Stochastic Demand and Revealed Preference Richard Blundell Dennis - - PowerPoint PPT Presentation

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

• f 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

• f 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

• f 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

• f 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: p0q = 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: p0q = x for all prices and total outlays x 2 R. ε 2 RJ1, 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: p0q = x for all prices and total outlays x 2 R. ε 2 RJ1, 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: p0q = x for all prices and total outlays x 2 R. ε 2 RJ1, 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: p0q = x for all prices and total outlays x 2 R. ε 2 RJ1, 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

• f 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

• f 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

• f 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

• f 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

• f 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

• f 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 qtRqs implies psqs psqt

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 qtRqs implies psqs psqt 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 qtRqs implies psqs psqt if qt is indirectly revealed preferred to qs then qs is not strictly preferred to qt 9 a well behaved concave utility function the data satisfy GARP

Blundell, Kristensen and Matzkin ( ) Stochastic Demand 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 f˜ xs, ˜ xt, ˜ xu, ...˜ xv, xw g = fp0

sqt(˜

xt), p0

tqu(˜

xu), p0

vqw (˜

xw ), 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 f˜ xs, ˜ xt, ˜ xu, ...˜ xv, xw g = fp0

sqt(˜

xt), p0

tqu(˜

xu), p0

vqw (˜

xw ), 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

• f 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

• f 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

• f 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

• utlay 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

• utlay x0?

Varian support set for d (p0, x0, ε) is given by: SV (p0, x0, ε) =

• q0 :

p0

0q0 = x0, q0 0 and

fpt, qtgt=0...T satis…es RP

• .

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

• utlay x0?

Varian support set for d (p0, x0, ε) is given by: SV (p0, x0, ε) =

• q0 :

p0

0q0 = x0, q0 0 and

fpt, qtgt=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

• utlay x0?

Varian support set for d (p0, x0, ε) is given by: SV (p0, x0, ε) =

• q0 :

p0

0q0 = x0, q0 0 and

fpt, qtgt=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: SV (p0, x0, ε) for consumer

• f type ε

Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 12 / 37

e-Bounds on Demand Responses

Can we improve upon SV (p0, x0, ε)?

Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 13 / 37

e-Bounds on Demand Responses

Can we improve upon SV (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 SV (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 e qt (ε) = qt (˜ xt, ε) by p0

0qt (˜

xt, ε) = x0

Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 13 / 37

e-Bounds on Demand Responses

Can we improve upon SV (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 e qt (ε) = qt (˜ xt, ε) by p0

0qt (˜

xt, 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, p0

0q0 = x0

fp0, pt; q0, e qt (ε)gt=1,...,T satisfy RP

• Blundell, Kristensen and Matzkin ( )

Stochastic Demand November 2010 13 / 37

e-Bounds on Demand Responses

Can we improve upon SV (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 e qt (ε) = qt (˜ xt, ε) by p0

0qt (˜

xt, 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, p0

0q0 = x0

fp0, pt; q0, e qt (ε)gt=1,...,T satisfy RP

• 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 SV (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 e qt (ε) = qt (˜ xt, ε) by p0

0qt (˜

xt, 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, p0

0q0 = x0

fp0, pt; q0, e qt (ε)gt=1,...,T satisfy RP

• 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 q1jx(t): d1 (x, t, τ) = F 1

q1(t)jx(t) (τjx) .

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 q1jx(t): d1 (x, t, τ) = F 1

q1(t)jx(t) (τjx) .

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 ρτ (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 ρτ (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) = x p1 (t) d1 (x, t, ε (t))

p2 (t)

• .

Blundell, Kristensen and Matzkin ( ) Stochastic Demand 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 ^ d (, t, τ) = arg min

dn2Dn

1 n

n

∑

i=1

ρτ (q1i (t) 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 ^ d (, t, τ) = arg min

dn2Dn

1 n

n

∑

i=1

ρτ (q1i (t) 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 RjKnj 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 ^ d (, t, τ) = arg min

dn2Dn

1 n

n

∑

i=1

ρτ (q1i (t) 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 RjKnj denote basis functions spanning the sieve Dn. Then ˆ d1 (x, t, τ) = ∑k2Kn ˆ πk (t, τ) Bk (x), where ˆ πk (t, τ) is a standard linear quantile regression estimator: ˆ π (t, τ) = arg min

π2RjKnj

1 n

n

∑

i=1

ρτ

• q1i (t) π0Bi (t)
• ,

t = 1, ..., T.

Blundell, Kristensen and Matzkin ( ) Stochastic Demand 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) d (, t, τ) jj2 = OP

• nm/(2m+1)

, pnΣ1/2

n

(x, τ) ˆ d1 (x, t, τ) 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: DT

C ,n = DT n \ 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: DT

C ,n = DT n \ fdn (, , τ) satis…es RPg .

We de…ne the constrained estimator by: ^ dC (, , τ) = arg min

dn(,,τ)2DT

C ,n

1 n

T

∑

t=1 n

∑

i=1

ρτ (q1,i (t) d1,n (t, xi (t))) .

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: DT

C ,n = DT n \ fdn (, , τ) satis…es RPg .

We de…ne the constrained estimator by: ^ dC (, , τ) = arg min

dn(,,τ)2DT

C ,n

1 n

T

∑

t=1 n

∑

i=1

ρτ (q1,i (t) 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 DT

C ,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 DT

C ,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 DT

C ,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 DT

C ,n (ε) as the set of demand functions

satisfying x (t) p (t)0 d (x (s) , s, τ) + ǫ, s < t, t = 2, ..., T, for some ("small") ǫ 0.

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 DT

C ,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 DT

C ,n (ε) as the set of demand functions

satisfying x (t) p (t)0 d (x (s) , s, τ) + ǫ, s < t, t = 2, ..., T, for some ("small") ǫ 0. Rede…ne the constrained estimator to be the optimizer over DT

C ,n (ε) DT C ,n.

Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 21 / 37

Under assumptions A1-A3 and that d0 2 DT

C , then for any ǫ > 0:

jj^ dǫ

C (, t, τ) d0 (, t, τ) jj∞ = OP(kn/pn) + OP

• km

n

• ,

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 DT

C , then for any ǫ > 0:

jj^ dǫ

C (, t, τ) d0 (, t, τ) jj∞ = OP(kn/pn) + OP

• km

n

• ,

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 DT

C , then for any ǫ > 0:

jj^ dǫ

C (, t, τ) d0 (, t, τ) jj∞ = OP(kn/pn) + OP

• km

n

• ,

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.

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 20 40 60 80 100 120 140 160 Demand bounds, τ = 0.50 price, food demand, food true 95% conf. band

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.

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 20 40 60 80 100 120 140 160 Demand bounds, τ = 0.50 price, food demand, food true 95% conf. band

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 BT

p0,x0 :

9V > 0, λ 1 : V (t) V (s) λ (t) p (t)0 (q (s) q (t))

• .

Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 24 / 37

Testing for Rationality

Test statistic: Given the vector of unrestricted estimated intersection demands, b q, we compute its distance from Sp0,x0: ρn (b q,Sp0,x0) := inf

q2Sp0,x0

kb qqk2

ˆ W test

n

, where kk ˆ

W test

n

is a weighted Euclidean norm, kb qqk2

ˆ W test

n

=

T

∑

t=1

(b q (t) q (t))0 ˆ W test

n

(t) (b q (t) q (t)) .

Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 25 / 37

Testing for Rationality

Test statistic: Given the vector of unrestricted estimated intersection demands, b q, we compute its distance from Sp0,x0: ρn (b q,Sp0,x0) := inf

q2Sp0,x0

kb qqk2

ˆ W test

n

, where kk ˆ

W test

n

is a weighted Euclidean norm, kb qqk2

ˆ W test

n

=

T

∑

t=1

(b q (t) q (t))0 ˆ W test

n

(t) (b q (t) q (t)) . Distribution under null: Using Andrews (1999,2001), ρn (b q,Sp0,x0) !d ρ (Z,Λp0,x0) := inf

λ2Λp0,x0

kλ Zk2 , 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 log d1,n(log x, t, τ) =

qn

∑

j=0

πj (t, τ) (log x)j +

rn

∑

k=1

πqn+k (t, τ) (log x νk (t))qn

+ ,

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 log d1,n(log x, t, τ) =

qn

∑

j=0

πj (t, τ) (log x)j +

rn

∑

k=1

πqn+k (t, τ) (log x νk (t))qn

+ ,

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

3.8 4 4.2 4.4 4.6 4.8 5 5.2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2

τ = 0.1 τ = 0.5 τ = 0.9

95% CIs

Figure: Unconstrained demand function estimates, t = 1983.

Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 28 / 37

RP Restricted Engel Curves

3.8 4 4.2 4.4 4.6 4.8 5 5.2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2

τ = 0.1 τ = 0.5 τ = 0.9

95% CIs

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

0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 20 40 60 80 100 120 140 160 180 200 T = 4 T = 6 T = 8

Figure: Demand bounds at median income, τ = 0.1.

Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 31 / 37

Demand Bounds Estimation

0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 20 40 60 80 100 120 140 160 180 200 T = 4 T = 6 T = 8

Figure: Demand bounds at median income, τ = 0.5.

Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 32 / 37

Demand Bounds Estimation

0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 20 40 60 80 100 120 140 160 180 200 T = 4 T = 6 T = 8

Figure: Demand bounds at median income, τ = 0.9.

Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 33 / 37

Demand Bounds Estimation

0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 50 100 150 200 250 T = 4 T = 6 T = 8

Figure: Demand bounds at 25th percentile income, τ = 0.5.

Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 34 / 37

Demand Bounds Estimation

0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 50 100 150 200 250 T = 4 T = 6 T = 8

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)

• r 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)

• r 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