Directory UMM :Data Elmu:jurnal:J-a:Journal of Econometrics:Vol95.Issue1.2000:

Journal of Econometrics 95 (2000) 97}116

Detection of change in persistence of a linear

time series

Jae-Young Kim

!

,

"

,

*

!Department of Economics, State University of New York, Albany, NY 12222, USA "Department of Economics, Hong Kong University of Science and Technology, Hong Kong

Received 1 January 1997; received in revised form 1 February 1999; accepted 1 April 1999

Abstract

This paper studies how to detect structural change characterized by a shift in persist-ence of a time series. In particular, we are interested in a process shifting from stationarity to nonstationarity or vice versa. A general linear process is considered that includes an ARMA process as a special one. We derive a statistic for testing the occurrence of such a change and investigate asymptotic behavior of it. We show that our test has power against fairly general alternatives of change in persistence. A Monte Carlo study shows that our test has reasonably good size and power properties in"nite samples. We also discuss how to estimate the unknown period of change. We apply our test to two examples of time series, the series of the U.S. in#ation rate and the series of U.S. federal government's budget de"cit in the postwar period. For these two series we have found strong evidence of structural change from stationarity to nonstationarity. ( 2000 Elsevier Science S.A. All rights reserved.

JEL classixcation: C1; C22; C5

Keywords: Change in persistence; Unknown change period

*Department of Economics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. Fax:#852-2358-2084.

E-mail address:jykim@cnsunix.albany.edu (J.-Y. Kim)

1. Introduction

Recent literature on nonstationary time series addresses the importance of information on di!erent degrees of persistence in linear time series for econometric inference. In this paper, we consider a stochastic process that may undergo a shift in persistence. In particular, we are interested in a process shifting from a lower level persistence to a higher level persistence, an example of which is a shift from stationarity to a unit root, or vice versa. We study a formal theory for detecting such a shift in this paper.

Examples of time series having change in persistence are found in De Long and Summers (1988), real output series of the U.S. and the European countries. De Long and Summers (1988) conjectured that these series shifted from station-arity to a unit root after WWII. They performed informal tests to "nd the evidence in favor of their conjecture. Another example is in Hakkio and Rush (1991) where they considered the U.S. federal government budget de"cit to"nd that the de"cit process underwent a change from stationarity to a unit root. However, Hakkio and Rush (1991)'s work is also based on informal inference. Bai (1992) stressed the importance of change in persistence of a process but did not discuss how to formally detect such a change.

Statistical inference on persistence change involves consideration of two fundamentally di!erent processes connected at a point of time, especially for change from stationarity to a unit root or vice versa. Di$culty arises in this case because usual inferential statistics do not have standard properties. Our ap-proach in this paper is based on residual-based tests for stationarity that are considered in Nyblom and Makelainen (1983), Rogers (1986), Nyblom (1986), Nabeya and Tanaka (1988), McCabe and Leybourne (1988) and Kwiatkowski et al. (1992). We derive a test statistic by applying the analyses of these authors to the case of persistence change with the date of change being unknown. We apply our statistic to the following three di!erent ways for handling the problem of the change period being unknown. First, a maximum-Chow-type test considered in Davies (1977), Hawkins (1987), Kim and Siegmund (1989), and Andrews (1993). Second, Hansen (1991)'s mean score test, and third, Andrews and Ploberger (1994)'s mean-exponential test.

We investigate asymptotic behavior of our test statistics. We show that our tests have power against fairly general alternatives of change in persistence. Also, we discuss how to consistently estimate the unknown period of change. A Monte Carlo study shows that our testing procedures have reasonably good size and power properties in "nite samples. A Monte Carlo study also shows that, among the three tests the mean score test has the smallest size distortion in our case.

from stationarity to nonstationarity. For quarterly data on the U.S. in#ation rate the estimated period of change is 1973:3 and for the U.S. federal govern-ment's budget de"cit it is 1968:4.

Section 2 discusses how to model a change in persistence of a linear process. In Section 3 we develop test procedures for detecting change in persistence of a linear process and investigate some asymptotic properties of the tests. Also, in Section 3 we discuss how to estimate the date of change. Section 4 provides some simulation results, and Section 5 applies our procedures to two time-series data.

2. Modeling change in the persistence

To study a stochastic processMy

tN that undergoes a shift in persistence, we consider the following hypotheses: The null hypothesis H

0 thatyt maintains stationarity of constant persistence throughout the sample period; An alterna-tive hypothesis H

1thatyt maintains stationarity of constant persistence until some period, after which it becomes a process of higher persistence such as a unit root. Or, an alternative hypothesis H@₁ that y

t is a process of relatively high persistence until some period, after which it becomes a process of lower persist-ence. The null hypothesis can be described as

H

0: yt"r0#zt, for t"1,2,¹, (2.1)

wherer₀is a constant, andz

t is a stationary variable satisfying the following (uniform mixing) regularity conditions:

Assumption 1. The processMz

tN=0 is such that

(i) E(z)"0;

(ii) EDzDc`e(Rfor somec'2; (iii) Mz

tN=0 isu-mixing with mixing coezcients umsuch that+=m/1u1~2@m c(R; (iv) The long-runvariancep2

z"+=j/0E[zj`1z@1]exists;

(v) lim T6

=var(¹~1@2+sTt/1zt)"sp2z; limT6

=var(¹~1@2+Tt/sT`1zt)"(1!s)p2z

eachs3(0, 1).

The above conditions allow for a broad class of weakly dependent time series and have been used by Phillips (1987), Phillips and Perron (1988) and Phillips and Solo (1992), among others, to derive limiting behavior of a stochastic process. Alternatively, for the null hypothesis, one may consider a trend station-ary process:

Hc₀: y

t"ct#r0#zt, fort"1,2,¹. (2.2)

Now, consider an alternative hypothesis that y

q¹, but aftert"[q¹] it becomes a process of higher persistence such as a unit root:

H₁: y

t"r0#zt,0, fort"1,2,[q¹],

y

t"r1#zt,1, fort"[q¹]#1,2,¹,

(2.3)

wherez

t,0is a stationary process satisfying conditions in Assumption 1;zt,1is

a process of higher persistence thanz

t,0, andr0andr1are constants.

On the other hand, consider an alternative hypothesis H@₁thaty

tis a process of relatively high persistence until t"[q¹], but after t"[q¹] it becomes a process of lower persistence:

H@₁: y

t"r1#zt,1, fort"1,2,[q¹],

y

t"r0#zt,0, fort"[q¹]#1,2,¹,

(2.3@)

wherez

t,i, i"0, 1, are as in (2.3).

Likewise, we can describe an alternative hypothesis for Hc₀ in (2.2) that corresponds to change in H

1(2.3) but with a trend:

Hc₁: y

t"ct#r0#zt,0, for t"1,2,[q¹],

y

t"ct#r1#zt,1, for t"[q¹]#1,2,¹.

(2.4)

Notice that y

t"ct#r1#zt,1 with zt,1"zt~1,1#ut is equivalent to

y

t"c#yt~1#ut, a unit root process with drift. Also, we can describe an

alternative hypothesis Hc@₁ for the null Hc₀that corresponds to the change in H@₁(2.3@) but with a trend:

Hc@₁: y

t"ct#rt#zt,1, fort"1,2,[q¹],

y

t"ct#r0#zt,0, fort"[q¹]#1,2,¹.

(2.4@)

3. Test for and estimation of structural change

In this section we discuss how to detect the occurrence of structural change characterized by the alternative hypothesis (2.3), (2.3@), (2.4), or (2.4@). We derive a test statistic and investigate asymptotic properties of the test statistic. In particular, asymptotic behavior of the test statistic is studied both under the null and under the alternative hypotheses. We"nd that our test has nontrivial power against fairly general alternative hypotheses of change in persistence. In addi-tion, we discuss how to estimate the unknown change pointq.

3.1.1. Test for structural change

We"rst discuss how to test H

0in (2.1) against H1in (2.3) or H@1in (2.3@). Later

(2.4@). Letz8_t, t"1, 2,2,¹be the residuals from the regression ofy_ton intercept, de"ne the following partial sum processes separately before and after [q¹]:

S

Now, consider the following statistic:

N

The statistic (3.3) is a key element in our testing procedure. In the above statisticN

T(q) the true value ofqis unknown. Under the situation of the true change period being unknown three di!erent ways for testing structural change can be derived based onN

T. First, a maximum-Chow-type test as is considered in Davies (1977), Hawkins (1987), Kim and Siegmund (1989), and Andrews (1993) for testing H

0against H1with unknown break pointt"q¹

whereT_{is a compact subset of (0, 1). Second, Hansen (1991)}'_{s mean score test is}

EN

T(q),

P

q|T

N

T(q) dq. (3.5)

Third, Andrews and Ploberger (1994)'s mean-exponential test statistic forN Tis

Theorem 3.1. Suppose that Assumption 1 is true forz

tunder the null hypothesisH0.

Then,underH₀it is true that

whereH())is the statistic in each of the procedures (3.4)}(3.6); <(r)is a standard

Brownian bridge: <_(r)"=_(r)!r=_{(q) for r}3[0,q]; <_(r!_q)"=_(r!_q)! (r!_q)=₍₁!_q) forr3[q, 1], where =_{()) is a Wiener process, and the symbol} Nsignixes weak convergence of the associated probability measures.

Upper tail and lower tail critical values of the limit processes of the test statistics (3.4)}(3.6) are provided in Table 1(a}c). The critical values are cal-culated via a direct simulation, using samples of sizes 100, 200, 500 and 2000 with 10,000 replications. In principleT_{can be any compact subset of (0, 1). In} practice, however, it needs to be small enough so that all of the statistics discussed in this paper can be calculated (see Gregory and Hansen, 1996). As in many earlier works, we considerT"[0.2, 0.8] for a reasonable choice.

The decision rule for testing H

0against H1or H@1is as follows: Reject H0in

favor of H

1(H@1) if the value of the test statistic (3.4), (3.5), or (3.6) is larger

(smaller) than an upper tail (lower tail) critical value.

Now, consider the two hypotheses of Hc₀in (2.2) and Hc₁in (2.4) or Hc@₁in (2.4@) that take a trend into account. With these latter hypotheses we need to consider the partial sum process of residuals from a regression of y

t on intercept and trend. Letz8

t, t"1, 2,2,¹be the residuals from the regression ofyon intercept

and trend. Also, letS

c0,t(q) fort"1,2,q¹andSc1,t(q) fort"q¹#1,2,¹be as de"ned above (3.2) with thisz8

t. Consider the following statistic:

N cT(q)"

((1!q)¹)~2+T_qT`1S

c1,t(q)2 (q¹)~2+q₁TS

c0,t(q)2

. (3.7)

Theorem 3.2. Suppose that Assumption 1 is true forz

tunder the null hypothesisHc0.

Then,underHc₀it is true that

1. N cT(q)N

(1!_q)~2:1

q<2(r!q)2dr

q~2:q₀<

2(r)2dr

,N

c,=(q) for each q3T,

2. H(N

cT)NH(Nc,=),

whereH())is the statistic in each of the procedures (3.4)}(3.6); <₂(r)is a

second-level Brownian bridge: <

2(r)"=(r)#(2r!3r2)=(q)#(!6r#6r2):q0=(s) ds

for r3[0,q] and <

2(r!q)"=(r!q)#(2(r!q)!3(r!q)2)=(r!q)#

(!6(r!_q)#6(r!_q)2):1

q=(s) dsforr3[q, 1].

Upper tail and lower tail critical values of the limit processes of the test statistics in Theorem 3.2 (2) are provided in Table 1(d}f ). The critical values are calculated via a direct simulation, using the same sample sizes and replications as for H(N

T).

The decision rule for testing Hc₀against Hc₁(or Hc@₁) is the same as the case with hypotheses H

Table 1

Sample sizes Percentiles

0.05 0.1 0.5 0.75 0.9 0.925 0.95 0.975 0.99

(a) Critical values of maximum-Chow (V)

¹"100 0.6437 0.8349 2.7995 5.2218 8.9760 10.4143 12.3985 16.0510 21.3910

¹"200 0.6631 0.8544 2.8534 5.3131 9.2282 10.6115 12.8015 16.8082 22.1015

¹"500 0.6794 0.8654 2.8744 5.4466 9.3102 10.6727 12.7522 16.8202 22.1502

¹"2000 0.6962 0.8742 2.9287 5.4959 9.4977 10.9415 13.0695 17.2395 22.9695

(b) Critical values of mean-Chow (V)

¹"100 0.1862 0.2296 0.7239 1.3081 2.2251 2.5382 3.0040 3.9122 5.0922

¹"200 0.1893 0.2331 0.7345 1.3255 2.2660 2.5684 3.0291 3.9650 5.2025

¹"500 0.1927 0.2368 0.7275 1.3219 2.2706 2.6164 3.1192 4.0506 5.2556

¹"2000 0.1916 0.2363 0.7372 1.3376 2.2838 2.6306 3.1323 4.0756 5.2856

(c) Critical values of exponential-Chow (V)

¹"100 !0.1819 !0.0904 0.9975 2.7169 5.9190 7.1830 9.0123 12.5990 17.6390

¹"200 !0.1854 !0.0911 1.0224 2.7655 6.0563 7.2813 9.3013 13.1563 18.2213

¹"500 !0.1701 !0.0875 1.0176 2.8343 6.0668 7.3343 9.2743 13.1643 18.1943

¹"2000 !0.1893 !0.0879 1.0395 2.8325 6.2202 7.5335 9.5452 13.3902 19.0652

(d) Critical values of maximum-Chow (V2)

¹"100 1.3825 1.6572 4.7580 8.4462 14.5587 16.5937 19.5237 25.2837 33.3537

¹"200 1.4743 1.7553 4.8288 8.6367 14.3124 16.2924 19.0424 24.7924 32.2224

¹"500 1.5192 1.8005 4.9186 8.7377 14.6477 16.9027 20.3110 26.3027 34.6077

¹"2000 1.5027 1.7897 4.9219 8.5918 14.4918 16.6143 19.7193 25.7493 33.7743

(e) Critical values of mean-Chow (V2)

¹"100 0.3187 0.3556 0.8683 1.3835 2.1001 2.3063 2.5794 3.1074 3.8744

¹"200 0.3231 0.3660 0.8635 1.3590 2.0005 2.1969 2.5283 3.0489 3.7569

¹"500 0.3299 0.3743 0.8723 1.3726 2.0118 2.2068 2.4704 3.0004 3.6626

¹"2000 0.3256 0.3665 0.8715 1.3552 1.9942 2.1962 2.5019 3.0405 3.7405

(f) Critical values of exponential-Chow (V2)

¹"100 !0.1232 0.1725 1.6777 4.3219 9.3152 11.2152 14.0002 19.2452 27.2952

¹"200 !0.1201 0.1885 1.6941 4.1933 8.8711 10.5478 13.1478 18.3578 24.8378

¹"500 !0.1136 0.2007 1.6919 4.2040 9.0090 9.0090 13.8473 18.9123 26.6623

¹"2000 !0.1298 0.1838 1.6773 4.0553 8.7595 10.5128 13.2128 18.9128 26.3328

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Remark 1. One may think of a test for testing structural change based on an a priori known change pointt"_q¹. However, a test conditional on the a priori known change point may cause a serious size distortion of the test, as was well explored by Christiano (1992), Banerjee et al. (1992), Zivot and Andrews (1992).

3.1.2. Asymptotic local power

The above testing procedures that are designed to test for persistence change in a stochastic process have power against a variety of alternatives. In particular, our procedures have power against fairly general alternatives of change in persistence. In the following we develop formal arguments to support this fact.

First, we investigate asymptotic behavior of H(N

T) under the alternative hypothesis H

1in (2.3) withzt,1being a unit root. We will show that under this

H

1, each of three statisticsH(NT) is of O1(¹2). Then it implies that the tests based

on H(N

T) are consistent under an alternative of change from stationarity to nonstationarity. The same analysis can be applied to H(N

cT) under Hc1 with

z

t,1being a unit root.

Theorem 3.3. (i)Letq6"maxMq₀,qNfor a givenq3T_{.Also,letq1be any point in(q6,} 1].Then under the alternative hypothesisH

1withzt,1"zt~1,1#ut,utbeing iid,it

(ii)Under the alternative hypothesisH

1withzt,1being a unit root,it is true that

H(N

T)"O1(¹2),

whereH())is the statistic in each of the procedures (3.4)}(3.6).

Notice that Theorem 3.3 (i) implies that our tests based on H(N T) are consistent under the alternative of a temporary change from stationarity to a unit root with the process returning to stationarity after the temporary period. This is becauseq1in the theorem is any point in (q(, 1].

Now, we investigate the asymptotic power of H(N

T) under the alternative hypothesis H₁in (2.3) withz

t,1being stationarity of higher persistence thanzt,0.

We assume thatz

t,1satis"es Assumption 1 with its long-run variance beingp21.

Theorem 3.4. Assume thatz

t,1inH1is a stationary process satisfying Assumption

1with+=

criticalvalue ofH())in (3.4)}(3.6). Then underH₁with the abovez_t,1it is true that

Therefore, the tests based on H(N

T) in (3.4)}(3.6) have nontrivial power against alternatives of change from stationarity of lower persistence to station-arity of higher persistence. The same results as those in Theorem 3.4 can be obtained forH(N

cT) under Hc1a stationary process of higher persistence than

z

t,0. ForH(NT) under H@1or forH(NcT) under Hc@1, on the other hand, the power

properties in Theorems 3.3 or in 3.4 are not attainable. However, in this latter case we can show that the statistic H()) converges to a limit under H@₁ or

Hc@₁ whose distribution has much smaller percentile values than under H())

under H

0or Hc0:

Corollary 3.1. Assume that z

t,1 in H@1 or Hc@1 is a stationary process satisfying

Assumption 1 with +=

j/0E[zj`1,1z1,1]"p21 where p21'p20. Then under H@1 or

Therefore, in the above caseN

T(q) converges to a limit which has nondegener-ate distribution only for q(q

0. We can see that even for eachq3(0,q0) this

nondegenerate distribution has smaller percentile values than the limit distribu-tion ofN

Tunder H0or Hc0. Similar implication in the case ofzt,1being a unit

root applies for the above result.

3.2. Estimation of structural change

In this subsection we consider how to estimate the unknown change point. Thus, letK

Assumption 2 describes a law of large numbers for a sequance of stationary z8 @s, which usually holds for OLS residuals likez8. Now, letq( be such that

q("

G

argmaxq|TKT(q) under H1or Hc1,

argmin

q|TK

T(q) under H@1or Hc@1.

The following theorem shows asymptotic properties ofq(:

Theorem 3.5. Under the alternative hypothesis H₁, H@₁, Hc₁, or Hc@₁ and under Assumption 2,it is true that

1. (q(!_q

0)"o1(1),

2. ¹(q(!_q

0)"O1(1).

Thus,q( is a consistent estimator with¹-rate of convergence.

In sum, inference on the possibility of a structural change implied in (2.3) (or (2.3@), (2.4),(2.4@)) can be made in two steps: First, test the null hypothesis (2.1) (or (2.2)) against the alternative hypothesis (2.3) (or (2.3@), (2.4),(2.4@)) based on the test (3.4), (3.5) or (3.6). Second, if the null hypothesis is rejected, then estimate

q₀byq(.

4. Simulation studies on the size and power

Our simulation study is based on samples of sizes 100, 200, and 500 with 2000 replications. To investigate the size of each test we consider an AR(1) process for z

t: y

t"r0#zt, (4.1)

where

z

t"ozt~1#et, t"1,2,¹,

wheree

t&iid N(0, 0.01), r0"0.1 ando"0.7. The results are provided in Table

2(a}d). Among the three tests (3.4)}(3.6) the mean score test (3.5) looks the best in terms of size distortion. For the maximum-Chow (MX) test and the mean-exponential (EX) test the amount of size distortion is substantial, especially for a small sample. Therefore, we recommend to use the mean score test (3.5) for a small sample analysis. For a su$ciently large sample, the amount of size distortion of MX and EX becomes small.

To investigate the power property of each test we consider the same model as above, allowing change in the parametero.

z

t"o(1)zt~1#et, t"1,2,q¹,

z

Table 2

¹"100 ¹"200 ¹"500

a MX MN EX MX MN EX MX MN EX

(a) Empirical sizes

0.010 0.114 0.038 0.108 0.073 0.022 0.0715 0.049 0.0165 0.044

0.025 0.177 0.069 0.1665 0.1155 0.0425 0.1065 0.087 0.0285 0.0725

0.050 0.2445 0.1155 0.2305 0.1845 0.083 0.1695 0.151 0.0565 0.131

0.100 0.3445 0.1935 0.325 0.2775 0.14 0.246 0.25 0.116 0.2125

(b) Power values whena"0.05

¹"100 ¹"200 ¹"500

Case q MX MN EX MX MN EX MX MN EX

0.25 0.818 0.671 0.8105 0.9075 0.776 0.9005 0.981 0.9295 0.9785

I 0.50 0.808 0.716 0.8055 0.895 0.854 0.8915 0.9805 0.967 0.9785

0.75 0.6785 0.5915 0.6725 0.7535 0.698 0.745 0.9035 0.893 0.9

0.25 0.729 0.5495 0.7215 0.796 0.623 0.79 0.878 0.743 0.8705

II 0.50 0.7165 0.573 0.711 0.7675 0.663 0.7585 0.8765 0.806 0.87

0.75 0.653 0.497 0.645 0.6515 0.5335 0.6425 0.723 0.651 0.718

(c) Power values whena"0.025

¹"100 ¹"200 ¹"500

Case q MX MN EX MX MN EX MX MN EX

0.25 0.7775 0.603 0.768 0.8755 0.721 0.8715 0.97 0.909 0.968

I 0.50 0.7655 0.6495 0.7635 0.86 0.812 0.8565 0.97 0.9555 0.969

0.75 0.615 0.515 0.6085 0.693 0.637 0.688 0.8845 0.864 0.881

0.25 0.681 0.4755 0.675 0.7425 0.55 0.7345 0.8405 0.6895 0.8355

II 0.50 0.668 0.4975 0.6605 0.7185 0.5955 0.7125 0.835 0.759 0.8325

0.75 0.5835 0.4335 0.5735 0.577 0.4685 0.5735 0.6675 0.581 0.659

(d) Simulation results of the estimate ofq

q ¹"100 ¹"200 ¹"500

0.25 23.8310(1.6893) 48.2510(2.8496) 122.1140(5.3649)

0.50 49.3965(2.9085) 99.7295(0.9909) 249.7965(0.7002)

0.75 75.0095(0.1882) 150.0125(0.1742) 375.0090(0.1730)

Note:a"theoretical size; MX"Maximum-Chow; MN"Mean-Score; EX"Mean-exponential. In (d), the number outside the parenthesis is equal to

the mean ofq(¹; The number inside the parenthesis is equal to the standard deviation ofq(¹.

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for each ofq"0.25, 0.5, 0.75, witho(2)"1. We consider two di!erent cases with

o(1)"0.85 ando(1)"0.95. The simulation results are provided in Tables 2(b)

and 2(c) for test sizes of 5 and 2.5%, respectively.1For di!erentq's power values are slightly di!erent. Withq"0.75 power values are the lowest in most cases. As ¹increases power values become higher and higher in each case of the tests. Since our tests are consistent, power values will converge to unity for all cases as ¹tends to in"nity.

To"nd the accuracy of the estimatork)"_q(¹we evaluate the mean and the standard deviation ofk) based on the simulation of the above model (4.2) with 2000 replications. The mean and the standard deviation of the estimateq( are provided in Table 2(d). The mean of the estimated change periodkK is within less than two-period distance from the true change periodk

0"q0¹in all the cases

under consideration except one. Also, the standard deviation of the estimate is quite small in all the cases.

5. Applications

In this section we apply our tests to two examples of time series data, data on the rate of in#ation and data on the federal government's budget de"cit. For U.S. postwar quarterly data, we"nd clear evidence of change from stationarity to nonstationarity in these two series.

5.1. The inyation rate

The GNP de#ator is used for the level of prices. We get the in#ation rate (n) from the quarterly data on GNP de#ator (p): n

t"(pt!pt~1)/(pt~1). Data

period is from 1948:2 to 1994:1. Table 3(a) provides test results based on the maximum-Chow, the mean score and the mean-exponential tests with the estimated date of change. For all three tests the null of the no structural change is rejected in favor of H

1at the 1% level. The estimated change period is 1973:3.

We checked whether or notn_tis stationary in the two periods of 1948:2}1973:3 and 1973:4}1994:1, respectively. The result is provided in Table 3(b). For both tests of Augmented Dickey}Fuller and Phillips}Perron, the null of a unit root is rejected at the 1% level in the"rst period, 1948:2}1973:3. On the other hand, in the period of 1973:4}1994:1 the null of a unit root is not rejected at the 10% level by both of the two tests. Therefore, we can conclude that in the postwar period the in#ation rate undergoes change from stationarity to a unit root around 1973:3.

Table 3

(a) Statistics for structural change in the in#ation rate of GD (Data period: 1948 : 2}1994 : 1)

MX MN EX q(¹

53.6379 7.3966 48.4580 102.00 (1973 : 3)

(b) Test for a unit root in the in#ation rate of GD

Test statistics Sample I (1}102) Sample II (103}186) ADFo-statistic !52.3141(!20.3) !5.1775(!11.2)

ADFt-statistic !4.0120(!3.46) !1.3955(!2.57)

Phillps}Perrono-statistic !69.5188(!20.46) !11.2389(!11.2) Phillps}Perront-statistic !7.2321(!3.46) !2.4254(!2.57) (c) Statistics for structural change in the government budget de"cit (Date period: 1947:1}1993:2)

MX MN EX q(¹

46.4259 10.8832 41.8676 88.00 (1968.4)

(d) Test for a unit root in the federal budget de"cit

Test statistics Sample I (1}88) Sample II (89}186)

ADFo-statistic !52.8554(!13.6) !0.8188(!5.7)

ADFt-statistic !4.2027(!2.58) !0.4173(!1.62)

Phillps}Perrono-statistic !23.5852(!13.6) !1.002(!5.7) Phillps}Perront-statistic !3.7545(!2.58) !0.5125(!1.62)

Note: In (b) and (d) the numbers inside the parentheses are 1% critical values for sample I and 10% critical values for sample II.

5.2. Government budget dexcit

The issue that U.S. government's spending and taxing policies began to violate the government's intertemporal budget constraint was raised by several researchers such as Hamilton and Flavin (1986), Wilcox (1987), Kremers (1989), Trehan and Walsh (1990), Smith and Zin (1988), and Hakkio and Rush (1991). The analyses of these authors are based on the assumption that the beginning period of the violation of the budget constraint is known. The results are mixed: Some of these authors argue that spending and revenue kept a well-balanced relationship in the long run. Others including Hakkio and Rush (1991) argue that the well-balanced relationship is true only at the early stage of the post-war era.

In this section we use Hakkio and Rush (1991)'s model for the federal government's intertemporal budget constraint to test for the occurrence of violation of the budget constraint. Hakkio and Rush (1991) derived a version of the government's intertemporal budget constraint given in the following:

GG

t"a#Rt#_j?lim₌bj`1Bt`j#et, (5.1)

where GG

t is the total government spending on goods and services, transfer payments, and interest on the debt,R

tis the government revenue,Btis the funds raised by issuing new debt, a is a constant, and b"1/(1#i) where i is the interest rate. As in Hakkio and Rush (1991) we can assume that the limit term in Eq. (5.1) is zero, which is true unlessB

tgrows at a faster rate than (1#i). Then, (5.1) can be rewritten as

GG

t!Rt"yt, (5.2)

wherey

t"a#et.

For the government budget constraint not to be violated,y

tin (5.2) should be a stationary process. We test whether there occurs a change iny

tfrom stationar-ity to nonstationarstationar-ity. If we decide in favor of such a change, we can conclude that the spending and taxing policies of the government became to violate after some period in the post-war era. If this is the case, then we further estimate the date of change.

Data period is from 1947:1 through 1993:2. The test results are provided in Table 3(c).2For all three tests of maximum-Chow, mean-Chow and exponen-tial-Chow the null of no structural change is rejected at the 1% level. The estimated change period is 1968:4. We checked whether or not the budget de"cit is stationary in the two periods of 1947:1}1968:4 and 1969:1}1993:2. The result is provided in Table 3(d). For both tests of Augmented Dickey}Fuller and Phillips}Perron, the null of a unit root is rejected at the 1% level in the"rst period, 1947:1}1968:4. On the other hand, in the period of 1969:1}1993:2 the null of a unit root is not rejected at the 10% level. Therefore, we can conclude that in the postwar period the government budget policy began to violate its intertem-poral budget constraint around 1968:4.

It would be more reasonable to think that the change from stationarity to nonstationarity in y

t of (5.2) is a temporary one because corrective measures would be instituted at some point of time. However, as explained in the paragraph right after Theorem 3.3 our test is consistent under an alternative of this type of change. Therefore, if our objective is to test for the occurrence of a change from stationarity to nonstationarity, not to fully identify the nature of change, then our procedure works even in the situation when the change is a temporary one.

Acknowledgements

The author would like to thank Cheng Hsiao (the editor), two anonymous referees and seminar participants at the 1997 ASSA meetings in New Orleans.

Appendix. Mathematical proofs

Proof of Theorem 3.1. (1) The partial sum process of residuals from a regression of a process satisfying Assumption 1 on intercept converges to a Brownian bridge <_{(r) as de}"_{ned in Theorem 3.1 as was shown in Phillips and Perron}

(1988). Then, by a functional central limit theorem and the continuous mapping theorem as in Chan and Wei (1988), we have

(q¹)~2q+T for eachq3¹under the null hypothesis, wherep2₀is the long-run variance of z

tunder the null. Then, (1) of Theorem 3.1 follows._{(2) It follows from the continuous mapping theorem and continuity of the} functionals.

Proof of Theorem 3.2. The partial sum process of residuals from a regression of a process satisfying Assumption 1 on intercept and trend converges to the second-level Brownian bridge<

2as de"ned in Theorem 3.2 as was shown by

MacNeill (1978) or Schmidt and Phillips (1989). Then, the same asymptotics as in (A.1) is applicable for (q¹)~2+q₁TS2

c0,tand ((1!q)¹)~2+TqT`1S2c1,tto have the same limit processes as in (A.1) with<_{(r) replaced by}<

2(r). The rest of the proof

is the same as in Theorem 3.1.

Proof of Theorem 3.3. (i) Consider"rst the case ofq*_q

0. Under the alternative

hypothesis, sincez8

wherea3[q, 1]. Then, by a functional central limit theorem and the continuous mapping theorem as in Chan and Wei (1988), it follows that

¹~3@2S

whereW(q,q6,s) is as de"ned in Theorem 3.3. Therefore, we have (ii) Consider"rst the special case ofz

t,1being a unit root as in (i). By the result of

part (i) above, we know thatN

T(q)"O1(¹2) forq)q0since the numerator of

denominator being nonzero in probability. Then, by the de"nition of the maximum, max

q|TN

T(q)"O1(¹2) under the alternative hypothesis. Also, by the

similar reason it follows that the mean-Chow test and the exponential-Chow test are of O

1(¹2) as well under the alternative. The same result as above applies

for a more general case of nonstationarity forz t,1.

Proof of Theorem 3.4. Letz

t,1be as in Theorem 3.4. Then, by the same method

as in the proof of Theorem 3.1 we can show that

N Theorem 3.4 follows for the maximum-Chow test under H

1as in the theorem.

Proof of Corollary 3.1. Letz

t,1be as in Corollary 3.1. Then, by the same method

as in the proof of Theorem 3.4 we have

N

Proof of Theorem 3.5. We considerq( only under H

1or Hc1. Similar analysis is

applicable for the proof of consistency ofq( under H@₁or Hc@₁and is omitted. Also, we consider only the case ofz

t,1being a unit root. The analysis for the case of

z

t,1being a stationary process of di!erent persistence fromzt,0is similar and is

omitted. (1) Let ;_(q

0) be an open neighborhood of q0 and U be a local base of

q₀ containing all such neighborhoods. To prove that (q(!_q

0)"o1(1) it is

1(1) with the probability limit

of each being nonzero.

uniformly ine. First, consider the case of q"q

0#e. RewritingKT(q) as in the

t in (A.3) is the dominant term, it follows that K

Now, consider the caseq"_q

esis. Also notice that the dominant term in (A.4) is

((1!_q)¹)~2 +T

q0T`1

z82

t"((1!q)¹)~2O1(((1!q0)¹)2).

But this dominant term is less than the numerator ofK T(q0):

uniformly ine3(0,d). Notice that under Assumption 2 the denominators of (A.4) and ofK

T(q0) are asymptotically equivalent. This completes the proof. h

(2) De"ne;

To prove (A.6) it is su$cient to show that there exists a "nite real number C'0 such that, for q¹"q

0¹#Morq¹"q0¹!MforM'C,

P[K

T(q)!KT(q0)*0](g.

uniformly inM for all¹'¹

0and for eachg'0. First, consider the case of

uniformly in q¹"_q

0¹#M where +qqT0T`1z82t"(q¹!q0¹)2O1(1) under the

alternative hypothesis. We can take C big enough so that K

T(q)(KT(q0) for

q¹"_q

Now, consider the caseq¹"_q

0¹!MforM'C. Notice that in this case

K T(q)"

((1!q)¹)~2M+q0T

qTz82t#+qT0T`1z82tN

(q¹)~1+q₁Tz82

t

"((1!q)¹)~2MO1((q0!q)¹)#O1(((1!q0)¹)2)N

O

1(1)

(A.8)

uniformly in M where +T_q0T`1z82_t"O

1(((1!q0)¹)2) under the alternative

hy-pothesis. Also notice that the dominant term in (A.8) is

((1!_q)¹)~2 +T

q0T`1

z82_t"((1!_q)¹)~2O₁(((1!_q

0)¹)2).

But this dominant term is less than the numerator ofK T(q0):

((1!q)¹)~2 +T

q0T`1

z82

t(((1!q0)¹)~2

T

+

q0T`1

z82

t

since q¹"_q

0¹!M for M'C. Then, it follows that KT(q)(KT(q0) for

q¹"_q

0¹!Muniformly inM'C. This completes the proof. h

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