finite-sample properties of modified unit root tests in the presence structural change

Applied Mathematics and Computation 149 (2004) 625–640

www.elsevier.com/locate/amc

Finite-sample properties of modified unitroot tests in the presence structural change

Steven Cook

Department of Economics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK

Abstract

Using Monte Carlo experimentation the finite-sample properties of modified unit

root tests are examined in the presence of structural breaks. It is found that previously

derived results concerning the size robustness of weighted symmetric and recursively

mean-adjusted unit root tests in the presence of structural breaks are achieved at the

expense of dramatic losses in power, thereby reducing the appeal of the tests to the

practitioner. The properties of the tests are also seen to be highly dependent upon

the position or timing of the observed break. Finally, bias in the estimation of the

autoregressive parameter is analysed. The results derived demonstrate that while re-

cursive mean adjustment reduces negative bias in the absence of a break, it leads to

increased positive bias when a break occurs.

� 2003 Elsevier Inc. All rights reserved.

Keywords: Unit root tests; Empirical power; Monte Carlo simulation; Structural change; Recursive

mean adjustment; Weighted symmetric estimation

1. Introduction

Testing the unit root hypothesis has become a familiar feature of appliedeconometric and statistical research, with the unit root tests of [1] frequently

E-mail address: [email protected] (S. Cook).

0096-3003/$ - see front matter � 2003 Elsevier Inc. All rights reserved.

doi:10.1016/S0096-3003(03)00167-X

mail to: [email protected]

626 S. Cook / Appl. Math. Comput. 149 (2004) 625–640

employed. The Dickey–Fuller (DF) sl test can be illustrated using the fol-

lowing first-order autoregressive, AR(1), testing equation: 1

1 Al

testing

focus u

approp

analys

yt ¼ lþ qyt�1 þ et et � i:i:d: ð0; r2Þ t ¼ 1; . . . ; T ð1Þ

where yt and yt�1 are the variable of interest and its one period lag, l is an

intercept parameter, q is the autoregressive parameter and et is an identically

and independently distributed innovation process with a zero mean and con-

stant variance r2. The sl statistic tests the unit root hypothesis H0 : q ¼ 1

against the alternative H1 : jqj < 1 using the pivotal statistic ðq̂0 � 1Þ=seðq̂0Þ,where q̂0 is the ordinary least squares estimator (OLSE) of q in (1). Under the

assumption of a unit root, the first two moments of fytg are time varying, being

given via repeated substitution as:

E½yt� ¼ y0 þ tl ð2Þ

V½yt� ¼ tr2 ð3Þ

Testing of the unit root hypothesis therefore requires the use of specifically

calculated critical values due to the non-standard distribution of sl. However,

despite its widespread application, the sl test is known to suffer from low

power when the autoregressive parameter (q) is close to 1. Additionally, the

results of [2] have shown the sl test to suffer from severe size distortion when a

unit root process experiences a structural break early in the sample period.

Therefore, when a break is present under the null, the DF test can spuriously

reject the unit root hypothesis.In response to the problems of the low power and size distortion of the sl

test, the properties of modified DF tests have been examined. Recently results

have been presented suggesting that the weighted symmetric DF (sw) test of [3]and the recursively mean-adjusted DF (sr) test of [4] may provide a solution to

both problems. Firstly, the Monte Carlo results of [4,5] show the sw and sr teststo possess greater power than the original sl test. Secondly, [6,7] have em-

ployed Monte Carlo analysis to examine the size properties of the sw and srtests when a break occurs under the null. The results derived show that themodified tests do not exhibit the oversizing and spurious rejection associated

with the sl test. However, while the modified tests possess improved size

properties when applied to unit root processes subject to a break, their power

properties when applied to stationary series (jqj < 1) subject to a break have

yet to be examined. This issue merits close attention as while the size correction

though the DF test can be employed with alternative deterministic terms included in the

equation, the model containing an intercept alone will be examined here. Given the present

pon the impact of mean breaks or level shifts on the properties of unit root tests, this is the

riate form to consider. This specification will be employed for all of the unit root tests

ed.

S. Cook / Appl. Math. Comput. 149 (2004) 625–640 627

offered by sw and sr tests would encourage their adoption when analysing series

subject to structural change, this would be negated should the tests suffer fromlow power in such circumstances. It is also important to analyse the properties

of the tests in the presence of structural change given the prevalence of

structural breaks in economic data (see, inter alia, [8,9]). This paper therefore

examines the power properties of the sl; sw and sr tests when data are sta-

tionary but subject to a break in mean. The results obtained allow it to be seen

whether the known increased power of the modified tests in the absence of a

break extends to the case where a break is present.

In addition to considering the power properties of the alternative tests, theestimated values of the autoregressive parameter (q) obtained using the alter-

native tests are noted. It is well documented that the OLSE q̂0 exhibits a

downward bias. 2 However, despite the analysis of [4] showing this bias to be

substantially reduced when recursive mean adjustment is employed, this result

is again based upon the absence of a break in the series examined. The present

study therefore compares the estimated values of q obtained using the sl, srand sw tests for stationary series subject to a break in mean.

This paper proceeds as follows. In section 2 the unit root tests to be em-ployed are presented. Section 3 contains the Monte Carlo experimental design

employed. Section 4 presents results for test power and the bias of the esti-

mators of the autoregressive parameter. Section 5 concludes.

2. Unit root tests

2.1. Dickey–Fuller test

The first test considered is the standard DF sl test. As explained above, thesl statistic is given as the t-statistic, or pivotal test, of H0 : q ¼ 1 in the testing

equation (1). In response to the observed low power of the sl test when q is

close to 1, alternative modified tests with greater power have been suggested.

The two modified tests considered here are outlined below.

2.2. Weighted symmetric Dickey–Fuller tests

The first modified DF test considered is the weighted symmetric DF test of[3]. Denoted as sw, this test has been shown to possess high power against

stationary alternatives, with the Monte Carlo results of [5] showing it to be the

most powerful of the unit root tests they consider, outperforming the genera-

lised least squares test of [12] amongst others. The sw test can be illustrated as

follows. Given the following model:

2 The downward bias of the OLSE of q in AR models is discussed by, inter alia, [10,11].


yt ¼ aþ qyt�1 þ ut ð4Þ

where ut is an innovation process, and its forward representation:

yt ¼ aþ qytþ1 þ ut ð5Þ

the weighted symmetric estimator of q is based upon a double-length regression

and is calculated as the value minimising:

QwðqÞ ¼XT�1

t¼1

wtðyt � qyt�1Þ2 þXTt¼2

ð1� wtþ1Þðyt � qytþ1Þ2 ð6Þ

where the weighting series wt is given as wt ¼ ðt � 1Þ=T , with T denoting the

sample size. The weighted symmetric estimator of q is then calculated as:

q̂w ¼PT

t¼2 ytyt�1PT�1

t¼2 y2t þPT

t¼1 y2t

ð7Þ

with the unit root hypothesis (H0 : q ¼ 1) tested using the pivotal statistic sw:

sw ¼ r�1w ðq̂w � 1Þ

XT�1

t¼2

y2t

þ T�1

XTt¼1

y2t

!1=2

ð8Þ

where r2w ¼ ðT � 2Þ�1Qwðq̂wÞ.

2.3. Recursively mean-adjusted Dickey–Fuller test

The second modified DF test examined is the recursively mean-adjusted DF

test of [4]. As noted in [4], the use of mean-adjusted observations ðyt � �yÞ in the

following DF regression results in correlation between the regressor ðyt�1 � �yÞand the error �t:

yt � �y ¼ qðyt�1 � �yÞ þ �t ð9Þ

The resulting downward bias of the OLSE q̂ has been derived by, inter alia,

[10,11] as:

Eðq̂� qÞ ¼ �T�1ð1þ 3qÞ þ oðT�1Þ ð10Þ

To reduce the bias of the estimated autoregressive parameter, it is suggested in

[4] that recursively mean-adjusted observations are employed. The recursive

mean (�yt) is calculated as:

�yt ¼ t�1Xt

i¼1

yi ð11Þ

with the recursively mean-adjusted version of (9) given as:

ðyt � �yt�1Þ ¼ qðyt�1 � �yt�1Þ þ �t ð12Þ


It can be seen that the use of (12) overcomes the problem of correlation be-

tween the regressor and error term, with the regressor and �t now independentwhen q ¼ 1. As noted above, the recursively mean-adjusted DF statistic in

pivotal form is denoted as sr and is calculated as ðq̂r � 1Þ=seðq̂rÞ, where q̂r is the

OLSE of q in (12). In a Monte Carlo analysis of test power, the easily applied

sr test was found in [4] to possess almost identical power to the more com-

plicated sw test. Additionally, it was found that q̂r was subject to substantially

less bias than the standard OLSE q̂0 when estimating q.

3. Monte Carlo design

As noted above, the sw and sr tests have been found via Monte Carlo ex-

perimentation to possess greater power than the sl test. In addition, the em-

pirical sizes of the modified tests have been shown by [6,7] to be robust to

breaks in mean under the null. This contrasts with the findings of [2] where the

sl test was found to exhibit severe oversizing for breaks in mean occurringearly in the sample period. However, while the sw and sr tests may have at-

tractive size properties for unit root processes subject to a break, it has yet to be

examined whether this is at the expense of a reduction in the power of the tests.

As the known higher power of the sw and sr tests in the absence of a break

cannot be guaranteed to extent to the case where a break occurs, this merits

close examination. 3

To analyse the tests in the presence of structural breaks, the experimental

design of [6] is modified, with the data generation process (DGP) given asbelow:

3 Th

the pre

yt ¼ astðjÞ þ nt t ¼ 1; . . . ; T ð13Þnt ¼ qnt�1 þ gt ð14Þgt � i:i:d: Nð0; 1Þ ð15Þ

a ¼ kffiffiffiffiT

pð16Þ

stðjÞ ¼0 for t6 jT1 for t > jT

�j 2 ð0; 1Þ ð17Þ

The error series fgtg is generated as a standard normal variate using the

RNDNS procedure in GAUSS. All experiments are performed over 10,000replications using sample sizes of T 2 ð100; 200Þ, these values being typical of

much economic research. For all experiments an additional initial 100 obser-

vations are generated and discarded to minimise the influence of initial

e importance of conducting further research to examine the power of the modified tests in

sence of a break in mean is recognised in [6,7].


conditions. Denoting the break fraction as j, a break in mean is imposed after

observation jT . In contrast to [2] where results for a limited number ofbreakpoints are tabulated, all possible breakpoints over the range j 2 ð0; 1Þ inincrements of 0.01 are considered here. Following [6], two values are employed

for the break parameter, k 2 f0:5; 1g, with the actual size of the break imposed

(a) being dependent upon the sample size also.

In contrast to [2,6,7] where size properties are considered, the power of the

tests are examined here by generating stationary series. To achieve this the

values q 2 f0:9; 0:95g are employed in (14), in contrast to previous studies

where unit root processes were generated by setting q ¼ 1. As noted above, thefirst objective of this paper is to examine the empirical power of the alternative

tests in the presence of a break in mean. This is performed at the 5% level of

significance using the critical values of [13,5,4] for the sl, sr and sw tests, re-

spectively. The secondary element of the present analysis is to examine the bias

of the estimated values of the autoregressive parameter q in (14). The mean

values of the estimators q̂0, q̂r and q̂w across replications are therefore recorded

for each of the experimental designs examined.

4. Monte Carlo results

4.1. Empirical power

Before considering the results for the alternative tests in the presence of a

break in mean, the powers of the tests in the absence of a break are examined.The above DGP is therefore employed without a break imposed, with the

empirical powers of the tests calculated for the four combinations of sample

size (T 2 f100; 200g) and the autoregressive parameter (q 2 f0:9; 0:95g). Theresults of this analysis are presented in Table 1. As expected, the power of all

tests improves as the sample size is increased or a lower value of q is consid-

ered. From inspection of Table 1 it can be seen that the known increased power

of the modified tests in the absence of a break is confirmed. For all of the

designs examined the modified tests exhibit greater than the sl test. While thesw test exhibits greater power than the sr test for three of the four designs,

the difference in power between the modified tests is slight.

With the established power results in the absence of structural change rep-

licated, it can now be seen whether the increased power of the modified tests

extends to the case of breaks in mean. Given the range of experimental designs

considered, analysis of the results obtained is simplified by presenting the re-

sults derived graphically. In Figs. 1–8 the empirical powers of the three tests are

provided for all possible combinations of the sample sizes (T 2 f100; 200g),break sizes (k 2 f0:5; 1:0g) and values of the autoregressive parameter

(q 2 f0:9; 0:95g) presented above. For each combination of design parameters,

Table 1

Empirical power of the unit root testsa

T q sl sr sw

100 0.90 34.31 52.52 53.08

100 0.95 12.24 19.67 19.87

200 0.90 87.38 96.89 96.65

200 0.95 32.56 50.96 52.61

a The empirical power of the alternative unit tests in the absence of a break in mean, calculated

over 10,000 replications.


the power of each test is reported over the full range of breakpoints (jT )considered.

In Figs. 1–4, the empirical powers of the sl, sw and sr tests are presented for

the sample size of T ¼ 100. In Figs. 1 and 2, results are depicted for the smaller

break size of k ¼ 5, with q ¼ 0:9 and 0.95, respectively. From inspection of

Figs. 1 and 2 it can be seen that for all tests power initially falls as the break

occurs later in the sample (higher values of j), but then increases as the break is

imposed in the final few observations of the sample period. In both Figures it is

clear that the sl test exhibits the most extreme behaviour, possessing the

highest power of the three tests for early breaks, but the lowest when the breakoccurs at other points in the sample period. Although the sw and sr tests havesimilar power, the sw test is slightly more powerful when the break occurs early

in the sample, but is less powerful when the break occurs later. With reference

to the power calculations for the tests in the absence of a break reported in

Table 1, it can be seen that the presence of a break in mean can increase the

power of the sl test, while it always reduces the power of the sw and sr tests.Considering the parameter values fT ;qg ¼ f100; 0:9g, the sl, sw and sr tests

Fig. 1. Empirical power (T ¼ 100, k ¼ 5, q ¼ 0:9).




have empirical powers of 34.31%, 52.52% and 53.08% in the absence of a

break. When a break of k ¼ 5 is imposed, the maximum powers of the sl, swand sr tests considering all breakpoints are 71.83%, 42.43% and 40.49%, re-

spectively. Similarly, for the parameter values fT ; qg ¼ f100; 0:95g, the sl, swand sr tests have empirical powers of 12.24%, 19.67% and 19.87%, without a

break, but maximum powers of 41.26%, 19.01% and 17.28% when a break of

k ¼ 5 is imposed. This indicates that when considering stationary series, the swand sr tests are less likely to correctly reject the null of unit root when a break ispresent than when it is absent. In contrast, the sl will reject the unit root hy-

pothesis more often if the break occurs early in the sample period, but less

often if it occurs later in the sample period. The results in Figs. 1 and 2



therefore show that the alternative tests possess differing behaviour, with their

properties highly dependent upon the point at which a break occurs.

In Figs. 3 and 4 the power of the unit root tests are presented for the larger

break of k ¼ 10. The results obtained follow a similar pattern to those for the

smaller break, but are more pronounced. With the larger break it can be seen

that the power of the sw and sr tests are further reduced, while the high power

of the sl test is increased for early breaks. For the q ¼ 0:9, the maximum power

of the tests with the larger break are fsl; sw; srg ¼ f98:55%; 30:07%; 25:01%g,while for q ¼ 0:95 the corresponding maximum powers are f84:19%;16:42%; 12:67%g. Again it is clear from Table 1 that the power of the sl test canbe increased when a break occurs, while this is never the case for the sw and srtests. However, it must be recognised that the power of the tests is highly de-

pendent upon the position of the break, with all tests exhibiting extremely low

power, often below 1%, for a break which occurs in the middle section of the

sample period.

For completeness, Figs. 5–8 present empirical power calculations for thetests using the larger sample size. These findings for T ¼ 200 mimic those for

the smaller sample. Again the properties of the tests depend upon the point at

which the break is imposed, with the sl test having the greatest variation in

power. As before the power of the sl test can be increased as a result of a break

occurring. This property does not hold for the sw and sr tests. For these

modified tests, power is always lower in the presence of a break than in the

absence of a break, irrespective of where the break occurs. To illustrate this, the

maximum power of the sl, sw and sr tests for fq; ag ¼ f0:9; 5g, f0:95; 5g,f0:9; 10g, f0:95; 10g are f99:04%; 85:24%; 79:47%g, f72:69%; 41:48%; 37:87%g,f100%; 50:43%; 39:24%g and f98:60%; 25:34%; 18:95%g, respectively. Refer-

ence to the values in Table 1 shows that the power of the modified tests for the

�no-break� case always exceeds the maximum power obtained when a break




occurs. Additionally, it must be noted that maximum values reported above

relate to breaks which occur in the tails of the sample, for breaks imposed at all

other points the powers of all tests are substantially lower.

To summarise the above results, it is clear that the high power of the

modified tests noted in the absence of a break is substantially reduced when a

break occurs. This finding is observed for all of the combinations of break size,

sample size and values of the autoregressive parameter employed.

4.2. Estimating the autoregressive parameter

Before discussing the values of q̂0, q̂w and q̂r observed in the presence of a

break in mean, values obtained for these estimators in the absence of a break




are presented in Table 2. The calculated values show a downward bias is

present in all estimators of q, although as argued in [4], q̂r is subject to less bias

than q̂0. The results also show q̂r is less biased than the weighted least squares

estimator q̂w, which takes very similar values to q̂0.

To examine the properties of the estimators when a break in mean occurs,

the results of the Monte Carlo analysis are presented graphically in Figs. 9–16.

A clear pattern is present in these graphs which chart the values of the alter-

native estimators over a range of breakpoints. For all sample sizes, break sizesand values of q considered, all estimators have an inverted U-shaped distri-

bution across the breakpoints. That is, as the breakpoint is initially increased

Table 2

Estimates of the autoregressive parametera

T q q̂l q̂r q̂w

100 0.90 0.859 0.892 0.861

100 0.95 0.906 0.938 0.909

200 0.90 0.880 0.898 0.880

200 0.95 0.929 0.946 0.930

a Estimated values of the autoregressive parameter using the alternative unit tests in the absence

of a break, calculated over 10,000 replications.

Fig. 9. Alternative estimators of q (T ¼ 100, k ¼ 5, q ¼ 0:9).






the values of the estimators increase. This is particularly noticeable for q̂0

where a very large downward bias is present for breaks in mean occurring early

in the sample period. However, as the break is imposed towards the end of thesample, the values of the estimators fall. Despite these similarities between

the estimators there are some clear differences. Of the estimators, q̂0 displays

the greatest variation, exhibiting negative bias for early and late breaks, but a

positive bias for breaks imposed at all other points. The weighted least squares

estimator q̂w displays similar behaviour, although the observed variation is less

pronounced. In contrast, the recursively mean-adjusted estimator q̂r exhibits

very different properties. In virtually all of the experiments performed, q̂r

overestimates q, the rare exceptions to this being when breaks occur very early




or very late in the sample period. Therefore, it can be concluded that while the

use of recursive mean-adjustment leads to reduced negative bias in the absence

of breaks, it leads to an increased positive bias when breaks occur.

5. Conclusion

It has been noted previously that the DF test suffers from low power when

applied to near unit root processes. It also has been shown that the test canexhibit severe size distortion when applied to unit root series subject to a

structural break. Recent results have suggested that the modified tests of [3,4]




overcome both of these problems, making them appear very attractive to the

practitioner. However, while these modified tests based upon weighted sym-

metric estimation and recursive mean adjustment have high power in the ab-

sence of structural change and attractive size properties when a break occurs,

the results derived in this paper show the tests to suffer from low power when

applied to stationary series subject to a break in mean. In contrast it has been

seen that the original DF test can have very high power when a break in mean

occurs at the start of the sample period.In addition to considering the power properties of alternative unit root tests,

Monte Carlo experimentation was also employed to examine bias in the estima-

tion of the autoregressive parameter q. Again results obtained in the presence of


a break contrasted with known results derived for the no-break case. In general

it was found that downward bias in the absence of a break was replaced withpositive bias in the presence of a break. The results obtained for the recursive

mean adjustment were of particular interest. Although this estimator is known

to reduce downward bias in the absence of a break, it was seen to exhibit the

greatest upward bias of all estimators when a break occurs.

In summary, the recent literature concerning the properties of modified unit

root tests has been extended. While the size of the tests may be robust to a

single break in mean, the practitioner should note that this is at the expense of

a dramatic loss in power. Given the prevalence of such breaks in economic timeseries, this result is of particular importance.

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finite-sample properties of modified unit root tests in the presence structural change

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