Revealed Preference and Expansion Paths Suppose we have a discrete price distribution, f p ( 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, f p ( 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, f p ( 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, f p ( 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, f p ( 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 p 0 ? Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 7 / 37
Revealed Preference and Expansion Paths Suppose we have a discrete price distribution, f p ( 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 p 0 ? 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 ( p t , q t ) satisfy GARP if q t R q s implies p s q s � p s q t 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 ( p t , q t ) satisfy GARP if q t R q s implies p s q s � p s q t � if q t is indirectly revealed preferred to q s then q s is not strictly preferred to q t 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 ( p t , q t ) satisfy GARP if q t R q s implies p s q s � p s q t � if q t is indirectly revealed preferred to q s then q s is not strictly preferred to q t 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 x v , x w g = f p 0 x t ) , p 0 x u ) , p 0 f ˜ x s , ˜ x t , ˜ x u , ... ˜ s q t ( ˜ t q u ( ˜ v q w ( ˜ x w ) , x w 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 x v , x w g = f p 0 x t ) , p 0 x u ) , p 0 f ˜ x s , ˜ x t , ˜ x u , ... ˜ s q t ( ˜ t q u ( ˜ v q w ( ˜ x w ) , x w g Proposition (BBC, 2003) Suppose that the sequence f q s ( x s ) , q t ( x t ) , q u ( x u ) ..., q v ( x v ) , q w ( x w ) 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, f p ( 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, f p ( 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, f p ( 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 , q t ( 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, f p ( 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 , q t ( 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 f q 1 , q 2 , ... q T g which record the choices made by a particular consumer ( ε ) when faced by the set of prices f p 1 , p 2 , ... p T 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 f q 1 , q 2 , ... q T g which record the choices made by a particular consumer ( ε ) when faced by the set of prices f p 1 , p 2 , ... p T g . What is the support set for a new price vector p 0 with new total outlay x 0 ? Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 12 / 37
Support Sets and Bounds on Demand Responses: Suppose we observe a set of demands f q 1 , q 2 , ... q T g which record the choices made by a particular consumer ( ε ) when faced by the set of prices f p 1 , p 2 , ... p T g . What is the support set for a new price vector p 0 with new total outlay x 0 ? Varian support set for d ( p 0 , x 0 , ε ) is given by: � � p 0 0 q 0 = x 0 , q 0 � 0 and S V ( p 0 , x 0 , ε ) = q 0 : . f p t , q t g t = 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 f q 1 , q 2 , ... q T g which record the choices made by a particular consumer ( ε ) when faced by the set of prices f p 1 , p 2 , ... p T g . What is the support set for a new price vector p 0 with new total outlay x 0 ? Varian support set for d ( p 0 , x 0 , ε ) is given by: � � p 0 0 q 0 = x 0 , q 0 � 0 and S V ( p 0 , x 0 , ε ) = q 0 : . f p t , q t g t = 0 ... T satis…es RP In general, support set will only deliver set identi…cation of d ( x , p 0 , 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 f q 1 , q 2 , ... q T g which record the choices made by a particular consumer ( ε ) when faced by the set of prices f p 1 , p 2 , ... p T g . What is the support set for a new price vector p 0 with new total outlay x 0 ? Varian support set for d ( p 0 , x 0 , ε ) is given by: � � p 0 0 q 0 = x 0 , q 0 � 0 and S V ( p 0 , x 0 , ε ) = q 0 : . f p t , q t g t = 0 ... T satis…es RP In general, support set will only deliver set identi…cation of d ( x , p 0 , ε ) . Figure 2(a) - generating a support set: S V ( p 0 , x 0 , 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 ( p 0 , x 0 , ε ) ? Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 13 / 37
e-Bounds on Demand Responses Can we improve upon S V ( p 0 , x 0 , ε ) ? Yes! We can do better if we know the expansion paths f p t , q t ( x , ε ) g t = 1 ,.. T . Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 13 / 37
e-Bounds on Demand Responses Can we improve upon S V ( p 0 , x 0 , ε ) ? Yes! We can do better if we know the expansion paths f p t , q t ( x , ε ) g t = 1 ,.. T . For consumer ε : De…ne intersection demands e q t ( ε ) = q t ( ˜ x t , ε ) by p 0 0 q t ( ˜ x t , ε ) = x 0 Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 13 / 37
e-Bounds on Demand Responses Can we improve upon S V ( p 0 , x 0 , ε ) ? Yes! We can do better if we know the expansion paths f p t , q t ( x , ε ) g t = 1 ,.. T . For consumer ε : De…ne intersection demands e q t ( ε ) = q t ( ˜ x t , ε ) by p 0 0 q t ( ˜ x t , ε ) = x 0 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 : � � q 0 � 0 , p 0 0 q 0 = x 0 S ( p 0 , x 0 , ε ) = q 0 : f p 0 , p t ; q 0 , e q t ( ε ) g t = 1 ,..., T satisfy RP Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 13 / 37
e-Bounds on Demand Responses Can we improve upon S V ( p 0 , x 0 , ε ) ? Yes! We can do better if we know the expansion paths f p t , q t ( x , ε ) g t = 1 ,.. T . For consumer ε : De…ne intersection demands e q t ( ε ) = q t ( ˜ x t , ε ) by p 0 0 q t ( ˜ x t , ε ) = x 0 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 : � � q 0 � 0 , p 0 0 q 0 = x 0 S ( p 0 , x 0 , ε ) = q 0 : f p 0 , p t ; q 0 , e q t ( ε ) g t = 1 ,..., T satisfy RP By utilizing the information in intersection demands, S ( p 0 , x 0 , 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 ( p 0 , x 0 , ε ) ? Yes! We can do better if we know the expansion paths f p t , q t ( x , ε ) g t = 1 ,.. T . For consumer ε : De…ne intersection demands e q t ( ε ) = q t ( ˜ x t , ε ) by p 0 0 q t ( ˜ x t , ε ) = x 0 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 : � � q 0 � 0 , p 0 0 q 0 = x 0 S ( p 0 , x 0 , ε ) = q 0 : f p 0 , p t ; q 0 , e q t ( ε ) g t = 1 ,..., T satisfy RP By utilizing the information in intersection demands, S ( p 0 , x 0 , 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 ( p 0 , x 0 , ε ) the identi…ed set of demand responses for p 0 , x 0 , 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, q i ( t ) = ( q 1 , i ( t ) , q 2 , i ( t )) = consumer i ’s demand, x i ( 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 ) = ( d 1 ( x ( t ) , t , ε ) , d 2 ( 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 ) = ( d 1 ( x ( t ) , t , ε ) , d 2 ( x ( t ) , t , ε )) . Given t , d 1 ( 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 d 1 ( 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 d 1 ( x , t , ε ) is invertible in ε and is continuously di¤erentiable in ( x , ε ) . Identi…cation Result: d 1 ( x , t , τ ) is identi…ed as the τ th quantile of q 1 j x ( t ) : d 1 ( x , t , τ ) = F � 1 q 1 ( t ) j x ( t ) ( τ j x ) . 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 d 1 ( x , t , ε ) is invertible in ε and is continuously di¤erentiable in ( x , ε ) . Identi…cation Result: d 1 ( x , t , τ ) is identi…ed as the τ th quantile of q 1 j x ( t ) : d 1 ( x , t , τ ) = F � 1 q 1 ( t ) j x ( t ) ( τ j x ) . Thus, we can employ standard nonparametric quantile regression techniques to estimate d 1 . 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 f y < 0 g � 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 f y < 0 g � τ ) y , τ 2 [ 0 , 1 ] , be the check function used in quantile estimation. The budget constraint de…nes the path for d 2 . We let D be the set of feasible demand functions, � � d � 0 : d 1 2 D 1 , d 2 ( x , t , τ ) = x � p 1 ( t ) d 1 ( x , t , ε ( t )) D = . p 2 ( t ) Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 17 / 37
Unrestricted Demand Estimation Let ( q i ( t ) , x i ( t )) , i = 1 , ..., n , t = 1 , ..., T , be i.i.d. observations from a demand system, q i ( t ) = ( q 1 i ( t ) , q 2 i ( t )) 0 . Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 18 / 37
Unrestricted Demand Estimation Let ( q i ( t ) , x i ( t )) , i = 1 , ..., n , t = 1 , ..., T , be i.i.d. observations from a demand system, q i ( t ) = ( q 1 i ( t ) , q 2 i ( t )) 0 . We then estimate d ( t , � , τ ) by n 1 ^ ∑ d ( � , t , τ ) = arg min ρ τ ( q 1 i ( t ) � d 1 n ( x i ( t ))) , t = 1 , ..., T , n d n 2D n i = 1 where D n is a sieve space ( D n ! D as n ! ∞ ). Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 18 / 37
Unrestricted Demand Estimation Let ( q i ( t ) , x i ( t )) , i = 1 , ..., n , t = 1 , ..., T , be i.i.d. observations from a demand system, q i ( t ) = ( q 1 i ( t ) , q 2 i ( t )) 0 . We then estimate d ( t , � , τ ) by n 1 ^ ∑ d ( � , t , τ ) = arg min ρ τ ( q 1 i ( t ) � d 1 n ( x i ( t ))) , t = 1 , ..., T , n d n 2D n i = 1 where D n is a sieve space ( D n ! D as n ! ∞ ). Let B i ( t ) = ( B k ( x i ( t )) : k 2 K n ) 2 R jK n j denote basis functions spanning the sieve D n . Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 18 / 37
Unrestricted Demand Estimation Let ( q i ( t ) , x i ( t )) , i = 1 , ..., n , t = 1 , ..., T , be i.i.d. observations from a demand system, q i ( t ) = ( q 1 i ( t ) , q 2 i ( t )) 0 . We then estimate d ( t , � , τ ) by n 1 ^ ∑ d ( � , t , τ ) = arg min ρ τ ( q 1 i ( t ) � d 1 n ( x i ( t ))) , t = 1 , ..., T , n d n 2D n i = 1 where D n is a sieve space ( D n ! D as n ! ∞ ). Let B i ( t ) = ( B k ( x i ( t )) : k 2 K n ) 2 R jK n j denote basis functions spanning the sieve D n . Then ˆ d 1 ( x , t , τ ) = ∑ k 2K n ˆ π k ( t , τ ) B k ( x ) , where ˆ π k ( t , τ ) is a standard linear quantile regression estimator: n � � 1 q 1 i ( t ) � π 0 B i ( t ) ∑ π ( t , τ ) = arg ˆ min ρ τ , t = 1 , ..., T . n π 2 R jK n j i = 1 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: � n � m / ( 2 m + 1 ) � jj ^ d ( � , t , τ ) � d ( � , t , τ ) jj 2 = O P , � ˆ � ! d N ( 0 , 1 ) , p n Σ � 1 / 2 ( x , τ ) d 1 ( x , t , τ ) � d 1 ( x , t , τ ) n where Σ n ( 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: D T C , n = D T n \ f d n ( � , � , τ ) satis…es RP g . 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: D T C , n = D T n \ f d n ( � , � , τ ) satis…es RP g . We de…ne the constrained estimator by: T n 1 ^ ∑ ∑ d C ( � , � , τ ) = arg min ρ τ ( q 1 , i ( t ) � d 1 , n ( t , x i ( t ))) . n d n ( � , � , τ ) 2D T t = 1 i = 1 C , n 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: D T C , n = D T n \ f d n ( � , � , τ ) satis…es RP g . We de…ne the constrained estimator by: T n 1 ^ ∑ ∑ d C ( � , � , τ ) = arg min ρ τ ( q 1 , i ( t ) � d 1 , n ( t , x i ( t ))) . n d n ( � , � , τ ) 2D T t = 1 i = 1 C , n 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 ^ d C will be on the boundary of D T 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 ^ d C will be on the boundary of D T 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 ^ d C will be on the boundary of D T 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 D T 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 ^ d C will be on the boundary of D T 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 D T 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 D T C , n ( ε ) � D T C , n . Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 21 / 37
Under assumptions A1-A3 and that d 0 2 D T C , then for any ǫ > 0: � � C ( � , t , τ ) � d 0 ( � , t , τ ) jj ∞ = O P ( k n / p n ) + O P jj ^ d ǫ k � m , 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 d 0 2 D T C , then for any ǫ > 0: � � C ( � , t , τ ) � d 0 ( � , t , τ ) jj ∞ = O P ( k n / p n ) + O P jj ^ d ǫ k � m , 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 d 0 2 D T C , then for any ǫ > 0: � � C ( � , t , τ ) � d 0 ( � , t , τ ) jj ∞ = O P ( k n / p n ) + O P jj ^ d ǫ k � m , 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. Demand bounds, τ = 0.50 true 160 95% conf. band 140 120 100 demand, food 80 60 40 20 0 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 price, food 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 160 95% conf. band 140 120 100 demand, food 80 60 40 20 0 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 price, food 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 S p 0 , x 0 denote the set of demand sequences that are rational given prices and income: � � 9 V > 0 , λ � 1 : q 2 B T S p 0 , x 0 = p 0 , x 0 : . 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 S p 0 , x 0 : q � q k 2 ρ n ( b q , S p 0 , x 0 ) : = inf k b , ˆ W test q 2 S p 0 , x 0 n where k�k ˆ is a weighted Euclidean norm, W test n T q ( t ) � q ( t )) 0 ˆ q � q k 2 W test ∑ k b = ( b ( t ) ( b q ( t ) � q ( t )) . ˆ W test n n t = 1 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 S p 0 , x 0 : q � q k 2 ρ n ( b q , S p 0 , x 0 ) : = inf k b , ˆ W test q 2 S p 0 , x 0 n where k�k ˆ is a weighted Euclidean norm, W test n T q ( t ) � q ( t )) 0 ˆ q � q k 2 W test ∑ k b = ( b ( t ) ( b q ( t ) � q ( t )) . ˆ W test n n t = 1 Distribution under null: Using Andrews (1999,2001), k λ � Z k 2 , q , S p 0 , x 0 ) ! d ρ ( Z , Λ p 0 , x 0 ) : = ρ n ( b inf λ 2 Λ p 0 , x 0 where Λ p 0 , x 0 is a cone that locally approximates S p 0 , x 0 and Z � N ( 0 , I T ) . 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 p 0 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 p 0 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 p 0 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 p 0 . 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 p 0 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 p 0 . 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 p 0 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 p 0 . 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 q n π j ( t , τ ) ( log x ) j ∑ log d 1 , n ( log x , t , τ ) = j = 0 r n π q n + k ( t , τ ) ( log x � ν k ( t )) q n ∑ + + , k = 1 where q n � 1 is the order of the polynomial and ν k , k = 1 , ..., r n , are the knots. Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 27 / 37
Estimation In the estimation, we use log-transforms and polynomial splines q n π j ( t , τ ) ( log x ) j ∑ log d 1 , n ( log x , t , τ ) = j = 0 r n π q n + k ( t , τ ) ( log x � ν k ( t )) q n ∑ + + , k = 1 where q n � 1 is the order of the polynomial and ν k , k = 1 , ..., r n , 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 4.2 τ = 0.1 τ = 0.5 4 τ = 0.9 3.8 95% CIs 3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 3.8 4 4.2 4.4 4.6 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 4.2 τ = 0.1 τ = 0.5 4 τ = 0.9 3.8 95% CIs 3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 3.8 4 4.2 4.4 4.6 4.8 5 5.2 Figure: Constrained demand function estimates, t = 1983. Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 29 / 37
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