In this note, we consider the contradiction between the fact that the best fit for the UK consumption data in Davidson et al. (1978) is obtained using an equation with an intercept but without an error correction term, whereas the equation with error correction and without the intercept has better post-sample forecasting properties than the former equation. This contradiction is explained and the two equations reconciled in a nonlinear framework by applying a smooth transition regression model to the data.consumption equation; model misspecification testing; nonlinearity; smooth transition regression

Eliasson, Ann-Charlotte

Teräsvirta, Timo

English

Research Papers in Economics

Error correction in DHSY
Ann-Charlotte Eliasson
Deutsche Bank, Fixed Income & Relative Value Research
Frankfurt am Main, Germany
Timo Teräsvirta
Department of Economic Statistics
Stockholm School of Economics,
Box 6501, SE-113 83 Stockholm, Sweden
SSE/SFI Working Paper Series in Economics
and Finance, No. 517
November 22, 2002
Abstract
In this note, we consider the contradiction between the fact that the
b e s tﬁ tf o rt h eU Kc o n s u m p t i o nd a t ai nD a v i d s o ne ta l .( 1 9 7 8 )i so b t a i n e d
using an equation with an intercept but without an error correction term,
whereas the equation with error correction and without the intercept has
better post-sample forecasting properties than the former equation. This
contradiction is explained and the two equations reconciled in a nonlinear
framework by applying a smooth transition regression model to the data.
Keywords: consumption equation, forecast failure, model misspeciﬁ-
cation testing, nonlinearity, smooth transition regression
JEL Classiﬁcation Codes:C 2 2
Acknowledgements: This research has been supported by Jan Wal-
lander’s and Tom Hedelius’s Foundation for Social Research. We wish to
thank David Hendry for allowing us to use his data set and for clarify-
ing discussions. The responsibility for any errors and shortcomings in the
paper remains ours. Material from this paper has been presented at the
conference on Multivariate Nonlinear Modelling, Rotterdam, September
1999.
11 Introduction
Davidson, Hendry, Srba and Yeo (1978), DHSY in short, is an inﬂuential article
on modelling the consumption-income relationship, and it has inspired a large
number of follow-up studies. It has contributed to the popularity of the concept
of error correction, introduced by Phillips (1957), for which statistical justiﬁcation
arrived later in the form of cointegration; see Granger (1981). Nevertheless, the
role of error correction in modelling the UK consumption 1958(2)-1970(4) in
DHSY appears somewhat ambiguous. When the equation is estimated without
the intercept, the error correction term, the logarithmic consumption/income
ratio, has a signiﬁcant coeﬃcient estimate. When the same model contains an
intercept, the error correction term is no longer signiﬁcant at the 10% level,
whereas the intercept is strongly signiﬁcant. The authors are aware of this. They
estimate a model without the intercept and another one with the intercept but
without the error correction term. The latter equation yields better ﬁt, but the
former one is favoured for theoretical reasons and because of the fact that the
former model has better out-of-sample forecasting performance of the two. DHSY
note that the two equations ”exhibit an interesting conﬂict between goodness of
ﬁt and parameter stability as criteria for model selection.”
In this note we reconsider the situation in the light of the period 1958(2)-
1970(4). In particular, we assume that the subsequent observations are com-
pletely unknown to us. We simply model the period 1958(2)-1970(4) as well
as we can using the data for this period. The resulting model reconciles the
model with an error correction term and the one with an intercept. This non-
linear model, however, while ﬁtting the data very well, breaks down in the sense
of Clements and Hendry (1998) when new information about the inﬂation rate
becomes available in the early 1970’s.
The note has the following structure. In Section 2 we introduce the data and
the notation. The problem is deﬁned in Section 3. In Section 4 we introduce our
model, the smooth transition regression model. Section 5 contains the modelling
results and some discussion. Section 6 concludes.
2 Data and notation
The data set used in the paper is the same as the one DHSY used. It has
since been extended but, at the same time, some of the data deﬁnitions have
been changed, so that the old and the revised data are not compatible. An
essential feature is that the estimation period is the same as in DHSY: 1958(2) to
1970(4). The following notation is used. Lowercase letters represent logarithmic
variables: xt =l nXt. Variable Ct is the consumption of nondurable goods, Yt is
the disposable income, Pt is the consumption price index, and D68t is a dummy
variable for a temporary switch in consumption between the ﬁrst and the second
2quarter. The variables in the ﬁnal consumption equation are the four-quarter
diﬀerences ∆4ct,∆4yt,∆∆4yt,∆4pt,∆∆4pt, the error correction term ct −yt, and
D68t. Augmented Dickey-Fuller tests reject the unit root hypothesis at the 10%
level of signiﬁcance for all variables except ct − yt.
3 The problem
In order to illustrate the situation, we begin by estimating the DHSY equation
(45)a u g m e n t e db yt h ei n t e r c e p t . I ti sb a s e do nf o u r -q u a r t e rd i ﬀ e r e n c e so ft h e
logarithmic series and has the following form:
∆4ct =0 .014
(0.0048)
+0 .37
(0.049)
∆4yt − 0.17
(0.047)
∆∆4yt − 0.21
(0.070)
∆4pt − 0.16
(0.13)
∆∆4pt
−0.041
(0.028)
(ct−4 − yt−4)+0 .0074
(0.0030)
∆4D68t + εt (1)
R2 =0 .81,  σL =0 .0055 pLJB =0 .66
pARCH(1) = 0.54 pARCH(4) = 0.77 pFRESET(1,43) = 0.066
pLMAC(1) = 0.56 pLMAC(4) = 0.81
where the ﬁgures in parentheses are estimated standard errors, pLJB is the p-
value of the Lomnicki-Jarque-Bera normality test, pARCH(j) is the p-value of the
LM test of no ARCH against ARCH of order j, pLMAC(j) i st h ep -v a l u eo ft h eL M
test of no error autocorrelation against autocorrelation of order j, pFRESET(1,43)
is the p-value of the RESET with the quadratic term. It is seen from (1) that
the intercept is highly signiﬁcant, whereas the estimated coeﬃcient of the error
correction term ct−4 −yt−4 is not. Removing it would be an obvious decision if a
model selection criterion such as AIC or BIC were used as a guidance. However,
DHSY omit the intercept instead for the reasons discussed in the Introduction.
We shall take another look at the situation using a particular nonlinear model,
the smooth transition regression (STR) model, as our tool. It turns out that with
our approach we are able to reconcile the two competing alternatives.
It should be noted that Cook, Holly and Turner (1999) and Cook (2000) have
also modelled the DHSY data using nonlinear (in variables, not in parameters)
models. The aims of their analysis are, however, diﬀerent from ours, and we do
not consider their results in our discussion.
4 Smooth transition regression model
The STR model is nonlinear model capable of describing asymmetries and other
types of nonlinearity in macroeconomic relationships; for a recent account, see
3Teräsvirta (1998). It can be deﬁned as follows:
yt = φ
zt+θ
ztG(γ,c,s t)+εt
= {φ + θG(γ,c,s t)}
zt + εt,t=1 ,...,T (2)
where zt =( w
t,x
t) with wt =( 1 ,y t−1,...,y t−p), xt =( x1t,...,x kt),x jt,j =
1,...,k, are exogenous to the parameters of interest, and m = p+k +1. Further-
more, φ =( φ0,φ 1,...,φp) and θ =( θ1,...,θm) are parameter vectors, c =( c1,...,cK)
is a vector of location parameters, c1 ≤ ... ≤ cK, and εt ∼iid(0,σ 2). Transition
function G(γ,c,s t) is a bounded function of st, continuous everywhere in the
parameter space for any value of st. The last expression in (2) indicates that the
model can be interpreted as a linear model with stochastic time-varying coeﬃ-
cients φ + θG(γ,c,s t). The logistic transition function has the general form
G(γ,c,s t)=( 1+e x p {−γ
K 
k=1
(st − ck)})
−1,γ>0 (3)
where γ>0 is an identifying restriction. Equation (2) jointly with (3) deﬁnes
the logistic STR (LSTR) model. The most common choices for K are K =1and
K =2 . For K =1 , the parameters φ + θG(γ,c,s t) change monotonically as a
function of st from φ to φ + θ. For K =2 , they change symmetrically around
the mid-point (c1 +c2)/2 where this logistic function attains its minimum value.
Slope parameter γ controls the slope and c1 and c2 the location of the transition
function. When γ →∞ , (3) becomes a step function for K =1and a ”double
step” function for K =2 .
In this note, the transition variable st is an element of vector zt. As we have
no theory for choosing st in advance, the choice will be data-driven and will be
discussed in the next section.
5R e s u l t s
We follow the model selection strategy outlined in Teräsvirta (1998) in order to
build our STR consumption equation for the DHSY data. Thus, we ﬁrst test
linearity of (1) against STR. This involves choosing K =1and approximating
the transition function by a third-order Taylor expansion around γ =0 , see
Teräsvirta (1998). The results can be found in Table 1. They show that linearity
Test statistic\Transition variable ∆4yt ∆∆4yt ∆4pt ∆∆4pt ct−4 − yt−4
F04 (3rd order Taylor expansion) 0.55 0.95 0.63 0.25 ..
F02 (1st order Taylor expansion) 0.44 0.72 0.067 0.076 0.51
Table 1: p-values of linearity tests of equation (1) against STR
4cannot be rejected at the conventional 5% signiﬁcance level for any of the potential
transitional variables. However, if linearity is tested directly against the logistic
STR model (K =1and the ﬁrst-order Taylor expansion) the test statistic when
the inﬂation variable ∆4pt−4 is assumed to be the transition variable equals p =
0.067. Furthermore, if the latter test is performed using equation (45) in DHSY,
the one without the intercept, as the null model, the corresponding p-value equals
0.0041. Therefore it seems fair to argue that equation (45)i sm i s s p e c i ﬁ e da n d
that the misspeciﬁcation involves the intercept, the error correction component
and the inﬂation rate. It should be noted that the asymptotic distribution theory
for the tests is only applicable when the series are stationary. The error correction
term appears not to be stationary, but due to short series it is unlikely that the
results are much aﬀected by that.
Based on the test results, we reject linearity and proceed with estimation of the
appropriate STR model with ∆4pt as the transition variable. After eliminating
redundant variables and imposing appropriate parameter restrictions, the ﬁnal
STR model has the following form:
∆4ct =0 .34
(0.038)
∆4yt − 0.15
(0.034)
∆∆4yt − 0.11
(0.078)
∆4pt − 0.35
(0.20)
∆∆4pt − 0.18
(0.027)
(ct−4 − yt−4)
+0.0093
(0.0025)
−{0.016
(0.0038)
− 0.35
(0.20)
∆∆4pt − 0.18
(0.027)
(ct−4 − yt−4)}
×[1 + exp{−12.4
(11.7)
(∆4pt − 0.022
(0.0018)
)]
−1 + εt (4)
R2 =0 .85,  σ =0 .0049  σL/ σ =0 .84 pLJB =0 .68
pARCH(1) = 0.94 pARCH(4) = 0.69 pFRESET(1,41) = 0.24
pLMAC(1) = 0.62 pLMAC(4) = 0.60 pFencomp(3,39) = 0.99
where pFencomp(3,39) is the p-value of the LM test of the restrictions on the inter-
cept and the coeﬃcients of ∆∆4pt and ct−4−yt−4. Fencomp(3,39) is an F-statistic
for testing the null hypothesis that (4) encompasses (1). This null hypothesis
cannot be rejected, and no other test suggests model inadequacy either. Besides,
the tests of no additive nonlinearity and parameter constancy (results not shown),
see Teräsvirta (1998), do not contain evidence against (4). The residual standard
deviation of the STR equation is only 84% of that of (1), which is a noticeable
improvement.
Ericsson and MacKinnon (in press), using a longer observation period ending
1975(4), argued that seasonal dummy variables are required in the linear DHSY
model. The coeﬃcient estimates of seasonal dummies are not signiﬁcant, however,
when these variables are added to (4).
The estimated nonlinear error correction relationship equals
et = −0.18(ct−4 − yt−4) −{ 0.016 − 0.18(ct−4 − yt−4)}
×{1+e x p {−12.4(∆4pt − 0.022)}
−1 (5)
5Figure 1: Error correction term (5) (solid line), the DHSY model (dashed-dotted
line), and the linear model with intercept (dashed line) graphed over time
The restriction between the ”linear” and ”nonlinear” coeﬃcient of ct−4 − yt−4 is
supported by the data. It is seen from (5) that the equilibrium error correction
occurs only at low levels of the inﬂation. The graph of (5) appears in Figure
1. When inﬂation is suﬃciently high, the intercept takes over, and the long run
relationship between consumption and income ceases to exist. Figure 2 depict-
ing the transition function as a function of the transition variable shows that
a majority of quarters actually belong to the set in which no error correction
takes place. The long-run relationship DHSY entertain is local in the sense that
it is supported by the data only at periods of low inﬂation. This explains the
presence of the intercept in (1), but retaining the error correction term in the
linear equation also receives partial support from (4). Figure 1 shows that the
error correction term in the equation of DHSY is trending, which explains the
result of the unit root test mentioned in Section 2. On the other hand, the error
correction of (4) is stationary so that the equation is a balanced one. The STR
equation thus reconciles the two competing linear models in DHSY, the one with
the intercept and no long-run relationship and the one with an error correction
term but no intercept.
6 Final remarks
The STR model (4) represents what the data tell us about the relationship be-
tween consumption and inﬂation in the UK for 1958(2)-1970(4). Nevertheless,
the model fails outside the estimation period in the sense that its forecast ac-
curacy is inferior to that of the error correction model in DHSY. This appears
contradictory as the inﬂation rate after the estimation period has been higher,
not lower, than during it. One would therefore expect the nonlinear equation do
6Figure 2: Transition function of model (4) graphed as a function of transition
variable. Each dot represents an observation
better than the linear error correction one because according to the STR model
the error correction is only operational when the inﬂation rate is suﬃciently low.
The main reason for this outcome is that during the estimation period the
inﬂation rate appears stationary. However, in the early 1970s there is a strong
upsurge in inﬂation suggesting that the inﬂation rate is better modelled as an
integrated variable. This is why equation (4) ceases to be a useful approximation
to the data-generating process, and consequently it may be argued that it ”breaks
down” in the 1970s. The new information about the inﬂation rate can then be
interpreted as a post-sample break in the model, as deﬁned in Clements and
Hendry (1998). It is quite conceivable, however, that the data-generating process
remains unchanged for a period including the years 1971-1975. The problem
is simply that the approximation to it that originally appeared fully plausible
turns out to be inadequate when drastically new information about the dynamic
behaviour of some of the time series included in the model becomes available.
One cannot expect the DHSY equation to be an adequate description of the
DGP either, but it happens to contain a linear combination of the three variables
that in the light of later information appear cointegrated; see Hendry, Muellbauer
and Murphy (1990). It thus has a chance of providing reasonable forecasts (in
fact the decisive criterion for DHSY to choose this equation was its post-sample
forecasting performance).
More generally, the results in this note suggest that when the time series ini-
tially available for macroeconometric modelling are short, the conclusions about
7the functional form of the model may change when more data become available.
In other words, with more information, breaks are likely to occur. This may
speak for parsimonious linear speciﬁcations. We have simply shown how ﬁtting
an STR model to the data from 1958(2)-1970(4) neatly solves the contradiction
between the goodness of ﬁt and parameter stability highlighted in DHSY.
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8

DHSY revisited: The role of asymmetries,

Econometric modelling of the aggregate time-series relationship between consumers’ expenditure and income in United Kingdom,

Forecasting Economic Time Series,

Frequency domain and time series properties of asymmetric error correction terms,

Modeling economic relationships with smooth transition regressions,

Some properties of time series data and their use in econometric model speciﬁcation,

Stabilization policy and the time forms of lagged responses,

The econometrics of DHSY,

Error correction in DHSY

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Error correction in DHSY

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