Hedonic Models & New Characteristics

Hedonic Models and New Characteristics
Ian Crawford
Oxford and IFS
J. Peter Neary
Oxford, CEPR and CESifo
Challenges and Opportunities in Price Measurement
The Shard
July 19, 2018
Crawford & Neary (Oxford) New Characteristics RES-ONS: July 19, 2018 1 / 41

Introduction
Outline
How to correct measured inﬂation for quality change?

Introduction
Outline
National statistical agencies, like the ONS or the BLS or Eurostat,
now routinely use “hedonic” methods to correct their measures of
inﬂation.

Introduction
Outline
inﬂation.
But the standard method does not account for changes to product
speciﬁcations on the extensive margin. This can lead to bias.

Introduction
Outline
inﬂation.
We show how to correct for it and provide an empirical example.

Introduction
Outline
inﬂation.
Our methodological innovation is to combine:

Introduction
Outline
inﬂation.
Linear characteristics model of Gorman (1956)

Introduction
Outline
inﬂation.
Linear characteristics model of Gorman (1956)
Sato (1976) - Vartia (1976) - Feenstra (1994) index number for new
goods

Introduction
Outline
1 Background Theory
2 Adjusting for New Characteristics: General
3 Adjusting for New Characteristics: A Structural Approach
4 An Application: New Cars, New Characteristics
5 Summary and Conclusion

Background Theory
Outline
1 Background Theory
Measuring Inﬂation
Hedonic Methods
The Linear Characteristics Model

Background Theory Measuring Inﬂation
The True Cost-of-Living Index
A utility-maximizing consumer:
x1
t , ..., xK
t = arg max
x1,...,xK
u(x1
, ..., xK
) ∑
k
pk
t xk
= yt

x1
t , ..., xK
t = arg max
x1,...,xK
u(x1
, ..., xK
) ∑
k
pk
t xk
= yt
The expenditure function:
c(pt, u) = min
x1,...,xK
∑
j
p
j
txj
u(x1
, ..., xK
) = u

x1
t , ..., xK
t = arg max
x1,...,xK
u(x1
, ..., xK
) ∑
k
pk
t xk
= yt
The expenditure function:
c(pt, u) = min
x1,...,xK
∑
j
p
j
txj
u(x1
, ..., xK
) = u
The constant-utility Kon¨us price index is deﬁned as:
PK(p0, p1, u) =
c(p1, u)
c(p0, u)
Characteristics New Characteristics

Observable Price Indexes
Standard index number theory then provides a number of ways of
either approximating or, subject to functional form assumptions,
exactly computing the true index.

For example:
PL = ∑
j
p
j
1
p
j
0
s
j
0 PP =

∑
j
p
j
1
p
j
0
−1
s
j
1


−1
PF = PLPP PT = ∏
j
p
j
1
p
j
0
1
2 (s
j
0+s
j
1)
where s
j
t =
p
j
tx
j
t
∑i pi
txi
t
.

For example:
PL = ∑
j
p
j
1
p
j
0
s
j
0 PP =

∑
j
p
j
1
p
j
0
−1
s
j
1


−1
PF = PLPP PT = ∏
j
p
j
1
p
j
0
1
2 (s
j
0+s
j
1)
where s
j
t =
p
j
tx
j
t
∑i pi
txi
t
.
Problem: No allowance for quality change

Background Theory Hedonic Methods
Hedonic Methods
There are two basic methods of correcting for quality change:

Hedonic Methods
1 the time dummy method - based on the repackaging model of Fisher
and Shell (1971) and Muellbauer (1974)

Hedonic Methods
1 the time dummy method - based on the repackaging model of Fisher
and Shell (1971) and Muellbauer (1974)
2 the indirect method - based on the linear characteristics model of
Gorman (1956) and Lancaster (1966).

There are K varieties of products with quantities denoted by xk and
unit prices by pk.

unit prices by pk.
These products are diﬀerentiated by J characteristics.

unit prices by pk.
These products are diﬀerentiated by J characteristics.
In the linear characteristics model the total amount of a given
characteristic present in a bundle of varieties x1
t , ..., xK
t observed in
period t is
z
j
t = ∑
k
a
k,j
t xk
t
where a
k,j
t represents the amount of characteristic j present in one
unit of product k according to the product speciﬁcation in period t.

Background Theory The Linear Characteristics Model
The consumer’s problem is:
max
x1,...,xK
v(z1
, ..., zJ
)
subject to: ∑
k
pk
t xk
= yt and zj
= ∑
k
a
k,j
t xk

max
x1,...,xK
v(z1
, ..., zJ
)
subject to: ∑
k
pk
t xk
= yt and zj
= ∑
k
a
k,j
t xk
Maximising behaviour implies the ﬁrst order conditions:
pk
t = ∑
j
a
k,j
t π
j
t ∀k, t
where π
j
t ≡ 1
ψt
v
j
t are the shadow prices of characteristics.

max
x1,...,xK
v(z1
, ..., zJ
)
subject to: ∑
k
pk
t xk
= yt and zj
= ∑
k
a
k,j
t xk
Maximising behaviour implies the ﬁrst order conditions:
pk
t = ∑
j
a
k,j
t π
j
t ∀k, t
where π
j
t ≡ 1
ψt
v
j
t are the shadow prices of characteristics.
This is the hedonic pricing equation.

The budget constraint “adds up” in both characteristics and product
space:
∑
k
pk
t xk
t = ∑
j
π
j
tz
j
t = yt

The budget constraint “adds up” in both characteristics and product
space:
∑
k
pk
t xk
t = ∑
j
π
j
tz
j
t = yt
The model can be thought of in terms of either preferences-
for-products or preferences-for-characteristics:
x1
t , ..., xK
t = arg max u(x1
, ..., xK
) ∑
k
pk
t xk
= yt
z1
t , ..., zJ
t = arg max v(z1
, ..., zJ
) ∑
j
π
j
tzj
= yt

z1
t , ..., zJ
t = arg max v(z1
, ..., zJ
) ∑
j
π
j
tzj
= yt
The hedonic expenditure function:
c(πt, v) = min
z1,...,zJ
∑
j
π
j
tz
j
t v(z1
, ..., zJ
) = v

z1
t , ..., zJ
t = arg max v(z1
, ..., zJ
) ∑
j
π
j
tzj
= yt
The hedonic expenditure function:
c(πt, v) = min
z1,...,zJ
∑
j
π
j
tz
j
t v(z1
, ..., zJ
) = v
The constant-utility Kon¨us hedonic price index is deﬁned as:
PK(π0, π1, v) =
c(π1, v)
c(π0, v)
CostOfLiving New Characteristics

All of these hedonic price indices depend on shadow price-relatives:
π
j
1
π
j
0

π
j
1
π
j
0
If a new feature is added to products in this category in period 1:
what is the denominator π
j
0?

π
j
1
π
j
0
If a new feature is added to products in this category in period 1:
what is the denominator π
j
0?
If an existing feature is withdrawn from the products in this category
in period 1: what is the numerator π
j
1?

Adjusting for New Characteristics: General
Outline
1 Background Theory

Rationing Theory
The solution comes from the theory of consumer behaviour under
rationing
(Hicks (1940), Rothbarth (1941), Neary and Roberts (1980))

Rationing Theory
rationing
Central idea: Use “virtual prices” ˜π

Rationing Theory
rationing
Prices that would induce the consumer to consume voluntarily the
constrained characteristic

Rationing Theory
rationing
Prices that would induce the consumer to consume voluntarily the
constrained characteristic
Here, the “constraint” is that a characteristic is unavailable in one
period
Skip Details

Rationing Theory
Consider the constrained hedonic cost function
˜c(πt, v, ¯z1
t ) = min
z2,...,zJ
π1
t ¯z1
t + ∑
j=2
π
j
tzj
subject to v(¯z1
, ..., zJ
) = v

Rationing Theory
Consider the constrained hedonic cost function
˜c(πt, v, ¯z1
t ) = min
z2,...,zJ
π1
t ¯z1
t + ∑
j=2
π
j
tzj
subject to v(¯z1
, ..., zJ
) = v
It satisﬁes Sheppard’s Lemma
π1 ˜c(πt, v, ¯z1
t ) = ¯z1
t
πj ˜c(πt, v, ¯z1
t ) = zj
(πt, v, ¯z1
t ) ∀j = 2, ..J

New Characteristics Bias
The relationship between the constrained and unconstrained hedonic
cost functions is given by
˜c(πt, v, ¯z1
t ) = π1
t ¯z1
t + ∑
j=2
π
j
tzj
(πt, v, ¯z1
t )
= π1
t z1
( ˜πt, v) + ∑
j=2
π
j
tzj
( ˜πt, v)
= c( ˜πt, v) + π1
t − ˜π1
t ¯z1
t
where ˜π1
t is the virtual price of the constrained characteristic and
˜πt = ˜π1
t , π2
t , ..., πJ
t is the shadow support price vector.

The shadow support prices are those which would precisely induce the
consumer to choose the constrained characteristics vector
¯z1
t = z1
( ˜πt, v)

¯z1
t = z1
( ˜πt, v)
The welfare cost of the constraint on the characteristic is
∂˜c(πt, v, ¯z1
t )
∂¯z1
t
=
∂c( ˜πt, v)
∂π1
t
∂π1
t (πt, v, ¯z1
t )
∂¯z1
t
−
∂π1
t (πt, v, ¯z1
t )
∂¯z1
t
¯z1
t + π1
t − ˜π1
t
= π1
t − ˜π1
t

¯z1
t = z1
( ˜πt, v)
The welfare cost of the constraint on the characteristic is
∂˜c(πt, v, ¯z1
t )
∂¯z1
t
=
∂c( ˜πt, v)
∂π1
t
∂π1
t (πt, v, ¯z1
t )
∂¯z1
t
−
∂π1
t (πt, v, ¯z1
t )
∂¯z1
t
¯z1
t + π1
t − ˜π1
t
= π1
t − ˜π1
t
Thus the change in welfare associated with a change in the constraint
is the diﬀerence between the shadow price of the characteristic and its
virtual counterpart.

Consider two periods t ∈ {0, 1} and suppose that the ﬁrst
characteristic is not available in the base period ¯z1
0 = 0 but is
added to the product speciﬁcation in the following period.

0 = 0 but is
Then
˜c(π0, v, ¯z1
0) = c( ˜π0, v) + π1
0 − ˜π1
t ¯z1
0
which becomes
˜c(π0, v, 0) = c( ˜π0, v).

0 = 0 but is
Then
˜c(π0, v, ¯z1
0) = c( ˜π0, v) + π1
0 − ˜π1
t ¯z1
0
which becomes
˜c(π0, v, 0) = c( ˜π0, v).
Consequently, the Kon¨us hedonic cost-of-living index can be expressed
conveniently as the ratio of unconstrained hedonic cost functions
where the base prices are given by the virtual prices:
PK( ˜π0, π1, v) =
c(π1, v)
c( ˜π0, v)
CostOfLiving Characteristics Sato-Vartia

The fact that everything can be expressed in terms of the
unconstrained cost function means that all of the standard index
number results are available.

But all of them require that the shadow price relative π1
1/ ˜π1
0 is
known.

1/ ˜π1
0 is
known.
Solutions:

1/ ˜π1
0 is
known.
Solutions:
Ignore the aﬀected characteristics and chain.

1/ ˜π1
0 is
known.
Solutions:
Estimate the model and solve 0 = z1(π, v) for ˜π1.

1/ ˜π1
0 is
known.
Solutions:
Estimate the model and solve 0 = z1(π, v) for ˜π1.
Something else ...

Adjusting for New Characteristics: A Structural Approach
Outline
1 Background Theory

The Sato-Vartia Index
To make progress, we need to impose more structure

We assume CES preferences over characteristics:
v(z) = ∑ βjz
σ−1
σ
j
σ
σ−1
= ∑ βjzθ
j
1
θ
c(π, u) = c(π)u = ∑ βσ
j π1−σ
j
1
1−σ
u

We assume CES preferences over characteristics:
v(z) = ∑ βjz
σ−1
σ
j
σ
σ−1
= ∑ βjzθ
j
1
θ
c(π, u) = c(π)u = ∑ βσ
j π1−σ
j
1
1−σ
u
where σ ≡ 1
1−θ and θ ≡ σ−1
σ .

The Sato-Vartia cost-of-living index is exact for CES preferences

This holds for preferences over characteristics as well as over goods:
PSV =
c(π1)
c(π0)
= ∏
j
πj,1
πj,0
ωj
where:
ωj =
µj
∑ µj
µj =
sj,1 − sj,0
ln sj,1 − ln sj,0

This holds for preferences over characteristics as well as over goods:
PSV =
c(π1)
c(π0)
= ∏
j
πj,1
πj,0
ωj
where:
ωj =
µj
∑ µj
µj =
sj,1 − sj,0
ln sj,1 − ln sj,0
Note that this corresponds exactly to the true hedonic index but does
not depend on unobserved coeﬃcients.
New Characteristics Sato-Vartia-Feenstra

The Sato-Vartia-Feenstra Index
Now consider two periods t ∈ {0, 1} and suppose that the set of
characteristics changes from J0 in the base period to J1 in the
comparison period.

comparison period.
The intersection J = J0 ∩ J1 = ∅ contains the set of characteristics
which are available in both periods.

comparison period.
We can show that:
PSVF =
c( ˜π1)
c( ˜π0)
=
λ1
λ0
1
σ−1
∏
j∈J
π
j
1
π
j
0
ωj
is the exact price index.

comparison period.
We can show that:
PSVF =
c( ˜π1)
c( ˜π0)
=
λ1
λ0
1
σ−1
∏
j∈J
π
j
1
π
j
0
ωj
It equals the Sato-Vartia index, times an adjustment factor

comparison period.
We can show that:
PSVF =
c( ˜π1)
c( ˜π0)
=
λ1
λ0
1
σ−1
∏
j∈J
π
j
1
π
j
0
ωj
It equals the Sato-Vartia index, times an adjustment factor
This needs an estimate of σ, but otherwise depends only on
observables.
Sato-Vartia

The adjustment factor
λ1
λ0
1
σ−1
where
λt =
∑j∈J π
j
tz
j
t
∑j∈Jt
π
j
tz
j
t
≤ 1
takes into account the changing set of characteristics on the extensive
margin.

λ1
λ0
1
σ−1
where
λt =
∑j∈J π
j
tz
j
t
∑j∈Jt
π
j
tz
j
t
≤ 1
margin.
If new characteristics are important, λ1 will tend to be small so the
hedonic price index will be lower.

λ1
λ0
1
σ−1
where
λt =
∑j∈J π
j
tz
j
t
∑j∈Jt
π
j
tz
j
t
≤ 1
margin.
If new characteristics are important, λ1 will tend to be small so the
hedonic price index will be lower.
The adjustment factor is less important the higher is the substitution
parameter σ, i.e., ignoring new characteristics matters less if they are
close substitutes for existing ones.

An Application: New Cars, New Characteristics
Outline
1 Background Theory

The Data
Data were collected by the UK Oﬃce for National Statistics from
What Car on the major new car models available from 1988 to mid
1995.

The Data
1995.
Data on market shares and sales are from the Competition
Commission.

The Data
1995.
Commission.
The data covers 36 main models.

The Data
1995.
Commission.
The data covers 36 main models.
Prices and speciﬁcations are measured quarterly (30 periods, 3160
observations)

Summary Statistics

Changes in Characteristics
Speciﬁcations generally change twice per year:

at the beginning of the fourth quarter, which marks the time when new
annual registration letters were issued in the UK;

and at the beginning of the second quarter, which normally coincides
with changes in vehicle excise duty and taxes on motor fuels.

Looking at the intensive margin ﬁrst, there were quality improvements
in terms of performance across the board:

0-60 mph acceleration times improved over the period by 6.5%

fuel economy improved by 2.8%.

fuel economy improved by 2.8%.
both torque and brake-horse-power improved by 5.2% on average
(torque and bhp are proportional at a ﬁxed engine speed).

The availability of features also improved:

central locking was present in a quarter of cars at the start of the
period; this rose to nearly two-thirds by the end,

manual sun-roofs went from being available on 17.6% of models to 25%

electric front-windows were uncommon at beginning with 8% market
share, but by the end 40% of new models had them.

electrically operated door mirrors went from 8% to 25%

electrically operated door mirrors went from 8% to 25%
split-fold rear seats increased from being speciﬁed in around half of
new cars to nearly three-quarters.

On the extensive margin there were a number of changes to the set of
characteristics over the period.

electronic seat and steering-adjustability both became available in the
fourth quarter of 1988;

electric sun-roofs were introduced in the ﬁrst quarter of 1990;

air conditioning was introduced in the fourth quarter of 1993 as were
powered (heated) seats and drivers’ airbags.

air conditioning was introduced in the fourth quarter of 1993 as were
powered (heated) seats and drivers’ airbags.
CD-players were introduced in the second quarter of 1994. By then
simple radio-only models had gone - they disappeared from the new car
market by the fourth quarter of 1992.

Other features were short-lived and simply came and went:

head-lamp cleaners were available in some cars (Volvos) at the start of
our period of observation, but disappeared at the start of 1990 only to
reappear brieﬂy from mid 1993 to mid 1994 (on Alfa Romeo’s and
Volvo’s again) before disappearing again.

Similarly early trip-computers became available (also on Alfa’s) but
disappeared only to be oﬀered brieﬂy by Rover before again
disappearing.

Similarly early trip-computers became available (also on Alfa’s) but
disappeared only to be offered briefly by Rover before again
disappearing.
Trip computers are now completely standard but they didn’t take off
until after the end of our period of observation.

The Proﬁle of New Car Prices
The density of the prices of new cars over time
5 10 15 20
0.000.050.100.150.20
£ '000's
Density

The Proﬁle of New Car Prices
The density of the prices of new cars over time
88 90 92 94
05101520
£'000's
88 89 90 91 92 93 94 95
200
200
200
200 200
400
400
400
400
400
400
600
600
600
600
600
800
800
1000
1000
1000
1200
1200
12001400
1400
1600
1600
1600
1600
1800
1800
1800
2000
2000
2000

Recap on Methodology
1 Estimate within period linear characteristics models for each period
using only extant characteristics to recover
{πt, zt}t=0,...,T

{πt, zt}t=0,...,T
2 Fit a value for σ.

{πt, zt}t=0,...,T
3 Compute the adjustment factors
{λt}t=0,...,T

{πt, zt}t=0,...,T
3 Compute the adjustment factors
{λt}t=0,...,T
4 Compute the variety-adjusted Sato-Vartia hedonic price index
c(πt)
c(π0)
=
λt
λ0
1
σ−1
∏
j∈J
πj,t
πj,0
ωj

Empirical Results: Price Levels
88 90 92 94
0.81.01.21.4
Index
88 89 90 91 92 93 94 95
Laspeyres
Paasche
Fisher
Sato−Vartia
Sato−Vartia−Feenstra
Repackaging
Notes: 1988q4: adjustable seats and adjustable steering introduced; 1990q1: electric sun roofs introduced; 1990q2: headlamp
cleaners withdrawn; 1991q1: trip computers introduced; 1992q1: trip computers withdrawn; 1992q3: trip computers
re-introduced for one quarter; 1992q4 radio-only withdrawn; 1993q2: headlamp cleaners re-introduced; 1993q4: air conditioning,
power seats and driver’s airbags introduced; 1994q2 CD players available; 1993q4: headlamp cleaners ﬁnally withdrawn.

Empirical Results: Inﬂation Rates
89 90 91 92 93 94 95
−0.3−0.2−0.10.00.1
Index
89 90 91 92 93 94 95
Laspeyres
Paasche
Fisher
Sato−Vartia
Sato−Vartia−Feenstra
Repackaging
Notes: 1988q4: adjustable seats and adjustable steering introduced; 1990q1: electric sun roofs introduced; 1990q2: headlamp
cleaners withdrawn; 1991q1: trip computers introduced; 1992q1: trip computers withdrawn; 1992q3: trip computers
re-introduced for one quarter; 1992q4 radio-only withdrawn; 1993q2: headlamp cleaners re-introduced; 1993q4: air conditioning,
power seats and driver’s airbags introduced; 1994q2 CD players available; 1993q4: headlamp cleaners ﬁnally withdrawn.

Conclusions
Without quality adjustment we see an annual average growth in prices
of new cars of about 5.37%.

Conclusions
Hedonic quality adjustment on the basis of overlapping characteristics
reduces this to an annual average growth rate of 3.45%.

Conclusions
Allowing for the changes on the extensive margin of characteristics
gives an annual average growth rate of about 2%.

Conclusions
Very roughly:

Conclusions
Very roughly:
if we take the Boskin Commission’s estimate of the bias from quality
change of 0.6 percentage points per year,

Conclusions
Very roughly:
if we take the Boskin Commission’s estimate of the bias from quality
change of 0.6 percentage points per year,
our results suggest that allowing for changes in the extensive margin
could add half as much again to that.

Summary and Conclusion
Outline
1 Background Theory

Summary
We introduce a new approach to correcting inﬂation for quality
change at the extensive margin
The theory builds on and extends:
The Gorman approach to preferences over characteristics
The Feenstra approach to correcting price indices for new goods
The method we propose is easy to apply in practice
An application to UK new car prices shows that accounting for
changes in characteristics matters:
It reduces inﬂation by 1.45% per annum
Half as much as accounting for quality change at the intensive margin

Practical Pros and Cons
The main drawback of our approach is that it is based on an
assumption.

assumption.
The generalised CES is “not-quite-superlative”, and therefore we
cannot just assume that the approximation to true preferences will be
good in the vicinity of the data (although see Hill (2006)).

assumption.
The generalised CES is “not-quite-superlative”, and therefore we
cannot just assume that the approximation to true preferences will be
good in the vicinity of the data (although see Hill (2006)).
Nonetheless it buys the ability to address the new-characteristics
problem without any of the diﬃcult econometric work involved in
solving estimated hedonic demand equations for reservation prices.

This is, we would argue, a practical plus.

This is because NSI’s are never keen to adopt regression-based
methods when producing their inﬂation data because of (entirely valid)
worries over robustness and reproducibility.

Extrapolation of estimated hedonic models/demand equations might
therefore be a step too far.

Extrapolation of estimated hedonic models/demand equations might
therefore be a step too far.
Our method has the major advantage of not requiring NSI’s to do any
econometric work over-and-above what they are doing already.

Hedonic Models & New Characteristics

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