A Time Series Aggregate Demand Model with Demographic and
Income Distribution Indices
C.L.F. Attfield†
University of Bristol
December 2005
Age demographic and income distribution indices, derived
from cross section data from 1973 to 2003, are incorporated
into an aggregate time series, cointegrating, “almost ideal”
demand system consisting of 12 commodity groups. The
demand equations are estimated using the single equation
FM-OLS procedure and a sub-system, consisting of the
commodity groups Food, Alcohol, Clothing and Transport,
is analysed using maximum likelihood techniques.
Estimates and forecast comparisons are made with the
standard model with the indices omitted.
1. Introduction
Recent research on aggregate demand systems, e.g., Lewbel & Ng (2003), emphasises
the importance of demographic variables which reflect the age structure of the
population. In the last 50 years there have been significant changes in the age
structure of the economy which may have profound effects on aggregate demand for
commodities, e.g., the impact of an “ageing population”. Standard demand models
are largely silent on the effect of such changes in the demographic structure. In this
paper it is shown that by assuming a fixed age effect at the household level,
aggregating households with heads of the same age and then aggregating across all
age groups, an aggregate demand model can be derived which includes demographic
effects. The estimation problem which arises is that the resulting model has upward
of 70 explanatory variables, i.e., age effects for age groups from ages 19 to 90, prices
and real income. While estimation of such a model is feasible in a cross section
context, in a time series analysis where demand equations are cointegrated such a
large number of variables with their corresponding lags makes estimation impractical.
In this paper, therefore, an index of the age demographic effect constructed by
Attfield (2005a, 2005b) from the Family Expenditure Surveys (FES), which collapses
the age effects into a single variable for various commodity groups, is incorporated
into the Almost Ideal (AI) demand model of Deaton and Muellbauer (1980). The AI
model gave superior results in a time series setting over the other most popular
flexible functional form model, the TRANSLOG of Jorgenson, Lau and Stoker (1982)
– see Attfield (2004).
In such models it is well known that aggregation over all households also results in a
term which depends on the distribution of income across households. If income were
distributed equally this term would be zero. In most aggregate models the term is
either ignored altogether or proxied in some fashion. Given the availability of
household data in the FES, the income distribution term can be calculated and
†
The research in this paper is part of the ESRC project "Demographic and Income Distribution Indices
for Demand Models", Ref: RES-000-22-1084.
included in an aggregate model. In the next section, section 2, we sketch the
derivation of the model with both demographic and income distribution indices. In
section 3 we give time series estimates and forecasts in a comparison of models with
and without the indices included. Section 4 concludes the paper.
2. Aggregate Demand Models
2(i). The Standard AI Demand Model
A good deal of recent empirical work on demand systems applies the flexible
functional form almost ideal demand model of Deaton & Muellbauer (1980). At the
household level such models imply a demand function in the form of budget shares:
whjt = α oj + ∑ γ ij ln pit + ln( xht / Pt* ) β j
(2.1)
i
where whjt = x hjt / x ht is the budget share of commodity j at time t for household h,
with x hjt the expenditure of the hth household on the jth commodity and x ht the total
expenditure on all commodities by the hth household. ln pit is the natural logarithm
of the price of commodity i and ln Pt* is Stone’s price index which is an
approximation which linearises the model.
In its aggregate form, the coefficients in (2.1) are usually estimated from:
 


*
+ ∑ γ ij ln pit + β j ln ∑ x ht  − ln Pt* − ln H t 
w jt = α ojt

  ∀h

i

(2.2)
∑ xhjt


, and ln  ∑ x ht  − ln Pt* − ln H t is the log of real expenditure per


∑ xht
 ∀h

where w jt = ∀h
∀h
*
is
household, with Ht the total number of households at time t. In most studies α ojt
either treated as a constant or proxied by macro variables.
But, aggregating (2.1) over all households gives:
∑ xhjt ∑ xht whjt
w jt = ∀h
∑ xht
∀h
= ∀h
∑ xht
∀h
∑ xht ln xht
= α oj + ∑ γ ij ln pit + β j ∀h
i
To reconcile this aggregation with (2.2) requires:
-2-
∑ xht
∀h
− β j ln Pt* .
α*ojt
= α oj + β j
∑ xht ln xht
∀h
∑ xht
∀h
Let:


− β j ln ∑ x ht  + β j ln H t .

 ∀h
∑ xht ln xht


ln k t = ln ∑ x ht  − ln H t − ∀h


∑ xht
 ∀h

∀h
 ∑ x ht ln x ht
 x

= −  ∀h
− ln t
xt
 Ht


x
∴ ln k t = −∑  ht
x
∀h  t
  x ht
 ln
  xt








(2.3)
where xt is the mean of total nominal expenditure over all households.
*
It follows that α oj
= α oj − β j ln k t so that (2.2) becomes:
 


w jt = α ojt − β j ln k t + ∑ γ ij ln pit + β j ln ∑ x ht  − ln Pt* − ln H t  .

  ∀h

i

Deaton & Muellbauer (1980, p.315) point out that the definition of lnkt in (2.3) is
identical to Theil’s (1972) entropy measure of income equality. If all households
have the same income so that x ht = xt then lnkt in (2.3) is zero. Of course, lnkt cannot
be constructed from aggregate macro data but can easily be computed from household
data on expenditures, as in the FES.
2(ii). The AI Demand Model with a Demographic Index
Suppose that all households are grouped into those with heads the same age at time t,
and that there are G such groups denoted by Γ gt , g = 1,..., G . Let H gt be the number
of households in age group Γ gt and H t the total number of households, and let the
constant α oj in equation (2.1) be the sum of a fixed effect for each age group in the
population, θ gj , which can be thought of as a taste parameter in the utility function,
-3-
and an overall constant across all age groups, θ oj so that:
α oj = θ oj + θ gj .
Then, budget shares of good j for all households in group g are:
∑ xht whjt
wgjt =
h∈Γgt
∑ xht
(2.4)
h∈Γgt
(
= θ oj + θ gj + ∑ γ ij ln pit + β j ln x gt − ln k gt − ln Pt* − ln H gt
)
i
which is equation (2.2) of the previous section but with the new definition of α oj .
Now, aggregating over all G age groups gives:
w jt =
x jt
xt
∑ x gjt ∑ x gt wgjt
=
g
∑ x gt
=
g
∑ x gtθ gj
∴ w jt = θ oj +
g
∑ x gt
g
g
∑ x gt
g
(
+ ∑ γ ij ln pit + β j ln x gt − ln k t − ln Pt* − ln H t
)
(2.5)
i
which is equation (2.2) with the change in definition of the intercept and the
incorporation of the demographic term ∑ x gt θ gj ∑ x gt , which captures the effects
g
g
of movements in the age structure of the population. Clearly, if the coefficients θ gj
are the same for all age groups, for commodity j, the index collapses to a constant and
(2.5) reverts to (2.2). If, on the other hand, there is a wide variation in the demand
behaviour for different age groups this variation will be reflected in the index and
have an impact on demand for commodity j. In the next section we include an
estimate of the demographic term in a set of time series aggregate demand equations.
3. Empirical Results for AI Models.
An empirical investigation of (2.5) is not feasible in a time series setting, using
current technology, because if the population is divided into adult age groups ranging
from 19 to 90 years of age there are 72 of the parameters θ gj . Fortunately, Attfield
-4-
(2005b), uses cross section techniques to estimate the index:
I θjt =
∑ x gtθ gj
g
∑ x gt
g
for 12 commodity groups using the FES from 1973 to 2003. The commodity groups
used are consistent with the COICOP definitions of groups, i.e., the commodity
definitions based on the European classification, the Classification Of Individual
COnsumption by Purpose (COICOP), which groups consumption under 12 main
headings, viz.,
Commodity Name
Reference Name
1.
2.
3.
4.
5.
Food
Alcohol
Clothing
Housing
Furnishings
Food and Non-Alcoholic Beverages
Alcoholic Beverages, Tobacco and Narcotics
Clothing and Footwear
Housing, Water, Electricity, Gas and Other Fuels
Furnishings, Household Equipment and
Routine Maintenance of House
6. Health
7. Transport
8. Communication
9. Recreation and Culture
10. Education
11. Restaurants & Hotel
12. Miscellaneous
Health
Transport
Communication
Recreation
Education
Restaurants
Miscellaneous
For most households, of course, expenditure on Education is concentrated in only a
few years so that most of the cells in the data set are empty for this variable. As we
are constructing indices and demand equations over all households across all years in
the sample the Education variable had to be omitted from the analysis reducing the
number of commodity groups to 11.
Attfield (2005b) also estimates the income distribution index in (2.3):
x
ln k t = −∑  ht
x
∀h  t
  x ht
 ln
  xt



using household data from the FES for the period 1973 to 2003. The model to be
estimated from aggregate time series data is then:
w jt = θ oj + ∑ γ ij ln p it + ϕ1 j Î θjt + ϕ 2 j ln k̂ t + β j ln x*t
(3.1)
i
where the parameters ϕ1 j ,ϕ 2 j are included to allow for any discrepancy in the
magnitude of the variables which are constructed from the FES samples and the
aggregate data series on the same variables collected from the Office of National
Statistics website. Notice that the total population size, N t , replaces the number of
(
)
households, H t , so that the variable ln x*t = ln xt − ln Pt* − ln N t is real expenditure
per capita. Since the figures in the national accounts are mainly grossed up from
-5-
household figures, this change in definition should only have an effect on the constant
term in equation (3.1).
To estimate the aggregate model, quarterly time series data for the period 1973Q2 to
2003Q2 were obtained from the ONS data bank1. Prior to estimating the equations all
variables were tested for unit roots using the procedures by Ng and Perron (1997) and
Perron and Ng (1996) which optimally choose the lag length for the ADF test. Their
DF-GLS test did not reject the null of a unit root for all variables except the
demographic indices for Food, Clothing, Health, Transport, Communications and
Restaurants at the 5% level but only Communications at around the 1% level. It was
therefore assumed that the system could be estimated as a set of I(1) variates.
Given the number of variables in the system (36) it is not practical to test for
cointegration using Johansen’s (1995) likelihood ratio system test. It is possible,
however, to consider each of the commodities with its own demographic index, all
prices, income and the income distribution index as a system consisting of 16
variables. Using PcGive (2001) the AIC was consistent in choosing one lag in each
system for both the 16 variable model, which includes the demographic and income
distribution indices, and for the standard model, without the indices, which consists of
14 variables. Using PcGive (2001) to obtain Johansen’s trace test statistic for
cointegration resulted in being able to accept (at the 5% level) between 8 and 10
cointegrating relations for the model with indices and between 5 and 6 cointegrating
relations for the standard model without the indices. The addition of the two index
variables therefore increases the number of cointegrating equations by between 3 and
4 in the systems.
3(i) Single Equation Estimates of Cointegrating Demand Equations
To obtain parameter identification in the systems we would need to impose at least 8
to 10 normalisations/restrictions in the index model and at least 5 to 6 in the standard
model. As these restrictions would remove coefficients of interest in the demand
equations, i.e., coefficients on some of the price and the income variables, we
estimated the demand equations using the single equation FM-OLS procedure for
cointegrated equations coded in the COINT module in GAUSS (2002) by Phillips and
Ouliaris (1995).
Tables 1(a), 1(b) Here
Table 1(a) gives FM-OLS estimates for the equations containing the indices and table
1(b) own price elasticities, income elasticities and the results of a Wald test for
homogeneity on the price coefficients. The demographic indices, I θjt , make a
significant contribution for the commodity groups Food, Alcohol, Clothing,
Communication, Recreation, and Miscellaneous while the income distribution index,
ln k t , has a significant coefficient for all commodity groups except Furnishings.
There is some evidence of serial correlation in the Clothing, Housing, Health,
Recreation and Miscellaneous equations which could be a result of the inadequate de1
Quarterly, seasonally unadjusted series on real and nominal expenditure on the 12 commodity groups
were obtained. Commodity price indices and total expenditure (income) variables were constructed
from these series. Prior to analysis, seasonal components were removed using seasonal dummies. The
annual series on the demographic and income distribution indices were interpolated to quarterly using
an interpolation procedure in GAUSS (2002) producing a quarterly series from 1973Q2 to 2003Q2.
-6-
seasonalising of the series [see the actual series in the within sample forecasts in Figs.
1(a)-4(a)]. Price elasticities in table 1(b) are of the correct sign for all commodity
groups except Housing and Transport while all income elasticities are of the correct
sign and classify Transport, Recreation, Restaurants and Miscellaneous as luxury
goods. Homogeneity is satisfied for all equations except Alcohol, Clothing, Housing,
Furnishings and Recreation.
Tables 2(a), 2(b) Here
As a comparison, tables 2(a) and 2(b) give FM-OLS estimates and Wald tests for the
equations for the standard AID model, excluding the indices. There is some evidence
of serial correlation in the Clothing, Health, Communication, Recreation and
Miscellaneous equations – which is similar to the index model. Price elasticities in
table 2(b) are of the correct sign for all commodity groups except Housing while all
income elasticities are of the correct sign and, as for the index model, classify
Transport, Recreation, Restaurants and Miscellaneous as luxury goods.
Homogeneity is satisfied for all equations except Food, Alcohol, Clothing, Housing,
Furnishing, Health and Miscellaneous.
In summary for this section, then, although both the demographic and income
distribution indices are significant determinants in a number of the demand equations
the inclusion of these variables only improves the homogeneity tests slightly in that, at
the 1% level, homogeneity cannot be rejected in 6 cases for the index model but for
only 5 in the standard AID model.
3(ii) ML Estimates of System for Food, Alcohol, Clothing and Transport
In this section we estimate a cointegrating AI demand system using Johansen’s more
efficient maximum likelihood procedure. As pointed out in the previous section,
however, the large number of variables make estimation of the full system virtually
impossible with current technology. Of the commodity groupings Food, Alcohol,
Clothing and Transport probably have the most consistent definitions for their
constituents over the sample period. We therefore model demand equations for these
four commodities plus an aggregate group of All Other Goods as a demand system
with the latter group dropped from the analysis to avoid the linear dependence
inherent in the model as we are working with budget shares.
The starting point for the estimation is the vector error correction model (VECM):
∆z t = π o + π 1∆z t −1 + ... + π 1∆z t −k + δα' z t −1 + ξ t
(
(3.2)
)
where the (15x1) vector, z t ' = w1 ,..., w4 ,ln p1t ,...,ln p5t , Î1θt ,..., Î 4θt ,ln k̂ t ,ln x*t ,
contains all the variables in the model. That is, budget shares for Food, Alcohol,
Clothing and Transport, w1 ,..., w4 ; demographic indices for the same four variables
Î1θt ,..., Î 4θt ; log price indices for Food, Alcohol, Clothing, Transport and All Other
Goods, ln p1t ,...,ln p5t ; the income distribution index, ln k̂ t ; and log real expenditure
per capita, ln x*t . In (3.2) the column dimension of δ and row dimension of α is r,
the number of cointegrating relationships in the system, which must lie between 0 and
15.
-7-
The number of cointegrating equations, r, was found using Johansen’s (1995)
procedure using COINT 2.0 (1994). The lag length in (3.2) was determined from an
unrestricted model in levels using PcGive (2001). The BIC, Hannan-Quinn and
Akaike information tests gave results for lag lengths of 1, 4, and 4 respectively in
levels (0, 3 and 3 in first differences) so tests and subsequent estimation were carried
out using 3 lags in first differences in the VECM. The trace test statistic for the null
of 11 cointegrating vectors is 50.329 with a 5% critical value of 47.21 so 11
cointegrating vectors can be rejected in favour of 12 or more, while the trace test for
the null of 12 cointegrating vectors is 27.568 with a 5% critical value of 29.68 so we
can’t reject 12 vectors in favour of 13 or more. On the other hand, the λ-max statistic
for the null of 7 vectors is 55.258 with a 5% critical value of 51.42 so that 7
cointegrating vectors can be rejected in favour of 8; the λ-max statistic for the null of
8 vectors is 38.232 with a 5% critical value of 45.28 so that the null of 8 vectors
cannot be rejected in favour of 9. We therefore took the more conservative estimate
of 8 cointegrating vectors from the λ-max statistic for the estimation of the model2.
With 8 cointegrating equations the dimension of α′, in (3.2), is (8x15) and we can
always find an (8x8) non-singular matrix, say G, such that:
δα' = δG −1Gα' = δ ∗α ∗' .
So that the matrix G can be used to place 8 linear normalisations on each of the 8
cointegrating equations to uniquely identify the coefficients. Without any loss of
generality we can take the first 4 equations as the budget share demand equations and
the next 4 as price equations. The demand equations have 4 natural linear
normalisations in the form of a coefficient of –1 on the “own” budget share and 0 on
the other shares. Homogeneity gives a further 1 normalisation and zero coefficients
on “alien” demographic indices give the final 3 normalisations. For normalisations on
the price equations we assume that each of the prices for the four commodities are a
function of the price of All Other Goods, the demographic indices, Î1θt ,..., Î 4θt , the
income distribution index, ln k̂ t and the log of real per capita income, ln x*t . The
normalised cointegrating matrix, α ∗' , then becomes:
w1
ln p1
ln p 2
ln p 5
Î 1θ
γ 41
γ 51
γ 42
γ 52
ϕ11
0
γ 43
γ 53
γ 34
0
γ 44
0
γ 54
α 15
α 25
α 35
0
0
α 16
α 26
α 36
−1
0
0
α 17
α 27
α 37
−1
α 18
α 28
α 38
w2
w3
w4
−1 0
0 −1
0
0
0
0
γ 11
γ 21
γ 31
0
0
γ 12
γ 22
γ 32
−1
0
0
γ 13
γ 23
γ 33
γ 14
−1
γ 24
0
0
0
0
−1
0
0
0
0
0
0
0
0
0
0
0
−1
0
0
0
0
0
0
0
0
0
ln p 3
ln p 4
Î 2θ
Î 3θ
Î 4θ
0
0
0
ϕ 21
β1
0
0
ϕ 22
β2
0
ϕ12
0
0
ϕ 23
β3
0
0
ϕ13
0
ϕ14
ϕ 24
β4
α 45
α 55
α 65
α 75
α 46
α 56
α 66
α 76
α 47
α 57
α 67
α 77
α 48
α 58
α 68
α 78
ln k̂
ln x ∗
.
So that there are seven –1/0 normalisations on each of the demand equations, which,
with homogeneity, i.e., ∑ γ ij = 0 ,∀j , gives the 8 normalisations required for
i
2
Critical values for both test statistics were obtained from Table 1 of Osterwald-Lenum (1992, p.468)
-8-
identification.
Tables 3(a), 3(b) Here
Tables 3(a) and 3(b) give the results of the estimation using PcGive (2001). For this
model the coefficients on all the demographic indices are significantly different from
zero as are the coefficients on the income distribution variables - except for clothing.
The equation errors are all white noise at the 1% level and own price elasticities are of
the correct sign – except for alcohol – as are income elasticities – except for transport.
For this system, homogeneity is imposed by normalisation and so cannot be tested.
Symmetry, however, does impose 6 testable restrictions, i.e., γ ij = γ ij ,∀i ≠ j . A
likelihood ratio test gave a chi-square test statistic of 17.87 which, with 6 degrees of
freedom, results in a p-value of 0.007 which is a rejection of simultaneity but close to
non-rejection at the 1% level.
Tables 4(a), 4(b) Here
Tables 4(a) and 4(b) give estimates of the standard AID model for comparison. In
this model there are 10 variables in the VECM and the number of lags from
information criteria were also chosen as 3. The trace test statistic for cointegration for
the null of 4 cointegrating vectors against 5 or more was 94.19 which compares with a
5% critical value of 94.15 while the λ-max statistic for the null of 3 against 4 vectors
is 38.65 with 5% critical value of 45.28. The λ-max statistic result implies that there
are less than 4 cointegrating equations so that the standard AID model does not form a
cointegrating system. However, for comparison we assumed 4 cointegrating vectors
to give the results in tables 4(a) and 4(b). As there are only 4 cointegrating equations
only 4 normalisations are required to just identify the coefficients so that, in this case,
homogeneity imposes 4 additional and testable restrictions on the model. A
likelihood ratio test of these restrictions resulted in a test statistic of 32.99. The
statistic is distributed as chi-square with 4 degrees of freedom so the test rejects
homogeneity at any sensible level of significance. In addition there is a good deal of
serial correlation in the clothing and transport equation errors and only alcohol and
clothing have the correct signs on the estimated price elasticity, although all the
income elasticities have the correct sign.
For this sub-model then, the demand system with the demographic and distribution
indices easily outperforms the standard model without the indices. In fact, the
standard model does not appear to consist of a cointegrating system.
Figures 1 to 4 Here
Turning to dynamic forecasts from the Johansen estimation procedure, generated by
PcGive (2001), figures 1(a)-4(a) give within sample forecasts, while figures 1(b)-4(b)
give out of sample forecasts for the period up to 2013. Judging by the root mean
square error criterion , given in figs1(a)-4(a), the inclusion of the index variables in
the categories Food and Alcohol improves forecasts for the within sample period3.
For the other two categories the inclusion of the indices does not improve forecasts,
although forecasts from the index model are within the 95% forecast intervals for the
standard model. It should be realised, however, that there are five more variables in
3
Data up to and including 1998Q2 was used to estimate the model and dynamic forecasts extrapolated
using these estimates.
-9-
the system with indices so that there is a great deal more variation and we are not
comparing like with like.
Figures 1(b)-4(b) depict the difference between out of sample forecasts for budget
shares for the two models. In this case the 95% forecast intervals for the index model
are graphed and while the point forecasts for Food, Alcohol and Transport are similar,
for Clothing, forecasts from the standard model lie outside the 95% confidence
interval for the index model within about 2 years4.
4. Conclusion
Time series age-demographic and income distribution indices, derived from a series
of FES and EFS cross section data from 1973 to 2003, are incorporated into the
standard almost ideal demand model of Deaton and Muellbauer (1980). Variables in
the system are found to be I(1) and cointegrated and appropriate estimation techniques
are applied firstly to single equation models for the commodities Food: food and nonalcoholic beverages; Alcohol: alcoholic beverages, tobacco and narcotics;
Clothing:clothing and footwear; Housing: housing, water, electricity, gas and other
fuels; Furnishings: furnishings, household equipment and routine maintenance of
house; Health; Transport; Communication; Recreation: recreation and culture;
Education; Restaurants: restaurants and hotels; and Miscellaneous. Secondly, a subsystem of feasible size consisting of Food, Alcohol, Clothing and Transport was
estimated using Johansen’s maximum likelihood procedure.
Overall, the index variables do make a significant contribution to the demand
equations and, in the sub-system, were found to form a cointegrating demand system
when the standard model, excluding the index variables, failed to do so. The index
model also removed all the serial correlation from the errors of the vector error
correction model used to obtain the estimates while the standard model did not.
One advantage of estimating the model in a VECM framework is that there may be
enough cointegrating equations to place normalisations on the demand system which
satisfy the theoretical propositions of demand theory. For example, in this research,
for the model with indices in section 3(ii), 8 cointegrating equations were found in the
system of 15 variables. With 8 cointegrating equations, 8 normalisations can be
placed on the cointegrating equations. Without any loss of generality we nominated
the first 4 equations as the demand equations and normalised them so that “alien”
demographic indices and budget shares were excluded and the demand equations
satisfied the property of homogeneity.
With the exception of Clothing, the standard model gives forecasts up to 2013 which
are within the 95% confidence interval for the model with indices. However, by their
very nature changes in the age structure of the population, reflected in the
demographic index and in the distribution of income, are very slow moving so we
might expect in the short to medium term, the standard model to perform fairly well.
This is borne out by the forecasts in section 3. For longer term forecasts however, the
index model is the appropriate tool for obtaining forecasts.
4
From the within sample forecasts in figure 3(a) the index model forecasts are superior for the first two
years.
- 10 -
References
Attfield, C.L.F., (2005a), “Extending COICOP Codes to FES Samples 1973-2003”,
Mimeo, University of Bristol, 2005.
http://www.ecn.bris.ac.uk/www/ecca/ESRC_Demand_Indices/Extending.COI
COP.pdf
Attfield, C.L.F., (2005b), ”Income Distribution and Demographic Indices from
Demand Equations”, Mimeo, University of Bristol, 2005.
http://www.ecn.bris.ac.uk/www/ecca/ESRC_Demand_Indices/Income_demog
raphic_indices.pdf
Attfield, C.L.F., (2004), “A Comparison of the Translog and Almost Ideal Demand
Models”, mimeo,
http://www.ecn.bris.ac.uk/www/ecca/ESRC_Demand/demand.pdf, University
of Bristol.
COINT 2.0, (1994), Sam Ouliaris and Peter C.B. Phillips, GAUSS Procedures for
Cointegrated Regressions, Aptech Systems Inc., 23804 Kent-Kangley road,
Maple Valley, WA 98038, USA.
Deaton, Angus and John Muellbauer, (1980), “An Almost Ideal Demand System”,
American Economic Review, 70, pp.312-326.
GAUSS, (2002), Version 5, Aptech Systems Inc., Mathematical and Statistical
Division.
Johansen, S., (1995), Likelihood Based Inference in Cointegrated Vector
Autoregressive Models, Oxford University Press.
Jorgenson, D.W, L.J. Lau, and T.M. Stoker, (1982), “The Transcendental Logarithmic
Model of Aggregate Consumer Behaviour”, in R. Basmann and G. Rhodes
(eds), Advances in Econometrics, Volume 1, JAI Press, Connecticut, pp.97238.
Lewbel, Arthur and Serena Ng, (2003), “Demand Systems with Nonstationary
Prices”, Mimeo, Department of Economics, Boston College.
Ng, Serena. and P. Perron, (1997), “Lag Length Selection and Constructing Unit Root
Tests with Good Size and Power,” Mimeo, Boston College and Boston
University. Boston College Working Paper 369.
Osterwald-Lenum, Michael, (1992), “A Note with Quantiles of the Asymptotic
Distribution of the Maximum Likelihood Cointegration Rank Test Statistics”,
Oxford Bulletin of Statistics, 54, pp461-472.
Perron, P. and Serena Ng, (1996), “Useful Modifications to Unit root tests with
Dependent Errors and their Local Asymptotic Properties”, Review of
Economic Studies, 63, pp.435-464.
PcGive, (2001), Doornik, Jurgen A. and David F. Hendry, Modelling Dynamic
Systems Using PcGive 10, Volume II, Timberlake Consultants, London.
Phillips, Peter C.B., and Sam Ouliaris, (1995), Coint 2.0a GAUSS Procedures for
Cointegrated Regressions, Aptech Systems Inc., Mathematical and Statistical
Division.
Theil, Henri, (1972), “The Information Approach to Demand Analysis”,
Econometrica, 33, pp.67-87.
- 11 -
Table 1(a) FM-OLS Estimates of Model with Demographic and Income Distribution Indices
Dependent
Variable
Explanatory Variables
Diagnostics
lnp1
lnp2
lnp3
lnp4
lnp5
lnp6
lnp7
lnp8
lnp9
lnp10
lnp11
lnp12
lnz
lnk
I θj
Box-Ljung
(11)
w1
0.0689
(7.408)
0.0296
(4.773)
0.0321
(4.094)
-0.0482
(-5.081)
-0.0323
(-1.544)
0.0087
(0.914)
0.0024
(0.189)
-0.0026
(-1.033)
-0.0357
(-2.688)
0.0318
(1.942)
-0.0819
(-4.697)
0.0215
(3.470)
-0.0871
(-10.927
0.1630
(6.369)
-2.1489
(-3.630)
24.6613
(pval=0.0102)
w2
-0.0029
(-0.387)
0.0444
(7.978)
0.0012
(0.243)
-0.0259
(-4.399)
-0.0109
(-1.029)
-0.0019
(-0.288)
-0.0003
(-0.047)
0.0019
(0.886)
-0.0013
(-0.144)
0.0207
(2.135)
-0.0311
(-2.923)
-0.0001
(-0.022)
-0.0238
(-4.877)
0.0469
(3.641)
8.5315
(1.677)
24.5666
(pval=0.0105)
w3
-0.0043
(-0.271)
0.0011
(0.128)
-0.0146
(-1.493)
-0.0089
(-0.923)
0.0620
(2.740)
-0.0054
(-0.537)
-0.0222
(-1.420)
-0.0210
(-3.493)
-0.0061
(-0.345)
0.0031
(0.128)
0.0056
(0.352)
0.0009
(0.096)
-0.0279
(-2.368)
-0.0963
(-2.596)
-6.2360
(-2.209)
50.7089
(pval=0.0000)
w4
-0.0161
(-1.040)
0.0059
(0.481)
-0.0114
(-1.015)
0.1619
(14.227)
-0.0590
(-2.205)
-0.0094
(-0.865)
0.0082
(0.523)
0.0014
(0.327)
0.0130
(0.843)
0.0008
(0.029)
-0.0380
(-1.268)
-0.0460
(-4.548)
-0.0891
(-7.565)
-0.0776
(-2.066)
-0.9589
(-0.928)
37.7869
(pval=0.0001)
w5
-0.0222
(-2.067)
-0.0194
(-2.174)
-0.0097
(-1.063)
-0.0115
(-1.200)
0.0485
(3.047)
-0.0111
(-0.995)
-0.0221
(-1.620)
-0.0120
(-3.536)
-0.0058
(-0.450)
-0.0102
(-0.541)
0.0509
(2.587)
0.0157
(2.608)
-0.0075
(-0.611)
0.0106
(0.342)
-6.8214
(-1.039)
20.2944
(pval=0.0415)
w6
-0.0066
(-2.575)
-0.0037
(-2.138)
-0.0033
(-1.831)
0.0024
(1.213)
-0.0066
(-1.805)
0.0031
(1.079)
-0.0033
(-1.272)
-0.0005
(-0.426)
0.0073
(2.273)
0.0177
(4.043)
-0.0065
(-1.401)
0.0003
(0.147)
-0.0077
(-4.379)
-0.0352
(-5.432)
1.5245
(0.554)
28.1629
(pval=0.0031)
w7
-0.0421
(-1.615)
-0.0497
(-1.933)
-0.0544
(-2.565)
0.0027
(0.102)
0.0434
(1.013)
-0.0635
(-2.836)
0.0692
(2.136)
0.0249
(3.191)
0.0415
(1.485)
-0.0733
(-1.643)
0.1491
(3.747)
-0.0443
(-3.118)
0.1351
(6.601)
0.1928
(2.731)
-1.1353
(-0.478)
19.2206
(pval=0.0572)
w8
-0.0039
(-1.474)
-0.0028
(-1.216)
-0.0020
(-1.033)
0.0020
(0.651)
-0.0019
(-0.376)
-0.0057
(-1.579)
0.0039
(1.086)
0.0128
(11.730)
-0.0299
(-7.967)
0.0072
(1.455)
0.0226
(5.575)
-0.0041
(-1.889)
-0.0034
(-1.168)
0.0253
(2.666)
4.3712
(1.793)
22.4839
(pval=0.0209)
w9
0.0188
(1.259)
-0.0012
(-0.148)
0.0210
(1.714)
-0.0046
(-0.426)
0.0061
(0.224)
0.0310
(2.187)
-0.0460
(-2.659)
-0.0103
(-1.519)
0.0077
(0.520)
-0.0367
(-1.820)
0.0159
(0.720)
0.0114
(1.175)
0.0365
(2.831)
0.1443
(3.287)
-11.9838
(-2.744)
28.6440
(pval=0.0026)
w10
0.0325
(1.553)
-0.0188
(-1.560)
-0.0337
(-2.529)
-0.0380
(-2.986)
-0.0393
(-1.774)
0.0040
(0.266)
0.0302
(2.372)
0.0034
(0.621)
0.0495
(2.079)
0.0361
(1.151)
-0.0397
(-1.964)
0.0083
(0.807)
0.0395
(2.638)
-0.1357
(-3.637)
-0.9164
(-0.422)
14.6158
(pval=0.2008)
w11
-0.0228
(-2.130)
0.0083
(0.946)
0.0292
(2.270)
-0.0443
(-3.474)
0.0191
(0.955)
0.0180
(1.469)
-0.0322
(-2.263)
-0.0033
(-0.820)
-0.0092
(-0.631)
0.0035
(0.165)
-0.0034
(-0.180)
0.0385
(4.119)
0.0610
(4.423)
-0.3431
(-9.378)
-30.4490
(-2.365)
28.6453
(pval=0.0026)
Commodity groups are: 1. Food, 2. Alcohol, 3. Clothing, 4. Housing, 5. Furnishings, 6. Health, 7. Transport, 8. Communication, 9. Recreation, 10. Restaurants, 11. Miscellaneous, 12. Education
Values in parenthesis are t-ratios unless described otherwise.
Table 1(b) Model with Demographic and Income Distribution Indices:
Calculated Elasticities and Tests for Homogeneity
Commodity
Group
Own Price
Coefficient
Price
Elasticity
Income
Coefficient
Income
elasticity
Wald Test for
Homogeneity
Food
0.0689
(7.408)
-0.1143
-0.0871
(-10.927)
0.3846
4.5665
(pval=0.0326)
Alcohol
0.0444
(7.978)
-0.0543
-0.0238
(-4.877)
0.5298
21.1594
(pval=0.0000)
Clothing
-0.0146
(-1.493)
-1.0288
-0.0279
(-2.368)
0.5993
7.3692
(pval=0.0066)
Housing
0.1619
(14.227)
0.3393
-0.0891
(-7.565)
0.4744
14.5503
(pval=0.0001)
Furnishings
0.0485
(3.047)
-0.2343
-0.0075
(-0.611)
0.8830
9.0068
(pval=0.0027)
Health
0.0031
(1.079)
-0.6875
-0.0077
(-4.379)
0.3881
0.2693
(pval=0.6038)
Transport
0.0692
(2.136)
0.2135
0.1351
(6.601)
1.9285
0.3300
(pval=0.5657)
Communication
0.0128
(11.730)
-0.3111
-0.0034
(-1.168)
0.8203
2.9156
(pval=0.0877)
Recreation
0.0077
(0.520)
-1.0078
0.0365
(2.831)
1.3439
10.5459
(pval=0.0012)
Restaurants
0.0361
(1.151)
-0.6175
0.0395
(2.638)
1.3562
1.8633
(pval=0.1722)
Miscellaneous
-0.0034
(-0.180)
-0.7068
0.0610
(4.423)
1.6043
0.2752
(pval=0.5998)
Values in parenthesis are t-ratios unless described otherwise.
Elasticities are computed at sample means.
Table 2(a) FM-OLS Estimates of Standard AID Model
Dependent
Variable
Explanatory Variables
lnp1
lnp2
lnp3
lnp4
lnp5
lnp6
lnp7
lnp8
w1
0.0705
(6.570)
0.0180
(2.371)
0.0079
(0.816)
-0.0446
(-4.051)
-0.0252
(-0.995)
-0.0034
(-0.345)
0.0457
(4.676)
-0.0051
(-1.588)
w2
-0.0009
(-0.102)
0.0434
(8.155)
-0.0077
(-1.518)
-0.0215
(-4.021)
-0.0038
(-0.365)
-0.0047
(-0.607)
0.0151
(2.694)
w3
-0.0071
(-0.466)
0.0072
(0.741)
0.0002
(0.021)
-0.0160
(-1.284)
0.0490
(1.861)
0.0026
(0.199)
w4
-0.0176
(-1.086)
0.0022
(0.174)
0.0031
(0.277)
0.1682
(15.355)
-0.0497
(-2.004)
w5
-0.0179
(-1.444)
-0.0205
(-2.330)
-0.0036
(-0.422)
-0.0098
(-0.918)
w6
-0.0067
(-2.184)
-0.0029
(-1.360)
0.0019
(0.717)
w7
-0.0427
(-1.555)
-0.0474
(-2.655)
w8
-0.0041
(-1.365)
w9
Diagnostics
lnp9
lnp10
lnp11
lnp12
lnz
-0.0486
(-4.121)
0.0477
(2.255)
-0.0956
(-5.532)
0.0171
(2.501)
-0.0982
(-11.149)
12.2275
(pval=0.3468)
0.0008
(0.396)
-0.0038
(-0.584)
0.0196
(1.703)
-0.0441
(-4.401)
-0.0008
(-0.133)
-0.0267
(-5.427)
23.3376
(pval=0.0158)
-0.0488
(-3.847)
-0.0204
(-3.184)
0.0074
(0.552)
-0.0117
(-0.417)
0.0289
(1.557)
0.0040
(0.373)
-0.0219
(-1.866)
39.0480
(pval=0.0001)
-0.0059
(-0.529)
-0.0166
(-1.164)
0.0081
(2.066)
-0.0008
(-0.062)
0.0174
(0.651)
-0.0439
(-1.456)
-0.0486
(-4.995)
-0.0812
(-7.333)
21.4968
(pval=0.0286)
0.0404
(2.606)
-0.0069
(-0.631)
-0.0183
(-2.119)
-0.0108
(-3.630)
-0.0166
(-2.036)
-0.0030
(-0.164)
0.0470
(3.003)
0.0117
(1.811)
-0.0168
(-1.683)
18.9457
(pval=0.0621)
0.0041
(1.686)
-0.0043
(-0.853)
0.0044
(1.419)
-0.0120
(-5.719)
0.0014
(1.427)
0.0044
(1.990)
0.0185
(3.448)
-0.0065
(-1.532)
-0.0006
(-0.282)
-0.0034
(-1.742)
28.9733
(pval=0.0023)
-0.0808
(-3.970)
-0.0134
(-0.482)
0.0219
(0.530)
-0.0667
(-2.774)
0.1181
(5.086)
0.0119
(1.636)
0.0652
(2.786)
-0.0931
(-2.357)
0.1593
(3.865)
-0.0367
(-2.615)
0.1084
(5.663)
9.3604
(pval=0.5887)
-0.0008
(-0.353)
-0.0061
(-3.002)
-0.0031
(-1.126)
-0.0069
(-1.044)
-0.0051
(-1.151)
0.0104
(2.362)
0.0097
(6.269)
-0.0212
(-7.200)
-0.0003
(-0.058)
0.0260
(6.967)
-0.0006
(-0.273)
-0.0074
(-2.669)
26.1796
(pval=0.0061)
0.0167
(1.005)
-0.0040
(-0.387)
-0.0062
(-0.396)
-0.0191
(-2.022)
-0.0065
(-0.209)
0.0279
(1.494)
-0.0012
(-0.069)
-0.0222
(-3.002)
0.0352
(1.831)
-0.0610
(-2.347)
0.0202
(0.845)
0.0245
(2.081)
0.0157
(1.115)
29.9284
(pval=0.0016)
w10
0.0318
(1.429)
-0.0141
(-1.090)
-0.0177
(-1.372)
-0.0340
(-2.213)
-0.0254
(-1.135)
0.0087
(0.607)
-0.0037
(-0.273)
0.0099
(1.993)
0.0425
(2.364)
0.0388
(1.572)
-0.0436
(-2.132)
0.0071
(0.704)
0.0569
(4.022)
16.2489
(pval=0.1321)
w11
-0.0206
(-1.044)
0.0071
(0.523)
0.0815
(5.987)
-0.0072
(-0.484)
0.0404
(1.258)
0.0325
(1.707)
-0.0997
(-5.534)
0.0186
(3.126)
-0.0523
(-3.228)
0.0203
(0.680)
-0.0238
(-0.884)
0.0193
(1.538)
0.0879
(6.169)
44.1668
(pval=0.0000)
Commodity groups are: 1. Food, 2. Alcohol, 3. Clothing, 4. Housing, 5. Furnishings, 6. Health, 7. Transport, 8. Communication, 9. Recreation, 10. Restaurants, 11. Miscellaneous
Values in parenthesis are t-ratios unless described otherwise.
Box-Ljung (11)
Table 2(b) Standard AID Model:
Calculated Elasticities and Tests for Homogeneity
Commodity
Group
Own Price
Coefficient
Price
Elasticity
Income
Coefficient
Income
elasticity
Wald Test for
Homogeneity
Food
0.0705
(6.570)
-0.4033
-0.0982
(-11.149)
0.3061
36.7202
(pval=0.0000)
Alcohol
0.0434
(8.155)
-0.1179
-0.0267
(-5.427)
0.4729
35.5385
(pval=0.0000)
Clothing
0.0002
(0.021)
-0.9745
-0.0219
(-1.866)
0.6847
1.8488
(pval=0.1739)
Housing
0.1682
(15.355)
0.0736
-0.0812
(-7.333)
0.5210
26.8111
(pval=0.0000)
Furnishings
0.0404
(2.606)
-0.3485
-0.0168
(-1.683)
0.7362
12.4880
(pval=0.0004)
Health
0.0044
(1.419)
-0.6419
-0.0034
(-1.742)
0.7261
12.7657
(pval=0.0004)
Transport
0.1181
(5.086)
-0.2968
0.1084
(5.663)
1.7453
0.7324
(pval=0.3921)
Communication
0.0097
(6.269)
-0.4810
-0.0074
(-2.669)
0.6076
4.0599
(pval=0.0439)
Recreation
0.0352
(1.831)
-0.6842
0.0157
(1.115)
1.1483
1.0657
(pval=0.3019)
Restaurants
0.0388
(1.572)
-0.7072
0.0569
(4.022)
1.5135
0.0108
(pval=0.9172)
Miscellaneous
-0.0238
(-0.884)
-1.3240
0.0879
(6.169)
1.8712
17.6208
(pval=0.0000)
Values in parenthesis are t-ratios unless described otherwise.
Elasticities are computed at sample means.
Table 3(a) ML Estimates of System with Demographic and Income Distribution
Indices (homogeneity normalised on the coefficient on lnp5)
Dependent
Variable
Explanatory variables
Diagnostics
Box-Ljung
(11)
lnp1
lnp2
lnp3
lnp4
lnp5
lnz
lnk
I θj
w1
0.0976
(8.200)
0.2585
(21.528)
0.0670
(10.259)
0.0574
(5.155)
-0.4805
-
0.0360
(3.015)
0.3297
(5.465)
-9.9604
(-7.886)
10.6223
(pval=0.4754)
w2
-0.0298
(-8.061)
0.0588
(15.112)
0.0142
(6.326)
0.0144
(3.640)
-0.0574
-
-0.0385
(-9.893)
0.0254
(1.999)
19.3200
(3.823)
8.2307
(pval=0.6925)
w3
0.0208
(2.820)
0.0893
(11.380)
0.0080
(1.762)
-0.0003
(-0.044)
-0.1178
-
-0.0078
(-0.992)
-0.0249
(-0.677)
5.0357
(1.556)
14.3766
(pval=0.2129)
w4
-0.1522
(-10.932)
-0.9309
(-61.919
-0.1689
(-18.714
-0.1610
(-9.842)
1.4130
-
-0.2662
(-7.066)
-0.6757
(-2.665)
-65.9580
(-7.113)
20.7203
(pval=0.0364)
Commodity groups are: 1. Food, 2. Alcohol, 3. Clothing, 4. Transport, 5. All Other Commodities.
Values in parenthesis are t-ratios unless described otherwise.
Table 3(b) System with Demographic and Income Distribution Indices:
Calculated Elasticities ((homogeneity normalised on the coefficient on lnp5))
Commodity Group
Own PriceCoefficient
Price Elasticity
Income Coefficient
Income elasticity
Food
0.0976
(8.200)
-0.3463
0.0360
(3.015)
1.2548
Alcohol
0.0588
(15.112)
0.1970
-0.0385
(-9.893)
0.2403
Clothing
0.0080
(1.762)
-0.8773
-0.0078
(-0.992)
0.8874
Transport
-0.1610
(-9.842)
-1.8404
-0.2662
(-7.066)
-0.8295
Values in parenthesis are t-ratios unless described otherwise.
Elasticities are computed at sample means.
Table 4(a) ML Estimates of Standard AID System for Commodities Food, Alcohol,
Clothing and Transport (Homogeneity Imposed)
Dependent
Variable
Explanatory Variables
Diagnostics
lnp1
lnp2
lnp3
lnp4
lnp5
lnz
Box-Ljung
(11)
w1
1.1227
(2.337)
1.4310
(3.611)
-0.0439
(-0.141)
-0.3943
(-1.092)
-2.1154
-
-0.0696
(-0.245)
17.0470
(pval=0.1065)
w2
-0.0678
(-2.375)
0.0099
(0.419)
0.0137
(0.742)
0.0619
(2.883)
-0.0177
-
-0.0332
(-1.963)
20.2141
(pval=0.0425)
w3
-0.0385
(-1.179)
0.0403
(1.494)
-0.0024
(-0.113)
-0.0140
(-0.570)
0.0147
-
-0.0447
(-2.311)
35.9968
(pval=0.0002)
w4
-3.3413
(-2.261)
-4.1340
(-3.391)
0.1956
(0.204)
1.6789
(1.512)
5.6008
-
0.0504
(0.058)
47.1792
(pval=0.0000)
Commodity groups are: 1. Food, 2. Alcohol, 3. Clothing, 4. Transport, 5. All Other Commodities.
Values in parenthesis are t-ratios unless described otherwise.
Table 4(b) Standard AID System: Calculated Elasticities for Food, Alcohol,
Clothing and Transport (Homogeneity Imposed)
Commodity Group
Own PriceCoefficient
Price Elasticity
Income Coefficient
Income elasticity
Food
1.1227
(2.337)
7.0062
-0.0696
(-0.245)
0.5078
Alcohol
0.0099
(0.419)
-0.7719
-0.0332
(-1.963)
0.3457
Clothing
-0.0024
(-0.113)
-0.9897
-0.0447
(-2.311)
0.3575
Transport
1.6789
(1.512)
10.4894
0.0504
(0.058)
1.3463
Values in parenthesis are t-ratios unless described otherwise.
Elasticities are computed at sample means.