Tests for a Mean Shift with Good Size and Monotonic
Power
Mohitosh Kejriwaly
Purdue University
July 3, 2007
Abstract
Commonly used LM tests for a shift in the mean of a dynamic time series su¤er from
the important problem of non-monotonic power in …nite samples. Wald tests, on the
other hand, can be seriously oversized with persistent errors. This paper proposes a new
hybrid estimator of the long-run variance which is based on residuals computed under
both the null and alternative hypotheses. It is shown that modi…ed Wald tests based
on this estimator can bypass the non-monotonic power problem while maintaining
adequate size properties. The proposed tests may be expected to su¤er some power
loss compared to the Wald tests in the case of mildly persistent errors, where the
latter are adequately sized. Our simulations show, however, that the power loss is not
signi…cant. An empirical illustration is provided.
1
Introduction
Testing for a shift in the mean of a series is an important aspect of the analysis of time
series data. Bai (1994) considers a mean shift at an unknown date in a linear process where
the unknown shift date is estimated by least squares. He establishes the consistency, rate of
convergence and the asymptotic distribution of the estimated change point assuming that
the magnitude of the shift is small.1 With respect to testing, Andrews (1993) derives the
limit distribution of the Sup-type tests, which is based on the maximal value of some statistic
over possible break dates within a pre-speci…ed set that excludes some proportion of the data
I am indebted to Pierre Perron for continuous guidance and support. Any remaining errors are my own
responsibility.
y
Krannert School of Management, Purdue University, 403 W.State Street, West Lafayette IN 47907
(mkejriwa@purdue.edu).
1
Bai (1997) generalizes these results to regression models involving stationary and/or trending regressors.
1
near the beginning or the end of the sample. Andrews and Ploberger (1994) develop tests in
a local asymptotic framework that are optimal in the sense that they maximize a weighted
average power criterion. More recently, Kim and Perron (2006) compare the asymptotic
relative e¢ ciency of structural break tests using a framework based on the approximate
Bahadur slope.
Vogelsang (1999) discusses sources of non-monotonic power for a wide variety of tests for
a shift in the mean of a dynamic time series.2 With serially correlated errors, the problem
lies in the behavior of the estimate of the long-run variance under the alternative hypothesis
of a shift in mean. In particular, if the error variance is estimated under the null (the LM
tests), non-monotonic power can result. The extent of such non-monotonicity can be quite
serious and power can even go to zero as the magnitude of the shift increases. Crainiceanu
and Vogelsang (2007) show that data dependent bandwidths used in the estimation of the
long-run variance (based on the misspeci…ed null model) are too large under the alternative
hypothesis. This in turn leads to a decrease in power as the magnitude of change increases.
A possible solution to this problem of non-monotonic power is to estimate the long-run
variance under the alternative hypothesis (the Wald tests). However, as documented by
Vogelsang (1999), this leads to size distortions in excess of 20% when the errors follow a …rst
order autoregression with autoregressive coe¢ cient 0.7. Andreou (2007) proposes a nearstationarity condition which can solve the non-monotone power problem of the CUSUM
test.3
This paper proposes a new hybrid estimator of the long-run variance which is based
on residuals computed under both the null and alternative hypotheses. In particular, the
estimate of the variance and bandwidth are based on the residuals calculated under the
alternative hypothesis while the covariances are estimated under the null hypothesis. We
show both theoretically and through Monte-Carlo experiments that modi…ed Wald tests
based on this estimator can not only bypass the problem of non-monotonicity and maintain
power comparable to the usual Wald tests but also retain adequate size properties. Our
analysis also suggests that the source of size distortions associated with the usual Wald tests
is the estimation of the covariances under the alternative hypothesis.
The rest of the paper is organized as follows. Section 2 presents the model and derives
2
Perron (1991) suggests that this non-monotonic feature is likely to be common to most tests for structural
change when lagged dependent variables are included as regressors.
3
Her results, however, are very sensitive to the value of a parameter which represents deviations of the
autoregressive parameter from unity. Moreover, the results are primarily based on Monte-Carlo experiments
with little theoretical justi…cation.
2
the main result of the paper. Section 3 o¤ers Monte-Carlo evidence to assess the adequacy of
the proposed test in terms of …nite sample size and power. Section 4 presents an application
of the proposed tests to the U.S. in‡ation rate. Section 5 concludes.
2
The Main Result
We consider the simple mean shift model
+ DUtc + ut
yt =
c
where DUtc = I(t > Tbc ); Tbc = [T
(1)
]. We make the following assumption on the errors:
Assumption A1: The errors fut g satisfy an invariance principle:
T
1=2
[T r]
X
t=1
ut ) W (r)
2
where W (:) is a standard Brownian motion,
1
= limT !1 T
P
E( Tt=1 ut )2 and ) denotes
weak convergence under the Skorohod topology.4 Let SSR0 denote the sum of squared
residuals under the null hypothesis of stability and SSR ( ) denote the sum of squared
residuals obtained using the break date Tb (= [T ]). Let ^ denote the break fraction which
minimizes the sum of squared residuals, that is, ^ = arg min
arbitrarily small positive number ;
2
=f :
;
g. For a given break fraction
1
, the LM and Wald test statistics for a mean shift are then given by
LM ( ) =
e
2
SSR0
SSR( )
2
e
T
X
= T 1
u
e2t + 2T
t=1
SSR0
1
T 1
X
j=1
SSR( )
^ ( )
T
X
u^2t ( ) + 2T
^2( ) = T 1
W ald( ) =
where u
et
u^t ( ) = yt
= yt
T
(T
Tb )
1
PT
w(j; m)
e
T
X
t=j+1
u
et u
et
j
2
1
t=1
4
SSR( ); where, for some
2
t=1 yt (t
P
T
1
t=Tb +1 yt
T 1
X
w(j; m(
^ ))
j=1
= 1; ::; T ); u^t ( ) = yt
T
X
u^t ( )^
ut j ( )
t=j+1
Tb
1
PTb
t=1
yt if t
[T ] and
otherwise. m
e is the bandwidth estimated using residuals
Such an invariance principle can be shown to hold under quite general conditions on the errors. See, for
example, Qu and Perron (2007).
3
under the null hypothesis and m(
^ ) is the bandwidth estimated using residuals that are
obtained assuming a break date Tb = [T ]. For example, when Andrews’ (1991) data
dependent method is used, we have m
e = c(e T )1=# where # depends on the kernel (e.g.,
# = 3 for the Bartlett kernel and # = 5 for the Quadratic Spectral) and e is, say, based on
an AR(1) approximation: e = 4e2 =(1
e2 )2 with e the OLS estimate from a regression of
u
et on u
et 1 . Similarly, m(
^ ) is based on ^ ( ) which is computed using residuals u^t ( ). We
make the following assumption on the kernel function:
Assumption A2: We assume that the kernel function w(j; m) satis…es the regularity
P
conditions stated in Andrews (1991), in particular the fact that Tj=11 jw(j; mj = O(m).
2
We propose the following hybrid estimator of
^ 2M
=T
1
T
X
u^t ( ^ )2 + 2T
1
t=1
T 1
X
:
w(j; m(
^ ^ ))
j=1
T
X
t=j+1
and the corresponding test statistic:
W ( )=
SSR0
u
et u
et
j
SSR( )
(2)
(3)
^ 2M
We consider the following three functionals:
Z
M ean-J =
J( )
2
Z
1
Exp-J = log
exp J( )
2
2
Sup-J = sup J( )
2
where J = LM; W ald; W . In the following Proposition, we will show that tests based on
the hybrid estimator (2) has power that is monotonic with respect to the magnitude of the
break j j.
Proposition 1 Assume that yt (t = 1; :::; T ) is generated by (1); where
limit of the test statistic (3) is given by
T
1+1=#
p
W ( )!
( c )2 (1
)
2
(Op ( ))
6= 0. Then the
2
(4)
We also have the following approximation:
T
1+1=#
W ( )
( c )2 (1
(T
4
1=#
2 2
)
+ Op (1))
(5)
Proof: It is easy to show that
T
1
1
SSR0 = T
T
X
u2t +
c
c
(1
2
)
+ op (1)
t=1
We also have
T
1
SSR( ) = T
1
T
X
c
u2t
+(
c
2
)
+ op (1)
t=1
We thus get
Let
j
( c )2 (1
T 1 (SSR0 SSR( )) =
P
=plimT !1 T 1 Tt=j+1 ut ut j . We can show that
T
1
T
X
t=j+1
We thus have
T
u
et u
et
1=# 2
^M
=T
p
j
!
j
1=# 2
c
+
+ 2T
(1
c
1=# 2
)
2
T 1
X
lim (1
T !1
)
2
+ op (1)
j=T ) uniformly in j
w(j; m(
^ ^ ))(1
j=T ) + op (1)
(6)
j=1
P
P
We have m(
^ ^ ) = c(^ T )1=# ; ^ = 4^2 =(1 ^)4 ; ^ = T 1 Tt=2 u^t ( ^ )^
ut 1 ( ^ )=T 1 Tt=2 u^t 1 ( ^ )2 .
p
Using the consistency of ^ for c (see Bai, 1994), we can show that ^ ! 1 = 0 . We thus get
^ = Op (1) and m(
^ ^ ) = Op (T 1=# ). This gives
T 1
X
w(j; m(
^ ^ ))(1
j=T ) = Op (T 1=# )
(7)
j=1
Hence, we have from (6);
T
1=# 2
^M
= Op ( 2 )
and the limit in (4) follows. The approximation (5) is also immediate from (6) and (7).
Equation (4) shows that the proposed tests diverge at rate Op (T 1
spectral kernel (# = 5), the rate of divergence is Op (T
4=5
1=#
). With the quadratic
) while it is Op (T 2=3 ) with the
Bartlett kernel (# = 3). Hence, using the Bartlett kernel might lead to some power loss
due to the reduced rate of divergence. While the LM tests diverge at the same rate as the
proposed tests, the Wald tests diverge at a faster rate (Op (T )) and hence are asymptotically
more e¢ cient. However, as we show in the next section, the Wald tests can be seriously
oversized with strong persistence in the errors. The approximation (5) shows that as the
magnitude of the break j j increases, the limit of W ( ) increases so that power is monotonic.
5
Remark 1 Deng and Perron (2007) show that e2 = Op (
4=#+2
T 1=# ). This implies that for
the LM tests the denominator has a higher order than the numerator thus inducing a nonmonotonic power function. As pointed out by Kim and Perron (2006), this feature of the
LM tests can also be observed in the case of general regressors Xt whose mean increases with
the sample size.
3
Monte-Carlo Evidence
In this section, we present Monte Carlo evidence to assess the adequacy of the proposed tests
in …nite samples as well as compare its performance to the LM and Wald tests. We consider
the following error processes:
ut = ut
1
ut = et
+ et ; u0 = 0 (AR(1) Errors)
et
1
(MA(1) Errors)
The innovations et are generated as i:i:d: N (0; 1) random variables. We consider 4 values of
the AR(1) parameter and 2 values of the MA(1) parameter: = 0; 0:2; 0:5; 0:7; = 0:5; 0:7.
The sample sizes considered are T = 120 and T = 240. The level of trimming used is = :15.
The long-run variance estimators are constructed using a quadratic spectral kernel with an
AR(1) approximation to the bandwidth.5 All experiments are based on 1000 replications.
Table 1 presents a comparison of the size ( = 0) of the proposed tests with the LM and
Wald tests. Consider …rst the case of AR(1) errors. For mildly persistent errors ( = 0; 0:2)
and T = 120, the Wald and W are slightly oversized although doubling the sample size
results in an exact size closer to the nominal size. With strong persistence in the errors
( = 0:5; 0:7), the Wald tests su¤er from substantial size distortions. While the size improves
when the sample size is doubled, the distortions continue to persist. For example, with
= 0:7 and T = 240, the size of the exp-Wald test is nearly 14%. On the other hand,
the modi…ed tests perform quite well with the size never exceeding 10%. The Sup-W test
performs the best among the W -tests with …nite sample size very close to nominal size. The
LM tests also perform adequately in terms of size. With MA(1) errors, all the tests are quite
conservative irrespective of the sample size and the extent of persistence. However, as we
will see in the power simulations, the conservative nature of the tests do not seem to have
5
We also tried an ARMA(1,1) approximation for the bandwidth. The results showed substantial size
distortions with MA(1) errors.
6
Table 1: Size of LM, Wald and modi…ed Wald Tests (Nominal Size = 5%)
AR(1) Errors
=0
= 0:2
T = 120 T = 240 T = 120
= 0:5
= 0:7
T = 240 T = 120 T = 240 T = 120
T = 240
M -LM
.057
.047
.057
.057
.068
.056
.061
.059
E-LM
.052
.041
.052
.052
.056
.050
.034
.047
S-LM
.030
.035
.030
.041
.028
.035
.008
.021
M -W ald
.075
.051
.075
.066
.126
.088
.171
.110
E-W ald
.076
.054
.076
.074
.160
.106
.222
.135
S-W ald
.066
.051
.066
.073
.140
.091
.203
.125
M -W
.075
.053
.075
.063
.091
.074
.091
.073
E-W
.068
.052
.068
.063
.090
.062
.084
.064
S-W
.056
.045
.056
.052
.053
.046
.045
.036
MA(1) Errors
= 0:5
= 0:7
T = 120 T = 240
T = 120 T = 240
M -LM
.004
.009
.000
.000
E-LM
.004
.007
.000
.000
S-LM
.002
.005
.000
.000
M -W ald
.007
.011
.000
.001
E-W ald
.011
.007
.000
.000
S-W ald
.009
.005
.001
.001
M -W
.006
.010
.000
.000
E-W
.008
.009
.000
.000
S-W
.006
.006
.001
.000
an adverse e¤ect on power.
Next, we present a power comparison of the LM, Wald and W -tests. We consider a
7
break in the middle of the sample -
c
= 0:5:6 The power functions of the LM and modi…ed
tests with AR(1) errors are plotted in Figures 1 and 2.7 The …gures clearly illustrate that
the LM tests su¤er from the problem of non-monotonicity irrespective of the sample size
and the degree of persistence. Among the LM tests, the Mean-test seems to perform the
best while the Sup-test performs most poorly (though the Sup-test is quite conservative).
On the contrary, all the W -tests exhibit monotonic power (except for the case = 0 which
shows slight non-monotonicity). The ranking among the W -tests is less evident - all tests
seem to perform similarly except that the Mean-test performs slightly better than the Exp
and Sup-tests when
= 0:7 and T = 120. Figures 3 and 4 present a power comparison
of the Wald tests and the proposed modi…ed tests. The power functions of the modi…ed
tests are only slightly lower than those of the Wald tests. Thus, the proposed tests o¤er
an improvement in size over the Wald tests with little loss in power. Figure 5 presents a
power comparison of the LM and modi…ed Wald tests with MA(1) errors. In this case, the
decline in power for the LM tests occurs at larger values of as compared to the AR(1) case.
Comparing the power functions of the Wald and the modi…ed tests we found that as in the
AR(1) case, the modi…ed tests su¤er very little power loss compared to the Wald tests.
Overall, the proposed modi…cation of the Wald test clearly outperforms all the LM tests
in terms of power while still maintaining adequate size properties. Since the Wald tests are
based on long-run variance estimates constructed using residuals under the alternative hypothesis only, our proposed tests may be expected to su¤er some power loss; our experiments
show, however, that the loss is not signi…cant.
4
Empirical Application
This section presents an application of the proposed test to the U.S. in‡ation rate. There is
a substantial debate in the macroeconomics literature regarding the possible nonstationary
behavior of the in‡ation rate. The nonstationarity of in‡ation has important economic
implications. Conventional sticky price models such as Dornbusch (1976) and Taylor (1979)
imply stationarity of the price level. Moreover, the nonstationarity of in‡ation a¤ects the
construction and evaluation of monetary policy rules. Hall (1984) and Taylor (1985), among
many others, have advocated that monetary policy should target either the level or growth
of national income. If in‡ation is nonstationary, then so is nominal income growth unless
Power experiments were also conducted for c = 0:3; 0:7. The results were qualitatively similar to
= 0:5 and hence not reported.
7
All power functions presented are size-unadjusted.
6
c
8
in‡ation and real income growth are cointegrated. Benati and Kapetanios (2003) …nds
evidence of breaks in in‡ation dynamics for 23 in‡ation series from 18 countries. Rapach
and Wohar (2005) present evidence that breaks in the mean in‡ation rate are correlated
with those in the real interest rate for 13 industrialized countries. Levin and Piger (2004)
apply both classical and Bayesian econometric methods to test the presence of a break in
the mean of in‡ation after 1984 for 12 industrial countries. They show that allowing for a
break in intercept, the in‡ation measures generally exhibit low persistence.
We apply the proposed tests to analyze whether there was a mean shift in the U.S. in‡ation rate as a result of Paul Volcker taking up Fed Chairmanship in 1979 and consequent
changes in the Federal Reserve’s operating procedures. We use monthly data on the seasonally adjusted CPI in‡ation rate obtained from the FRED website. The sample period used is
1970:1-1987:7. The start date corresponds to Arthur F. Burns taking up Fed Chairmanship
while the end date corresponds to the end of the Volcker-era. Figure 6 presents a plot of
the in‡ation rate over the sample period. The plot suggests a change in the mean sometime
around the late ’70s/early 80’s. Table 2 presents results of the W - tests as well as the LM
tests.
Table 2: Empirical Results for the U.S. In‡ation Rate (1970:1-1987:7)
Test
Value 5% Critical Value
M ean-LM
2.13
2.88
Exp-LM
1.78
2.06
Sup-LM
6.85
8.85
M ean-W
2.81
2.88
Exp-W
2.61
2.06
Sup-W
9.05
8.85
The empirical results show that the LM tests cannot reject the null hypothesis of a
constant mean in the in‡ation rate over the sample period considered. In contrast, all the W tests except M ean-W reject the null of stability at the 5% level. The estimated break date
(that which minimizes the sum of squared residuals) is 1982:6 which falls within the period of
Volcker disin‡ation (1979:10-1982:10). To ensure the absence of a unit root, we estimated an
AR(p) model on the residuals where p was selected using the Bayesian Information Criterion
(BIC). The sum of the AR coe¢ cients was estimated at 0.66. The AR(p) models were also
estimated separately using residuals in each regime - the AR coe¢ cients summed to 0.72 in
9
the …rst regime and 0.36 in the second regime. These estimates fall within the range of values
of the persistence parameter used in the simulation experiments. which shows the bene…ts
of using the W -tests over the LM and Wald tests. This simple analysis demonstrates that
while the LM tests may lack power in detecting shifts in the mean of a time series, the W tests are easily able to detect such shifts and hence should be used in practical applications.
5
Conclusion
This paper proposes a solution to the size-power trade-o¤ between the LM and Wald tests
by proposing a new estimator of the long-run variance of the errors which is based on
estimated residuals under both the null and alternative hypotheses. Our results show that
structural break tests modi…ed with this estimator are able to bypass both the problem of
non-monotonicity of the LM tests as well as the size distortions of the Wald tests. The
proposed tests may be expected to su¤er some power loss compared to the Wald tests in
the case of mildly persistent errors, where the latter are adequately sized. Our simulations
show, however, that the power loss is not signi…cant. Hence, we expect that the proposed
tests will provide a useful addition to the existing battery of tests for structural change.
References
[1] Andreou, E. (2007): “Restoring Monotone Power in the CUSUM Test,” Manuscript,
Department of Economics, University of Cyprus.
[2] Andrews, D.W.K. (1991): “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation,”.Econometrica, 59, 817-858.
[3] Andrews, D.W.K. (1993): “Tests for Parameter Instability and Structural Change with
Unknown Change Point,”Econometrica, 61, 821-856.
[4] Andrews, D.W.K. and Ploberger, W. (1994): “Optimal Tests when a Nuisance Parameter is Present only under the Alternative,”Econometrica, 62, 1383-1414.
[5] Bai, J. (1994): “Least Squares Estimation of a Shift in Linear Processes,” Journal of
Time Series Analysis, 15, 453-472.
[6] Bai, J. (1997): “Estimation of a Change Point in Multiple Regression Models,”Review
of Economics and Statistics, 79, 551-63.
[7] Benati, L. and Kapetanios, G. (2003): “Structural Breaks in In‡ation Dynamics,”Computing in Economics and Finance, No. 169.
10
[8] Crainiceanu, C. and Vogelsang, T.J. (2007): “Non-monotonic Power for Tests of Mean
Shift in a Time Series,”Journal of Statistical Computation and Simulation, forthcoming.
[9] Deng, A. and Perron, P. (2007): “A Non-local Perspective on the Power Properties of
the CUSUM and CUSUM of Squares Tests for Structural Change,” Journal of Econometrics, forthcoming.
[10] Dornbusch, R. (1976): “Expectations and Exchange Rate Dynamics,”Journal of Political Economy, 84, 1161-1176.
[11] Hall, R. (1984): “Monetary Strategy with an Elastic Price Standard,” Price Stability
and Public Policy, Federal Research Bank of Kansas City, 137-159.
[12] Kim, D. and Perron, P. (2006): “Assessing the Relative Power of Structural Break
Tests Using a Framework Based on the Approximate Bahadur Slope,” Manuscript,
Department of Economics, Boston University.
[13] Levin, A.T. and Piger, J.M. (2004): “Is In‡ation Persistence Intrinsic in Industrial
Economies?,”European Central Bank Working Paper No. 334.
[14] Perron, P. (1991): “A Test for Changes in a Polynomial Trend Function for a Dynamic
Time Series,”Manuscript, Department of Economics, Princeton University.
[15] Qu, Z. and Perron, P. (2007): “Estimating and Testing Structural Changes in Multivariate Regressions,”Econometrica, 75, 459-502.
[16] Rapach, D.E. and Wohar, M.E. (2005): “Regime Changes in International Interest
Rates: Are They a Monetary Phenomenon,” Journal of Money, Credit and Banking,
37, 887-906.
[17] Taylor, J. (1979): “Staggered Wage Setting in a Macro Model,” American Economic
Review, 69, 108-113.
[18] Taylor, J. (1985): “What would Nominal GNP Targeting do to the Business Cycle,”
Carnegie-Rochester Conference Series on Public Policy, 22, 61-84.
[19] Vogelsang, T.J. (1999): “Sources of Nonmonotonic Power when Testing for a Shift in a
Dynamic Time Series,”Journal of Econometrics, 88, 283-299.
11
Figure 1: Power Functions with AR(1) Errors: A Comparison of LM and W* Tests
Figure 2: Power Functions with AR(1) Errors: A Comparison of LM and W* Tests
Figure 3: Power Functions with AR(1) Errors: A Comparison of Wald and W* Tests
Figure 4: Power Functions with AR(1) Errors: A Comparison of Wald and W* Tests
Figure 5: Power Functions with MA(1) Errors: A Comparison of LM and W* Tests
Figure 6: The U.S. In‡ation Rate - 1970:1-1987:7