cointegration and direct tests of the rational expectations hypothesis

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Cointegration and direct tests of the rationalexpectations hypothesisMichael McAleer a b , C.R McKenzie c & M. Hashem Pesaran da Department of Economics, University of Western Australia,b Institute of Economic Research, Kyoto University,c Faculty of Economics, Osaka Universityd Cambridge Department of Economics, Trinity College, Los AngelesUniversity ofCalifornia

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To cite this article: Michael McAleer, C.R McKenzie & M. Hashem Pesaran (1994): Cointegration and direct tests of therational expectations hypothesis, Econometric Reviews, 13:2, 231-258

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ECONOMETRIC REVIEWS, 13(2), 23 1-258 (1994)

COINTEGRATION AND DIRECT TESTS OF THE RATIONAL EXPECTATIONS HYPOTHESIS

Michael McAleer

Department of Economics, University of Western Australia Institute of Economic Research, Kyoto University

C.R. McKenzie

Faculty of Economics, Osaka University

M. Hashem Pesaran

Trinity College, Cambridge Department of Economics, University of California, Los Angeles

Key Words and Phrases: rational expectations hypothesis; direct tests; orthogonality tests; cointegration; generated regressors.

JEL Classification: C32J C51J C52.

ABSTRACT

The paper is concerned with direct tests of the rational expectations

hypothesis (REH) in the presence of stationary and non-stationary variables.

Alternative methods of converting qualitative survey responses into

quantitative expectations series are examined. Testing of orthogonality and

the issue of generated regressors for models estimated by two step methods

are re-evaluated when the variable to be explained is stationary. A

methodological approach for testing the REH is provided for models using

Copyright 0 1994 by Marcel Dekker, Inc

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232 McALEER, McKENZIE, AND PESARAN

qualitative response data when there are unit roots and cointegration, and

alternative reasons are examined for rejecting the null hypothesis of

orthogonality. The usefulness of cointegration analysis for both the

probability and regression conversion procedures is also analysed.

Cointegration is found to be directly applicable for the probability conversion

approach with uniform, normal and logistic distributions of expectations and

for the linear regressicn conversion approach. In the light of new techniques,

an existing empirical example testing the REH for British manufacturing

firms is re-examined and tested over an extended data set.

"It is of the highest importance in the art of detection to be able to recognise out of a number of facts which are incidental and which vital."

Sherlock Holmes to Colonel Hayter in The Adventure of the Reigate Squire

1. INTRODUCTION

Qualitative series obtained through the use of survey data may be

converted to what may be termed "direct" measures of expectations through

the use of specific probability functions or specific regression models.

Pesaran (1987) provides a survey of the literature, and Smith and McAleer

(1992a) compare the robustness of alternative procedures for converting

qualitative responses to quantitative expectations. When such direct

measures of expectations are available, it is possible to test some of the

implications of the rational expectations hypothesis (REH), specifically

whether the expectations errors in aggregate at time t can be explained by

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COINTEGRATION AND DIRECT TESTS OF REH 233

publicly available information at time t-1. This procedure is commonly

known as the "orthogonality" test.

Although qualitative measures of expectations may be converted to

quantitative series, the conversion procedures which are used rely on an

auxiliary equation, namely a specific probability function or regression

model. If the underlying series to be investigated are stationary, it is

possible that the use of an auxiliary equation, and the consequent use of

generated regressors, will yield measurement errors which may render bias in

the conventionally programmed ordinary least squares (OLS) standard errors

used for the orthogonality test. However, if the underlying series are

non-stationary, it is possible that the use of generated regressors will not

bias the conventionally programmed standard errors.

When survey response expectations are of a qualitative nature, the

variables of interest are the series under investigation (I$), the percentages

of survey respondents reporting a rise (Rt) or fall (Ft) in IIt, and the

percentages of survey respondents expecting a rise (R:) or fall (5':) in IIt.

When TIt is either stationary or non-stationary, the issue of whether Rt, Ft ,

R: and F: are stationary or non-stationary, and whether there is any

cointegrating relation among (II,, Rt, Ft) and/or between (R:, F:) needs to

be investigated before a test of orthogonality can be conducted.

The plan of the paper is as follows. Alternative methods of converting

qualitative survey responses into quantitative expectations series are

examined in Section 2. Tests of orthogonality and the issue of generated

regressors for models estimated by two step methods are re-evaluated in

Section 3 when IIt is stationary. In Section 4, a methodological approach for

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testing the REH is provided for models using qualitative response data when

there are unit roots and cointegration, and alternative reasons are examined

for rejecting the null hypothesis of orthogonality. The usefulness of

cointegration analysis for both the probability and regression conversion

procedures is also analysed. Cointegration analysis is found to be directly

applicable for the probability conversion approach with uniform, normal and

logistic distributions of expectations across the respondents and for the linear

regression conversion approach. As there does not yet seem to be a

satisfactory literature for cointegration analysis in non-linear models, the

cointegration approach is not applied to the non-linear regression conversion

approach. In the light of new techniques, an existing empirical example

testing the REH for British manufacturing firms is re-examined and tested

over an extended data set in Section 5. Some concluding remarks are given

in Section 6.

2. ALTERNATIVE CONVERSION PROCEDURES

The probability approach and the regression method are illustrated in

Pesaran (1987). A brief discussion of both methods is given below since it

will be necessary in the following sections. The probability approach

assumes there is some indifference interval around zero, (-a, b), within which

respondents report the expected change in ITt as being zero, and outside

which they report the expected change in ITt as having changed. Defining Rt

as the union of individual respondent's information sets, it follows that

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where Ht(.) is the (aggregate) cumulative density function of IIt. Equations

(1) and (2) can be solved to yield estimates of ll;+l = E(IIt+l ( a t ) .

Three common probability distributions are considered below to

illustrate the difference arising from linear and non-linear relations between

TI: and functions of R: and F:. For the standard normal distribution, given e values of a and b, IIt+l is given by

e 1 e in which rt+l = $-l(1 - R:+~), = $- (Ft+l) and $-I(.) is the

inverse of the cumulative standard normal distribution. When the

cumulative function is the logistic, II:+~ is again given by (3), but now <+1

= log{(l - e F;+~)/F;+~I and r t f l = l o g { ~ ; + ~ / ( l - R;+~)I . Although

equation (3) represents the solution for 11:+1 as a function of a and b, the e unknown parameters cannot be estimated by replacing IIt+l by llt+l -

4+1, where (t+l is a zero mean, independently and identically distributed

random variable, because is not known at time t . However, a and b

can be estimated from

where rt = $-l(l - Rt), ft = $-l(Ft), and the OLS estimates a and b may

be substituted into (3) to obtain forecasts given by

When Ht(.) is the uniform distribution with range 2q, a solution for is

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in which a = 2q(q-a)/(2q-a-b) and ,8 = 2q(q-b)/(2q-a-b). Forecasts

f$+, are given as

where a and p are OLS estimates obtained from

In the regression conversion method, the quantitative expectations are

a function of a specific regression model rather than a specific probability

distribution. The percentage change in nt is assumed to comprise a

weighted combination of pt respondents who say that IIt increased (denoted

+ II . ) and mt who say IIt decreased (denoted II: ), so that 1 ,t ~ , t

+ where w ,t w. ) is the weight on the i'th (j'th) respondent reporting an ( ;,t

increase (decrease) during period t. Pesaran (1987, p. 222) assumes that

+ n+ = a + v ~ , ~ and IIj,t = -0 + V- +

i ,t j,t' where v ,t (v;,~) represents the

overall effect of respondent-specific factors assumed to be distributed

2 2 randomly with zero means and constant variance o+ (oJ, with v;,~ and v- j,t

independently distributed. Then TIt is given precisely by equation (8) (but

p t + m t

with different interpretations for a and P), Rt = E w ,t, Ft = E W: , and i = l j=1 J,t

tt is given by

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Depending upon the weights used in (9), which may vary from one time

period to the another, the disturbance term Et in (8) is potentially

heteroscedastic and serially correlated, which suggests the relation between

e e n t , Rt and Ft, as well as between llt+l, R:+~ and Ft+l , may be

non-linear. Pesaran (1987) discusses several such non-linear relations, and

Smith and McAleer (1992a) compare the empirical performance of several

linear and dynamic non-linear functional forms based on both the

probability and regression conversion approaches.

3. TESTING ORTHOGONALITY AND THE GENERATED REGRESSOR

PROBLEM

The orthogonality test examines whet her the expectations error,

nt - n:, is orthogonal to publicly known information at time t-1. Thus, the

regression model has the expectations error as the dependent variable and

variables in the information set as regressors, namely

* where Wt-l is a vector of variables publicly available at time t-1, and

t=2, ..., T. The null hypothesis of 4 = 0 is tested using the standard F test if

the assumption of homoscedasticity for vt in (10) is maintained. However,

the REH does not necessarily imply that vt is homoscedastic (see Pesaran

(1987, p. 239)).

Since is not observed but is obtained from an expectations

generating equation, the uncritical use of the conventional formula based on

OLS will generally yield biased standard errors. The expectations generating

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equations are given as (6) and (8)) which may be rewritten in matrix form as

where X = [R : F], Z = [ R ~ : F ~ ] and y' = (a , -@). Given the discussion of

the properties of Ct in (9)) it is worth noting that Ct may exhibit serial

correlation. This issue is considered in Section 4.

Consider the test of orthogonality given by

which, under the REH of I#I = 0, is equivalent to subtracting (12) from (11). *

Equating (13) to the matrix version of (10) yields ll - ne = W-1q5 + V,

where v = < + (X-Z) y = < + Q(R - R ~ ) - P(F - F ~ ) . Assuming that the

aggregate distribution and the indifference intervals are the same for

realisations and expectations, an estimate of ne may be obtained by

substituting the OLS estimate of 7 from ( l l ) , namely 7 = ( x ' x ) - ~ x ' ~ ) into

(12) to yield fIe= Z; = z(x'x)-'X'll (see Pesaran (1987, p. 226)).

Substituting the estimated expectations variable ie for lle in (13) gives

where lle - fIe is the measurement error associated with the conversion of

qualitative responses to quantitative observations and u = v + ( n e - ne) =

v + (ZY - z;) = v - z(x'x)-lx'(. Maintaining the assumption that 2 6 - D(0, a I), the covariance matrix of u in (14) is given by E

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where E(vttt) = Vt and E(vtFS) = 0 for t#s. Since Var(u) is not given

2 gv t

by uvI, even if vt and tt are uncorrelated, the conventionally programmed

standard errors of the OLS estimate of 4, 4, in (14) will be biased. The 2 direction of bias is generally unknown, but the use of uvI will yield a

downward bias in the variance of the OLS estimate of 4 in (14) when u = v t

0, as in the more standard examples examined in Pagan (1984) and

McKenzie and McAleer (1992).

The covariance matrix of u given above indicates that u is both serially

correlated and heteroscedastic. Newey and West (1987) suggest a simple

method for calculating a positive semi-definite covariance matrix which is

intended to be robust to unknown forms of serial correlation and

heteroscedasticity. Smith and McAleer (1992b) examine, using Monte Carlo

experiments, whether the Newey-West approach can approximate the known

covariance structure of u, such as that given above, in the context of testing

orthogonality of quantitative expectations derived from qualitative responses

in Smith and McAleer (1992a). They found a tendency for the Newey-West

procedure to not only over-reject a true null hypothesis but also to have

sizes close to those (incorrectly) obtained using the conventionally

programmed OLS standard errors.

A central issue to be addressed prior to testing for orthogonality in *

(lo), and in its operational version (14), is whether nt, If:, fit and W t are

stationary (integrated of order zero, I(0)) or non-stationary (integrated of

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^ e - order one, I(1)). Even if both and lIt I(1), i t is possible that the

forecast errors Ilt - IIf! - I(O), if there is a cointegrating relation between IIt

and fI:, CI(1,1), with cointegrating vector given by (1, -1) (see Engle and

Granger (1987)). Since TIt is given as a linear function of Rt and Ft in (8), it

is also crucial to test for unit roots in Rt and Ft. If Rt and Ft - I(1), it may

be necessary to test for the existence of one or more cointegrating relations

between I$, Rt and Ft using, for example, the method of maximum

e likelihood estimation developed in Johansen (1988, 1991). As IIt+l is a

e linear function of Rt+l and F:+~ in ( 6 ) , a cointegrating relation between R:

and F: may render lly stationary in the event that both R: and F: - I(1).

Two issues related to the nature of the underlying variables and the

appropriate functional forms for the expectations generating equations need

to be addressed prior to testing orthogonality. First, from a general

perspective, since Rt, Ft, R: and F: are positive fractions, i t may well be

argued that they cannot be 1(1) variables. However, using relatively short

time series it is possible that the outcomes of tests will indicate the variables

behave like 1(1) variables. The situation becomes more complicated when

the probability of observing values of the four variables on the boundaries of

the interval [0, 11 is non-zero, since tests of unit roots and cointegration in

such cases do not yet seem to have been developed. In this regard, it is

worth noting that the use of the standard normal or logistic distributions, as

in (3) and (4), does not constrain the explanatory variables to lie between 0

and 1.

The second issue concerns the role of non-linear functions. Although

in (8) (lI:+, in (6)) is expressed as a linear function of Rt and Ft (R:+~

and the use of varying coefficients in (8) is place of the constant

parameters a and could render both (8) and (6) non-linear functions, as

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COINTEGRATION AND DIRECT TESTS OF REH 24 1

would the use of appropriate weights in equation (9) (for further details, see

Smith and McAleer (1992a)). Moreover, the use of the standard normal or

logistic distributions renders TIt ( l l t+l) a linear function of variables that

e e are non-linear transformations of Rt and Ft (Rt+l and FtC1). I t is not

clear how I(0) or I(1) underlying variables are to be treated after being

transformed by known non-hear functions, as used in the standard normal

and logistic probability distributions (for a useful Monte Carlo analysis of

some simple non-linear transformations, see Granger and Hallman (1991)).

In addition, the literature on cointegration seems to he restricted at present

to linear relations between non-stationary variables, so that the distinction

between linear and non-linear expectations generating equations (namely,

variations of (6) and (8)) would seem to be crucial in conducting tests of

orthogonality.

The decision tree for obtaining Il: varies according to whether IIt is

determined to be I(0) or I ( l ) , and these are given in Figures 1 and 3,

respectively. Procedures for testing orthogonality also vary according to the

A e order of integration of TIt and TIt, and these are given in Figures 2 and 4 for

stationary and non-stationary TIt, respectively. In all Figures, the branches

depict mutually exclusive choices. In Figures 1 and 3, the distinction

between linear and non-linear functions refers to the equation for ll:, and

consequently the expectations generating equation for lIt in terms of Rt and

F , rather than to non-linearities arising from possible serial correlation in t

the error structure. These two Figures are based on equations (6) and (8),

that is, the use of the uniform probability conversion approach or the linear

regression conversion method. They may be reinterpreted in terms of

equations (3) and (4) for the standard normal and logistic probability

conversion methods in a straightforward manner. The logic behind the

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Figures is that, for the linear model: (a) it is necessary that it - I(0) for (8)

to be a balanced model; (b) given it - I(O), it is necessary to determine

whether ll: is of the same order of integration as IT,; (c) given lit and ll: are

of the same order of integration, it is necessary to determine whether vt - *

I(0) in (10) ; and (d) for the test of 4 = 0 in (14), the variables used in W-l

are presumed to be I(0) for the purpose of single-equation estimation and

testing.

A. STATIONARY nt A.l Decision Tree

Commencing with lit- I(0) in Figure 1, the relation between lit, Rt

and Ft can be linear or non-linear. In the linear case, such as (8), three

cases are possible, namely:

1. Either Rt or Ft - I(1), or both

Since lit - I(O), the equation relating lit to Rt and Ft is not balanced

and needs to be respecified.

2. The C(1,l) coefticients for Rt and Ft are not proportional to the

coefficients in the equation for ITt

If there is a cointegrating relation between Rt and Ft, say Rt + XFt '

CI(1,1), the equation for lit is not balanced and needs to be respecified unless

X is proportional to p/a in (8).

3. Either (a) Rt or Ft - I(O), or (b) the CI(1,l) coefficients for Rt and Ft

are proportional to the coe£Kcients in the equation for nt In case (a) or (b), the equation for ITt is balanced and OLS yields

fl-consistent estimates of cr and p, which may be used in (6) to provide

estimated expectations it. Three further cases are possible, namely:

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COINTEGRATION AND DIRECT TESTS OF REH

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3(i) Either R: or F: - I(1), or both

With IIt - I(0) and TI: - 1(1) in (6), the null hypothesis of

orthogonality is rejected.

3(ii) R: and F: - CI(1,l)

Let the CI(1,l) coefficients for R: and F: be such that R: + OF: - I(0).

(a) If 0 is not proportional to B/a, then ily in (6) will not be I(0).

Therefore, the null hypothesis of orthogonality is rejected.

(b) If Bis proportional to p/o, II: in (6) (and fI: in (7)) is I(0).

3(iii) R: and I?: - I(0)

Given the linear relation between II:, R: and F: in (6), it follows that

II: in (6) (and in (7)) is I(0).

When IIt is non-linear in Rt and Ft, two cases are possible, namely:

4. Rt and Ft - I(0)

Equation (8) for IIt is balanced and attention shifts to equation (6) for

TI:, where two further cases are possible, namely:

4(i) R: and F: - I(o)

This is the same as in 3(iii) above, so that II: - I(0).

4(ii) Either R: or F: - I(1), or both

A non-linear relation between ll Rt and Ft implies a similar t ' non-linear relation between II:, R: and F:. Since R: and/or F: - I( l) ,

it is not clear how to proceed since the literature on cointegration does

not presently deal with non-linear relations.

5. Either Rt or Ft - I( l) , or both

Since the equation for IIt is non-linear and is (apparently) unbalanced,

as in the case 4(ii) above it is not clear how to proceed.

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Use standard F tests correcting for generated regressors

Figure 2: Testinn Orthononality for Stationary IIt

A.2 Testing Orthogonality

There are three paths for obtaining II: - I(O), namely 3(ii)(b) and 3(iii)

when IIt is linear in Rt and Ft, and 4(i) for its non-linear counterpart. It is

clear that II: - I(0) when Rt, Ft, R: and F: are all I(0) variables, but this

condition is not necessary. The alternative methods of testing orthogonality

for nt - '(0) are given in Figure 2. Since IIt and fit are both I(O), the

forecast error IIt - II; - I(0). Suppose the variables available at time t to be

used in the orthogonality test are Wlt and WZt, where it is assumed that

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Wlt - I(0) and W2t - I(1). Define W; = [Wit: AWZt : AW,,], where A is

a matrix such that AWZt - CI(1,l). Given the stationarity of IIt, $ and

Wit, as well as AWZt and any CI(1,l) relation among the elements of WZt

(that is, maximum likelihood estimates of A), the OLS estimate of $ in (14),

$, is JT-consistent. Since the analysis of Section 3 applies in this case, the

null hypothesis $ = 0 of orthogonality may be tested incorporating the

generated regressor correction in computing the covariance matrix of $. The

specific choice of Wit, AWzt or any CI(1,l) relation of W2t will affect the

power of the test of 4 = 0.

B. NON-STATIONARY ITt

B.l Decision Tree

The relation between IIt, Rt and Ft may be linear or non-linear when

IIt - I(1), as seen in Figure 3. When there is a non-linear relation between

IIt - I(1) and stationary or non-stationary Rt and Ft , the literature does not

seem clear as to how to proceed. The subsequent discussion, therefore,

concentrates on the linear relation.

1. Rt and Ft - I(0)

The equation for JJt is unbalanced and, therefore, needs to be

respecified.

2. Either Rt or Ft - I(1)

When two or three of the variables in the equation for IIt are I ( l ) , the

issue of whether there is a cointegrating relation among them arises as

follows:

2(i) nt and either Rt or Ft CI(1,l)

The absence of a cointegrating relation between IIt and one of Rt and

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McALEER, McKENZIE, AND PESARAN

Ft implies the equation for lit is unbalanced, and so must be

respecified.

2(ii) Either (a) nt and either Rt or Ft - CI(1,1),

or (b) n,, R~ and F~ - c1(1,1)

Option (a) arises from either Rt or Ft being I(1) and the existence of a

cointegrating relation between IIt and one of Rt or Ft . Therefore, the

equation for lit is balanced. Three further cases are possible, namely:

(I) If R: and F: - I(O), lI: and are I(0) variables. Since IIt - 1(1),

the null hypothesis of orthogonality is rejected.

(II) If R: and F: - CI(1,1), say R: + $F:, and $ is proportional to

B/a, then IIf! - I(0) which leads to a rejection of the null hypothesis of

orthogonality However, if $is not proportional to 4/01, then in (6)

(and fI: in (7)) is I(1).

(ID) If either R: or F: - I(1), or both, then lI: and II: - I(1).

3(i) Either (a) nt and either Rt or Ft - CI(1,1),

or (b) I$, Rt and Ft - CI(1,l)

Option (a) arises when Rt and/or Ft - I(1) and there is a cointegrating

relation between I l t and one of Rt and Ft . Option (b) arises when I l t ,

Rt and Ft - I(1)) together with a cointegrating relation among them.

The three further cases are identical to those given as 2(ii)(I), (11) and

(111) above.

3(ii) Rt or Ft ' CI(1,l)

Since all three variables are 1(1) but there is no cointegrating relation

among them, the error tt in equation (8) is not I(0). Therefore, the

equation must be respecified.

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P

n; - I(1)

e A ^ e n, - n, - I(O) nt - nt - I(I)

Reject null

(a) Use standard F tests correcting for generated re essors if either Rt F or t - I(1)

(b) Use standard F tests if Rt and Ft - I(1)

Figure 4: Testing Onho~onalitv for Non-stationary TI,

B.2 Testing Orthogonality

Thus, when IIt is linear in Rt and Ft, there are four paths for obtaining

ll; - I(1), namely 2(ii)(II) (given non-proportionality), Z(ii)(III), 3(i)(II)

(given non-proportionality) and 3(i)(III). Each of these paths involves a

cointegrating relation among IIt and Rt and/or Ft. The alternative methods

of testing orthogonality for IIt - 1(1) are given in Figure 4. Since both IIt

and ll: - I(1), the forecast error ilt - ll; can be I(0) or 1(1), depending upon

whether or not there is a cointegrating relation with cointegrating vector

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^ e - given by (1, -1). In the case where IIt - IIt I ( l ) , the null hypothesis of

orthogonality is rejected because the forecast error can be predicted using

lagged forecast errors. An examination of equation (13) indicates that IIt -

II: - I(0) requires (X-Z) y - I ( 0 ) Recall that (X-Z)? = ~(R-R~)-@(F-F~)

from (13). Suppose that R, Re, F and Fe are all I(1). What is needed for

(X-Z)y - I(0) is that there be a cointegrating relationship among these

variables with cointegrating vector (1, -1, -A, A), where A = @/a and a and

,b are parameters in (6) and (8). A sufficient condition for (X-Z)? - I(0) is

that both R-R~ and F-Fe - I(0). If both R - R~ and F - Fe - I(1), a

sufficient condition for (X-Z)? - I(0) is that R - R~ and F - Fe - CI(1,l)

with cointegrating vector (1, -A). If (say) F~ - I(O), then the appropriate

cointegrating vector is (1, -1, -A) among R, R~ and F. If, in addition, F -

I(O), it is necessary that R - Re - I(0).

When IIt - I$ - I(O), the options are identical to those given in Figure

2. However, the properties of u in (14) depend on the properties of the OLS

parameter estimates of a and p in ( l l ) , namely a and p. These in turn

depend on the order of integration of Rt and Ft. If Rt and Ft - I(1) and n t ,

Rt and Ft - CI(1,1), then a and P are T-consistent (see Stock (1987)).

Therefore, the generated regressor problem discussed in Section 3 does not

arise in the estimation of (14), regardless of whether Wit, AWZt or any

CI(1,l) relation among the elements of Wat is used. When (say) Rt - I(1),

Ft - I(0) and IIt and Rt - CI( l , l ) , then a is T-consistent and P is

O-consistent (see Park and Phillips (1989)). As a result, a generated

regressor problem associated with P will arise when (14) is estimated by OLS.

In this case, using the results in Park and Phillips (1989, Lemma 2.1)

relating to the weak convergence of the sample second moments of I(0) and

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COINTEGRATION AND DIRECT TESTS OF REH 25 1

1(1) regressors and the cross products of these regressors with the

disturbance, the correct asymptotic covariance matrix of f l ( 4 - 4) for the * *

problem considered here is given by plim [QW-iVW-lQ/T], where Q * * -1

= ( W W l ) and

When the orders of integration of Rt and Ft are reversed, a similar problem

arises with respect to cr but not ,f3. As in the case of stationary I$, the

specific choice of variables to include in the orthogonality test will affect the

power of the test.

5. TESTING REH FOR BRITISH MANUFACTURING FIRMS

In the light of the discussion in Section 4, Pesaran's (1985, 1987)

empirical example testing the REH for British manufacturing firms is

re-examined. Pesaran (1987) used a non-linear variation of equations (7)

and (8) to obtain estimates of II: using the regression conversion approach. *

The REH was rejected using the orthogonality test when W in equation

(14) contained the following variables: the rate of change of the effective

exchange rate (et), the rate of change of the index of manufacturing output

(qt), the rate of change of the price index of material and fuels (ft), the

overall rate of unemployment (seasonally adjusted) (Rut) , and the rate of

change of manufacturing prices (I$). In Pesaran's (1987) data set for

1958Q1-1985Q2, it would appear that Wlt = [ft, qt, et] and W2t = [nt, (details are available on request).

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Pesaran's data set was updated for the period 1958Q2-1989Q4 using

data provided in Lee (1991). As a result, the data used here differs from that

in Pesaran's (1987) Tables A.2 and A.3 for TIt for 1972Q1-1977Q2 and

1981Q1-Q2, and the data for ft are given to four decimal places after 197261

rather than the original two. Observations for Rt, Ft , R: and F: for 198391

were also updated. The unemployment rate could not be used since a

consistent series were not available over the period 1958 to 1989 (see Lee

(1991) for further details).

Given the discussion in Section 4, all variables were tested for unit

roots using the Augmented Dickey-Fuller (ADF) test with a time trend

included. An initial lag length of four was used and the fifth lag was tested

for significance using the asymptotic t-ratio. The fifth lag was always

insignificant and the lag length was successively reduced until a significant

lag was obtained. Following Perron (1988), the ADF procedure was then

used to test whether there was any evidence for rejecting the null hypothesis

of a unit root. If rejection was possible, the variable was determined to be

I(0). If the null hypothesis could not be rejected, Dickey and Fuller's (1981)

42 test was used to test the joint null hypothesis of no intercept, no time

trend and a unit root. If this hypothesis was accepted, the unit root

hypothesis was then conducted using an ADF test in a model where the time

trend was excluded (see also Campbell and Perron (1991)). The results of

the ADF tests are presented in Table 1 and suggest that both Ilt and Rt -

1(1) and Ft - I(O), namely that the branch "Either Rt or Ft - I(1)" in Figure

3 is the relevant starting point. The caveat discussed in Section 4 regarding

the order of integration and the range of variation of fractions should be

borne in mind when interpreting the ADF test statistics for Rt, Ft , R: and

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TABLE 1: Order of Integration of Variables

Variable Description Name

ADF Suggested Order

n Rate of change of manufacturing prices R Fraction of firms reporting a price rise F Fraction of firms reporting a price fall

Re Fraction of firms expecting a price rise

F~ Fraction of firms expecting a price fall

e Rate of change of effective exchange rate (I Rate of change of index of manufacturing

output f Rate of change of price index of

materials and fuels

R-Re Difference in the fraction of firms reporting and expecting a price rise

Note: For ll and Re, the calculated statistics without trend; for R and F, the numbers are statistics without

trend; for R-Re, the number is an ADF(1) statistic with trend; and for the remaining variables, the numbers are DF statistics with trend. The respective 95% critical values are -2.88, -2.88, -3.45 and -3.45. In the terminology of Section 4, Wlt = [etl qt, ft] and W2t = [I$].

Cointegration between lit and Rt was tested using both the

Engle-Granger (EG) and Johansen procedures. The procedure to choose the

appropriate lag length was as for the ADF test discussed above. Lag lengths

of zero and one were chosen for the respective procedures. The EG test

statistic was -5.73 with a 95% critical value of -3.39, indicating that and

R - C I ( l 1 ) Table 2 contains the results for the trace and maximal

eigenvalue test statistics given in Johansen (1988) for a vector autoregression

with no trends. These two statistics also indicate that IIt and Rt - CI(1,l).

On the basis of the arguments in Section 4, this finding of

cointegration, together with the results of Table 1, indicate that ll: - I(1).

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TABLE 2: Cointegration Tests for ll and R

Test Null Alternative Test 95% Critical Statistic Hypothesis Hypothesis Statistic Value

Trace r = O r > l 35.99 19.96 r i l r = 2 6.57 9.24

Maximal r = O r = l 29.41 15.67 Eigenvalue r < 1 r = 2 6.57 9.24

Note: r is the number of cointegrating relationships.

Since i t was found in Table 1 that Ft , F: and R~-R: - I(O), this suggests

that IIt - II: - I(0). Using the uniform probability or linear regression

A e conversion procedure to generate IIt , the following estimates of (8), assuming

that Et follows a second4rder autoregressive process, were obtained:

where tt = IIt - 0.0856Rt + 0.0337Ft; the figures in round parentheses are

2 the absolute values of the t-ratios; R is the coefficient of determination; SE

is the estimated standard error; LL denotes the maximized log-likelihood

value; LMSC, LMF and LMH are Lagrange multiplier tests for fourth-order

serial correlation, for functional form misspecification and for

2 2 heteroscedasticity, which are distributed as x ( ~ ) , x ( ~ ) and x ( ~ ) , respectively,

under the appropriate null hypotheses (see Pesaran and Pesaran (1991) for

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details). The diagnostics for serial correlation and functional form

misspecification are not significant at the five percent level. Since LMH is

significant at the one percent level, White's standard errors are given in

square parentheses.

Using the estimates in (16), it is generated as

* - f Setting W t = [et-l, t-l, qt-l, Ant-,], the OLS estimates of (14) are

given as

The figures in round parentheses are the absolute values of unadjusted

t-ratios. For the reasons discussed in McAleer and McKenzie (1991), the

generated regressor problem will bias the tests for serial correlation and

functional form misspecification in an unknown direction but will not affect

the test for heteroscedasticity. Owing to the statistically significant

heteroscedasticity , the figures in square parentheses are the absolute values

of t-ratios doubly-adjusted for heteroscedasticity and the generated

regressor problem caused by the fl-consistent estimators in (16), namely

(the estimated coefficient of Ft) and the estimates of the autoregressive

parameters. Unadjusted diagnostic tests are given for both serial correlation

and functional form misspecification, with the doubly-adjusted diagnostic

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tests given in square parentheses. The unadjusted and doubly-adjusted

Wald test statistics for the joint hypothesis that the parameters associated

with the four explanatory variables in (17) are zero are calculated to be 23.51

and 10.93, respectively, both of which are distributed as X 2 variates under (4)

the null hypothesis. Such evidence regarding orthogonality, serial correlation

and functional form misspecification leads to the rejection of the null

hypothesis of rationality.

6. CONCLUSION

The paper examined the orthogonality test of the rational expectations

hypothesis (REH) in the presence of stationary and non-stationary variables.

Alternative linear and non-linear methods of converting qualitative survey

responses into quantitative expectations series were examined for use in the

orthogonality test. When a stationary variable is to be explained, the

orthogonality test based on a two step method of estimation was

re-evaluated in the light of generated regressors. A novel methodological

approach was provided for testing the REH for models using qualitative

response data in the presence of unit roots and cointegration. In particular,

alternative reasons were examined for rejecting the null hypothesis of

orthogonality. Cointegration analysis was applied to the probability

conversion approach with a uniform distribution of expectations and to the

linear regression conversion method, with a straightforward extension to the

normal and logistic probability conversion approaches. The REH was tested

for British manufacturing firms over the period 1958Q2-1989Q4 and

rationality was rejected.

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ACKNOWLEDGEMENTS

The authors would like to thank Michio Hatanaka, David Hendry, Soren

Johansen, Les Oxley, Jeremy Smith, Hiro Toda and seminar participants at

the Kansai Quantitative Analysis Study Group - Kyoto University, Meiji

Gakuin University, Reserve Bank of Australia, Institute of Mathematical

Statistics - University of Copenhagen, University of Cambridge, Institute of

Economics and Statistics - University of Oxford, University of Melbourne,

University of Western Australia, and the 1991 Australasian Meeting of the

Econometric Society in Sydney, for helpful comments and discussions. We

are also grateful to Kevin Lee for assistance with updating the data. The

first author wishes to acknowledge the financial support of the Australian

Research Council and a Japanese Government Foreign Research Fellowship

at Kyoto University, the second author wishes to thank The Kikawada

Foundation and the University of Western Australia for financial support,

and the third author wishes to acknowledge the financial support of the

ESRC and the Isaac Newton Trust of Trinity College, Cambridge.

REFERENCES

Campbell, J.Y. and Perron, P . , (1991). Pitfalls and opportunities: What macroeconomists should know about unit roots. NBER Macroeconomics Annual, Volume 6, 1991 (O.J. Blanchard and S. Fischer, Eds.) Cambridge: MIT Press, 141-201.

Dickey, D.A. and Fuller, W. A., (1981). Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica, 49, 1057-1072.

Engle, R.F. and Granger, C.W.J., (1987). Co-integration and error correction: Representation, estimation, and testing. Econometrica, 55, 251-276.

Granger, C.W. J. and Hallman, J., (1991). Nonlinear transformations of integrated time series. Journal of Time Series Analysis, 12, 207-224.

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Johansen, S., (1988). Statistical analysis of cointegrating vectors. Journal of Economic Dynamics and Control, 12, 231-254.

Johansen, S., (1991). Estimation and hypothesis testing of cointegration vectors in gaussian vector autoregressive models. Econometrica, 59, 1551-1580.

Lee, K.C., (1991). Formation of price and cost inflation expectations in British manufacturin industries: A multisectoral analysis. Unpublished B paper, Department o Applied Economics, University of Cambridge.

McAleer, M. and McKenzie, C.R., 1991). Keynesian and new classical models of unemployment revisite . Economic Journal, 101, 359-381.

McKenzie, C.R. and McAleer, M., (1992). On efficient estimation and correct inference in models with generated regressors: A general approach. Unpublished paper, Faculty of Economics, Osaka University.

Newey, W.K. and West, K.D., (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55, 703-708.

Pagan, A., (1984). Econometric issues in the analysis of regressions with generated regressors. International Economic Review, 25, 221-247.

Park, J.Y. and Phillips, P.C.B., (1989). Statistical inference in regressions with integrated processes: Part 2. Econometric Theory, 5, 95-131.

Perron, P., (1988). Trends and random walks in macroeconomic time series: Further evidence from a new approach. Journal of Economic Dynamics and Control, 12, 297-332.

Pesaran, M.H., (1985). Formation of inflation expectations in British manufacturing industries. Economic Journal, 95, 948-975.

Pesaran, H.M., (1987). The Limits to Rational Expectations. Oxford: Basil Blackwell.

Pesaran, M.H. and Pesaran, B., (1991). Microfit 3.0: An Interactive Econometric Software Package. Oxford: Oxford University Press.

Smith, J . and McAleer, M., (l992a). Alternative procedures for converting qualitative response data to quantitative expectations: An application to Australian manufacturing. Unpublished paper, Department of Economics, University of Western Australia.

Smith, J . and McAleer, M., (1992b). Newey-West covariance matrix estimates for models with generated regressors. Unpublished paper, Department of Economics, University of Western Australia.

Stock, J.H., (1987). Asymptotic properties of least squares estimators of cointegrating vectors. Econometrica, 55, 1035-1056.

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cointegration and direct tests of the rational expectations hypothesis

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