Unit Root Tests in the Presence of Markov Regime-Switching
Charles R. Nelson, Jeremy Piger and Eric Zivot* Department of Economics, Box 353330 University of Washington Seattle, WA 98195-3330
January 1999 Revised: May 1999
Preliminary; Not to be Quoted
Abstract We investigate the performance of a battery of standard unit root tests when the true data generating process has a Markov-switching trend growth rate and variance. Regime switching under both the null hypothesis of a unit root and the alternative hypothesis of trend stationarity is considered. In contrast to the case of a single break in trend growth rate, multiple Markov-switching breaks under the null hypothesis do not create size distortions in the Augmented Dickey-Fuller test. Markov-switching in variance under the null hypothesis does not adversely affect standard unit root tests but can lead to overrejection in tests which allow for structural change. All tests have very low power when regime switching occurs under an alternative hypothesis which is stationary in the periods between the switching. Key words: Unit Root, Stochastic Trends, Deterministic Trends, Markov Switching, Regime Switching, Structural Change, Heteroskedasticity J.E.L classification: C22
Address correspondence to: Charles R. Nelson
[email protected] (206) 685-1382 *
Nelson and Piger acknowledge support from the Van Voorhis endowment at the University of Washington. Nelson and Zivot acknowledge support from the National Science Foundation under grant SBR-9711301. The authors are grateful to James Morley, Chris Murray and Dick Startz for helpful comments. Responsibility for errors are entirely the authors’.
Nelson, Piger and Zivot, Unit Roots and Markov-Switching
1) Introduction In a 1989 Econometrica paper, James Hamilton proposed a model in which the growth rate of the trend function of U.S. GNP switches between two different states according to a first order Markov process. His paper spawned a sizable literature modeling structural change using “Markov-switching”. A recent search yielded no less than 250 citations of Hamilton’s paper, many employing some version of his original model. Examples include investigations of housing prices (Hall, Psaradakis and Sola, 1997), the effects of inflation on UK commercial property values (Barber, Robertson and Scott, 1997), the natural rate of unemployment (Bianchi and Zoega, 1997), the effects of oil prices on U.S. GDP growth (Raymond and Rich, 1997), an inflation targeting rule (Dueker and Fischer, 1996), labor market recruitment (Storer, 1996), the nominal exchange rate (Engel, 1994), the dividend process (Driffill and Sola, 1998), government expenditure (Rugemurcia, 1995) and the level of merger and acquisition activity (Town, 1992). There are also a growing number of theoretical models which give rise to regime switching of the type considered by Hamilton, for example Bonomo and Garcia (1991), Cooper (1994), Evans, Honkapahja and Romer (1998) and Howitt and Mcafee (1992). Several recent studies have also used Markov-switching as an alternative to ARCH and GARCH to model heteroskedasticity in financial and macroeconomic time series. As Kim and Kim (1996) and the references therein make clear, this recent popularity mainly stems from the inability of GARCH type models to capture quick regime changes in volatility such as that observed around the 1987 stock market crash. Markov-switching enables the econometrician to model these regime changes by allowing the unconditional variance of the process to switch. Examples of where this technique has been used in the literature include Schwert (1989b, 1996)
3
Preliminary
and Turner, Startz and Nelson (1989) for stock returns, Kim (1993) for the determinants of monetary growth uncertainty and Garcia and Perron (1996) for the inflation and real interest rate. A separate area of interest in the time series literature, represented in work such as Nelson and Plosser (1982) and summarized nicely in Phillips and Xiao (1998), involves testing whether stochastic innovations to an economic series have permanent effects on the level of the series. If innovations have permanent effects, the series is said to have a unit root or be I(1) while if innovations do not have permanent effects the series is said to be I(0). Using standard diagnostic tests for a unit root, such as the Dickey-Fuller, hereafter DF, and Augmented Dickey-Fuller, hereafter ADF, tests [Dickey and Fuller(1979, 1981), Said and Dickey(1984)], many researchers are unable to reject the unit root null hypothesis for macroeconomic and financial time series such as GDP, interest rates and exchange rates. Some of these results have been challenged, beginning with Perron (1989), as spurious artifacts of the presence of structural breaks. Perron argued that standard unit root tests have very low power against alternatives with structural breaks in the level or growth rate of the trend function. Perron remedied this problem by augmenting DF tests with dummy variables to account for one structural break in the series. Christiano (1992), Banerjee, Lumsdaine and Stock (1992) and Zivot and Andrews (1992) extend the Perron methodology to endogenous estimation of the date the structural break occurs while Lumsdaine and Papell (1997) consider a model with two structural breaks. Leybourne, Mills and Newbold (1998) demonstrate that the converse issue also poses a problem for unit root tests. They find that standard unit root tests can generate spurious rejections when there is a single structural break in trend under the null hypothesis. In this case, an I(1) series which undergoes structural change may appear to be I(0).
4
Nelson, Piger and Zivot, Unit Roots and Markov-Switching
Given the extensive use of unit root tests on economic time series and the seemingly good fit which Markov-switching models provide to many of these same series, it is natural to ask what effects Markov-switching regime change may have on standard unit root tests. In particular, if a series exhibits switches in either innovation variance or the growth rate of its trend function, might these switches affect inference about whether the series contains a unit root? Examples of where this issue may be relevant in the literature are not hard to find. Evans and Wachtel (1993) perform unit root tests on the price level and based on the failure to reject they suggest an I(1) Markov-switching trend model for prices. Garcia and Perron (1996) argue for an I(0) Markovswitching trend and variance model of the inflation and real interest rates based on unit root tests performed by Perron (1990) suggesting these series were I(0) if one break in the level of the trend function is allowed. Evans and Lewis (1993) provide Monte Carlo evidence that tests of whether two series are cointegrated will incorrectly conclude that the two series have separate I(1) components if one of the series has a Markov-switching trend growth rate. Finally, many studies which employ a Markov-switching variance or trend function simply assume a unit root in the series of interest without any pre-testing, most likely because unit root tests from previous studies suggest the series are I(1). Examples include Hamilton’s original paper for GNP, Engel (1994) for the nominal exchange rate, Gray (1996) for the short term interest rate and Cecchetti and Mark (1990) for consumption and dividends. The literature surrounding structural breaks and unit root tests provides some insight into the effects of a pre-specified number of breaks in trend growth rate on standard unit root tests. However, it is not clear that these results generalize to the case of endogenous, Markovswitching breaks in trend. Perhaps the closest to addressing this question is Balke and Fomby (1991) who demonstrate that standard unit root tests continue to have very low power
5
Preliminary
when a series has endogenous, probabilistic breaks in trend growth rate and level. However, the process driving their breaks is an independent Bernoulli process, not a Markov-switching process. In addition, they do not consider structural breaks under the null hypothesis of a unit root nor do they consider the effects of switching variances. Several authors have considered the effects of heteroskedasticity on unit root tests, for example Kim and Schmidt (1993) and Hecq (1995) consider GARCH type heteroskedasticity for various tests of a unit root. However, the effects of a variance process which undergoes Markov-switching regime changes on unit root tests has not been considered. In this paper we investigate the effects of Markov-switching trend growth rates and Markovswitching heteroskedasticity on a battery of standard unit root tests. In section 2 we evaluate the performance of unit root tests when the true data generating process is I(1) and follows a simple Markov-switching variance and trend growth rate model. To our knowledge this is the first time the effects on unit root tests of endogenous, probabilistic structural breaks in trend growth rate under the null hypothesis of a unit root have been evaluated. We find that unlike the single break case considered by Leybourne, Mills and Newbold (1998), Markov-switching breaks in trend growth rate do not cause significant size distortions for the ADF test. Also, the Markovswitching variance causes essentially no size distortions for the standard DF and ADF tests. This is in slight contrast to Kim and Schmidt who find small size distortions induced in the DF test by GARCH errors. However, Markov-switching variance does cause significant over-rejection in tests which allow for structural breaks in trend growth rate and level. This is similar to the finding of Hecq (1995) for the case of GARCH errors. In section 3 we evaluate the performance of the same battery of tests when the true data generating process is I(0) and has a Markovswitching variance and trend growth rate. In line with the existing literature we find that unit root
6
Nelson, Piger and Zivot, Unit Roots and Markov-Switching
tests do a poor job of distinguishing an I(0) model with structural change from an I(1) model. Interestingly, this continued to be true even for tests which allow a single structural break in the trend function. In section 4 we explore the effects on unit root tests of a model with more complex Markov-switching dynamics. Section 5 questions some of the existing literature regarding the effects of additive outliers on unit root tests. Section 6 concludes.
2. Breaks Under the Null Hypothesis and Unit Root Tests 2.1 Specification and Statistical Properties of the I(1) Model To begin, consider the following model of a time series motivated by Hamilton (1989): q
∆y t = µ t + ∑ψ k ∆y t − k + vt j =1
vt ~ N (0, σ vt2 )
(1)
µ t = µ1 * S t + µ 0 (1 − S t ) σ vt2 = σ v21 * S t + σ v20 * (1 − S t ) S t is a stationary, ergodic unobserved state variable which takes on the value 0 or 1 in accordance with some probability process, as yet unspecified. In words, the process follows an AR(q) process in first differences with a switching growth rate and variance, where the switching is controlled by some process driving S t . The switching occurs between only two states, meaning the structural breaks in the series are constrained between two values. The model could be augmented to include additional states, but this would subtract clarity while not changing the theme of the derivations below. It is not difficult to show that the switching growth rate and variance model can be written as a model with a constant growth rate and a serially correlated, heteroskedastic error term. To see this consider the following alternative representation of (1):
7
Preliminary
q
∆y t = µ + ∑ψ k ∆y t − k + et j =1
(2)
et = ( µ1 − µ ) S t + ( µ 0 − µ )(1 − S t ) + vt Here, the process is written with a constant growth rate, the case for which standard unit root tests were derived. Thus, if the switching growth rate and variance are to have any effect on unit root tests it must be due to the error term, et . To further investigate et consider its autocovariance function: Cov(et , et − k ) = ( µ 1 − µ 0 ) 2 ( E ( S t S t − k ) − p 2 )
(3)
where p is equal to the unconditional probability that S t = 1 . If we make the assumption that S t and S t − k are independent as k grows large we have: lim Cov(et , et − k ) = ( µ1 − µ 0 ) 2 ( E ( S t ) E ( S t − k ) − p 2 ) = ( µ1 − µ 0 ) 2 ( p 2 − p 2 ) = 0 k →∞
(4)
Therefore, under this independence assumption the switching process is written with constant growth rate and errors exhibiting serial correlation which dies out over time. Also, et is heteroskedastic due to the switching variance in (1): E (et2 ) = ( µ 1 − µ 0 ) 2 ( p − p 2 ) + σ vt2
(5)
Some interesting cases are worth mentioning. First of all, if S t is driven by a process independent of time, such as a Bernoulli process, (3) suggests that the switching model can be characterized with constant growth rate and serially uncorrelated, heteroskedastic, errors. A second interesting case is that of first order Markov-switching. In this case the value of S t at time t depends only on its value at time t-1, such that P( S t = 1 | S t −1 = 1) = p11 and P( S t = 0 | S t −1 = 0) = p00 . What will be the properties of et in this case? A result from the theory of Markov processes tells us that the transition probabilities p11 and p 00 converge to the
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Nelson, Piger and Zivot, Unit Roots and Markov-Switching
unconditional probabilities p and (1- p ) at a geometric rate. Then, noting that E ( S t S t − k ) = p * P( S t = 1 | S t − k = 1) we have for the case of Markov-switching that Cov(et , et − k ) = ( µ 1 − µ 0 ) 2 ( p * P( S t = 1 | S t − k = 1) − p 2 ) → 0 geometrically. Thus, the model in (1) can be written as a model of constant growth rate with errors exhibiting serial correlation which dies off geometrically. It should be noted that this result is entirely due to the modeling of breaks in the trend function as endogenous, probabilistic events. It does not hold true in models assuming a pre-determined number of structural breaks in trend growth rate such as the cases considered by Perron (1989) and Zivot and Andrews (1992) among others. Several previous studies, for example Schwert (1989a), investigate the properties of unit root tests under various forms of ARMA innovations. Therefore, we will find an ARMA process a useful alternative representation of et in the Markov-switching case. Consider the following stationary AR(1) representation of S t given by Hamilton (1989): S t = (1 − p 00 ) + λS t −1 + ξ t
(6)
λ = −1 + p 00 + p11
where, conditional on S t −1 = 1 , ξ t = (1 − p11 ) with probability p11 and ξ t = p11 with probability 1 − p11 and conditional on S t −1 = 0 , ξ t = ( p00 − 1) with probability p 00 and ξ t = p 00 with probability 1 − p 00 . Using (6) note that: et − λet −1 = d + bξ t + vt − λvt −1 d = − pb + b(1 − p00 ) + pbλ
(7)
b = ( µ1 − µ 0 ) The term on the right hand side of (7) shares a similar autocovariance function to an MA(1) in that it is zero after the first lag. However, the autocovariance function cannot be matched to an MA(1) because of the Markov-switching heteroskedasticity in vt . Thus, et is the sum of a
9
Preliminary
stationary AR(1) and a process which approximates an MA(1). These approximate ARMA dynamics in the error term are added to the AR(p) dynamics in ∆y t already present in the model. 2.2 Simulation Results and the I(1) Model In this section we present Monte Carlo evidence regarding the size behavior of various unit root tests when the data generating process is (1) and S t is Markov-switching. For simplicity we set q equal to zero for all simulations. Without the switching, (1) is then simply a random walk with constant drift and i.i.d. errors, a case for which most unit root tests perform relatively well. We demonstrated in the previous section that (1) can be re-written with a constant drift term and serially correlated, heteroskedastic error terms. Thus, the only difference between the model for which unit root tests perform well, and therefore the driving factor behind any size distortions in these tests, is the heteroskedasticity in the errors and the serial correlation induced by the switching trend growth rate. Recognizing this, the Monte Carlo simulations are parameterized for cases where the induced serial correlation and amount of heteroskedasticity are big and cases where they are small. To see the key parameters determining the amount of serial correlation consider the k-lag autocorrelation of et : ( µ 1 − µ 0 ) 2 ( p * P( S t = 1 | S t − k = 1) − p 2 ) ( µ1 − µ 0 ) 2 ( p − p 2 ) + σ vt2
(8)
Ignoring terms which depend only on the transition probabilities and rearranging the remaining expression yields a ratio determining the relative size of the autocorrelations: (µ1 − µ 0 ) 2 σ vt2
10
(9)
Nelson, Piger and Zivot, Unit Roots and Markov-Switching
Equation (9) can be interpreted as a signal to noise ratio. The larger the variance of the error term, the larger the noise masking the serial correlation induced by the difference in the drift terms. The larger this ratio the larger the autocorrelations of et while smaller ratios yield smaller autocorrelations. Likewise, the key parameters determining the amount of heteroskedasticity are summarized in the ratio of (5) when S t = 0 to that when S t = 1 : ( µ1 − µ 0 ) 2 ( p − p 2 ) + σ v20 ( µ1 − µ 0 ) 2 ( p − p 2 ) + σ v21
(10)
Clearly there are numerous combinations of the relative size of (9) and (10) which could be formed. We choose to focus on two cases, one in which (9) is large and the amount of heteroskedasticity given by (10) is small and one in which (9) is small and (10) is large. This helps us to separate out the effects of the switching drift term and the switching variance. For the remainder of the paper we will refer to the first situation as the low correlation, high heteroskedasticity case and the second as the high correlation, low heteroskedasticity case. For each unit root test 1000 Monte Carlo simulations were performed with two sample sizes, 200 and 500 and values of p11 and p 00 ranging between 0.1 and 0.9. The low correlation, high heteroskedasticity case uses ( µ1 − µ 0 ) = 3 and a value of (10) equal to 4 when p = 0.5 . The high correlation, low heteroskedasticity case uses ( µ1 − µ 0 ) = 9 and a value of (10) equal to 1.1 when p = 0.5 . 2.2.1 Dickey-Fuller Test The standard DF test of a unit root has the null hypothesis that the series of interest follows a random walk with drift: y t = c + y t −1 + η t
(11)
11
Preliminary
where η t is a martingale difference sequence with common variance σ η2 . The alternative hypothesis is a stationary process around a deterministic time trend: y t = c + βt + η t The DF test nests these two hypotheses in the following test regression: y t = c + βt + ρy t −1 + η t
(12)
and tests the null hypothesis that ρ = 1 . Under the null hypothesis, Rubin (1950) has shown that the OLS estimate of ρ , ρˆ , is still consistent. However, the distribution of the t-statistic on ρ = 1 is not asymptotically normal. Instead, the limit distribution of this t-statistic is a function of de-meaned and de-trended Brownian motion. Dickey and Fuller (1979) employed a large number of Monte Carlo simulations to obtain critical values for the standard t-statistic t =
( ρˆ − 1) . se( ρˆ )
Figures 1-4 present the finite sample size results of a 5% DF test when the true data generating process is (1)1. Figures 1 and 3 are the small correlation, large heteroskedasticity case for sample sizes of 200 and 500 while figures 2 and 4 are the large correlation, small heteroskedasticity case results. The figures demonstrate that the serial correlation induced by the switching trend growth rate creates size distortions in the DF test. When the serial correlation is small (Figures 1 and 3) the test performs fairly well. However, when the serial correlation is large (Figures 2 and 4), the test is severely oversized. This is not surprising given that previous investigations, for example Schwert (1989a), have documented the deleterious size effects of serial correlation on the standard DF test. This is precisely the reason for the development of
1
To smooth the graphs a nine dimensional response surface was fitted to the actual rejection probabilities obtained from the Monte Carlo experiments. However, when discussing the results in the text we refer to the actual rejection probabilities, not the smoothed versions.
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Nelson, Piger and Zivot, Unit Roots and Markov-Switching
tests robust to serially correlated errors such as the parametric ADF test and the non-parametric Phillips-Perron test. Perhaps more interesting is the pattern of the size distortions in Figures 1-4. First of all, in all cases where p11 + p 00 = 1 the actual size of the test is very close to its nominal size. The reason lies in the fact that when p11 + p00 = 1 the unconditional expectation of S t , given by p , is equal to P( S t = 1 | S t −k = 1) for all k. Thus, in this special case the states are independent which from (3) tells us that the switching trend growth rate will induce no serial correlation. The error term is still heteroskedastic but the DF test does not seem to be affected by its presence in this case. The role of heteroskedasticity will be discussed in more detail in section 2.2.2. The test performs most poorly for large average values of (9) and values of p11 and p 00 which increase the size of (8). These values occur for extreme values of p11 and p 00 , that is when p11 = p 00 = 0.1 and p11 = p 00 = 0.9 . In the former case the test is oversized, sometimes rejecting over 70% of the time, while in the latter it is undersized, rejecting less than 1% of the time in some cases. Thus, the failure of the DF test in the presence of a switching drift is simply a function of the severity of the serial correlation introduced by the probabilistic trend breaks. 2.2.2 Heteroskedasticity and the Dickey-Fuller Test Figures 1-4 demonstrate that Markov-switching heteroskedasticity does not cause significant size distortions for the DF test. This is interesting given that the DF test is based on the assumption that the sequence of innovations are i.i.d [Dickey and Fuller (1979, 1981)]. The adverse affects of non-i.i.d. innovations are pointed out by Phillips (1987) and Phillips and Perron (1988), hereafter PP, who demonstrate that the DF test is not asymptotically justified when innovations follow general forms of serial correlation and heteroskedasticity.
13
Preliminary
However, there are many situations when the DF test has the correct asymptotic distribution if the innovations are only heteroskedastic and not serially correlated. For example, Pantula (1988) shows the DF test has the correct asymptotic distribution under ARCH(1) innovations. Godfrey and Tremain (1988) provide Monte Carlo evidence that the DF test has approximately the correct size when innovations are taken as independent drawings from chi-square distributions. Also using Monte Carlo techniques, Peters and Veloce (1988) and Kim and Schmidt (1993) provide Monte Carlo evidence that standard DF tests are oversized in the face of GARCH errors, but not significantly. The one exception is Hamori and Tokihisa (1997) who show that the DF test can be significantly oversized when the innovations undergo a onetime shift in variance. The Monte Carlo evidence presented in Figures 1-4 suggest that, absent of other forms of breaks, Markov-switching heteroskedasticity is one case where the DF test is justified. In this section we demonstrate this analytically. Consider the following process: yt = α + ρyt −1 + ε t
(13)
PP show that under fairly general conditions regarding the process governing ε t the OLS tstatistic on the null hypothesis that ρ = 1 converges in distribution to the standard DF distribution plus a term which depends on the following two nuisance parameters: T
γ 0 ≡ lim T −1 ∑ E (ε t2 ) T →∞
2
λ = lim T T →∞
t =1
−1
E (ε 1 + ε 2 + K + ε T )
(14) 2
When γ 0 = λ2 the second term in the PP distribution disappears and the distribution of the OLS t-statistic collapses to the DF distribution. It is clear from the definition of the two nuisance parameters that they are equal when the ε t are independent. Thus, when faced with
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Nelson, Piger and Zivot, Unit Roots and Markov-Switching
heteroskedastic but independent innovations that meet the conditions given by PP the Dickey-Fuller distribution is asymptotically justified. As pointed out by Kim and Schmidt (1993), because the main interest in the PP paper is on serial correlation the conditions given by PP are more stringent than is needed for the simple independent heteroskedastic case. Kim and Schmidt note that these conditions can be avoided through the use of a Martingale Functional Central Limit Theorem. Specifically, we need the following convergence result for the sequence of partial sums: rT
1
∑ ε t → B(r ) γ 0T t =1
(15)
where B(r ) is an independent Brownian Motion. The relevant Functional Central Limit Theorem can be found in Davidson (1995): Let ε t be the martingale difference sequence with mean zero, T
variance σ t2 < ∞ and lim T −1 ∑ σ t2 = σ 2 < ∞ . Then, if ε t satisfies the following conditions, T →∞
T
t =1
T −1 ∑ ε t2 → σ 2
(16)
ε p sup t t → 0 t
(17)
p
t =1
rT
lim T −1 ∑ E (ε t2 ) = rσ 2
T →∞
(18)
t =1
the convergence described in (15) occurs. Here we are interested in the case where: ε t ~ N (0, σ t2 )
(19)
σ t2 = σ 12 S t + σ 02 (1 − S t )
15
Preliminary
and S t is two state Markov-switching. The Markov-switching variance process is certainly a T
martingale, as the innovations are independent. Also, lim T −1 ∑ σ t2 = (σ 12 + σ 22 ) p + σ 22 T →∞
t =1
where p is the unconditional probability that S t = 1 . Using this it is clear that both (16) and (18) hold, while (17) can be proven by invoking a more stringent condition, sup t E (ε t ) 4 < ∞ . Given that E (ε t ) 4 = 3(σ t2 ) 2 and both σ 12 and σ 22 are finite this condition holds. Therefore, an innovation with two state Markov-switching in variance satisfies the conditions needed for the DF t-statistic to converge to the PP distribution. Also, given that the innovations are independent the PP distribution collapses to the simple DF distribution asymptotically. Hence, the DF test is justified asymptotically under Markov-switching heteroskedasticity. With higher order autoregressive dynamics in y t the ADF methodology of Dickey and Fuller (1979) can be employed in the test regression to capture the effects of this extra serial correlation. Simulation results presented in the next section suggest the ADF test is also robust to the Markov-switching heteroskedasticity. 2.2.3 Augmented Dickey-Fuller Test The ADF test builds on the DF test by allowing for richer dynamics in the series of interest. In particular, the ADF test has the null hypothesis that the series being tested follows an AR(k) process in first differences: k
y t = c + y t −1 + ∑ φ j ∆y t − j + η t
(20)
j =1
where η t is as defined in (11). The alternative hypothesis is again that y t is stationary around a deterministic time trend. The test regression nesting both hypotheses is then:
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Nelson, Piger and Zivot, Unit Roots and Markov-Switching
k
y t = c + ρy t −1 + βt + ∑ φ j ∆y t − j + η t
(21)
j =1
Under the null hypothesis the φ ' s are coefficients on the stationary first difference of I(1) variables. Thus, standard asymptotic results apply for the distribution of φˆ j . However, ρ$ is the estimate of a coefficient on an I(1) variable and has a different asymptotic distribution. Dickey and Fuller (1979) show that the t-statistic associated with the null hypothesis ρ$ = 1 has an asymptotic distribution equal to that of the t-statistic estimated on (12). Said and Dickey (1984) demonstrate that the ADF test is asymptotically valid for more general ARMA processes of unknown order provided that the lag length k increases at a suitable rate with the sample size. A problem with the ADF test is that the lag length k is rarely known in practice and thus must be chosen by the researcher in some way. There is ample evidence, see for example Hall (1994) and Perron and Ng (1995), that procedures which estimate the lag length from the data are superior to those which impose some arbitrary k. In this experiment two methods are used to estimate k. The first is a backward selection procedure posited by Campbell and Perron (1991). In this procedure a maximum lag length k is chosen and used to estimate (21). A standard t-test on the significance of φ k is performed and, if it is significant, k is set equal to k . If not, k is reduced by one and the process is repeated. Perron and Ng demonstrate for the ARMA case that if k is chosen greater than the true value of k the t-test generated by this procedure has an asymptotic distribution equal to that derived by Dickey and Fuller. The alternative procedure used is the Schwarz information criterion (SIC). The SIC chooses k to minimize a function of the least squares residuals from estimating (21) over all possible values of k up to some arbitrary maximum k. The SIC introduces a penalty factor for larger values of k thus tending to yield a more parsimonious model than the Campbell-Perron procedure.
17
Preliminary
Note that the ADF procedure approximates the serial correlation in the data with a finite order autoregressive representation. Recall from section 2.1 that the model given in (2) has errors which follow a process similar to an ARMA(1,1) and that these errors die off at a geometric rate. Given this relatively fast rate of decay we might expect a relatively short order autoregression would provide an adequate approximation. Figures 5-8 show the size results of the Monte Carlo experiment when the ADF test is performed using the Campbell-Perron procedure and k is set equal to the lower integer bound of T 1 / 3 in accordance with Said and Dickey (1984). The performance of the test is quite good in all cases. The test usually rejects between 4% and 8% of the time, and in general is slightly oversized. This is likely due to the Campbell-Perron pre-testing procedure which has been documented by Hall (1994) to cause slight over-rejection. Thus, the ADF autoregressive representation of the serial correlation is adequate. Also, as with the standard DF test, the ADF test does not seem affected by the heteroskedasticity introduced by the switching variance. Figures 9-12 show the size results of the Monte Carlo experiment when the ADF test is performed using the SIC lag selection procedure. Again, the maximum k evaluated, k , is set equal to the lower integer bound of T 1 / 3 . The test performs reasonably well, rejecting in a range similar to the ADF test with the Campbell-Perron procedure. However, the test does not approximate the serial correlation in the data as well as the Campbell-Perron based test as is evidenced by the slope of the rejection surface for different values of p11 and p 00 . This slope indicates that different values of the transition probabilities, which lead to different patterns of serial correlation, are affecting the test. The reason for this may be the SIC, because it penalizes added lags, tends to yield a smaller value of k than the Campbell-Perron procedure and thus is not capturing as much of the correlation in the residuals. However, regardless of the lag selection
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Nelson, Piger and Zivot, Unit Roots and Markov-Switching
procedure used, the ADF test provides a reasonably sized test of the presence of a unit root in a series with a switching trend growth rate and heteroskedasticity. 2.2.4 Modified Phillips-Perron Tests Phillips (1987) and PP (1988), provide a non-parametric alternative to the ADF test in situations where the errors in a unit root process are non-i.i.d. Phillips (1987) demonstrated that when the error term in (11) follows a general stationary time series process the distributions of the statistic T ( ρˆ − 1) and the t-statistic on the null hypothesis that ρ = 1 are not that tabulated by Dickey and Fuller. Instead the distributions are augmented by the nuisance parameters given in (14). PP suggest modifying these statistics in order to remove the nuisance parameters from the distribution. Despite the promise of the PP tests they do not perform well in practice. In a revealing simulation investigation Schwert (1989a) demonstrated that in cases where the error term has large negative moving average errors the PP tests based on standard asymptotic critical values can be extremely oversized even in relatively large samples. Perron and Ng (1996) demonstrate the poor size performance of the PP test extends to large negative AR(1) errors. They suggest a modified statistic based on work in Stock (1990). Let the error term in (11) follow the following process: ηt = ψ ( L)ε t ∞
∞
ψ ( L) = ∑ψ j and ∑ j ψ j < ∞ j =0
j =0
ε t ~ i.i.d .(0, σ ) 2 ε
∞
Define γ j = E (η tη t − j ) and λ = σ ε ∑ψ j . The PP test statistic based on T ( ρˆ − 1) is then: j =0
Z ρ = T ( ρˆ − 1) − (1 / 2)(T 2σˆ ρ2 / s 2 )(λˆ2 − γˆ 0 )
19
Preliminary
and the modified PP statistic is: MZ ρ = Z ρ + (T / 2) * ( ρˆ − 1) 2 where γˆ 0 =
T 1 T 2 ∑ηˆ t , ηˆ t = y t − ρˆy t −1 , s 2 = (T − 2) −1 ∑ η t2 , σˆ ρ is the least squares standard error t =1t T t =1
for ρ , and λˆ is a consistent estimator of λ . Perron and Ng conclude that the test performs best when a parametric autoregressive estimator of λ is used based on the autoregression given by (21). Details are given in Perron and Ng (1996). Figures 13-16 demonstrate that the MZ t test is undersized, sometimes severely. For a sample size of 200 and the large correlation, low heteroskedasticity case, the 5% nominal sized test has actual size closer to 2% in most cases. These results confirm Perron and Ng’s finding that the test is somewhat undersized when the innovation process follows an AR(1). Recall from (7) that the error term in these simulations contains an AR(1) component which may explain the test’s low rejection frequency. 2.2.5 Unit Root Tests Which Allow for Structural Breaks Since the influential work of Perron (1989) a large number of unit root tests which allow for different sorts of structural breaks under both the null and alternative hypotheses have been developed. The choice to use one of these break tests usually occurs either because visual inspection of the data suggests a break or theory tells the researcher that the data in question has a break. The data generated by (1) could appear to have two kinds of breaks. Clearly, the switching drift term will create breaks in the growth rate of the trend function. Less obviously, because the series is I(1) the Markov-switching variance may appear as a break in the level of the trend function. It is possible that tests which allow for a break in level will spuriously interpret these breaks in variance as breaks in level. This was first pointed out by Hecq (1995), who
20
Nelson, Piger and Zivot, Unit Roots and Markov-Switching
presented Monte Carlo evidence that tests which allow for a change in mean can generate spurious break detection when the true process is I(1) with certain parameterizations of IGARCH errors. Here we investigate the effects of Markov-switching variance and trend breaks on two unit root tests which assume a single break in the growth rate and the level of the trend function, one detailed in Perron (1994) and the other developed by Zivot and Andrews (1992). Labeling Tb as the date the break occurs, the Perron test has the null hypothesis: t
t
t
j =1
j =1
j =1
y t = c + y t −1 + γDU t + δD(TB ) t + η t = y 0 + ct + γ ∑ DU t +δ ∑ D(TB ) t + ∑η t 1, if t > Tb DU t = 0, if t ≤ Tb 1, if t = Tb + 1 D(TB ) t = 0, otherwise t
∑ DU j =1
t
t − Tb , if t > Tb = DTt = 0, if t ≤ Tb
t
∑ D(TB) j =1
(22)
t
= DU t
where the error term η t is assumed to follow a general ARMA process, A(L) η t =B(L) ε t , ε t ~ i.i.d .(0, σ ε2 ) and the order of the lag polynomials A(L) and B(L) are possibly unknown. The alternative hypothesis is: y t = c + βt + θDTt + λDU t + η t
(23)
Note that (22) can be written in the same specification as (23) except the innovation term will be I(1). The test is then performed by estimating the model according to (23), forming the least squares residuals from the regression, and performing an ADF test for a unit root in these residuals. A different critical value, given in Perron (1994), must be used due to the preestimation of parameters and the break date. As in Banerjee, Lumsdaine and Stock (1992), the
21
Preliminary
time of the break, Tb is estimated by sequentially performing the test for all possible values of the break date and choosing the one which provides the most evidence against the null hypothesis. The Zivot-Andrews test is similar to the Perron test except it does not allow for breaks under the null hypothesis. The null hypothesis is then: y t = c + y t −1 + η t while the alternative hypothesis is the same as in the Perron test. Changing the null hypothesis allows a one stage testing procedure based on the estimated value of ρ in the test regression: k
y t = c + βt + γDTt + λDU t + ρyt −1 + ∑ c j ∆yt − j + ε t . The break date is estimated as in the Perron j =1
test. The two tests have the same asymptotic distribution, however, they may perform differently in finite samples due to the differing null hypothesis. Figures 17-20 present the results of the Perron break test when applied to data generated by (1) while Figures 21-24 present the results for the Zivot-Andrews test. Both tests were performed using the Campbell-Perron lag length procedure. As the small correlation, large heteroskedasticity figures demonstrate, the Markov-switching heteroskedasticity can cause significant over-rejection in these tests. For example, figures 17 and 21 demonstrate that when the low variance state dominates (high values of p11 relative to p 00 ) the tests are significantly oversized, sometimes rejecting at rates as high as 30%. Even where there is only a slight amount of heteroskedasticity, as in figures 18 and 22, the tests are still oversized, rejecting at a rate higher than 15% in some cases. These Monte Carlo results may help explain some puzzling results in the empirical literature. Perron (1989) found that a unit root in common stock prices is strongly rejected if a single
22
Nelson, Piger and Zivot, Unit Roots and Markov-Switching
exogenous structural break in level and trend is allowed. After endogenizing the estimation of the break date Zivot and Andrews (1992) were still able to reject the null hypothesis at the 10% level and only just failed to reject at the 5% level. This is a striking result considering that most models of the stock price assume not only that it contains a unit root, but that it follows a random walk. The idea that stock prices follow a deterministic time trend with stationary deviations from that trend has profound implications for models of asset pricing. One possibility is that Perron’s results are a spurious artifact of the well-documented existence of heteroskedasticity in the innovations to the stock price. Several studies have modeled the innovations to the random walk stock price as having a variance which Markov-switches between a low and high variance state, with the low variance state dominating [see for example Turner, Startz, Nelson (1989)]. This is precisely the sort of parameterization for which the tests considered above performed worse. This is consistent with Hecq (1995) who finds that spurious rejections in unit root tests which allow for shifts in level are worse for parameterizations of IGARCH processes often found in asset prices. The evidence presented here and by Hecq suggests these tests should be used with care in the presence of both conditional heteroskedasticity, (ARCH, GARCH) and structural changes in variance, (Markov-switching).
3. Breaks under the Alternative Hypothesis and Unit Root Tests Perron (1989) pointed out that standard unit root tests tend to have very low power against an I(0) alternative which undergoes a single structural break in trend. Balke and Fomby (1991) extend this result to a model with a shifting level where the shifting is driven by a Bernoulli process. They show this lack of power occurs because the shifting trend introduces a unit root into an otherwise stationary process. In this section we investigate this issue when the true
23
Preliminary
process is I(0) conditional on a Markov-switching variance and trend growth rate. We write this process as: yt = τ t + ηt τ t = µ1S t + µ 0 (1 − S t ) + τ t −1 ηt ~ N (0, σ η2t )
(24)
σ η2t = σ η21S t + σ η20 (1 − S t ) S t is again two-state Markov-switching. The model given by (24) is much like that in Lam (1990) with the added feature of Markov-switching heteroskedasticity. Innovations do not have permanent effects in the periods between shifts in the growth rate of the time trend. For some intuition into how unit root tests may perform at distinguishing this model from the I(1) model, consider the alternative representation of the Markov trend function, τ t : τ t = µ 0 * t + ( µ1 − µ 0 ) ∑ S t . Then: yt = DTt + RTt + ηt DTt = µ 0 t
(25)
RTt = RTt −1 + ( µ1 − µ 0 ) * S t Thus, yt can be written as the sum of a deterministic trend DTt , a stochastic trend RTt , and heteroskedastic noise. The stochastic trend is introduced because the effects of shocks introduced by the state variable are permanently reflected in the level of the series, meaning the model is technically I(1). So tests for a unit root should have very low “power” in the sense that the alternative given by (24) isn’t really an alternative at all, it actually is consistent with the null hypothesis. Intuitively, one would think that unit root tests would perform better when the variance of the error term in the stochastic trend, given by ( µ1 − µ 0 ) 2 * ( p − p 2 ) , is smaller
24
Nelson, Piger and Zivot, Unit Roots and Markov-Switching
rather than larger. Notice that this variance is smaller not when there are a smaller number of breaks, but instead when one transition probability dominates another. For example, ( p − p 2 ) is smaller when p11 = 0.9, p00 = 0.1 than when p11 = 0.9, p 00 = 0.9 , the case when there are the fewest breaks. One would also expect that as the sample size increases the test would detect the permanent shocks in the model more, thereby reducing power. Monte Carlo simulations were performed on a subset of the tests described in section 2 to determine their ability to distinguish the I(0) model from the I(1) model. We consider two versions of (24), using the same parameterizations of µ1 , µ 0 , σ η21 , and σ η20 as in section 2. In all cases we consider size adjusted power, meaning the correct finite sample 5% critical values were obtained for use in the power simulations. To save space, we present only those results for a sample size of 200 and the parameterization labeled the large correlation, small heteroskedasticity case in section 2. 3.1 Augmented Dickey Fuller Tests The ability of the ADF tests to distinguish the alternative model was quite poor. Figure 25 is an example, showing the Campbell-Perron version of the test. The test never rejects more than 30% of the time and usually rejects in the 5-10% range. This is what one would expect given that we have shown that the model in (24) is actually I(1). In essence, the ADF test is doing a good job of recognizing the permanent shocks introduced by the breaking growth rate. As mentioned above, one would also expect the test to perform best when one transition probability dominates the other. The pattern of the rejection probabilities suggests this is the case. Interestingly, the version of the test based on the SIC selection procedure performs much worse than the Campbell-Perron version, never rejecting at a rate greater than 5%. As expected, as the sample
25
Preliminary
size increased to 500 (not shown), the tests became worse in terms of power, rejecting around 5% for all values of the transition probabilities. 3.2 Modified Phillips-Perron Test The modified Phillips-Perron test also did a very poor job at distinguishing the two data generating processes, rarely rejecting more than the nominal size of the test, 5%. Again, this is likely due to the fact that the alternative model contains a unit root. The test is performing well in that it rejects close to its nominal size given that the data generating process is consistent with the null. 3.3 Break Tests Tests which allow for a single break under the alternative were developed to deal with the low power of standard unit root tests against such alternatives. It is of interest to see whether these tests still yield increased power when the true data generating process has multiple, probabilistic switching breaks. Again, when employing a test allowing for structural breaks the choice of test is often based on visual inspection of the data. Data generated by the process in (24) will appear to have breaks in trend growth rate. However, the Markov-switching heteroskedasticity will not appear as shifts in level as in section 2 because the innovations in (24) do not have permanent effects on the level of the time series. Therefore we employ versions of the Perron (1994) and Zivot and Andrews (1992) tests which allow for a single break in trend growth rate only. There has been some argument in the literature, for example Garcia and Perron (1996), that when there are multiple structural breaks in a series it may be enough to simply account for the largest of these breaks when performing unit root tests. Figures 27 and 28, which present the results of the tests for a sample size of 200 and the large correlation, small heteroskedasticity
26
Nelson, Piger and Zivot, Unit Roots and Markov-Switching
case, suggest that this is not the case in this situation. While the tests which allow for one break do improve somewhat on the ADF test in terms of distinguishing the I(0) from the I(1) model, the improvement is not great. The Zivot-Andrews test does a better job than the Perron test, likely due to the fact that breaks under the null hypothesis are not allowed under the ZivotAndrews test. A rejection under the Zivot-Andrews test may not be signaling rejection of the I(1) model but instead rejection of an I(1) model without breaks. Nonetheless, neither test consistently distinguishes the two models more than 50% of the time. The pattern of the rejection probabilities bears further comment. In line with prior intuition, the tests have higher power when one transition probability dominates. However, because the tests allow for a single break in the growth rate of the process, one might think that the tests would perform better when the truth was closer to this single break case, that is when p11 = p00 = 0.9 . The tests instead do very poorly in this case, suggesting that the effect of the increased variance of the stochastic trend RTt as p11 and p00 increase dominates any benefits of allowing for a single break.
4. A Markov-Switching Data Generating Process with Complex Dynamics The original Hamilton model only captures non-linearities in the trend function of the series of interest. Kim and Nelson (1998), (hereafter KN) find that a model that allows for Markovswitching heteroskedasticity and other non-linearities in the cyclical component of a process does a good job of characterizing U.S. GDP. Here we present a simple version of the KN model and investigate the effects of this generating process on one of the unit root tests investigated above, the ADF test. The model we will consider is the sum of a trend and cyclical component as follows:
27
Preliminary
y t = ξ t + ct ξ t = µ + ξ t −1 + η t , η t ~ N (0, σ η2t ) ct = φ1 ct −1 + φ 2 ct − 2 + ν t + τS t , vt ~ N (0, σ vt2 ) σ
2 vt
= σ S t + σ (1 − S t )
σ
2 ηt
= σ η21 S t + σ η20 (1 − S t )
2 v1
(26)
2 v0
The trend component, ξ t , follows a random walk, introducing a unit root into (26). Again, S t follows a Markov process as described above. Markov-switching enters in three ways. First, the variance of innovations to the trend component, ξ t have a high and low variance state. Secondly, innovations to the cyclical component ct have a high and low variance state independent of the trend component. Finally, asymmetry in the cyclical component is incorporated through a negative shock, or “pluck”, given by τ which occurs when the cyclical component is in the high variance state. The model does not allow any shifts in the trend component. Monte Carlo experiments were calibrated using the parameter estimates obtained by KN for US postwar quarterly GDP. The ADF test did well with this generating process having approximately the correct size. Increasing the depth of τ leaving everything else constant did yield a test which was oversized. This would be expected since increasing the depth of the pluck increases the variance of the stationary component, causing it to stand out as the dominant process. Further experiments suggested the key relation was the size of τ in relation to σ η2t in the high variance state. However, parameterizations indicative of actual GDP data suggest unit root tests performed on postwar quarterly GDP are likely reasonably sized, even if the underlying process involves some probabilistic switching. While unit root tests may be justified for postwar quarterly GDP this does not mean they are necessarily justified for longer GDP series. Ben-David and Papell (1995), Cheung and Chinn (1997) and Diebold and Senhadji (1996) reject the null hypothesis of a unit root using data
28
Nelson, Piger and Zivot, Unit Roots and Markov-Switching
extending back to 1870. Others, for example Murray and Nelson (1999), have argued that these results are an artifact of a one time outlier – the Great Depression. To investigate this issue the KN model was calibrated to have a large downward shock occurring at a frequency of every 120 years on average. Other moments were set to match the 1870-1994 U.S. real GDP series compiled by Maddison (1995). ADF tests were oversized in this case, rejecting between 18 and 20 percent of the time. The reason for this may be that the large catastrophic outliers accentuate the stationary component, causing unit root tests to be oversized in finite samples. The ADF test has been documented to have difficulties in such a situation, see for example Murray and Nelson (1999).
5. A Digression on Additive Outliers A related literature to that considering the effects of structural breaks on unit root tests is that considering the effects of additive outliers (AO’s) on these same tests. An AO is an aberration in the data that affects certain observations in isolation from others. Franses and Haldrup (1994), hereafter FH, point out that such outliers, if they occur in an otherwise integrated process, will lend evidence against a unit root because the outliers will be stationary. As an example, consider the autoregressive process: y t = φy t −1 + ε t
(27)
ε t ~ i.i.d . N (0, σ ε2 )
Because of additive outliers which make their way into the series the econometrician observes: xt = y t + θδ t
(28)
where δ t is a Bernoulli random variable taking on the value 0 or 1 with probability π . The existence of θ tends to bias φ toward zero. FH demonstrate through Monte Carlo simulations
29
Preliminary
that such a process will have deleterious effects on standard DF tests. They advocate a procedure which statistically estimates the location of the outliers and corrects for them. However, based on the evidence we have seen regarding endogenous structural breaks in section 2 we question whether this is necessary. Under the null hypothesis φ = 1 , (28) may be rewritten in first differences as: ∆xt = ε t + θ (δ t − δ t −1 )
(29)
This is simply an MA(1) process plus a noise term. Thus, the observed size distortions FH observe with the DF test are likely simply a result of the serial correlation induced by the AO’s. We might think that if the econometrician uses ADF tests instead of DF tests the size biases would be eliminated. To investigate this possibility, data was generated according to (27) and (28) with θ = 3 , π = .05 , φ = 1 , ε t ~ N (0, 1) and a sample size of 100. For this parameterization FH found that a 5% DF test rejected 18.1% of the time. However, we find that if the ADF test is used the test only rejects at a rate of 7.7%. Therefore, the ADF test seems well equipped to handle serial correlation induced by probabilistic occurring AO’s.
6. Conclusion For the past 20 years researchers have shown great interest in whether economic time series are I(1) or I(0). The primary tools in these investigations are a battery of unit root tests. At the same time there is a growing consensus that many economic and financial time series undergo structural breaks in trend growth rate and variance. These have been modeled with much success by an endogenous Markov-switching state variable. We have investigated the performance of standard unit root tests when the true process is I(1) in the periods between Markov-switching
30
Nelson, Piger and Zivot, Unit Roots and Markov-Switching
breaks in trend growth rate and variance and when it is I(0) between such breaks. We summarize our main findings below: 1) When the true process is I(1) and undergoes Markov-switching in both growth rate and variance, standard DF tests and Modified Phillips-Perron tests are size distorted, seemingly as a result of serial correlation induced by the switching trend growth rate. However, ADF tests have approximately the correct size. 2) Tests which allow for a single break in growth rate and level over-reject the null hypothesis when there are Markov-switching breaks in growth rate and variance. The main reason for this appears to come from the Markov-switching heteroskedasticity and, as in Hecq (1995), the problem is worse for parameterizations reminiscent of many financial time series. This may help explain some puzzling results in the literature such as Perron’s (1989) conclusion that stock prices are stationary around a broken trend. 3) In line with previous literature, all the unit root tests did a poor job of distinguishing an I(0) process with Markov-switching breaks from an I(1) process. This included tests which allow for a single structural break in trend growth rate suggesting that only accounting for one of several breaks is not enough to significantly increase power. This is likely because the Markov-switching trend breaks add a unit root to the otherwise I(0) process. Unit root tests are ill-equipped to distinguish whether permanent shocks in a process are coming from innovations every period or infrequent shocks to the trend function.
31
Preliminary
References Balke, N.S., and Fomby, T.B. (1991), “Shifting Trends, Segmented Trends and Infrequent Permanent Shocks,” Journal of Monetary Economics, 28, 61-85. Banerjee, A., Lumsdaine, R.L., and Stock, J.H. (1992), “Recursive and Sequential Tests for a Unit Root: Theory and International Evidence,” Journal of Business and Economic Statistics, 10, 271-287. Barber, C., Robertson, D., and Scott, A. (1997), “Property and Inflation: The Hedging Characteristics of UK Commercial Property, 1967-1994,” Journal of Real Estate Finance, 15, 59-76. Ben-David, D., and Papell, D.H. (1995), “The Great Wars, the Great Crash, and Steady State Growth: Some New Evidence About an old Stylized Fact,” Journal of Monetary Economics, 36, 453-475. Bianchi, M., and Zoega, G. (1997), “Challenges Facing Natural Rate Theory,” European Economic Review,” 41, 535-547. Bonomo, M., and Garcia, R. (1991), “Consumption and Equilibrium Asset Pricing: An Empirical Assessment,” CRDE Working Paper No. 2991, Université de Montréal (1991). Campbell, J.Y., and Perron, P. (1991), “Pitfalls and Opportunities: What Macroeconomics Should Know about Unit Roots,” NBER Macroeconomics Annual, 141-201. Cecchetti, S.G., and Mark, N.C. (1990), “Evaluating Empirical Tests of Asset Pricing Models – Alternative Interpretations,” American Economic Review, 80, 48-51. Cheung, Y-W., and Chinn, M. D. (1996), “Further Investigation of the Uncertain Unit Root in GDP,” Journal of Business and Economic Statistics, 15, 68-73. Christiano, L.J. (1992), “Searching for a Break in GNP,” Journal of Business and Economic Statistics, 10, 237-250. Cooper, R. (1994), “Equilibrium Selection in Imperfectly Competitive Economies with Multiple Equilibria,” Economic Journal, 104, 1106-1122. Davidson, J.E.H. (1995), Stochastic Limit Theory, 2nd ed. Oxford University Press. Dickey, D.A., and Fuller, W.A. (1979), “Distribution of the Estimators for Autoregressive Time Series with a Unit Root,” Journal of the American Statistical Association, 74, 427-31. Dickey, D.A., and Fuller, W.A. (1981), “Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root,” Econometrica, 49, 1057-1072.
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Diebold, F.X., and Senhadji, A.S. (1996), “The Uncertain Unit Root in Real GDP: Comment,” American Economic Review, 86, 1291-1298. Driffill, J., and Sola, M. (1998), “Intrinsic Bubbles and Regime-Switching,” Journal of Monetary Economics, 42, 357-373. Dueker, M., and Fischer, A.M. (1996), “Inflation Targeting in a Small Open Economy: Empirical Results for Switzerland,” Journal of Monetary Economics, 37, 89-103. Engel, C. (1994), “Can the Markov Switching Model Forecast Exchange Rates?,” Journal of International Economics, 36, 151-165. Evans, G.W., Honkapohja, S., and Romer, P. (1998), “Growth Cycles,” American Economic Review, 88, 495-515. Evans, M.D.D., and Lewis, K.K. (1993), “Trend in Excess Returns in Currency and Bond Markets,” European Economic Review, 37, 1005-1019. Evans, M.D.D., and Wachtel, P. (1993), “Were Price Changes During the Great-Depression Anticipated? Evidence from Nominal Interest Rates,” Journal of Monetary Economics, 32, 3-34. Franses, P.H., and Haldrup, N. (1994), “The Effects of Additive Outliers on Tests for Unit Roots and Cointegration,” Journal of Business and Economic Statistics, 12, 471-478. Garcia, R., and Perron, P. (1996), “An Analysis of the Real Interest Rate under Regime Shifts,” Review of Economics and Statistics, 78, 111-125. Godfrey, L.G. and Tremayne, A.R. (1988), “On the Finite Sample Performance of Tests for Unit Roots,” (Unpublished manuscript, University of New York). Gray, S.F. (1996), “Modeling the Conditional Distribution of Interest Rates as a Regime Switching Process,” Journal of Financial Economics, 42, 27-62. Hall, A. (1994), “Testing for a Unit Root in Time Series with Pretest Data-Based Model Selection,” Journal of Business & Economic Statistics, 12, 461-470. Hall, S., Psaradakis, Z., and Sola, M. (1997), “Switching Error-Correction Models of Housing Prices in the United Kingdom,” Economic Modeling, 14, 517-527. Hamilton, J.D. (1989), “A New Approach to the Economic Analysis of Non-stationary Time Series and the Business Cycle,” Econometrica, 57, 357-384. Hamori, S., and Tokihisa, A. (1997), “Testing for a Unit Root in the Presence of a Variance Shift,” Economics Letters, 57, 245-253.
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Hecq, A. (1995), “Unit Root Tests with Level Shift in the Presence of GARCH,” Economics Letters, 49, 125-130. Howitt, P., and Mcafee, R.P. (1992), “Animal Spirits,” American Economic Review, 82, 493-507. Kim, C.-J. (1993), “Sources of Monetary Growth Uncertainty and Economic Activity – The Time-Varying Parameter Model with Heteroskedastic Disturbances,” Review of Economics and Statistics, 75, 483-492. Kim, C.-J., and Kim, M.-J. (1996), “Transient Fads and the Crash of ’87,” Journal of Applied Econometrics, 11, 41-58. Kim, C.-J., and Nelson, C.R. (1998), “Friedman’s Plucking Model of Business Fluctuations: Tests and Estimates of Permanent and Transitory Components,” forthcoming Journal of Money, Credit and Banking. Kim, K., and Schmidt, P. (1993), “Unit Root Tests with Conditional Heteroskedasticity,” Journal of Econometrics, 59, 287-300. Lam, P.-S. (1990), “The Hamilton Model with a General Autoregressive Component,” Journal of Monetary Economics, 26, 409-432. Leybourne, S.J., Mills, T.C., and Newbold, P. (1998), “Spurious Rejections by Dickey-Fuller Tests in the Presence of a Break Under the Null,” Journal of Econometrics, 87, 191-203. Lumsdaine, R.L., and Papell, D.H. (1997), “Multiple Trend Breaks and the Unit Root Hypothesis,” The Review of Economics and Statistics, 79, 212-218. Maddison, A. (1995) “Monitoring the World Economy: 1820-1992,” Paris: OECD. Murray, C.J., and Nelson, C.R. (1999), “The Uncertain Trend in U.S. GDP,” forthcoming in Journal of Monetary Economics. Nelson, C.R., and Plosser, C.I. (1982), “Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implications,” Journal of Monetary Economics, 10, 139-162. Pantula, S.G. (1988), “Estimation of Autoregressive Models with ARCH Errors,” Sankhya B, 50, 119-138. Perron, P. (1989), “The Great Crash, the Oil Price Shock and the Unit Root Hypothesis,” Econometrica, 57, 1361-1401. Perron, P. (1990), “Testing for a Unit Root in a Time-Series with a Changing Mean,” Journal of Business and Economic Statistics, 8, 153-162.
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Perron, P. (1994), “Trend, Unit Root and Structural Change in Macroeconomic Time Series,” In Cointegration for the Applied Economist, ed. B.B. Rao, New York: St. Martin’s Press, pp. 113-146. Perron, P, and Ng. S. (1995), “Unit Root Tests in ARMA Models with Data-Dependent Methods for the Selection of the Truncation Lag,” Journal of the American Statistical Association, 90, 268-81. Perron, P., and Ng. S. (1996), “Useful Modifications to Some Unit Root Tests with Dependent Errors and Their Local Asymptotic Properties,” Review of Economic Studies, 63, 435-463. Peters, T.A., and Veloce, W. (1988), “Robustness of Unit Root Tests in ARMA Models with Generalized ARCH Errors,” (Unpublished manuscript, Brock University). Phillips, P.C.B. (1987), “Towards a Unified Asymptotic Theory for Autoregression,” Biometrika, 74, 535-547. Phillips, P.C.B., and Perron, P. (1988), “Testing for a Unit Root in Time Series Regression, Biometrika, 75, 335-346. Phillips, P.C.B., and Xiao, Z. (1998), “A Primer on Unit Root Testing,” (Unpublished manuscript, Yale University). Raymond, J.E., and Rich, R.W. (1997), “Oil and the Macroeconomy: A Markov State-Switching Approach, Journal of Money, Credit and Banking, 29, 193-213. Rubin, H. (1950), “Consistency of Maximum Likelihood Estimates in the Explosive Case”. In Statistical Inference in Dynamic Economic Models, ed. T.C. Koopmans, New York: Wiley, pp. 356-64. Rugemurcia, F.J. (1995), “Credibility and Changes in Policy Regime.” Journal of Political Economy, 103, 176-208. Said, S.E., and Dickey, D.A. (1984), “Testing for Unit Roots in Autoregressive-Moving Average Models of Unknown Order,” Biometrika, 71, 599-607. Schwert, W.G. (1989a), “Tests for Unit Roots: A Monte Carlo Investigation,” Journal of Business and Economic Statistics, 7, 147-159. Schwert, W.G. (1989b), “Why Does Stock Market Volatility Change Over Time?” Journal of Finance, 44, 1115-1153. Schwert, W.G. (1996), “Business Cycles, Financial Crises and Stock Volatility”. In Stock Market Crashes and Speculative Manias, ed. E.N. White, Cheltenham, U.K.: Elgar, pp. 231-73.
35
Preliminary
Stock, J.H. (1990), “A Class of Tests for Integration and Cointegration,” (manuscript, Harvard University). Storer, P. (1996), “Separating the Effects of Aggregate and Sectoral Shocks with Estimates from a Markov-Switching Search Model,” Journal of Economic Dynamics and Control, 20, 93-121. Town, R.J. (1992), “Merger Waves and the Structure of Merger and Acquisition Time-Series,” Journal of Applied Econometrics, 7, S83-S100. Turner, C.M., Startz, R., and Nelson, C.R. (1989), “A Markov Model of Heteroskedasticity, Risk, and Learning in the Stock Market,” Journal of Financial Economics, 25, 3-22. Zivot, E., and Andrews, D.W.K. (1992), “Further Evidence on the Great Crash, the Oil Price Shock, and the Unit Root Hypothesis,” Journal of Business and Economic Statistics, 10, 251-270.
36
Nelson, Piger and Zivot, Unit Roots and Markov-Switching
Dickey Fuller Test and the I(1) Model
Figure 2 Sample Size = 200, Large Correlation, Small Heteroskedasticity Case
Figure 1 Sample Size = 200, Small Correlation, Large Heteroskedasticity Case 0.08
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Figure 4 Sample Size = 500, Large Correlation, Small Heteroskedasticity Case
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Figure 3 Sample Size = 500, Small Correlation, Large Heteroskedasticity Case
p11
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37
Preliminary
Augmented Dickey Fuller Test and the I(1) Model (Campbell-Perron Lag Selection Procedure)
0.07
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p11
0.00 0.5
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Figure 8 Sample Size = 500, Large Correlation, Small Heteroskedasticity Case
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0.7 0.9
38
0.8
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Figure 6 Sample Size = 200, Large Correlation, Small Heteroskedasticity Case
Figure 7 Sample Size = 500, Small Correlation, Large Heteroskedasticity Case
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Figure 5 Sample Size=200, Small Correlation, Large Heteroskedasticity Case
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Nelson, Piger and Zivot, Unit Roots and Markov-Switching
Augmented Dickey Fuller Test and the I(1) Model (SIC Lag Selection Procedure)
Figure 9 Sample Size = 200, Small Correlation, Large Heteroskedasticity Case
Figure 10 Sample Size = 200, Large Correlation, Small Heteroskedasticity Case
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Figure 12 Sample Size = 500, Large Correlation, Small Heteroskedasticity Case
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Figure 11 Sample Size = 500, Small Correlation, Large Heteroskedasticity Case
p11
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39
Preliminary
Modified Phillips-Perron Test and the I(1) Model
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Figure 16 Sample Size = 500, Large Correlation, Small Heteroskedasticity Case
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40
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Figure 15 Sample Size = 500, Small Correlation, Large Heteroskedasticity Case
p11
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Figure 14 Sample Size = 200, Large Correlation, Small Heteroskedasticity Case
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Figure 13 Sample Size = 200, Small Correlation, Large Heteroskedasticity Case
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0.2
0.1
Nelson, Piger and Zivot, Unit Roots and Markov-Switching
Perron (1993) Break Test and the I(1) Model
Figure 17 Sample Size = 200, Small Correlation, Large Heteroskedasticity Case
Figure 18 Sample Size = 200, Large Correlation, Small Heteroskedasticity Case
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Figure 20 Sample Size = 500, Large Correlation, Small Heteroskedasticity Case
Figure 19 Sample Size = 500, Small Correlation, Large Heteroskedasticity Case 0.14
0.1 0.09
0.12 0.08 0.1
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41
Preliminary
Zivot-Andrews (1992) Break Test and the I(1) Model
Figure 21 Sample Size = 200, Small Correlation, Large Heteroskedasticity Case
Figure 22 Sample Size = 200, Large Correlation, Small Heteroskedasticity Case
0.3
0.14
0.25
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0.1 0.2
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size 0.06
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Figure 23 Sample Size = 500, Small Correlation, Large Heteroskedasticity Case
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Figure 24 Sample Size = 500, Large Correlation, Small Heteroskedasticity Case
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42
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Nelson, Piger and Zivot, Unit Roots and Markov-Switching
Unit Root Tests and the I(0) Model (All Graphs are for Sample Size = 200, Large Correlation, Small Heteroskedasticity Case)
Figure 26 ADF Test, SIC Lag Selection Procedure
Figure 25 ADF Test, Campbell-Perron Lag Selection Procedure 0.25
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Figure 28 Perron (1993) Break Test
Figure 27 Modified Phillips-Perron Test
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0.03 0.2
0.025 0.15 size
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43
Preliminary
Unit Root Tests and the I(0) Model (All Graphs are for Sample Size = 200, Large Correlation, Small Heteroskedasticity Case)
Figure 29 Zivot-Andrews (1992) Break Test 0.6
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44
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