The ARAR Error Model for Univariate Time
Series and Distributed Lag Models
R. A. L. Carter
University of Western Ontario and University of Calgary
A. Zellner
University of Chicago
Abstract
We show that the use of prior information derived from former empirical
findings and/or subject matter theory regarding the lag structure of the
observable variables together with an AR process for the error terms
can produce univariate and single equation models that are intuitively
appealing, simple to implement and work well in practice. Key words:
time series analysis, model formulation. JEL Classification: C11, C22.
1
Introduction
“None of the previous work should be construed as a demonstration of the inevitability of MA disturbances in econometric models.
As Parzen’s proverb1 reminds us the disturbance term is essentially
man-made, and it is up to man to decide if some of his creations are
more reasonable than others.” Nicholls, Pagan and Terrell (1975)
p.117.
Much theoretical and empirical work in economics features low order autoregressions in endogenous variables, often accompanied by exogenous variables.
For example Caplin and Leahy (1999) and Sargent (1987, Chapter IX) present
AR(1) models. AR(2) models for cycles appear in: Samuelson(1939a,b), Burns
and Mitchell (1946), Goodwin (1947), Cooper and Johri(1999) and Sargent
(1987, Chapter IX). Papers by Metzler (1941), Garcia-Ferrer et. al. (1987),
Geweke (1988), Zellner and Hong (1989), Zellner, Hong and Min (1991) and
Zellner and Chen (2001) feature AR(3) models. Gandolfo (1996) provides many
examples of models which can be expressed as autoregressions, plus exogenous
variables including: cobweb models, foreign trade and taxation multipliers,
multiplier-accelerator models, market adjustment models and inventory models.
1 “God made X (the data), man made all the rest (especially, the error term).” quoted in
Nicholls et al (1975). We add that in many instances humans, not God, construct the data.
1
Thus, economists building time series models often have quite strong prior
beliefs, based on theoretical and applied work, about the lag structure for an
observable variable. However, they usually have quite weak prior beliefs about
the lag structure for the error in a model. The AR model for the error was
popular until Box and Jenkins (1976) methods became dominant in univariate
time series analysis and introduced the MA model without much theoretical
justification. This produced complicated likelihood functions and estimation
procedures and implied infinite AR processes for observed variables. We show
here how the use of prior information about the lag structure of the observable
variables and a return to the AR model for the error is intuitively appealing,
yields finite AR processes for observed variables and simplifies inference procedures.
2
Univariate Models
A reasonable starting point for an empirical analysis is to specify that the
random variable of interest has been generated by an autoregressive process
that reflects prior beliefs about the lag structure. This gives
φ(L)(Yt − µ) = Ut ,
(2.0.1)
where φ(L) is a polynomial of degree p in the lag operator L, µ is the origin
from which Yt is measured (the mean if Yt is stationary) and Ut is a covariancestationary error with zero mean. Researchers may have quite strong beliefs
about the value of p and any restrictions on φ(L) necessary to make Yt stationary. In such cases the parameters of interest are the coefficients of φ(L), φi ,
or some function of them, such as the roots of φ(L). However, it is rare for
researchers to have such strong prior beliefs about the error process and thus a
variety of models for Ut may be entertained: see e.g. Fuller and Martin (1961),
Zellner et. al. (1965) and Zellner and Geisel (1970).
One possible model for the error is
Ut = ε t ,
(2.0.2)
where εt is white noise with zero mean and variance σ 2 . But this rather
restrictive model implies that a shock to the subject-matter portion of the
model, φ(L)(Yt − µ), has an impact in only the current period. Of course,
if φ(L) is invertible the MA representation of Yt ,
Yt = µ +
1
εt ,
φ(L)
(2.0.3)
has infinite length.
2.1
The ARMA model
A popular alternative model for Ut is the MA model of Box and Jenkins(1976).
Ut = θ(L)εt ,
(2.1.1)
where θ(L) is an invertible polynomial of degree q in L. This model was introduced “To achieve greater flexibility in fitting actual time series, . . . ”2 rather
than from any explicit prior knowledge about the behavior of the error. But
this model is only slightly less restrictive than the white noise model because it
implies that the effect on φ(L)(Yt − µ) of a shock εt dies out completely after q
periods. Also strong restrictions on the values of the autocorrelations of Y t are
needed for θ(L) to be invertible. Now the infinite MA representation
Yt = µ +
θ(L)
εt
φ(L)
(2.1.2)
allows somewhat richer behavior than does the white noise case. However, the
AR representation,
φ(L)
(Yt − µ) = εt ,
(2.1.3)
θ(L)
is also infinite, which may be undesirable from a subject matter viewpoint and
which complicates inference about the φi .
Of course, an MA model for Ut is desirable on a priori grounds if the observations on Yt were the result of temporal aggregation or if the model features
unobserved states. However, in many cases an MA model for Ut arises from an
examination of sample autocorrelations and is not justified by any prior theory.
In these cases the model described in the next section is worthy of consideration.
2.2
The ARAR model
We propose an alternative, AR model for Ut which is parsimonious, allows rich
time series behavior and simplifies inference procedures:
ω(L)Ut = εt ,
(2.2.1)
where ω(L) is a polynomial of degree r in L. Assuming ω(L) is invertible, the
infinite MA representation of Ut is
Ut =
1
εt .
ω(L)
(2.2.2)
Substituting (2.2.2) into (2.0.1) gives the “structural form” of the model, which
we label3 ARAR(p,r), as
φ(L)(Yt − µ) = Ut =
2 Box
1
εt .
ω(L)
(2.2.3)
and Jenkins (1976) page 11.
(1982) introduces a model with a similar label. His ARARMA model consists of
two filters applied sequentially. The first filter is a nonstationary AR with a stationary output
which is input to a second ARMA filter with a white noise output. However, his model does
not partition the second filter into a portion containing prior information about the dynamics
of Yt and a portion modeling the error dynamics.
3 Parzen
In (2.2.3) the impact on φ(L)(Yt − µ) of a shock εt may die out very slowly,
rather than being cut off abruptly, so our model can produce rich behavior in
Ut with values of r as small as two. Also, (2.2.2) is consistent with Wold’s
(1938) Decomposition Theorem, in contrast to MA models for Ut which impose
a truncation4 on the Wold representation of Ut . If Yt is covariance-stationary,
then the MA representations of all the above models for Yt are infinite and,
therefore, consistent with Wold’s theorem. However, (2.2.2) has the advantage
of being the simplest model that also imposes this consistency on Ut .
The “reduced form” of the model is, from (2.0.1) and (2.2.2),
ω(L)φ(L)(Yt − µ) = εt ,
(2.2.4)
α(L)Yt = α0 + εt
(2.2.5)
or
where α(L) = ω(L)φ(L) is a polynomial in L of degree p + r with restricted
coefficients and α0 = ω(1)φ(1)µ. If φ(L) and ω(L) are invertible the MA representation of Yt is
1
εt .
(2.2.6)
Yt = µ +
ω(L)φ(L)
The restrictions that the ARAR model imposes on the coefficients of α(L) will
affect the rate at which the impulse response coefficients in (2.2.6) decline, but
that decline will be more rapid than in models with fractional differencing.
In contrast to (2.1.3), both (2.0.1) and (2.2.3) are finite AR processes. Thus
the ARAR model is simpler than the ARMA model and it should be assigned
higher prior probability according to the “Simplicity Postulate” of Wrinch and
Jeffreys (1921). Also, few scientific theories are framed as infinite AR processes,
but many are in the form of finite difference or differential equations. Finally,
the form of (2.2.4) makes it simpler to estimate than either (2.1.2) or (2.1.3).
To see the parsimony of (2.2.2) as compared to (2.1.1) consider an example
in which ω(L) = 1 − ω1L − ω2 L2 = (1 − ξ1 L)(1 − ξ2 L) with ξ1 = .7 and ξ2 = −.5.
Then the coefficient of εt−5 in (2.2.2) is .111 so it would take an MA process
with at least five parameters to approximate the inverse of the AR with two
parameters. Next assume the roots are complex conjugate with ξ1 = .7 + .5i
and ξ2 = .7 − .5i. Now the coefficient of εt−5 is .242 so, again, an approximating
MA would have to have at least five parameters, compared to two in ω(L).
The only behavior that is allowed by the ARMA model but ruled out by
the ARAR model is an infinite AR in Yt ; although if p and r are high (2.2.4)
will be a very long AR. Seasonal effects are modeled by expressing φ(L) or
ω(L) as products of nonseasonal and seasonal polynomials. Because some prior
information about φ(L) is assumed to be available, the polynomials φ(L) and
ω(L) can be analyzed separately.
Some of the coefficients of α(L) may be quite small in absolute value, even
though no φi or ωi is small. This can result in small and imprecise estimates of
4 Of course, it is possible for the Wold decomposition to yield a finite MA form in some
cases. However, we regard a model which requires a finite MA form for every case to be less
desirable.
these αi if the restrictions implied by the ARAR structure are ignored, tempting
researchers to impose invalid zero restrictions on α(L). As we show below, direct
estimation of φ and ω is easy so there is no need to focus solely on (2.2.5).
Before considering inferences about φ(L) and ω(L) we note that, without
further information, they are not identified. To see this, interchange φ(L) and
ω(L) in (2.2.3) which would yield the same unrestricted AR(p + r) as (2.2.5) so
the likelihood function based on (2.2.5) would be unchanged. Also, multiplying
both sides of (2.2.3) by a nonzero scalar υ0 would leave (2.2.5) unchanged but
it would result in υ0 φ(L) = υ0 − υ0 φ1 L − . . . υ0 φp Lp . We avoid this identification problem by adopting the common assumption that the model has been
normalized to have the first term in φ(L) equal to 1.0.
Identification failure also arises if both sides of (2.2.3) are multiplied by the
invertible lag polynomial υ(L) to give
υ(L)φ(L)Yt = υ(1)µ +
υ(L)
εt ,
ω(L)
(2.2.7)
leaving (2.2.5) and the likelihood function unchanged. This is analogous to
model multiplicity in ARMA models containing the products of seasonal and
nonseasonal lag polynomials. Following Box and Jenkins (1976), we assume that
all common factor polynomials like υ(L) have been canceled out of the model.
Even after normalization and common factor cancellations, an identification
problem remains. Write φ(L) and ω(L) in terms of terms of their roots as
φ(L) = (1 − λ1 L)(1 − λ2 L) . . . (1 − λp L)
(2.2.8)
ω(L) = (1 − ξ1 L)(1 − ξ2 L) . . . (1 − ξr L).
(2.2.9)
α(L) = φ(L)ω(L)
= (1 − λ1 L) . . . (1 − λp L)(1 − ξ1 L) . . . (1 − ξr L).
(2.2.10)
and
Thus
Now assume the λi and ξi are unknown. Then write
α(L) = (1 − η1 L)(1 − η2 L) . . . (1 − ηp+r L)
(2.2.11)
and assume we know the values of the roots ηi , i = 1, 2, . . . , p + r. The identification problem lies in deciding which terms (1 − ηi L) are part of φ(L) and
which are part of ω(L). The solution is to use the same prior information that
was used to specify φ(L) to make this decision. In many cases φ(L) will be
specified to be of degree two giving a damped cycle with a period in some a
priori most probable range. Thus the roots ηi belonging to φ(L) must be a
complex conjugate pair with modulus less than one and a period lying in the
most probable range. Any roots ηi which are real or complex conjugate with a
smaller period must belong to ω(L).
In practice the values of the ηi are unknown. But once we specify p and r we
can fit an AR(p + r) by ordinary least squares (OLS) and find the roots of the
resulting α̂(L). They can be used to find initial values of the φ̂i and ω̂i for use
in nonlinear least squares (NLS) estimation of ω(L)φ(L). This idea can even be
used on a long AR model produced by someone else. Using their estimated AR
coefficients we can obtain estimated roots, moduli and period to help us judge
whether their model is really the reduced form of an ARAR model.
Alternatively, a priori plausible initial values of φi and ωi can be used in
NLS to obtain estimates ω̂(L) and φ̂(L). Then their roots can be compared to
those of α̂(L) from (2.2.5) as a check that the specification is correct.
After ruling out under identified models, the total number of free parameters
in φ(L) and ω(L) may be equal to the number of free parameters in α(L). Such
models are just identified so, under normality, ML estimates of the φi and ωi
can be derived from OLS estimates of the αi . Alternatively, if NLS estimates of
the φi and ωi are used to derive estimates of the αi they should equal the OLS
estimates. However, if the number of parameters in α(L) is greater than that
in φ(L) and ω(L) (e.g. if some of the φi or ωi are specified to be zero without
reducing the value of p or r) the model is over identified so it is not generally
possible to solve for unique estimates of the φi and ωi from OLS estimates of
the αi . In such cases it will be necessary to estimate the φi and ωi directly by
NLS.
An interesting simple case is obtained by assuming that p and r are both
one so that (2.2.4) becomes
ω(L)φ(L)(Yt − µ) = (1 − ωL)(1 − φL)(Yt − µ)
(2.2.12)
2
= (1 − [ω + φ]L + ωφL )Y − (1 − ω)(1 − φ)µ
= εt
from which
Yt
= (1 − ω)(1 − φ)µ + (ω + φ)Yt−1 − φωYt−2 + εt
= α0 + α1 Yt−1 + α2 Yt−2 + εt .
(2.2.13)
(2.2.14)
We believe this ARAR(1,1) model is a useful alternative to the popular
ARMA(1,1) model with the same number of parameters.
Imposing stationarity and invertibility on the ARMA(1,1) model results in
very strong restrictions on the range of admissible values for the autocorrelations
of Yt at lags 1 and 2, ρ1 and ρ2 : see Box and Jenkins(1976), Figure 3.10(b). The
analogous restrictions on the ARAR(1,1) model result in restrictions on ρ1 and
ρ2 which are much weaker. Assuming that −1 < φ < 1 and −1 < ω < 1, results
in the feasible set of α1 and α2 values being a subset of that for an unrestricted
AR(2) which excludes the subset associated with complex roots. This is shown
in Figure 1 which should be compared with Box and Jenkins (1976), Figure
3.9. These restrictions also restrict the ρ1 , ρ2 space to a subset of that for
an unrestricted AR(2). This space is shown in Figure 2a which should be
compared with Box and Jenkins (1976) Figures 3.3(b) and, especially, 3.10(b),
which is reproduced below as Figure 2b. The admissible parameter space shown
in Figure 2a is considerably larger than that shown in Figure 2b and negative
values of ρ2 are excluded. This last point should aid in model formulation as
sample autocorrelations which are negative at both lags one and two would be
evidence against an ARAR(1,1) model being appropriate. We note that for
many economic time series the first two sample autocorrelations are positive so
that the ARAR model is not ruled out.
Since both φ(L) and ω(L) have been normalized and there are no common
factors, all that is needed for identification is some prior information on the roots
of α(L), φ and ω. For example we might identify φ as the (absolutely) larger root
of α(L). Since this model is just identified, we could obtain estimates of φ and
ω (plus µ) from estimates of the αi by OLS, ML, MAD, traditional Bayes and
Bayesian Method of Moments (BMOM): see Zellner (1996, 1997), Zellner and
Tobias (2001) and Green and Strawderman (1996). The stationarity restrictions
imply that −2 < α1 < 2 and −1 < α2 < 1. Note that if, for example, φ > 0 and
ω < 0, α1 may be quite small in absolute value but imposing α1 = 0 would be a
specification error. Alternatively, one could estimate φ and ω by NLS and use
the results to form estimates of the αi , which should be the same as the OLS
estimates. Posterior densities for φ and ω can be obtained from the unrestricted
posterior for α1 and α2 by simulation. If the stationarity restriction is to be
imposed draws which violate this restriction would be discarded.
Now assume a quarterly seasonal model for the error, ω(L) = 1 − ω4 L4 , with
the nonseasonal, subject-matter structure φ(L) = 1 − φL, as before. This gives
a restricted AR(5) for the reduced form,
α(L)
= (1 − φL − ω4 L4 + φω4 L5 )
4
(2.2.15)
5
= (1 − α1 L − α4 L − α5 L ),
with only three parameters in contrast to an unrestricted AR(5) with five. However, this model is over identified so it would be necessary to obtain estimates
of φ and ω4 directly using NLS. To ensure that ω(L) is invertible impose the
uniform prior, p(ω4 ) = .5 over the range −1 < ω4 < 1. To impose stationarity
on Yt use the prior p(φ) ∝ (1 − φ)a−1 (1 + φ)b−1 for −1 < φ < 1: see Zellner
(1971 p.190). Finally, adopt a uniform prior on log(σ) so that p(σ) ∝ σ −1 .
Then, given a T × 1 vector y of data and assuming εt to be independent normal,
σ can be integrated out analytically and the joint posterior density for φ and
ω4 can be analyzed using bivariate numerical integration.
The first step in building a general ARAR model is to select a degree p for the
polynomial φ(L) based on prior knowledge. Data plots, sample autocorrelations
and partial autocorrelations may also be useful at this stage, e.g. to confirm the
presence of unit roots or cycles which suggest a p ≥ 2. A tentative AR(r) model
for the error Ut should now be specified with r believed large enough to make εt
white noise. This will imply that the degree of α(L) is p + r. The suitability of
r can be gauged from the autocorrelations of the residuals from OLS applied to
(2.2.5). This procedure differs from the Box and Jenkins (1976) method which
bases the lag lengths primarily on the sizes of data autocorrelations and partial
autocorrelations. Their procedure may be suitable if the researcher has no prior
beliefs about φ(L), although it involves considerable pretesting that can have
adverse effects upon subsequent inference; see Judge and Bock(1978). Of course,
models initially based on features of a sample can in the future be rationalized
by theory and confirmed with new samples.
In Bayesian analysis the choice of p and r should be guided by posterior
odds ratios. If the odds analysis leads to several favored models results regarding
parameters and predictions can be averaged over the models leading to forecasts
which are often superior to non-combined forecasts. If the prior odds ratio is
set to one, the posterior odds ratio becomes the Bayes factor, B1,2 , which can
be calculated exactly for small models: see Monahan (1983). For large samples
Schwarz (1978) provides the approximation
B1,2 ' T q/2 elr ,
(2.2.16)
where: the log-likelihood ratio lr = log[l(Θ̂1 /y)/l(Θ̂2/y)], Θ̂i is the maximumlikelihood estimate of Θ from model i, q = q2 − q1 and qi is the number of
parameters in model i.
We note that the regression equation commonly used to test for the presence
of a unit root versus a linear trends can be written as
φ(L)Yt = µ0 + µ1 t + [ω(L)]−1 εt ,
(2.2.17)
α(L)(Yt − µ0 − µ1 t) = εt :
(2.2.18)
or as
see Schotman and van Dijk (1991) and Zivot (1994). These are both examples
of the ARAR models (2.2.3) and (2.2.4) with µ a linear function of the trend
variable t and φ(L) of degree 2 or more. If the unit root is believed to be in
φ(L), repeated draws can be made from the joint posterior of its coefficients
and the roots calculated for each draw. The proportion of draws for which
the roots are complex versus real and with modulus below 1.0 versus 1.0 or
more are estimates of the posterior probabilities of these properties of φ(L): see
Geweke(1988), Schotman and van Dijk (1991) and Zivot (1994).
2.3
Empirical example: Housing starts
One of the leading indicators published by the U.S. Department of Commerce is
the number of permits issued by local authorities for the building of new private
housing units. Pankratz (1983) has described it as “an especially challenging
series to model” with Box-Jenkins procedures5 . This makes it an attractive
series to use in illustrating the ARAR technique.
First we formed a prior belief about the degree of φ(L). Since this series is
thought to lead the business cycle, we believed it should have a cycle. Therefore,
5 Pankratz, (1983) p.369. We suspect that the absence of exogenous variables, such as real
per capita income, the stock of housing and the real price of housing, from the univariate time
series model accounts for much of the difficulty in modeling and, especially, forecasting this
series.
we specified6 p = 2 and we chose a proper prior for the parameters of φ(L)
which placed a modest amount of probability on the region corresponding to
conjugate complex roots. Because the data are quarterly, we chose an AR(4)
as our tentative model for Ut . This allows for imperfections in the seasonal
adjustment and for the presence in the error of omitted variables which are
seasonally unadjusted. Thus our structural model is
(1 − φ1 L − φ2 L2 )(Yt − µ) = Ut ,
(2.3.1)
(1 − ω1 L − ω2 L2 − ω3 L3 − ω4 L4 )Ut = εt .
(2.3.2)
with
From (2.3.1) and (2.3.2) we obtained the restricted AR(6) reduced form
Yt
= µφ(1)ω(1) + (φ1 + ω1 )Yt−1 + (φ2 + ω2 − ω1 φ1 )Yt−2 +
(ω3 − ω1 φ2 − ω2 φ1 )Yt−3 + (ω4 − ω2 φ2 − ω3 φ1 )Yt−4 −
(ω3 φ2 + ω4 φ1 )Yt−5 − ω4 φ2 Yt−6 + εt ,
(2.3.3)
which is just identified and implies the unrestricted reduced form AR(6)
Yt
= α0 + α1 Yt−1 + α2 Yt−2 + α3 Yt−3 + α4 Yt−4 +
α5 Yt−5 + α6 Yt−6 + εt .
(2.3.4)
Note that if φ(L) has a unit root then φ(1) = 0 and if ω(L) has a unit root then
ω(1) = 0. Thus, given a strong prior belief that µ 6= 0 or a large sample mean
for Yt , an estimate of α0 close to 0 would lead us to question the stationarity of
the both Yt and Ut .
Our sample of 191 observations was obtained from the U.S. Department of
Commerce, Bureau of Economic Analysis, Survey of Current Business, October
1995 and January 1996. These data are seasonally adjusted index numbers
(based on 1987) extending from January 1948 until October 1995. We converted
them to quarterly form by averaging over the months of each quarter: they
are plotted in Figure 3. Neither Figure 3, nor the sample autocorrelations
and partial autocorrelations7, in Table 1, display the pattern typical of a unit
root but they do show the cyclical pattern rather clearly. These results, and
those which follow, used the observations from the third quarter of 1949 until
the third quarter of 1990, 165 observations. The earlier observations provided
lagged values and the observations from the fourth quarter of 1990 to the third
quarter of 1995 were used for post-sample forecasts.
If our specification in (2.3.1) and (2.3.2) is adequate OLS applied to (2.3.4)
should yield serially uncorrelated residuals and the roots of α̂(L) should contain
a conjugate complex pair with a modulus less than one and a period between
12 and 24 quarters. These are estimates of the roots of φ(L) giving a damped
cycle with a period of three to six years. There may also be additional conjugate
6 In this example we are abstracting from the problem of analyzing multiple cycles, such as
long cycles in construction activity.
7 For all the calculations reported in Tables 1 to 4 we used TSP Version 4.3A for OS/2.
complex pairs of roots with moduli less than one but with shorter periods which
model the dynamics of Ut .
The estimated lag polynomial, with standard errors in parentheses, is
α̂(L) = 1 − 1.189L
(07685)
− .2700L5
(.1228)
− .002388L2
(.1208)
+ .2014L6.
(.07871)
+ .4343L3
(.1236)
− .01077L4
(.1228)
(2.3.5)
The sample autocorrelations and partial autocorrelations for the OLS residuals
are shown in the second and third columns of Table 1: they indicate a lack of
serial correlations. The roots of α̂(L) plus their moduli and periods are given
in Table 2. There are three complex conjugate pairs all of which have moduli
less than one. The pair in the first line have a period of about 23 quarters
which corresponds to our prior belief about the period of the business cycle.
The other two pairs have shorter periods which we attribute to dynamics of
the error. Thus on the basis of these results it appears that our specification is
adequate.
Unit root and trend analysis was carried out using (2.2.18) assuming a
normal likelihood and α0 = α(1)µ0 , β1 = α(1)µ1 and ρ = φ1 + φ2 . The
results are in Table 3 where the means of the posteriors are denoted by ˆ and
the standard deviations by “Std Dev”. PD (ρ ≥ 1) is the posterior probability
that there is a root of one or more when a diffuse prior is used while PJ (ρ ≥ 1)
is obtained8 when Jefferys’ prior is used: see Phillips (1991). These results led
us to infer that there is neither a unit root in Yt nor a linear trend9 .
If the parameters φi and ωi are not of explicit interest and all that is wanted
are forecasts of future values of Yt they can easily be obtained by both Bayesian
and frequentist analysis of the unrestricted AR(6). Our primary interest here is
to draw inferences about φ1 and φ2 with µ, ω1 , ω2 , ω3 , ω4 and σ being nuisance
parameters. For this purpose we used NLS, which also gives the means and
standard deviations of large-sample normal approximations to the posteriors
for the φi and ωi , assuming diffuse priors and normal likelihood. The results
are in the second column of Table 4. The approximate posterior mode for σ
is denoted by σ̂. Since this model is just identified, these estimates could have
been derived from those given in (2.3.5).
The posterior means of ω2 and ω3 are small compared to their standard
deviations so we imposed ω2 ≡ ω3 ≡ 0 which over identified the model. Thus
NLS was used to obtain the results in the third column of Table 4. The approximation in (2.2.16) gave a Bayes factor in favor of the restricted model against
the unrestricted model of 32.5.
The results in Table 4 can be used to find approximate posterior probabilities
for various aspects of the dynamic behavior of Yt . The roots of φ(L) are complex,
8 These
values were calculated using the COINT routines for GAUSS which scale the trend
value to range from 1 to 1/T.
9 We drew the same inference from the results of augmented Dickey-Fuller (Said and Dickey
(1984)), Phillips-Perron (1988) and weighted symmetric (Pantula et.al. (1994)) tests.
Table 1: Housing Starts; Autocorrelations and Partial Autocorrelations
Data
AR(6) Residuals
Autocorr. Partial Autocorr.
Partial
Lag
(S.E.)
(S.E.)
(S.E.)
(S.E.)
1
.911
.911
.00213
.00213
(.0765)
(.0765)
(.0778)
(.0778)
2
.765
-.376
-.0273
-.0273
(.125)
(.0765)
(.0779)
(.0778)
3
.581
-.242
-.0238
-.0237
(.150)
(.0765)
(.0779)
(.0778)
4
.397
-.0111
.0146
.0139
(.162)
(.0765)
(.0780)
(.0778)
5
.230
-.0207
.0481
.0468
(.168)
(.0765)
(.780)
(.0778))
6
.0751
-.126
-.0628
-.0630
(.144)
(.0765)
(.0781)
(.0778))
7
-.0459
.0382
.00562
.00920
(.170)
(.0765)
(.0785)
(.0778)
8
-.141
-.0471
.0232
.0221
(.170)
(.0765)
(.0785)
(.0778)
9
-.210
-.0548
.0322
.0283
(.171)
(.0765)
(.0785)
(.0778)
10
-.267
-.106
-.0387
-.0384
(.172)
(.0765)
(.0786)
(.0778)
Ljung-Box (1978) portmanteau statistics at lags 5 & 10
Q5
344
19.3
Q1 0
370
30.3
Data source: U.S. Department of Commerce,
Survey of Current Business
Table 2: Housing Starts; Roots of α̂(L) by OLS
Roots
.860 ± .242i
−.624 ± .382i
.358 ± .586i
Modulus
.893
.731
.687
Period
23.0
2.42
6.14
Table 3: Housing Starts; Posteriors for Unit Root and Trend
No Trend Linear Trend
α̂0
15.4
15.2
Std Dev
3.39
3.43
β̂1
—.00568
Std Dev
—.0123
ρ̂
.836
.833
Std Dev
.0352
.0359
PD (ρ ≥ 1)
.000
.000
PJ (ρ ≥ 1)
.000
.000
Table 4: Housing Starts; Asymptotic Posterior Moments
Unrestricted Restricted
Unrestricted
Restricted
ARAR(2,4) ARAR(2,4)
ARMA(2,2) ARMA(2,2)
φ̂1
1.72
1.63
1.04
1.22
Std Dev
.0727
.0628
.189
.0771
φ̂2
-.798
-.722
-.217
-.371
Std Dev
.0699
.0623
.176
.0741
ω̂1
-.531
-.461
——Std Dev
.0985
.0805
——ω̂2
-.114
——–
—Std Dev
.118
———ω̂3
-.206
———Std Dev
.110
———–
ω̂4
-.252
-.149
——Std Dev
.0882
.0743
——θ̂1
——-.175
—Std Dev
——.177
———-.398
-.329
θ̂2
Std Dev
——.0910
.0845
σ̂
7.38
7.41
7.52
7.48
ln likelihood
-560
-562
-565
-565
Q5
.658
4.80
2.37
3.96
Q10
1.89
5.79
6.96
7.39
Table 5: Housing Starts; Approx. Posterior Moments, Functions of φ1 and φ2
Unrestricted
Restricted
ARAR(2,4)
ARAR(2,4)
Dynamic
Method of
Std.
Std.
Property
Approx.
Mean Dev. Skew. Kurt. Mean Dev. Skew.
Kurt.
cycle
φ21 + 4φ2
Normal
Simulated
-.233
-.228
.0631
.0648
0.0
.138
3.0
3.12
-.232
-.227
.0815
.0810
0.0
.0600
3.0
2.91
modulus
√
−φ2
Normal
Simulated
.893
.893
.0391
.0392
0.0
-.120
3.0
3.03
.850
.849
.0366
.0365
0.0
-.123
3.0
2.91
period
2π/ arctan(ϑ)
Normal
Simulated
23.0
24.0
2.75
4.75
0.0
7.91
3.0
175
21.9
23.7
3.41
8.87
0.0
21.0
3.0
831
leading to a cycle in Yt , if φ21 + 4φ2 √
< 0. If a cycle is present, it will be
damped if the modulus of the roots,
−φ2 , is less than 1.0 and its period
p
will be 2π/ arctan(ϑ), where ϑ = −φ21 − 4φ2 /φ1 . One way to approximate
the posterior probabilities of these properties is to take the first two terms
in a Taylor series expansion of these functions about φ̂1 and φ̂2 which have
asymptotically normal posterior distributions. The moments of these normal
approximations are given in Table 5.
An alternative procedure is to simulate the asymptotic joint posterior for the
φi and ωi and at each replication compute the relevant functions of φ1 and φ2 .
Estimates of posterior moments can be calculated from these simulated values
and the relative frequency curves are estimates of the posterior densities. An
estimate of the posterior probability of the existence of a cycle is the proportion
of the φ√21 + 4φ2 values which are negative and the proportion of those values for
which −φ2 < 1.0 is an estimate of the conditional posterior probability that
the cycle is damped. Also, estimates of the conditional posterior probability for
interesting ranges of the period can be calculated from the simulated values of
2π/ arctan(ϑ). We performed 5000 replications of these simulations for both the
unrestricted and restricted ARAR(2,4) models. The relative frequency curves
for φ21 +4φ2 are shown in Figure 4 for the unrestricted model and Figure 7 for the
restricted model. Their moments are in the top block of Table 5. Although this
is a nonlinear function of φ1 and φ2 , the two methods of approximation are very
close to one another. The estimated posterior probability of the existence
√ of a
cycle is greater than .99 for both models. The relative frequency curves for −φ2
are in Figures 5 and 8 and their moments are in the second block of Table 5.
The estimate of the conditional posterior probability of the cycles being damped
is 1.0 for both models. In this case too, the two methods of approximation gave
very similar results. The relative frequency curves for 2π/ arctan(ϑ) are in
Figures 6 and 9 and their moments are in the bottom block of Table 5. Now
the two methods of approximation were very different: the simulated values
are positively skewed and leptokurtic. Still, our estimate of the conditional
probability of the period being between 16 and 30 quarters is more than .90 for
both models.
While the ARAR model is best suited to obtaining inferences about the
φi , it is also useful to compare its forecasting performance with that of a
traditional ARMA model. An ARMA model might be suggested here by the
conversion of the data from monthly to quarterly form by averaging. Application of Box-Jenkins model identification techniques to the results in Table 1
led to an ARMA(2,2) model. This is one of the models found by Pankratz10
to be adequate and it retains the form of φ(L) suggested by our prior beliefs.
The NLS results for this model are in the last two columns of Table 4. The
restriction θ1 ≡ 0, imposed to produce the results in the last column, was also
imposed by Pankratz. Using a starting point based on the sample autocorrelations the ARMA routine took 22 iterations to converge. In contrast, the
ARAR routine started from 0.0 and took five iterations to converge. Also, as a
vehicle for learning about φ1 and φ2 the restricted ARMA(2,2) is less successful
than the restricted ARAR(2,4): the Q10 is larger for the restricted ARMA(2,2),
the absolute ratios of posterior means to standard deviations are larger for the
restricted ARAR model and the approximate Bayes factor is 1.38 in favor of the
restricted ARAR. Also, the ARMA(2,2) results have quite different dynamic
properties relative to those of the ARAR(2,4) models: the approximate posterior probability of cycles for the unrestricted ARMA(2,2) is only .276 while that
for the restricted ARMA(2,2) is only .401.
We obtained one-step-ahead forecasts by the restricted and unrestricted
ARAR(2,4) model and by the restricted and unrestricted ARMA(2,2) for 20
quarters beyond the end of the sample. For the first forecast the coefficients
were set at the sample period posterior means. Then for subsequent forecasts
they were updated for each period. Since data for this period had been left
out of the sample used in estimation, we were able to calculate the forecast
errors for each of the 20 quarters in the forecast period. Summary statistics
for these forecast errors, as percentages of the actual future values of Y t , are
in Table 6. The main difference between the methods is the size of the mean
percentage forecast errors, which were larger for the ARAR models. This made
the RMSE for the unrestricted ARAR(2,4) larger than that for the unrestricted
ARMA(2,2). However, the standard deviation of the restricted ARAR percentage forecast errors was smaller than for the restricted ARMA so that in this
case ARAR has a smaller RMSE. These forecasts are rather imprecise because,
as noted above, obvious exogenous variables have been omitted and no account
has been taken of long cycles in building activity.
This example demonstrates several advantages of the ARAR model over
the ARMA model. NLS estimates of the ARAR parameters took many fewer
iterations than ARMA estimates. There was more serial correlation in the
10 The
other model which Pankratz found adequate was an ARMA(3,2) with φ 2 set to 0.
For our data that model has slightly higher residual autocorrelation, was less parsimonious
and had larger forecast errors than did the ARAR(2,2).
Table 6: Housing Starts; Percentage Forecast Error Summary Statistics
Unrestricted Restricted Unrestricted
Restricted
ARAR(2,4) ARAR(2,4) ARMA(2,2) ARMA(2,2)
Mean
2.92
2.52
1.85
1.86
Std Dev
7.25
6.68
7.25
6.98
RMSE
7.82
7.14
7.49
7.22
ARMA residuals than in the ARAR residuals and the Bayes factor favored
the ARAR model. The estimates of the ARAR parameters and the dynamic
properties of the estimated ARAR model were in closer accord with our prior
beliefs than were the ARMA results. Finally, the restricted ARAR model had
percentage forecast errors with smaller RMSE than did the restricted ARMA
model.
Although a sample of size 165 may seem large enough to justify largesample approximate posteriors, it is still useful to consider the calculation of
exact posteriors for the φi and ωi . We adapted the procedure in Zellner (1971)
Chapter IV to obtain exact posteriors for φ1 , φ2 , ω1 and ω4 from the restricted
ARAR(2,4) model of column 3 in Table 411 .
Let φ0 = [φ0 , φ1 , φ2 ], where φ0 = µφ(1), and ω 0 = [ω1 , ω4 ]. Our prior12 on φ,
ω and σ was
p(φ, ω, σ) = p(φ|σ)p(ω|σ)p(σ).
(2.3.6)
An inverted gamma distribution was used for p(σ) with parameters s2 = 1.0
and v = 1 giving a mode of .7071. We set p(ω|σ) ∝ 1.0 to reflect our lack of
prior knowledge regarding the behavior of Ut .
We chose a normal form for p(φ|σ) with
E(φ|σ)0 = φ0 = [0, 1.0, −.5]
and covariance matrix V (φ|σ) = σ 2 W −1 with


400 0
0
16 −8  .
W −1 =  0
0 −8 16
This prior is centered in the region corresponding to complex roots for φ(L)
but it is still rather uninformative with respect to φ1 and φ2 . For example,
conditional on σ = .7071, the prior probability of obtaining complex roots is
only about .12.
11 For these calculations the data was measured as deviations from the estimated sample
mean divided by the standard deviation of these deviations. This had no effect on the φ̂1 , φ̂2 ,
ω̂1 or ω̂4 but it served to eliminate overflow and underflow errors in the numerical evaluations
of the integrals discussed below.
12 This prior, and the improper prior used in section (3.3), are quite different to the smoothness priors introduced by Shiller (1973) because they assign prior probability to individual
parameters while the smoothness prior assigns prior probability only to the differences between
parameters
Let the vector of initial values of Yt be y00 = [y1 , . . . , yp+r ] and the vector of
the T remaining values be y0 = [yp+r+1 , . . . , yp+r+T ]. For this example p = 2,
r = 4, T = 165 and all inferences are conditional on y0 . Define y(ω) as the
T × 1 vector with elements yt (ω) = ω(L)yt and z(ω) as the T × k matrix with
rows zt0 (ω) = [ω(1), yt−1 (ω), yt−2 (ω)], where k = p + 1. Assume εt ∼ IN (0, σ 2 ).
Then the joint posterior, after completing the square on φ, can be written as
(
)
g 0 Q(ω)[φ − φ(ω)])
g
2 (ω) + [φ − φ(ω)]
(ṽ sg
(ṽ−+k+1)
p(φ, ω, σ|y, y0 ) ∝ σ
exp −
2σ 2
g = [W + z0 (ω)z(ω)]−1 [W φ + z0 (ω)y(ω)],
where: ṽ = T + v; φ(ω)
0
and
g [W + z0 (ω)z(ω)]φ(ω)
g
2 (ω) = vs2 + φ0 W φ + y0 (ω)y(ω) − φ(ω)
ṽ sg
Q(ω) = [W + z0 (ω)z(ω)].
The use of the proper prior p(φ|σ) in (2.3.6) ensures that Q(ω) will be nonsingular even if ω(1) = 0, thus avoiding the need to confine attention to models
with µ ≡ 0 or to impose the restriction ω(1) 6= 0: see Zellner (1971) page 89.
Integrating out σ 2 gives
h
i−(ṽ+k)/2 n
o−(ṽ+k)/2
g 0 H(ω)[φ − φ(ω)]
g
2 (ω)
p(φ, ω|y, y0 ) ∝ sg
ṽ + [φ − φ(ω)]
h
i−1
(2.3.7)
2 (ω)
Q(ω).
where H(ω) = sg
Integrating (2.3.7) over φ gives the joint posterior density for ω1 and ω4 .
h
i−ṽ/2
−1/2
2 (ω)
.
|W + z0 (ω)z(ω)|
p(ω|y, y0 ) ∝ ṽ sg
(2.3.8)
However, the lack of identification mentioned above means that (2.3.8) could
also be the posterior density for φ1 and φ2 . This composite (scaled) posterior
is shown in Figure 10. Identification was achieved by imposing our prior belief
that φ(L) has conjugate complex roots leading to cyclical behavior in yt . Thus
we identified the higher hill in Figure 10, in the region leading to complex roots,
as the joint posterior for φ1 and φ2 and the lower hill, in the region leading to
real roots, as the joint posterior for ω1 and ω4 .
Since (2.3.8) is difficult to integrate analytically, the marginal posteriors for
ω1 , ω4 , φ1 and φ2 were obtained numerically. To obtain the marginal posterior
for either ω1 or ω4 we numerically integrated13 over the other ωi . Since conditional on ω, (2.3.7) is multivariate Student-t, conditional posterior densities of
the φi are univariate Student-t which can be obtained analytically. Then the
products of these conditional densities and the joint density for ω in (2.3.8) are
the conditional densities of the φi , given ω.
13 All the numerical integrations reported here were done by Gaussian quadrature using
GAUSS 386 version 3.2.13
Table 7: Housing Starts; Exact Posterior Moments, Restricted ARAR(2,4)
Coeff.
φ1
φ2
ω1
ω4
Mean
1.62
-.706
-.440
-.137
Std. Dev.
.0680
.0671
.0860
.0768
Skewness
-.0000186
-.000210
-.00176
-.00848
The marginal posterior densities of the φi were obtained by integrating these
products over ω1 and ω4 14 . The marginal posteriors for ω1 and ω4 are plotted
in Figures 11 and 12, those for φ1 and φ2 are in Figures 13 and 14. In each
figure the exact density is graphed with a solid line while the large-sample
normal approximate density, is graphed with a dashed line. The exact posterior
moments are given in Table 7. The exact and approximate posterior densities
differ because they are based on different priors and because the sample is only
moderately large. However, they are quite close in the case of φ1 and φ2 , which
are the parameters of most interest.
To summarize, the exact posterior densities for the ARAR parameters were
easily obtained and were close to the asymptotic normal posteriors derived from
the NLS estimates. In the next section these techniques will be extended to
single equation, distributed lag models.
3
3.1
Distributed Lag Models
Early distributed lag models
Jorgenson (1966) considered the general distributed lag model
Yt = µ + λ(L)xt + Vt
(3.1.1)
where xt is the value of an exogenous variable Xt , the lag polynomial λ(L) is
infinitely long and the error was modeled as Vt = εt . Then λ(L) was approximated as δ(L)/φ(L) with δ(L) = δ0 + δ1 L + . . . + δm Lm and φ(L) as in (2.0.1),
having no factors in common with δ(L). These assumptions lead to
φ(L)Yt = µ0 + δ(L)xt + φ(L)εt
(3.1.2)
where µ0 = φ(1)µ. The parameters of interest are the coefficients of the lag
polynomials φ(L) and δ(L). As in the previous section, the first disadvantage
of this model for Vt is that it confines the effect of a shock εt on Yt − µ − λ(L)xt
to only the current period. Secondly, it implies an MA form for the error term
in (3.1.2) leading to complicated inference procedures15 .
14 Gibbs sampling procedures could also have been employed to compute these integrals.
Marginal densities for φ1 or φ2 can also be obtained by integrating the portion of (2.3.8)
which we have identified as the joint posterior of φ1 and φ2 over the other φi .
15 These complications are avoided if the error term φ(L)ε is white noise; see Zellner and
t
Geisel (1970).
A more general model for Vt is the stationary ARMA
π(L)Vt = θ(L)εt .
(3.1.3)
This leads to the Box and Jenkins(1976) transfer function model
π(L)φ(L)Yt = π(1)µ0 + π(L)δ(L)xt + φ(L)θ(L)εt .
(3.1.4)
Now it’s even more difficult to draw inferences about the φi and δi because of
the long, restricted MA structure of the error. Also it’s important not to over
parameterize the pairs δ(L), φ(L) and π(L), θ(L) to avoid introducing common
factors which destroy identification.
The ARMAX model can be obtained by either imposing π(L) ≡ φ(L) on
(3.1.3) or by adding a distributed lag in xt to the univariate ARMA model
(2.1.2) leading to
φ(L)Yt = µ0 + δ(L)xt + θ(L)εt .
(3.1.5)
This model is somewhat simpler to analyze than the transfer function model
but it retains the inconvenient MA error structure.
3.2
The ARAR distributed lag model (ARDLAR)
Our general distributed lag model is the univariate ARAR model (2.0.1) extended
by the addition of a distributed lag in xt to obtain
φ(L)Yt = µ0 + δ(L)xt + Ut ,
(3.2.1)
We assume that the degrees of φ(L) and δ(L) can be specified a priori, that
φ(L) is invertible and that it has no factors in common with δ(L). Then we can
write the model as
Yt
µ0
δ(L)
1
+
xt +
Ut
φ(1) φ(L)
φ(L)
= µ + λ(L)xt + Vt .
=
(3.2.2)
This model retains the same infinite lag structure on xt as Jorgenson’s model
(3.1.1), the transfer function model (3.1.4), and the ARMAX model (3.1.5).
It also approximates this infinite lag structure by the ratio of the two finite
lag polynomials: λ(L) = δ(L)/φ(L). However, in contrast to these models,
we specify the same stationary AR process for Ut as in the univariate case:
ω(L)Ut = εt and φ(L)Vt = Ut . Then (3.1.1) becomes the structural
ARDLAR(p,m,r) model16
Yt = µ +
1
δ(L)
xt +
εt .
φ(L)
φ(L)ω(L)
(3.2.3)
We can rewrite (3.2.3) as the reduced form
ω(L)φ(L)Yt = α0 + ω(L)δ(L)xt + εt ,
16 This
(3.2.4)
produces the same results as the technique introduced by Fuller and Martin (1961).
where α0 = µω(1)φ(1).
Our ARDLAR model is simpler than the transfer function model but it
still produces very rich behavior in both Yt and Vt because both λ(L) and
1/(ω(L)φ(L)) are infinite in length. Thus it retains the infinite lag structure on
xt and there is no need to assume that Ut is white noise to facilitate estimation. Our model is especially convenient because it allows seasonal behavior to
be included in ω(L), which could be the product of seasonal and nonseasonal
polynomials.
The ARDLAR lag model requires the same sort of prior information to
identify the φi and ωi as does the ARAR model. Write the reduced form (3.2.4)
as
α(L)Yt = α0 + β(L)xt + εt ,
(3.2.5)
where α(L) = ω(L)φ(L) and β(L) = ω(L)δ(L). In contrast to the univariate
case, interchanging φ(L) and ω(L) in (3.2.4) will change the likelihood function
except in the special case that φ(L) = ω(L). Similarly, interchanging φ(L)
and δ(L) will change the likelihood function, unless φ(L) = δ(L), and so will
interchanging ω(L) and δ(L), unless ω(L) = δ(L). Thus, so long as δ0 6= 1.0,
a sufficient condition for identification is that φ(L) and ω(L) be of different
degrees. Two other sources of identification failure are removed by assuming
that φ(L) has its first term equal to 1.0 and that there are no common factors
in φ(L) and δ(L).
Now assume that α(L) and β(L) are known but φ(L), ω(L) and δ(L) are
unknown and write α(L) in terms of its roots, ηi . Given the values of ηi we
would use prior information in the same way as with the ARAR model to decide
which (1 − ηi L) are part of φ(L) and which are part of ω(L). Once ω(L) had
been identified in this way it could be factored out of β(L) to leave δ(L). In
practice, the roots ηi and the polynomial β(L) are unknown but estimates of
them may provide some guidance as to identification.
In general, the number of free parameters in α(L) and β(L) are p + r and
r + m + 1, respectively. Unrestricted OLS applied to (3.2.5) gives p + m + 2r + 1
point estimates which exceeds the number of free parameters in φ(L), ω(L) and
δ(L) by r: thus ARDLAR models are generally over identified.
Note that (3.2.5) is the autoregressive-distributed-lag (ARDL) model much
favored by Hendry: see inter alia Hendry et al (1986). However, the two models
are built in different ways. The ARDLAR model begins with fairly precise prior
beliefs about the dynamics represented by φ(L) and δ(L). It then explicitly
introduces the common factor ω(L) as the model for the error Ut . Prior beliefs
about ω(L) are often diffuse. The parameters of interest are the φi and δi while
the ωi and σ are nuisance parameters.
In contrast, the ARDL model begins with the unrestricted polynomials α(L)
and β(L) and tries, through COMFAC analysis, to discover if they contain any
common factors, like ω(L). There are no prior beliefs regarding φ(L) and δ(L).
Any common factors that are discovered are then put into an AR model for the
error. In this procedure the parameters of interest are initially the αi and βi
but if common factors are discovered interest presumably switches to the φi and
δi . Since this procedure uses the same data over several rounds of testing, it is
subject to the pre-test problems mentioned above.
Assuming identification, the common factor ω(L) in (3.2.4) imposes a restriction
linking the parameters in α(L) and β(L). To exploit this restriction, first
condition on the coefficients of ω(L) to obtain
φ(L)[ω(L)Yt ] = δ(L)[ω(L)xt ] + εt .
(3.2.6)
Then condition on the elements of φ(L) and δ(L) to obtain
ω(L)[φ(L)Yt ] = ω(L)[δ(L)xt ] + εt .
(3.2.7)
Equations (3.2.6) and (3.2.7) can be used to generate NLS estimates or to
provide the basis for application of the Gibbs sampler.
An interesting simple example assumes that the degrees of δ(L) and φ(L)
are both one, with | φ |< 1 and δ1 /δ0 6= −φ, so that
Yt = µ +
(δ0 + δ1 L)
xt + V t .
(1 − φL)
(3.2.8)
If observations are taken quarterly without seasonal adjustment an attractive
AR model for Ut is the product of a nonseasonal AR(1) and a seasonal AR(1):
ω(L)Ut = (1 − ω1 L)(1 − ω4 L4 )Ut = εt
(3.2.9)
with | ω1 |< 1 and | ω4 |< 1. Thus our model for Ut is a restricted AR(5) with
two free parameters and our model for Vt is a restricted AR(6) with three free
parameters (φ, ω1 and ω4 ) plus σ 2 .
Since δ(L) 6= δ0 and the degree of ω(L) is different from that of φ(L), φ, ω1
and ω4 are identified, as are δ0 and δ1 . The model for Yt can be written as
(1−ω1 L)(1−ω4 L4 )(1−φL)Yt = α0 +(1−ω1 L)(1−ω4 L4 )(δ0 +δ1 L)xt +εt (3.2.10)
or as the unrestricted ARDL
Yt = α0 + Σ6j=1 αj Yt−j + Σ6j=0 βj xt−j + εt .
(3.2.11)
The over identification of this model is easily seen from the restrictions:
α0 = µφ(1)ω(1)
α1 = ω 1 + φ
α2 = −ω1 φ
α3 = 0
α4 = ω 4
α5 = −ω4 (ω1 + φ)
α6 = ω 1 ω4 φ
β 0 = δ0
β 1 = δ 1 − ω 1 δ0
β2 = −ω1 δ1
β3 = 0
β4 = −ω4 δ0
β5 = ω4 (ω1 δ0 − δ1 )
β6 = ω 1 ω4 δ 1 .
(3.2.12)
To take advantage of the common factors in (3.2.11) first condition on ω1
and ω4 , then on φ, δ0 and δ1 to obtain equations analogous to (3.2.6) and (3.2.7).
The first step in building a general ARDLAR model is to select p, the degree
of φ(L), and m, the degree of δ(L). These selections should reflect prior beliefs,
based on subject matter knowledge and previous research, about the presence
of cycles or trends and about the pattern of the coefficients in the ratio λ(L) =
δ(L)/φ(L). A tentative AR model of degree r for the error Ut should now be
chosen. This will imply that α(L) is of degree p + r and β(L) is of degree m + r.
The adequacy of the choice of r can be checked by examining the residuals from
the OLS fit of (3.2.5). If the aim of the model is only to produce forecasts one
may wish to stop at this point. However, if the aim is to gain knowledge about
the parameters of φ(L) and δ(L) the procedures discussed above and illustrated
below should be followed.
Error correction models, partial adjustment models and adaptive expectations models can all be written as ARDLAR models. A summary of these model
is given in Table 14 below.
3.3
Empirical example: growth of real GDP
The AR(3) model has often been used to forecast real GDP or its rate of growth,
as in Geweke(1988). However, Garcia-Ferrer et al (1987), Zellner and Hong
(1989), Hong (1989) and Zellner et al (1991) have shown that AR(3) models
give poor forecasts of turning points which can be improved by converting
them to distributed lag models including lagged exogenous variables as leading
indicators. We show in this example how a further extension to an ARDLAR
is useful.
Let Yt be the first difference of the log of real GDP. We begin by building univariate AR and ARAR models for Yt to further illustrate the ARAR
technique and to provide a basis for comparison with the subsequent ARDLAR
model. We then introduce two leading indicators and use them as the exogenous
variables in an ARDLAR model.
We follow earlier literature in specifying the first of our univariate models as
an unrestricted AR(3). A priori we believe that Yt follows a damped cycle so
the first step in building our ARAR model was to specify p = 2. Then setting
r = 1 led to an α(L) of degree three, as in the unrestricted AR(3).
(1 − ω1 L)(1 − φ1 L − φ2 L2 )Yt = α0 + εt .
(3.3.1)
Yt was constructed from quarterly observations on U.S. GDP in millions of
1987 dollars, seasonally adjusted at annual rates, from the fourth quarter of
1949 to the fourth quarter of 1990; 165 observations. Observations from the
first quarter of 1991 to the third quarter of 1995 were held back for use in
calculating forecast errors. The results of trend and unit roots analyses using
three lags (the same as in (3.3.1)) appear in top panel of Table 8 which uses
the same notation as Table 3. We concluded from these results that Yt is free
of unit roots and trends.
Table 8: Unit Roots and Trends; Real GDP, M2 and Stock Returns
No Trend Linear Trend
Rate of Growth of Real GDP
α̂0
.04604
.006394
Std Dev
.001045
.001776
β̂1
—-.003146
Std Dev
—.002529
ρ̂
.3894
.36375
Std Dev
.09417
.09600
PD (ρ ≥ 1)
.00000
.00000
PJ (ρ ≥ 1)
.00000
.00000
Rate of Growth of Real M2
α̂0
.002594
.0032332
Std Dev
.0008632
.001585
β̂1
—-.001253
Std Dev
—.002611
ρ̂
.5922
.5910
Std Dev
.06351
.06352
PD (ρ ≥ 1)
.00000
.00000
PJ (ρ ≥ 1)
.00000
.00000
Rate of Growth of Real Stock Returns
α̂0
.006503
.01626
Std Dev
.005782
.01163
—-.01928
β̂1
Std Dev
—.01996
ρ̂
.1447
.1370
Std Dev
.07708
.07726
PD (ρ ≥ 1)
.00000
.00000
PJ (ρ ≥ 1)
.00000
.00000
OLS results for the unrestricted AR(3) model are in the second column of
Table 9. The roots of α̂(L), their moduli and periods are in the top panel of
Table 10. They support our a priori belief in a damped cycle. Because the
ARAR(2,1) model is just identified estimates of its parameters could have been
obtained from the OLS results or directly by NLS: see column three of Table 9.
Of course, the ln likelihoods, residual diagnostics, estimated roots, moduli and
periods are the same for the two methods. The value of α̂3 obtained by OLS
is small compared to its standard deviation but the values of both φ̂2 and ω̂1
are large compared to their standard deviations. Thus setting α3 to 0 would
be a specification error. Table 11 uses the same notation as Table 5 to present
posterior results for the dynamic properties of the estimated φ(L) from Table 9.
The asymptotic posterior probability of the presence of a cycle is .72.
We now introduce two leading indicator variables which have been used in
the past by Garcia-Ferrer et al (1987), Zellner and Hong (1989), Hong (1989)
and Zellner et al (1991); the rate of growth of real M2 and the real rate of return
on stocks. Monthly data on M2 in billions of 1987 dollars and on an index of the
prices of 500 common stocks were taken from U.S. Department of Commerce,
Bureau of Economic Analysis, Survey of Current Business, October 1995 and
January 1996. To convert these monthly series to quarterly series we used the
value for the third month in each quarter as the value for that quarter. The rate
of growth of real M2, x1,t , is the first difference of the log of M2 and the real
rate of return on stocks, x2,t , is the first difference of the log of common stock
prices which were deflated by the consumer price index. The results of trend
and unit roots analyses for x1,t and x2,t are in the second and third panels of
Table 8. The lag length was chosen to minimize the BIC for an unrestricted AR
model: it was a lag of one in both cases. Here too, all the posterior probabilities
of stochastic nonstationarity were zero to eight significant digits.
To obtain an ARDLAR model we added distributed lags in x1,t and x2,t
to (3.3.1). In doing so we specified the lag polynomials so as to: reflect the
character of x1,t and x2,t as leading indicators; account for the fact that the
reporting lag before data on the growth of M2 are publicly available is much
longer than the reporting lag for real stock returns; take advantage of the way in
which monthly values of x1,t and x2,t were converted to quarterly values. This
produced a generalization of the restricted reduced form (3.2.4) which we label
ARDLAR(p, m1 , m2 , r)
ω(L)φ(L)Yt = α0 + ω(L)δ1 (L)x1,t + ω(L)δ2 (L)x2,t + εt
(3.3.2)
where: ω(L) and φ(L) are specified as above; δ1 (L) = δ1,2 L2 ( so m1 = 2 with
δ1,0 = δ1,1 = 0 ) and δ2 (L) = δ2,1 L (so m2 = 1 with δ2,0 = 0). Thus, there is a
delay of three months before x1,t has an impact on Yt and a delay of only one
month before x2,t has an impact.
We can also write (3.3.2) as the partially restricted reduced form ARDL(3,3,2)
model
Yt
= α0 + α1 Yt−1 + α2 Yt−2 + α3 Yt−3 +
β1,2 x1,t−2 + β1,3 x1,t−3 + β2,1 x2,t−1 + β2,2 x2,t−2 + εt , (3.3.3)
Table 9: GDP Growth; Asymptotic Posterior Moments
Restricted
AR(3)
ARAR(2,1) ARDLAR(2,2,1,1) ARDL(3,3,2)
Eqn (2.2.5) Eqn (3.3.1)
Eqn (3.3.2)
Eqn (3.3.3)
By OLS on
ARDL(3,3,2)
Eqn (3.3.3)
α̂0
Std Dev
α̂1
Std Dev
α̂2
Std Dev
α̂3
Std Dev
.004884
.001103
.3403
.07883
.1257
.08267
-.08642
.07841
.004884
.001103
——————-
.004049
.001056
——————-
.004049
.001056
.2091
.07766
.1357
.07206
-.05417
.06628
.004023
.001056
.1993
.07826
.1158
.07693
-.01090
.07346
β̂1,2
Std Dev
β̂1,3
Std Dev
β̂2,1
Std Dev
β̂2,2
Std Dev
————————-
————————-
————————-
.1100
.05283
.04623
.02255
.04137
.008879
.01739
.005981
.1328
.07869
.003707
.07698
.03639
.009529
.02818
.01047
φ̂1
Std Dev
φ̂2
Std Dev
ω̂1
Std Dev
δ̂1,2
Std Dev
δ̂2,1
Std Dev
——————————-
.7676
.1443
-.2023
.1279
-.4273
.1364
————-
.6295
.1450
-.1289
.1208
-.4203
.1410
.1100
.05283
.04137
.008879
——————————-
——————————-
σ̂
.09761
.009761
.008921
.008921
.08919
ln(L)
Q5
Q10
531.7
.459
6.78
531.7
.459
6.78
547.3
.803
9.41
547.3
.803
9.41
548.7
.537
7.96
where the restrictions, which over identify the model, are
α0 = µφ(1)ω(1)
α 1 = ω1 + φ1
α 2 = φ 2 − ω 1 φ1
β1,2 = δ1,2
α3 = −ω1 φ2
β1,3 = −ω1 δ1,2
β2,1 = δ2,1
β2,2 = −ω1 δ2,1 .
NLS results for this model are in the fourth column of Tables 9. They show
that the addition of the distributed lags in x1,t and x2,t was useful because the
posterior means δ̂1,2 and δ̂2,1 are large compared to their standard deviations
and because the residual variance was reduced. Also the approximate Bayes
factor is 36100 in favor of the ARDLAR(2,2,1,1) over the ARAR(2,1).
The fifth column of Table 9 shows the results for the reduced form ARDL(3,3,2)
implied by the restrictions shown below (3.3.3), while those in the sixth column
were obtained by unrestricted OLS on (3.3.3). A researcher who focussed on
these OLS results might be tempted to set α3 and β1,3 to zero. But the results
in columns four and five of Table 9 suggest that these restrictions would be inappropriate. Also the approximate Bayes factor in favor of the ADLAR(2,2,1,1)
model over the partially restricted ARDL(3,3,2) is 40.69. The estimated roots
of α̂(L) implied by the ADLAR(2,2,1,1) model, plus their modulus and period,
are given in the third panel of Table 10. They agree closely with those from
the AR(3) and ARAR(2,1) models but are much different to those from the
partially restricted ARDL(3,3,2) model in the fourth panel.
As in the housing example, we approximated the posterior distributions of
the dynamic properties of φ(L) from the ARAR(2,1) and ARDLAR(2,2,1,1)
models in two ways: by a normal distribution using the first two terms in a
Taylor series expansion of the asymptotic NLS distributions or by simulating
the nonlinear functions 5000 times. The moments of these two approximations
are in Table 11 and their relative frequency curves are in Figures 15 to 20.
From the simulations the estimated posterior probability of there being a
cycle is .73 for the ARAR(2,1) model and .62 for the ARDLAR(2,2,1,1) model.
Conditional on there being a cycle, the estimated probability of it being damped
was 1.0 for both models. As with the housing example, the two approximations
were quite close to one another for the functions showing the existence of a cycle
and its modulus. However, the simulations of the period were skewed right and
leptokurtic especially for the ARAR(2,1) model. For both models the estimated
probability of the period being between six quarters and 30 quarters was greater
than .96.
Next we obtained forecasts of the level of GDP for 16 periods beyond the
end of the sample used in estimation. For this purpose the ARAR(2,1) model
was written in its restricted AR(3) reduced form and the ARDLAR(2,2,1,1) in
its restricted ARDL(3,3,2) reduced form. Then forecasts were calculated from
these two models and from the partially restricted ARDL(3,3,2) and converted
to forecasts of the level of GDP. Summary statistics for the percentage forecast
Table 10: GDP Growth; Roots of Estimated α̂(L)
Roots
Modulus
Period
Unrestricted AR(3) by OLS
.3838 ± .2344i
-.4273
.4497
.4273
11.46
α̂(L) From ARAR(2,1) by NLS
.3838 ± .2343i
-.4273
.4497
.4273
11.46
α̂(L) From ARDLAR(2,2,1,1) by NLS
.3147 ± .1727i
-.4203
.3590
.4203
12.52
α̂(L) From ARDL(3,3,2) by OLS
.4151
-.3026
.08681
.4151
.3026
.08681
Table 11: GDP Growth; Approx. Posterior Moments, Functions of φ1 and φ2
ARAR(2,1)
ARDLAR(2,2,1,1)
Dynamic
Method of
Std.
Std.
Property
Approx.
Mean Dev. Skew. Kurt. Mean Dev. Skew. Kurt.
cycle
φ21 + 4φ2
Normal
Simulated
-.220
-.199
.334
.329
0.0
.138
3.0
2.95
-.119
-.0928
.334
.336
0.0
.256
3.0
3.15
modulus
√
−φ2
Normal
Simulated
.450
.500
.142
.0925
0.0
.179
3.0
2.75
.359
.438
.168
.0922
0.0
.194
3.0
2.66
period
2π/ arctan(ϑ)
Normal
Simulated
11.5
13.2
6.02
15.7
0.0
16.5
3.0
362
12.5
12.8
13.1
13.2
0.0
8.59
3.0
103
Table 12: GDP Growth; Percentage Forecast Error Summary Statistics
Partially Restricted
ARAR(2,1) ARDLAR(2,2,1,1)
ARDL(3,3,2)
Mean
-.0434
-.180
-.170
Std Dev
1.65
1.74
1.74
RMSE
1.66
1.75
1.75
errors are shown in Table 12. The forecasting performance of the ARAR model
was the best of the three models, in spite of its lack of exogenous variable. On
RMSE grounds the ARDLAR and ARDL models were nearly equal.
To calculate exact posterior results for the ARDLAR(2,2,1,1) model, let
x0t = [x1,t−2 , x2,t−1 ] and γ 0 = [µ0 , φ1 , φ2 , δ1,2 , δ2,1 ]. In contrast to the housing
model we used a uniform, diffuse prior for ω1 , γ 0 and log σ.
Let s = max(p + r, m1 + r, m2 + r), and write the initial values of Yt and xt
as y00 = [y1 , . . . , ys ] and x00 = [x01 , . . . , x0s ]. Write the T remaining observations
as y0 = [ys+1 , . . . , ys+T ] and x0 = [xs+1 , . . . , xs+T ]. In this example s = 3
and T = 165. All inferences were conditional on x00 and y00 . Define y(ω) as
the T × 1 vector with elements yt (ω) = yt − ω1 yt−1 and x(ω) as the T × 2
matrix with rows x0t − ω1 x0t−1 . Then let z(ω) be the T × k matrix with rows
z0t (ω) = [ω(1), yt−1 (ω), yt−2 (ω), x0t (ω)], where k = 1 + p + 2 = 5. Assume
εt ∼ IN (0, σ 2 ). Then the joint posterior, after completing the square on γ, is
)
(
g 0 [z0 (ω)z(ω)][γ − γ(ω)])
g
2 (ω) + [γ − γ(ω)]
(v sg
−(T +2)
p(γ, ω, σ|y, z, y0 , x0 ) ∝ σ
exp −
2σ 2
where;
g = [z0 (ω)z(ω)]−1 z0 (ω)y(ω)
v = T − k; γ(ω)
and
g 0 [z0 (ω)z(ω)]γ(ω).
g
2 (ω) = y0 (ω)y(ω) − γ(ω)
v sg
Integrating out σ 2 gives
h
i−(v+k)/2 n
o−(v+k)/2
g 0 H(ω)[γ − γ(ω)]
g
2 (ω)
p(γ, ω|y, z, y0 , x0 ) ∝ sg
v + [γ − γ(ω)]
h
2 (ω)
where H(ω) = sg
i−1
(3.3.4)
0
[z (ω)z(ω)].
Integrating (3.3.4) over γ gives the marginal posterior density for ω1 .
h
i−v/2
−1/2
2 (ω)
|z0 (ω)z(ω)|
.
p(ω1 |y, z, y0 , x0 ) ∝ v sg
(3.3.5)
This density is plotted, together with the large-sample approximation, in Figure
21. Posterior moments for ω1 are in Table 13. Since a diffuse prior was used, the
difference between the two densities is due entirely to the approximation error
resulting from the small sample size. Conditional on ω1 , (3.3.4) is a multivariate
Table 13: GDP Growth; ARDLAR(2,2,1,1) Model, Exact Posterior Moments
Coeff.
φ1
φ2
ω1
δ1,2
δ2,1
Mean
.4678
-.02723
-.4203
.1472
.03836
Std. Dev.
.1842
.1190
.1898
.06145
.008749
Skewness
-.4976
-.02367
.5911
.3166
-.02477
Student-t density so the conditional posterior densities of the γi are univariate
Student-t. To obtain the marginal posterior density for an individual γi we
numerically integrated ω1 out of the product of the conditional Student-t density
and the marginal density for ω1 given in (3.3.5). The resulting posterior densities
are plotted in Figures 22 to 25 and their moments are in Table 13. Clearly, the
asymptotic approximations are not nearly so close to the exact densities as was
the case in the housing example above.
4
Conclusions
We have shown in this paper that a useful model for the error in univariate
and single equation distributed lag models is a finite, stationary AR. In the
past researchers have often used a white noise or MA model for the error without much prior or other information to support their choice. The ARAR and
ARDLAR models allow for very rich behavior of the error process and yet are
usually easier to implement empirically than models with MA errors. Other
appealing features of our ARAR model are its parsimony and the fact that all
its components obey the Wold decomposition. We have shown theoretically and
empirically how a researcher’s prior beliefs about the autoregressive structure of
the observable variable can be used to solve the identification problem inherent
in the model. In cases with restrictions on the parameters of the lag polynomials or if exogenous variables are present the models become over identified.
The extension of these ideas to dynamic multivariate time series, simultaneous
equations models and nonlinear models displaying chaos are the subjects of our
current research.
Table 14: Examples of Other ARDLAR Models
Simple Error Correction Model
Banerjee et al (1993), Hendry et al (1986)
φ(L) = 1 − φL, ω(L) = 1 − ωL,
δ(L) = δ0 + δ1 L,
∆Yt = α0 + (ω + φ − 1)(Yt−1 − xt−1 ) − ωφ(Yt−2 − xt−2 ) + δ0 ∆xt +
[ω + φ + δ1 + (1 − ω)δ1 − 1]xt−1 − ω(φ + δ1 )xt−2 + εt
Partial Adjustment Model
Harberger(1960), Nerlove(1958), Pagan(1985)
∆Yt = µ∗ + φ∗ (yt∗ − Yt−1 ) + Ut ,; 0 ≤ φ∗ < 1
yt∗ = ψ0 + ψ1 xt
φ(L) = 1 − φL, δ(L) = δ0 , ω(L) = 1
φ = 1 − φ ∗ , δ 0 = φ ∗ ψ 1 , µ0 = µ ∗ + φ ∗ ψ 0
(1 − φL)Yt = µ0 + δ0 xt + εt
Adaptive Expectations Model
Zellner and Geisel (1970), Zellner (1971)
Yt = µ + ψx∗t+1 + Vt ,
x∗t+1 = φx∗t + (1 − φ)xt , 0 ≤ φ ≤ 1
δ0 = ψ(1 − φ)
ω(L)(1 − φL)Vt = εt
ω(L)(1 − φL)Yt = δ0 ω(L)xt + εt
Finite Distributed Lag Model
φ(L) ≡ 1.0
ω(L)Yt = ω(L)δ(L)xt + εt
Acknowledgements
Carter acknowledges the generous hospitality of the Graduate School of Business,
The University of Chicago during the conduct of this research. Zellner’s research
was financed in part by the National Science Foundation and by the H.G.B.
Alexander Endowment Fund, Graduate School of Business, University of Chicago.
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