Approximate p-values of predictive tests for structural stability

Economics Letters 63 (1999) 245–253

Approximate p-values of predictive tests for structural stability

*Amit Sen

University of Missouri–Rolla, Department of Economics, 102 Harris Hall, Rolla, MO 65409-1250, USA

Received 8 June 1998; accepted 13 January 1999

Abstract

We approximate the p-values for the non-standard asymptotic null distribution of predictive tests proposed byGhysels et al. (Journal of Econometrics,1997;82:209–233). The p-value response surfaces are approximated bya parametric function using the methodology proposed by Hansen (Journal of Business and Economic Statistics,1997;15:60–67). We illustrate the use of the approximate p-values through an application to consumption basedasset pricing models. 1999 Elsevier Science S.A. All rights reserved.

Keywords: GMM; Predictive test; Structural stability

JEL classification: C1

1. Introduction

Within the generalized method of moments (GMM) framework (see Hansen, 1982, for details),Ghysels et al. (1997), henceforth GGH, propose predictive tests (PR) to ascertain if the underlyingmodel exhibits structural instability. Numerous studies have used the predictive test to uncoverstructural instability in the known breakpoint case (for example, see Ghysels and Hall, 1990b;Hamori, 1992; Bufman and Leiderman, 1993; Fauvel and Samson, 1991). The GMM estimator of aparameter vector, u, is based on the following set of moment conditions derived from the underlyingmodel

E f(x , u ) 5 0 (1)f gt 0

where f(x ,u ) is a (q31) vector of known functional form, u is the true value of the ( p31)t o 0ˆparameter vector (with q.p), and x is a set of forcing variables. The GMM estimator, u , is definedt T

as

*Tel.: 11-573-3416688; fax: 11-573-3414866.E-mail address: [email protected] (A. Sen)

0165-1765/99/$ – see front matter 1999 Elsevier Science S.A. All rights reserved.PI I : S0165-1765( 99 )00031-2

246 A. Sen / Economics Letters 63 (1999) 245 –253

T T1 1ˆ ] ]u 5 argmin O f(x , u ) 9W O f(x , u ) (2)F G F GT u t T tT Tt51 t51

21where T is the sample size, and W is a consistent estimator of S whereT

T1]]S 5 lim var O f(x , u ) (3)F G] t 0ŒT →` T t51

Consistent estimators of S have been proposed (see Andrews, 1991,1993; Gallant, 1987; Newey andWest, 1994).

The predictive test is designed to uncover structural instability characterized by a single break sothat some aspect of the underlying model changes at a discrete point in the sample. The breakpoint isdenoted by p [(0, 1) which corresponds to [pT ] in the sample indexed by h1,2, . . . , T j.

Although, the exact location of the breakpoint is unknown, it is assumed to lie in some closedinterval P#(0, 1). GGH propose three versions of the predictive test, namely, SupPR , AvPR , andT T

ExpPR and derive the limiting distribution under the null hypothesis of structural stability. TwoT

features of the limiting null distributions stand out. First, the limiting null distribution are non-standard. Second, the distribution depends on the number of parameters estimated ( p), the number ofoveridentifying restrictions (q2p), and the interval over which the researcher searches for thebreakpoint (P). For these tests to be operational in empirical research, critical values for a set offrequently used significance levels must be tabulated. In addition to the critical values, appliedresearchers find it informative to look at the p-value associated with the calculated test statistics.

Due to the non-standard nature of the limiting distributions, both the critical values and the p-valuescannot be determined analytically. GGH tabulate the critical values for numerous combinations of p,q2p, and P based on Monte Carlo simulations. However, they do not provide approximate p-valuesresponse surfaces. In this paper, the methodology proposed by Hansen (1997) to approximate p-valueresponse surfaces for statistics with non-standard limiting distributions.

We found the critical values reported in Tables 3–8 of GGH (pp. 224–229) are incorrect. Thisdiscovery has been brought to the attention of the authors of GGH, and they are in the process ofpreparing a corrigendum which tabulates the correct critical values. We have calculated the correctcritical values for all versions of the PR tests, and used these for the calculation of approximateT

p-values. The complete set of critical values used in this paper are available in the corrigendum toGGH or from the author upon request.

The rest of the paper is organized as follows. In Section 2, we present the asymptotic nulldistributions of predictive test. We summarize the methodology used to calculate approximate criticalvalues and p-values in Section 3. In Section 4, we illustrate the use of the approximate p-valuesthrough an application to the consumption based asset pricing model (CBAPM) presented in Hansenand Singleton (1982). All calculations were done in GAUSS.

2. The predictive test

In order to understand the basic idea behind the construction of PR , it is instructive to firstT

consider the case when the breakpoint is known, say p. First, the data from before and after the

A. Sen / Economics Letters 63 (1999) 245 –253 247

breakpoint are collected into two sub-samples: T (p) 5 h1, 2, . . . , [pT ]j and T (p) 5 h[pT ] 1 1, . . . ,1 2

T j. If the underlying model does not exhibit structural instability, the moment conditions from T (p)2

evaluated at the estimated parameter vector from T (p) should be close to zero. The GMM estimate of1

u based on T (p) is analogous to (2). Formally, the null and alternative hypothesis are respectively1

H :E f(x , u ) 5 0, t [ T (4)f g0 t 0

E f(x , u ) 5 0, t [ Tf gt 0 1H (p): (5)HA E f(x , u ) ± 0, t [ Tf gt 0 2

Under H (p), some aspect of the underlying model has changed in the sub-sample after the break,A

T (p). Ghysels and Hall (1990a) define PR (p) as2 T

T T1 21ˆ ˆ ˆ]]]]PR (p) 5 O f(x , u (p)) 9 V (p) O f(x , u (p)) (6)F G F GT t 1 2 t 1(T 2 [pT ]) t5[pT ]11 t5[pT ]11

21ˆwhere V (p) is a consistent estimator of V (p) 5 S (p) 1 dF F (p)9S (p)F (p) F (p)9, d 5 (1 22 2 2 2f 1 1 1 g 221

p) /p, S (p) are the sub-sample analog of S, andi

≠f(x , u )1 t] ]]]F (p) 5 O .i T ≠u 9i t[Ti

Expressions for the consistent estimator of V (p) are given in Ghysels and Hall (1990a).2

When the breakpoint is unknown, the relevant alternative hypothesis is H 5hH (p):p [Pj.A A

The strategy to test for an unknown break is to construct the sequence of PR (p), indexed byT

p [P, and apply some functional to this sequence. GGH show that under certain regularityconditions, the limiting null distribution of the process PR (p) is given byT

PR (p) ⇒ BW(p) 1 BO2(p) (7)T

where BW(p) 5 [B (p) 2 pB (1)]9[B (p) 2 pB (1)] /p(1 2 p), BO2(p) 5 [B (1) 2 B (p)]9[B (1) 21 1 1 1 2 2 2

B (p)] /(1 2 p), B is a ( p31) vector of independent Brownian motions and B is a (q2p31) vector2 1 2

of independent Brownian motions. Based on the limiting distribution of the process PR (p), GGHT

propose three functionals of PR (p) as test statistics, SupPR 5 Sup PR (p), ExpPR 5T T p [P T T

log e exp 0.5PR (p) dJ(p) , and AvPR 5 e PR (p)dJ(p), where J(p) is a prior weightingf gs dP T T P T

distribution for p. The limiting distribution of SupPR , AvPR , and ExpPR are obtained by applyingT T T

the respective functional to the limiting process in (7). It is clear that the limiting null distribution ofeach statistic depends on the number of parameters estimated ( p), the number of overidentifyingrestrictions (q2p), and the pre-specified interval P. As mentioned above, the critical values reportedin Tables 3–8 (pp. 224–229) of GGH are incorrect. In personal correspondence with the authors, wehave been assured that a corrigendum will be prepared to clarify this mistake. For the purposes of ourpaper, we generate correct critical values and then use these to calculate approximate p-values. Wenow turn our attention to the calculation of approximate p-values for all versions of the predictive test.


3. Approximate p-value calculations

In this section, we present a brief overview of Hansen’s (1997) procedure used to obtainapproximate p-values for SupPR , AvPR , and ExpPR . Approximate p-value response surfaceT T T

calculations are carried out for the SupPR , AvPR , and ExpPR for all combinations arising fromT T T

setting p51, 2, . . . , 8; q2p51, 2, . . . , 8; and P5[b, 12b] with b50.15, 0.20, 0.30, 0.35, 0.40,0.45. We begin with a summary of the methodology proposed by Hansen (1997) to approximate thetrue p-value function corresponding to the non-standard distribution of a random variable X. The truep-value function, p(x) 5 Pr(X $ x), is approximated by

2 2 np xuj, h 5 1 2 x j 1 j x 1 j x 1 ? ? ? 1 j x uh (8)s d s d0 1 2 n

2where x ( ? uh) is a chi-squared distribution with h degrees of freedom. Let n(xuj) be the n th order2 npolynomial in x, that is, a (xuj ) 5 j 1 j x 1 j x 1 ? ? ? 1 j x . We denote the inverse of the truen o 1 2 n

21p-value function by Q(?), that is, Q(?)5p (?). Then we have the following identity

q 5 p Q(q) ; q [ [0, 1] (9)s d

The parameters j and h are chosen so as to make p(xuj,h) as close as possible to p(x) for all values ofx in the range of the random variable X. This is equivalent to choosing j and h so that q is as close aspossible to p(Q(q)uj,h) for all values of q[[0, 1]. To obtain a higher degree of precision for smallerp-values, Hansen (1997) proposes choosing j and h so as to minimize a weighted cumulative errordue to the parametric approximation of the true p-value function. Specifically, Hansen (1997) positsthe following criterion function

1 1 / r

j * r5argmin E u p Q(q)uj,h 2 qu w(q) (10)s dS Dh* 3 4( j 9,h)

0

where r is some large positive integer and w(q) is a function that assigns large weights to smallp-values and a small weight to large p-values. For further details regarding the choice of theweighting function, the reader is referred to Hansen (1997). The particular choice of w(q) used byHansen (1997) is

0 ,0.8 , q # 120.8 2 q

w(q) 5 (11)F]]]G ,0.1 , q # 0.80.751 ,0 # q # 1

The criterion function in (10) cannot be completely specified, since the true quantile function, Q(?), isunknown. Therefore, we replace the integral in (10) by a sum over a sufficiently fine grid on [0,1], say

M ˆhq j , and replace Q(q ) by the approximate critical value (Q(q )) that corresponds to thei i50 i i


1(12q )3100% significance level. This is equivalent to minimizing the weighted cumulative absoluteiMerror for the set of equally spaced p-values, namely, hq j . Following Hansen (1997), we use r58,i i50

q 5 i /1000, and M5999 to determine the optimal value of (j 9,h) by solving the followingi

optimization problem

M 1 / 8j * 8ˆu u5argmin O psQ(i /1000)uj, hd 2 (1 /1000) w(i /1000) (12)S D F Gh* ( j 9,h) i50

In addition, n is chosen so that the maximum error in the fitted p-values over all q values,iˆMax u p(Q(q )uj, h) 2 q u, is less than 0.01. The approximate p-value function is now obtained1#i#1000 i i

by setting (j 9, h)9 equal to (j *9, h*)9 in the expression of p( ? uj,h) given in Eq. (8).In Tables 1–3 we report the optimal parameters, (j *9, h*)9, indexing the approximate p-value

functions for SupPR , AvgPR , and ExpPR for P5[0.15, 0.85]. The set of optimal parameters forT T T

Table 1Approximate distribution for Sup PR (p) with P5(0.15, 0.85)p [P T

p51 p52 p53 p54 p55 p56 p57 p58

q2p51 j 21.28 21.97 22.76 23.54 23.72 24.13 24.83 25.240

j 1.04 1.05 1.02 1.01 1.10 1.09 1.09 1.111

h 4.51 5.94 6.69 7.47 9.91 11.08 11.88 13.46q2p52 j 21.66 22.28 23.04 23.32 24.10 24.21 24.81 25.590

j 1.06 1.08 1.05 1.11 1.09 1.13 1.13 1.111

h 5.88 7.34 8.11 10.16 10.81 13.01 14.03 14.44q2p53 j 21.88 22.72 23.32 23.74 24.20 24.23 25.48 25.220

j 1.07 1.07 1.08 1.08 1.12 1.16 1.05 1.141

h 7.27 8.32 9.55 10.79 12.59 14.85 13.32 16.60q2p54 j 22.45 22.70 23.68 23.77 24.51 25.59 25.47 27.020

j 1.04 1.06 1.04 1.12 1.09 1.03 1.11 1.031

h 7.87 9.62 10.06 12.67 13.18 12.56 15.67 14.12q2p55 j 22.60 23.52 23.60 24.16 24.64 24.48 26.90 27.190

j 1.06 1.04 1.09 1.10 1.11 1.20 1.02 1.021

h 9.36 9.96 12.26 13.35 14.64 18.02 13.78 15.02q2p56 j 22.71 23.77 23.79 25.07 25.54 25.95 26.43 26.530

j 1.09 1.03 1.08 1.04 1.03 1.08 1.08 1.141

h 11.09 10.87 13.24 12.77 13.75 15.78 16.73 19.58q2p57 j 23.41 24.36 24.32 25.32 26.07 26.69 26.64 27.630

j 1.05 1.00 1.07 1.07 1.05 1.01 1.08 1.041

h 11.29 11.14 13.88 14.41 14.87 14.82 17.95 17.47q2p58 j 23.79 23.89 24.38 25.25 25.60 26.93 27.79 27.130

j 1.02 1.07 1.11 1.10 1.08 1.04 1.01 1.101

h 11.84 14.05 15.81 16.40 17.35 16.39 16.35 20.68

1 ˆThe approximate critical values Q(?) can be obtained by simulating the distribution of X. In our case, the random variableX corresponds to SupPR , AvPR , and ExpPR . We simulated R550 000 draws from the respective distributions using aT T T

grid of 3600 equally spaced points on [0, 1]. The critical value for the test with (1003a)% level of significance is given bythe (12a)3100 percentile of the vector of random draws sorted in ascending order.


Table 2Approximate distribution for AvgPR (p) with A5(0.15, 0.85)T

p51 p52 p53 p54 p55 p56 p57 p58

q2p51 j 20.74 21.20 21.70 21.98 22.37 22.92 23.49 23.880

j 1.46 1.51 1.48 1.58 1.65 1.57 1.59 1.581

h 2.17 3.32 4.26 5.90 7.49 8.06 9.24 10.37q2p52 j 20.98 21.38 21.91 22.38 22.77 23.18 23.41 24.060

j 1.37 1.45 1.49 1.47 1.51 1.52 1.62 1.591

h 3.13 4.43 5.52 6.50 7.79 9.00 11.19 11.77q2p53 j 21.18 21.76 22.04 22.58 23.15 23.69 24.07 24.560

j 1.36 1.39 1.46 1.47 1.45 1.47 1.48 1.451

h 4.24 5.21 6.74 7.68 8.47 9.56 10.69 11.43q2p54 j 21.54 21.90 22.37 22.93 23.33 23.61 24.24 24.830

j 1.33 1.34 1.38 1.36 1.45 1.47 1.48 1.481

h 5.11 6.16 7.30 8.02 9.75 11.12 12.05 12.89q2p55 j 21.72 22.35 22.15 22.71 23.90 23.60 25.10 24.420

j 1.32 1.32 1.47 1.44 1.34 1.49 1.37 1.541

h 6.22 6.89 9.58 10.22 9.49 12.67 11.35 15.65q2p56 j 21.80 22.44 22.82 23.58 23.77 24.60 24.89 25.150

j 1.34 1.31 1.33 1.36 1.40 1.39 1.40 1.481

h 7.55 8.02 9.16 10.00 11.63 12.07 13.37 15.54q2p57 j 22.34 22.79 22.83 23.91 23.82 24.86 24.69 25.860

j 1.29 1.30 1.36 1.36 1.45 1.34 1.44 1.391

h 7.95 8.97 10.75 10.96 13.56 12.61 15.48 15.04q2p58 j 22.47 23.28 23.41 23.28 24.24 24.58 25.45 24.930

j 1.29 1.24 1.33 1.42 1.37 1.43 1.38 1.481

h 9.17 9.14 11.21 13.76 13.62 15.45 15.28 18.68

other choices of P are available from the author upon request. In Table 4, we report the maximum andmedian error due to approximation over all combinations of p, q2p, and P, for a selected set ofp-values. An inspection of these tables shows that for most distributions the maximum error is thelargest for p-value equal to 0.80 (the error ranges from 0.0015 to 0.0066). For all other p-values, themaximum error due to approximations is less than 0.0035.

4. An illustration using CBAPM

We illustrate the calculation of approximate p-values through an application to CBAPM. We followthe treatment of this model as described in Hall (1993). The model is estimated using monthly data onpersonal consumption of nondurables and services from National Income and Product Accounts, andtwo choices of returns from the NYSE, namely, the equally weighted returns (EWR) and the valueweighted returns (VWR) over the period 1960:01 to 1991:12. The moment conditions are

g 21R Ct11 t11]] ]]E b 2 1 z 5 0 (13)FH S D J GtP Ct t


Table 3Approximate distribution for ExpPR (p) with A5(0.15, 0.85)T

p51 p52 p53 p54 p55 p56 p57 p58

q2p51 j 20.61 20.97 21.44 21.80 22.24 22.56 23.09 23.410

j 2.14 2.16 2.06 2.12 2.14 2.10 2.11 2.141

h 2.08 3.15 3.84 4.96 5.94 6.80 7.64 8.76q2p52 j 20.84 21.24 21.66 22.00 22.42 22.71 23.00 23.560

j 2.12 2.13 2.12 2.12 2.17 2.19 2.26 2.201

h 3.09 4.09 4.98 6.02 7.11 8.29 9.74 10.18q2p53 j 20.96 21.55 21.95 22.26 22.82 23.01 23.79 23.550

j 2.16 2.11 2.09 2.12 2.11 2.19 2.03 2.201

h 4.30 4.96 5.85 6.97 7.67 9.26 8.82 11.42q2p54 j 21.37 21.52 22.21 22.53 22.85 23.48 23.67 24.570

j 2.06 2.11 2.05 2.09 2.15 2.09 2.16 2.071

h 4.89 6.21 6.59 7.80 9.07 9.38 10.96 10.71q2p55 j 21.53 22.17 21.95 22.60 23.41 23.28 24.83 24.770

j 2.09 2.03 2.24 2.15 2.04 2.25 1.97 2.041

h 5.99 6.45 8.92 9.17 9.02 11.80 9.63 11.58q2p56 j 21.78 22.19 22.81 23.29 23.53 24.26 24.55 24.470

j 2.10 2.04 1.96 2.03 2.06 2.06 2.05 2.231

h 7.02 7.67 7.92 9.02 10.20 10.80 11.73 14.51q2p57 j 22.12 22.72 22.91 23.43 24.01 24.59 24.63 25.720

j 2.08 1.98 2.02 2.14 2.11 2.00 2.11 1.981

h 7.79 8.01 9.26 10.72 11.33 11.21 13.27 12.30q2p58 j 22.47 22.88 23.15 23.39 24.08 24.65 25.66 25.140

j 1.99 2.01 2.11 2.16 2.06 2.07 1.93 2.141

h 8.19 9.10 10.75 12.12 12.10 12.81 11.87 15.49

where b is the discount factor, R is the payoff of the asset in time t11, P is the price of the assett11 tgat time t, C is the consumption at time t, the utility function is U(C ) 5 C /g, g is the coefficient oft t t

relative risk aversion, and z is a vector of instruments. In this exercise, we set z 5 h1, C /C , r ,t t t t21 t21

C /C , r j9, where r 5 R /P . The matrix S in (3) is estimated ast21 t22 t22 t t11 t

T1ˆ ] ˆ ˆS 5 O(e z )(e z )9T t t t tT t51

g21ˆ ˆˆ ˆwhere e 5 br C /C 2 1, and (g, b ) are the estimated parameters. For each return index, wes dt t t11 t

estimated the model using the entire sample and calculated all versions of PR using P5[0.15, 0.85].T

These results are reported in Table 5 below. Hansen’s overidentifying restrictions test, denoted by J ,T2is asymptotically distributed as x -distribution with three degrees of freedom. Therefore, J for VWRT

is insignificant at the 10% significance level and J for EWR is significant at the 1% significanceT

level. Based on the full sample estimates, it is a natural next step to check if structural instability canexplain the rejection of the model for EWR. We calculate the approximate p-values for each statisticusing the results in Tables 1–3. For example, the calculated SupPR statistic with EWR is 18.7517T

and with VWR is 15.19. From Table 1, we find that the appropriate value of (j , j , h) is0 1

(22.72,1.07,8.32) with p52 and q2p53. The approximate p-value for SupPR with EWR isT


Table 4Absolute error in fitted distributions

p-value SupPR distributions AvgPR distributions ExpPR distributionsT T T

Median Maximum Median Maximum Median Maximumerror error error error error error

0.00 0.0001 0.0007 0.0001 0.0006 0.0001 0.00060.01 0.0003 0.0014 0.0004 0.0015 0.0003 0.00160.02 0.0005 0.0017 0.0005 0.0020 0.0005 0.00180.03 0.0005 0.0018 0.0005 0.0020 0.0005 0.00180.04 0.0005 0.0019 0.0005 0.0021 0.0005 0.00200.05 0.0005 0.0019 0.0005 0.0018 0.0005 0.00210.06 0.0005 0.0017 0.0005 0.0021 0.0006 0.00210.07 0.0005 0.0018 0.0005 0.0023 0.0005 0.00210.08 0.0005 0.0019 0.0005 0.0018 0.0005 0.00210.09 0.0005 0.0022 0.0005 0.0024 0.0005 0.00220.10 0.0005 0.0018 0.0005 0.0018 0.0005 0.00200.15 0.0005 0.0018 0.0006 0.0022 0.0005 0.00200.20 0.0005 0.0023 0.0005 0.0019 0.0005 0.00220.25 0.0006 0.0021 0.0005 0.0023 0.0005 0.00250.30 0.0006 0.0024 0.0006 0.0023 0.0006 0.00240.40 0.0005 0.0021 0.0006 0.0022 0.0006 0.00200.50 0.0006 0.0023 0.0006 0.0022 0.0006 0.00260.60 0.0006 0.0024 0.0006 0.0024 0.0006 0.00240.70 0.0008 0.0026 0.0007 0.0025 0.0007 0.00350.80 0.0015 0.0045 0.0016 0.0066 0.0016 0.0066

212x (22.7211.07318.7517u8.32)50.0315, and the approximate p-value for SupPR with VWR isT212x (22.7211.07315.19u8.32)50.1081. Hence, we find that the SupPR statistic is quite close toT

being significant at the 10% level. From Table 2, with p52 and q2p53, we have (j , j ,0 1

h)5(21.76, 1.39, 5.21). AvPR with EWR is equal to 9.0587 and so the approximate p-value isT212x (21.7611.3939.0587u5.21)50.0619. From Table 3, with p52 and q2p53, (j , j , h)50 1

(21.55, 2.11, 4.96), and the calculated ExpPR statistic with EWR is 5.6356. So the approximateT2p-value is 12x (21.5512.1135.6356u4.96)50.0646. We find that the calculated AvPR andT

ExpPR statistic with EWR is very close to being significant at the 5% level. Lastly, the approximateT2p-values for AvPR and ExpPR with VWR are respectively 12x (21.7611.3933.7716u5.21)5T T

20.6536 and 12x (21.5512.1133.3125u4.96)50.3597. Therefore, the availability of p-values allows

Table 5aSupPR , AvPR , ExpPRT T T

J SupPR AvPR ExpPRT T T T

VWR 2.9538 15.1900 3.7716 3.3125(0.0315) (0.0619) (0.0646)

EWR 11.6810 18.7517 9.0587 5.6356(0.1081) (0.6536) (0.3597)

a J is Hansen’s overidentifying restrictions test for the full sample; the values in the parenthesis are the associatedT

p-values. a, b, c denote significance at the 1, 5, and 10% levels, respectively.


the researcher to draw his /her own conclusion about the significance or insignificance of thecalculated test statistics.

Acknowledgements

I am grateful to Alastair Hall, Bruce Hansen, and V. Samarnayake for helpful comments. BruceHansen provided a copy of the program used to calculate the approximate p-values. All remainingerrors are our own.

References

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Approximate p-values of predictive tests for structural stability

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