Dimitrios Papaikonomou

Jacinta Pires

English

Research Papers in Economics

Are US Output Expectations Unbiased? A Cointegrated
VAR Analysis in Real Time
by
Dimitrios Papaikonomoua and Jacinta Piresb,∗
aMinistry of Finance, Greece
bChrist Church, University of Oxford, UK
Abstract
Real-time output and direct measures of expectations at diﬀerent time horizons are analysed
within a cointegrated VAR. We generally ﬁnd expectations to be unbiased in the long-run
with stationary expectational errors that persist between 4 and 14 quarters.
JEL Classiﬁcations: C15, C32, D84.
Keywords: Expectations, Survey Data, Real-Time Data, Long Run Structural
VAR, Bootstrap Methods.
∗We are most grateful to Kevin Lee and David Fielding for their invaluable help and guidance.
Financial support from the portuguese institution ”Fundacao para A Ciencia e Tecnologia”, is also
gratefully acknowledged.
11I n t r o d u c t i o n
Direct measures of expectations provide the means by which the expectations formation
process can be explicitly studied. However, in most applications the time-series properties
of the data have either been ignored or have not been explicitly accounted for. In particular,
most tests of the Rational Expectations Hypothesis (REH) have been carried out under the
implicit assumptions of stationarity of the expectational errors and long-run unbiasedness.1
Furthermore, rationality tests on output expectations, for instance, are frequently carried
out using the most recently revised data on actual output, which can be quite dissimilar to
the series available at the time the expectations were formed.2
This paper addresses both these issues in an analysis of the expectations formation
process of US output. We account for the I(1) properties of the series by utilizing the
multivariate cointegration framework of Johansen (1988) and its subsequent generalization
in Pesaran, Shin and Smith (2000), in order to simultaneously model US actual output
and direct measures of expected output over the period 1969q2-2001q2. Actual output
is measured in real time, thus avoiding the criticisms associated with data revisions. Our
direct measures of output expectations consist of the current-period, the one, two and three-
period-ahead forecasts provided by the Survey of Professional Forecasters. We revisit the
issues of (i) stationarity of the expectational errors, (ii) long-run unbiasedness and further
investigate (iii) the short-run dynamics of output expectations and (iv) gain an insight as
to how long the long run actually is.
2 Modelling Framework
2.1 Deﬁnition of the Expectational Errors
We consider the expectational errors ηi,t, i =1 ,2,3,4, t =1 ,2,...,T,g i v e nb y
η1,t = ty
e
t −t+1 yt, η2,t =t y
e
t+1 −t+2 yt+1, η3,t =t y
e
t+2 −t+3 yt+2,
η4,t = ty
e
t+3 −t+4 yt+3, t =1 ,2,...,T, (1)
where syp denotes actual output at time p available at time s and the superscript ”e”
denotes expectation, so that sye
p is the expectation formed at time s on the value of output
1One exception is Lahiri and Chun (1989).
2Patterson and Heravi (1991), for example, illustrate that diﬀerent vintages of UK GDP
components need not even be cointegrated.
2in time p.3 The expectational errors in (1) may alternatively be written as
ηt = β
0
0zt − A(L)∆zt , t =1 ,2,...,T, (2)
where ηt =[ η1,t, η2,t, η3,t, η4,t]0, zt =[ tyt−1, tye
t, tye
t+1, tye
t+2, tye
t+3]0 , A(L)=A1L−1 +
A2L−2 + A3L−3 + A4L−4, L being the lag-operator,
A1 =
⎡
⎢ ⎢ ⎢
⎣
10000
10000
10000
10000
⎤
⎥ ⎥ ⎥
⎦, A2 =
⎡
⎢ ⎢ ⎢
⎣
00000
10000
10000
10000
⎤
⎥ ⎥ ⎥
⎦, A3 =
⎡
⎢ ⎢ ⎢
⎣
00000
00000
10000
10000
⎤
⎥ ⎥ ⎥
⎦,
A4 =
⎡
⎢ ⎢ ⎢
⎣
00000
00000
00000
10000
⎤
⎥ ⎥ ⎥
⎦,and β
0
0 =
⎡
⎢ ⎢ ⎢
⎣
−11000
−10100
−10010
−10001
⎤
⎥ ⎥ ⎥
⎦. (3)
2.2 Expectational Errors in a Cointegrating VAR Framework
Provided that zt =[ tyt−1,t ye
t,t ye
t+1,t ye
t+2,t ye
t+3]0 is an I(1) process with linear determin-
istic trending behaviour, it can be approximated by the following VECM4
∆zt = a0 + a1t + Πzt−1 +
p−1 X
i=1
Γi∆zt−i + et,t =1 ,2,...,T, (4)
where
a0 = −Πµ +[ Π + Γ(1)]γ,( 5 )
a1 = −Πγ,( 6 )
et ∼ IN(0,Ω), Ω positive-deﬁnite, µ and γ are 5 × 1 vectors of unknown coeﬃcients,
Γ(1) = I5 −
Pp−1
i=1 Γi, Γi, i =1 ,...,p− 1 and Π are 5 × 5 coeﬃcient matrices and
0 ≤ rank[Π]=r<5,s ot h a tΠ = αβ
0 with α, β being 5 × r,f u l lc o l u m nr a n k .
The frequent assumption of stationarity of the expectational errors ηt deﬁn e di n( 2 ) ,
requires that β = β0. Under this hypothesis the expected value of ηt takes the form of
E[ηt]=β
0
0µ +( β
0
0γ)t − A(1)γ , t =1 ,2,...,T, (7)
3Actual output is GNP up to 1991q4 and GDP thereafter, following a change in the
survey’s questionnaire. We use the 15-day vintage as it reﬂects more accurately the infor-
mation available to the respondents of the survey when the questionnaires are sent out. All
variables are in natural logarithms.
4See for example Pesaran, Shin and Smith (2000).
3where A(1) =
P4
i=1 Ai and Ai, i =1 ,2,3,4 were deﬁned in (3). Unbiasedness requires
E[ηt]=0, which in the current framework corresponds to β
0
0γ = 0,also known as the co-
trending hypothesis, and A(1)γ = β
0
0µ. Since the VECM in (4) does not provide estimates
of γ and µ, the restrictions A(1)γ = β
0
0µ cannot be tested in general. However, provided
that β = β0 and for γ =[ γ1, γ2, γ3, γ4, γ5]0, co-trending is equivalent to γ =[ γ1, γ1, γ1,
γ1, γ1]0,w h e r eγ1 is the mean of ∆tyt−1. In other words, the variables in zt are not only
driven by a common stochastic trend, but they also share the same deterministic trend with
tyt−1. Therefore, given β = β0, β
0
0γ = 0 a n da ne s t i m a t eo fˆ γ1, long-run unbiasedness
imposes through (5) the following set of testable restrictions
A(1)ˆ γ =(α
0α)
−1α
0[Γ(1)ˆ γ − a0]. (8)
3 Empirical Analysis
Our data on zt =[ tyt−1,t ye
t,t ye
t+1,t ye
t+2,t ye
t+3]0 are quarterly and cover the period 1968q4-
2001q2. The empirical analysis is based on the VECM given by (4), augmented by the
dummy vector Dt =[ d71q4t, d74q4t, d81q1t, d96q1t, d99q4t]0,w h e r ed71q4t takes the
value of one in 1971q4 and zero otherwise and the remaining variables are similarly deﬁned.
The dummies d71q4t and d74q4t are intended to capture the slowdown in economic activity
at the end of 1971 and 1974, while d81q1t, d96q1t andd99q4t are controlling for the eﬀects of
the comprehensive revisions in GDP that took place in December 1980, December 1995 and
August 1999. The employment of multivariate ADF tests and the LR statistics for testing
the restriction of the trend coeﬃcients according to (6) (available on request) indicated that,
at conventional signiﬁcance levels the vector zt can indeed be described as an I(1) process
with linear deterministic trending behaviour, irrespective of the choice of r. The lag-length,
p, was set equal to 2 after testing for signiﬁcance of additional lags within an unrestricted
VA R (8) in the level of zt, as well as with the use of the AIC and SBC.
3.1 Properties of the Expectational Errors
Table 1 reports the λ − trace and maximal eigenvalue cointegration rank statistics. In
general, the simulated critical values verify the presence of a ﬁnite-sample bias in favour of
rejection reported, for example, by Harris and Judge (1998). Nevertheless, this appears to
make no qualitative diﬀerence to the outcome of the tests, as all the evidence appear to be
clearly in favour of our economic priors that r =4 .
Table 2 reports the tests of three sets of over-identifying restrictions on β
0
∗ =[ β
0,−β
0γ].
The ﬁrst set, denoted ROV1, corresponds to the test of β = β0, which cannot be rejected
even asymptotically. The set of restrictions denoted as ROV3 corresponds to the joint
hypothesis β = β0 and β
0
0γ = 0. Asymptotically this joint hypothesis is clearly rejected.
4With reference to the simulated ﬁnite-sample distribution, however, rejection is avoided at
the 5% level, although not at the 10% level.5 Stronger evidence is obtained for the hypothesis
that the trend coeﬃcients are absent from the cointegrating relations corresponding to η1,t,
η2,t and η3,t alone, denoted as ROV2. These results indicate that the joint hypothesis
β = β0 and β
0
0γ = 0 appears to be consistent with the data as far as η1,t, η2,t and η3,t are
concerned. Regarding η4,t the evidence appear to be less conclusive and raise the suspicion
that it could possibly follow a near trend-stationary process. With reference to expression
(7), such a property would imply a particularly disappointing performance on the part of
economic agents in predicting the tree-period-ahead level of output, as the mean forecast
error would be a linear function of time. In the light of this and considering the evidence
reported in, inter alia, Johansen (2000) regarding the ﬁnite-sample bias of χ2 tests, we
have decided to put more weight on the simulation results and maintain the co-trending
hypothesis.
The diagnostic and descriptive statistics of the VECM estimated subject to β = β0 and
β
0
0γ = 0 (available on request) revealed no signs of model misspeciﬁcation at conventional
levels of signiﬁcance. The restrictions in (8) were tested for ˆ γ1 =0 .0075176 and the
relevant Wald statistic was found to be 2.691 with an asymptotic 5% critical value of 9.49,
thus verifying that expectations are indeed unbiased. However, it should be stressed that
this result rests on the joint assumption β = β0 and β
0
0γ = 0, for which the evidence were
less convincing as far as η4,t is concerned.
3.2 Dynamic Behaviour of Output Expectations and Expectational Er-
rors
Figure 1 plots the Generalized Impulse Responses (GIR) for zt to a one-standard error
innovation in tyt−1, which in this case are exactly equivalent to the more standard Orthogo-
nalized Impulse Responses (OIR).6 The impact eﬀect of the shock is to raise real-time output
by 0.62% and output expectations between 0.668% and 0.697%. After approximately 14
quarters all variables stabilize at 0.756% higher than their pre-shock level. This illustrates
the I(1) properties of the variables that cause a given shock to have a permanent eﬀect, as
well as the stationary nature of the expectational errors that eventually eliminates the gap
between actual and expected output.
T h er e s p o n s eo ft h ee x p e c t a t i o n a le r r o r sηt is explicitly illustrated in Figure 2. On
impact, the rise in tyt−1 results in underprediction which is greater the longer the forecast
horizon and ranges between -0.10% and -0.16%. The expectational errors persist between 4
and 14 quarters, with the adjustment process being faster the shorter the forecast horizon.
5Non-parametric simulation is based on Fachin (2000).
6For a proof of the fact that GIR = OIR when considering a shock in the ﬁrst variable
of the system see Pesaran and Shin (1998).
54 Concluding Remarks
Our ﬁndings have conﬁrmed the quite frequent assumption of stationary expectational
errors. Furthermore, all expectations may reasonably be argued to be unbiased in the
long run, although the evidence is weaker in the case of the three-period ahead forecast.
Expectational errors are found to persist between 4 and 14 quarters, with the adjustment
process being faster the shorter the forecast horizon.
6References
Fachin, S., 2000, Bootstrap and Asymptotic Tests of Long-Run Relationships in Coin-
tegrated Systems, Oxford Bulletin of Economics and Statistics, 62, 4, 543-551.
Harris, R.I.D. and G. Judge, 1998, Small Sample Testing for Cointegration using the
Bootstrap Approach, Economics Letters, 58, 31-37.
Johansen, S., 1988, Statistical Analysis of Cointegration Vectors, Journal of Economic
Dynamics and Control, 12, 231-254.
Johansen, S. , 2000, A Bartlett Correction Factor for Tests on the Cointegrating
Relations, Econometric Theory 740-778.
Lahiri, K. and T.S. Chun, 1989, Some Tests for Unbiasedness in the Long-Run Using
Survey Data, International Economic Journal, 3, 27-42.
Patterson, K.D. and S.M. Heravi, 1991, Are Diﬀerent Vintages of Data on the Com-
ponents of GDP Co-integrated? Some Evidence for the United Kingdom, Economics
Letters, 35, 409-413.
Pesaran, M.H. and Y. Shin, 1998, Generalised Impulse Response Analysis in Linear
Multivariate Models, Economics Letters, 58, 17-29.
Pesaran, M.H., Y. Shin and R.J. Smith , 2000, Structural Analysis of Vector Error
Correction Models with Exogenous I(1) Variables, Journal of Econometrics, 97, 293-
343.
7  ￿
 
Table 1 – Cointegration Rank Statistics  
  λ-Trace  Maximum Eigenvalu  
0 H  
1 H   Statistic  5% cv  Statistic  5% cv 
r = 0  r = 1  223.03 
87.17 a 
87.48 p 
91.77 n 
80.88 
37.86 a 
40.98 p 
41.12 n 
r ≤ 1  r = 2  142.15 
63.00 a 
60.69 p 
66.44 n 
65.74 
31.79 a 
33.55 p 
34.56 n 
r ≤ 2  r = 3  76.41 
42.34 a 
44.15 p 
44.21 n 
43.93 
25.42 a 
27.65 p 
27.74 n 
r ≤ 3  r = 4  32.48 
25.77 a 
24.72 p 
25.90 n 
24.23 
19.22 a 
19.37 p 
20.01 n 
r ≤ 4  r = 5  8.24 
12.39 a 
12.78 p 
12.52 n 
8.24 
12.39 a 
12.78 p 
12.52 n 
Notes:  Small-sample  results  are  based  on  a  bootstrap  with  20,000  simulations.  “p”  and    “n”  indicate parametric and non -
parametric respectively, while “a” indicates asymptotic critical value. 
 
 
          Table 2 – Tests of Restrictions on the Cointegrating Parameters 
Restrictions  LR-statistic Asymptotic  Small  Sample 
    5% cv  10% cv  5% cv  10% cv 
1 OV R   5.28 9.49  7.78  _ _ 
2 OV R   9.78 14.07  12.02  _ _ 
3 OV R   19.08 15.51  13.36  21.17 p 
21.39 n 
18.29 p 
18.36 n 
Notes:  OVi R , i=1,2,3, are defined in the text. Small-sample results are based on a bootstrap with 10,000 simulations. “p” and “n” 
indicate parametric and non-parametric respectively. 
 
 
 

Are US Output Expectations Unbiased? A Cointegrated VAR Analysis in Real Time

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