2 R 3 = 6R3 2 24rwR rwR 2 4 αrr 2 2 αrr 3 α2 R 3 r w αr 2 wr α 2 wr 3 α2 R 3 4R 3 ( 3rwR R 3 ) /rw 2 6R ( 3rwR R 3 ) /r w ã 6R ( 3rwR R 3 ) /r w ã 2 [4R ( 2r w ) R] 2 /ã 2 We can see that ˆβ converges at rate 3/2 to a Gaussian variable 3/2 ˆβ.ψ () L h 3 N ( 0, σ 2 Ψ ), 38

39 where Ψ = 2 [4R ( 2r w ) R 2 ] (6R r 2 wr 3 + 9r 2 wr 4 ) α 2. For the su ADF test, R = r w, R 3 = r 3 w, Ψ = 2 [4r w + 3 ( 2r w )] r 3 w (25r w 24) α 2. For the ADF test, R =, R 3 = ; Ψ =2/α 2.herefore, all three tyes of the ADF statistics can be comared with the usual t for an asymtotically valid test. 39