Stochastic Demand and Revealed Preference
Richard Blundell
Dennis Kristensen
Rosa Matzkin
UCL & IFS, Columbia and UCLA
November 2010
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
1 / 37
Introduction
This presentation investigates the role of restrictions from economic
theory in the microeconometric estimation of nonparametric models
of consumer behaviour.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
2 / 37
Introduction
This presentation investigates the role of restrictions from economic
theory in the microeconometric estimation of nonparametric models
of consumer behaviour.
Objective is to uncover demand responses from consumer expenditure
survey data.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
2 / 37
Introduction
This presentation investigates the role of restrictions from economic
theory in the microeconometric estimation of nonparametric models
of consumer behaviour.
Objective is to uncover demand responses from consumer expenditure
survey data.
Inequality restrictions from revealed preference are used to improve
the performance of nonparametric estimates of demand responses.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
2 / 37
Introduction
This presentation investigates the role of restrictions from economic
theory in the microeconometric estimation of nonparametric models
of consumer behaviour.
Objective is to uncover demand responses from consumer expenditure
survey data.
Inequality restrictions from revealed preference are used to improve
the performance of nonparametric estimates of demand responses.
Particular attention is given to nonseparable unobserved heterogeneity
and endogeneity.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
2 / 37
Introduction
New insights are provided about the price responsiveness of demand
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
3 / 37
Introduction
New insights are provided about the price responsiveness of demand
especially across di¤erent income groups and across unobserved
heterogeneity.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
3 / 37
Introduction
New insights are provided about the price responsiveness of demand
especially across di¤erent income groups and across unobserved
heterogeneity.
Derive welfare costs of relative price and tax changes.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
3 / 37
The Problem
Assume every consumer is characterised by observed and unobserved
heterogeneity (h, ε) and responds to a given budget (p,x ), with a
unique, positive J-vector of demands
q = d(x, p, h, ε),
where demand functions d(x, p, h, ε) : RK++ ! RJ++ satisfy
adding-up: p0 q = x for all prices and total outlays x 2 R.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
4 / 37
The Problem
Assume every consumer is characterised by observed and unobserved
heterogeneity (h, ε) and responds to a given budget (p,x ), with a
unique, positive J-vector of demands
q = d(x, p, h, ε),
where demand functions d(x, p, h, ε) : RK++ ! RJ++ satisfy
adding-up: p0 q = x for all prices and total outlays x 2 R.
ε 2 RJ
1,
J
1 vector of (non-separable) unobservable heterogeneity.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
4 / 37
The Problem
Assume every consumer is characterised by observed and unobserved
heterogeneity (h, ε) and responds to a given budget (p,x ), with a
unique, positive J-vector of demands
q = d(x, p, h, ε),
where demand functions d(x, p, h, ε) : RK++ ! RJ++ satisfy
adding-up: p0 q = x for all prices and total outlays x 2 R.
ε 2 RJ
1,
J
1 vector of (non-separable) unobservable heterogeneity.
Assume ε? (x, h) for now.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
4 / 37
The Problem
Assume every consumer is characterised by observed and unobserved
heterogeneity (h, ε) and responds to a given budget (p,x ), with a
unique, positive J-vector of demands
q = d(x, p, h, ε),
where demand functions d(x, p, h, ε) : RK++ ! RJ++ satisfy
adding-up: p0 q = x for all prices and total outlays x 2 R.
ε 2 RJ
1,
J
1 vector of (non-separable) unobservable heterogeneity.
Assume ε? (x, h) for now.
The environment is described by a continuous distribution of q, x and
ε, for discrete types h.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
4 / 37
The Problem
Assume every consumer is characterised by observed and unobserved
heterogeneity (h, ε) and responds to a given budget (p,x ), with a
unique, positive J-vector of demands
q = d(x, p, h, ε),
where demand functions d(x, p, h, ε) : RK++ ! RJ++ satisfy
adding-up: p0 q = x for all prices and total outlays x 2 R.
ε 2 RJ
1,
J
1 vector of (non-separable) unobservable heterogeneity.
Assume ε? (x, h) for now.
The environment is described by a continuous distribution of q, x and
ε, for discrete types h.
Will typically suppress observable heterogeneity h in what follows.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
4 / 37
Non-Separable Demand
For demands q = d(x, p, ε):
One key drawback has been the (additive) separability of ε assumed
in empirical speci…cations.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
5 / 37
Non-Separable Demand
For demands q = d(x, p, ε):
One key drawback has been the (additive) separability of ε assumed
in empirical speci…cations.
We will consider the non-separable case and impose conditions on
preferences that ensure invertibility in ε (which corresponds to
monotonicity for J = 2 which is our leading case). Assume unique
inverse structural demand functions exists - Fig 1a.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
5 / 37
Non-Separable Demand
For demands q = d(x, p, ε):
One key drawback has been the (additive) separability of ε assumed
in empirical speci…cations.
We will consider the non-separable case and impose conditions on
preferences that ensure invertibility in ε (which corresponds to
monotonicity for J = 2 which is our leading case). Assume unique
inverse structural demand functions exists - Fig 1a.
Here we consider the case of a small number of price regimes and use
revealed preference inequalities applied to d(x, p, ε) to improve
demand predictions
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
5 / 37
Non-Separable Demand
For demands q = d(x, p, ε):
One key drawback has been the (additive) separability of ε assumed
in empirical speci…cations.
We will consider the non-separable case and impose conditions on
preferences that ensure invertibility in ε (which corresponds to
monotonicity for J = 2 which is our leading case). Assume unique
inverse structural demand functions exists - Fig 1a.
Here we consider the case of a small number of price regimes and use
revealed preference inequalities applied to d(x, p, ε) to improve
demand predictions
In other related work Slutsky inequality conditions have been shown
to help in ‘smoothing’demands for ‘dense’or continuously distributed
prices
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
5 / 37
Related Literature
Nonparametric Demand Estimation: Blundell, Browning and
Crawford (2003, 2007, 2008); Blundell, Chen and Kristensen (2007);
Chen and Pouzo (2009).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
6 / 37
Related Literature
Nonparametric Demand Estimation: Blundell, Browning and
Crawford (2003, 2007, 2008); Blundell, Chen and Kristensen (2007);
Chen and Pouzo (2009).
Nonseparable models: Chernozhukov, Imbens and Newey (2007);
Imbens and Newey (2009); Matzkin (2003, 2007, 2008), Chen and
Laio (2010).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
6 / 37
Related Literature
Nonparametric Demand Estimation: Blundell, Browning and
Crawford (2003, 2007, 2008); Blundell, Chen and Kristensen (2007);
Chen and Pouzo (2009).
Nonseparable models: Chernozhukov, Imbens and Newey (2007);
Imbens and Newey (2009); Matzkin (2003, 2007, 2008), Chen and
Laio (2010).
Restricted Nonparametric Estimation: Blundell, Horowitz and
Parey (2010); Haag, Hoderlein and Pendakur (2009), Kiefer (1982),
Mammen, Marron, Turlach and Wand (2001); Mammen and
Thomas-Agnan (1999); Wright (1981,1984)
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
6 / 37
Related Literature
Nonparametric Demand Estimation: Blundell, Browning and
Crawford (2003, 2007, 2008); Blundell, Chen and Kristensen (2007);
Chen and Pouzo (2009).
Nonseparable models: Chernozhukov, Imbens and Newey (2007);
Imbens and Newey (2009); Matzkin (2003, 2007, 2008), Chen and
Laio (2010).
Restricted Nonparametric Estimation: Blundell, Horowitz and
Parey (2010); Haag, Hoderlein and Pendakur (2009), Kiefer (1982),
Mammen, Marron, Turlach and Wand (2001); Mammen and
Thomas-Agnan (1999); Wright (1981,1984)
Inequality constraints and set identi…cation: Andrews (1999,
2001); Andrews and Guggenberger (2007), Andrews and Soares
(2009); Bugni (2009); Chernozhukov, Hong and Tamer (2007)....
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
6 / 37
Revealed Preference and Expansion Paths
Suppose we have a discrete price distribution, fp (1) , p (2) , ...p (T )g.
Observe choices of large number of consumers for a small (…nite) set
of prices - e.g. limited number of markets/time periods
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
7 / 37
Revealed Preference and Expansion Paths
Suppose we have a discrete price distribution, fp (1) , p (2) , ...p (T )g.
Observe choices of large number of consumers for a small (…nite) set
of prices - e.g. limited number of markets/time periods
Market de…ned by time and/or location.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
7 / 37
Revealed Preference and Expansion Paths
Suppose we have a discrete price distribution, fp (1) , p (2) , ...p (T )g.
Observe choices of large number of consumers for a small (…nite) set
of prices - e.g. limited number of markets/time periods
Market de…ned by time and/or location.
Questions to address here:
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
7 / 37
Revealed Preference and Expansion Paths
Suppose we have a discrete price distribution, fp (1) , p (2) , ...p (T )g.
Observe choices of large number of consumers for a small (…nite) set
of prices - e.g. limited number of markets/time periods
Market de…ned by time and/or location.
Questions to address here:
How do we devise a powerful test of RP conditions in this
environment?
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
7 / 37
Revealed Preference and Expansion Paths
Suppose we have a discrete price distribution, fp (1) , p (2) , ...p (T )g.
Observe choices of large number of consumers for a small (…nite) set
of prices - e.g. limited number of markets/time periods
Market de…ned by time and/or location.
Questions to address here:
How do we devise a powerful test of RP conditions in this
environment?
How do we estimate demands for some new price point p0 ?
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
7 / 37
Revealed Preference and Expansion Paths
Suppose we have a discrete price distribution, fp (1) , p (2) , ...p (T )g.
Observe choices of large number of consumers for a small (…nite) set
of prices - e.g. limited number of markets/time periods
Market de…ned by time and/or location.
Questions to address here:
How do we devise a powerful test of RP conditions in this
environment?
How do we estimate demands for some new price point p0 ?
In this case Revealed Preference conditions, in general, only allow
set identi…cation of demands.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
7 / 37
Revealed Preference and Expansion Paths
How do we devise a powerful test of RP?
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
8 / 37
Revealed Preference and Expansion Paths
How do we devise a powerful test of RP?
Afriat’s Theorem
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
8 / 37
Revealed Preference and Expansion Paths
How do we devise a powerful test of RP?
Afriat’s Theorem
Data (pt , qt ) satisfy GARP if qt Rqs implies ps qs
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
ps qt
November 2010
8 / 37
Revealed Preference and Expansion Paths
How do we devise a powerful test of RP?
Afriat’s Theorem
Data (pt , qt ) satisfy GARP if qt Rqs implies ps qs
ps qt
if qt is indirectly revealed preferred to qs then qs is not
strictly preferred to qt
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
8 / 37
Revealed Preference and Expansion Paths
How do we devise a powerful test of RP?
Afriat’s Theorem
Data (pt , qt ) satisfy GARP if qt Rqs implies ps qs
ps qt
if qt is indirectly revealed preferred to qs then qs is not
strictly preferred to qt
9 a well behaved concave utility function
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
the data satisfy GARP
November 2010
8 / 37
Revealed Preference and Expansion Paths
Data: Observational or Experimental - Is there a best design for
experimental data?
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
9 / 37
Revealed Preference and Expansion Paths
Data: Observational or Experimental - Is there a best design for
experimental data?
Blundell, Browning and Crawford (Ecta, 2003) develop a method for
choosing a sequence of total expenditures that maximise the power of
tests of RP (GARP).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
9 / 37
Revealed Preference and Expansion Paths
Data: Observational or Experimental - Is there a best design for
experimental data?
Blundell, Browning and Crawford (Ecta, 2003) develop a method for
choosing a sequence of total expenditures that maximise the power of
tests of RP (GARP).
De…ne sequential maximum power (SMP) path
fx̃s , x̃t , x̃u , ...x̃v , xw g = fps0 qt (x̃t ), pt0 qu (x̃u ), pv0 qw (x̃w ), xw g
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
9 / 37
Revealed Preference and Expansion Paths
Data: Observational or Experimental - Is there a best design for
experimental data?
Blundell, Browning and Crawford (Ecta, 2003) develop a method for
choosing a sequence of total expenditures that maximise the power of
tests of RP (GARP).
De…ne sequential maximum power (SMP) path
fx̃s , x̃t , x̃u , ...x̃v , xw g = fps0 qt (x̃t ), pt0 qu (x̃u ), pv0 qw (x̃w ), xw g
Proposition (BBC, 2003) Suppose that the sequence
fqs (xs ) , qt (xt ) , qu (xu ) ..., qv (xv ) , qw (xw )g
rejects RP. Then SMP path also rejects RP. (Also de…ne Revealed
Worse and Revealed Best sets.)
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
9 / 37
Revealed Preference and Expansion Paths
- great for experimental design but we have Observational Data
continuous micro-data on incomes and expenditures
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
10 / 37
Revealed Preference and Expansion Paths
- great for experimental design but we have Observational Data
continuous micro-data on incomes and expenditures
…nite set of observed price and/or tax regimes (across time and
markets)
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
10 / 37
Revealed Preference and Expansion Paths
- great for experimental design but we have Observational Data
continuous micro-data on incomes and expenditures
…nite set of observed price and/or tax regimes (across time and
markets)
discrete demographic di¤erences across households
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
10 / 37
Revealed Preference and Expansion Paths
- great for experimental design but we have Observational Data
continuous micro-data on incomes and expenditures
…nite set of observed price and/or tax regimes (across time and
markets)
discrete demographic di¤erences across households
use this information alone, together with revealed preference theory to
assess consumer rationality and to place ‘tight’bounds on demand
responses and welfare measures.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
10 / 37
Revealed Preference and Expansion Paths
So, is there a best design for observational data?
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
11 / 37
Revealed Preference and Expansion Paths
So, is there a best design for observational data?
Suppose we have a discrete price distribution,
fp (1) , p (2) , ...p (T )g.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
11 / 37
Revealed Preference and Expansion Paths
So, is there a best design for observational data?
Suppose we have a discrete price distribution,
fp (1) , p (2) , ...p (T )g.
Observe choices of large number of consumers for a small (…nite) set
of prices - e.g. limited number of markets/time periods. Market
de…ned by time and/or location.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
11 / 37
Revealed Preference and Expansion Paths
So, is there a best design for observational data?
Suppose we have a discrete price distribution,
fp (1) , p (2) , ...p (T )g.
Observe choices of large number of consumers for a small (…nite) set
of prices - e.g. limited number of markets/time periods. Market
de…ned by time and/or location.
Given t, qt (x; ε) = d(x, p(t ), ε) is the (quantile) expansion path of
consumer type ε facing prices p(t ).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
11 / 37
Revealed Preference and Expansion Paths
So, is there a best design for observational data?
Suppose we have a discrete price distribution,
fp (1) , p (2) , ...p (T )g.
Observe choices of large number of consumers for a small (…nite) set
of prices - e.g. limited number of markets/time periods. Market
de…ned by time and/or location.
Given t, qt (x; ε) = d(x, p(t ), ε) is the (quantile) expansion path of
consumer type ε facing prices p(t ).
Fig 1b
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
11 / 37
Support Sets and Bounds on Demand Responses:
Suppose we observe a set of demands fq1 , q2 , ...qT g which record the
choices made by a particular consumer (ε) when faced by the set of
prices fp1 , p2 , ...pT g .
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
12 / 37
Support Sets and Bounds on Demand Responses:
Suppose we observe a set of demands fq1 , q2 , ...qT g which record the
choices made by a particular consumer (ε) when faced by the set of
prices fp1 , p2 , ...pT g .
What is the support set for a new price vector p0 with new total
outlay x0 ?
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
12 / 37
Support Sets and Bounds on Demand Responses:
Suppose we observe a set of demands fq1 , q2 , ...qT g which record the
choices made by a particular consumer (ε) when faced by the set of
prices fp1 , p2 , ...pT g .
What is the support set for a new price vector p0 with new total
outlay x0 ?
Varian support set for d (p0 , x0 , ε) is given by:
S V (p0 , x0 , ε) =
Blundell, Kristensen and Matzkin ( )
q0 :
p00 q0 = x0 , q0 0 and
fpt , qt gt =0...T satis…es RP
Stochastic Demand
.
November 2010
12 / 37
Support Sets and Bounds on Demand Responses:
Suppose we observe a set of demands fq1 , q2 , ...qT g which record the
choices made by a particular consumer (ε) when faced by the set of
prices fp1 , p2 , ...pT g .
What is the support set for a new price vector p0 with new total
outlay x0 ?
Varian support set for d (p0 , x0 , ε) is given by:
S V (p0 , x0 , ε) =
q0 :
p00 q0 = x0 , q0 0 and
fpt , qt gt =0...T satis…es RP
.
In general, support set will only deliver set identi…cation of
d(x, p0 , ε).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
12 / 37
Support Sets and Bounds on Demand Responses:
Suppose we observe a set of demands fq1 , q2 , ...qT g which record the
choices made by a particular consumer (ε) when faced by the set of
prices fp1 , p2 , ...pT g .
What is the support set for a new price vector p0 with new total
outlay x0 ?
Varian support set for d (p0 , x0 , ε) is given by:
S V (p0 , x0 , ε) =
q0 :
p00 q0 = x0 , q0 0 and
fpt , qt gt =0...T satis…es RP
.
In general, support set will only deliver set identi…cation of
d(x, p0 , ε).
Figure 2(a) - generating a support set: S V (p0 , x0 , ε) for consumer
of type ε
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
12 / 37
e-Bounds on Demand Responses
Can we improve upon S V (p0 , x0 , ε)?
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
13 / 37
e-Bounds on Demand Responses
Can we improve upon S V (p0 , x0 , ε)?
Yes! We can do better if we know the expansion paths
fpt , qt (x, ε)gt =1,..T .
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
13 / 37
e-Bounds on Demand Responses
Can we improve upon S V (p0 , x0 , ε)?
Yes! We can do better if we know the expansion paths
fpt , qt (x, ε)gt =1,..T .
For consumer ε: De…ne intersection demands q
et (ε) = qt (x̃t , ε) by
0
p0 qt (x̃t , ε) = x0
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
13 / 37
e-Bounds on Demand Responses
Can we improve upon S V (p0 , x0 , ε)?
Yes! We can do better if we know the expansion paths
fpt , qt (x, ε)gt =1,..T .
For consumer ε: De…ne intersection demands q
et (ε) = qt (x̃t , ε) by
0
p0 qt (x̃t , ε) = x0
Blundell, Browning and Crawford (2008): The set of points that are
consistent with observed expansion paths and revealed preference is
given by the support set:
S (p0 , x0 , ε) =
Blundell, Kristensen and Matzkin ( )
q0 :
q0 0, p00 q0 = x0
et (ε)gt =1,...,T satisfy RP
fp0 , pt ; q0 , q
Stochastic Demand
November 2010
13 / 37
e-Bounds on Demand Responses
Can we improve upon S V (p0 , x0 , ε)?
Yes! We can do better if we know the expansion paths
fpt , qt (x, ε)gt =1,..T .
For consumer ε: De…ne intersection demands q
et (ε) = qt (x̃t , ε) by
0
p0 qt (x̃t , ε) = x0
Blundell, Browning and Crawford (2008): The set of points that are
consistent with observed expansion paths and revealed preference is
given by the support set:
S (p0 , x0 , ε) =
q0 :
q0 0, p00 q0 = x0
et (ε)gt =1,...,T satisfy RP
fp0 , pt ; q0 , q
By utilizing the information in intersection demands, S (p0 , x0 , ε)
yields tighter bounds on demands.
These are sharp in the case of 2 goods. (BBC, 2003, for RW bounds
for the many goods case).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
13 / 37
e-Bounds on Demand Responses
Can we improve upon S V (p0 , x0 , ε)?
Yes! We can do better if we know the expansion paths
fpt , qt (x, ε)gt =1,..T .
For consumer ε: De…ne intersection demands q
et (ε) = qt (x̃t , ε) by
0
p0 qt (x̃t , ε) = x0
Blundell, Browning and Crawford (2008): The set of points that are
consistent with observed expansion paths and revealed preference is
given by the support set:
S (p0 , x0 , ε) =
q0 :
q0 0, p00 q0 = x0
et (ε)gt =1,...,T satisfy RP
fp0 , pt ; q0 , q
By utilizing the information in intersection demands, S (p0 , x0 , ε)
yields tighter bounds on demands.
These are sharp in the case of 2 goods. (BBC, 2003, for RW bounds
for the many goods case).
Figure 2b, c - S (p0 , x0 , ε) the identi…ed set of demand responses for
p0 , x0 , ε given t = 1, ..., T .
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
13 / 37
Unrestricted Demand Estimation
Observational setting: At time t (t = 1, ...,T ), we observe a random
sample of n consumers facing prices p (t ).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
14 / 37
Unrestricted Demand Estimation
Observational setting: At time t (t = 1, ...,T ), we observe a random
sample of n consumers facing prices p (t ).
Observed variables (ignoring other observed characteristics of
consumers):
p (t ) = prices that all consumers face,
qi (t ) = (q1,i (t ) , q2,i (t )) = consumer i’s demand,
xi (t ) = consumer i’s income (total budget)
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
14 / 37
Unrestricted Demand Estimation
We …rst wish to recover demands for each of the observed price
regimes t,
q (t ) = d(x (t ), t, ε), t = 1, ..., T ,
where d is the demand function in price regime p (t ).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
15 / 37
Unrestricted Demand Estimation
We …rst wish to recover demands for each of the observed price
regimes t,
q (t ) = d(x (t ), t, ε), t = 1, ..., T ,
where d is the demand function in price regime p (t ).
We will here only discuss the case of 2 goods with 1-dimensional error:
ε 2 R,
d(x (t ), t, ε) = (d1 (x (t ), t, ε), d2 (x (t ), t, ε)) .
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
15 / 37
Unrestricted Demand Estimation
We …rst wish to recover demands for each of the observed price
regimes t,
q (t ) = d(x (t ), t, ε), t = 1, ..., T ,
where d is the demand function in price regime p (t ).
We will here only discuss the case of 2 goods with 1-dimensional error:
ε 2 R,
d(x (t ), t, ε) = (d1 (x (t ), t, ε), d2 (x (t ), t, ε)) .
Given t, d1 (x (t ), t, ε) is exactly the quantile expansion path (Engel
curve) for good 1 at prices p (t ).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
15 / 37
Unrestricted Demand Estimation
Assumption A.1: The variable x (t ) has bounded support, x (t )
2 X = [a, b ] for ∞ < a < b < +∞, and is independent of
ε U [0, 1].
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
16 / 37
Unrestricted Demand Estimation
Assumption A.1: The variable x (t ) has bounded support, x (t )
2 X = [a, b ] for ∞ < a < b < +∞, and is independent of
ε U [0, 1].
Assumption A.2: The demand function d1 (x, t, ε) is invertible in ε
and is continuously di¤erentiable in (x, ε).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
16 / 37
Unrestricted Demand Estimation
Assumption A.1: The variable x (t ) has bounded support, x (t )
2 X = [a, b ] for ∞ < a < b < +∞, and is independent of
ε U [0, 1].
Assumption A.2: The demand function d1 (x, t, ε) is invertible in ε
and is continuously di¤erentiable in (x, ε).
Identi…cation Result: d1 (x, t, τ ) is identi…ed as the τth quantile of
q1 jx (t ):
d1 (x, t, τ ) = Fq1 1(t )jx (t ) (τ jx ) .
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
16 / 37
Unrestricted Demand Estimation
Assumption A.1: The variable x (t ) has bounded support, x (t )
2 X = [a, b ] for ∞ < a < b < +∞, and is independent of
ε U [0, 1].
Assumption A.2: The demand function d1 (x, t, ε) is invertible in ε
and is continuously di¤erentiable in (x, ε).
Identi…cation Result: d1 (x, t, τ ) is identi…ed as the τth quantile of
q1 jx (t ):
d1 (x, t, τ ) = Fq1 1(t )jx (t ) (τ jx ) .
Thus, we can employ standard nonparametric quantile regression
techniques to estimate d1 .
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
16 / 37
Unrestricted Demand Estimation
We propose to estimate d using sieve methods.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
17 / 37
Unrestricted Demand Estimation
We propose to estimate d using sieve methods.
Let
ρ τ (y ) = (I fy < 0g
τ ) y , τ 2 [0, 1] ,
be the check function used in quantile estimation.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
17 / 37
Unrestricted Demand Estimation
We propose to estimate d using sieve methods.
Let
ρ τ (y ) = (I fy < 0g
τ ) y , τ 2 [0, 1] ,
be the check function used in quantile estimation.
The budget constraint de…nes the path for d2 . We let D be the set of
feasible demand functions,
D= d
0 : d1 2 D1 , d2 (x, t, τ ) =
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
x
p1 (t ) d1 (x, t, ε (t ))
p2 (t )
November 2010
.
17 / 37
Unrestricted Demand Estimation
Let (qi (t ) , xi (t )), i = 1, ..., n, t = 1, ..., T , be i.i.d. observations
from a demand system, qi (t ) = (q1i (t ) , q2i (t ))0 .
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
18 / 37
Unrestricted Demand Estimation
Let (qi (t ) , xi (t )), i = 1, ..., n, t = 1, ..., T , be i.i.d. observations
from a demand system, qi (t ) = (q1i (t ) , q2i (t ))0 .
We then estimate d (t, , τ ) by
1 n
^
d ( , t, τ ) = arg min
∑ ρτ (q1i (t )
d n 2Dn n i =1
d1n (xi (t ))) ,
t = 1, ..., T ,
where Dn is a sieve space (Dn ! D as n ! ∞).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
18 / 37
Unrestricted Demand Estimation
Let (qi (t ) , xi (t )), i = 1, ..., n, t = 1, ..., T , be i.i.d. observations
from a demand system, qi (t ) = (q1i (t ) , q2i (t ))0 .
We then estimate d (t, , τ ) by
1 n
^
d ( , t, τ ) = arg min
∑ ρτ (q1i (t )
d n 2Dn n i =1
d1n (xi (t ))) ,
t = 1, ..., T ,
where Dn is a sieve space (Dn ! D as n ! ∞).
Let Bi (t ) = (Bk (xi (t )) : k 2 Kn ) 2 RjKn j denote basis functions
spanning the sieve Dn .
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
18 / 37
Unrestricted Demand Estimation
Let (qi (t ) , xi (t )), i = 1, ..., n, t = 1, ..., T , be i.i.d. observations
from a demand system, qi (t ) = (q1i (t ) , q2i (t ))0 .
We then estimate d (t, , τ ) by
1 n
^
d ( , t, τ ) = arg min
∑ ρτ (q1i (t )
d n 2Dn n i =1
d1n (xi (t ))) ,
t = 1, ..., T ,
where Dn is a sieve space (Dn ! D as n ! ∞).
Let Bi (t ) = (Bk (xi (t )) : k 2 Kn ) 2 RjKn j denote basis functions
spanning the sieve Dn .
Then d̂1 (x, t, τ ) = ∑k 2Kn π̂ k (t, τ ) Bk (x ), where π̂ k (t, τ ) is a
standard linear quantile regression estimator:
π̂ (t, τ ) = arg min
π 2RjKn j
Blundell, Kristensen and Matzkin ( )
1 n
ρτ q1i (t )
n i∑
=1
Stochastic Demand
π 0 Bi ( t ) ,
t = 1, ..., T .
November 2010
18 / 37
Unrestricted Demand Estimation
Adapt results in Belloni, Chen, Chernozhukov and Liao (2010) for
rates and asymptotic distribution of the linear sieve estimator:
jj^
d ( , t, τ )
p
d ( , t, τ ) jj2 = OP n
nΣn 1/2 (x, τ ) d̂1 (x, t, τ )
m/(2m +1 )
,
d1 (x, t, τ ) !d N (0, 1) ,
where Σn (x, τ ) ! ∞ is an appropriate chosen weighting matrix.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
19 / 37
RP-restricted Demand Estimation
No reason why estimated expansion paths for a sequence of prices
t = 1, ..., T should satisfy RP.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
20 / 37
RP-restricted Demand Estimation
No reason why estimated expansion paths for a sequence of prices
t = 1, ..., T should satisfy RP.
In order to impose the RP restrictions, we simply de…ne the
constrained sieve as: DCT ,n = DnT \ fdn ( , , τ ) satis…es RPg .
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
20 / 37
RP-restricted Demand Estimation
No reason why estimated expansion paths for a sequence of prices
t = 1, ..., T should satisfy RP.
In order to impose the RP restrictions, we simply de…ne the
constrained sieve as: DCT ,n = DnT \ fdn ( , , τ ) satis…es RPg .
We de…ne the constrained estimator by:
^
dC ( , , τ ) = arg
Blundell, Kristensen and Matzkin ( )
min
dn ( , ,τ )2DCT ,n
1 T n
∑ ρτ (q1,i (t )
n t∑
=1 i =1
Stochastic Demand
d1,n (t, xi (t ))) .
November 2010
20 / 37
RP-restricted Demand Estimation
No reason why estimated expansion paths for a sequence of prices
t = 1, ..., T should satisfy RP.
In order to impose the RP restrictions, we simply de…ne the
constrained sieve as: DCT ,n = DnT \ fdn ( , , τ ) satis…es RPg .
We de…ne the constrained estimator by:
^
dC ( , , τ ) = arg
min
dn ( , ,τ )2DCT ,n
1 T n
∑ ρτ (q1,i (t )
n t∑
=1 i =1
d1,n (t, xi (t ))) .
Since RP imposes restrictions across t, the above estimation problem
can no longer be split up into T individual sub problems as the
unconstrained case.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
20 / 37
RP-restricted Demand Estimation
Theoretical properties of restricted estimator: In general, the RP
restrictions will be binding. This means that ^
dC will be on the
boundary of DCT ,n . So the estimator will in general have non-standard
distribution (estimation when parameter is on the boundary).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
21 / 37
RP-restricted Demand Estimation
Theoretical properties of restricted estimator: In general, the RP
restrictions will be binding. This means that ^
dC will be on the
boundary of DCT ,n . So the estimator will in general have non-standard
distribution (estimation when parameter is on the boundary).
Too hard a problem for us....
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
21 / 37
RP-restricted Demand Estimation
Theoretical properties of restricted estimator: In general, the RP
restrictions will be binding. This means that ^
dC will be on the
boundary of DCT ,n . So the estimator will in general have non-standard
distribution (estimation when parameter is on the boundary).
Too hard a problem for us....
Instead: We introduce DCT ,n (ε) as the set of demand functions
satisfying
x (t )
p (t )0 d (x (s ) , s, τ ) + e,
for some ("small") e
Blundell, Kristensen and Matzkin ( )
s < t,
t = 2, ..., T ,
0.
Stochastic Demand
November 2010
21 / 37
RP-restricted Demand Estimation
Theoretical properties of restricted estimator: In general, the RP
restrictions will be binding. This means that ^
dC will be on the
boundary of DCT ,n . So the estimator will in general have non-standard
distribution (estimation when parameter is on the boundary).
Too hard a problem for us....
Instead: We introduce DCT ,n (ε) as the set of demand functions
satisfying
x (t )
p (t )0 d (x (s ) , s, τ ) + e,
for some ("small") e
s < t,
t = 2, ..., T ,
0.
Rede…ne the constrained estimator to be the optimizer over
DCT ,n (ε) DCT ,n .
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
21 / 37
Under assumptions A1-A3 and that d0 2 DCT , then for any e > 0:
p
jj^
dCe ( , t, τ ) d0 ( , t, τ ) jj∞ = OP (kn / n) + OP kn m ,
for t = 1, ..., T . Moreover, the restricted estimator has the same
asymptotic distribution as the unrestricted estimator.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
22 / 37
Under assumptions A1-A3 and that d0 2 DCT , then for any e > 0:
p
jj^
dCe ( , t, τ ) d0 ( , t, τ ) jj∞ = OP (kn / n) + OP kn m ,
for t = 1, ..., T . Moreover, the restricted estimator has the same
asymptotic distribution as the unrestricted estimator.
Also derive convergence rates and valid con…dence sets for the
support sets.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
22 / 37
Under assumptions A1-A3 and that d0 2 DCT , then for any e > 0:
p
jj^
dCe ( , t, τ ) d0 ( , t, τ ) jj∞ = OP (kn / n) + OP kn m ,
for t = 1, ..., T . Moreover, the restricted estimator has the same
asymptotic distribution as the unrestricted estimator.
Also derive convergence rates and valid con…dence sets for the
support sets.
In practice, use simulation methods or the modi…ed bootstrap
procedures developed in Bugni (2009, 2010) and Andrews and Soares
(2010); alternatively, the subsampling procedure of CHT.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
22 / 37
Demand Bounds Estimation
Simulation Study: Cobb-Douglas demand function.
Demand bounds, τ = 0.50
true
95% c onf. band
160
140
demand, food
120
100
80
60
40
20
0
0.93
0.94
0.95
0.96
0.97
0.98
price, food
0.99
1
1.01
1.02
Figure: Performance of demand bound estimator.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
23 / 37
Demand Bounds Estimation
Simulation Study: Cobb-Douglas demand function.
95% con…dence bands of demand bounds.
Demand bounds, τ = 0.50
true
95% c onf. band
160
140
demand, food
120
100
80
60
40
20
0
0.93
0.94
0.95
0.96
0.97
0.98
price, food
0.99
1
1.01
1.02
Figure: Performance of demand bound estimator.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
23 / 37
Testing for Rationality
Constrained demand and bounds estimators rely on the fundamental
assumption that consumers are rational.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
24 / 37
Testing for Rationality
Constrained demand and bounds estimators rely on the fundamental
assumption that consumers are rational.
We wish to test the null of consumer rationality.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
24 / 37
Testing for Rationality
Constrained demand and bounds estimators rely on the fundamental
assumption that consumers are rational.
We wish to test the null of consumer rationality.
Let Sp0 ,x0 denote the set of demand sequences that are rational given
prices and income:
Sp0 ,x0 =
q 2 BpT0 ,x0 :
Blundell, Kristensen and Matzkin ( )
V (t )
9V > 0, λ 1 :
V (s ) λ (t ) p (t ) 0 (q (s )
Stochastic Demand
q (t ))
November 2010
.
24 / 37
Testing for Rationality
Test statistic: Given the vector of unrestricted estimated
intersection demands, q
b, we compute its distance from Sp0 ,x0 :
ρn (q
b,Sp0 ,x0 ) :=
inf
q2Sp0 ,x0
b qk2Ŵ test ,
kq
n
where k kŴ ntest is a weighted Euclidean norm,
b qk2Ŵ test =
kq
n
Blundell, Kristensen and Matzkin ( )
T
∑ (qb (t )
t =1
q (t ))0 Ŵntest (t ) (q
b (t )
Stochastic Demand
q (t )) .
November 2010
25 / 37
Testing for Rationality
Test statistic: Given the vector of unrestricted estimated
intersection demands, q
b, we compute its distance from Sp0 ,x0 :
ρn (q
b,Sp0 ,x0 ) :=
inf
q2Sp0 ,x0
b qk2Ŵ test ,
kq
n
where k kŴ ntest is a weighted Euclidean norm,
b qk2Ŵ test =
kq
n
T
∑ (qb (t )
t =1
q (t ))0 Ŵntest (t ) (q
b (t )
q (t )) .
Distribution under null: Using Andrews (1999,2001),
ρn (q
b,Sp0 ,x0 ) !d ρ (Z ,Λp0 ,x0 ) :=
inf
λ2Λp0 ,x0
kλ
Z k2 ,
where Λp0 ,x0 is a cone that locally approximates Sp0 ,x0 and
Z
N (0, IT ).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
25 / 37
Estimating e-Bounds on Local Consumer Responses
For each household de…ned by (x, ε), the parameter of interest is the
consumer response at some new relative price p0 and income x or at
some sequence of relative prices. The later de…nes the demand curve
for (x, ε).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
26 / 37
Estimating e-Bounds on Local Consumer Responses
For each household de…ned by (x, ε), the parameter of interest is the
consumer response at some new relative price p0 and income x or at
some sequence of relative prices. The later de…nes the demand curve
for (x, ε).
A typical sequence of relative prices in the UK:
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
26 / 37
Estimating e-Bounds on Local Consumer Responses
For each household de…ned by (x, ε), the parameter of interest is the
consumer response at some new relative price p0 and income x or at
some sequence of relative prices. The later de…nes the demand curve
for (x, ε).
A typical sequence of relative prices in the UK:
Figure 4: Relative prices in the UK and a ‘typical’relative price
path p0 .
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
26 / 37
Estimating e-Bounds on Local Consumer Responses
For each household de…ned by (x, ε), the parameter of interest is the
consumer response at some new relative price p0 and income x or at
some sequence of relative prices. The later de…nes the demand curve
for (x, ε).
A typical sequence of relative prices in the UK:
Figure 4: Relative prices in the UK and a ‘typical’relative price
path p0 .
Figure 5: Engel Curve Share Distribution
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
26 / 37
Estimating e-Bounds on Local Consumer Responses
For each household de…ned by (x, ε), the parameter of interest is the
consumer response at some new relative price p0 and income x or at
some sequence of relative prices. The later de…nes the demand curve
for (x, ε).
A typical sequence of relative prices in the UK:
Figure 4: Relative prices in the UK and a ‘typical’relative price
path p0 .
Figure 5: Engel Curve Share Distribution
Figure 6: Density of Log Expenditure.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
26 / 37
Estimation
In the estimation, we use log-transforms and polynomial splines
qn
log d1,n (log x, t, τ ) =
∑ πj (t, τ ) (log x )j
j =0
rn
+
∑ πq +k (t, τ ) (log x
n
νk (t ))q+n ,
k =1
where qn 1 is the order of the polynomial and νk , k = 1, ..., rn , are
the knots.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
27 / 37
Estimation
In the estimation, we use log-transforms and polynomial splines
qn
log d1,n (log x, t, τ ) =
∑ πj (t, τ ) (log x )j
j =0
rn
+
∑ πq +k (t, τ ) (log x
n
νk (t ))q+n ,
k =1
where qn 1 is the order of the polynomial and νk , k = 1, ..., rn , are
the knots.
In the implementation of the quantile sieve estimator with a small
penalization term was added to the objective function, as in BCK
(2007).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
27 / 37
Unrestricted Engel Curves
Unconstr. food demand est., t=1
4.2
τ = 0.1
τ = 0.5
τ = 0.9
4
3.8
95% CIs
log-food exp.
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
3.8
4
4.2
4.4
4.6
log-total exp.
4.8
5
5.2
Figure: Unconstrained demand function estimates, t = 1983.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
28 / 37
RP Restricted Engel Curves
Constr. food demand est., t=1
4.2
τ = 0.1
τ = 0.5
τ = 0.9
4
3.8
95% CIs
log-food exp.
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
3.8
4
4.2
4.4
4.6
log-total exp.
4.8
5
5.2
Figure: Constrained demand function estimates, t = 1983.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
29 / 37
Estimation
conditional quantile splines - 3rd order pol. spline with 5 knots
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
30 / 37
Estimation
conditional quantile splines - 3rd order pol. spline with 5 knots
RP restrictions imposed at 100 x-points over the empirical support x.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
30 / 37
Estimation
conditional quantile splines - 3rd order pol. spline with 5 knots
RP restrictions imposed at 100 x-points over the empirical support x.
1983-1990 (T=8).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
30 / 37
Estimation
conditional quantile splines - 3rd order pol. spline with 5 knots
RP restrictions imposed at 100 x-points over the empirical support x.
1983-1990 (T=8).
Figures 9-11: Estimated e-Bounds on Demand Curve
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
30 / 37
Estimation
conditional quantile splines - 3rd order pol. spline with 5 knots
RP restrictions imposed at 100 x-points over the empirical support x.
1983-1990 (T=8).
Figures 9-11: Estimated e-Bounds on Demand Curve
Demand (e-)bounds (support sets) are de…ned at the quantiles of x
and ε
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
30 / 37
Estimation
conditional quantile splines - 3rd order pol. spline with 5 knots
RP restrictions imposed at 100 x-points over the empirical support x.
1983-1990 (T=8).
Figures 9-11: Estimated e-Bounds on Demand Curve
Demand (e-)bounds (support sets) are de…ned at the quantiles of x
and ε
tightest bounds given information and RP.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
30 / 37
Estimation
conditional quantile splines - 3rd order pol. spline with 5 knots
RP restrictions imposed at 100 x-points over the empirical support x.
1983-1990 (T=8).
Figures 9-11: Estimated e-Bounds on Demand Curve
Demand (e-)bounds (support sets) are de…ned at the quantiles of x
and ε
tightest bounds given information and RP.
varies with income and heterogeneity
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
30 / 37
Demand Bounds Estimation
Demand bounds, τ = 0.1
200
T= 4
180
T= 6
T= 8
160
demand, food
140
120
100
80
60
40
20
0
0.9
0.92
0.94
0.96
0.98
price, food
1
1.02
1.04
Figure: Demand bounds at median income, τ = 0.1.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
31 / 37
Demand Bounds Estimation
Demand bounds, τ = 0.5
200
T= 4
180
T= 6
T= 8
160
demand, food
140
120
100
80
60
40
20
0
0.9
0.92
0.94
0.96
0.98
price, food
1
1.02
1.04
Figure: Demand bounds at median income, τ = 0.5.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
32 / 37
Demand Bounds Estimation
Demand bounds, τ = 0.9
200
T= 4
180
T= 6
T= 8
160
demand, food
140
120
100
80
60
40
20
0
0.9
0.92
0.94
0.96
0.98
price, food
1
1.02
1.04
Figure: Demand bounds at median income, τ = 0.9.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
33 / 37
Demand Bounds Estimation
Demand bounds, τ = 0.5
T=4
T=6
T=8
250
demand, food
200
150
100
50
0
0.9
0.92
0.94
0.96
0.98
price, food
1
1.02
1.04
Figure: Demand bounds at 25th percentile income, τ = 0.5.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
34 / 37
Demand Bounds Estimation
Demand bounds, τ = 0.5
T=4
T=6
T=8
250
demand, food
200
150
100
50
0
0.9
0.92
0.94
0.96
0.98
price, food
1
1.02
1.04
Figure: Demand bounds at 75th percentile income, τ = 0.5.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
35 / 37
Endogenous Income
To account for the endogeneity of x we can utilize IV quantile
estimators developed in Chen and Pouzo (2009) and Chernozhukov,
Imbens and Newey (2007).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
36 / 37
Endogenous Income
To account for the endogeneity of x we can utilize IV quantile
estimators developed in Chen and Pouzo (2009) and Chernozhukov,
Imbens and Newey (2007).
Chen and Pouzo (2009) apply to exactly this data using the same
instrument as in BBC (2008).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
36 / 37
Endogenous Income
To account for the endogeneity of x we can utilize IV quantile
estimators developed in Chen and Pouzo (2009) and Chernozhukov,
Imbens and Newey (2007).
Chen and Pouzo (2009) apply to exactly this data using the same
instrument as in BBC (2008).
Our basic results remain valid except that the convergence rate stated
there has to be replaced by that obtained in Chen and Pouzo (2009)
or Chernozhukov, Imbens and Newey (2007).
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
36 / 37
Endogenous Income
To account for the endogeneity of x we can utilize IV quantile
estimators developed in Chen and Pouzo (2009) and Chernozhukov,
Imbens and Newey (2007).
Chen and Pouzo (2009) apply to exactly this data using the same
instrument as in BBC (2008).
Our basic results remain valid except that the convergence rate stated
there has to be replaced by that obtained in Chen and Pouzo (2009)
or Chernozhukov, Imbens and Newey (2007).
Alternatively, the control function approach taken in Imbens and
Newey (2009) can be used. Again they estimate using the exact same
data and instrument. Specify
ln x = π (z, v )
where π is monotonic in v , z are a set of instrumental variables.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
36 / 37
Summary
Objective to elicit demand responses from consumer expenditure
survey data.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
37 / 37
Summary
Objective to elicit demand responses from consumer expenditure
survey data.
Inequality restrictions from revealed preference used to produce tight
bounds on demand responses.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
37 / 37
Summary
Objective to elicit demand responses from consumer expenditure
survey data.
Inequality restrictions from revealed preference used to produce tight
bounds on demand responses.
Derive a powerful test of RP conditions.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
37 / 37
Summary
Objective to elicit demand responses from consumer expenditure
survey data.
Inequality restrictions from revealed preference used to produce tight
bounds on demand responses.
Derive a powerful test of RP conditions.
Particular attention given to nonseparable unobserved heterogeneity
and endogeneity.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
37 / 37
Summary
Objective to elicit demand responses from consumer expenditure
survey data.
Inequality restrictions from revealed preference used to produce tight
bounds on demand responses.
Derive a powerful test of RP conditions.
Particular attention given to nonseparable unobserved heterogeneity
and endogeneity.
New (empirical) insights provided about the price responsiveness of
demand, especially across di¤erent income groups.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
37 / 37
Summary
Objective to elicit demand responses from consumer expenditure
survey data.
Inequality restrictions from revealed preference used to produce tight
bounds on demand responses.
Derive a powerful test of RP conditions.
Particular attention given to nonseparable unobserved heterogeneity
and endogeneity.
New (empirical) insights provided about the price responsiveness of
demand, especially across di¤erent income groups.
Derive welfare costs of relative price and tax changes across the
distribution of demands by income and taste heterogeneity.
Blundell, Kristensen and Matzkin ( )
Stochastic Demand
November 2010
37 / 37