16

3. The Economic Theory-Model

As noted above, it is important to distinguish autonomous from
derived relationships in a non-constant world; yet in practice it is
exceptionally difficult to do so. The main basis for any asserted status
of an equation must be its correspondence (or otherwise) to a theoretical
relationship. Thus, to guide the data analysis, we present a suggestive,
if somewhat simplistic, theory-model which highlights what dependencies
between variables might be anticipated.

Most house builders are small in terms of the fractions of the
markets they supply (housing) and from which they demand inputs (labour,
capital, land, materials and fuel). In the longer run, competitive forces
might be expected to operate in such conditions so that only normal
profits are earned (the going rate of return on capital), with builders
who fail to minimise costs eventually being eliminated. Consequently,
despite its artificiality, insight can be gained by analysing the decision
processes of a single builder who produces homogeneous units, faces given
costs, uses best practices, and seeks to optimise his expected long-run
return. Those assumptions would allow one to formulate an "optimal-
control.” model yielding linear, intertemporal decision rules which
maximise the expected value of the postulated objective function
conditional on costs and demand, by using the certainty-equivalence
principle (e.g., see Theil [1964, pp. 52ff.]). An analytical solution can
be obtained only by postulating known and constant stochastic processes
for the uncontrolled variables. Such an assumption in effect removes the
uncertainty from the problem, and will be interpreted here as narrowing
the applicability of the resulting theory to an equilibrium world: that
is, one which is stationary, essentially certain, and devoid of problems

like evolving seasonality, adverse weather, changes in legislation or


30b

“(y°y) Sup IewT4Se WoIy sUOTZBTEII0D [eTIABd |YyQ 21k SZeyDLIG UT soan3ta,

TE*O- 8°0 8L°0 €0°0 6T*O- 8z°0- £0°0 FIO 90°0 (stro-) T1701 742-1} ‘oT
6€°0- ¥T°0 19°0- 80°0- S0°O 0Z°0 €0°O0 €0°0 (61°0-) 90°0 T3(Kn) +6
7S°0 9670 9TO~ TZ*O- BI°O- ZT°O- 60°0- (E¥0) EOV0- = *(d-90) +g
TS*0- 90°0- 61°O0- LE°O 90°0- 9€°0 (LT*O-) T¥°0 T3(aydy +z
ST°O- 91°0- 1S*0- ETt0- zS*0- (Eyro-) ssto- *(yd-ud) +9
z9°0 «L€°0-60°0 ZEW (yz +) ‘Teo Tagly ec
12"0 60°0 --2Z*0~—(94*0-) 900 tantly sy
SovO 1St0 (6z70) TS*0 = *(d-oayEy +e
6t'o (ze*0 ) ~TE*0 *ud)y z
(Z4L°0 ) S8°0 Gudlyy%y +1
6 8 L 9 s ’ € z I Analy

ef?) UF SOTQETILA SYQ AOJ XFAQeW VOFReTSAIOD eIeq *E STGEL


28

interpretable functtons of the regressors in Table 2 and marginalising
with respect to all the other potential variables. En route, a

distributed lag in the mortgage interest rate Rm j = 1,...,4) also was

tj 6
allowed for, and that is reflected in the finally selected specification.
Since it illustrates the relative roles of all the potential

explanatory factors, the following equation was selected as being the most

interesting for present purposes:

ph.) + 0.19 Mpa

+ 0.10 A,(ce-p), - 0-16 A
[-05]

1 [-04] [-03]

t-2 1 t-2

(4.4) ipa, 0.67 A,(A
[-06]

+ 0.09 A,y,_, ~ 1.9 D° - 0.25 (pn-ph), _, - 0.027 (ph-p), _

r.04) 2? [.5y [-06] tl F015] tel

+ 0.13: (ce-p),_4 ~ 0-022 (ury)

~ 0.23 {Rm(1-t)},_, - 0.073
[-03] [-012]

tl 5.13) [-049]
T= 9% R2= 0.86 = 0.944% (6,76) = 0.9 n4(2,80) = 0.9
BCL) = 0.2 602) = 15 ,

3
t

distributed lag; t is the standard tax rate, so Rm(1-t) is the after-tax

where A(x) = 0.1 } =0 (4-1) xy a normalised linearly declining
interest rate; and D° = D/100, so that its coefficient represents 1.9%.
The first six regressors represent disequilibrium or growth factors, all
of which vanish in a static state; and the last six represent levels which
persist in the equilibrium solution. Within those sets, variables are
organised by influences from house prices, costs, net demand or supply,
and other factors.

To analyse (4.4), we first derive its static solution by setting all

growth rates equal to zero.

(4.5) pa-ph = -0.10(ph-p) + 0.52(cce-p) - 0.09(u-y) - 0.92Rm(1-t) - 0.29
(.05) (.09) (.05) (.71) (.18)

The quoted standard errors are asymptotic approximations for non-linear


35

Appendix A. A Simple Theoretical Model of the Market for New Housing

Consider a builder constructing dwellings subject to a Cobb-Douglas
production function of the form

(A.1) c= K+ MeL? a+6<l,

where a, 8, and Ky are positive constants; M denotes direct inputs
(workers and materials, with a price tndex per unit dwelling given by CC);
and L denotes land (with a price per plot of P&). Ignoring any fixed

costs, profits are given by
(A.2) m = PneC - CCM - P&eL ;

and the objective is to maximise 1 subject to (A.1), where CC, P&, and Ph
are taken as given (i.e., the builder is small relative to the whole
market). The assumption that a+ 8 <1 is made to rule out increasing
returns as the scale of production grows. In practice, there are elements
of local monopolistic power, so the demand function facing the builder is

postulated to be
(A.3) c=K- (Pn/Ph)Y y<o,

where Ky is a positive constant.

The algebra of maximising 1 subject to (A.1) and (A.3) is tedious but
well-known (for an excellent introduction, see Smith [1982]) and yields
the solution that pn is a weighted average of ph, cc, and pk, with the
individual weights being dependent on a, 8, and y, and where (e.g.)
doubling all nominal prices would leave decisions about quantittes
unaltered. In the important special case that a+ B = 1 (so constant
returns prevail), pn only depends on cc and p& with weights a and l-a.

However, Ph would then proxy P&, since any increase in land prices would


10

Xe if XL is available when forecasts are made; however, if Ze is
contemporaneous with Ye? then LE will rarely be known at time t-1. Also,
T,need not be a "good" information set for explaining Yp> nor need the
{B, } in (2.4) bear sensible economic interpretations or be constants
across different states of the world. Consequently, while these data
eriteria are necessary, they are not sufficient to justify a given model
for inference, forecasting, or policy analysis. Indeed, three further

criteria are of equal importance.

(D): The first of these, criterion (v), is theory consistency, which

is also standard in econometrics and requires that an empirical model
should reproduce the theory from which it is ostensibly derived under the
hypothetical conditions relevant to that theory. That may sound weak, but
some published equations violate (v), and finding a model form which is
theory-consistent in several different but relevant hypothetical states of
the world can be non-trivial.

(E): Next, data admissibility, criterion (vi), entails that a model

should be unable to predict data values which violate definitional
constraints. For example, that prices are always non-negative or that
houses not started cannot be completed are data requirements and so should
be satisfied automatically (i.e., with probability one) by the model.”
Clearly, (vi) is closely related to the choice of functional form.

(F): Finally, and perhaps most importantly, encompassing (labelled
criterion (vii)) requires that any model F(*) claimed to adequately
represent the data generation process D(*) should be able to account for
the results obtained by other models of that process. That follows
because if one knew the mechanism generating all the data (as in a Monte
Carlo study, say), which here would be Dw, see wy l¥, 3 ¥), then by formal

reductions equivalent to those which produced F(*), one could deduce what


43

Mizon, G.E. (1984) "The encompassing approach in econometrics”, Chapter 6
in D.f. Hendry and K.F. Wallis (eds.), Econometrics and Quantitative
Economics. Oxford: Basil Blackwell, 135-172.

Mizon, G.E. and J.-F. Richard (1983) “The encompassing principle and its
application to testing non-nested hypotheses"; CORE discussion paper
8330, Université Catholique de Louvain, Louvain-la-Neuve, Belgiun,
forthcoming in Econometrica.

Naylor, Th.H., T.G. Seaks, and D.W. Wichern (1972) “Box-Jenkins methods: An
alternative to econometric models", International Statistical Review,
40, 2, 123-137.

Nellis, J.G. and J.A. Longbottom (1981) “An empirical analysis of the
determination of house prices in the United Kingdom", Urban Studies,
18, 9-21.

Nelson, C.R. (1972) "The prediction performance of the FRB-MIT-PENN model
of the U.S. economy”, American Economic Review, 62, 5, 902-917.

Pesaran, M.H. (1974) “On the general problem of model selection", Review of
Economic Studies, 41, 2, 153-171.

Prothero, D.L. and K.F. Wallis (1976) "Modelling macroeconomic time
series", Journal of the Royal Statistical Society, Series A, 139, 4,
468-500 (with discussion). -

Rao, C.R. (1948) “Large sample tests of statistical hypotheses concerning
several parameters with applications to problems of estimation”,
Proceedings of the Cambridge Philosophical Society: Mathematical and
Physical Sciences, 44, 50-57.

Schumpeter, J. (1933) "The common sense of econometrics", Econometrica, 1,
1, 5-12.

Schwarz, G. (1978) "Estimating the dimension of a model", Annals of
Statistics, 6, 2, 461-464.

Smith, A. (1982) A Mathematical Introduction to Economics. Oxford: Basil
Blackwell.

Theil, H. (1964) Optimal Decision Rules for Government and Industry.
Amsterdam: North-Holland Publishing Co.

Trivedi, P.K. (1984) "Uncertain prior information and distributed lag
analysis", Chapter 7 in D.F. Hendry and K.F. Wallis (eds.),
Econometrics and Quantitative Economics. Oxford: Basil Blackwell,
173-210.

White, H. (1980) "A heteroskedasticity-consistent covariance matrix
estimator and a direct test for heteroskedasticity”, Econometrica, 48,
4, 817-838.

Zellner, A. and F. Palm (1974) “Time series analysis and simultaneous
equation econometric models", Journal of Econometrics, 2, l, 17-54.


41

Engle, R.F. (1982) “Autoregressive conditional heteroscedasticity with
estimates of the variance of United Kingdom inflation", Econometrica,
50, 4, 987-1007.

Engle, R.F., D.F. Hendry, and J.-F. Richard (1983) "“Exogeneity",
Econometrica, 51, 2, 277-304.

Ericsson, N.R. (1978) Modelling the Market for Owner-Occupied Housing in
the United Kingdom: An Exercise in Econometric Analysis. M.Sc.
thesis, London School of Economics.

Ericsson, N.R. and D.F. Hendry (1985) “Conditional econometric modeling: an
application to new house prices in the United Kingdom", Chapter 11 in
A.C. Atkinson and S.E. Fienberg (eds.) A Celebration of Statistics:
The ISI Centenary Volume. New York: Springer-Verlag, 251-285.

Florens, J.-P. and M. Mouchart (1980) “Initial and sequential reduction of
Bayesian experiments", CORE discussion paper 8015, Université
Catholique de Louvain, Louvain-la-Neuve, Belgium.

Florens, J.-P. and M. Mouchart (1985) "Conditioning in dynamic models",
Journal of Time Series Analysis, 6, 1, 15-34.

Godfrey, L.G. (1978) “Testing against general autoregressive and moving
average error models when the regressors include lagged dependent
variables", Econometrica, 46, 6, 1293-1301.

Granger, C.W.J. (1983) “Forecasting white noise” in A. Zellner (ed.),
Applied Time Series Analysis of Economic Data. Washington: U.S.
Bureau of the Census, 308-326 (with discussion).

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Cycle Research. Minneapolis: Federal Reserve Bank of Minneapolis,
7-21.

Granger, C.W.J. and A.A. Weiss (1983) "Time series analysis of error-
correction models" in S. Karlin, T. Amemiya, and L.A. Goodman (eds.)
Studies in Econometrics, Time Series, and Multivariate Statistics.
New York: Academic Press, 255-278.

Haavelmo, T. (1944) “The probability approach in econometrics",
Econometrica, 12, supplement, i-viii, 1-118.

Harvey, A.C. (1981) The Econometric Analysis of Time Series. Oxford:
Philip Allan.

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Economists Advisory Group.

Hendry, D.F. (1984) “Econometric modelling of house prices in the United
Kingdom", Chapter 8 in D.F. Hendry and K.F. Wallis (eds.) Econometrics
and Quantitative Economics. Oxford: Basil Blackwell, 211-252.



20

in such an equilibrium pn is determined by ce and the factors affecting
profitability. Since the system evolves so long as a disequilibrium
persists, pn, will reflect current and past values of construction costs.
Nevertheless, “explaining” pn, by conditioning on lee, _ys i= 0,..-,2}
alone would not necessarily produce a useful model.’
One way to see such an argument is to consider (3.1) and (3.3) being
d

equated instantaneously by Poe adjusting so that ct = ce Then

(3-8) pay = CrytBy)[Crg-Bq) + Yyph, + Byce, + YY, - BY - Y4R,]
For large values of Y3> we have pn. * ph, and the influence of ce, becomes
small.® A generalisation of (3.8) is the basis for the empirical model
presented below, which turns out to yield estimates consistent with the
view that Y3 is indeed large and y * 1. However, there is evidence that

c® # ct in general, using data on a series for the stock of unsold

t
completions, Us. (which was collected only up until 1978). From the

identity

(3-9) US, = US.) +C,- Se,

where S2 denotes sales of completions, then Ayus, reflects changes in
net demand relative to the outstanding stock.” As shown in Figure 2, US
has fluctuated substantially. A model for Us, is developed at the end of
Section 4.

As a consequence of these considerations, our model class was
required to reproduce (3.8) only under equilibrium assumptions, but
otherwise was determined empirically by commencing from an unrestricted
autoregressive-distributed lag equation in which all variables entered
with up to four lags. Additive seasonal intercepts were included, since
none of the data series was seasonally adjusted, but these did not prove

significant. The log-linear specification was retained because Lt ensured


31

{Rm(1-t)},_45 and since that variabie plays a small role in the model, it
would seem sensible to further simplify the model by omitting interest
rates altogether. Doing so, G Increases to 0.95% and (u-y) 1 ceases to
be significant, whereas (ph-P) 1 becomes better determined, with most of
the remaining coefficients being unaltered.

Since the data behave very differently before and after 1971, and
since the sample is large enough to produce sensible estimates from each
half, we tested the model by fitting it separately to the two sub-samples.
That is a demanding test in the present context, since “success” (i.e,
non-re jection of parameter constancy) would imply an ability to track
turbulent data from estimates based on a quiescent period. Conversely,
rejection would imply a need to revise the specification, though without
clarifying precisely how. In the event, the sub-sample estimates of o (in
models without interest rates) were 0.80% and 0.90% respectively, against
the whole-period figure of 0.95%. An F-test of constancy across sub-
samples yielded n(10,73) = 3.1, thus rejecting the null. Most of the
estimates were in fact fairly similar between the sub-periods and the
whole sample, but those for (u-y) 4 changed sign, as did that for
(co-P) 4 in the first sub-period. Otherwise, the secondperiod estimates
were similar to those for the whole period, with rather more rapid
adjustment, consistent with the need for builders to respond more rapidly
to large disequilibria and/or substantial changes.

The last stage of the analysis is to examine the evidence for market
disequilibria using the short series on unsold completions qs.) for
1967-1978. As before, we First record the unrestricted log-linear

representation in which us, is explained by up to two lags of y, r,

t
(pn-p), and (ph-pn) (see Table 4). There, y denotes the sum over j of

coefficient estimates for a given variable, from which a derived long-run


Kendall [1969]) and often yields forecasts outside the estimation sample
which are poorer than the data-based within-sample fit. "Theory-driven"
models tend to manifest symptoms of dynamic mis-specification (e.g.,
residual autocorrelation afflicts many equations in large-scale
econometric models) and often fit poorly due to being excessively
restricted. Since, in principle, econometric specifications nest t:ime-
series formulations (see Zellner and Palm [1974] and Prothero and Wallis
[1976]), the complementarity of information from data and theory bears
stressing and argues for an integrated approach.

In practice, a complete spectrum of views exists concerning the
"best" combinations of theory and data modelling, and most practitioners
blend both elements to produce a mixture often labelled “iterative model
building”. The statistical properties of such mixed approaches have
proved hard to analyse, especially since the initial theory may be revised
in the light of any anomalous data evidence. However, some Monte Carlo
evidence is available (see Kiviet [1981, 1982]), highlighting the
difficulties of selecting models from noisy data. Moreover, pre-test
theory indicates severe inferential problems in simplifying models by
using sample evidence (see Judge and Bock [1978]). Nevertheless, little
empirical research in economics commences from fully pre-specified models
which adequately represent all salient features of the data.

Consequently, by default, many important aspects of most models have to be
selected from the observed sample, including the choices of alternative
potential explanatory variables, lag reaction profiles, functional forms,
error properties, seasonal variations, and even the evolution over time of
parameters of interest.

In the present approach, the data analysis is strongly guided by

prior economic theory. The theory suggests the form that the model class


26a

Table 2: An Unrestricted Model for 4,pn,*

Variable j=0 1 2 3 4
Ayer, -1 -05(.16) +25(.16) -04(.15) .13(.11)
4yphe_y +35( .09) -05(.13) -.06(.13) -04(.12) -.08(.12)
aM e-3 - -00(.07) -.18(.08) -01(.08) ~.11(.08)
‘ey -05( .08) +21(.14) -21(.12) -15(.12) -05(.10)
4 (comp) py +14( 11) -11(.12) +14(.11) -05(.11) -.03(.10)
iP ey -07( .18) +20(.19) +12(.19) -12(.18) +04(.17)
Qs +41(1.21) .003(.008) -002(.010) -001(.008) -
Phys - ~.09(.07) - - -
(uy) py - -.01(.02) - - -
Very - -.05(.13) - - -
(ee-P) 4 - - - - -18(.10)
Peay - - - - +11(.05)
(pn-ph),_, - ~.48(.15) - - -

a7 = 94, R2 = 0.87, G = 1.092%, &(12) = 9.7, ng (11,45) = 1.8, &,(1) = 0.00.
{Qs j=0,...,3} denote a constant and three seasonal shift dummy variables.
§ (12) is Box and Pierce's [1970] statistic based on the residual

correlogran.


17

tastes, and so on. Nevertheless, the resulting equations help to
constrain the equilibrium solutions of the empirical model as well as to
indicate relevant variables and parameterisations of interest.

Builders, as location-specific suppliers of new dwellings, have some
element of monopolistic power and can influence sales somewhat by (say)
advertising. In the medium term, they can determine the volume or the
price of their new construction (or possibly some combination thereof);
usually, their supply schedule reflects a willingness to supply more
houses with higher profitability of construction. Conversely, final
purchasers demand more housing as its price falls relative to that of
(e-g-) goods and services, and choose between new and second-hand units
on the basis of their costs (other factors being constant). Ina
schematic formulation which deliberately abstracts from dynamics,
completions of new houses in period t (denoted by c.) are produced from a
stock of uncompleted dwellings Wy)» with variations in the rate of
completions depending on changes in new house prices (Pa,) and in

construction costs (cc). We use a log-linear representation
8. - =
(3-1) ef = By + Byu,_1 7 Bgce, + Bapn, (8, 2 0, 1 = 1,2,3),

where lowercase variables denote logarithms of the corresponding
capitalized variables and C® denotes the planned supply of completions.

Letting S. denote starts of new dwellings, then

(3.2) UL, =U +8. -¢, .

t-1
Thus, UL is the (end-of-period) integral of past starts less completions,
and stock-flow ratios (e.g., ee) are crude measures of the average lag

between starting and ending construction. Since adjustment costs (hiring

and firing workers, paying overtime or idletime, etc.) suggest that change


22

Table 1. A Comparison of G for Different Models of Aypny

Univariate Bivariate Econometric
autoregression model model
Eq. (4.1) Eq. (4.2) Eq. (4.4)
é 1.55% 1.29% 0.94%
k 3 ‘ 6 12

2.38% respectively. As can be seen, three variables effect a 35%
reduction in 6 over the unconditional standard deviation, whereas it takes
twelve to effect a 60% decrease.

A stronger justification than goodness-of-fit is needed for the
larger size of the econometric specification. The two natural arguments
are the resulting understanding of how the aarket functions (of obvious
importance for predicting the complicated indirect effects flowing from
changing government policies) and the feedback to improve economic
analysis of markets in general (so that theories can commence from
corroborated models, rather than from a priori assertions). The following

results should help in judging the realism of such justifications.


19

house price Ph, is determined primarily by the demand for housing in

relation to the pre-existing stock, H Given Phi» (3.3) then

t-1°
determines the demand for new houses which is confronted with a supply of
ce dwellings. In general, cé and ce will not be equal, and builders will
either experience unsatisfied demand or end up holding unsold houses.

Either way, they must adjust by changing output or price. However, if ¥3

is large, disequilibrium will persist until Pn is fully adjusted to Ph.

Simultaneously, 4H. must be altering, given the equation

(3.4) HL = (1-6, By +C, + 0,

where 5 is the rate of destruction of houses and 0, is other net sources
of housing supply (e-g., from the governmental or rental sectors). The
whole stock-flow system evolves until an appropriate combination of stock

and price results, with flows in balance.

If we consider a static equilibrium defined by ca =c®

t and all change

ceasing, then, from (3.4),
(3.5) c= %&nd+h,

using log-linear equations where 5 = 6 and O- 0 for simplicity.
Further, we assume that the function for the total demand for housing can
be written as

= - -p) - = 3!
(3.6) h Apt Ay ~ Ay(ph-p) ABR UZ >
and the function for the volume of work in progress is

= - '
(3.7) u K + Ky (pa cc) + 2 >

where z denotes other exogenous influences (such as technology in (3.7))
and P is the overall price level of goods and services. Together with
(3.1) and (3.3), we obtain five equations to determine the equilibrium

values of c, h, u, pn, and ph, given y, 6, p, R, cc, and z; Consequently,


27

regressors have "t-statistics" in excess of 2, although that should not be
interpreted as entailing their irrelevance, given the generally high
correlations between successive lags of economic variables.
Notwithstanding their standard errors, many of the coefficients are
negligibly small. However, three of the variables in levels omitted from
both (4.1) and (4.2) have large coefficients, highlighting the role of the
“equilibrating” mechanisms postulated in Section 3.

To an economist, both (4.1) and (4.2) have fatal flaws as claimed
autonomous relationships, even though neither is an unacceptable data
description on (i)-(iv) of Section 2. Concerning (4.1), the rate of
inflation of new house prices is modelled as independent of inflation of
goods and services; and, if Aypn, became constant at uw, the model would
restrict y to the single value 2.6% (the sample mean). Both implications
are implausible, although only prolonged changes from the sample behaviour
would reveal the empirical inadequacy of the model. Similar difficulties
afflict (4.2) even though 4yphe_y is used: for example, under a constant
growth rate iv in second-hand house prices (and averaging over the
seasonals), yw * 0.8u" + 0.005. Thus, the prices Pn and Ph would diverge
indefinitely unless ri = 2.62.

The significance of (pn-ph) in Table 2 can now be seen in

t-1
perspective: any divergence of the house prices alters the conditional
growth rate of pn relative to ph so as to bring the relative price (Pn/Ph)
into line (see Granger and Weiss [1983] for a discussion on the
relationship between time-series models, error-correction models like the
one in (4.4), and the existence of long-run relationships between
variables). Consistent with (3.8), the long-run relative price varies

with real construction costs. That evidence is sufficiently favourable to

the theory in Section 3 to merit parsimonious modelling by forming


14

adding Zop to (2.5) produces an insignificant improvement in fit, then

t
(2.5) parsimoniously encompasses the model embedding (2.5) and (2.6).

This aspect of simplicity, therefore, remains important in establishing
credible models. For an empirical attempt at encompassing a range of
disparate models using an embedding strategy, see Davidson et al. [1978]
and the follow-up in Davidson and Hendry [1981]; conversely, Bean [1981]
investigates encompassing using the Cox approach. Note, however, from
(2.9b), that only one direction of testing is really worthwhile and that
all models in Bean's study that are variance-dominated are rejected.

Given the best available theoretical formulation of any problem, it
seems sensible to design models to satisfy (i)-(vii) as far as possible,
recognising the possibility of conflict between criteria for any limited
class of models under consideration. In particular, data admissibility is
remarkably difficult to achieve in practice without simply asserting that
errors are drawn from truncated distributions with bounds which
conveniently vary over time; and theory consistency and variance dominance
also may clash, especially for parsimonious formulations. How a
compromise is achieved must depend on the objectives of the analysis
(e-g-, forecasting, policy advice, testing economic theories, etc.) as
well as on creative insights which effect a resolution.

Since (i)-(vii) are to be satisfied by appropriate choice of the
model given the data, relevant “test statistics" are little more than
selection criteria, since “large” values on such tests would have induced
a re-designed model. Genuine tests of a data-based formulation then occur
only if new data, new forms of tests, or new rival models accrue. Such an
approach is similar in spirit to the data-based apsect of Box and
Jenkins's [1976] methods for univariate time-series modelling, but

emphasises the need to estimate the most general model under consideration


where xi

~j

of (2.2) depends on the actual irrelevance of the variables excluded from

= Cy eee xp and x, = XX x) for T > {> i> 1. The usefulness

D(+), on the suitability of the parameterisation 8 (which may include
“transients” relevant to sub-periods only) and on the adequacy of the
assumed form of F(+|9). Aspects of those conditions are open to direct
testing against the observed data. Although the choice of functional form
is of considerable importance, it depends intimately on the nature of the
problem. Thus, for this general analysis, we assume that the time series
Xe has been appropriately transformed so that only linear models need to
be considered (so X, may involve logarithms, ratios, etc. of the original
variables). Finally, x is partitioned into > Z's where Xe is to be

explained conditional on Zee corresponding to the claimed factorisation:

(2.3) Fy.

5) = FQylags *3 OFC] &) >

where $' = g(8)' = C34 3 ) € @,x%,, and all the parameters of interest

can be obtained from a alone. If such conditions are fulfilled, then Ze

is said to be weakly exogenous for d and only the conditional model

Fy lZes 3 t) needs to be analysed, greatly simplifying the modelling

exercise if there are many variables in Zee This formulation is discussed

more fully in Engle et al. [1983] and Florens and Mouchart [1980, 1985],

and builds on the work of Koopmans [1950] and Barndorff-Nielsen [1978].
Adding the further assumption that the maximum lag length of

dependence in (2.2) is fixed at 2 < T periods, then the conditional linear

model can be written as

&
(2.4) Ye ~ Bote + Rite + & t=1,..., T,

ignoring transients to simplify notation. That provides the general


25

model with up to fifth-order lags on pn and ph yielded the following

equation:

(4.2) Opry = 0.55 Ayph,_, + 0.29 Aypn,_» - 0.004
[08] [.07] [004]

+ 0.012 Q,. + 0.012 Q,. + 0.008 Q
r-004) 2 y.004) 2%  [.005) 3

T=94 R220.72 G= 1.29% n, (6,82) = 1-2 n,(8,80) = 0.6

ng (12,76) = 2.3 n, (13,75) = 4.0 &,(1) = 0.02 &,(2) =6.0 .

[*] denotes White's [1980] heteroscedasticity-consistent estimate of the

standard error. Manifestly, 4) ph is a highly significant predictor of

t-1
jpn: thus, (4.2) is as usable for one-step-ahead predictions as (4.1),
but has a significantly smaller residual variance. Although (4.1) and
(4.2) are not nested, (4.1) is a special case of the unrestricted version
of (4.2) which the latter parsimoniously encompasses. Directly testing
the significance of lags of ph in the unrestricted version of (4.2) yields
Ny (5,80) = 8.6, so that the white-noise residual of (4.1) is far from
being an innovation on the joint information set generated by (pn, ph).
Conversely, the errors on (4.2) are accepted as being an innovation
process on that information set. However, ns (+) reveals residual
heteroscedasticity in (4.2), and ng(*) indicates the possibility of non-
constant parameters.

That last problem is probably due primarily to measurement errors.
During late 1981, commercial banks began to rapidly expand their loans for
house purchases in competition with building socteties. !* Banks lent
mainly against more expensive dwellings, and in larger than average loans.
Thus, they attracted a distinctly biased sample of house purchasers.
However, the data series are based on returns for the average prices of

houses sold with a mortgage from a building society. Consequently, the


0.08

0.06:

0.04

0.02

0.00,

0.02

23b

——A\(pn- p),
5 |
M4 "
ad \i
Ah \ Hac
vt VA |) !
a | VLA \\ ANS it
1960 1962 1884 | | 1986 " 38 wo) 1972 1914 1971 1978 at

y i] h

Figure 4. Rate of change of real new house prices [A,(pn — p),].


The topic of our application naturally determines the formulation of
the theory-model. Casual observation suggests that a vast complex of
social, economic, and demographic factors influence house prices. Here we
are concerned with modelling the average price of newly completed private
dwellings in the United Kingdom (from 1959-1982), denoted by Pn, and shown
in Figure 1. A separate model of the price at which the existing stock of
housing is transacted (denoted by Ph) is presented in Hendry [1984].
Heuristically, we consider the joint density of (Pn, Ph) as being
factorised into a conditional density for Pn, given Ph, and a marginal
density for Ph, with our models for Pn and Ph corresponding to the latter
two densities. The economic justifications for the resulting parameters
of the conditional density being of interest are presented in Section 3 in
the context of a theory-model of the decisions of the construction
industry.? Section 3 also briefly considers alternative approaches to
modelling Pn based on construction costs and on (marginal) models derived
from the supply of and demand for new dwellings.

In fact, the general formulation of earlier models of new house
prices has been of an interaction between supply and demand, with prices
implicitly determined by the level which equates supply and demand for new
housing. The validity of this market-clearing paradigm is not obvious in
the United Kingdom, where the volume of new housing is small compared to
the total volume of transactions in existing houses. Also, there is clear
evidence of large changes over time in the stock of completed but unsold
houses, indicative of a non-clearing market. That aspect therefore
requires appraisal, so the available data are described in Section 3.

Section 4 reports the various empirical results obtained and
discusses the light they throw on understanding the determination of

housing prices. Section 5 briefly concludes the study.


should have in order to satisfy a number of reasonable properties likely
to obtain in any static-equilibrium state of the world. The functional
form is specified to ensure invariance to a range of transformations, and
the length of the longest lag in the maintained model is pre-assigned a
value such that we would be surprised if even longer lags were needed to
make the model adequately characterise the data. Such an approach
produces a general maintained model, which is usually heavily over-
parameterised. Reduction of the general model by data-based sequential
simplifications in the light of the prior theory yields a parsimonious
summary which aims to be both data-coherent and theory-consistent, with
interpretable parameters corresponding to nearly orthogonal variables (see
Trivedi [1984] for a general discussion of this strategy for model
selection).

At this stage, the model has been designed to satisfy a range of
statistical and economic criteria. Since the criteria may conflict, some
of the “art” apparent in modelling remains, perhaps necessitating
appropriate compromises between tractability, coherency, and credibility.
Moreover, no unique path for simplification exists, so the final model may
vary with the investigator. However, by sharing a common initial model
and subjecting selected simplifications to testing on later data and
against rival models, the strategy has some in-built protection against
choosing poor or non-constant representations. It is important to stress
that the prototypes of the model presented below were first developed in
1978 and have altered rather little since 1980 despite several later
tests. Section 2 presents a more extensive discussion of the empirical
econometric modelling methodology to establish terminology and exposit the

main concepts.


15

to establish the innovation variance. Moreover, existing empirical models
and available subject-matter theory play a larger role, while being
subjected to a critical examination for their data coherency on

(4)-(vil)-

Indeed, econometric analysis always has involved a close blend of
economic theory and statistical method (e.g., see Schumpeter [1933]).

That economic analysis should be used is unsurprising, but the role of
statistics has proved more problematical in terms of a complete
integration of the economic and statistical aspects of model formulation,
even though Haavelmo [1944] stressed the necessity of carefully
formulating the statistical basis of an economic theory-model. He also
showed the dangers of simply “adding on” disturbance terms to otherwise
deterministic equations and asserting convenient properties to justify
(say) least-squares estimation.

Conversely, massive difficulties confront any purely data-based
method, since the interdependence of economic variables entails a vast
array of potential relationships for characterising their behaviour. That
aspect is discussed more fully in Hendry et al. [1984], but the theory-
model in Section 3 highlights the existence of many derived equations from
a small set of “autonomous” relationships. Since economic systems are far
from being constant, and the coefficients of derived equations may alter
when any of the underlying parameters or data correlations change, it is
important to identify models which have reasonably constant parameters and
which remain interpretable when some change occurs. That puts a premium
on good theory.

While our paper remains far from resolving these fundamental issues,
it seeks to link the two aspects by using considerations from both

Sections 2 and 3 in formulating the empirical equations of Section 4.


2. Econometric Modelling

In this section, we discuss relevant aspects of our statistical
approach for modelling new house prices: see Hendry and Richard [1982,
1983] for more detailed discussion and bibliographical information.
Modelling is viewed here as an attempt to characterise data properties in
simple parametric relationships which remain reasonably constant over
time, account for the findings of pre-existing models, and are
interpretable in the light of the subject matter. The observed data
wy eee Wp) are regarded as a realisation from an unknown dynamic economic

mechanism represented by the joint density function:

(2.1) Dw +s wlw ;w,

where T is the number of observations on w

Wee Wo denotes the initial

conditions, and » is the relevant parameterisation. D(+) is a function of
great complexity and high dimensionality, summarising myriads of disparate
transactions by economic agents and involving relatively heterogeneous
commodities and prices, as well as different locations and time periods
(aggregated here to quarters of a year). Limitations in data and
knowledge preclude estimating the complete mechanism.

A model for a vector of observable variables {x,} can be
conceptualised as arising by first transforming wy, 80 that Xe isa
sub-vector, then (implicitly) marginalising the joint density D(+*) with
respect to all variables in % other than x (i.e., with respect to those
variables not considered in the analysis). That produces the reduced

; density F(x) eee EplXq3 8), where 8 is the induced function of y Next,

one sequentially conditions each X, on past observables to yield:

mA

(2.2) FOIE @ = 1 FOIE as ® geQ,

1

t


20a

3.0

Us,

26

2.2

1.8

1.4

1.0

0.6

1968 1970 1972 1974 1976 1978

Figure 2. Logarithm of the stock of unsold completions (us,).


International Finance Discussion Papers

Number 254

Revised: June 1985

CONDITIONAL ECONOMETRIC MODELLING: AN APPLICATION
TO NEW HOUSE PRICES IN THE UNITED KINGDOM

by

Neil R. Ericsson and David F. Hendry

NOTE: International Finance Discussion Papers are preliminary materials
circulated to stimulate discussion and critical comment. References in
publications to International Finance Discussion Papers (other than an
acknowledgment by a writer that he has had access to unpublished materials)
should be cleared with the author or authors.


7

Appendix B. Data Definitions

VARIABLE? DEFINITION sourcr>
c Private sector housing completions (GB) E.T., M.D.S.
cc Index of the cost of new construction H.C.S., M.D.S.

(1975 = 100)
D Dummy variable for 1981(111)-1982(ii1) See text

P General index of retail prices E.T.
(1975 = 100)

Ph Index of prices of comparable dwellings D.O.E., B.S.A.
(second-hand houses) on which transactions
were completed

Pn Average price of new dwellings on which new E.T., B.S.A-
building-society mortgages were completed
(1975 = 100)

R Bank of England's minimum lending rate to E.T., M.D.S.
the market (the base rate from 1981(1ii))

Rn Building-society interest rate on new B.S.A., F.S.
mortgages

s Private sector housing starts (GB) E.T., M.D.S.

T Standard rate of income tax A.A.S.

U Uncompleted houses (GB): constructed from

the identity UE =U, -1 + Ss, - Ces with
several benchmark surveys as checks

us Stock of unsold completions H.C.S.
("dwellings completed not sold")

Y Real personal disposable income (1975 prices) E.T., M.D.S.

4411 variables are quarterly, seasonally unadjusted.

ba.a.s., Annual Abstract of Statistics, H.M.S.0.; B.S.A., Building Societies
Association Bulletins and Compendium of Statistics; D.0.E., Department of
the Environment; E.T., Economic Trends, Annual Supplements, 1980-1985,
H.M.S.0.; F.S., Financial Statistics, H.M.S.0.; H.C.S., Housing and
Construction Statistics, H.M.S.0.; M.D.S., Monthly Digest of Statistics,
H.M.S.0.



the calculated residuals are consistent with tg} being such an innovation
process.

Several particular hypotheses can be investigated as follows.
Firstly, defining white noise by the second-order property that, for
E(g,) = Q, BCE, Sy) = Q for all k#0 (i.e., & is unpredictable from its
own past: alone), one could test for residual autocorrelation. For
example, suppose an investigator postulated a model with a maximal lag
length * If * were less than & and/or the elements of {B,, im0,...,2"}
were inappropriately restricted, then the residuals might manifest serial
correlation. Hence, criterion (i) of model adequacy is that the residual
process (i.e., that which is left unexplained after modelling is ended)
should be empirical white noise (see Granger [1983]). Note that an
autocorrelated error can be “explained” in part (e.g., by Box-Jenkins
methods). Also, {g,} need not be homoscedastic in (2.4), so that
inferences may have to allow for potential heteroscedasticity.
Fortunately, heteroscedastic-consistent covariance matrices can be
constructed with ease (see White [1980], Domowitz and White [1982], and
Messer and White [1984]), and a variety of tests for residual
heteroscedasticity is available.

Next, the assertion that & is an innovation relative to {z,> xD
entails that lags longer than & are redundant in (2.4) and that selecting
ace is invalid. Thus, the residuals in (2.4) should have the smallest
generalised variance (adjusted for degrees of freedom) in this class of
constructed error processes. That property is called parsimonious
variance dominance and provides criterion (ii). If a model did not have
white-noise residuals, it could te variance-dominated by a corresponding
model which also “mopped up" the residual autocorrelation parsimonivusly.

Thus, (1) 1s a necessary condition for (ii), but is not sufficient,


maintained model and is estimable by a variety of methods. Rather
clearly, however, many drastic a priori assumptions have been made in
order to formalise (2.4), and such assumptions need not be valid
empirically. Consequently, we now consider how to evaluate such models.

Any postulated model can be evaluated by comparing its claimed
properties with its actual behaviour. As formulated, (2.4) entails
restrictions relative to six different sources of information, which are
summarised as:

(A) the history of the {x} process, denoted by Xe-1 (namely, only

xt is relevant if (2.4) is valid);

(B) the current value of x (namely, it is valid to condition XY, on

Z,)5
(C) the future of the {x,} process (namely, the parameters remain
constant on x),

(D) the subject-matter theory (so that (2.4) is consistent with the

available theory);

(E) the structure of the measurement system (e.g., definitional

constraints must not be violated); and

(F) rival models (which should not contain additional information

relevant to explaining fy, })-
We now consider empirical model selection criteria derived from each of
those information sources.

(A): A crucial aspect of evaluating the empirical validity of (2.4)
concerns the properties of {g,}. If the assumptions underlying (2.4) are
a good approximation, then a cael EW IL) where i, = Gh» Xe) Thus
{g.} is a derived (rather than an autonomous) process, which by

construction is uncorrelated with iL and hence is an innovation relative

to Le One set of tests of (2.4) seeks to evaluate the extent to which


33

All of the test criteria are acceptable, but, although both n+) and
ng+) are smaller than in Table 4, neither is greatly favourable to the
model. Unsold completions seem to be extremely sensitive to all of the
demand factors and to adjust fairly rapidly, but still to reveal that
disequilibria persist for around about three quarters to a year on

average.

5. Conclusion

A complete statistical analysis of the model-building procedures
applied above is bound to indicate that the finally selected model is
subject to considerable uncertainty in its specification. Most Monte
Carlo studies hold the model specification fixed as the sample size varies
and still yield large uncertainty regions. Moreover, in dynamic
equations, nominal and actual test sizes can depart radically for relevant
sample sizes, and test powers often are unimpressive. When the equation
formulation is itself data-based, models can have but a tentative status.
In Monte Carlo terms, large variability seems likely to arise from the
selection process.

Some protection against "spurious" estimates is provided by having a
pre-defined maintained hypothesis embodying subject-matter knowledge;
Sections 2 and 3 above discussed the principles underlying the model of
Table 2. However, the reparameterised equation (4.4) which summarises the
salient featuves of that table reflects a larger element of judgment; the
reported coefficient standard errors are much smaller in (4.4) than in
Table 2, primarily because of the reduced collinearity, but also partly
because the imposed restrictions are acceptable through being apparent in

the unrestricted model. Different samples and/or investigators could


24

All statistics except nC) are viewed as Lagrange-multiplier or efficient
score statistics (see Rao [1948]). Under the relevant null, ny is
distributed in large samples as a central F; similarly, &() is

asymptotically x2(j,0). The statistics are

ny(*) = Lagrange-multiplier statistic for testing against residual
autocorrelation (see Godfrey [1978], Harvey [1981, p. 173]),

ng(*) = Wald statistic for testing against the relevant unrestricted
maintained model (e.g., in (4.1), a fifth-order autoregression

with seasonally shifting intercepts),

ng(*) = Chow's [1960, pp. 594-595] statistic for testing parameter
constancy,

ng(*) = White's [1980] statistic for testing against residual
heteroscedasticity,

E6(*) = Engle's [1982] ARCH statistic (i-e., for testing against first-
order autoregressive conditional heteroscedasticity),

&7(*) = Jarque and Bera's [1980] statistic for testing against non-
normality in the residuals (based on skewness and excess
kurtosis).

From mr), (4.1) has white-noise residuals, and ng) confirms it is
an acceptable simplification of the autoregression. The parameters are
not significantly non-constant over the last two observations (despite
measurement problems noted below) or indeed over several longer test
samples; !? There is no evidence of residual heteroscedasticity, but the
residuals are highly non-normal, reflecting the marked failure of (4.1) to
predict the large changes in pn observed during the boom periods.

An indirect check on the usefulness of the theory-model of Section 3
is that lagged values of ph should be informative about present pn, since
it is assumed that the market for the stock of houses nearly clears each
period whereas that for the flow adjusts more slowly while builders adapt
to disequilibria. Thus, a bivariate model of pn on lagged values of pn

and ph should perform better than (4.1). Simplifying from a bivariate


38

Footnotes

lNote that few agents buy a weighted average of new and second-hand
housing, hence our desire to model both variables rather than an overall
“house price index”. In any case, the relative price of new to existing
dwellings is crucial to the construction industry.

2 aypical models of completions have them determined as a distributed lag
on starts. Even if the weights on the lags sum to unity, any (e.g.-)
positive stochastic disturbance to the equation implies houses completed
which were not started.

Spormally, variance dominance refers to the underlying (and unknown) error
variances. In practice, we often say a model variance-dominates another
if the estimated residual variance of the former is smaller than that: of
the latter.

4Note that we have abstracted from weather effects, etc.

Spamilies already housed may nevertheless wish to switch to a new house.
oan practice, it would be desirable to allow for changes in their
attributes and composition.

"hn particular, being a derived relationship, its parameters need not: be
very constant.

8, opendix A sketches an alternative theoretical derivation for a profit-
maximising builder which yields a solution similar to that in (3.8).
nefining the lag operator L as Lx, = Xp then we let A = (I-L)x, +
More generally, ae = a-ytx,. If i and/or j is undefined, it is taken
to be unity.

por comparison with Tables 1 and 2 and equation (4.4), Ericsson [1978]

obtains G = 1.14% (1958(1)-1974(iv)), and Hendry [1980, p. 31], d= 0.93%

(1958(4v)-1976(4it)).


23c

0.06

“ \ul\I
: | 1976 1980,

1960 1962 [1964 1966 1968 1970 1972 1974 Vw
~0.02

——(pn - ph),

°

0.10

0.14

0.18

Figure 5. Logarithm of the relative price of new to second-hand houses {(pn — ph),].


4a

220: PRICES
Pn,
180
-~--Ph,
140
100
60:
20:

1960 1965 1970 1975 1980

Figure 1. Selected United Kingdom price indices. Dates on this figure and Figures 2-5 mark the first quarters of the respective years.


18

is costly and costs are incurred by maintaining the inventory Uv a
builder minimising costs should aim for a constant rate of production.“
Thus, 8 = 1 seems likely in (3.1), corresponding to C = KU in equilibrium
(K a positive constant).

Since (3.1) is conditional on the pre-existing stock of work in
progress, the role of ce, and pn. ig to alter the mean lag around
exp(-By)s with the main impact of changes in long-run profitability being

via the level of u, Thus, it seems reasonable to expect Bo td 8, with

t-1"
both being relatively small.
On the demand side, population, income, interest rates, and the

relative price of new to second-hand housing are the main determinants of
purchases of completions. Again we use a log-linear equation
(3.3) ef = 7% + C7), + Yon, - Yg(pn-Ph), — YR, >
where of is the demand for completions, XN, is the total number of families
in the relevant geographical region,» Yy, is total real personal disposable
income, and R. is the interest rate. It is unclear how significantly
demographic factors should influence the relative price of new to existing
housing (since much of their effect will be reflected in the conditioning
variable Ph). So, to a first approximation, we assume yy" VY) * 1; hence
n, can be dropped, leaving ¥,_ to capture both scale changes (e.g., via
population size) and changes in real personal disposable income per
capita. Since this abstract analysis assumes homogeneous housing unite,®
a very large value of %3 might be anticipated, reflecting a willingness to
switch freely between otherwise identical new and almost new dwellings,
depending on their respective prices.

The overall demand for housing relates to the national stock, LA)

and, as c is a small fraction of that stock, the average second-hand


30

S), constant parameters (rejected if D° is omitted), and approximately
normal errors. Thus, the equation offers a reasonable data description
and is consistent with the economic analysis of Section 3. Figure 6 shows
the track of Mpa, from (4.4) against that of A,pn,.

Next, we consider the dynamics of (4.4). The reaction of pn, to a
change in ph, is quite rapid initially (e.g., Pa, would change by over
0.6% by the end of six months in response to a 1% change in Ph)» followed
by an oscillatory convergence. For construction costs, however, a much
slower adjustment pattern is observed, the main feedback in (4.4) being
lagged by one year. Such lag responses are also consistent with the
theory-model of Section 3 and show the need to model Pn given Ph and
solve for the long-run role of construction costs rather than to model Pn
directly, given CC (omitting Ph). Note that, because Table 2 includes
the “construction cost" hypothesis as a special case, the present
approach automatically encompasses that rival model (although Appendix A
should clarify that several different interpretations are possible). The
dynamic impacts of changes in u and y have appropriate signs and seen
more important quantitatively than their equilibrium impacts. The
interest-rate coefficient is small and not well determined.

The above interpretations of the individual coefficients of (4.4)
rest upon an implicit assumption of relatively orthogonal regressors.
Table 3 reports the matrix of correlations for the whole sample. The
figures in brackets are the partial correlations from estimating (4.4),
and it is noteworthy that five of these have the opposite sign to the
corresponding simple correlation, highlighting the difficulty of
interpreting simple correlations directly when in a multivariate context.
Of the regressor intercorrelations, 36 are smaller than 0.5 in absolute

value and only two are larger than 0.75. Since both involve the term


30a

(pp) Uonenbs wos) pany pur

njae ‘said dsnoy Mou Jo aMuRYD Jo HEY “9 aandLy

"i861

"eL6l

"61"

$/6l"

Teceit

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Hud'y) sanjen peng —----
NY

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23

tration of the methodology described in Sections 2 and 3,

we mode determinants of new house prices and present evidence for
Markei cisequilibria using the series on unsold completions. Before doing
so, we i .vst- develop simple time-series models of Pn: such models

establish a useful baseline against which to evaluate our econometric
model of ®n. Further, those models will allow insight into the role of
economic theory in econometric modelling.

Tne nistorical quarterly time series for the rates of change of pn
and (pn-p) (i-e., nominal and real new house prices respectively) are
shown in Figures 3 and 4. The series are highly volatile, and two large
“booms” in 1971-1974 and 1977-1979 are evident. Also, Aypn, has tended to
be positive fin almost every quarter), whereas 4, (pa-p), often has been

negative. @?recise data definitions are recorded in Appendix B. Next, the

relative price (pn-ph), is shown in Figure 5 and reveals substantial

ie

swings: zenerally, (pn-ph) falls (rises) when ph is rising (falling) most

t

rapidly, suggestive of pn adjusting to lagged (and possibly current) ph.

A fth-order autoregression for pn suggested the following

simplifi2d model:

(4.1) Bopa, = 0.50 Ajpn,_, - 0.24 Appn._, + 0.006
Vt lo6) 2 fF Coy F 83 C003)

Ro = 0.59 6 = 1.55% 1, (6,85) = 0.7 4 (6,85) = 0.15

1.0 ng(4,87) = 0.7 (1) = 0.04 €,(2) = 27.9

iclent standard errors are shown in parentheses, T denotes

, 22 is the squared multiple correlation coefficient, and &

is the © sicua

ndard deviation. The nyCr) are test statistics

labelled ax

possible to correspond to the order of the criteria in

Sectio , acd the figures in parentheses are their degrees of freedom.


1. Introduction

The feature which distinguishes econometric modelling from time-
series analysis is the integral role of economic theory in orienting the
former. At one extreme, a univariate time-series model is inherently
mechanistic and has little or no need for subject-matter knowledge.
Often, the procedure for choosing a model can be automated so as to
satisfy appropriate criteria, such as minimising a residual variance
adjusted for degrees of freedom. Even a bivariate model needs little more
than common sense in selecting a relevant covarying series. At the other
extreme, prior to data analysis a formal intertemporal optimization model
can be developed for the behaviour of rational economic agents who fully
account for all relevant costs and available information. The data
evidence is then used to calibrate the unknown parameters of the theory-
model. Any required data transformations are derived from the theory
(e.g., moving averages might represent “permanent” components, or
residuals from auxiliary regressions might act as “transitory” or
“surprise” effects).

When formulated as such, the "data-driven" and "theory-driven”
approaches to modelling have been viewed as competitive rather than
complementary (see, for example, Naylor et al. [1972] and Granger and
Newbold [1977]). Confrontations between the rival strategies in terms of
forecasting accuracy have not generally been kind to supporters of
“theory-driven” modelling (see Nelson [1972]), although that is neither
surprising nor definitive in view of the choice of a mean-squared-error
criterion; moreover, such results depend on the length of the forecast
horizon (see the discussion in Kmenta and Ramsey [1981]}).

Each extreme also has severe drawbacks. The first approach is open

to such difficulties as spurious correlations (witness Coen, Gomme, and


36

be reflected in increased prices for existing housing. Thus, an
alternative representation would be pn = Acc + (1-\)ph (+ demand factors),
which is closely similar to (3.8). Note that the functional form of the
model is log-linear, given the postulated production and demand functions.
Also, if (A.3) is “inverted” to express Pn/Ph as a function of C, the
latter can be eliminated from the schedule for the supply of new housing,
as in the main text. Conversely, since u in (3.8) is endogenous to the
construction sector, a formulation dependent only on “outside” influences

would necessitate substituting in its determinants.


produce simplifications different from (4.4). The alternative of not
basing the selection of the model upon the data is even less appealing.

Partial protection is offered by testing the selected model for its
ability to encompass rival hypotheses as well as by checking parameter
constancy on later data. Models which are encompassing and remain
constant over time are useful tools for later applications. The present.
equation for new house prices fits substantially better than any pre-
existing models of house prices (e.g., see the estimates discussed in
Nellis and Longbottom [1981]). However, it is an explanation conditional
on contemporaneous second-hand house prices Phi? marginalising with
respect to Phe would increase the residual standard deviation from 6 to
& = (82 + a2G2)1/2, where 4 is the estimated coefficient of Ph, in (4.4)
(1.e., 0.27) and @? is the estimated error variance of the model for
Ph, (e-8+5 @ = 1.43% in Hendry [1984, eq. (18)]). Thus, 6 * 1.02% ts
implied, which remains smaller than in existing models which do not
include Ph, in modelling Pais Nevertheless, we intend to conduct direct:
tests between the various marginal models of pn in due course, and hope to
account for the sub-sample variation in 6.

A mixed outcome, in which some developments have been implemented
while others remain to be carried out, is fairly typical of empirical
econometric research. Viewed as part of a research strategy in which
anomalies point towards new research areas, remaining problems become a
future stimulus rather than a major drawback. They are also a caution to
the limitations of a model rather than a definitive rejection, and a sign
that stringent evaluation criteria are being demanded rather than that

econometric modelling is not worth undertaking.


1

parameter values should be found in other models of the mechanism (at
least in large samples). Consequently, if the selected model is claimed
to characterise the data process adequately, it too should satisfy that
requirement and allow the results of rival models to be derived. ‘Should
the estimated parameters of rival models which are in fact obtained differ
significantly from those derived using the selected model, then that would
contradict the assertion that the selected model 2dequately described the
data generation process. Thus, encompassing requires that any model F(*)
of the mechanism generating {x} should mimic that property of D(*) and be
able to account for the empirical results reported by rival models of {x,}-
Before concluding this section, we consider the implications of this
concept and its relation to testing non-nested hypotheses, which can be

seen most easily for two rival linear models:

(2-5) By t EQ, lz.) = 4148)
and

(2.6) Hy + EG, |25,) = 248 >

where each hypothesis separately asserts Ys = ~ IN(O, ag,

=z!
ct eee)
In (2.5) and (2.6), & and 5 are k xt and kyxl vectors of unknown

(generic symbols for sets of regressors) have

1)

parameters, and Zit and Zoe

(at least) some variables which are not in common. For simplicity, we

assume they have none in common. Formally, the joint density of Yer Zap

and Z

Zop can be factorised as FO lap Zoe Fae lzoe Foes +).

Note that, given the joint density, both (2.5) and (2.6) must be derived
representations; hence, while separate, they are also inter-related.
Here, (2.5) entails the conditional irrelevance of Zoe in explaining Ye

given Zee Under joint normality, z

t Ze and Zoe can be linked using:

(2-7) ze 7 Bort Sie?


13

More generally, an encompassing strategy suggests trying to
anticipate problems in rival models of which their proponents may be
unaware. For example, (2.5) may correctly predict that {vo} is not white
noise, or that Ss is not constant over t = 1,...,T (e-g., if & is
constant but Il varies as t does, then (2.5) predicts that vy should vary
with t). Corroborating such phenomena adds credibility to the claim that
the successful model reasonably represents the data process, whereas
disconfirmation clarifies that it does not.

For the specific class of linear models, the propositions in (2.9)
are testable by a large range of tests. Of these, perhaps the best known
belong to the class of one-degree-of-freedom tests proposed by Cox [1961,
1962] using a modified likelihood-ratio statistic and implemented by
Pesaran [1974] for models like (2.5) versus (2.6). That class seems to
test ay which is necessary but not sufficient for HL when kyl. (See the
discussion following the survey by MacKinnon [1983].) Mizon and Richard
[1983] and Mizon [1984] present equations for generating a very large
class of tests of either a, or Hy or other functions of parameters for

which encompassing is deemed relevant. Under 4), EQ zie 6.

te) = Bey
so that EC, Zoe) = 0. Consequently, if &> S> and J are separately
estimated under their own assumptions and used to construct an estimate of
as & - & (so a, becomes a = Q), then a minor transformation of the
Wald test of a= Q yields the conventional F-test on the marginal
significance of adding (the non-redundant elements of) Zoe to (2.5) (see
Atkinson [1970] and Dastoor [1983]). It is unsurprising that in the
present linear context there should be no sharp dichotomy between nested
and non-nested approaches to testing (2.5) against (2.6). However, the

union of (2.5) and (2.6) must always encompass both, so parsimonious

encompassing is essential to avoid vacuous formulations. For example, if


CONDITIONAL ECONOMETRIC MODELLING: AN APPLICATION

TO NEW HOUSE PRICES IN THE UNITED KINGDOM

Neil R. Ericsson
International Finance Division
Federal Reserve Board
Washington, D.C. 20551

and

David F. Hendry*
Nuffield College
Oxford OX1 1NF England

Revised: June 1985

*this paper was prepared at the invitation of the International Statistical
Institute and appears in a slightly modified form in Ericsson and Hendry
[1985]. It is based on work initially undertaken in Ericsson [1978] and
extended for the U.K. Department of the Environment in Hendry [1980]. We
are indebted to Frank Srba for invaluable help in carrying out the
preliminary analyses; to Jon Faust for his excellent research assistance;
and to Anthony Atkinson, Julia Campos, John Muellbauer, Andrew Rose, and
two anonymous referees for their helpful comments. Recent research has
been supported by grants from the Economic and Social Research Council to
the MIME programme at the London School of Economics, and by

E.S.R.C. grants HR8789 and B 00 220012 to Nuffield College. We are
grateful for the financial assistance from the E.S.R.C., although the views
expressed in this paper are solely the responsibility of the authors and
should not be interpreted as reflecting those of the E.S.R.C., the Board of
Governors of the Federal Reserve System, the Federal Reserve Banks, or
other members of their staffs.


29

functtons of estimated parameters, based on the covariance matrix of the
estimates in (4.4). The coefficients in (4.5) can be interpreted in the
light of (3.8) (assuming y= By = 1). That entails By = 6.0, 83 = 1.2,
Y3 = 10,2. The last two of those three coefficients are of the
anticipated size and sign, but the first is so much larger than expected
as to be somewhat implausible (suggesting that supply is very elastic in
response to changes in real costs, rather than in response to
profitability). However, in (4.4) the coefficients of (ph-p) ,-7 and
(comp), 4 would have to be equal in magnitude and opposite in sign for Bo
to equal By when (4.5) is interpreted as (3.8). For comparison, the

static solution implied by the unrestricted model in Table 2 is
(4.6) pn-ph = -0.19(ph-p) + 0.37(cc-p) - 0.0l(u-y) - 0.1lly + 0.03p .

Given the uncertainty inherent in the unrestricted model, the two derived
equilibria are acceptably similar.
Of course, direct estimation of (3.8) (i-e., omitting all dynamics)

is not overly enlightening; but, for completeness, we record such

results. 5

-—TM
(4.7) (pa-ph), = ~ 0-35 ph, ~ 0.02 u, + 0.23 y, + 0.42 (corp), + 0-32 p,

+ 0.81 {Rm(1-t)}, + 0.021 Q), + 0.012 Qo, + 0.004 Q,, - 2.09
T=94 G= 1.75% 1, (6,78) = 9.6 «

While (4.7) does not even fit as well as (4.1) (which phenomenon does not
entail that equilibrium economic theory is vacuous), a coherent pattern of
estimates emerges across the empirical counterparts of (3.8), with the
last coming close to implying that Bo * By (but Y # 1).

The test statistics of (4.4) suggest white-noise errors (with (4.4)

parsimoniously dominating the model in Table 2 in terms of the value of


21

positive predictions of prices, yielded parameters which were elasticities
and so could in principle be constant over time, and allowed freedom to
switch between any of a number of sensible alternatives for the dependent
variable [such as pas (pn-ph),, (parce), (pn-P) > (pn-p-y) .» and changes
in all of these] without altering the specification of the model.
Moreover, a constant percentage residual standard error also seemed a
reasonable requirement.

The basic difficulties inherent in econometric modelling have now
been introduced, and prior to empirical implementation it is worth
considering: why bother? A straightforward answer is that anything less
than a properly specified, autonomous relationship may “break down"
whenever the statistical properties of any of the actually relevant
variables alter. In practice, such events occur with monotonous
regularity. Seen from this perspective, time-series models are simply
very special cases of econometric equations in which all (for univariate
models) or most (for “transfer function" models) covariates are ignored.
Consequently, they too should regularly “break down” or mis-predict; and,
in practice, they do indeed do so. However, tests of predictive failure
of time-series models may have low power because the models themselves fit
poorly, in which case large mis-predictions are needed to obtain
“significant” outcomes. Econometric equations also often badly mis-
predict, revealing their inadequacy; but they remain susceptible to
progressive improvement using the approach discussed in Section 2.

Nevertheless, much of the reduction in the error variance may derive
from only a few additional factors, as the residual standard deviations
@ in Table 1 tllustrate. 1° There, k is the total number of regressors in
each model, the sample size is 94 (1959(1)-1982(11)), and the mean and

standard devLlation of {ay pa, } (the cegr -sand in every case) are 2.59% aad


ABSTRACT

The statistical formulation of the econometric model is viewed as a
sequence of marginalising and conditioning operations which reduce the
parameterisation to manageable dimensions. Such operations entail that the
“error” is a derived rather than an autonomous process, suggesting
designing the model to satisfy data-based and theory criteria. The
relevant concepts are explained and applied to data modelling of UK new
house prices in the framework of an economic theory-model of house
builders. The econometric model is compared with univariate time-series

models and tested against a range of alternatives.


39

lit should be noted that n,(*) will usually reflect changes in o as well

as in the regression coefficients.

12ohese last are friendly (non-profit-making) societies whose primary
function is to act as financial intermediaries between savers and
potential house owners seeking mortgages. For econometric analyses of
their behaviour, see Hendry and Anderson [1977] and Anderson and Hendry
[1984].

the last observation lies outside our data period and hence entails a
testable prediction of the model.

4 computer-program limitations precluded additional lags or the inclusion
of further variables, so Ro. and its lags were omitted.

96 be comparable with (4.5), p is included and the interest rate is

after tax. The substantial residual autocorrelation in (4.7) precludes

calculation of sensible estimates of standard errors.


26

series were distorted for a period until building societies attracted back
a representative selection of house purchasers. For second-hand house
prices, the main biases appear to have been in 1981(iv) and 1982(i) and
(ii) (the termination of our data period). Here, the model is conditional
on lagged ph; and as the relative distortions between the price series for
new and existing house purchases is not known, it is difficult to assert
any precise pattern for the residuals consequent on the measurement
problem. For example, if the bias in observing 4yph. as Oph is a, and

~
that for 4jpn, as 4)pn. ise then

t tv’

—_ —~ ~
(4.3) Bjpn, = 0.554)ph,_, + 0.294; pn, _y + {e, - 0.29e,_) - 0.554,_1} te,

where the innovation error is denoted by € and we assume (4.2) holds for

t
the correctly measured series. (Seasonal factors are ignored for

simplicity.) The restriction that e, = 4a, with the latter measured as

t
-2.1%, -2.1%, and +4.2% in the three relevant quarters (from Hendry
[1984]) could be rejected. This finding is consistent with the observed
predictions from (4.1), which suggest a smaller but more prolonged
distortion, but is also interpretable as evidence against the hypothesis
that the large value of n3(*) here (or the corresponding test statistic
for the ph model) is mainly due to mismeasurement. Until later data
become available to clarify the issue, some doubt must remain concerning
how distorted the series are over the last four observations. Below,
however, we will continue to act as if the hypothesis were valid and, from
the patterns of the residuals in (4.1) and (4.2), construct a dummy
variable D with the values (110-1 -1) from 1981(1ii) to 19s2(iti). 23
The unrestricted fourth-order autoregressive-distributed lag
representation with (3.8) as its static-equilibrium solution is shown in

Table 2, in a reparameterisation intended to aid interpretability of this

highly over-parameterised equation. 14 Unsurprisingly, few of the


32

Table 4: An Unrestricted Model for us ,7

Variable 5 =0 1 2 l
US 5 -1 (-) 0.74 (0.21) -0.15 (0.16) -0.41
Thy 0.49 (0.15) -0.11 (0.22) 0.46 (0.18) 0.84
Ye-4 -0.71 (1.12) -1.98 (0.93) -0.30 (1.23) -2.99
(po~P)p-4 1.10 (1.65) 1.50 (2.52) -0.90 (1.42) 1.70
(ph-pn),_y  -1-81 (1.36) 0.09 (1.70) -0.17 (1.92) -1.89
Wert -0.13 (0.08) -0.14 (0.06) -0.07 (0.08) -
Constant 32.4 (10.8) - - -

a7 = 45, R2 = 0.988, G = 9.0%, (6,21) = 2.01, m(6,21) = 2.23
E(1) = 2.3. , MLC uo 8 ;

solution analogous to equation (4.6) above can be derived. The mean and

standard deviation of us, and Ajus are (2.1, 63%) and (-1%, 21%)

t

respectively; so, even if Ajus were the dependent variable in Table 4, R2

t
would still exceed 80% despite the large value of G. All of the effects
of r, y, (pn-p), and (ph-pn) are sensibly signed, the last of these
yielding a static relative price elasticity in excess of 4.5 (though some
care is required in interpreting these magnitudes, as us, is conditioned
on pn,)+

A simplified representation of the model in Table 4 is given below:

Tio 7 3.07 Yen

3
rt ze -09] [-30]

<—
(4.8) Ajus, = 0.51A,r_ + 1.36 4, (pn-p),_) + 9-
[-08] [-75] [

+ 1.43 (pn-ph), + 1.64 (pnp), - 0.36 us,_, + 33.4
[-22

[-68] ] t-o7) ‘+t [3.1]

- 0.09 Q,, - 0.15 Q,, - 0.08 Q
f.04} 1 .o4) 2% —¢.05] 2

T= 45 R2=0.87 G= 8.6% nj (6, 28) = 1.6 Ng (7,27) = 0.6
ng(6, 28) = 1.6 &6(1) = 1.9 &(2) =0.3 .


23a

——Apm
0.08

0.06 TM

: Na a

1960 1962 1964 1966 1970 1972 1974 1976 1978 1980
ide

0.02

Figure 3. Rate of change of new house prices (Apn,).


emphasising that white-noise residuals are a minimal requirement for model
adequacy, whether or not modified to account for parsimony (e.g., see
Schwarz [1978]). Models derived by sequentially simplifying unrestricted
representations such as (2.4) tend to have innovation errors. Conversely,
the previously noted drawback of “theory-driven” modelling (that the
associated errors are ‘not innovations) is easily understood if the theory
is not sufficiently general to posit the “correctTM value & of *

a priori.

(B): The validity of the assertion of weak exogeneity is criterion

(iii). Unfortunately, weak exogeneity per se is not easily tested in a
class of models like (2.4); and to do so may require modelling {z,},
thereby defeating the main purpose of the conditioning assumption.
However, if the data generation process of {z, does not stay constant
over the sample, yet oy is constant in (2.3), then this enhances the
credibility of the weak exogeneity assertions underlying (2.3). When dy
is invariant to changes in cy and (2.3) is valid, then zy is said to be
super exogenous for dy:

(C): Parameter constancy (after duly incorporating all relevant
transients) provides criterion (iv). The formulation in (2.4) explicitly
defines certain parameters (Bo eee By)s changes in which would invalidate
the model. It seems natural to seek models with constant parameters,
whatever the purpose of the modelling exercise, and to test assertions of
constancy as a check on the usefulness of the model.

Summarising, (i) + (11), (111), and (iv) respectively relate to the
validity of assumptions concerning lagged, contemporaneous, and leading
data relative to any given observation at time t. In econometrics, (1)-
(iv) are reasonably conventional criteria for selection and evaluation of

models. A model which satisfies (1)-(iv) will be useful for forecasting


40

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12

where [lis defined by (2) [Zo¢) = Boe (80 E(Zoe eit) = Q), and (again for

expositional simplicity) we assume EC ie Ste) = Q. From (2.5) and (2.7),
= 6t 1 =

(2-8) ye = Soe t Me + Gee) = Beoe + Moe .

Consequently, (2.8) is what the model (2.5) predicts the model (2.6)

should find, so that if (2.5) is to encompass (2.6), it must be the case

that

(2.9a) HL: 6, = E'S.
and

(2.96) Hy:

The hypothesis in (2.9a) is called parameter encompassing, and that in
(2.9b) variance encompassing, where a least-squares notion of encompassing
is beiag employed. In passing, there seems little point in testing (2.5)
against (2.6) or vice versa unless both models do satisfy their claimed
formulations, which first requires evaluating both on criteria (i)-(iv) at
least. If so, then (in large samples) the non-negative definiteness of 2
entails that H, cannot hold unless Oy < On9° iee., Hy variance-dominates
Hy Thus, variance dominance is necessary, but not sufficient, for
variance encompassing. That in turn entails that encompassing is
asymmetric in the present context: if (2.5) encompasses (2.6), the
converse is false. Also, as A, is sufficient for a> it is readily
established that least-squares encompassing is transitive. Neither of
(2.5) or (2.6) may encompass the other, in which case a more general model
is necessary. Thus, encompassing defines a partial ordering over models,
an ordering related to that based on goodness-of-fit; however,
encompassing is more demanding. It is also consistent with the concept of
a progressive research strategy (e.g., see Lakatos [1970]), since an
encompassing model is a kind of “sufficient representative” of previous

empirical findings.