Testing for a unit root in time series with trend breaks

JIM LEE Fore Hays State U~livct'sity

Hays, Kzmsas

Testing for a Unit Root in Time Series with Trend Breaks

This paper examines the implications of structural breaks tor the unit-root property of some post-World War II macroeconomie time series. Based on sequential methods, trend breaks are found ibr 8 out of the 17 series considered. Conditional on a single break, the augmented Dickey-Fuller test rejects the no-break unit-root null for 5 series. Further, 'allowing for multiple breaks as opposed to a single break yields little gain against the null. The e~idence against the null is much stronger based on recursive instead of sequential tests and weaker based on Phillips-Perron instead of augmented Dickey-Fuller statistics.

1. Introduction Since Nelson and Plosser's (1982) finding of unit roots in a wealth of

U.S. t ime series, it has become a consensus that secular growth ill economic activity can best be characterized by a stochastic as opposed to deterministic trend, so that macroeconomic data are difference instead of t rend stationary. Their test procedure, which follows Dickey and Fuller (1981), however, has been scrutinized.

F rom the perspect ive that a dynamic economy occasionally encounters exogenous shocks that appreciably change the courses of a wide variety of economic processes, studies have begun to examine the extent to which such events affect inferences on the unit-root proper ty of these t ime series. For instance, Balke and F o m b y (1991a) and Reichlin (1989) illustrate that a trend-stat ionary series with infrequent t rend breaks is observationally equiv- alent to a difference-stat ionary series without breaks. In effect, conventional unit-root tests entail bias toward accepting the latter.

In empirical work, Perron (1989, 1990a), and Rappopor t and Reiehlin (1989) attr ibute historical events, such as the onset of the Great Depression in 1929 and the oil price shock in 1973, to statistical bias in favor of the unit-root null hypothesis based on the conventional Dickey-Fuller testing method. However, the evidence against the null is weakened by Banerjee, Lumsdaine, and Stock (1992), Perron (1990b), Perron and Vogelsang (1992), and Zivot and Andrews (1992), who replace the selection ofbreakpoints using [)retest information with da ta -dependent algorithms. '

I For instance, at the 5% level, Perron (1990b) finds {mly 8, instead of the 11 series in his original (1989) paper, to be trend stationau'; and Zivot and Andrews (1992) find only 6 series.

Journal of Macroeconomics', Sunnner 1996, Vol. 18, No. 3, pp. 503~519 Copyright © 1996 by Louisiana State Universitx Press 0164-0704 / 96 / $1.50

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J im Lee

Based on recursive or sequential estimation, these recent studies eval- uate the presence of a one-time, or the most prominent, trend break bx locating the test statistic that lends the least support fi)r the unit-root nul]. One drawback of this approach is the possible occurrence of muhiple breaks as opposed to a single break. This argument is supported by Balke and Foml)y (1991a, 1994), who find a large number of outliers in some post-World War II data. Their findings fiarther strengthen the argument against the unit-root null in favor of a breaking trend alternative.

The objective of the present study, thereibre, is to explore the extent to which structural breaks in the postwar period affect inferences on the presence of unit roots in some representative time series. To this end, we treat the issues of structural breaks and unit-root tests sequentially. First, Chris- tiano's (1992) framework is used to detect breaks in a deterministic trend. Next, unit-root tests are performed conditional on these trend breaks. The test results are further used to assess the perfornmnce of the recently developed statistics conditional on a one-time trend break.

The paper is organized as follows. The next section discusses the models, test hypotheses, and finite-sample critical values. Sections 3 and 4 discuss the test results and the results particularly for the output data, respectively. The last section summarizes and concludes the paper.

2. Models, Tests, and Critical Values One way to test for the null hypothesis of a unit root (or random walk

process) against the alternative of a trend stationary process in a sequence {Yt} is based on the following augmented Dickey-Fuller (1981) regression:

L

y t = g + ~ t + C ~ y t - l + E c t A y t ~ + e t , t = l . . . . . T; (1) f=l

where A is a difference operator such that Ag t = Yt - Y t - 1, and the disturbance term e t is white noise. A popular test focuses on testing whether the estimated coefficient of ~ = 1 using its t-statistic, hereafter ADF t a. In addition, a unit root with drift implies a = 1 and g ¢ 0. As an "alternative to the use of lagged first-differenced terms, Phillips and Perron (1988) suggest a transformation of t~ which requires a nonparametric estimate of the spectral density of Ay t

at frequency zero, measured relative to the variance of Ay t. The resulting statistic, hereafter PP G, has a limiting distribution that can be compared with Dickey and Fuller's (1981) critical values. To ensure robustness and over- come the criticisms of any individual testing techniques, both ADF and PP procedures are considered in this paper. Further, to select the appropriate lag length L for the ADF test, the Akaike Information Criterion (AIC) is

504

Testing fi~r a Unit Root in Time Series with Trend Breaks

used. ̀ ) For the PP test, the autoregressive lag order of Ayt used to estimate the long-run variance is fixed at 6, which is consistent with Said and Diekey's (]984) T I/:~ rule.

To explore the significance of structural change on unit-root test statistics, we compare two alternative approaches. The first is developed bv Banerjee, Lumsdaine, and Stock (1999~), involving recursive estimation of the ADF t~-statistic. The second approach involves sequential estimation. In particular, we examine, first, a tu-test (Banerjee, Lumsdaine, and Stock 199fi; Zivot and Andrews 199fi) and, second, an F-test (Christiano 1999~). While the t¢,-tests underscore the inference on the unit-root h~pothesis conditional on the most prominent trend break, the F-test can be applied to detect multiple breakpoints.

Under the recursive approach, a sequence of ADF t~-statistics is obtained through estimating Equation (1) recursively over periods 1 through 1: where I: = 5 . . . . . T - 1 (without both end points)) The recursive test [or a unit root is per~brmed by searching ~br the minimum tc~ = win ta(;L), where k = k/T is the proportional location of the break date in the data sample (that is, break fraction). The lag length L in each recursive test is treated as data-dependent and is determined by the AIC.

The recursive method builds on the idea that the relevant data series 1nay be trend stationary over some observations but not the entire sample. If a structural break is present at period k, then a trend-stationary process will be detected before this period, and thereafter, the tu-statistic will indicate the presence of a unit root. The no-break unit-root null hypothesis is rejected if win t~(K) exceeds the corresponding finite-sample critical value (in absolute level) reported in Table 1 (panel A). The critical values Ibr a random walk process (with or without drift) are generated using Monte Carlo simulations followed by Banerjee, Lumsdaine, and Stock (1992, Table 1).4 More specifically, the minimum value of a sequence of"/~-statistics obtained through estimating Equation (1) with artificial data following a random walk (with or without drift) constitutes one point of the simulated statistics. As fi)r all other finite-sample critical values presented below, the Monte Carlo simulation is repeated 1,500 times and the distributions are obtained from the sorted vector of replicated statistics.

eOther data-dependent methods for selecting the lag length L have been suggested bv Campbell and Perron (1991) and Perron (199{)). flowever, since there is no consensus on the appropriate strategy,', we adopt the AIC for its popularity.

:~Zivot and Andrews (199fi) show that there is no unique solution for ~dlowing the end points. IThe distributions of the critical values tabulated in this paper do not appear to va~ in any

systeinatie way, if lags are included in the estimating equation.

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Jim Lee

Alternatively, under the sequential approach, a one-time dumn,'~ regressor is employed to explore the timing of structural change. The ADF equation augmented with a single trend break can be parameterized as:

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Yt = g + lit + 7d(k)t + Od(k)tt + ayt-~ + 2 ceAyt-~ + 6 , ,%1

(2)

where the dummy regressor d(k)t = 1 for t _> k, and 0 otherwise. By con- struction, Equation (2) encompasses three types of trend breaks. First, d(k), allows for a discontinuous jump in the linear trend line t at period k, and is referred to as a mean-shift model (model A). Second, there is a segmented trend (model B) which depicts a kink, or a change in the slope, of the trend function. Model B can be depicted by setting 7 = - k 0 such that the two dummy regressors collapse into (t-k)d(k) t. Taken together, Equation (2) accommodates the simultaneous occurrence of both events, that is, a shift in the trend function immediately followed by a change in its slope (model C).

Using an intervention framework, as depicted by Equation (2), Perron (1989) performs unit-root tests with a priori knowledge about the location of a breakpoint. His approach, however, has drawn criticisms from Banerjee, Lnmsdaine, and Stock (1992), Christiano (1992), and Zivot and Andrews (1992), who argue that structural breaks should be treated as unknown a priori. Alternatively, they devise data-dependent algorithms to endogenize the selection of a single breakpoint which has plausible impacts on the unit-root test statistics. Given the a priori unknown nature of trend breaks, unit-root tests are perlbrmed conditional on the three hypothesized breaking- trend alternatives: (1) 3'¢ 0, ¢ = 0 for model A (mean shift); (2) 1( = -k0, 0 ¢ 0 for model B (trend shift); and (3) 7 ~ 0, ~ ~ 0 fbr model C (both mean and trend shifts).

This data-dependent breakpoint selection method involves sequential, rather than recursive, estiination of Equation (2) for each possible break date k = 5 . . . . . T - 1 using the full T observations. The lag truncation parameter L for each sequential estimation is selected by the AIC. In particular, we examine two statistics. The first is the minimum to:statistic, rain ta*()~), fbr testing the estimated 0¢ = 1 over the T - 5 regressions. The no-break unit-root null is rejected if any rain tc,*()~) exceeds the corresponding finite-sample critical value (in absolute level) reported in Table 1 (panel A). Following Zivot and Andrews (1992), the critical values are tabulated using Monte Carlo simulations in which the true data-generating process follows a random walk (with or without drift), and the regression equation is augmented with a single trend break in the form of model A, B, or C (as depicted by Equation [2]) that occurs randomly over .03 < ~, < .99 (corresponding to k = 5 . . . . . T - 1). More specifically, from each estimation of Equation (2) with alternatiw~

506

Testing.for a Unit Root in Time Series with Trend Breaks

TABLE 1. Critical Values of rain t~ and max F-Tests

Without Drift With Drift

1% 5% 10% 1% 5% 10%

A. min t{~-Test Reeursive rain ta -4 .21 -3 .96 -3 .55 -4.31 -4.(}5 -3 .65 Sequential rain t~* Model A -5 .40 -4 .74 -4 .53 -5 .35 -4 .82 -4 .56 B -5 .00 -4 .47 -4 .13 -4 .68 -4 .40 -4 .23 C -5 .62 -5 .03 -4 .78 -5 .72 -5 .07 -4.81 B. nmx F-Test ,~lodel A 25.18 18.96 16.64 22.34 18.16 16.15 B 24.12 17.62 14.93 20.65 16.18 14.18 C 16.18 13.00 11.72 15.91 12.07 11.38

NOTES: The critical values are compnted using Monte Carlo simulations with 152 artificial data points generated by the true model: A y t = p + et, where e, ~nid(0,1) and Yo = O. The intercept la = 0 for the model without drift, and p. = 1 for the model with drift. The reeursive rain t~,-statistics are obtained b~tsed on Equation (1) following the procedures described in Banerjee, Lmnsdaine, and Stock (1992). The miifimum value of the sequence of t , ' s constitutes one point of the simulated statistics. Following the procedures described in Zivot and Andrews (1992), and Christiano (1992), the sequential rain t~* and m a x F-statistics are obtained from the ADF regression of Equation (2) over 5 < k -< 151 with no lags of dxyt. The n m x F-statistics are the mzaimnm of sequence F(r, T-r -3) with the corresponding number of parameter restrictions (r equals i for model A or B, and 2 for model C) and the degrees of freedom (152 observations less the number of parameters in the unrestricted regression, that is, 148 for model A or B, and 147 for model C). The simulations involve 1,500 replications aml the finite-sample distributions are obtained from the sorted vector of replicated statistics.

interventions (models A, B or C) and artificial data that follow a random walk (with or without drift), a rain t,*-statistie is retrieved to compile the distributions of Monte Carlo critical values. The figures are close to the asymptotic distributions tabulated by Zivot and Andrews (1992, Tables 2 to 4).

Next, we extend Christiano's (1992) framework to detect multiple stnmtural breaks. It involves a sequence of F statistics for testing a no-break ,lull hypothesis (i.e., 7 = 0 = 0) against each of the three trend-break alternatives. The F(r, T-r-L-3) statistics are computed sequentially over .03 _< < .99 with the numerator r = 1 for one parameter restriction associated with model A or B, and r = 2 for model C; and the denominator equal to the degrees of freedom for the unrestricted model. The null hypothesis is rejected if any F-statistic in the sequence exceeds the corresponding critical value reported in Table 1 (panel B). Critical wdues are generated using Monte

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Carlo simulations similar to those fbllowed by B a u e r j e e , Lumsdaine, and Stock (1992) and Christiano (1992) : More specifically, fronl each estimation of Equat ion (2) with alternative interventions (model A, B or C) x~qth artificial data following a random walk (with or x~4thout drift), the ln~bxinlnln F-statistic tor testing the no-break null hypothesis is selected to tabulate the distributions of the Monte Carlo critical values. In fine with their figures, the 5% critical values, for instance, are above 10, which are substantially higher than the asymptotic value of approximately 3.1.

In the F-test, the search for a breakpoint amounts to searching ibr the maximum statistic, ,~ulx F. A breakpoint at )v* is a (strict) global maximum if the F-statistic evaluated at 1~* is larger than the F-statistic at any other )~, that is, K* U [.03, .99] and F()v*) > F(k) for all ~, E [.03, .99] and ~, ;e k*. In addition, the F-test allows for the possibility of more than one break. I f more than one F-statistic above the 10% critical value exist, a breakpoint at 1~** is a (strict) local maxilnum if the F-statistic evaluated at 1~** is larger than the F-statistic at any other ~, sufficiently close to it, that is, )~** E [.03, .99] and F0~**) > F(K) for all k E [.03, .99], ~, ~ )~**, and )v n Ns()v**) where Na(~,**) is a &neighborhood of )~** for some positive (but small) value of 8. In practice, 8 is set to eliminate the F-statistics immediately adjacent to a max F-statistic. fs

Once breakpoints are identified, unit-root tests can be per formed within the f ramework of an intervention model:

I j L

i=l .1=] #=1

where the d u m m y regressor d(/~)i, is now indexed by a p rede te rmined break date ~:. Equat ion (2') allows for I mean shifts and j t rend shifts. Following Banerjee, Lumsdaine, and Stock (1992, Tables 4 and 7), we tabulate finite- sample critical values for the G-statistics based on Equation (2') with a single breakpoint that occurs randomly over .03 < ~ < .99. As displayed in Table 2 (panel A), the distributions are close to the distributions for sequential rain ta-statisties (Table 1).

For multiple breaks, we consider a simple two-break case. In addition, due to reasons apparent below, we consider only the case in which a specific

5Banerjee, Lumsdaine, and Stock (1992) provide critical values for models A and B only, and Christiano (1992) tabulates the values fbr model C with bootstrapping. Our simulations are performed without bootstrapping.

~s As a referee points out, since the max F-test proceeds as if the ~dtemative hypothesis is a single break, it m W not be an effective way for searching multiple breaks. On the other hand, the computational burden would be enormous if all possible combinations of break points were searched simultaneously. For example, for a simple case of two breaks and N observatiol*s, there are already N(N - 1) possible combinations.

508

Testing for a Unit Root in Ti~ru~ Series" with Trend Breaks

TABLE 2. Critical Values of Unit-Root t~ Tests with Breaks

Without Drift With D~ft

1% 5% 10% 1% 5% 10%

A. One Break Model A -5.25 -4.75 -4.50 -5.31 -4.72 -4.53 t?. -5.01 -4.47 -4.15 -4.67 -4.46 -4.20 C -5.66 -5.05 -4.75 -5.56 -5.07 -4.77 B. Two Breaks Model A -5.58 -5.03 -4.68 -5.46 -4.97 -4.60 B -5.20 -5.90 -4.60 -5.37 -4.82 -4.50 C -6.03 -5.54 -5.16 -6.14 -5.48 -5.12

NOTES: The critical values are computed using Monte Carlo simulations with 152 data points generated by the true model: Ay t = ~t + e t, where e t ~nid(0,1) and y() = 0. The intercept

= 0 for the model without drift, and ~t = 1 for the model with drill. The regression is based on Equation (2') with no lags of A!h. The break fractions are chosen randomly within the interval )~E [.03, .99]. The entries are the t~-statistics ewdnated at the selected vtdne of k associated with the critical values of the m a x F-statistics reported in Table 1. The simulations involve 1,500 replications and the finite-sample distributions are obtained fi'om the sorted vector of replicated statisties.

model is applied to both breakpoints. As in the single break case, tile breakpoints are selected randomly. As shown in panel B of Table 2, while the true model remains to be a random walk process without break, accounting for one more breakpoint (with additional parameters) in the regression equation raises the critical values. 7

To recapitulate, the maintained hypothesis throughout this study is the presence of a unit root without any trend break. This hypothesis is tested against a stationary process along three breaking-trend alternatives. In addition, an F-test is used to directly test for the presence of multiple trend breaks as an alternative to single-break tests developed in the literature. Monte Carlo simulations are constructed to derive finite-sainple critical values for unit-root testing purposes.

7Our methods are in contrast with those followed by Balke and Fomby (1991b) who consider structural change in the true data-generating process as opposed to the regression model. Hence, their results appear to be the opposite of those presented in Table 2, that is, the critical values decrease with an increasing number of breaks.

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Jim Lee

3. Data and Test Results Nelson and Plosser's (1982) original annual data of 14 nmcroecono,nic

time series have been reexamined by Perron (1989, 1990), Perron and Vogelsang (1992), Rappoport and Reichlin (1989), and Zivot and Andrews (1992), among others. These studies tbcns on a single break associated with the onset of the Great Depression in 19'29 or the oil price shock in 1973. Motivated by Balke and Fomby's (1991a, 1994) finding of numerous structural shocks in some postwar time series, we examine the quarterly data of Nelson-Plosser series covering the period of 1955:i-1992:iv (152 observations), s For the interest rate series, we examine both the T-bill rate and the 30-year bond yield. For the money snpply and velocity series, we consider both M1 and M2, along with their corresponding velocities. As such, the sample consists of 17 series. Except the data for the T-bill rate, bond ~eld, and unemployment which are in their natural levels, the log levels are used.

Table 3 reports the ADF and PP t~-statisties for testing the estimated = 1 based on Equation (1). Next to the t~-statistics are the lag orders of

the first difference term. The null hypothesis of a unit root cannot be rejected for all 17 series at the 5% confidence level. At the 10% level, it is rejected only for the M2 velocity series (PP test), Even though the ADF and PP statistics difl~r substantially for the majority of the series, their qualitatiw, results remain similar.

To filrther investigate the possibility that the unit-root null can be rejected within a subsample as opposed to the full sample of the data series, reeursive t,:tests are performed. The rain t,:statistics and their corresponding periods are reported in the last colunln of Table 3. Based on the 10% or higher critical values displayed in Table 1 (panel A), the unit-root null can be rejected for the subsamples of 9 series.

Table 4 summarizes the results of sequential tests. Based on mi~ t*- statistics, the no-break unit-root null hyl)othesis is rejected for 5 series at the 10% confidence level or higher. In particular, the statistics for the employment series are above the critical values (in absolute level) across all three models. As a comparison, the sequential to:test provides less evidence against the null than the reeursive test. Furthermore, except fbr two cases (model B of the M1 velocity series and model C of the stock price series), the periods associated with the sequential mi~ ta*-statistics differ from those with the reeursive rain to:statistics.

SExeept those for the selies of real and nominal wages, all data are obtained from Kolb and Wilson (1995). The data for real and nominal wages are obtained from various issues of'National Economic Trends published by the Federal Reserve Bank of St. Louis. All data are seasonally adjusted at the sourees. The trend component of MI w~riables is a linear trend. Although earlier data are available, it is instructive to avoid potentitfl structural changes during the Korean War and the price control periods in the early 1950s which could coinplieate our analysis.

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Based on max F-statistics, the no-break null hypothesis is rejected fbr 8 series. When local max F-statistics in addition to global max F-statistics are admitted, two break'points are detected for 4 series (all under model C). However, most of the break dates associated with local max F-statistics are located within 6 quarters of the corresponding global maxima. For tile T-bill rate series, on the other hand, the break date (1979:iv) associated with the rain tc~-statistic under model C coincides with a local instead of a global mar F-statistic.

As a comparison, the max F-test suggests evidence of trend breaks for more series than the amount realized using the sequential rain t ,* test, but fewer than those using the recursive test. At the 10% confidence level or higher, the no-break hypothesis is consistently rejected across the three tests for 3 series (employment, T-bill rate, and bond yield). The statistics f'or 5 series (real per capita GNP, unemployment, consumer prices, M2 velocity, and stock prices) that reject the no-break unit-root null under the recursive rain tu-test are not significant under the sequential rain t,~*-test.

Most breakpoints associated x~4th the 10% or higher rain t,*-statistics coincide with break'points identified using max F-statistics. In addition, the break dates corresponding to the sequential statistics appear to cluster around some particular periods or major economic events. To discern their historical patterns, Table 5 presents multiple break dates in chronological order. First, the employment series encounters structural shocks around the recession in the early 1960s; and the real wages series has a trend break associated with the wage and price controls during the 1970s.

In contrast with Perron's (1989) argument, the oil shock in 1973 had no significant impact on the output series. A break in the nomiiml GNP series is fbund to lye associated with the second oil shock in 1978, perhaps due to price rather than output changes. Furthermore, a major Fed policy shift in 1979 and financial deregulation in the early 1980s induced permanent shocks on a number of financial series. Finally, as tbund in Poole ( 1994)~ the M 1 series encountered structural change after the October 1987 stock market crash.

Table 5 further shows that various trend breaks occur around a peak or trough of a business cycle. Such observations appear to support Simkin's (1994) attempt to augment unit-root tests x~Sth breakpoints associated with the business cycle chronology. Yet, it is apparent that not every business cycle turning point constitutes a structural break. Furthermore, most multiple breaks cluster around major historical ewmts, fbr example, the first major oil shock (for the real wages series) and the Fed policy shi|} (for the M1 velocity, T-bill rate, and bond yield series) in the 1970s. One findings of predominantiy one-time breaks in the postwar sample are at odds with Balke and Fombv (1991a, 1994) who find far more [requent structural shocks in the tbrm ot outliers in similar time series.

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Next, we augment the ADF and PP tests with trend breaks in the form of an intervention model represented by Equation (2"). Since the specific nature of structural change is assumed to remain unknown a priori, we alternatively compute three sets of test statistics, each of which corresponds to a possible breaking-trend alternative. The test results with predetermined break dates are reported in Table 6. The ADF ta-statistics coincide with the sequential rain ta*-statistics (Table 4) because their break dates are identical.

The qualitative results using breakpoints predetermined by max F-statistics are similar to those under the sequential t,, method. Taken together, the sequential statistics indicate that not every trend break results in the rejection of the unit-root null. Three series (nominal GNP, M 1, and M2) are identified with trend breaks using the max F-test but no change in their unit-root behavior even after the breaks are acconnted for. Furthermore, Ibr the series with two breaks, the test statistics conditional on both breaks are qualitatively the same as those conditional on a single break associated with the global maximum.

In contrast with the above results, the PP t,, statistics in all cases show no evidence of rejecting the no-break unit-root null. As Campbell and Perron (1991) explain, the discrepancies between ADF and PP test statistics may be attributable to dift~rences in using the lagged difference terms to handle serial correlation. Nevertheless, such discrepancies appear to prevail with alternative lag specifications (not reported here). 9 Furthermore, as Shiller and Perron (1985) point ont, the power of the ADF test is positively related to the span of the data (i.e., the number of years the sample covers). Hence.. the evidence against the null would be stronger for Nelson and Plosser's (1982) original data set which covers twice the time span of our sample.

4. Results for the Output Data Since Nelson and Plosser's (1982) study of U.S. aggregate activity, the

stochastic behavior of postwar output data has received much attention. In particular, Perron (1989, 1990a) shows that the historical mean of the real CNP series has a shift in 1973, and incorporating such information yields test statistics that reject the unit-root null in favor of a stationary process along a breaking trend. In this section, therefore, we discuss our results oil the postwar output data, along with other findings in the literature.

First, the recursive rain ta-statistic for the real CNP series is -3 .28 (see Table 3). Banerjee, Lumsdaine, and Stock (1992) report a corresponding

"All ADF tests have been conducted with fixed lag Lengths of 4 and 6 instead of using the AIC; and for PP tests, 12 it/stead of 6. The qnalitative results for all series remain the same as those reported in this paper.

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figure of -2 .60 for the period of 1947:i-1989:ii with the truncation lag parameter L fixed at 4. The discrepancy is perhaps due to the different time spans and the lengths of lagged difference term included. Nevertheless, based on the 5% critical value (with drift) of -4.06, the null hypothesis of a unit root across the full sample can not be rejected in either case. Despite these results, the statistic for the real per capita GNP series is -4.12, which is in favor of trend stationarity over some subsample.

Similarly, the rain tct*-statistic for the real GNP series cannot reject the no-break unit-root null hypothesis. More specifically, the figures for models A, B, and C are -3 .63 (1964:i), -3.0:3 (1968:iii), and -3 .78 (1965:i), respectively. The figures are consistent with those reported by Banerjee, Lnms- daine, and Stock (1992) for models A and B, and Christiano (1992) for model C. On the other hand, Zivot and Andrews (1992) report a slightly higher (in absolute value) rain tc~*-statistic of -4 .08 lbr model B (1972:ii). In any case, however, there is no significant e~Sdence against the unit-root null.

In line with Christiano's (1992) figures (with bootstrapping), the finite- sample critical valnes for the max F-statistics are much higher than the asymptotic values, for example, 3.1 at the 5% level. In effect, no statistically significant trend breaks are found for the real GNP series. Taken together, the above results are in conflict with Perron's selection of the outbreak of the first oil shock in 1973 as a breakpoint for the output series. Furthermore, they contradict Balke and Fomby (1991a, 1994), who use iterative methods to reveal a large nmnber of structural breaks in the series.

5. Summary and Concluding Remarks In light of Perron's (1989) controversial study, which shows that the

power of unit-root tests is sensitive to structural change in a data series, a one-time trend break has been routinely considered in testing for unit roots. This approach has been challenged by Balke and Fomby (1991a, 1994) who find multiple breaks in many postwar time series. In this paper, we Mve extended Christiano's (1992) max F-test to reexamine the possibility of multiple breaks and their impacts on inferences of the unit-root behavior.

Out of the 17 series considered, 8 have been found to have a breaking trend. In contrast with Balke and Fomby's (1991a, 1994) findings, we have found far fewer trend breaks. Furthermore, allowing for an additional break does not yield additional evidence against the unit-root mill hypothesis in am~ of those 4 series. Even though not eve~ breakpoint results in the rejection of the unit-root hypothesis, the break dates identified using the max F-test are consistent with those using the sequential G test. As such, the mi, G*-statistie, which focuses only on a single trend break, is rather reliable (and cost effective) in testing for unit roots with the possibility of structural change.

5IT

jim Lee

Nevertheless, our inferences have been complicated by a number of factors and, thus, should be inteq)reted with caution. In particular, first, we have followed previous studies to consider the null hypothesis of a random walk process without any change in the drift. Simulated critical values have been generated accordingly. However, one might consider a shift in the drift, especially under the plausible alternative of a change in the slope of the deterministic trend (model B).

Second, alternative breakpoint selection procedures (sequential vs. recursive) and test statistics (ADF vs. PP) yield different results. Since there is no unique way to select break dates or the most reliable test method, our results should be viewed as indicative rather than conclusive. This problem is particularly important when the PP statistics indicate much weaker evidence against the unit-root unll hypothesis than the ADF counterparts. Despite these limitations, the evidence of structural breaks found in this paper warrants further investigation of their impacts on the long-run property of macroeconomic processes using other test statistics or models.

Received: March 1994 Final vers'ion: September 1995

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Testing for a unit root in time series with trend breaks