Crossing-Rate Tests for Unit Root in Autoregressive Series*

Hashem Dezhbakhsh Department of Economics

Emory University Atlanta, GA 30322-2240

econhd@emory.edu

and

Daniel Levy Department of Economics

Emory University Atlanta, GA 30322-2240

econdl@emory.edu

July 1997

Preliminary and Incomplete, Comments Welcome.

*We are grateful to Benjamin Kedem for helpful discussions and suggestions, to the participants of the unit root test session participants at the Western Economic Association meeting in Seattle, WA, and to Yihong Xia for excellent research assistance. The usual disclaimer applies.

Crossing-Rate Tests for Unit Root in Autoregressive Series

Abstract

The widely used unit root tests have several well known power and robustness limitations

which, in the absence of viable alternatives, practitioners often overlook. We propose two unit

root tests that are promising for their power and robustness properties. The test statistics use

crossing rates, obtained by counting the number of times a series crosses its arithmetic mean

level, to detect the presence of a unit root. We derive some of the statistical characteristics of

the proposed tests. We also conduct sampling experiments to estimate the finite sample

quantiles of one of the proposed tests and compare its power to a popular unit root test proposed

by Dickey and Fuller. Results suggest that the test is more powerful than the Dickey-Fuller test

and more robust to outliers as well as to a structural shift in the autoregressive parameter. The

better performance of the proposed test is more pronounced in small samples, which are typical

for postwar annual series, and in near unit root cases. The computational simplicity of the tests

add to their appeal.

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1. Introduction

Until 1980s, much of the empirical time series analyses in economics assumed stationarity

and occasional departures from this assumption were all based on heuristic inspection of data.1

In absence of any rigorous test for nonstationarity, this statistically expedient practice

appeared justified, particularly since methods of inference for nonstationary series were

nonexistent or rudimentary at best. Dickey (1976) and Dickey and Fuller (1979 and 1981)

developed several least squares based and likelihood ratio tests for nonstationarity. The

Dickey-Fuller (DF) tests, which are known as unit root tests in econometric parlance, examine

the null hypothesis that a series follows a random walk against the alternative that the series is

stationary. Unit root tests have now become an integral part of time series econometrics. 2

Unfortunately, the DF tests as well as their modified versions proposed by Said and

Dickey (1984), Phillips (1987) and Phillips and Perron (1988) have several limitations, three

of which are well documented and widely known.3 First, the tests have low power, so the

preponderance of unit root sightings might well be due to the tests failure to reject an incorrect

unit root null. The power problem is particularly serious against autoregressive (AR) series

with roots close to one. Indeed, the power of the tests in such cases could be as low as their

nominal size. Moreover, the performance of the tests in terms of maintaining size and power

is particularly poor in small samples--those with fifty or fewer observations. Many economic

time series, however, are reliable only for the period after WWII. The annual frequency of

these series limits the available observations to about fifty, thus confining the applicability of

these tests. Second, the Dickey-Fuller tests are based on the least squares estimation which is

1 For example, in traditional Box-Jenkins analysis casual inspection of correlogram is the basis for ascertaining whether or not a series is stationary; See, e.g., Box and Jenkins (1976).

2 See, e.g., Fuller (1984), Dickey et al. (1986), Diebold and Nerlove (1990), and Campbell and Perron (1991) for a survey of this literature.

3 For detailed discussions of these shortcomings see Schwert (1989), Perron (1989), Diebold and Nerlove (1990), Diebold and Rudibusch (1991), Kwiatkowski, et al. (1992), Harris (1992), DeJong, et al. (1992b), and Hassler and Wolters (1994).

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known to be sensitive to outliers. Even one or two outliers in a small sample can exert undue

influence on the outcome of the test. Third, the tests can not distinguish between unit root

series and stationary series which contain a parameter shift during the sampling period. A

structural change in a parameter, resulting from incidents such as oil price shocks, can,

therefore, be mistakenly identified as unit root evidence.

Given the above shortcomings, it merits to develop alternative tests for unit root,

particularly tests which are not derivatives of the Dickey and Fullers’s least squares or

likelihood ratio tests. We develop two new unit root tests based on an approach other than the

least squares or likelihood ratio approach. This approach, which is widely used to develop

statistical procedures in engineering, has so far remained unnoticed in econometrics. The

general idea is to use a binary representation of a time series—also known as hard limiting or

clipping a series—to make inference about its distributional characteristics [Hinich (1967)].

Applying this approach to unit root testing allows us to draw on a distinctive characteristic of a

random walk process. Unlike a stationary autoregressive series that crosses its mean

frequently, a unit root process has no attractor and its return to any specific point is expected

to take a very long time. This characteristic of the random walk process has been known for

some time but never exploited for testing purposes.4

We use this intuitive distinction to obtain a statistical measure for distinguishing between

a process containing a unit root and a stationary (or trend stationary) autoregression. The test

formulation depends on the null and alternative hypotheses considered. For example, to test

the null of a random walk with no trend against a stationary AR alternative, we compute the

number of times a series crosses its arithmetic mean to obtain the crossing rate—defined as the

number of crossings divided by the sample size minus one.5 The crossing rate is small under

the null but increases with a reduction in the autoregressive parameter (a drop in the

4 For example, Feller (1957, p. 81) and more recently Granger (1991, p. 66) note that the expected time before the process returns to any given point is very long.

5 It must also be noted that crossing rate has been used in the statistics literature to estimate parameters of stationary autoregressive models. Kedem's (1980b) and (1993) work in this area is particularly notable.

3

correlation between consecutive observations). The maximum value is attained for series

consisting of independent observation (e.g., a white noise series). We provide preliminary

simulation results for this test consisting of the empirical quantiles under the null (critical

values) and power comparisons with the DF test. Our results show that this intuitive test is

simple, effective in small samples and near unit root cases, more powerful than the

corresponding DF test, and robust to outliers and to a shift in the autoregressive parameter.

We also develop a crossing rate test for the null hypothesis that a series is random walk

with drift against the alternative that the series is stationary AR with a deterministic trend. In

this case the crossing rate statistic is computed after estimating the drift parameter under the

null and de-drifting the series using this estimate. Again, a small value of the resulting

statistic supports the null, while a large value leads to a rejection of the null. Simulation

results for this test are in progress and not reported here.

The remaining sections of the paper are organized as follows. Section 2 states several

forms of the unit root hypothesis and discusses for each the Dickey-Fuller tests and their

variants. Section 3 introduces the crossing rate test for unit root when the series does not

contain a trend and presents some of its statistical properties. Section 4 reports empirical

quantiles of the proposed test statistic and its null properties. This section also contains

sampling experiments that compare the proposed test to a similar DF test in terms of power

and robustness. Section 5 develops a crossing rate test for the case where the series is trended.

Section 6 offers some concluding remarks and a synopsis of possible extensions of the paper.

2. Unit Root Hypotheses and Existing Tests

Consider a stochastic process {yt} generated by the linear model

y t yt t= + + +− utα β ρ 1 , t = 1, 2,…, (1)

where 0 ≤ ρ ≤ 1, yo = 0, and the innovations ut are i.i.d. normal (0, σ2). Dickey (1976) and

Dickey and Fuller (1979, 1981) offer several tests of the unit root restriction ρ = 0. The

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difference between these test stems from the difference in the restrictions imposed on α and β

under the null and alternative hypotheses. Using their notation, the tests which are relevant for

our purpose are as follows.

No. Test Null Restrictions Restrictions Under Alternative 1 $ρ α β ρ= = =0 1, α β ρ <0 1, = = 2 $τ α β ρ= = =0, 1 α β ρ= = <0 1, 3 $ρμ α β ρ= = =0, 1 β ρ= <0 1, 4 $τμ α β ρ= = =0, 1 β ρ= <0 1, 5 Φ1 α β ρ= = =0, 1 β α ρ= ≠ ≠0 0, ( )or 1 6 $ρτ β ρ= =0, 1 ρ < 1 7 $ττ β ρ= =0, 1 ρ < 1 8 Φ 3 β ρ= =0, 1 β ρ≠ <0 1or

Instead of reproducing the statistics for the above tests we confine to the following brief

description, referring the interested reader to the original references. The statistics for tests 1

and 2 are, respectively, the least squares estimate of ρ and the corresponding t statistic in a

regression of yt on yt-1. The statistics for tests 3 and 4 are, respectively, the least squares

estimate of ρ and the corresponding t statistic in a regression of yt on a constant and yt-1. The

statistics for tests 6 and 7 are, respectively, the least squares estimate of ρ and the

corresponding t statistic in a regression of yt on a constant, time trend, and yt-1. Test 5 is based

on a likelihood ratio statistic comparing sum of (yt − yt-1)2 with the sum of squared residuals

from regressing yt on a constant and yt-1. Test 6 is also based on a likelihood ratio statistic

comparing sum of (yt − yt-1)2 with the sum of squared residuals from regressing yt on a

constant, time trend, and yt-1. The distribution of these statistics under the corresponding null

hypotheses are nonstandard. Dickey and Fuller derive limit representations for these

distributions.

The first five tests are applicable to series that contain no trend and the last three tests are

for trended series. The tests closely associated, such as, tests 1 and 2 tests 3 and 4, and tests 6

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and 7, have similar powers. Since the current draft of this paper reports simulation results only

for unit root tests for nontrended series, we consider the first 5 tests. Tests 1 and 2 are the most

powerful among these when the constant term is zero under the alternative. Obviously, this

condition is not verifiable in practice, making tests 3 and 4 more useful than tests 1 and 2. The

higher power reported for tests 1 and 2 is an artifact of simulation designs which set constant to

zero. To give the DF tests the most advantage, we follow this practice in designing our

simulations. Given the aforementioned association, we evaluate test 2 rather than both tests 1

and 2. We draw on results reported in Dickey (1976) for evaluating tests 3 and 4. We do not

consider test 5 because of the composite nature of the alternative hypothesis it tests. Results of

comparing tests 2, 3, and 4 with our first test are reported in section 4. Tests 6-8 will be

compared with our second test in the next draft of the paper.

We note that few extensions of the DF tests have also been developed for more general

schemes. Among these are tests by Said and Dickey (1984), Phillips (1987), and Phillips and

Perron (1988). These tests appear to share the limitations of the DF tests. This is hardly

surprising given the common approach of these tests.6

3. Crossing Rate Test for Unit Root (no Trend)

Statistical tests often exploit the differences between the characteristics of the data

generating process under the null hypothesis and those under the alternative hypothesis. For

example, several of the DF tests use the difference in the null and alternative likelihood

functions to establish a rejection rule. The test we propose relies on a well known property of

6 Parenthetically, it is worth mentioning a promising alternative to the traditional unit root inference. The approach which has been suggested recently involves testing the above hypotheses in a reverse order--α < 1 as the null and α = 1 as the alternative. Reversing the order would give the null hypothesis of α < 1 a conservative treatment, thus allowing stationarity to remain the maintained hypothesis even if the test is not powerful. The composite nature of the null hinders a straight forward application of this approach. DeJong, et al. (1992a) develop a test for a single restriction that is consistent with the null--for example α = .85. Their parameterization is criticized because of its arbitrariness; see, e.g., Kwiatkowski, et al. (1992). This approach has remained fairly unexplored but it is promising.

6

the unit root process, namely its long passage time. Unlike a stationary autoregressive process

or a white noise process that cross their means frequently, a unit root process has no attractor,

and its return to any specific point is expected to takes a very long time. This characteristic,

which is refereed to as long passage time, has been well known in probability theory but never

exploited for testing purposes. The classical gambler's ruin problem in coin tossing, for

example, deals with the passage time for the resulting discrete-valued random walk. Other

applications can be found in probability textbooks; see, e.g., Feller (1957, ch. 13) or Ross

(1985, ch. 4).

We use this intuitive distinction to obtain an operational measure for distinguishing

between an autoregressive process containing a unit root and a stationary autoregression. By

computing the number of times a series crosses its arithmetic mean, we can obtain the crossing

rate--defined as number of crossings divided by the sample size minus one. The crossing rate

is small for unit root series and large for stationary series. It increases with a drop in the

correlation between consecutive observations and attains a maximum value when a series

consists of independent observation (e.g., a white noise series). We use the crossing rate of a

time series, ξ1, as the statistic for testing the unit root hypothesis against a general stationary

alternative. The rejection criterion for the test is given by the rule: reject if ξ1 > c , where ξ1

is the crossing rate statistic and c is chosen so that prob(g > c) equals the designated

significance level. A formal exposition for testing nontrended series follows.

Let {yt} be a time series generated by

y yt t= + +− utα ρ 1 , t = 1, 2,…, (2)

where the parameters and variates are as defined in (1), except that the distribution of u’s does

not need to be normal. The null hypothesis is H and0 0: 1α ρ= = and the alternative

hypothesis is H1 1: ρ < . Consider a realization of yt, t = 1:T, and its corresponding binary

representation ψt defined as ψt = 1 if yt ≥ m and ψt = 0 if yt < m ∀t, where m is the arithmetic

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mean of the realized yt series. This binary representation, which is sometimes referred to as a

clipped sequence, is widely used in applied time series analysis, particularly in engineering

areas like signal processing; see, e.g., Kedem (1994,1980a, and 1980b).

The number of times yt crosses its arithmetic mean is given by

( ) . (3) ( )DT tt

T

y = − −=∑ ψ ψ 1

2

2t

DT(y) denotes the number of mean-crossings. Note that in a similar manner we can compute

the number of times the series crosses any level other than its arithmetic mean. In such cases

the binary sequence is redefined to indicate whether yt is above this level or below it. The

proposed test statistic is

( )

ξ1 = −DT

T y1

. (4)

Some key observations related to the statistical properties of the ξ1-statistic are as follows.

The sufficiency of any statistics which is a function of the binary process ψt is demonstrated in

Kedem (1980, ch. 1), and it extends to ξ1. Moreover, using the results in He and Kedem

(1990), it can be easily shown that regardless of the variance of the innovation process, σ2,

ξ , 1a.s.⎯ →⎯ 0

when ρ = 1. Almost sure convergence implies convergence in probability limit; therefore, the

degenerate value of ξ1 is concentrated on zero under the null.

The distribution of ξ1 under the null does not converge to a known type. The next draft of

this paper will include a representation for the asymptotic distribution of ξ1 under the null.

The simulation experiments conducted in the next section estimate the small sample

distribution of the ξ1-statistic under the null and the alternative. The resulting empirical

quantiles provide some preliminary critical values. The proposed test involves counting the

number of times a series crosses its arithmetic mean, computing ξ1, and then comparing ξ1

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with the tabulated critical value corresponding to a designated significance level. If ξ1 is

larger than the critical value, then the unit root hypothesis is rejected in favor of stationarity.

We need to emphasize that the proposed test is applicable both in cases where the

alternative involves a constant term and where it does not. More specifically, the test statistic

ξ1 is a function of the binary sequence obtained by subtracting the series from its arithmetic

mean. Since the arithmetic mean is a consistent estimator of the mean of the series under the

alternative, αρ1−

, the statistic ξ1 is invariant with respect to α under the alternative.7 When

α = 0, obviously, the mean is zero and its estimator also tends toward zero. So unlike DF’s

tests 1 and 2, the power of the proposed test is robust to the choice of α.

Another advantage of the proposed test is its resilience to aberrant data. For example,

outliers affect ξ1 through the binary series ψt which bounds their influence, as any observation

far away from the mean of the series yt has no more influence on ψt than does an observation

in close proximity of the mean.

4. Critical Values and Power Comparisons in Finite Samples

Critical Values:

To implement the proposed test, we need to obtain the critical values corresponding to

various significance levels. The critical values are constructed from the null distribution

quantiles of the test statistic. The limiting distribution of the ξ1-statistic, however, does not

have a well known form. We therefore estimate the quantiles from the simulated distribution

of the ξ1-statistic.

The data for simulations are generated according to the process

y y , (5) ut t= +−1 t

7 Note that under the null α is zero, otherwise the series will be trended.

9

where ut are i.i.d. normal (0, σ2). The experiments are conducted for several parameter values.

We use T = 25, 50, 100, and 250 for the sample size. Except for T = 25, these values are the

same as those used by Dickey and Fuller. Since we retain these values for estimating power

functions in section 5, similarity of our designs with theirs makes a comparison of findings

easy. We choose an additional value T = 25 to ascertain how the tests perform in very small

samples. The finding will be useful since most postwar annual series and much of

international annual series have fewer than 50 observations. Although the limiting distribution

of ξ1 does not depend on σ, its finite sample distribution might. We, therefore, choose several

values for σ, σ = 1, 2,3, and 5. The initial value of yt is set to zero in all cases.

For each of the sixteen combinations of T and σ we generate a series according to the

above random walk process, discard the first 100 observations to reduce the effect of

initialization, and collect a sample of size T. Then, we count the number of times the series

crosses its arithmetic mean and compute the corresponding ξ1-statistic. The experiment is

then repeated 10,000 times to obtain a sufficiently accurate estimate of the distribution of the

ξ1-statistic.8 These 10,000 values are used to tabulate the empirical density and obtain the

estimated quantiles. This procedure is repeated for all combinations of T and σ.

Results of these simulations are presented in Figures 1-4 and Table 1. Figure 1 presents

empirical density functions of the ξ1-statistic for σ = 1 and sample size of 25, 50, 100, and

250. Figures 2-4 contain similar information for σ equal to 2, 3, and 5, respectively. The

graphs highlight several characteristic of the ξ1-statistic. First, the shape of the density does

not seem to be affected by variance of the innovation process even in various size samples as

the densities are remarkably similar across σ's. Second, in all cases much of the mass is

concentrated close to zero, confirming the notion that random walk series hardly cross a given

level (e.g., their arithmetic mean). Third, as the sample size increases the density becomes

more condensed as it approaches its degenerate limiting value. This holds true irrespective of

8 The Gauss Software System version 3.0 is used to conduct the simulations.

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the variance of the innovation process. For example, when the sample size is 100, the

likelihood that crossing rate exceeds .08 is less than 12 percent in all cases. Even when the

sample size is as small as 25, the likelihood that crossing rate exceeds .16 is around 8 percent.

Table 1 contains estimates of the distribution quantiles for the ξ1-statistic. It is evident

that these quantiles are highly invariant with respect to σ. Also, the lower quantiles are all

zero reflecting that much of the density is concentrated at zero. The higher quantiles have a

noticeable pattern; they take a larger value in cases with smaller samples. For example, the

99th quantile varies from .28 for T = 25 to .11 for T = 250. This is in accord with the fact that

the test is consistent so the distribution of the ξ1-statistic takes a more degenerate shape as

sample size increases.

The empirical quantiles presented in Table 1 can be used to implement the proposed test

using approximate significance levels of 1, 5, and 10 percents. We can simply compute the

crossing rate with respect to the arithmetic mean of the series and compare it with the value

given in the table. If the calculated rate in the data exceeds the tabulated value, we reject the

null hypothesis that the process has a unit root. Given that the quantile function appears to be

fairly invariant to σ, one does not need to estimate the variance of the innovation sequence to

be able to implement the test.9 The quantile estimates reported in Table 1 also reveal that

simple linear interpolation can provide reliable critical values for sample sizes that are not

included in these simulations.

Power Comparisons:

We conduct Monte Carlo experiments to evaluate the power of the proposed test and

compare it with the DF's test 2 ( $τ test). The data for these experiments are generated by

equation (2) and the following set of parameters. We set T equal to 25, 50, and 100, and σ

equal to 1, 2,3, and 5. To highlight the power differentials, we set ρ equal to .99, .95, .90, .80.

9 Note that under the null this variance can be easily estimated from first difference of the series.

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We choose 1% , 5%, and 10% as the significance level. Dickey and Fuller only report

empirical quantiles that correspond to these three significance levels, so to be able to evaluate

the power of the DF test, we are limited to these significance levels. We set the initial value of

yt and α to zero in all cases.

For each of the parametric specifications we generate a series according to equation (2),

discard the first 100 observations to reduce the effect of initialization, and collect a sample of

size T. Then, we count the number of times the series crosses zero and compute the

corresponding ξ1-statistic, which is then compared with the appropriate empirical quantile

(critical value). The null hypothesis of unit root is rejected if ξ1 exceeds this value. To force

the size of the test to be as close to the requisite significance level as possible, so that

comparison with the DF test is meaningful, we randomize the decision rule. Randomization is

a procedure by which the rejection region of a test that is based on a statistic with a discrete

distribution is adjusted to achieve the designated significance level.10 The experiment is

repeated 3,000 times. The power is estimated by the number of rejections divided by 3000.

The DF test is also applied to each of these 3000 series, but only in the case where σ = 1. This

is to avoid redundant computation and also because of the fact that the DF test is invariant

with respect to σ. The power of the DF test is estimated as the percentage of times that $τ test

reject the null. The above procedure is repeated for all parameter sets. Results are

summarized in Tables 2-4. These tables also include power estimates for two other DF tests, tests 3 and 4 denoted by $ρμ and $τμ , respectively. Results for these two tests are from Dickey

(1976) which has a similar simulation design.

10 Randomization procedure involves supplementing a test with an auxiliary experiment designed so that its outcome in conjunction with the test outcome leads to (e.g.) a 10% level test. For example, if a discretely distributed statistic takes values corresponding to the 9% and 11% tail probabilities but not to the 10%, then a 10% level test can not be performed directly. So an adjusted decision rule is adopted which rejects all the time when the p-value is 9% or below and 50% of the time when the p-value is 11%. The resulting significance level is .09+.5(.11-.09) or 10%. An auxiliary experiment with 50-50 outcomes determines when to reject if the statistic takes an 11% probability value; see, e.g., Lehman (1986, ch. 1).

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The most salient feature of the tests can be summarized as follows. First, the power

functions for all four tests are increasing in ρ as well as the sample size. Power tends to

increase considerably as the sample gets larger. This reflects the fact that all tests are

consistent. The ξ1-test performs much better than all three DF tests. The differential is more

considerable for tests 3 and 4, which unlike test 2 do not know that α is zero and have to

estimate it. Note that, as indicated earlier, practitioners usually do not know whether or not α

is zero under the alternative. So tests 3 and 4 have more practical value than test 2. Here,

however, we use test 2 as a maximum benchmark for the power of the DF tests, given that α is

set to zero. Third, only in a few cases the power of the ξ1-test is marginally less than that of

the DF tests. These are mostly the cases with a smaller ρ. On the other hand, the ξ-test has its

edge in cases with high ρ, e.g. ρ = .90 or above; these cases have a more practical relevance

because of the difficulty involved in discriminating between random walk and near random

walk series.11 The power differential is in favor of the ξ1-test by as much as 80 percent or

more in some of these cases. This differential is particularly pronounced for T = 25. Fourth,

the power of the ξ1-test appears to be resilient to a change in σ, regardless of the significance

level of the test.

Perron (1989) shows that the DF tests is highly sensitive to a structural shift in model

parameters, and the test confounds such shifts with nonstationarity. The proposed test,

however, does not rely on estimation of model parameters, and therefore it seems to be robust

to such structural shifts. We conducted a series of experiments to ascertain the extent of such

robustness. Our preliminary results, which are not reported here, suggest that unlike the DF

tests, the ξ1-test is not affected by shift in parameters.

A final point that needs to be emphasized is related to the dependency of the DF tests on

the initial value of the process. The test assumes that the initial value of the series is zero.

11 These are the values that are also the most relevant for macroeconomic research since empirical studies of aggregate time series suggests that many of these series may be characterized by an autoregression with a root near unity; see, e.g., Nelson and Plosser (1982).

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Results reported by Dejong, et al. (1992b) suggest that the power of the DF test is

considerably lower if the initial value is not zero. The performance of the proposed test,

however, should not depend on the initial value. An increase, for example, in the initial value

leads to a parallel shift in the series as well as its arithmetic mean, leaving the crossing rate

unchanged. Therefore, it is reasonable to conjecture that if we allow the initial value to

deviate from zero, then the power differential will be even more in favor of the ξ1-test. Here,

we conduct all our simulation with zero as the initial value, thus giving the DF test a

substantial advantage. This issue will also be explored in the next draft.

5. Crossing Rate Test for Unit Root (with Trend)

In this section we propose a crossing-rate test for trended series. Consider a series

generated by equation (1) under the null hypothesis H0 0 0: , , 1α β ρ≠ = = , where the

parameters and variates are as defined for (1). This yields

y yt t= + +− utα 1 , t = 1, 2,…, (6)

which is a random walk with drift. If the drift parameter α is positive the series trends upward

and if α is negative it trends downward. The equation in (6) can be equivalently expressed as

y t . t = 1, 2,…, (7) yti

t

= + +=∑α 0

1ui

u

1 1

where the term αt captures the cumulative effect of the drift while is the nontrended

component of the random walk.

y ii

t

01

+=∑

The alternative hypothesis is H 0: ,β ρ≠ < . The series is difference stationary under

the null and trend stationary under the alternative. Also, note that the trend is stochastic under

the null but deterministic under the alternative. The test we propose below is also applicable

in cases where the alternative hypothesis does not impose any restriction on β—the case

considered by DF tests 6 and 7.

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The proposed test involves two steps: (1) removing the drift, as specified under the null,

and (2) applying the crossing-rate statistic to the de-drifted series. One can de-drift a series by

differencing it, estimating the mean of the differenced series, and cumulatively subtracting the

estimated mean from the original series. For example, the differenced series,

yt − yt-1, equals α + ut under the null and its arithmetic mean, denoted by ~α , is a consistent

estimate of α, given that u’s are white noise. The de-drifted series wt obtains from cumulative

subtraction, which is apparent from equation (7), that is

w y tt t= − ~α , t = 1, 2,…, T. (8)

This is equivalent to subtracting ~α from each observation in the differenced series and

cumulating these adjusted differences to obtain the de-drifted random walk series.

Once the series is de-drifted, the procedure described in section 3 can be applied to the new

series. The resulting statistic is

ξψ ψ

2

12

2

1=

−

−

−=∑ ( )t tt

T

T , (9)

where ψt is the binary representation of the de-drifted series wt. Values of the statistics that are

larger than a designated critical value will lead to a rejection of the unit root hypothesis. These

critical values, distributional characteristics under the null and alternative hypotheses, and

small sample power comparisons with DF’s tests 6 through 8 will be reported in the next draft

of the paper. The draft will also evaluate these tests in terms of their robustness to outliers,

initial values of the series, and a shift in the autoregressive parameter.

6. Concluding Remarks

Testing for unit root has become an integral part of time series inference in economics.

Unit root tests are routinely applied in a diagnostic context whereby the test outcome

determines the subsequent choice of filtering, estimation, and forecasting procedures. The

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tests are also applied to financial data to examine efficient market models through their unit

root implications. Moreover, unit root tests constitute the building block of cointegration

inference which provides a statistical framework for examining long run economic

relationships. It is therefore critical to have reliable tests for unit root. This due importance is

exemplified by a multitude of studies that test economic series for unit root and examine its

consequences.

Unfortunately, the existing tests perform poorly in many relevant situations. In addition

to low power, the popular Dickey-Fuller tests are sensitive to the initial value of the series,

presence of outliers, and a structural shift in the autoregressive parameter. We propose two

tests for unit root using a different approach than the least squares and likelihood ratio

approach of the DF tests. One test is applicable to trended series and the other to nontrended

series. The proposed tests are based on crossing rate of the series--measured by the number of

times a series crosses its arithmetic mean. We also report finite sample distribution quantiles

(critical values) for one of the tests and compare its power to the corresponding DF tests.

These quantiles appear to be independent of the variance of the error process. This along with

simplicity of the test adds to its practical appeal.

16

References Box, G.E.P. and G.M. Jenkins, 1976, Time Series analysis: Forecasting and Control, (San

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DeJong, D.N., J.C. Nankervis, 1992b, N.E. Savin, and C.H. Whiteman, "The Power Problems of Unit Root Tests in Time Series with Autoregressive Errors" Journal of Econometrics, 53, 323-343.

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Dickey, D., and W. Fuller, 1981, "Likelihood Ratio Tests for Autoregressive Time Series with a Unit Root," Econometrica, 49, 1057-1072.

Diebold, F.X., and M. Nerlove, 1990, "Unit Roots in Economic Time Series: A Selective Survey," in T.B. Fomby and G.F. Rhodes (eds.), Advances in Econometrics, 8, 3-70, (Greenwich, CT.: JAI Press).

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Engle, R.F. and C.W.J. Granger, 1987, "Co-integration and Error Correction: Representation, Estimation, and Testing," Econometrica, 55, 251-276.

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Hamilton, J. D., 1994, Time Series Analysis, (Princeton, NJ: Princeton University Press). Harris, R.I.D., 1992, "Testing for Unit Root Using the Augmented Dickey-Fuller Test: Some

Issues Relating to the Size, Power and the Lag Structure of the Test," Economics Letters, 38, 381-386.

Hassler, U. and J. Wolters, 1994, "On the Power of Unit Root Tests Against Fractional Alternatives," Economics Letters, 45, pp. 1-5.

17

He, S. and B. Kedem, 1990, "The Zero-Crossing Rate of Autoregressive Processes and Its Link to Unit Roots," Journal of Time Series Analysis, 11, 201-213.

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of Stationarity against the Alternative of a Unit Root," Journal of Econometrics, 54, 159-178.

Lehman, E.L., 1986, Testing Statistical Hypotheses, (New York: John Wiley). Lucas, Andre, 1995, “An Outlier Robust Unit Root Test with an Application to the Extended

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Economic Prespectives, 2, 147-174.

18

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

a. Sample Size = 25

0.00

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0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

b. Sample Size = 50

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

c. Sample Size = 100

0.00

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1.00

d. Sample Size = 250

Figure 1. Empirical Density of the ξ1 Statistic Under the Null (σ = 1)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

a. Sample Size = 25

0.00

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0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

b. Sample Size = 50

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

c. Sample Size = 100

0.00

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0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

d. Sample Size = 250

Figure 2. Empirical Density of the ξ1 Statistic Under the Null (σ = 2)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

a. Sample Size = 25

Figure 3. Empirical Density of the ξ1 Statistic Under the Null (σ = 3)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

b. Sample Size = 50

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

c. Sample Size = 100

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

d. Sample Size = 250

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

a. Sample Size = 25

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

b. Sample Size = 50

Figure 4. Empirical Density of the ξ1 Statistic Under the Null (σ = 5)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

c. Sample Size = 100

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

d. Sample Size = 250

Table 1. Empirical Quantiles of g Under the Null

Sample Size σ 1% 5% 10% 25% 50% 75% 90% 95% 99%

T = 25 1 0.00 0.00 0.00 0.00 0.00 0.04 0.12 0.16 0.28

2 0.00 0.00 0.00 0.00 0.00 0.00 0.12 0.16 0.28

3 0.00 0.00 0.00 0.00 0.00 0.00 0.12 0.16 0.28

5 0.00 0.00 0.00 0.00 0.00 0.00 0.12 0.16 0.28

T = 50 1 0.00 0.00 0.00 0.00 0.00 0.04 0.10 0.14 0.22

2 0.00 0.00 0.00 0.00 0.00 0.04 0.10 0.14 0.22

3 0.00 0.00 0.00 0.00 0.00 0.04 0.10 0.14 0.20

5 0.00 0.00 0.00 0.00 0.00 0.04 0.10 0.14 0.22

T = 100 1 0.00 0.00 0.00 0.00 0.00 0.04 0.08 0.11 0.16

2 0.00 0.00 0.00 0.00 0.00 0.04 0.08 0.11 0.17

3 0.00 0.00 0.00 0.00 0.00 0.04 0.08 0.11 0.16

5 0.00 0.00 0.00 0.00 0.00 0.04 0.09 0.12 0.16

T = 250 1 0.00 0.00 0.00 0.00 0.01 0.04 0.06 0.08 0.11

2 0.00 0.00 0.00 0.00 0.01 0.04 0.06 0.08 0.11

3 0.00 0.00 0.00 0.00 0.01 0.04 0.06 0.08 0.11

5 0.00 0.00 0.00 0.00 0.01 0.04 0.06 0.08 0.11

Table 2. Estimates of Power Functions, Significance Level = 0.10

Sample α σ = 1 σ = 2 σ = 3 σ = 5

Size ______________ _______ _______ _______DF g g g g

T = 25 0.99 0.145 0.167 0.173 0.163 0.1650.95 0.218 0.354 0.384 0.345 0.3580.90 0.335 0.545 0.543 0.554 0.5240.70 0.821 0.899 0.906 0.903 0.8930.50 0.978 0.985 0.985 0.984 0.9850.30 0.999 0.999 0.998 0.998 0.9970.00 1.000 1.000 1.000 1.000 1.000

T = 50 0.99 0.168 0.156 0.173 0.155 0.1740.95 0.337 0.403 0.451 0.416 0.4510.90 0.583 0.659 0.689 0.667 0.6900.70 0.997 0.986 0.987 0.986 0.9830.50 1.000 1.000 1.000 0.999 1.0000.30 1.000 1.000 1.000 1.000 1.0000.00 1.000 1.000 1.000 1.000 1.000

T = 100 0.99 0.190 0.212 0.204 0.209 0.1790.95 0.582 0.595 0.587 0.578 0.5770.90 0.942 0.885 0.869 0.866 0.8660.70 1.000 1.000 1.000 1.000 1.0000.50 1.000 1.000 1.000 1.000 1.0000.30 1.000 1.000 1.000 1.000 1.0000.00 1.000 1.000 1.000 1.000 1.000

Table 3. Estimates of Power Functions, Significance Level = 0.05

Sample α σ = 1 σ = 2 σ = 3 σ = 5

Size ______________ _______ _______ _______DF g g g g

T = 25 0.99 0.066 0.083 0.082 0.080 0.0900.95 0.118 0.191 0.208 0.190 0.2130.90 0.172 0.321 0.335 0.312 0.3350.70 0.628 0.744 0.760 0.734 0.7530.50 0.923 0.934 0.934 0.937 0.9310.30 0.994 0.987 0.992 0.986 0.9900.00 1.000 1.000 0.999 0.999 1.000

T = 50 0.99 0.079 0.085 0.088 0.080 0.0860.95 0.181 0.254 0.250 0.256 0.2730.90 0.370 0.482 0.474 0.450 0.4800.70 0.978 0.939 0.943 0.943 0.9490.50 1.000 0.997 0.997 0.996 0.9980.30 1.000 1.000 1.000 1.000 1.0000.00 1.000 1.000 1.000 1.000 1.000

T = 100 0.99 0.101 0.103 0.098 0.104 0.0940.95 0.362 0.392 0.365 0.401 0.3350.90 0.790 0.719 0.705 0.719 0.6940.70 1.000 0.998 1.000 0.999 0.9970.50 1.000 1.000 1.000 1.000 1.0000.30 1.000 1.000 1.000 1.000 1.0000.00 1.000 1.000 1.000 1.000 1.000

Table 4. Estimates of Power Functions, Significance Level = 0.01

Sample α σ = 1 σ = 2 σ = 3 σ = 5

Size ______________ _______ _______ _______DF g g g g

T = 25 0.99 0.018 0.016 0.015 0.014 0.0150.95 0.022 0.056 0.048 0.039 0.0450.90 0.039 0.091 0.091 0.079 0.0950.70 0.214 0.329 0.330 0.329 0.3450.50 0.601 0.652 0.654 0.624 0.6650.30 0.901 0.851 0.849 0.842 0.8640.00 0.997 0.977 0.974 0.980 0.981

T = 50 0.99 0.015 0.020 0.019 0.022 0.0150.95 0.037 0.057 0.073 0.072 0.0690.90 0.097 0.147 0.157 0.184 0.1580.70 0.743 0.658 0.685 0.694 0.6760.50 0.994 0.935 0.958 0.962 0.9520.30 1.000 0.997 0.997 0.998 0.9970.00 1.000 1.000 1.000 1.000 1.000

T = 100 0.99 0.018 0.031 0.018 0.028 0.0200.95 0.093 0.127 0.104 0.131 0.1140.90 0.352 0.357 0.315 0.351 0.3220.70 0.999 0.968 0.965 0.971 0.9620.50 1.000 1.000 1.000 1.000 1.0000.30 1.000 1.000 1.000 1.000 1.0000.00 1.000 1.000 1.000 1.000 1.000