A cointegration approach to estimating preference parameters

Journal

ELSWIER

of Econometrics

82 (1997)

107-134

A cointegration approach to estimating parameters

Abstract

In this paper, we cstimatc durable consumption, which moments (GMM). The GMM uidity constraints, aggregation

the (long-run) intertemporal elasticity of substitution of nonhas often been estimated with the generalized methods of estimator, however, is not cons&c& in the presence of liq-

over heterogeneous consumers, unknown preference shocks, form of time-nonseparability. We use Englc and Granger’s cointegration in order to develop an estimator which is consistent even in the presence

or a general methodology

of these factors. We then form a formal test that compares the estimates obtained using cointegration techniques with those obtained using GMM. @ I997 Elsevier Science S.A. Kay lrorck Consumption-based asset pricin g: Intertemporal JEi. ch.w$u~tiorr: E2 I: C22; C32

elasticity of substitution

1. Introduction In this paper, we develop an econometric method to estimate preference parameters by utilizing the information in stochastic and deterministic time trends. The first-order condition that equates the relative price and the

* Corresponding

author.

We are grateful to seminar participants North American Winter Meetings of the

at Carnegie Econometric

@eens University, University of Rochester, University conference on empirical applications of structural models too many pcoplc have provided valuable assistance in like to thank those who have provided us with written Hansen, Lars Hansen, Adrian Pagan, Robert Porter, aad Stanley Engerman for his help in obtaining data and to All remaining

shortcomings

0304-4076/97/Sl7.00

PIISO304-4076(97)00053-5

8

Mellon Society

University. in 1989.

of Toronto, the Wisconsin/E~~?to~~t~r~i~~~ in May 1990 for helpful comments. Though this project to mention by name, we would comments, including John Campbell, Brucs three anonymous referees. We are grateful to Meg McConnell for her research assistance.

are our own. 1997 Elsevier

Science

S.A.

All

Cornell University, the Northwestern University.

rights

reserved

contemporaneous marginal rate of substitution of two goods is used to derive the restriction that the relative price and consumption of the two goods are cointegrated. ’ The cointegrating vector involves preference parameters, which are estimated with a cointegrating regress&. In our application, we estimate the (long-run) intertemporal elasticity of substitution (IES) of nondurable consumption, which is a key parameter in a consumption-based asset pricing model (CCAPM). The parameter can also be estimated by Hansen’s (1982) GMM in a C-CAPM. The C-CAPM is rejected strongly by Hansen and Singleton ( 1982) when stock returns and Treasury Bill rates are used together. Possible reasons for the rejection of the C-CAPM have been pointed out. These include liquidity constraints (see, e.g., Hayashi, 1985a; Zeldes, 1989), unknown preference shocks (e.g., Garber and King., 1983), time-nonseparable preferences (e.g., Eichenbaum et al., 1988; Constantinides, 1990; Eichenbaum and Hansen, 1990; Ferson and Constantinides. 1991: Ferson and Harvey, 1992; Cooley and Ogaki 1996; Heaton, 1995), and small infonnation cost (e.g., Cochrane, 1989). GMM estimation of nonlinear Euler equations also assumes that there are no measurement errors. The main purpose of this paper is to develop an estimator, which is consistent even in the presence of factors such as liquidity constraints, aggregation over heterogeneous consumers, unknown preference shocks, a general form of timenonseparability, measurement errors, and the possibility that consumers do not know the true stochastic law of motion of the economy. The GMM estimator is not consistent in the presence of these factors, but the cointegrating regression estimator is consistent under certain assumptions. It is important to develop such an estimator because much recent research simulates economies with features that accounts for GMMs rejections of the C-CAPM such as liquidity constraints in recent works (see, e.g., Deaton, 1991, Marcet and Singleton, 1991; Heaton and Lucas, 19% j. An estimator, which is consistent in the presence of liquidity constraints, can be used to guide the choice of parameters for these simulations. We form a formal test that compares the estimates obtained using cointegration techniques with those obtained using GMM in the spirit of Hausman’s (1978) specification test. Since the GMM estimator is not consistent but the cointegrating

’ In this paper. we use Houthakker’s (1960) addilog utiliry function. The cointcgration approach can also bc used to estimate the curvature parameters of the extended addilog utility function as in Atkeson and Ogaki (1996). and the CES utility function as in Ogaki and Reinhart (IYYS). De&on and Wigley (1971). Deaton (1974). Miron (1986). and Ball (lY9Gj. among addilog utility functions. Ogaki ( 1988) introduces the cointegration approach parameters of the addilog utility function. Ogaki (1992) uses the cointegration income elasticities for credit goods: Coolcy Amano and Wirjanto demand; and Amano independently. Clarida elasticities fo,, imported

others. have estimated to estimate preference approach to estimate

food and other goods; Braun (1994). to estimate a utility function for cash and and Ogaki (1996). to estimate a utility function for consumption and leisure; (1996) and Amano, Ho. and Wirjanto (1906) to estimate models of import and Wirjanto (1997)‘to estimate a model of government spending. Working (1993. 1994) estimates addilog utility functions to estimate price and income goods with cointegrating regressions.

regression estimator is consistent in the presence of factors such as iiquidi~ constraints, this test can be interpreted as a test for the C-CAPM against an alternative hypothesis that such factors are present. The rest of this paper is organized as follows. In Section 2, the preferences of the representati.*.e consumer are specified, and a restriction on the trends of the relative price and consumption of two goods is derived. We examine the implications of the restriction in terms of stochastic and deterministic cointegration. We discuss the intuition behind the cointegration approach and the robustness of our approach as compared to GMM. Section 3 describes our econometric procedures, and Section 4 discusses the data. In Section 5, the results of the cointegrating regressions are presented, and our estimates of the long-run IES of nondurable consumption are compared with those obtained by GMM. Section 6 contains concluding remarks. 2. The cointegration approach In this section, we derive a restriction on the trends of economic variables from a first-order condition which equates the relative price and the marginal rate of substitution. This restriction implies that the relative price and real consumption expenditures are cointegrated. We discuss the intuition and robustness of our results.

The present paper employs the addilog utility function proposed and estimated by Houthakker (1960). The addilog utility function assumes that preferences are represented by an isoelastic form for each good and that goods are additively separable Specifically, consider an economy with n goods. Suppose that a representative consumer maximizes the lifetime utility function U = E.

[ 1 .&h(t) t-tJ

(2.1)

in a complete market at period 0, where E,(e) denotes expectations conditional on the information available in period t.2 The intra-period utility function is assumed to be of the addilog form (2.2)

I! The existence of a representative consumer under complete markets is discussed by Ogaki ( 1997) for the general concave utility function and by Atkeson and Ogaki ( 1996) for the extended addilog utility function. WC will discuss aggregate conditions for the cointegmtion approach under incomplete markets.

where xi>0 for i = I,..., II and 0;‘s represent preference shocks. Here the stochastic process {[o,(t), . . . , o,,(t)]‘: - 00
Si(t)=

l)+.--+uiCi(t

-k)}exp(Ujt)

(2.3)

for i = 1, . . . .I?, where Cl(t) is real consumption expenditure for good i in period t. Following Eichenbaum and Hansen (1990), we allow for the possibility of technological progress in the transformation of purchases of good i into Si(t) in (2.3) via the exponential deterministic trend exp(Qgt). Below, we will consider the case in which the 0;‘s are known to be zero as well as the case in which the 0;‘s are unknown. Note that the purchase of one unit of good i at period t increases .S;(t + T) by ai, exp(U;(t + T)) units for nonnegative TGk. This type of method of specifying time-nonseparability is used by Hayashi (1985b), Eichenbaum et al. ( 1988), Eichenbaum and Hansen ( 1987), and Heaton ( 1995), among others. In our empirical work, we take a measure of nondurable consumption as one good (say good I ) and interpret the curvature parameter for nondurable consumption (XI) as the long-run intertemporal elasticity of substitution (IES) for the nondurable consumption. 3 As we will discuss in Section 2.4, this interpretation relies on the assumption of additive separability across the goods. It should be noted that this separability assumption is already made in Hansen and Singleton ( 1982) and Ferson and Constantinides ( 1991), both of which use the GMM approach and are closely related to this paper. Let Pi(t) be the purchase price of consumption good i. We take good 1 as a numeraire for each period: P,(t) 3 1. The first-order condition which equates the relative price between good i and good l(fi(t) = pi(t)//‘,(t)) with the marginal rate of substitution of these goods is Pi(t) =

cU/(;Ci(t) xJ/c’;C,(t)

Et [~~,$‘i~(t = Et &#rCu(t

+ r)/‘ZCi(t)] + 7)/W,(t)

E, C~=“/Y~i(~ + T)u,’ exp(OJf(t + = E, [‘&#r6~(t

T)){+Si(r

1 +

+ r)at exp(o,{(t + r)){&(t

T)}-“]

+ r)}-“]

’

(2.4)

3 This parameter is the long-run IES for nondurnhle consumption when we allow current and past consumption to adjust. When preferences are time nonseparable. the short-run IES is different from the long-run

IES because

we take

past

consumption

to be fixed

in the short-run.

This first-order condition forms the basis of the cointegration approach and summarizes the information needed from the demand side. In order to m supply side in the simplest way, let us consider an endowment econom production. Let C:(l) be the endowment of good i and c:(t) = log(CT(t)). In equilibrium, c,(t) = log(Ci(t)) = c;(t). In a production economy, we require that equilibrium consumption satisfy the trend properties we assume for C,*(I). The trend properties of equilibrium consumption are likely to be closely related to those of the technology shock to the good i industry in a production economy. We consider three alternative assumptions about the trend properties of C;(I). In each of the three assumptions, C~(t)/C~(t - 1) is stationary for all i. This insures that Sj(t)/{Cj(t)exp(U~t)} is stationary in equilibrium. To see this, let SF(t) be the St(t) implied by Cr and note that CT
= {&i”(f

+ T)/Ci”(f) + ai,Ci”(t + T - 1)/c;(f)

+niCi*(t + T - k)&+(f)}

exp(Uf7)

(2.5)

is stationary. We also make an extra assumption that the expectation of a stationary variable conditional on the consumer’s information set is equal to the expectation conditional on the stationary variables included in his information set. Then ~~(t)exp(~~f)[C~(t)exp(~l~~f)]~“’/{exp(~~t)[C~(f)exp(U,“t)]~~~} is stationary because the right-hand side of

E, [ ,&PW = E,f=JPfq(f

+ r) oi,exp((1[7){S~(t + s)/[C;(l)exp(0,r)]}-‘~~ -t r)4 exp((&t){S;(t

+ T)/[C;(f)exp(4f)]}-tI] (2.6)

is stationary. The right-hand side of (2.6) is the ratio of conditional expectations of the functions of stationary variables. Taking the natural log of the left hand side, we conclude that p;(f) - xlc;(f) + xjctF(f) + (1 - xl)tl;f - (1 - xi)Oft is stationary, where m(t) = log(P;(t)), c;(r)= log(C~(r)) for i = i....,n. We shall call this restriction the stationarity restriction. This restriction implies that pi(t) - ztcT(r) + x;cT(t) is trend stationary in general, and is stationary if and onlyif(l-rt)Of-(I-xi)(,“‘=O.

2.2.

Cointqrution

This section defines notions of stxhastic cointegration, the deterministic cointegration restriction, and cotrending. 4 These notions are useful for discussing implications of the stationarity restriction under ahcmative assumptions. When a scalar stochastic process is stationary after first differencing UIZCIhas positive spectral density at frequency zero, the process is said to be djfference stationary. A trend stationary process is also stationary after first differencing but has zero spectral density at frequency zero. A scalar difference stationary process l,(t) and a vector difference stationary process X(t) are said to be cointegrated with a normalized cointegrating vector Y.~if up - &Y(t) is stationary. Let X(f) be a k-dimensional difference stationary process: X(t) - X(2 - 1) =

/I,

+

(2.7)

t+(t)

for t > 1 where lr, is a k-dimensional vector of real numbers and I:.\ with mean zero. Then recursive substitution in (2.7) yields X(r)

=

j1.J

is stationary (2.8)

-t-P(r),

where X0(r) is x”(t)

= X(0) + &f).

(2.9)

r=I

Relation (2.5) decomposes the difference stationary process X(t) into deterministic trends arising from drift 11.rand a difference stationary process without drift, X’(f). Suppose that J’(I) is a scalar difference stationary process with drift jc,.. Decomposing JY(~)into a deterministic trend icxt and a difference stationary process without drift v,“, as in (2.8), yields y(t) = pyt + yyt).

(2.10)

Difference stationary processes y(t) and X(t) are said to be .stoclttrsticc~ll~ with a nornwlixcl cointcyrrrting uector ;‘.\ when there exists a k-dimensional vector ;‘,r such that y’(r) - ;.$Y”(f) is stationary. Stochastic cointegration only requires that the stochastic trend components of the series are cointegrated. We may write u”(t) - $X0(r) = UC+xJt), assuming that v”(f) - $X0(f) has mean UC. Here c&t) is stationary with mean zero. Then by using (2.7) and (2.10), we obtain

coinlqruli’crted

y(1)

J In Ogaki

=

o< +

(1988).

/lJ

+

stochastic

;$Y(t)

+

(2.11)

cc(t),

cointcgration

is called

the stochastic

part

of cointegration

and

the

deterministic cointegration restriction is called the deterministic part of cointegration. Campbell and Perron (I991 ) cxrend 3gki and Park’s (1989) definition of the deterministic cointcgration rcstriction. and Chapman and Ogaki (1993) extend the dclinitton of cotrwding to more gcncral forms of deterministic trends.

(2.12) Suppose that a vector ;$ satisfies Cl! = ;$‘/I,.

(2.13)

Then Y(t) - $X(t) does not possess any deterministic trend, and Y(r) and X(t) are cotwncld with a norndixd cotrending cector ;f. If k > 1 and if one of the components of Al.,.is nonzero, there are infinitely many cotrending vectors. Consider an extra restriction that the normalized cointegrating vector 3 is a cotrending vector. This restriction, which we call the cktermini.stic cointqpution restriction, requires that the cointegrating vector eliminate both the stochastic and deterministic trends. In this case, y(f)

=

Oc + 7.:X(t)

(2.14)

+ t:Jtj.

Consider a cointegrated system involving a trend stationary process z(t): z(t)

=

oz +

p;t

(2.15)

-I- c,(t),

where E,(t) is stationary with zero mean and ;r, #O. Suppose that an economic model implies the restriction that y(t) - $X(t) - Liz is stationary. Since y(t)-fX(t)-

y-z(t)=

--;g1:

$(jlJ

-;&

-;‘=p=)t+

{y”(t)-y.;xo(t))

-;‘,E(6),

this restriction implies that y(t) and X(t) are stochastically cointegrated with a normalized cointegrating vector ;‘.r and that y(t) and [X(t)‘,z(t)]’ are cotrended with a cotrending vector [;$;,J: jLr

= ;‘.&

+

;‘=/l:.

(2.16)

From (2.16) and (2.12), we obtain (2.17)

;‘: = pcJpz.

Relation (2.17) can be used to estimate 7:. 2.3. hpliccrtions

of the strrtioncrrity restriction

In this section, we study the implicattons of the stationarity restriction. We consider only the pair of good I and good 2 since our results generalize to any pair of goods. The stationarity restriction is a result of the assumption of the long-run stability of preferences. Preference parameters can be identified from the stationarity restriction if the supply side is substantially more volatile than the demand side in the long-run. This requires the assumption that at least one

of c;(z) and c;(t) has a stochastic trend. 5 Stable preferences and technological shocks with stochastic trends seem to be plausible assumptions for identification. First, consider the case in which both CT(~) and c,*(t) are difference stationary: ” Aswuptim

ICI: The process {c;(t): t 30) is difference stationary for i = 1,2.

Assttmptiotz lb: The processes {c;(t): t >O} and {c;(r): t 20) are not stochastitally cointegrated.

Assumption I b will be satisfied for equilibrium consumption in a production economy if the technological shock in the good I industry has a different stochastic trend component from the technological shock in the good 2 industry. Under Assumption I, the stationarity restriction implies that pz(t) - xlcT(t) + x&(t) is trend stationary. Thus (pr(t).c;(t),c,“(t))’ is stochastically cointegrated with a cointegrating vector ( I, --XI, x2)‘. However, the deterministic cointegration restriction is not necessarily satisfied under Assumption I. The stationarity restriction implies that y?(t) - x,cT(t) + x&(t) is stationary under the condition that there is no technological progress in the transformation technology from consumption purchases to service tlows (namely, Oj = 0 for i = I, 2). Hence, consider the following assumption: A.s.sunption 2.

Assumption I is satisfied and (c = 0 for i = I, 2.

Under Assumption 2, (pz(r), cf(t), c;(t))’ is stochastically cointegrated with a cointegrating vector ( I, -XI, x2 )’ and satisfies the deterministic cointegration restriction. Second, consider the case where the log of the endowment of good I is trend stationary and that of good 2 is difference stationary. Asst~ntption 3r1. The process {c;(t) : t Z 0) is difference stationary and the process {c;(t): t >O} is trend stationary with a nonzero linear trend.

Assumption 3a will be satisfied for equilibrium consumption in a production economy if the technological shock in the good I industry is dil‘ference stationary and the technological shock in the good 2 industry is trend stationary.

5 Ogaki (1988) develops an econometric method based on GMM which uses the information deterministic trends to estimate the preference parameters of the addilog utility fimction when of C;(I) and c;(r) are trend srationary. ‘I.4

C:(I)

specinl case is that c:(f) is lognormally distributed.

and c’;(t)

arc martingale

when

the real interest

rate

is constant

in both and

Under Assumption 3a, the stationarity restriction implies that p?(t) and c;(r) are stochastically cointegrated with a cointegrating vector ( 1 - xl )‘. Assumption 3a is enough to identify XI. In order to identify x2 as well as ~1, we need Assumption 3b. Under Assumption 3 (3a and 3b), the stationarity restriction implies that (~l(t),cT(f ),cz(f))’ is cotrended with a cotrending vector (1, --zl, x2)‘.

We have shown that the first-order condition (2.7) implies a stationarity restriction and that this restriction leads to cointegration restrictions. These restrictions can be nsed to estimate preference parameters and to test the first-order condition. In this section, we discuss the intuition behind these results and compare our approach with the GMM approach. 2.4.1. The Cobb-Dmgirs

utility firnction

We provide an intuitive explanation for the stationarity restriction by comparing the addilog utility function with the Cobb-Douglas utility function. First consider the addilog utility function with x1 = x2 = . . . = x,~= 1. in this case, the utility function is a monotone transformation of the Cobb-Douglas utility function and is homothetic. The stationarity restriction implies that pz(t) - c;(t) + c:(t) is stationary. The Euler equation literature for multiple-good models often uses the Cobb-Douglas utility function that is not additively separable (see, e.g., Dunn and Singleton, 1986). Eichenbaum and Hansen ( 1990) and Ogaki ( 1992) have shown that the first-order condition for the Cobb-Douglas utility function implies that pz(t) - c;(t) + c:(t) is stationary even without the separability assumption. ’ We depart from homothetic preferences, by using the addilog utility function, which is nonhomothetic if xi # Zj for some i and j. In the addilog utility function in Eq. (2.2), the curvature parameters govern total expenditure elasticities in the following sense. Imagine that a consumer with this utility function is able to trade service flows in spot markets and let e:‘(f) be the price of Si(t)e Fix all S;(f) @CO, I,...) except for Si(t) for i = 1,. . . , N. Consider the problem of maximizing (2.2) subject to a constraint c:= , e”(t)Si(t) = E(t). Define total expenditure elasticity, ei, by ei = dS,(t)/SE(t). It is easy to show that 4 = c/x, for some constant c which depends on t in general. Thus, Xi/Xj is the ratio of total expenditure elasticities of good j and good i. To develop the intuition for the stationarity restriction, imagine that the relative price p?(r) is stationary and 0; = 0; = 0 for simplicity. If xl =x2 = 1, then pz(l)cl(t) + c?(t) does not possess any time trends, and therefore cl(i) and q(t)

‘The curvature paramctcrs cstimakd intertemporal elasticities of substitution scparuble.

from cointcgnting regressions with the Cobb-Douglas utility

cannot function

be iuterpreted as the that is not additively

must grow at the same rate in the long run. On the other hand, the addilog utility function implies that p?(r) - rlcl (t) + x&t) is stationary. When ZI > x2, good I has a lower total expenditure elasticity than good 2 and in the long-run consumption of good 1 grows at a slower rate than does consumption of good 2. The curvature parameters that are estimated from cointegrating regressions can be interpreted as preference parameters which govern total expenditure elasticities without the additive separability assumption (see Ogaki, 1992). On the other hand, the interpretation of an estimate for I/xl from a cointegrating regression as the long-run IES depends on the assumption that nondurable consumption is additively separable from other goods across time. Intuitively, we identify the long-run IES because it is harder to make an intertemporal substitution for a necessary good than that for a luxury good. ’ We perform a test of this interpretation based on the separability assumption in Section 5.2. Note that the separability assumption is often made for GMM estimation (see, e.g., Hansen and Singleton, 1982; Ferson and Constantinides, 199 1).

One remarkable feature of cointegration techniques is that structural parameters can be estimated consistently without the assumption that the regressors are econometrically exogenous. This property is important because in most stochastic and dynamic rational expectations equilibrium models, few economic variables are econometrically exogenous. We can allow for measurement error without assuming that the regressors are uncorrelated with the measurement error of the regressand. The only assumption needed is that the multiplicative measurement error for each good is stationary. In contrast, Hansen and Singleton’s (1982) GMM approach does not allow for measurement errors unless they are of very special form. Garber and King ( 1983) point out that unknown preference shocks can explain the empirical rejections of the C-CAPM by the GMM approach. The cointegration approach allows for preference shocks since the stationarity restriction is robust to stationary unknown preference shocks. ’ The cointegration approach does not allow for permanent preference shocks with a unit autoregressive root, however. At least for the aggregate level of consumption such as nondurable consumption, permanent preference shocks are unlikely relative to permanent shocks in productivity. Though permanent preference shocks are assumed away, we do allow for habit formation through time-nonseparability. Constantinides (1988) shows that the

*There Ogaki

is no thcorctical

( 1996)

L, Recent Stockman

provide

r~‘usun

for

this

some

empirical

evidcncc

empirical

work

on equilibrium

and Tcsar

(1995).

intuition

unless

for this mu&Is

with

prefcrcnccs

are rcstrictcd.

Atkcson

and

intuition. prefcrcncc

shocks

include

Pnrkin

( 1988).

addition of habit formation could help to explain asset market data. Dumbi~i~ of goods is a source of time-nonseparability as in Mankiw (1982) and Dunn and Singleton (1986), among others. Even goods that are usuaily labeled as nondurables may have durability (see, e.g., Hayashi, 1985b, Eichenba~m and Hansen, 1987; Eichenbaum et al., 1988). Heaton (1995) investigates interactions between durability and habit formation for the C-CAPM. When time-nonseparability is allowed in the GMM procedure, as in Eichenbaum and Hansen (1987) and Eichenbaum et al. ( 1988) a particular form of time-nonseparability must be employed to obtain estimates of service flows. The form of the time-nonseparability should be limited so that the number of free parameters is not too large. The cointegration approach allows the estimation of the curvature parameters without requirin, ” the estimation of the form of timenonseparability. The cointegration approach permits aggregation over heterogeneous consumers under certain conditions. A sufficient condition is that the curvature parameters, the cx[.s,are identical across the consumers and that the difference between the log of the average consumption and the average of the log of consumption is stationary. In order to see this, suppose that there are N consumers in the economy and that the consumption expenditures of each consumer satisfies the stationarity restriction. Let C;(r) be consumption of good i by consumer j and assume that we observe equilibrium consumption, CT(r) = (l/N) xy=, C/(t). Then log(Pz(r)) - rl(l/N)~~~=, log(C{(t)) + x2( l/N)xT=, log(C{(t)) is trend stationary. This process is dtfferent from observable pz(t) - xtc;(t) + x&(t) only by the difference between the log of the average consumption, log(C,“(r)), and the average of the log of consumption, ( l/N) x,y=, log(C,‘(r)). A sufficient condition for this difference to be stationary is that the ratio of the aggregate consumption to individual consumption, C~(r)/C,!(t), is stationary. This ratio is actually constant under complete markets (see, e.g., Cochrane ( 1988) and Mace (1988) for conditions under which this holds for time-nonseparable preferences). The assumption of interior solutions for consumption is stringent at the individual level for durable goods. However, if we interpret c:(t) for durable consumption in our model as the target variable and if the departure of measured consumption from the target variable is stationary as in Caballero’s (1993) model, then the stationarity restriction will be satisfied by the measured consumption. It should be noted that the first-order condition S(t)= (ilU,@C~(t)}/{Z~/iCt (t)} is not affected by intertemporal market imperfections such as liquidity constraints when we assume interior solutions for consumption (Ci(r ) > 0 and Ct (t ) > 0). This first-order condition must be satisfied as long as good i and good I can be exchanged at the rate S(t) in period t regardless of the shapes of the intertemporal budget constraint. Hence the cointegration approach allows for liquidity constraints PS long as the ratio of the aggregate consumption to individual consumption, C~(r)/C/(t), is stationary, as argued above. This ratio is stationary

in a stationary Markov equilibrium in Marcet and Singleton’s (1991) single good economy with liquidity constraints. The first-order condition (2.4) equates the relative purchase price of consumption goods with the marginal rate of substitution for purchases of consumption goods. Eq. (2.4) does not focus on the relationship between user costs and the marginal rate of substitution for services. Liquidity constraints may contaminate the relationship between user costs and the intraperiod marginal rate of substitution for service flows. The relationship between the relative purchase price and the marginal rate of substitution for purchases of consumption goods is more robust. Also note that the stationarity restriction only involves the relative purchase price and the purchases of consumption goods. Consequently, the cointegration approach only requires data on purchase prices and purchases of two consumption goods and does not need data on user costs and service Rows that are not directly observable.

3. Econometric procedures In this section, we explain econometric procedures in this paper. We use Park’s ( 1992) canonical cointegrating regressions (CCR) to estimate cointegrating vectors because the OLS estimator is not efficient. lo The basic idea of the CCR is to estimate long-run covariance parameters and to transform the regressand and the regressors in order to remove the endogeneity problem while maintaining the cointegration relationship. The CCR estimators ate asymptotically efficient and have asymptotic distributions that can be essential!y considered as normal distributions conditional on the realizaticq of X(t), so that their standard errors can be interpreted in the usual way. There are other asymptotically efficient estimators. One reason to use the CCR is that Monte Carlo simulations in Park and Ogaki ( 199lb) have shown that the CCR estimators have better small sample properties than Johansen’s (1988) ML estimators in terms of the mean square error even when the Gaussian VAR structure assumed by Johansen is true. Another advantage to the CCR is that Monte Carlo simulations in Han and Ogaki (1997) have shown that Park’s (1990) tests of the null of stochastic cointegration and of the deterministic cointegration restriction have reasonable size

“‘For

an intuitiveexplanationof

the CCR

procedure,

set Ogaki

( 1993a),

whose

notationsfor

test

statistics WC follow. OLS is not &Gent because the errtir turm. the log of the right-hand side of (2.6) plus any stationary deviations caused by factors such as measurement errors. is correlated with the first dilTerencc of regressors at leads and lags as well ns contempnrily. All estimation and testing in the present paper except for those reported in Section 6 is done by Ognki’s ( 1993b) GAUSS CCR package.

and power. Park’s tests are based on Wald tests for spurious deterministic tren in the CCR procedure. ” In particular, we test the null of the deterministic cointegration restriction by testing if a linear trend is significant in a cointe~ating regression. It should be noted that cointegration is taken as the null hypothesis. In the standard procedures testing for cointegration (see, e.g., Engle and Granger, 1987) no cointegration is taken as the null hypothesis. Failure to reject the null of no cointegration is often interpreted as evidence against economic models which imply cointegration. However, these tests are known to have very low power against some alternatives and may fail to reject the null of no cointegration with high probability when the economic model is actually consistent with data. It is therefore preferable to test the null of cointegration in order to control the probability of rejecting a valid economic model. We apply (egression (2.11) to (,~(t).r’;(t),c~(t)) under Assumption I and to (pz(t),cT(t)) under Assumption 3. We apply regression (2.14) to (p#),cf(t),c,* (t)) under Assumption 2 since in this case the model implies the deterministic cointegration restriction. We impose the deterministic cointegration restriction by removing a time trend, as in (2.14). Efficiency gains from the imposition of the deterministic cointegration restriction have been discussed by West (1988) for the special case of one stochastic trend in the regressors and by Hansen ( 1989) and Park ( 1990a) for more general cases. The mode1 leads to a cointegrated system involving a trend stationary process under Assumption 3. in this case, we apply Park and Ogaki’s ( 199 I ) seemingly, unrelated canonical cointegrating regressions (SUCCR) to a system consisting of (2.11) and (2.15) to estimate ;:r, [lc, and [tI=. Then (2.17) is used to obtain an estimate for ;rz from these estimates of jtr, and ~1~. In cointegrating regressions, any variable can be used as the regressand alrd estimates will depend on the choice of the regressand in finite samples (see, e.g., Engle and Granger, 1987). Phillips and Park’s (1988) results indicate that the regressand should be chosen so that the parameters of interest are estimated linearly. For our purpose, the long-run IES for nondurable consu tion, l/xl, is the most important parameter. Under Assumption 1 and 2. we have ;‘.V-(-l/zt,r&t~) with Y(t)=,,(t) and X(t)=(&t).cz(t))‘. Under Assumption 3, we have ;*.r= I/xl and ;‘== &xl with Y(t) = ct(t),X(t) = pz(t), and z(t) =q(t). Under this assumption Q/XI =cI,//L,. Thus under any of the three assumptions, l/xl is estimated linearly when ct is used as the regressand. Hence we report results with this choice of the regressand and report some sensitivity analyses with respect to the choice of the re?ressand.

” These tests . . have power tcgration. it should be noted in small samples Pcrron, I991 ).

when

against misspeciticd orders of deterministic trends that virtually all unit root tests have either size

the orders

of dctsrministic

trends

are misspccified

(see,

as well as no coinor power problems e.g.,

Campbell

and

The CCR and SUCCR procedures need estimates of long-run covariance parameters. For this purpose, we use Park and Ogaki’s (1991 b) VAR prewhitening method with Andrews’s ( 1991) automatic bandwidth parameter. ” Following the recommendation of Park and Ogaki ( 1991b), we report the CCR estimators based on the third stage. Test statistics for the null of stochastic cointegration and the null of deterministic cointegration from the fourth stage CCR are used, as recommended by Han and Ogaki ( 1991). I3 As recommended by Park and Ogaki ( 199la) for SUCCR, long-run covariance parameters are estimated by the third stage CCR for ( 13). 4. Trend properties oi the data

In this section, we test the empirical validity of Assumptions 2 and 3. In the first subsection, we explain the data. In the second subsection, WC report test results.

We use seasonally adjusted quarterly data in the nationrc! income and product accconts (NIPA). Real consumption expenditures are those in constant 1982 dollars. TWO measures of nondurable consumption are used alternatively as good I. These are nondurables plus services (NDS) and nondurables (ND). Two mea-

sures of durable consumption are used. These are durable consumption in the NlPA (DN) and real durable consumption based on Gordon’s (1990) estimates for durable goods prices (DG). Gordon (1990) adjusts NIPA data for quality improvements in durable goods. Since Gordon’s data are annual, we construct quarterly data from 1947 to 1983 by applying Chow and Lin’s ( 1971)

method for distribution

to the level of real annual DG with a constant, time,

time squared, and the level of real quarterly DN as the regressors. Durables consumption is used as good 2 when NDS is taken as good I; durable consumption and services (SER) are alternatively used as good 2 when ND is taken as good 1. ‘) For C;(t) and C;(t) in the model, real per equivalent adult consumption

expenditures are constructed by dividing personal consumption expenditures in

“The prcwhitening is based on VAR of order based on AR( I ) for each disturbance term. are

I3 For this fourth stage. the singular bounded by 0.99 and the bandwidth

Monte

Carlo

valces of parameter

one and

the automatic

the VAR co&icient is bounded by fi

bandwidth matrix as in

for Han

pammctcr

is

prcwhitening and Ogaki’s

simulations.

tJ Real annual DG is obtained by dividing annual nominal durables in the NIPA by the implicit dellatar for durables in the NlPn zwltiplied by column (5) of Gordon’s ( 1990) Table 12. I I. We thank Robert Gordon for suggestive Chow and Lin’s method to us.

constant 1982 dollars by a measure of the equivalent adult population. This measure is constructed by P16 + OS( POP-PI 6) where P 16 is the civilian noninstitutional population I6 years of age and over and POP is the total population including armed forces overseas. The implicit deflator is used as the price for each consumption series. The implicit deflator for each series is constructed by dividing personal consumption expenditure in current dollars by that in constant 1982 dollars. The sample period extends from the first quarter in 1947 to the first quarter of 1989. except in the cases involving DG, when it runs from the first quarter in 1947 to the fourth quarter of 1983.

In order to confirm a well-known result that the null of difference stationarity cannot be rejected for real consumption (see, e.g., Eichenbaum and Hansen, 1990) for our data, we apply Park’s (1990) J( I, 5) test to the log of real per equivalent adult consumption expenditures. The J( I, 5) test does not require the estimation of the long-run variance and has an advantage over Phillips and Perron’s (1988) test and Said and Dickey’s ( 1984) test in that neither the bandwidth parameter nor the order of autoregression needs to be chosen. Park and Choi’s (1988) Monte Carlo experiments show that the J( I, 5) test has stable size and is not dominated by Phillips-Perron and Said-Dickey tests in small samples in terms of power. We also report results for the Said-Dickey test because it is often used in the literature. The first panel of Table I reports test results. The J( I, 5) test does not reject the null of difference stationarity for any of the five series of real consumption. Let SD(r) denote the Said-Dickey test with r lags. The SD(l), SD(4), and SD(7) tests are not significant at the 10% level for NDS, ND, DN, and SER. though the SD(4) and SD(7) are close to the 10% critical value for DN. Actually, SD(5)= -3.292 and SD(6)= -3.173 are significant at the 10% level for DN. The SD(l), SD(4) and SD(7) tests reject the null at the 10% level for DG. We apply Park’s ( 1990) G( I, q) test for the null of trend stationarity. Based on their Monte Carlo simulations, Kahn and Ogaki ( 199 I ) recommend small (1 when the sample size is small; we use 4 =2 and 3. For the estimation of the long-,un variance. we use Andrews’s (1991) QS kernel with the automatic bandwidth parameter estimator based on AR( I ). I5 The second panel of Table I reports results for the null of trend stationarity. Except for DN and DC, at least one of G( I, 2) or G( l,3) is significant at the 10% level for NDS, ND, and SER.

Is Kahn method 0.

for

and these

Ogaki tests,

(1991)

do

which

tends

not

recommend

to reduce

Andrews

the power.

and The

Monahan’s

bandwidth

(1992)

parameter

prewhitening is bounded

by

Tahlc Tests

I for

trend

propcrties

of red

consumption

NDS Tests

for the null

J(l.5)”

for the null

G(l.3)C

NDS,

ND,

from the NIPA, “Critical values

of trend

I.158 -2.078 -2.499 -2.408

DG

SER

0.686 -2.880

0.750 -3.246d

5.820 - I .455

-3.014 -3.099

-3.1 IId -3.341”

- I .O85 -1.151

stationarity

3.454” (0.063)

(0.045)

I .596 (0.120)

I .522 (0.217)

4.354’ (0.045)

5.091d (0.078)

4.233 (0.120)

I .634 (0.442)

3.816 (0.148)

(0.025)

SER.

DN.

4.0 I 6C

and DG stand

and durablcs for the I%.

from Park and Choi ( 1988). h SD(r) denotes the Said-Dickey lcvcls arc -4.01, -3.44, and c P-values are in parenthcscs. d Significant u Significant

DN

stationarity

- I .887 - I .996 -2.1 I4

G( 1.2)C

Note:

of diffcrcnce 2.1 13

SD( I )h SD(4)” SD(7)h Tests

ND

for nondurables

and services,

from Gordon (I 990). respectively. So/b, and 10% significance levels arc 0.123,

nondurablcs. 0.295,

7.396”

services, and 0.452.

durables These

arc

test with I’ lags. Critical values for the 1%). 5%. and IO% significance -3.14. Thcsc are from MacKinnon (1990) for T = 170.

at the 10% level. at the 5% level.

In light of these results, we employ the following specifications for the remainder of the empirical results reported in this paper. We specify the log of real consumption of NDS, ND and SER to be difference stationary. We try both the difference and trend stationarity specifications for DN and DG as a sensitivity analysis. When the specification that both cl and c2 are difference stationary is used, as in Assumptions I and 2, assumption I b needs to be satisfied. Table 2 reports test results for assumption lb while maintaining Assumption la. We use Park’s ( 1990) I( I, 5) test and Engle and @anger’s (1987) Augmented Dickey-Fuller (ADF) test for the null of no stochastic cointegration. The I( I, 5) test and the ADF test basically apply the J( I, 5) test and the Said-Dickey test to the residual from a static OLS cointegrating regression. The asymptotic properties of the ADF test are studied in Phillips and Ouliaris (1990) and MacKinnon (1990) provides critical values for the tests yielded by extensive Monte Carlo simulations. We include a constant and a trend term in the regression and ct is used as the regressand. These tests do not reject the null of no stochastic cointegration at the 5% level. Thus, the results are in favor of Assumption lb when Assumption la is maintained.

Table Tests

2 for no stochastic

I( 1.5)” ADF( I tb ADF(4)b ADF(7)b

cointqration

of ct and Q

DN/NDS

DG/NDS

DNiND

DG’ND

SER,ND

3.204 - 1.757 - 1SO6 - I .862

1.313 -3.561’ -2.646 -3.204

1.942 -2.087 - I .903 -2.153

0.796 -3.494 -2.553 -2.978

0.84X -3.164 -3.142 -2.819

Note: DNiNDS stands for et = NDS and cz = DN, etc. No test statistic reported in this table significant at the 5% level, which is cvidencc in favor of Assumption lb (2b) when Assumption (2a) is maintained. ‘I Critical Values at the 1%. 5%. from Park et al. (1988). b ADF(r)

denotes

Augmented

significance levels are -4.42, for T= 170. c Significant at the 10% lcvcl.

and

10% significance

Dickey-Fuller -3.84,

and

test with -3.54.

lcvcls

arc 0.103.

r lags. Critical

These

critical

0.251

values

values

and 0.384.

at the

are from

These

1%. 5%. and MacKinnon

is la are 10%

(1990)

5. Empirical results

Table 3 reports results from cointegrating regressions for five pairs of goods. For each of these pairs, results under alternative assumptions are reported. Since the CCR estimation and testing procedure is relatively new in the literature,. WC also report ADF test results for the null of no stochastic cointegration based on OLS with a constant and a time trend, and OLS estimates for selected specifications for each pair of goods. I6 The K statistic is the Wald test for the hypothesis xt = x2 = I, which is implied by the CobbDouglas utility function, I7 This hypothesis is decisively rejected by the K test in all casts for ND, for NDS and DN under the assumption that both consumption goods are ditference stationary (Assumptions I), and under the assumption that nondurable consumption is difference stationary and durable consumption is trend stationary (Assumption 3). It is also decisively rejected for NDS and DG under Assumption I, and under the assumption that both consumption goods are difference stationary with no technological progress in the technology

” Under

Assumotion

I, the ADF

is applied

to a regression

the I?b, 5%. and I;% signifcant lcvcls arc -4.78, -4.19, (1990) for T= 170. For Assumption 2. we use the same Assumption 3. the ADF is applied in a footnote for table 2.

to a regression

“The K statistic diverges under no cointqration because the Cobb-Douglas utility function implies

with

with

two

one regressor

as shown cointqration.

regressors.

Critical

and -3.8Y. These are from ADF as that for Assumption

by Park

and critical

values

et al. ( 1988).

This

values

for

MacKinnon I. Under ax

reported

is desimble

124

hf.

O{gaki,

J. I’. Pm? I Jorrrttrtl

q/’ Ecottotttctrics

82 (1997)

107.-134

which converts durable good purchases into service flows (Assumption 2). Much weaker evidence is found against this hypothesis for NDS and DN under Assumption 2. and for NDS and DC under Assumption 3.1x and the Cobb-Douglas utility function is rejected decisively for ND. The evidence for NDS is somewhat more ambiguous. Eichenbaum and Hansen (1990) tests this restriction on the deterministic trends for NDS and DN assuming trend stationary consumption. Their test results are ambiguous because the results depend on the sample period and the lag truncation number used in their estimation of the long-run covariance matrix for GMM. Table 3 Canonical

cointcgrating 1 !z,i’

Assumption c,: NDS

regression

results ;(?il,n

Kb

0.493 (0.057)

O.lY6 (0.027)

XXI .737 (0.000)

2

0.430 (0.254)

0.750 (0.111)

5.134 (0.077)

3

0.51 I (0.075)

0.804 (0.040 )

0.740 (0.162)

NDS

H( 1.2)”

ff(l.3)”

.

3.50’) (0.061 )

5.X60 (0.053)

14.230 (0.000)

4.375 (0.036)

I I .654 (0.003)

Q: DN

I

q: I

H(O. I )c

q:

42.Y77 (0.000)

.

0.865 (0.352)

I.41 I (3.494)

0.266 (0.034)

474.181 (0.000)

...

0.012 (0.YZ.l)

0.074 (0.963)

284.268 (0.000)

32.35’) (0.000)

3.8Y6 (0.048)

43.844 (0.000)

..

3.565 (0.050)

25.82 I (0.000)

DG

2

-0.237 (0.739)

0.265 (0.044 )

3

- 1.066 (0.X70)

-0.201 (0.4Y7)

6.2Y3 (0.043)

I

0.40x (0.071)

0. IO6 (0.040)

486.959 (0.000)

...

0.03 I ( 0.860)

6.498 (0.039)

2

0.470 (0.122)

0.47Y (0.039)

5 13.307 (0.000)

4.114 (0.043)

0.00 I (0.971)

13.095 (0.001)

3

0.386 (0.082)

0.45’) (0.03 I )

114.980 (0.000)

,..

0.543 (0.461 )

I .783 (0.410)

q:

ND

q:

DN

Ix We can test this hypothesis by applying G(p, DN, G(0, I ) is 0.003 with the p-value of 0.956 and ND and DN, G(0. I ) is IO.57 with the p-value of 0.005. For ND and SER. G(0. I) is II.286 with the y-value of 0.003.

q) tests to /I?(/) G(O.2) is 0.081 0.001 and G(0.2) p-value of0.001

- cl(/) + C,(I). For NDS and with the p-value of 0.060. For is IO.57 with the p-value of and G(0.2) is 11.33 with the

I/r,”

12,kQa

St3

H(0.

0.595 (0.125)

0.269 (0.036)

1277.604 (0.000)

2

0.204 (0.080)

0.354 (0.042 )

3

0.456 (0.085)

0.494 (0.048)

60.420 (0.000)

0.595 (0.125)

0.269

(0.036)

1277.604 (0.000)

2

0.204 (0.080)

0.354 (0.042)

811.263 (0.000)

3

0.456 (O.OSS)

0.494 (0.048)

60.420 (0.000)

I

0.!73 (0.044)

1.067 (0.095)

369.842 (0.000)

2

0.014 (0.080)

0.430 (0.042 )

” Standaid errors ’ Chi-square test parentheses. c Chi-square test f-values are in rl Chi-square test

are in parenthcscs. statistic with two degrees

Assumption c,: ND ~2: DG I

cl: ND

N(1.2)”

H(i.3$

.. .

3.653 (0.056)

4.707 (0.095)

I.516 (0.218)

0.005 (0.956)

3.114 (0.21 I)

0.0003 (0.987)

1.043 (0.594)

3.653 (0.056)

4.707 (0.095 )

0.005 (0.956)

3.114 (0.211)

0.0003 (0.987)

I .043 (0.594)

4.186 (0.041)

7.645 (0.022)

I.177 (0.278)

2.936 (0.230)

.

(r: DG

I

q:

811.263 (0.000)

1)’

I.516 (0.218)

.

ND (2: SER

3206.334 (0.000)

of freedom

statistics with one degree of freedom parentheses. statistics for stochastic cointegration.

4.694 (0.030)

for the restriction for the deterministic P-values

xl = 22 = I. P-values cointegration

are in

restriction.

are in parentheses.

For the model with NDS and DN, ali point estimates of I/xl and XZ/XI have the theoretically correct positive sign. We estimate xz/xl to be statistically significantly smaller than one under all assumptions. The H(0, 1) test decisively rejects the deterministic cointegration restriction and the H( 1,3) rejects stochastic cointegration at the 1% level under Assumption 2. There is much weaker evidence against the model in terms of the H(p, q) tests under Assumption 1 and under Assumption 3. Our estimate of l/xi under Assumption 1 is similar to that obtained under Assumption 3. Under Assumption 1, the ADF( I), ADF(4), and ADF(7) are -4.02, -3.73, and -3.66, respectively. Among these, only the ADF( 1) is significant at the 10% level. Under Assumption 3, the ADF( I), ADF(4) , and

ADF(6) are -2.53, -2.8 1, and -2.77, respectively. None of these is significant at the 10% level. I9 For the model with NDS and DG, the H( I, 2) and H( 1,3) statistics under Assumption 1 provide much less evidence against stochastic cointegration than for the model with NDS and DN under Assumption 1. ‘O Under Assumption 1, the point estimates of l/z, and Q/XI are also positive for DG and they are within two standard errors from the point estimates for PN. On the other hand, we find more evidence against the model with NDS and DG than the model with NDS and DN: the H( I, 3) test rejects stochastic cointcgration decisively and the point estimates of I/x, are negative under these assumptions. Again we estimate rxz/zl to be significantly smaller than one under all assumptions. Under Assumption I, the ADF( I), ADF(4), and ADF(7) are -3.22, -2.97, and -2.98. None of these is significant at the 10% level. 2’ For the model with ND and DN, all point estimates of I/xl and XZ/XI have the theoretically correct positive sign. We estimate Q/XI to be significantly smaller than one. Under Assumption I, the H( I, 3) statistic is significant at the 5% level though the H( 1, 2) is not. Under Assumption 2, the H(0, 1) is significant at the 5% level and the H( 1, 3) is significant at the 1% level. Under Assumption 3, the H( I, 2) and H( 1, 3) do not reject the model. The ADF( l), ADF(4), and ADF(7) zre -2.54, -2.97, and -2.77 under Assumption 3. None of these is significant at the 10% level. ” For the model using ND and DG, all point estimates of I/XI and XI/XI are positive and none of the H(p.q) test statistics are significant at the 5% level. The point estimates with DG are similar to those with DN under Assumptions 1 and 3. Under Assumption 2, the point estimate of I/xl is somewhat smaller for DG than for DN. Under Assumptions 2 and 3, the H(p,q) tests providc less evidcncc against the model for ND and DG than the model for ND and DN.‘3 However, the H( 1, 2) test provides stronger evidence against the model for ND and DG under Assumption I. We estimate Q/XI to be signifi-

“I The OLS estimate for I :x1 is 0.362 under Assumption I and 0.286 under Assumption 3. With (‘1 as the rcgrcssand for the CCR under Assumption 1, the cstimutc fix l/xl is 0.609 with the standard error

of 0.093.

and the H( 1.2)

and

H( I. 3) arc 0.023

and 0.5 IX with

the p-vaiuc

of 0.879

and 0.772.

?” III order to confirm that this diflkrcnce is not coming from the dillixcncc in the satnplc period, WC apply the CCR to NDS and DN with the same sample period 19471.-1983:lV as that for NDS and DC under Assumption I. We obtain I/r, =0.443. H( l,Z)= 3.262, and t1( 1,3)=h.l9S. 2’ The OLS estimate fOr II’XI under Assumption I is 0.313. With Q as the rcgrcssand in the CCR under Assumption I. tbc estimate 1% I 1x1 is 0.679 with a standard error of 0.135. and the H( I. 3) and H( I. 3) arc 0.404 atid I.207 with the p-values of 0.535 and 0.547. respcctivcly. 22 The

OLS

cstimatu

for

I ,‘r under

Assumption

3 is 0.267.

23 In order to confirm that these diffcrcnccs xc not coming from the dilkrcncc in the smnplc period. we apply the CCR to NDS and DN with the same sample period 1947: I - I983:IV us is used for NDS and DC under Assumption 2. We obtain I/XI =0.3X9 with a standard error of 0.107, H(0, I ) = 5.369, H( I. 2) = 3.676. and H( I, 3) = 7.620.

cantly smaller than one. The ADF( I ), ADF(4), and ADF(7) are -3.500, -3.42, and -3.41 under Assumptions 1 and 2. The ADF(l), ADF(4), and ADF(7) are - 1.70, -2.12, and -2.26 under Assumption 3. None of these is significant at the 10% level. ” For the model which uses ND and SER, all point estimates of i/xl and xz/xr are positive. Under Assumption 1, the point estimate of x2/x1 is greater than one but not significantly so. Under Assumption 2, we estimate Q/XI to be significantly smaller than one. Under Assumption 1, both H( 1,2) and H( 1,3) are significant at the 5% level. Under Assumption 2, the H(0, 1) test is significant at the 5% level, but the H( I. 2) and H( 1,3) do not reject stochastic cointegration. With cl as the regressand for the CCR under Assumption 1, the estimate for 1,‘~: is 0.216 with a standard error of 0.035, and the H( I, 2) and H( 1,3) are 6.179 and 6.980 with p-values of 0.013 and 0.030, respectively. With cz as the regressand for the CCR under Assumption 2, the H(0, I ), H( 1,2) and H( 1,3) are 61.74, 15.07, and 15.21, which provide overwhelming evidence against the model under Assumption 2. Under Assumption 1, the ADF( I), ADF(4), and ADF(7) are -4.13, -3.85, and -3.24. The ADF(1) is significant at the 10% level and the ADF(4) is close to being significant at the 10% level.25 Thus under at least one of Assumptions I, 2, and 3, the point estimates of l/q and xz/xt are positive, and the H( y, y) tests fail to reject the model at the 5% level for all pairs of the goods examined, except for one case in which they fail to reject at the 3% level. On the other hand, the ADF statistics are often not significant. These results are consistent with three situations. First, the dominant autoregressive root of the disturbance of r:,.(t) in (13) or (14) may be close to one even if it is smaller than one. It should be noted that the ADF test is not powerful, and hence there is a high probability that the ADF statistics will not be significant even though the variables are cointegrated. Since the model implies cointegration, we choose to focus on the test results for the null of cointegratio~ therefore avoid the low power problem of the tests for the null of no cointegation. Second, cc.(t) may be difference stationary with its random walk component (in the sense of Cochrane 1988) small enough not to make the H(p,q) statistics extremely large. Third, all of Assumptions l-3 may be misspecified because the dominant autoregressive root of each series is smaller than, but close to, one. In the latter two cases, the model is misspecified but cointegration can bc a g approximation. For ND, we can estimate the system with durable consumption and the system with SER simultaneously, using SUCCR. In this simultaneous estimation, there

?’ The OLS estimate of I/xl is 0.305 under Assumption 1, 0.020 under Assumption 2. and 0.228 under Assumption 3. With c2 us the regressand for the CCR under Assumption 2, the estimate of ljrl is 0.449 with a standard error of 0.154. and the H(O.1). H( 1.2). and H( 1.3) are 0.276, 0.044, and 0.081 with p-values of 0.599, 0.834. and 0.960. rcspcctivcly. 25 The

OLS

estimate

of I/r

is 0.194

under

Assumption

I.

Table

4

Seemingly

unrclatcd

Model ND. ND.

DN. DC.

SERh SERC

” Lagrange-multiplier h We USC Assumption and SER. C WC USC Assumption

canonical

cointegrating

regression

results

l/q

SC.

L”

p-v;‘lL,.

0.26 I 0.175

0.043 0.039

4.496 6.333

0.034 0.012

test for the cross-equation restriction 3 for the system with ND and DN

that is explained in the text. and Assumption 3 for the system

with

ND

2 for the system

and Assumption

with

ND

with

ND

and DC

3 for the system

and SER.

is a cross-equation restriction that the element of the cointegrating vector which corresponds to l/21 in the former system is the same as the element of the cointegrating vector which corresponds to I/r, in the latter system. A test this cross-equation restriction is of interest. Table 4 reports restricted estimates for I/r, and the Lagrange multiplier test (I,,) statistics. We USCAssumption 3 for the system with ND and DN, Assumption 2 for the system with ND and DC, and Assumption 3 for the system with ND and SER. There is some evidence against the cross-equation restriction, but the evidence is not strong.

In this section, we compare our estimates of I/xl from the cointegrating regressions with GMM estimates from the asset pricing equations of Ferson and Constantinides’s ( 1991). Ferson and Constantinides add one-period time nonseparability to Hansen and Singleton’s (1982) model, and show that this modification is very helpful in making asset pricing equations consistent with multiple returns data. The difference between Ferson and Constantinides’s work and ours is that we use Cooley and Ogaki’s ( 1996) two-step procedure. In the second step, the GMM procedure, we restrict I/XI in the asset pricing equation to be the point estimate from the cointegrating regression of the first step. The asymptotic properties of GMM procedure in the second step is not affected by the first step estimation because the estimator for l/xl from cointegrating regressions converge faster than the GMM estimators. For the GMM estimation, we assume that there is no measurement error, liquidity constraints, or preference shocks and that the real asset returns used for the estimation are stationary. We also impose the restriction that a: = 0 for 5 > 2 in (2.3). On the other hand, the interpretation of the estimates of I/xl from cointegrating regressions as the long-run IES requires the separability assumption for preferences as discussed above. Thus, we expect the two sets of estimates to bc different when these assumptions are violated in an important way. The asset

129

pricing equation then states that fi*E,[{S,(t + I)-” E,[S,(t)--

+/j*n$,(t + 2)-“}I?@ + pkz$*(t + I))-“‘]

+ I)]

(6.1)

= 1

for any gross real asset return R(t). Here I{* = exp( -XI 4)/L Comparing (6.1) with the asset pricing equation in Ferson and Constantinides ( 1991) and Cootey and Ogaki (1996), we see that 9 only affects the interpretation of the estimated /I. We initially estimflie /I” and al with I/XI restricted to be an estimate from a cointegrating regression reported above. Then we perform unrestricted estimation and form the likelihood ratio type test statistic, CT(see, e.g., Ogaki (1993~) for an explanation of this test). The null hypothesis is that the probability limit of the GMM estimatorz6 is equal to the true value of the preference parameter, I/x,, and the alternative hypothesis is that it is not. The null hypothesis is satisfied when the additional assumptions (such as no liquidity constraints) made for GMM estimation are satisfied. *’ We maintain the assumptions made for the cointegration approach. When the true value were known, this test with the cointegrating regression estimator replaced by the true value wou!d be the standard likelihood ratio type test. Because the cointegrating regression estimator is super consistent, the asymptotic properties of our test statistic are the same as those of the standard test statistic. Under the null hypothesis, the test statistic has an asymptotic chi-square distribution with one degree of freedom. When the additional assumptions are not true, the cointegrating regression still consistently estimates the long-run IES, but the GMM estimation does not. Hence we can expect the CT test statistic to be large under the alternative hypothesis. For NDS, we use the cointegrating regression estimate for I/xl for DN under Assumption 3 reported in Table 3. For ND, the cointegration regression estimate for DN and SER reported in Table 4 is used. For asset returns, we use the

16 Here we have made a technical assumption that the probability limit exists even when the additional assumptions for GMM are not satisfied. For example. in the case in which the number of moment restrictions is equal to three. the probability limit exists. and is equal to the values of the parameters that make the population values of the preference parameters is made for expositional simplicity. additional assumptions are true. ” If the pobability

limit

GMM criterion function zero. This limit coincides with the true when the additional assumptions are satisfied. This assumption If the limit does not exist, then the null hypothesis is that the

of the GMM

estimator

exists

when

the additional

satisfied. then the null hypothesis is that the probability limit of the GMM true value of the preference parameter, I/rl, and the alternative hypothesis

assumptions

estimator is that

is equal it is not.

are not to the In the

case in which the number of moment restrictions is equal to three. the probability limit exists, and is equal to the value of the parameter that makes the population GbfM criterion function zero. This limit coincides with the true value of the preference parameter when Ihe additional assumptions are satisfied. If the probability limit does not exist when the additional assumptions are not satisfied. then the null hypothesis is that the additional assumptions are satisfied and the alternative hypothesis is that they are not.

130 Table GMM

5 results

for asset prices

Consumption” Conventional NDS

(U)

NDS

(R)

instrumental 0.188 (0.090) 0.51 I

(u)

0.367 (0.173)

ND

(R)

0.261

instrumental

d.f.

variables

ND

Financial

J.r I.015 (0.01 I)

-0. I86 (0.090)

6.047 (0.109)

3

. ..

I.001 (0.002)

-0.357 (0.132)

7.748 (0.101 )

4

1,702 (O.lY2)

0.9YY (0.003)

-0.10x (0.142)

6.507 (0.08’))

3

. ..

7.258

4

0.75 I (0.386)

I .oo I

-0.041

(0.002)

(0.09‘)

)

(0.123)

varioblc.\

NDS

(U)

0.060 (0.033 )

I .03 I (0.017)

-0.50 I (0.086)

12.238 (0.007)

3

NDS

(R)

0.s I I

I .003 (0.00-1)

-0.704 (0.1 13)

13.804

4

(0.008)

ND

ND

(U)

(R)

Note: For the restricted Assumption 3 reported model

with

ND.

DN,

0. I72

0.905 (0.007)

-0.537 (0.204)

I I.546 (0.00’))

3

(0.17”)) 0.261

0.903 (0.005)

-0.61 I (0.060)

I I.661 (0.020)

4

estimation, in Table and SER

we use the CCR estimate 4 ;md for the model with reported

in Table

I.566 (0.21

I)

...

0.115 (0.734)

for the model with NDS and DN under NDS and the SLCCR estimate for the

4.

a In this column, we report the measure of nondurable consumption used and whether or not the estimation is restricted (indicated by R) or unrestricted (indicated by U). h Standard errurs are in parentheses. cThe likelihood ratio rype test with one degree of freedom for the restriction that I:131 in the asset pricing equations is the same as the cointegrating rcgrcssion estimate.

three-month Treasury bill yields from the Center for Research in Security Prices (CRSP) risk-free file and the value-weighted average of stock returns on the New York Stock Exchange and the American Stock exchange obtained form the CRSP. Table 5 reports our GMM results.‘” The first panel of Table 5 reports results when Hansen and Singleton’s (1982) conventional instruments of a constant, lagged consumption growth, and a lagged zx We use Hansen-Hcaton-Ogzki 3512371 for the GMM estimation. to (6.2).

GAUSS GMM package that was supported See Ogaki ( 1993cd) for details about how

by NSF Grant SESthe GMM is applied

real asset return are used. The second panel reports results obtained when financial variables of a constant, a dividend yield, and a yield spread are used as instruments. The dividend yield is the average dividend yield on the CRSP value-weighted stock index, seasonally adjusted by removing the dete~inistic seasonal components estimated from a regression with seasonal dummies. The yield spread is the yield to maturity of corporate bonds rated Baa by Moody’s Investor Services minus that of the Aaa corporate yield. In all cases, the likelihood ratio type test (CT test) does not reject the restriction that the GMM estimate of I/xl from the asset pricing equation is consistent with the cointegrating regression estimate of I /zt . For ND, the GMM point estimates of l/r, are within one standard deviation of the cointegrating regression estimate. On the other hand, the GMM point estimates are more than three standard deviations away from the cointegrating regression estimate for NDS, and thus the Wald test rejects the restriction that the Cr test does not. Because Monte Carlo evidence suggests that the likelihood ratio type test is typically more reliable than the Wald test for noniinear models.. this evidence against the restriction for NDS should not be taken seriously (see Ogaki ( 1993~) for discussions about the likelihood ratio type and Wald tests.) In terms of Hansen’s J test for the overidentifying restrictions, the asset pricing equation (6.1) is not rejected at the 5% level when conventional instrumental variables are used. According to the p-values reported, the asset pricing equation is typically rejected at the 1% level when financial instrumental variables are used. The evidence against the model is not overwhelming, however., because the J-test tends to over-reject in small samples (see, e.g., Ogaki, 1993~). 6. Conclusions

This paper has proposed an approach to estimate preference parameters by using the information in stochastic and deterministic trends. Under at least one of the alternative assumptions considered, the point estimates of the preference parameters have the theoretically correct sign and the tests fail to reject the restrictions on trends at the 5% level for all pairs of goods except for one case in which they fail to reject at the 3% level. We have formed a formal test to compare estimates of the long-run ES from the cointegrating regressions with estimates obtained using GMM. For GMM estimator to be consistent for the long-run intertemporal elasticity of substitution, additional assumptions such as no liquidity constraints are required. Because the cointegrating regression estimator is consistent while the GMM estimator is inconsistent, we can expect a large value of the test statistic when the additional assumptions are significantly violated. The test does not reject the null hypothesis that the GMM estimator also consistently estimate the preference parameter. This test result is consistent with two possibilities. One possibility is that the additional assumptions are true. However, it is hard to believe that the additional

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