Cointegration Tests in the Presence of Structural Breaks

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 440 February 1993

COINTEGRATION TESTS IN THE PRESENCE OF STRUCTURAL BREAKS

Julia Campos, Neil R. Ericsson, and David F. Hendry

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.

ABSTRACT

Structural breaks in stationary time series can induce apparent unit roots in those series. Thus, using recently developed recursive Monte Carlo techniques, this paper investigates the properties of several cointegration tests when the marginal process of one of the variables in the cointegrating relationship is stationary with a structural break. The break has little effect on the tests’ size. However, tests based on estimated error correction models generally are more powerful than Engle and Granger’s two-step procedure employing the Dickey-Fuller unit root test. Discrepancies in power arise when the data generation process does not have a common factor.

Key words and phrases: cointegration, econometrics, error correction, Monte Carlo, parameter nonconstancy, power, regime shifts, size, statistical inference, structural breaks, unit roots.

the marginal process being estimated, and of the error-correction-based test with no break in the marginal process. Appendix B presents auxiliary graphical evidence on the finite sample properties of the cointegration test statistics, including means and histograms of the statistics and means of the corresponding estimated coefficients.

2. The Data Generation Process, the Test Statistics, and Some Analytical Properties

This section describes the data generation process for the Monte Carlo simulation (Section 2.1), the test statistics studied (Section 2.2), and some analytical properties of those statistics (Section 2.3). The data generation process (DGP) is a first-order bivariate vector autoregressive process with a possible structural break in one of the variables’ processes. The test statistics are Dickey-Fuller (static regression) and error correction t-ratios, and each is considered with and without knowledge of the cointegrating vector (if it exists). Together, the DGP and the test statistics delimit the scope of the analysis. Some analytical properties of the test statistics prove helpful in interpreting the Monte Carlo evidence.

2.1. The Data Generation Process

The DGP is a linear first-order vector autoregression with normal disturbances, Granger causality in only one direction, and a possible structural break in the strictly exogenous process. For expositional convenience, this DGP is written as a conditional error correction model (1) and a marginal process (2):

Ay, = aAzt bly —Az)ia te (1) Zz = px1+é6D,+u, (2) where Et 0 o2 0 _ Fe) ~ (fo) [F al) ter and

0 otherwise.

L is the lag operator, A is the first-difference operator 1 — £, T is the sample size, and Ty + 1 and 7, are the beginning and the end of the break (1 < Typ < T; < T). For Y = exp(y) and Z = exp(z), a is the short-run elasticity of Y with respect to Z, and ) is the long-run elasticity (provided b # 0). The parameter 6 is the error correction coefficient in the conditional model of y;, given lagged y and current and lagged z; and e; and u, are the disturbances in this conditional-marginal factorization.

Cointegration Tests in the Presence of Structural Breaks

Julia Campos, Neil R. Ericsson, and David F. Hendry!

1. Introduction

Structural breaks in stationary time series can induce apparent unit roots, as shown by Perron (1989) analytically and empirically and by Hendry and Neale (1991) via Monte Carlo. Consequently, tests of unit roots have low power when applied to series with structural breaks. Conversely, such series mimic series with actual unit roots. Using Monte Carlo, this paper investigates the power of several cointegration tests when the marginal process of one of the variables in the cointegrating relationship contains a structural break. The detectability of the structural break itself is also examined, both by classical constancy tests and by recently introduced tests for invariance. The data include stationary and non-stationary series with breaks. Calculation and analysis of results employ recently developed Monte Carlo techniques, as described in Hendry (1984) and Hendry and Neale (1987, 1990).

A structural break has little effect on the size of the cointegration tests studied. However, the break does affect the power of cointegration tests when the process generating the data does not have a common factor. Specifically, tests based on estimated error correction models generally are more powerful than Engle and Granger’s (1987) commonly used “two-step” procedure employing the Dickey-Fuller unit root test. The error-correction-based test uses available information more efficiently than the Dickey-Fuller test, paralleling Kremers, Ericsson, and Dolado’s (1992) results without a structural break.

Section 2 describes the data generation process for the Monte Carlo study, the test statistics considered, and some of their analytical properties. Section 3 presents the experimental design and simulation techniques. Section 4 examines the detectability of stationarity and of the break in the marginal process. Section 5 interprets the Monte Carlo results on the cointegration tests. Section 6 concludes. Appendix A derives asymptotic properties of the unit root estimator in the presence of a break in

'The first and third authors are professors at the Universidad Auténoma de Barcelona, Barcelona, Spain, and at Nuffield College, Oxford, England respectively; the second author is a staff economist in the Division of International Finance, Federal Reserve Board, Washington, D.C., U.S.A. This paper was prepared for the conference “Recent Developments in the Econometrics of Structural Change” , C.R.D.E., University of Montreal, Montreal, Canada, October 2-3, 1992. The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as reflecting those of the Board of Governors of the Federal Reserve System or other members of its staff. We are grateful to Peter Phillips and Carmela Quintos for helpful comments, and to the U.K. Economic and Social Research Council for financial support to the third author under grant R000231184. All numerical results were obtained using PCNAIVE Version 6.01 and PcGive Version 7.00; cf. Hendry and Neale (1990) and Doornik and Hendry (1992).

case. 5

Before describing the test statistics in detail, it should be emphasized that either Case I or Case IV may characterize empirical time series, and it may be extremely difficult to distinguish between the two cases in practice. To illustrate, consider real per capita expenditure on non-durables and services (CN) and real per capita disposable income (J NC) in Venezuela, graphed in logs in Figure la as cn and inc. Initially, both series grow smoothly. Then, income jumps by over 30% in 1974 from increased petroleum revenues, remains relatively constant through 1981, and during the LDC debt crisis of the early 1980s falls to approximately its 1973 level. Expenditure parallels or lags behind income through 1981, but falls only slightly during 1981-1985. Campos and Ericsson (1988) find that each series appears I(1) empirically, and that expenditure and income are cointegrated, provided that inflation and liquidity effects are properly accounted for.

Still, the expenditure and income series in Figure la easily could have been generated by a stationary process with breaks in 1974 and 1981. Figure 1b plots the series y; and z, generated by (1)-(4) under Case IV,” and these series’ resemblance to cn and inc in Figure la is striking. Yet, the power of a Dickey-Fuller test to detect stationarity in such z; is less than its size; and the power of the Chow test to detect the break in z; is about 10% (at a nominal 5% level), even when the break point is known. Such similarities in time series motivate the Monte Carlo study below.

2.2. The Test Statistics

This subsection describes the test statistic for a unit root in the marginal process (2) and the test statistics for the cointegration of y, and z.

2.2.1. Marginal Processes

Frequently, investigators pre-test for unit roots in univariate autoregressive representations. Many tests exist: see Dickey and Fuller (1979, 1981), Phillips and Perron (1988), and Banerjee, Dolado, Galbraith, and Hendry (1993) inter alia. While only the Dickey-Fuller t-statistic (denoted tp) is considered in this paper, some of its properties are generic to unit root statistics. The asymptotic distribution of tpr is well-known for difference-stationary processes (such as Case I), but is shown in (15) to differ when a break occurs.

2.2.2. Cointegrated Processes

Using the dynamic bivariate process (1)—(4), this paper focuses on the relative merits of the two-step Engle-Granger and single-step dynamic-model procedures for testing for the existence of cointegration. See Engle and Granger (1987) on the former and Banerjee, Dolado, Hendry, and Smith (1986) inter alia on the latter. The former is characterized by a Dickey-Fuller (DF) statistic used to test for the existence of

Specifically, a= 0, b= —0.1, p= 0.8, 6 =1, 0. = oy = 1, T = 100, Tp = 25, and T; = 75.

By a suitable scaling of z; (and without loss of generality), 4 = 1: that is, the cointegrating vector for (y; : z:)’ is (1 : —1) if y, and z are cointegrated. Knowing that \ = 1 and imposing 4 at that value for estimation is with loss of generality, so Section 2.2 considers tests that utilize and that ignore this information.

The parameter space is restricted to {0 <a <1, —1 <6 <0}. In many empirical studies, a © 0.5 and b & —0.1, with 02 > o?. That is, the short-run elasticity (a) is smaller than the long-run elasticity (unity), adjustment to remaining disequilibria is slow, and the innovation error variance for the regressor process is larger than that of the conditional process. Also, z; is assumed weakly exogenous for the parameters in the conditional model (1); see Engle, Hendry, and Richard (1983) and Johansen (1992a). Section 3.1 gives the precise experimental design.

Four types of process for z; can arise from (2), and they are denoted Cases I-IV.

Case I: z has a unit root (p = 1) but no break (6 = 0). The variables y,; and z, are integrated of order one [denoted I(1)] and are cointegrated if —1 < 6 < 0. Banerjee, Dolado, Hendry, and Smith (1986) and Kremers, Ericsson, and Dolado (1992) inter alia analyze the properties of various cointegration estimators and test statistics under this DGP, both analytically and by Monte Carlo.

Case II: z is stationary (|p| < 1) with no break (6 = 0). Then, y is I(1) if b = 0, and y% and z are jointly stationary with an error correction representation if —1 < 6 <0. Davidson, Hendry, Srba, and Yeo (1978) and Davidson and Hendry (1981) provide some asymptotic and Monte Carlo evidence on the properties of the error correction test statistic.

Case III: z is I(1) (9 = 1) and has a break (6 # 0). If the break is large enough, z, May appear to be an I(2) process. While of potential empirical interest for nominal variables, this paper only briefly considers Case III, in Section 4. However, see Johansen (1992b, 1992c) for theoretical and empirical discussions.

Case IV: z, is stationary (|p| < 1) and has a break (6 # 0). This case is the primary focus of this paper. From extensive Monte Carlo evidence in Hendry and Neale (1991), z; may well appear to have a unit root when standard unit root tests are applied, and the break may be difficult to detect (see also Section 4 below). With z, behaving like a unit root process with no break (Case I), we conjectured that cointegration tests involving such a z, process would behave as in Case I. For the most part, this conjecture appears correct. However, the common factor restriction in Engle-Granger tests of cointegration plays an even larger role than anticipated, as shown both analytically and in the Monte Carlo (Section 5).

As implied by Kremers, Ericsson, and Dolado (1992, Section 5), the logical issues arising from common factor restrictions apply to processes more general than (1)-(4). Specifically, the cointegrating vector or vectors may enter more than one equation (i.e., no weak exogeneity); and a constant term, seasonal dummies, additional variables, and additional lags may be included. Some statistics’ distributions are more complicated with such generalizations, so this paper focuses on the bivariate

a unit root in the residuals of a static cointegrating regression. The latter is based upon the t-ratio of the coefficient on the error correction term in a dynamic model reparameterized as an error correction mechanism (ECM), noting that cointegration implies and is implied by an ECM. This t-ratio is denoted the ECM statistic. Each statistic may utilize or ignore knowledge about the value of the cointegrating vector (i.e., that A = 1).2 This subsection describes these four test statistics and clarifies analytical relationships between the test statistics; Section 2.3 presents some of their asymptotic properties.

The variables y; and z; are cointegrated or not, depending upon whether b < 0 or b= 0. Thus, tests of cointegration rely upon some estimate of 6. The four statistics considered here are all t-ratios on regression estimates of b. Let us denote those tratios by tecmt, tecMu, torr, and tpry, where the subscripts ECM and DF denote error correction model and Dickey-Fuller, and k and u indicate that the cointegrating vector is assumed known or unknown. These t-ratios on b are derived from the regressions:

Ay: = «taAzn+ b(y _ z)t-1 + Ext; (5) Ay: = K«ktaAyt+ b(y _ Z)t-1 + cz-1 + Eut, (6) Yt = Ko + 2 + Wke A(y—ko—Zz)t = Kit bly — Ko — Z)t-1 + Cnt, (7) and Yt . = Ko + A2¢ + Wut

respectively, where k, Ko, 1, and c are constants; and a tilde denotes an estimate from the static regression in (8). The regression errors for tgcmx and tgcmu are Ext and €,; respectively. Those for tpr, and tp, are ex, and eyz, with wy, and wy, being the associated static-regression residuals on which tpr, and tpr, are based.

The statistics tpr, and tecme are used to test the null hypothesis that b = 0 in (1), ie., that y and z are not cointegrated with a cointegrating vector (1 : —1). The statistics tpr, and tgcomu ignore that A = 1, and so are (implicitly or explicitly) used to test the null hypothesis that y and z are not cointegrated with an arbitrary cointegrating vector (1 : —A).

Both asymptotic and finite sample properties of the statistics can be better understood by examining the relationship between the DF regression equation (7) and the ECM regression equation (5). To do so, subtract Az; from both sides of the conditional DGP (1) and rearrange:

3A priori knowledge of the cointegrating vector frequently arises in economic modeling: for instance, of (logs of) consumers’ expenditure and disposable income, of wages and prices, of money and income, and of the exchange rate and foreign and domestic price levels.

1978 igsv7s 1986 i985 year

Figure la. The logarithms of real per capita consumers’ expenditure on nondurables and services (cn) and of real per capita disposable income (z7c).

i6

12 ? 5

L”. J 2a ae T 6a 8a 1806

Figure lb. A typical pair of series (y, and z;) from the cointegrated process with a break but without a common factor.

4a

Under Case III, z; has a unit root (o = 1) and a break (6 # 0). Suppose an investigator, unaware of the break, estimates

Am = pt¢uyt+ és (14)

in order to test for a unit root (¢ = 0). The probability limit of the least squares estimator (fi: $) of (u : ¢) is: fi 6K(M + K/6) plim = 4H" ; (15) Té 1—(K +2M)

where K is the length of the break (T, — To)/T, M is the time after the break (1 —1,)/T, and H = (1-M-— K)M + K(4 —3K)/12. As found in additional Monte Carlo simulations not reported below, the estimated means of ji and ¢ appear to match closely the analytical values in ( 15).

Three implications of (15) are of interest. First, the estimated intercept has a nonzero population value, which is proportional to 6. Second, the unit root estimator ¢+1 differs from unity by only O,(T~!). Third, the corresponding discrepancy does not depend on 6 (to 0,(T~!)) and is relatively negligible, especially when (K + 2M) is close to unity.

2.3.2. Cointegrated Processes

Asymptotic distributions of all four cointegration test statistics are known for Case I (z; with a unit root and no break), providing a baseline for the Monte Carlo study. Distributions under Cases II-IV are viewed as variants, with Case IV being the particular focus of Section 5. Because (1)-(4) has a unit root under Case I, distributional results involve Wiener processes. That said, trome, is approximately normally distributed for large q (when a # 1). Derivation of the asymptotic distributions appear in Dickey and Fuller (1979, 1981) and Phillips (1987b, 1988) (for tox); Banerjee, Dolado, Hendry, and Smith (1986) and Kremers, Kricsson, and Dolado (1992) (for tec); Engle and Granger (1987) and Phillips and Ouliaris (1990) (for tpru); and Park and Phillips (1988, 1989), Kiviet and Phillips (1992), Banerjee and Hendry (1992), Boswijk (1992), Banerjee, Dolado, Galbraith, and Hendry (1993), and Appendix A herein (for tECMu)-

For expositional convenience, we adopt certain notational conventions concerning Brownian motion (or Wiener) processes. Consider a normal, independently and identically distributed variable 1,,t = l,...,T: that is, m ~ 1N(0, oa). Here, m is usually either e;, €:, or wu. Define By»(r) as the partial sum ae ne/(To2)'/*, where r lies in [0,1], and [Tr] is the integer part of Tr. As discussed in Phillips (1987b), Br,,(r) converges weakly to a standardized Wiener process, denoted B,(r). For simplicity of notation, the argument r is suppressed, as is the range of integration over r when that range is [0,1]. Thus, integrals such as So By(r)?dr are written as J B?. The symbol

Aty—z)t = Wy—z)a+[(a-1)Ax +e] = Wy-z)ate, (9)

where the disturbance €_ IS:

Equation (7) is (9) with eg; = e, and an explicit constant term. The statistic tp, ignores potential information contained in Az. Equivalently, (7) imposes the restriction that the short- and long-run elasticities are equal (a = 1), or that there is a common factor in the relationship between y; and z;. A useful measure of the ignored information is:

q = —(a-1)s, (11) where s = o,,/0,. That is, g? is the variance of (a —1)Az, relative to that of €;. Also, q’ = R?/(1 — R?), where R? is the population R? for Awz: (or e; in (10)) regressed on Az, when b = 0. The value of q directly affects the distribution of tecmk; see Section 2.3. See Kremers, Ericsson, and Dolado (1992) for details on the restriction a = 1, and Hendry and Mizon (1978) and Sargan (1964, 1980) on common factors.

The relationship between (8) and (6) (the regressions with unknown A) parallels that between (7) and (5). Equation (1) can be transformed to:

which is (8) where the disturbance e,, is:

and the constant in (12) is implicit. Thus, tpp,, is affected not only by the common factor restriction, but also by the discrepancy between estimated and actual values of the cointegrating vector. While “super consistent”, the static-regression estimate \ may have poor finite sample properties, thereby affecting the properties of tp;,,; see Banerjee, Dolado, Hendry, and Smith (1986).

2.3. Properties of the Test Statistics

This subsection describes properties of the test statistic for a unit root in the marginal process and properties of the cointegration test. statistics.

2.3.1. Marginal Processes

Dickey and Fuller (1979, 1981), Phillips (1987b, 1988), and Banerjee, Dolado, Galbraith, and Hendry (1993) derive the asymptotic distributions of tpr under Cases I and IJ; and they are identical to the distributions of torr, as described in Section 2.3.2 below. Appendix A derives properties of tpr under Case III, providing a baseline for interpreting the Monte Carlo results, which are for Case IV.

conditional on the process for u;. Under the null hypothesis, (22) simplifies to: tecmte => N(0, 1)+ O,(q7'). (23)

The approximation in q is “small-o” in nature; cf. Kadane (1970, 1971). Thus, as q varies from small to large, the asymptotic distribution of tecmr shifts from the DF distribution to the normal distribution.

The asymptotic powers of tpr, and tgcm, are determined by (18) and (21). When q = 0, the two tests have the same power. When q is sufficiently large, tecmk has (arbitrarily) greater power than tpr,. That discrepancy arises because tprz is ignoring substantial information on Az, whereas tecom, uses that information to obtain a more precise estimate of b.

The asymptotic distributions of tp, and tgcy, resemble those of tprz and teomk, but are somewhat more complicated because the hypothesized cointegrating vector is estimated. Phillips and Ouliaris (1990) derive the asymptotic distribution of tpr, under the null hypothesis that 6 = 0. Appendix A derives the (null) asymptotic distribution of tgc¢uu, which is:

J B.dB, — (f BeBu)(f B2)(f BudB) f B?-(f B.B,)?(f B2)—

Because the asymptotic distribution of tgcy, in (24) does not depend on a, oy, or de,

tecMu => (24)

it is invariant to those parameters, and hence tests based on tgcy, are similar. Kiviet

and Phillips (1992), Banerjee and Hendry (1992), and Banerjee, Dolado, Galbraith,

and Hendry (1993) derive similarity without obtaining the asymptotic distribution of

tecMu. Equation (24) is implicit in derivations by Park and Phillips (1988, 1989),

and it is obtained in a different but isomorphic representation by Boswijk (1992). An alternative expression for (24) is enlightening. Consider the regression

Yo = Batee (25)

“linking” the levels of y; and z;. Let R denote the limiting form of Yule’s correlation for that regression when a = 6 = 0 (i.e., under the null of no relation between y, and zt) and that correlation is not adjusted for sample means. From Phillips (1986), R is: = —LBBy (26) (f Bz)(S B?)

Thus, schematically (24) is: trom. = DF —R- N(0,1) where DF and N(0,1) denote random variables with Dickey-Fuller and standardized normal distributions. This demonstrates that tgcy, and the Dickey-Fuller statistic

(27)

have different limiting distributions, so their critical values need separate tabulation.

“ =» ” denotes weak convergence of the associated probability measures as the sample size T’ — oo. See Billingsley (1968) and Banerjee, Dolado, Galbraith, and Hendry (1993) for further discussion. Mann and Wald’s (1943) order notation is used where needed.

The null hypothesis is no cointegration (6 = 0). The alternative hypothesis is cointegration (b < 0), and is characterized as a local alternative with:

b = eVT_1 & 4Y/T, (16)

where 7 is a negative fixed scalar. Equation (16) parallels the usual Pitman-type local alternative except that, in order to obtain statistics of O,(1), 6 differs from the null by O,(7'~') rather than by O,(T~!/?). Conveniently, distributions under the null hypothesis are obtained by setting y = 0. The generalization of B,(r) under this local alternative is the diffusion process: K,(r) = fo eB, (3)

= Bir) +7S5 eB, (3) dj, (17) where K,,(r) is an implicit function of +; see Phillips (1987b). If y = 0, then K,,(r) = B,(r). As with B,, the argument r in K,(r) and the limits of integration are dropped if no ambiguity arises from doing so.

Under the local alternative, tpr, is distributed as:

K.dB 2)1/2 J KedBe torr => Y(f K2) + [K? ’

which simplifies to the Dickey-Fuller distribution: Jf B.dB.

JIB (19)

Under the local alternative, the ECM statistic tgcoyp is distributed as: teome => YL + @°)?(f KZ)? (a—1)f K,dB, +57! f K.dB, (a— 1)? f K2 4 2(a—l)s' f KK. + sf K2 When a = 1, (20) simplifies to the DF distributions (18) (for 7 # 0) and (19) (for y = 0).

For a # 1, (20) can be reparameterized in terms of 7 and q exclusively: f Ku.dB.-—q"' f K.-dB. (21) For large q, (21) is approximately a standardized normal distribution:

tecmMk => N (o(1 +P )Pf K2)1/?,1) + O,(q7"), (22)

tprFk =>

under the null hypothesis.

(20)

teome > Y(14+q?)/2(f 2) 4

Critical values are all at the 5% level. For tpr, and tpry, the values are calculated from MacKinnon’s (1991, Table 1) response surfaces with N = 1 and N = 2 respectively (MacKinnon’s “N”), “with a constant but no trend” for both. The correct critical values for tgcm~ depend upon q as well as T, and those for tgomu depend upon T. Both sets of critical values could be simulated. However, g may not be known in practice, and even asymptotic critical values for tgcmu have not yet been simulated, so “safe” critical values may be constructed by assuming that tecme and tecMy have pure Dickey-Fuller-type distributions (i.e, ¢ = 0). Thus, the critical values for tecmz and tgcmy used here are the same as those for tpr,; and tpru.

Because qg is such an important parameter and because q < | in (28) is relatively small by empirical standards, the second set of experiments considers the effect of q = 3, albeit in a more limited design:

a = 0.0 [no common factor: q = |

b = (0.0 {no cointegration], —0.1 [cointegration])

s = 3.0

p = 0.8 [stationarity]

§ = 3.0 [a break of size s]

T = 10,11,12,...,98,99, 100

To = 25

T, = 75, (29)

resulting in 182 experiments. The values of s and 6 are equal in order to keep the time series properties of z the same as in (28).

3.2. Simulation

Simulation proceeded as follows. Random numbers for €; and u; were generated by multiplicative congruential generators and transformed to a normal distribution by Box and Muller’s (1958) method. The first twenty observations of each replication from (1)—(4) were discarded in order to attenuate the effects of initial values in stationary relations (such as in y; — z, when 6 < 0). For a particular experiment, P replications were generated, with a statistic lying in its critical region 5 of P times ( dependent upon the statistic). The fraction of rejections S/P is an unbiased Monte Carlo estimate of the underlying rejection frequency (e.g., of size or power).

Recursive algorithms exist for the statistics tprz, tecmk, and teomu, providing a computationally efficient means for their calculation over the full range of sample sizes T = 10,11,...,99,100 for any given set of values of the other experimental design variables.4 Thus, a replication of size T = 100 was generated, and the statistics were

calculated recursively on that sample for all sample sizes. Such re-use of the sam-

4The statistic tp ry, cannot be calculated recursively, so its properties are considered for T = 100 only. Also, D; was perturbed by a small error in order to permit recursive estimation for T < 25 of equations including D;.

ll

3. Experimental Design and Simulation

This section presents the experimental design and Monte Carlo simulation of the cointegration test statistics.

3.1. Experimental Design

To analyze the size and power of the cointegration tests in the presence of a structural break, two sets of Monte Carlo experiments were conducted with (1)-(4) as the DGP. The first set is a “broad” design, aimed at highlighting the effects of common factors, cointegration, and breaks over a range of sample sizes. The second set focuses on how the lack of a common factor affects the cointegration tests. Without loss of generality, 7? = 1 and A = 1. Thus, the experimental design variables are the parameters (a, b,s,p,6), the sample size T, and the break points To and T), noting that s now is dy.

The first set of experiments is a full factorial design of:

a = (1.0 [a common factor: q = 0], 0.0 [no common factor: q = s])

b = (0.0 [no cointegration], —0.1 [cointegration])

s = 1.

p = (1.0 [integration], 0.8 [stationarity])

6 = (0.0 [no break], 1.0 [a break of size s])

T = 10,11,12,...,98,99, 100

To = 25

T, = 75, (28)

resulting in 1456 experiments. For both sets of experiments, new z’s were generated for each replication, and the number of replications per experiment was P = 10,000.

The parameter values were chosen with the following in mind. For a = 1 (and so q = 0), the common factor restriction holds, so the distributions of the DF and ECM statistics should resemble each other. For a = 0, the common factor restriction is violated, but ¢ = s = 1, which is a “moderately small” value. The two values of b, 0.0 and —0.1, imply lack of and existence of cointegration respectively, although, in the latter case, the corresponding root of the system is still large: 0.9. The root of the marginal process (p) is either unity or large but stationary. The break in the marginal process is either zero or unity (i.e., 1-o,), where the latter value is rather small by empirical standards. However, for the purposes of this paper, unity seemed appropriate because it is small enough to make its detection by standard Chow tests difficult.

The sample size includes all values in {10, 100], providing small, medium, and large values. To ensure that most sample sizes included a break, To = 25, with T) = 75 so as to maximize the power of constancy tests over the full sample; see Hendry and Neale (1991). For a given length of break (7, — To)/T’, the particular choice of To and T, matters little for the power of the full-sample constancy tests.

10

T 60@ 7@ 8oe 90 1860

Figure 2a. Estimated rejection frequencies of tpr for the marginal process (Cases

I-IV).

I IIL __e | © © Gee Iv

so T 6e 7o@ so sea 100

Figure 2b. Estimated rejection frequencies of the split-sample Foyow (N |) for the marginal process (Cases I-IV).

12a

ple greatly reduces Monte Carlo variation for different values of T; see Hammersley and Handscomb (1964). Further, calculation of all test statistics on the same sample reduces Monte Carlo variation for the differences in properties across statistics. Graphical (rather than tabular) analysis of the Monte Carlo rejection frequencies is highly desirable, given the large number of experiments; cf. Ericsson (1991). Graphical analysis also corresponds to a (pseudo-) nonparametric estimation of the size and power functions of the tests.

4. Post-simulation Analysis: the Marginal Process

This section briefly examines one unit-root test statistic and two constancy test statistics on the marginal process for z;. The unit root statistic is the Dickey- Fuller statistic with a constant term (tp), which is the ¢-ratio on ¢ in (14), noting that the break dummy D, is explicitly excluded. The first constancy statistic is Chow’s (1960) predictive failure statistic applied to (14) and is denoted Foyow. The second constancy statistic is the t-ratio on the least squares estimate of 6 in the correctly specified marginal process (albeit with a constant term estimated):

and is denoted ts. This last statistic is also a statistic for testing the invariance of (14) to D,, as discussed in Engle and Hendry (1993). Critical values are at the 5% level and are taken from MacKinnon (1991) for tp, the F distribution for Fouow, and the t distribution for ts.

Figures 2a, 2b, and 2c plot the estimated rejection frequencies of tpr, Foxow, and ts respectively. The four lines on each graph correspond to Case I (p = 1, 6 = 0: —), Case II (p = 0.8, 6 = 0: — -), Case II] (p = 1, 6 = 1: «++: ), and Case IV (p = 0.8, 6=1: ---).

From Figure 2a, the estimated size of tpy (Case I) is close to 5% for all sample sizes. When p = 0.8 without a break (Case II), the power increases monotonically from about 7% (at T = 10) to 90% (at T = 100). When a break is added to that stationary series (Case IV), the power falls from 13% at T’ = 25 to less than 1% by I’ = 40, increasing to only around 23% by T = 100. As examined in greater detail by Hendry and Neale (1991), even small breaks can dramatically reduce the power of unit root tests. The size is also affected by breaks, noting that the rejection frequency for Case III varies between 0% and 25%, depending upon what fraction of the sample includes the break.

From Figure 2b, the estimated size of Foyow (Cases I and IT) is about 7-9% for small samples, tending to its nominal 5% value by T = 100. For stationary z, with a break, the power is never higher than 12%, even for a sample split at T’ = 25 where the first break occurs. The power is somewhat higher for non-stationary z;, but still never exceeds 30%. In essence, a break of one standard deviation over half the sample is small and hard to detect, in spite of its consequences on the unit root test.

12

The statistic ts should provide a highly powerful test, given that the dates and nature of the break are treated as known. However, controlling its size is problematic, as Figure 2c documents. Intuitively, D; behaves like an integrated process and so the distribution of ts is affected by the correlation between z,_; and D; in (30). This effect is lessened but not eliminated when z; has a stationary root. The “power” of ts appears impressive, but must be treated with caution, given the large distortion to size.

To summarize, the break in the marginal process is difficult to detect with the Chow statistic, yet it dramatically reduces the power of the Dickey-Fuller statistic for detecting a stationary root. A stationary process with a break is virtually observationally equivalent to a unit root process with no break.®

5. Post-simulation Analysis: Tests of Cointegration

Figures 3 and 4 plot rejection frequencies by the four cointegration tests for the first set of experiments. These rejection frequencies are under the hypotheses of no cointegration (Figure 3) and cointegration (Figure 4), and correspond to size and power, provided the correct critical values are used. Figures 3a-3d plot estimated sizes for (a=1, 6=0), (a=1, 6=1), (a =0, 6 =0), and (a = 0, 6 = 1) respectively: that is, for DGPs with and without a common factor in the conditional process and with and without a break in the marginal process. Figures 4a—4d present the corresponding plots for estimated powers. The primary interest here is in discerning the differences between Cases I and IV, so p = 1 when 6 = 0 and p = 0.8 when 6 = 1. Appendix B documents other finite sample distributional aspects of the test statistics and their associated estimators.

From Figures 3a and 3b, the estimated sizes for tgcm, and tpr, are both approximately 5%, which follows from the asymptotic equivalence of the two statistics when there is a common factor (a = 1). The size of tgcmu is around 3% because of the conservative choice of using MacKinnon’s critical values. All three sizes are virtually unaffected by the sample size T, confirming the accuracy of MacKinnon’s response surfaces for the critical values. The estimated size of tpr, is available at only T = 100, and is approximately 5%.

Invalidity of the common factor restriction (Figures 3c and 3d) clearly affects tecmr and tpr,. As anticipated from the asymptotics with no break, the rejection frequencies for tgcmr are below 5% (typically, between 2% and 4%), while those for tecm.z are unchanged (at 3%) from simulations with a common factor. The rejection frequency of tpr,; is about 5% in Figure 3c, in line with its invariance to the existence or lack of a common factor when there is no break; cf. Kremers, Ericsson, and Dolado ~~ 5Note, however, that if the root of 2 were treated as known, the distribution of ts; would be exactly a t.

SFaust (1992) formally establishes the near observational equivalence of trend-stationary and difference-stationary processes. His framework also may help establish a parallel result here.

13

10e0

9a

7a

60

40

30a

10

20a 30a 4@ soe T 60 7a 8a 90 ieee

Figure 2c. Estimated rejection frequencies of the invariance statistic ts for the marginal process (Cases I-IV).

12b

torx trcmu — -- tporu*

tecmxk

20 30 40 3o@ T 6@ 7@ soa soa 100

Figure 3c. Estimated sizes of four test statistics (tprr, tpru, tecmk, tecMu)- The experiments have: no common factor, no break, and no cointegration.

tecmx torK 2 tecmu ——- toru*

2@ 30a ae 3a T 6a 7@ soe 9o i1e0

Figure 3d. Estimated sizes of four test statistics (tprr, tpru, tecmk, tecMu)- The experiments have: no common factor, a break, and no cointegration.

13b

tremu - — - toru*

trom

2oe 30 4c so T 60 7oa 8a 9a 166

Figure 3a. Estimated sizes of four test statistics (tor, tpru, tecmk, tecMu)- The experiments have: a common factor, no break, and no cointegration.

tyre treemu — _— toru*

tecmx

2a 30 ao EZ °. T 60 70 soa 9o 100

Figure 3b. Estimated sizes of four test statistics (torr, tpFu, tecmk, tecMu)- The experiments have: a common factor, a break, and no cointegration.

13a

tecmx

tyorx —-- tromu - - - toru*

Figure 4c. Estimated powers of four test statistics (tors, tpru, tecmk, tecMu)- The experiments have: no common factor, no break, and cointegration.

tecmx torn tecmu - -- tpru*

180

7e@

6G

x

42ce

3a

2e 30 aoe 3a T 60 7@ soa se 100

Figure 4d. Estimated powers of four test statistics (tore, tpru, tecmk, tecMu)- The experiments have: no common factor, a break, and cointegration.

13d

trcee ——_ LSS trcmu - - - tora”

20 30 4G soa T 606 7e@ sa 9oe 160

Figure 4a. Estimated powers of four test statistics (torr, toru, tecmk, tecMu)- The experiments have: a common factor, no break, and cointegration.

tecmx

tork tremu - -- toru* 100

9oe

7oe

4a 20 20

10

2e@ 30a ao 35o T 60 70 8a 9a 1060

Figure 4b. Estimated powers of four test statistics (tprx, tpru, tecmk, tecMu)- The experiments have: a common factor, a break, and cointegration.

13c

tork- seseneessneseveeese teemu --—-— toru*

tecemx

2@ 3a 42a 3s@ T 6o 7@ 8o 9a 108080

Figure 5a. Estimated sizes of four test statistics (tprz, tpru, tecmk, tecMu). The experiments have: no common factor, a large break (6 = 3), and no cointegration.

tecomx

EME ooo tromu - - - toru* 109

90a sea 7oe

6a

30 zoe

106

z2e 30 aa so T 60 7o so oa iee

Figure 5b. Estimated powers of four test statistics (tprk, tpru, tecmk, tecMu). The experiments have: no common factor, a large break (6 = 3), and cointegration.

l4a

(1992). However, its rejection frequency is not invariant to the lack of a common factor when there is a break. The residual in the estimated equation for tpr, involves Az, which includes D; and a stationary error: see (9). The average size for tp, is about 63%, which is substantially higher than the sizes for tecy, and tecm,. The lack of invariance of tpr,% to a break when there is no common factor is even more apparent for powers and for larger qg, as seen below.

In Figures 4a and 4b, the DGP has a common factor, and y; and 2; are cointegrated. As under the null of no cointegration, the presence or lack of a break has no effect on the test statistics; and the estimated powers for tp, and tgom, are virtually identical, ranging from 5% at T = 10 to 33% at T = 100. The rejection frequency of tecMu is somewhat less, and unsurprisingly so because its rejection frequency under the null is less than 5% and because it ignores \ being unity.

When there is no common factor and no break (Figure 4c), the power of tecu, substantially dominates that of tpr,, with the former increasing to 70% by T = 100. Even tgcmz does better than tpr, at moderate to large samples, in spite of ignoring the value of the cointegrating vector. In fact, the power of tp; is invariant across Figures 4a, 4b, and 4c. By contrast, the powers of tecyy, and tgqmy, increase when the common factor is invalid, as predicted by theory.

With a break but no common factor (Figure 4d), the power of tgcm, exceeds that of tp, except for very small samples, where both powers appear approximately equal to size. Discrepancies between their powers at larger sample sizes appear smaller than without a break, but this is probably a spurious result due to inadequate control of the rejection frequency of tpr, under the null hypothesis (see Figure 3d).

Figures 5a and 5b plot estimated sizes and powers for the second set of experiments. The DGP has no common factor and a large break (6 = 3), so these figures are qualitatively similar to Figures 3d and 4d, but effects of the break are more pronounced from having a larger g. The rejection frequency of tecyx in Figure 5a is even smaller than in Figure 3d, as predicted by the asymptotic distribution of tec: shifting towards a normal distribution as q increases. Rejection frequencies for tor, resemble those in Figure 3d, but with a larger range, 2-11%. The distribution of tecmy still appears invariant to the break.

The powers for tec, and tgcmy in Figure 5b increase more rapidly with T than their powers in Figure 4d because of the greater information content in Az. The “power” of tprx has more pronounced dips after the breaks occur than in Figure 4d, and is somewhat inflated because its rejection frequency under the null hypothesis is inadequately controlled. Even so, the power of tec, dominates that of tp, for all sample sizes, as does the power of tgcy., for T > 40. By contrast, the power of tpry is less than its size at T = 100.

While this Monte Carlo study is limited by a relatively small experimental design for a, s, and b, both asymptotic theory and these simulations point to the advantages of the ECM statistics for empirically common values of a and s. Control of size

14

breaks anywhere in the DGP, properly accounting for the dynamic relationship between variables can be critical in testing for a long-run relationship between them.

Appendix A. Analytical Results

This Appendix derives asymptotic properties of the unit root estimator in the presence of a break in the marginal process being estimated (Section I) and of tgcmy with no break in the marginal process (Section II). The asymptotic distribution of tprr With no break was solved by Dickey and Fuller (1979), and the distribution of tecMu With a break is the focus of the Monte Carlo study in the body of the paper.

I. Breaks and the Distribution of the Unit Root Estimator Consider the DGP for z; in (2) under Case III:

where 2 = 0. Let K, L, and M be the length of the break (T; — To)/T, the time

until the end of the break T;/T, and the time after the break 1 — L respectively, all

relative to the time period T.. An investigator, unaware of the break, estimates

Az = pt+ouit& (A2)

in order to test for a unit root (¢ = 0). This section derives large sample properties of:

T -1 T ji T 2-1 > Az 1 1 = ; (A3) d T T 2 T 2 t-1 274-1 2 2-1 Aer

the least squares estimator of (u : ¢) in (A2), when (A1) holds for fixed nonzero 6 and kK. Here and below, summations are over t unless otherwise indicated.

Evaluation of the four different summations in (A3) is required. Without loss of generality, set o? = 1, so 6 is measured in standard deviations of u;. Also, all summations utilize an explicit representation for z;:

t j=1 t t j=1 j=l = 6th, (A4)

appears relatively straightforward for the ECM statistics in the presence of breaks. Tests with tgomu are insensitive to breaks under the null hypothesis, and MacKinnon’s Dickey-Fuller critical values provide a “safe” choice for treomr and tgeomy. Further, the power of the ECM statistics commonly exceeds that of DF statistics for empirically interesting parameter values.

Recursive algorithms helped reduce Monte Carlo imprecision across statistics and across (econometric) sample sizes, with graphical analysis providing a clear, simple summary of a vast array of estimated sizes and powers. Recursive procedures are also appealing empirically. Because statistics are affected by the accrual of information over time, full-sample and partia]-sample inferences may differ, especially with breaks. Recursive estimation and testing offer a window on those effects.

6. Summary and Remarks

Testing for cointegration has become an important facet of the empirical analysis of economic time series. Various tests have been proposed and widely applied, but most distributional results rest on the assumption of unit root processes with no structural breaks. Even so, regime shifts and structural breaks are empirically and economically plausible, as indicated by extensive discussion of the Lucas (1976) critique. Using Monte Carlo methodology, this paper examines the finite sample properties of four common tests of cointegration in the presence of a structural break.

When conditioning is valid, Dickey-Fuller statistics used to test for cointegration have no particular advantage over their ECM counterparts; and there is much to gain from using the latter when the common factor restriction is invalid, and especially so if a break occurs as well. These differences arise because the DF statistic ignores potentially valuable information by imposing a possibly invalid common factor restriction. Because common factor restrictions are generic to univariate-based tests of cointegration, these results should hold for the augmented Dickey-Fuller statistic, Sargan and Bhargava’s (1983) statistic, Phillips’s (1987a) Z, and Z;, statistics, and generalizations thereon by Phillips and Perron (1988) and Gregory and Hansen (1992). The problem is in using these univariate-based statistics to test a multivariate hypothesis, and not in the statistics themselves.

Conversely, maximum likelihood procedures such as those developed by Johansen (1988, 1991, 1992a), Johansen and Juselius (1990), Phillips (1991), and Boswijk (1992) do not impose common factor restrictions and so can have more desirable properties. Some caveats apply in practice. First, systems procedures may require modeling the break itself, and that may be difficult. Second, while conditional modeling is often simpler than dealing with complete systems, the assumed weak exogeneity may be invalid, implying trade-offs between conditional and systems modeling. Third, even in conditional models, general dynamics may not be sufficient to account for breaks. If breaks occur in the cointegrating vector itself, the Lagrange multiplier statistic of Quintos and Phillips (1992) may help detect them. With or without

15

2 = PPP 4K +26 LA 2; Lli + Uke + de dhe

It

a, 2 T T T T Ply P+ V+ DAZ +26, Lae + TK Y hi] To+1 T4+1 1 To+1 ™%4+1

lI

TK | . T TK | TM ely s + T?K?TM] + dhe + 2620 jhte+i +TK Pz Arti) JF J= j=

= 1973K?[K +3M] O,(T*)

TK _TM ; + 26[ 20 Jhte+i +TK > hr, 45] + O,(T ) (A9) jJ= jJ= O,(T*/?)

Because z# is O,(T?), the summation 77 z?_, is (A9), to O,(T?). Noting (Al), the summation DT z_,Az; is obtained by evaluating each of the summations y Z4-1U, and > 21-1D:

T T T 2s 2t-1U1 = BY Teale + Yo eae

TK | TM T = AL junes + TK Lunt) + Eheim + O,(T¥?), (A10) j= j= O,(T*/?) 0,(T)

and T T T 2 21-1 = bx T,-4D_+ 2 he De TK TK = 6 >) ht+j;-1 + O,(T) j= j= TK = 56T?K? + 2 ht+j~1 + O,(T). (All) j=

O,(T”) O,(T*”) From (A10) and (A11), it follows that:

18

where

TK ift=7,+1 T and ' t he = Dy; t=1,...,T, (A6) j=l

with hy being a random walk. Evaluating (A4) at ¢ = T obtains the summation a Ave

e An = 27

6TK + hr, (A7) | OAT) — O,(T"/?) where orders of magnitude appear below the component terms in the last. line. While terms such as 67K are non-stochastic, it is convenient (and legitimate) to use probabilistic orders throughout.

The summation yt Z4-1 1s (x? 2) — z7 + 2%, where yw z, can be obtained by using (A4) and evaluating the summation of J; over the three subsamples.

T T eA =. byt Uke

1

T = 0+6 > i,+6 > f+ Uh

To+1 T\+1 ee T TT = §S(t-T)+6 S TKEDK To+1 T,+1 a |

TK | TM T. = §VGt6UTK+ LA jI= j=

a = 1§T?K[(K +T-1) 42M] +h

1

T s6T?K[K +2M] + dhe + O,(T). (A8) O,(T”) O,(T°/?) Because zr is Op(T’)) and 2 = 0, the summation D7 z%_, is (A8), to O,(T).

The summation rH ze is obtained in a similar fashion.

17

Case III, p = 1, so the error is comprised of a break (a — 1)6D; and white noise (a — 1)u; + €, paralleling the regression (A2) under the DGP (Al). Under Case IV, | p |< 1, so the error also includes z~1, which has a break in it. Thus, whenever the marginal process has a break and the conditional process has no common factor, the regression for tpr, induces a break in the error, which may reduce the power of the corresponding cointegration test. By contrast, cointegration tests based on tecmr and tecmyz do not suffer from this problem because their corresponding regressions are still properly specified.

Il. The Distribution of tgcy, When There Is No Break

The test of b = 0 using tgcm, is known to be similar when the conditioning variable z, is strongly exogenous for the regression parameters (a, b,c, 02); see Kiviet and Phillips (1992) and Banerjee and Hendry (1992). The null limiting distribution of tgcMu is implicit as a special case of Park and Phillips (1988, Theorem 4.1a; 1989, Theorem 4.1b) and Boswijk (1992). The explicit representation provides insights not apparent from their general formulae, so it is derived here.

The DGP is (1)-(4) with b = 0 under Case I (ie., with p = 1 and 6 = 0):

Ay: = aAz + €: é, ~ IN (0 ao a Al5 Au = ur ~ IN (0,02). (a9) The estimated model is: Ay, = ada, + b(y — z)i-1 + C21 + %, (A16) which is (6) without a constant term. The rescaled parameter estimates from (A16) are: T T . T -1 T Su? T7932 wy, TO 9/? Dzau 1 1 7 VT (a — a) r r r Tb = T~3/2 > wy ut T? Yow? To? SO wii 21-1 Ta 1 1 1 iI T T T~ 3/2 zyue TO? Dwi TO zy 1 r 1 _T TOU > UtEt 1 iZ T- 2 Wr-16t (A17)

T T- L Zt-1Et

20

I

dz A% py 2t-1Ut + 622-1 D; = PT? K? ; O,(T?)

+ 5135 (hss + jUT +5) + TK Y wns + O,(T) — (Al2) 7 OT) The probability limit of (A3) can now be evaluated. Pre-multiplying (A3) by ; and substituting (A7), (A8), (A9), and (A12) into that equation obtains:

-1

Tt ji 1 Tv ee 6K + O,(T-1/?) plim = plim phim Te T~ > a1 T? » zy 36K? + O,(T-1/?) 1 1 _ 1 15K(K+2M) ] [ 6K ~ | 26K(K 42M) 1862(K +3M) 58°?

H-1 | 6K(M + K/6) | .

1—(K +2M) (A13)

wie

whete H = (L— K)M+K(4-3K)/12. The limiting distribution of (A3) is somewhat complicated. Because (jt : T¢) has a nonzero plim, the stochastic components of the first as well as the second matrix on the right-hand side of (A3) must be taken into account. We plan to derive that limiting distribution in due course. A break in the marginal process has similar effects on the unit root statistic tpr and the cointegration statistic tpr, when the common factor restriction is not valid - for the latter (a 4.1). For tpr,, the equation being estimated is:

A(y—z): = b(y—z)i-1 + ene

b(y — z)i1 + [(a- 1) Az + 1]

bly — 2)e-1

+[(a — 1)8D, +(a- 1p — 1)zy-1 + (a — 1)ue + &] (Al4)

where the constant is implicit and the term in square brackets is the error. Under

19

Appendix B. Estimated Finite Sample Properties

This Appendix documents certain distributional aspects of the test statistics and their associated estimators. With the exception of the means of the statistics themselves, there is little to distinguish the properties of the test statistics across DGPs (e.g., with or without a break) and even across test statistics for a given DGP.

Figures B.1-B.5 plot the Monte Carlo mean of the estimate of b for each test statistic and each DGP, estimated recursively over T. The values reported are “biases” relative to a value of zero, even under the alternative of cointegration. The estimated biases are graphed with plus-or-minus twice their (average) estimated standard error (ESE), as would be computed by a regression package, and plus-or-minus twice the Monte Carlo standard deviation (MCSD), reflecting the actual sampling distribution of the estimator of 6. The MCSD is always larger than the ESE for these experiments.

Figures B.6—B.10 plot the Monte Carlo mean of each statistic, with plus-or-minus twice its MCSD. As predicted by theory, the means of tpr; and tgcmy appear invariant to g when there is no break. The mean of tgcy,, shifts toward zero as q increases (under the null), and becomes more negative as q increases (under the alternative).

Figures B.11-B.15 plot the histograms of all four statistics at T’ = 100. While 10,000 replications is only a moderate number of replications for examining full distributional properties of the statistics, their distributions overall appear normal, in line with Banerjee and Dolado’s (1988) result that the Dickey-Fuller distribution is well approximated by a normal distribution with a negative mean.

22

where, from (A15), w; is the random walk: We = Yi 4

= W-i + Ct, (A18)

and e; = (a—1)u:4+ €, as in (10). Asymptotic distributions of the elements in (A17) involve the Brownian motion processes B,, B., and B., where the last is related to the first two by:

o0eB. = o-B.+(a—1)o.By. (A19)

Also, 0? = a? + (a— 1)*o2. Hence, by partitioned inversion in (A17): 5 Oe (f B2y(f B.dB.) _ (f BBe)(f B,,dB.) a A20 a (LB B2) -U BiB. (420) Thus, the limiting distribution of tgcmy is: J BedB. —(f BeBu)(f Bi)'(f BudBe)

tECMu => f B2-(f B.B.)?(f B2)-}

__[BedBe = ([ BB) f BY)" Bud Be) (a2 JB? -(f BeBu)(f Bd) using (A19). Because the asymptotic distribution of tgc¢yy does not depend on a, oy, or d;, it is invariant to those parameters, and hence tests based on tgcmu are similar. An alternative expression for (A21) is enlightening. Consider the regression

yo = Patt (A22)

“linking” the levels of y; and z;. Let R denote the limiting form of Yule’s correlation for that regression when a = 6 = 0 (1.e., under the null of no relation between y; and z,) and that correlation is not adjusted for sample means. From Phillips (1986), R is: — _JBBy (A23) (f BUY B?) Thus, schematically (A21) is: DF —R- N(0,1) 1-R2 where DF and N(0,1) denote random variables with Dickey-Fuller and standardized normal distributions. This demonstrates that tecy, and the Dickey-Fuller statistic have different limiting distributions, so their critical values need separate tabulation. Also, (A24) mirrors Park and Phillips’s (1988) Lemma 5.6, in which a related problem is addressed and R is a constant.

tECcMu => (A24)

21

ALS -2C Blas +2 SE(@eta) = ——— RLS -2C Bias 22 MCSD(Bias) Borkez

ze ET} «a se se 7 rr} 3@ 1¢0

Figure B.2a : tpr, (no break)

RLS -ZC Bias #2 SE(Beta) = RLS -ZC Bias +2 MCSD(Bias) BECHKO2

= —=——_ 1.5m-aa;

1. er arene 1 4 20 38 C 38 os 7 se rr 100

Figure B.2b : tgcms (no break)

RLS -2C Bias #2 SE(Beta) = BOCHneZ

RLS -2C Bias +2 MCSD(Bias)

Figure B.2c : tgcmu (no break)

Figure B.2. Biases of estimated 6 for three statistics (tprz, tecmk, tECMu), L2ESE

"RLS -2C Bias +2 SE(Beta) = ——— BECHuO6

ALS -2C Bias +2 SE(Beta) = ——— BLS -2C Bies #2 NCSB(Bias) SDFKO6

ze 3 « se oe ve se 20 100

Figure B.2d : tpr, (with break)

RLS -2C_ Bias +2 SE(Beta) = ——— BLS -ZC Bias 22 ACSD(Bias) BECHKOG

—~—_ at 4 n a a ror 28 3a Pry ery 6a 7 ea ery 108

Figure B.2e : tgome (with break)

BLS -2C Bias +2 MCSBCBias)

n a Sc cs eae oe 1 20 ae ery se rT} 7 oo 7 190

Figure B.2f : tgcmu (with break)

’

and +2MC'SD. The experiments have no common factor and no cointegration.

22b

RLS -2C Bias +2 NCSDGias)

' RLS -2C_ Bias #2 SE(@Beta) = ——— BOFkO1

1 4 “ 2 n 20 ery «a 30 oo 7 ae se

Figure B.la: tprx (no break)

RLS -ZC Bias #2 SE(Beta) =

RLS -2C Bias +2 MCSD(Bias)

Figure B.1b : tgcmyx (no break)

" RLS -2C Bias 22 SE(Beta) = RLS -ZC Bias +2 MCSD(Bias) BECha@1

Figure B.1c : tgcmu (no break)

198

" “BLS -2C Bias sZ SE(Beta) = BDFKOS

BLS -2C Bias +2 MCSD(Bias)

Figure B.1d : tor, (with break)

RLS -2C Bias +Z SE(Beta) = ——— RLS -2C Bias +2 MCSD(Bias) BECHKOS

Oo

a ee —— rn rn 2 1. L 2a 38 pry sa se 78 30 98 100

Figure B.le : tecms (with break)

RLS -ZC Bias +2 SE(Beta) = ——— RLS -2C Bias +2 MCSD(Bias) BECMuOS

ry aoe ee _ q oe . —_— <a ee ° — 7 7? 9 aa 1.3 oo, ae 30 . ] 3 ee 738 so 98 19

Figure B.1f : tgcmzu (with break)

Figure B.1. Biases of estimated b for three statistics (torx, tecmk, tecMu), L2ESE, and +2MC'SD. The experiments have a common factor but no cointegration.

22a

ALS -2C «Bias #2 SE@eta) = ——— RLS -2C Bias +2 NCSDCBias) BDrkO4

20 3 - se se ve se 368 ieee

Figure B.4a : tp, (no break)

RLS -2C Bias +2 SE(Beta) = RLS -2C Bias +2 MCSD(Bias) BECHKO4

— SSS -.k $$ re rs oe 7 ae rT) a 38 cif 73 se 9s 1eeo

Figure B.4b : tgcmx (no break)

"ALS ~ZC Blas +2 SE(Beta) = ——— RLS -2C Bias +2 MCSD(Bias) BECHuOt

ae 28 a se se 7 ae Er} 100

Figure B.4c : tecmu (no break)

RLS -2C Bias +2 3E(Beta) = ————- ss BLS -2C Bias +2 MCSB(Bias) EDFKOS

a1

ze 3a «o se oe ET) se er 198

Figure B.4d : tor, (with break)

ALS -2C_ Bias +2 SE(Beta) = TEQKeS

BLS -ZC Bias *2 MCSD(Bias)

}

:

20 30 ery se rT} 7. so er 198

Figure B.4e : tecms (with break)

ALS -2C Bias +2 SE(Beta) = ———— BLS -ZC Bias #2 MCSB(Bias) BECHuC8

20 ae ae se eo 7 Lad 2s 108

Figure B.4f : tecmu (with break)

Figure B.4. Biases of estimated 6 for three statistics (tprx, tacmk, tecmu), L2ESE, and +2MCSD. The experiments have no common factor but have cointegration.

22d

ALS -ZC Bias #2 SE(@eta) = ——— RLS -2C Bias 42 MCSD(Bies) RLS -2C Bias +2 SE(Beta) = ———— BLS -2C: Bias #2 MCSD(Bias) BOrkes BDFKG?

a1

ze 38 « 38 se 7 eo 38 ee ze 3 « se oe 7 se cr) 190

Figure B.3a: tpr (no break) Figure B.3d : tprx (with break)

RLS -ZC_ Bias #2 SE(Beta) = ———- RLS -2C Bias +2 MCSD(Bias) RLS -2C Bias +2 SE(Beta) = ——— RLS -2C_ Bias #2 MCSD(Bias) BECHK63 BECHKO?

See a

—_—

n " 1 a ao 20 «a 38 se 7 se os 108

et oo 7° 3a er 1ee

Figure B.3b : tecme (no break) Figure B.3e : tgcme (with break)

ALS -2C_ Blas #2 SE(Beta) = RLS -2C Bias +2 MCSD(Bias) ALS -2C Bias +2 SE(Beta) = ——— WLS -2C Blas #2 MCSD(Bias) BEChe3

BECHuO?

—_———__ _-———

+. __1 Ey y 190

ae a a ss ss 7 oe se 1900

Figure B.3c : tgcMu (no break) Figure B.3f : tecmu (with break)

Figure B.3. Biases of estimated 6 for three statistics (tprx, tecmk, tecMu), E2ESE, and +2MCSD. The experiments have a common factor, with cointegration.

22c

Figure B.6a : tprx (no break)

TECHKOL RLS -2C Mean ‘t’ + 2 MCSD = ———

4 ss 7@ 38 se 196

Figure B.6b : tgcm« (no break)

TEChHa@1 RLS -2C Mean ‘t’ + 2 MCSD =

ee

et}

-af

L —

~2 [

-3 f+

-« le ae 28 ee se 7 rT) rT) 108

Figure B.6c : tgcmu (no break)

TDFKGS BLS -2C Mean ‘t’ ¢ 2 NCSD = ———

20 ey as 3e oe 7° ve 30 198

Figure B.6d : tprx (with break)

TECHKOS BLS -ZC Mean ’t’ + 2 MCSD =

Figure B.6e : tacme (with break)

TEQ&ES MLS -ZC Mean ‘t’ + 2 MCSD =

ae 2a se se es 72 os s 198

Figure B.6f : tecmu (with break)

Figure B.6. Means of three test statistics (tors, tecmk, tecMu), With 42MC'SD of the statistics. The experiments have a common factor but no cointegration.

22f

RLS -ZC = Bias #2 SECBeta) = ———— RLS -ZC Bias +2 WCSB(Bias) ALS -2C = Bias #2 SE(Beta) = ——— RLS -ZC Bies #2 MCSD(Bies) BOrke? BDFkie

-~.2 -.2 —— -.4 -.4 _ 6 -.6 -.8 -.8 ze 38 a. 3° ee 7 ee 98 ieee 20 38 a se ce 73 ee sd) 198 Figure B.5a : tprx (no cointegration) Figure B.5d : tprx (with cointegration) RLS -2C Bias #2 SE(Beta) = ——— RLS -ZC Bias +2 NCSD(Bias) ALS -2C Bias +2 SE(Beta) = RLS -2C Bies #2 NCSD(Bias) BECHke9 BECHK10

ee -.3 wal -.3 -.4 i ze 30 o se so oy eo 30 100 20 30 « se se 78 rT) co) 100

Figure B.5b : tecmk (no cointegration) Figure B.5e : teom« (with cointegration)

RLS -2C Bias #2 SE(Beta) = ——— RLS -2C Bias +2 MCSD(Bias) RLS -ZC = Bias #2 SE(Beta) = ——— RLS -2C Bias 22 MCSD(Bias) BECriue3 BECMULO

ze 30 a se se 7 ee ET} 1¢8

Figure B.5c : tgcmy (no cointegration) Figure B.5f : tecmu (with cointegration)

Figure B.5. Biases of estimated 6 for three statistics (tprk, tecmk; tecMu), t2ESE, and +2MCSD. The experiments have a large break (6 = 3) and no common factor.

22e

TOrkeG RLS -2C Mean ‘t’ + 2 NCSD = ———— , TDPKG? BLS -2C Mean ‘t’ + 2 NCSD =

Figure B.8a : tprx (no break) Figure B.8d : tor (with break)

Figure B.8b : tecm« (no break) Figure B.8e : tecm« (with break)

TEChu83 RLS -2C Mean ‘t’ + Z NCSD = ——-—— TEQ&O? BLS -ZC Mean ‘t’ ¢ 2 NCSD =

Figure B.8c : tgcmu (no break) Figure B.8f : tecmu (with break)

Figure B.8. Means of three test statistics (tor, tecmk, tecMu), With +2MCSD of the statistics. The experiments have a common factor, with cointegration.

22h

20 38 - se so ve ee 38 1e0

Figure B.7a : tpr, (no break)

TECHkEZ RLS -20 Mean ‘t’ + Z MCSD = ———

1 n n n n 1 ae ETy a 38 oe 7 se cry 190

Figure B.7b : tecm« (no break)

TECHa@2 RLS -2C Mean ‘t’ + 2 NCSD = ————

1 a | n re ao 4 ao Fry aa se so 7 ae ery 1968

Figure B.7c : tgcmu (no break)

TDFKOG RLS -2C Mean ‘t' + 2 MCSD = ———

Figure B.7d : tprx (with break)

TECHKO6 ALS -ZC Mean ‘t’ + 2 MCSD =

n [ae a ee ST cer Cres ae 30 ery 38 oa 7 aa 2° 188

Figure B.7e : tacm. (with break)

TEC&06 RLS -2C Mean ’t’ + 2 ACSD =

Figure B.7f : tecmu (with break)

Figure B.7. Means of three test statistics (tprz, tacmk, tecmu), With 42MCSD of the statistics. The experiments have no common factor and no cointegration.

22g

3 -2 -2.5 3 ——— a eed -2.8 ze 308 -a se se we se 398 168

Figure B.10a : tp, (no cointegration)

ze ET) a) se se ve eo ery 100

Figure B.10b : tgcm, (no cointegration)

TEChaGS RLS -2C Mean ’t’ + 2 MCSD = ——-—

Figure B.10c : tgcmy (no cointegration)

YTOFKi6 RLS -2C Mean ‘t’ + 2 MCSD = ————

Figure B.10d : tprx (with cointegration)

TECHK16 RLS -2C Mean ‘t’ » 2 MCSD = ————

Figure B.10e : tgom« (with cointegration)

TECHu16 RLS -2C Mean ‘'t’ * 2 MCSD = ———

—

~~

Figure B.10f: tecmu (with cointegration)

Figure B.10. Means of three test statistics (tprx, tacomk, tecMu), With 42MCSD of the statistics. The experiments have a large break (6 = 3) and no common factor.

22}

TOFKO4 RLS -2C Mean 't’ + 2 MCSD = ——— TOPKOS BLS -ZC Mean ‘t’ + 2 MCSD =

i Figure B.9a : tprx (no break) Figure B.9d : tpr, (with break)

28 Fry ery ET) rT) 7a [Ty ery 1900

Figure B.9b : tecms (no break) Figure B.9e : tgoms (with break)

TECMa84 RLS -2C Mean 't’ + 2 MCSD = ———— TECM@uOB BLS -2C Mean ‘t’ + 2 MCSD =

Figure B.9c : tecmz (no break) Figure B.9f : tecmu (with break)

Figure B.9. Means of three test statistics (tore, tacms, tecmu), With +2MCSD of the statistics. The experiments have no common factor but have cointegration.

221

Figure ‘B. 12a: tprk (no break) Figure B. 12e: ‘tore (with break)

HecnKez {WPS t from ZERO: Frequency HECHOS €06¢ t from ZERO: Frequency

Hecreez + from ZERO: Frequency HECMUeS t from ZERO: Frequency

Figure B. 12d: tom. (no break) Figure B. 12h: tion (with break)

Figure B.12. Histograms of four test statistics (torr, tECMk; tpFu, tEcMu) at T = 100. The experiments have no common factor and no cointegration.

221

Horke1 “YA +t from ZERO: Frequency . HDPrkeS -Y¥a t from ZERO: Frequency

Figure B.1la: tpx (no break) Figure B.1le: tpr, (with break)

Ecnke1 420 t from ZERO: Frequency HECHKOGS eof t from ZERO: Frequency

* . = *

Figure B.11c: tpr, (no break) Figure B.11g : tpru (with break)

HECHae1 t from ZERO: Frequency HECHueS t from ZERO: Frequency

rs

Figure B.11d: tgcmz (no break) Figure B.11h : tgcmu (with break)

Figure B.11. Histograms of four test statistics (tprk, tecmk, toru, tecMu) at T = 100. The experiments have a common factor but no cointegration.

22k

Figure B. Ida: : LDP (no break) Figure | B. l4e: tre

—~

with break)

HECHKest wel t from ZERO: Frequency . HECHOS ‘‘meF ¢ from ZERO: Frequency

Figure I B. 3 14b : tnome (no break) Figure B. 14 : trom (with break)

Horues Bickey-Fuller Frequency HDFuss Dickey-Fuller Frequency

Figure B.14c : tpru (no break) Figure B. lg: ‘tors (with break)

HEChee4 t fron ZERO: Frequescy HECTues t from ZERO: Frequency

Figure I B. 314d : tecMu (no break) Figure B. 14h : tecmy (with break)

Figure B.14. Histograms of four test statistics (tprz, taomk, tpFu, tecmu) at T = 100. The experiments have no common factor but have cointegration.

22n

twrKe7 “YA % from ZERO: Frequency

HECNKe? 4LG@@ t from ZERO: Frequency

Figure B.13b : tgcm, (no break) Figure B. 13f : tome (with break)

HDFue3 Dickey-Fuller Frequency HoFuo? Dickey-Puller Frequency

Hecrues t from ZERO: Frequency HECHuO7 t from ZERO: Frequency

Figure B. 13d: tiscMs (no break) Figure B. 13h: tocate (with break)

Figure B.13. Histograms of four test statistics (tprx, tecmk, tpFu, tecmu) at T = 100. The experiments have a common factor, with cointegration.

‘22m

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HorKes -YA t from ZERO: Frequency

= —= ° = < *

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HECrmes t from ZERO: Frequency HeChu1e t from ZERO: Frequency

Figure B.15. Histograms of four test statistics (tpr., tecmk, tpFu, tecmu) at T = 100. The experiments have a large break (6 = 3) and no common factor.

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24

IFDP NUMBER

440

439

438

437

436

435

434

433

432

431

430

429

428

International Finance Discussion Papers TITLES 1993

Cointegration Tests in the Presence of Structural Breaks

1992

Life Expectancy of International Cartels: An Empirical Analysis

Daily Bundesbank and Federal Reserve Intervention and the Conditional Variance Tale in DM/$-Returns

War and Peace: Recovering the Market's Probability Distribution of Crude Oil Futures Prices During the Gulf Crisis

Growth, Political Instability, and the Defense Burden

Foreign Exchange Policy, Monetary Policy, and Capital Market Liberalization in Korea

The Political Economy of the Won: U.S.-Korean Bilateral Negotiations on Exchange Rates

Import Demand and Supply with Relatively Few Theoretical or Empirical Puzzles

The Liquidity Premium in Average Interest

Rates

The Power of Cointegration Tests

The Adequacy of the Data on U.S. International Financial Transactions: A Federal Reserve Perspective

Whom can we trust to run the Fed? Theoretical support for the founders views

Stochastic Behavior of the World Economy under Alternative Policy Regimes

AUTHOR (s )

Julia Campos Neil R. Ericsson David F. Hendry

Jaime Marquez

Geert J. Almekinders Sylvester C.W. Eijffinger

William R. Melick Charles P. Thomas

Stephen Brock Blomberg

Deborah J. Lindner

Deborah J. Lindner

Andrew M. Warner

Wilbur John Coleman II Christian Gilles Pamela Labadie

Jeroen J.M. Kremers Neil R. Ericsson Juan J. Dolado

Lois E. Stekler Edwin M. Truman

Jon Faust

Joseph E. Gagnon Ralph W. Tryon

Please address requests for copies to International Finance Discussion Papers, Division of International Finance, Stop 24, Board of Governors of the Federal

Reserve System, Washington, D.C.

20551.

27

Park, J.Y. and P.C.B. Phillips (1988) “Statistical Inference in Regressions with Integrated Processes: Part 1”, Econometric Theory, 4, 3, 468-497.

Park, J.Y. and P.C.B. Phillips (1989) “Statistical Inference in Regressions with Integrated Processes: Part 2”, Econometric Theory, 5, 1, 95-131.

Perron, P. (1989) “The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis”, Econometrica, 57, 6, 1361-1401.

Phillips, P.C.B. (1986) “Understanding Spurious Regressions in Econometrics”, Journal of Econometrics, 33, 3, 311-340.

Phillips, P.C.B. (1987a) “Time Series Regression with a Unit Root”, Econometrica, 55, 2, 277-301.

Phillips, P.C.B. (1987b) “Towards a Unified Asymptotic Theory for Autoregression” , Biometrika, 74, 3, 535-547.

Phillips, P.C.B. (1988) “Regression Theory for Near-integrated Time Series”, Econometrica, 56, 5, 1021-1043.

Phillips, P.C.B. (1991) “Optimal Inference in Cointegrated Systems”, Econometrica, 59, 2, 283-306.

Phillips, P.C.B. and S. Ouliaris (1990) “Asymptotic Properties of Residual Based Tests for Cointegration”, Econometrica, 58, 1, 165-193.

Phillips, P.C.B. and P. Perron (1988) “Testing for a Unit Root in Time Series Regression”, Biometrika, 75, 2, 335-346.

Quintos, C.E. and P.C.B. Phillips (1992) “Parameter Constancy in Cointegrating Regressions”, mimeo, Department of Economics, Yale University, New Haven, Connecticut.

Sargan, J.D. (1964) “Wages and Prices in the United Kingdom: A Study in Econometric Methodology” in P.E. Hart, G. Mills, and J.K. Whitaker (eds.) Econometric Analysis for National Economic Planning, Colston Papers, Vol. 16, London, Butterworths, 25-63 (with discussion); reprinted in D.F. Hendry and K.F. Wallis (eds.) (1984) Econometrics and Quantitative Economics, Oxford, Basil Blackwell, 275-314.

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Sargan, J.D. and A. Bhargava (1983) “Testing Residuals from Least Squares Regression for Being Generated by the Gaussian Random Walk”, Econometrica, 51, 1, 153-174.

26

IFDP NUMBER

427

426

425

424

423

422

421

420

International Finance Discussion Papers TITLES 1992 Real Exchange Rates: Measurement and

Implications for Predicting U.S. External Imbalances

Central Banks’ Use in East Asia of Money Market Instruments in the Conduct of Monetary Policy

Purchasing Power Parity and Uncovered Interest Rate Parity: The United States 1974 - 1990

Fiscal Implications of the Transition from Planned to Market Economy

Does World Investment Demand Determine U.S. Exports?

The Autonomy of Trade Elasticities: Choice and Consequences

German Unification and the European Monetary System: A Quantitative Analysis

Taxation and Inflation: A New Explanation for Current Account Balances

28

AUTHOR(s)

Jaime Marquez

Robert F. Emery

Hali J. Edison William R. Melick

R. Sean Craig Catherine L. Mann

Andrew M. Warner

Jaime Marquez

Gwyn Adams Lewis Alexander Joseph Gagnon

Tamim Bayoumi Joseph Gagnon