Measuring Substitution Patterns in
Differentiated Product Industries
Amit Gandhi
UW-Madison & Microsoft
Jean-François Houde
Wharton School & NBER
May 2, 2016
Measuring Substitution Patterns
1 / 48
Twenty-One Years of BLP!
175
Over 3,300 citations and 175 applications
0
25
Number of articles
50
75
100
125
150
Cumulative
Random coefficient
1995
2000
Measuring Substitution Patterns
2005
Year
2010
Introduction
2 / 48
Motivation
The Beauty of BLP: Flexible estimation of substitution patterns
with many products, aggregate data, and unobserved attributes.
Achieving this flexibility can be difficult in practice...
I
I
Precision: Often rely on external restrictions (e.g. supply, survey, etc.)
Numerical: Multiple solutions and/or poor convergence properties
Rising skepticism concerning the identification of the model...
I
I
Angrist & Pischke: Unclear mapping from instruments to parameters.
Despite the fact that the model IS identified (Berry & Haile, 2014)
Measuring Substitution Patterns
Introduction
3 / 48
What explains the difficulties in practice?
Is the variation in data simply too weak?
Or is it weakness of the instruments (IVs)?
I
i.e., Are we using the variation in the data in the optimal way?
Our paper argue that many empiricists’ problems could be caused by
weak IVs
Show how to construct strong IVs using a new approximation to the
optimal instruments of the model.
Measuring Substitution Patterns
Introduction
4 / 48
Key Takeaways
Differentiation IV: Capture the relative position of each product in
the characteristic space
I
I
Approximate optimal IV without requiring initial estimates
Simple to construct and test
Powerful in practice:
I
I
I
10+ improvement in precision
Fast convergence + Numerically stable
Flexible substitution: Multiple dimensions + Correlated heterogeneity
Related work:
I
I
Weak IV in BLP: BLP (1999), Conlon (2013), Reynaert & Verboven
(2013), Metaxoglou and Knittel (2014)
Differentiation IV: Nested-Logit (e.g. Berry (1994), Verboven (1996),
Bresnahan et al. (1997)), and Spatial Differentiation (e.g. Davis
(2006), Thomadsen (2005), Manuszak (2012), Houde (2012))
Measuring Substitution Patterns
Introduction
5 / 48
Outline
1
Description of the Model
2
Identification Problem
3
Optimal IV
4
Illustration: Exogenous Characteristics
Experiment 1: Exogenous characteristics
Experiment 2: Correlated Heterogeneity
Alternative Approach: BLP (1999)
5
Illustration: Endogenous Characteristics
Experiment 3: Endogenous prices
Experiment 4: Natural Experiments
6
Testing for Weak IVs
Measuring Substitution Patterns
Introduction
6 / 48
Market for Automobiles in the U.S.
Year
Models
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
92
89
86
72
93
99
95
95
102
103
116
110
115
113
136
130
143
150
147
131
Price
× 1,000
7.87
7.98
7.53
7.51
7.82
7.79
7.65
7.64
7.60
7.72
8.35
8.83
8.82
8.87
8.94
9.38
9.97
10.07
10.32
10.34
Measuring Substitution Patterns
Euro.
Asia
0.08
0.07
0.03
0.06
0.06
0.04
0.05
0.04
0.04
0.04
0.05
0.05
0.05
0.04
0.05
0.05
0.05
0.05
0.05
0.04
0.06
0.04
0.04
0.05
0.08
0.08
0.11
0.11
0.16
0.19
0.21
0.23
0.21
0.18
0.19
0.22
0.25
0.24
0.26
0.28
HP/WT
/ 100
4.90
3.91
3.64
3.47
3.37
3.38
3.40
3.46
3.48
3.50
3.49
3.47
3.51
3.61
3.72
3.79
3.95
3.96
4.06
4.19
Size
× 10,000
1.50
1.51
1.53
1.51
1.48
1.51
1.47
1.40
1.34
1.30
1.29
1.28
1.28
1.29
1.26
1.25
1.25
1.25
1.26
1.27
Description of the Model
$/Miles
/10
5.60
5.63
5.77
7.26
7.06
6.55
6.03
5.52
6.36
7.16
6.77
5.77
5.08
4.95
5.15
3.67
3.72
3.55
3.71
4.09
7 / 48
Substitution Patterns: Direct Approach
Data: Prices and quantities across a panel of products and markets.
Log-linear specification:
qjt = α0 +
nt
X
αjk pkt + ujt
k=1
Curse of dimensionality: Even with the ideal experiment, it is not
feasible to estimate αjk for all j and k.
I
The number of parameters grows with the number of products...
Solution: Model products as a collection of attributes (x), rather
than individual options.
Measuring Substitution Patterns
Description of the Model
8 / 48
The Characteristic Approach
Two Assumptions:
1
Preferences are linear in xjt ,
uijt
= U (xjt , pjt , ξjt ; βi )
= xjt βi − αi pjt + ξjt + εij
2
Common taste distributions (across t)
(βi , αi ) ∼ F ((β, α); θ) and εij ∼ T1EV(0, 1)
Aggregate demand function:
sjt = σj (pt , xt , ξt ; θ) = Pr (uijt > uikt , ∀k 6= j)
where θ ∈ RK for K << J.
Econometric challenge: Non-linear and non-separable error ξjt
Measuring Substitution Patterns
Description of the Model
9 / 48
BLP versus McFadden
What distinguish “BLP” models from other discrete-choice models?
I
Market/product unobserved quality index ξjt
This is essential to address standard omitted variable problems:
I
I
I
High quality cars are more expensive and more popular
Good schools attract the best students
Wealthy households live next to good public schools
This also complicates identification...
I
I
θ is not identified from the model fit
For any θ, there exists a unique quality vector ξt that explains st
Identification problem: How to distinguish between different quality
assignments (indexed by θ)?
Measuring Substitution Patterns
Description of the Model
10 / 48
A Motivating Example
Linear random-coefficient model with exogenous characteristics:
(2)
Z
exp δjt + σx vi xjt
(2)
φ(vi )dvi
σj δ t , x t ; σx
=
P
(2)
1 + nj 0t=1 exp δj 0 t + σx vi xj 0 t
(1)
(2)
where δjt = β0 + β1 xjt + β2 xjt + ξjt .
Invert demand system to obtain model residual:
(2)
(1)
(2)
ξjt (θ) = σj−1 s t , x t ; σx − β0 − β1 xjt − β2 xjt
Berry-Haile: There exists a simultaneity problem associated with σx
due to endogenous shares, even without prices.
Measuring Substitution Patterns
Identification Problem
11 / 48
Identification and Estimation
Conditional moment restriction
E [ξjt (θ)|x t ] = 0
iff
θ = θ 0 (i.e. true).
Non-linear IV estimation:
E ξjt θ 0 × Aj (x t ) = 0.
For any Aj (·) with
dim (Aj (x t )) ≥ dim (θ)
Does the choice of instruments Aj (x t ) matter?
Measuring Substitution Patterns
Identification Problem
12 / 48
Illustration: Exogenous Characteristics
Random coefficient model:
(1)
(2)
(2)
uijt = β0 + β1 xjt + β2 xjt + ξjt + σx vi xjt + εijt .
Unobserved heterogeneity:
I
I
vi ∼ N(0, 1)
Numerical integration: 10 points Gauss-Hermite quadrature.
Data:
I
I
I
Panel structure: 100 markets × 15 products
Characteristics: (ξjt , xjt ) ∼ N(0, I).
Dimension: |xjt | = K + 1
Measuring Substitution Patterns
Identification Problem
13 / 48
Identification in a Picture
Market IV: Sum of rivals’ characteristics
(A) Residual quality at σx = σx0
-4
2
0
-2
-4
-2
0
2
IIA Residual Quality (σ=0)
True Residual Quality (σ0)
4
4
(B) Residual quality at σx = 0
-15
-10
-5
0
5
(residual) Sum of rivals' characteristics
Note: Regression R2=0.
10
-15
-10
-5
0
5
(residual) Sum of rivals' characteristics
Note: Regression R2=0.
10
Moment restriction: Independence of ξjt and the sum of rivals
characteristics cannot distinguish (A) from (B)
Weak identification: The moment restrictions are “almost” satisfied at
σx 6= σx0 (Stock and Wright, 2000).
Measuring Substitution Patterns
Identification Problem
14 / 48
Identification in a Picture
(2)
Differentiation IV: Number of competitors located within 1 std-dev of xjt
(A) Residual quality at σx = σx0
-4
2
0
-2
-4
-2
0
2
IIA Residual Quality (σ=0)
True Residual Quality (σ0)
4
4
(B) Residual quality at σx = 0
-10
-5
0
5
(residual) Number of local competitors
Note: Regression R2=0.
10
-10
-5
0
5
(residual) Number of local competitors
Note: Regression R2=.11.
10
Moment restriction: Independence of ξjt and the number of close
competitors rules out (B).
I Testable implication: Strong instruments must reject H : σ̂ = 0
0
x
Let’s formalize this intuition for the general model:
(2)
βi
Measuring Substitution Patterns
= β (2) + v i ,
Pr(v i ) = Fv (v i ; Σ)
Identification Problem
15 / 48
Optimal IV
The “optimal” IV for Σ is (Amemyia (1977), Chamberlain (1987)):


(2)
−1
0 ∂σ
s
,
x
;
Σ
t
t
j
x t 
Dj,k x t ; Σ0 = E 
∂Σk
How? Approximate the optimal IV by estimating a non-parametric
first-stage regression (Newey, 1990)
Curse of Dimensionality: Dj,k (x t ) is a j-specific function of the
entire menu of characteristics.
I
I
First order: Dj,k (x t ; Σ0 ) ≈
Impossible: BLP (1995)
Pnt
j=1
x jt a j
Bottom line: If we can’t estimate the first-stage, how can we
construct powerful instruments?
Measuring Substitution Patterns
Optimal IV
16 / 48
What does the characteristic structure imply for the
reduced-form of the model?
Market-structure facing product j (dropping t):
(2)
(2)
(w j , w −j ) ≡ δj , x j
, δ −j , x −j
Properties of the linear-in-characteristics model:
I
Symmetry:
σj (w j , w −j ) = σk (w j , w −j )
I
∀k 6= j
Anonymity:
σ (w j , w −j ) = σ w j , w ρ(−j) ∀ρ
I
Translation invariant: for any c ∈ RK
σ (w j + (0, c) , w −j + (0, c)) = σ (w j , w −j )
Measuring Substitution Patterns
Optimal IV
17 / 48
Re-Express the Demand System
Express the “state” of the market in differences relative to j and treat
the outside option just like any other product.
I
Characteristic differences:
(2)
(2)
(2)
d j,k = x k − x j
I
New normalization:
τj =
I
1+
exp(δ )
P j
, ∀j = 0, . . . , n.
0
j 0 exp(δj )
(2)
Product k attributes: ω j,k = τk , d jt,k
Demand for product j is a fully exchangeable function of ω j :
σ (w j , w −j ) = D(ω j )
where ω j = {ωj,0 , . . . , ωj,j−1 , ωj,j+1 , . . . , ωj,n }.
Measuring Substitution Patterns
Optimal IV
18 / 48
Main Theory Result
Define the exogenous state of the market facing product j:
d j,k
dj
= xk − xj
= (d j,0 , . . . , d j,j−1 , d j,j+1 , . . . , d j,n )
Theorem
If the distribution of {ξj }j=1,...,n is exchangeable, then the reduced form
becomes
" −1
#
(2) ; Σ0 ∂σ
s,
x
j
Dj,k x; Σ0 = E
x = hk (d j )
∂Σk
where hk is a symmetric function of the state vector.
Implication: hk is a vector symmetric function (see Briand 2009)
Measuring Substitution Patterns
Optimal IV
19 / 48
Why is it useful?
1
Curse of dimensionality: In any fixed order approximation the
number of basis functions is independent of the number of products.
I
2
A vector generalization of Pakes (1994), Pakes and McGuire (1994).
Examples: Basis functions
I
First-order (i.e. market):
h(dj ) ≈
X
ak djt,k


X
X
=a×
djt,k  ≡
k6=j
I
k6=j
Second-order (i.e. distance):
2
X
k
d
,
0
jt,j
0
j 6=j
I
j 0 6=j
x jt
∀k = 1, . . . , K
Histogram basis (i.e. nested-logit or hotelling):
X
k
1(|djt,j
∀k = 1, . . . , K
0 | < κk )d jt,j 0 ,
0
j 6=j
Measuring Substitution Patterns
Optimal IV
20 / 48
Closing the loop
Let Aj (x t ) be an L vector of basis functions summarizing the
empirical distribution of characteristic differences: {d jt,k }k=0,...,nt .
Differentiation IV: These functions are moments describing the
relative isolation of each product in characteristic space.
Donald, Imbens, and Newey (2003): Using basis functions directly
as IVs, is asymptotically equivalent to approximating the optimal IV.
I
Recommended practice is to use low-order basis functions (Donald,
Imbens, and Newey 2008).
Alternatively use Lasso methods with higher order polynomials (e.g.
Belloni, Chernozhukov, and Hansen (2012))
Measuring Substitution Patterns
Optimal IV
21 / 48
Side Note: Demographic Panel
In many settings, product characteristics are fixed across markets, but
the distribution of consumer types vary (e.g. Nevo 2001).
Focus on a single characteristics xj
Assumption: Consumer heterogeneity ηit follows distribution of a
demographic can be decomposed as
ηit = µt + σt νi
where (µt , σt ) are known, and Pr(νi < x) = F (x) is common across t.
Preferences
uijt
= δjt + θηit xj + εijt
= δ̃jt + θνi x̃jt + εijt
where x̃jt = σt xj , and δjt = xj (β + θµt ) + ξjt = xj βt + ξjt .
x = x̃ − x̃ .
Differentiation IVs can be constructed as before: djt,k
jt
kt
Measuring Substitution Patterns
Optimal IV
22 / 48
Experiment 1: Exogenous Characteristics
Random coefficient model:
uijt = δjt +
K
X
(2)
vik xjt,k + εijt ,
vi ∼ N(0, σx2 I ).
k=1
Data:
I
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Panel structure: 100 markets × 15 products
Characteristics: (ξjt , x jt ) ∼ N(0, I).
Dimension: |x jt | = K + 1
Monte-Carlo replications = 1,000
Instruments (K + 1):
I
I
Pn
Market IVs (first-order): Aj (x t ) = l6=t j xlt,k , ∀k = 1, . . . , K
2
Pn
k
Diff. IVs (2nd-order): Aj (x t ) = j 0t=1 djt,j
, ∀k = 1, . . . , K
0
Measuring Substitution Patterns
Illustration: Exogenous Characteristics
23 / 48
Result 1: Weak IV Problem
0
0
1
Fraction
.05
2
Kernel density
3
.1
4
Specification: One dimension of heterogeneity
0
2
4
6
Random coefficient parameter
Market IV
Measuring Substitution Patterns
8
10
Diff. IV
Illustration: Exogenous Characteristics
24 / 48
Result 2: Weak IVs with Multiple Dimensions
VARIABLES
σ1
σ2
σ3
σ4
Nb. IVs
Dimensions of consumer heterogeneity
1
2
3
4
2.012 1.931
1.919
2.055
(1.3) (1.423) (1.383)
(1.467)
1.978
1.934
2.055
(1.302) (1.313)
(1.412)
1.992
2.088
(1.353)
(1.564)
2.106
(1.494)
2
3
4
5
Parenthesis: RMSE. Starting at true parameter = 2.
Measuring Substitution Patterns
Illustration: Exogenous Characteristics
25 / 48
Result 3: Differentiation IVs with Multiple Dimensions
VARIABLES
σ1
σ2
Dimensions of
1
2
2.002 1.998
(.107) (.109)
1.995
(.108)
σ3
σ4
σ5
σ6
Nb. IVs
2
3
consumer heterogeneity
3
4
6
1.994 1.994 1.991
(.105) (.111) (.115)
1.996 1.992 1.993
(.105) (.111) (.115)
1.995 1.994 1.991
(.109) (.113) (.116)
1.995 1.990
(.11) (.115)
1.997
(.118)
1.993
(.118)
4
5
7
Parenthesis: RMSE. Starting at true parameter = 2.
Measuring Substitution Patterns
Illustration: Exogenous Characteristics
26 / 48
Result 4: Weak IVs and Numerical Problems
Simulation: 1,000 samples and 10 random starting values each
Differentiation IV
Est.
RMSE
σ1
σ2
σ3
σ4
Algorithm
Nb. IVs
CPU time (sec)
Local optima (fraction)
Global min/Local min
Measuring Substitution Patterns
1.994
1.992
1.994
1.995
0.111
0.111
0.113
0.110
Simplex
5
0.718
0.002
46.6
Market IV
Est. RMSE
2.107
2.053
2.088
2.131
1.536
1.477
1.618
1.640
Simplex
5
33
0.53
53
Illustration: Exogenous Characteristics
27 / 48
Result 4: Weak IVs and Numerical Problems
Simulation: 1,000 samples and 10 random starting values each
Differentiation IV
Est.
RMSE
σ1
σ2
σ3
σ4
Algorithm
Nb. IVs
CPU time (sec)
Local optima (fraction)
Global min/Local min
Measuring Substitution Patterns
1.994
1.992
1.994
1.995
0.111
0.111
0.113
0.110
Simplex
5
0.718
0.002
46.6
Market IV
Est. RMSE
2.107
2.053
2.088
2.131
1.536
1.477
1.618
1.640
Simplex
5
33
0.53
53
Illustration: Exogenous Characteristics
28 / 48
Experiment 2: Correlated Random Coefficients
Consumer heterogeneity:
(2)
βi
∼ N(β (2) , Σ)
Note: 4 dimensions ⇒ 10 non-linear parameters (choleski)
Panel structure:
100 markets × 50 products
Differentiation IVs: Second-order polynomials (with interactions):
nt X
k
djt,j
0
2
j 0 =1
and
nt X
l
k
djt,j
0 × djt,j 0
j 0 =1
for all characteristics k and l.
Measuring Substitution Patterns
Illustration: Exogenous Characteristics
29 / 48
Simulation Results: Unrestricted Covariances
Estimated parameters:
Σ1.
Σ2.
Σ3.
Σ4.
True parameters:
Σ1.
Σ2.
Σ3.
Σ4.
CPU time (sec)
Σ.1
Σ.2
Σ.3
Σ.4
4.066
(.214)
-2.080
(.132)
1.993
(.143)
-1.147
(.123)
4.268
(.242)
0.697
(.154)
-0.405
(.124)
4.502
(.212)
-0.458
(.135)
3.050
(.204)
4.270
0.689
-0.398
4.505
-0.462
3.071
4.064
-2.083
1.995
-1.152
3.151
Parenthesis: RMSE. Starting values = true. Algorithm: Gauss-Newton.
Measuring Substitution Patterns
Illustration: Exogenous Characteristics
30 / 48
Alternative Approach: BLP (1999)
Berry et al. (1999) recognized that we cannot “regress”
∂σjt−1 (st , xt ; θ0 )/∂θ on xt to compute Dj,k x t ; Σ0 .
Instead they propose the following heuristic:
1
2
Estimate θ̂ using valid instruments
Compute the Jacobian of the quality assignment at ξjt = 0:
!
∂σj−1 (st , xt |θ0 ) ∂σj−1 (st , xt |θ̂) E
xt ≈
∂θ
∂θ
ξt =0
This approach works well in practice: Sovinsky (2008), and Reynaert
and Verboven (2013).
Drawbacks: Require relevant/consistent instruments to estimate θ̂
I
Question: Can any starting values for θ̂ yield strong IVs?
Measuring Substitution Patterns
Illustration: Exogenous Characteristics
31 / 48
Starting Values and BLP (1999)’s IVs
Experiment 1: Single dimension
I
Inconsistent starting values:
IVjt =
I
∂σj−1 (st , xt |θ̂ = θ0 + ∆) for ∆ ∈ {−4, 2, 0, 2, 4}.
∂θ
ξt =0
Three differentiation models:
F
F
F
uijt = δjt + θηi xjt + ijt
uijt = δjt + θ log(ηi )xjt + ijt
uijt = δjt − θ(ηi − xjt )2 + ijt
Experiment 2: Correlated heterogeneity
I
Random starting values:
IVjt =
∂σj−1 (st , xt |θ̂ = θ0 + ∆) for ∆ ∼ N(0, I ).
∂θ
ξt =0
Measuring Substitution Patterns
Illustration: Exogenous Characteristics
32 / 48
BLP (1999)
Simulation Results: Biased Starting Values
Diff.
IV
µij = θηi xj
θ̂ bias rmse
µij = θ exp(ηi )xj
θ̂
bias rmse
µij = θ(ηi − xj )2
θ̂
bias rmse
0
1
2
3
4
-2
-1
0
2
4
-4
-2
0
1
2
0.00
0.00
0.00
0.00
0.00
0.07
0.07
0.07
0.07
0.07
0.00
0.08
0.00
0.00
0.00
-0.01
0.00
0.45
0.43
0.37
0.30
0.30
0.00
0.40
0.00
0.01
0.07
0.12
0.14
0.18
0.17
1.02
1.80
2.38
-0.04
0.69
True parameter value = 2.
Takeaway: Starting values matter when the sign of the coefficient
determines which products are close substitutes.
Measuring Substitution Patterns
Illustration: Exogenous Characteristics
33 / 48
Simulation Results: Random Starting Values
Estimated parameters:
Σ1.
Σ2.
Σ3.
Σ4.
True parameters:
Σ1.
Σ2.
Σ3.
Σ4.
CPU time (sec)
Singular matrix (fraction)
Σ.1
Σ.2
Σ.3
Σ.4
4.491
(3.136)
-2.047
(1.379)
2.059
(.735)
-1.243
(.497)
4.394
(1.57)
0.747
(.41)
-0.389
(.307)
4.558
(.355)
-0.484
(.264)
3.089
(.256)
4.270
0.689
-0.398
4.505
-0.462
3.071
4.064
-2.083
1.995
-1.152
191.690
0.24
Takeaway: BLP-99 IVs are weak with correlated random-coefficients.
Measuring Substitution Patterns
Illustration: Exogenous Characteristics
34 / 48
Experiment 3: Endogenous Prices
Demand: Mixed-logit with vertical differentiation
I
Indirect utility
uijt = δjt − αi pjt + εijt
where αi = σp yi−1 , and log(yi ) ∼ N(µy , σy2 ) (known).
Supply: Bertrand-Nash with multi-product firms
I
Marginal cost: log (mcjt ) = γ0 + γx xjt + γξ ξjt + ωjt
Measuring Substitution Patterns
Illustration: Endogenous Characteristics
35 / 48
How to incorporate endogenous prices?
Price instruments: w t = {wjt }j=1,...,nt
Reduced-form of the model:
Dj (x t , p t , w t ; σp0 )
=E
∂σj−1 (s t , x t , p t |σp0 ) x t , w t
∂σp
!
6= h(d xj , d pj )
Solution: Use insights from BLP (1999) and Reynaert & Verboven
(2013) to take expectation for price inside:
p
Dj (x t , p t , w t ; σp0 ) ≈ Dj (x t , p̂ t , w t ; σp0 ) = h d xj , dˆj
p
p
I.e., djt,k
= pkt − pjt is replaced by dˆjt,k
= E (pkt |w kt ) − E (pjt |w jt ).
Measuring Substitution Patterns
Illustration: Endogenous Characteristics
36 / 48
BLP (1995) Instruments
BLP’s basis function:




X
X
zjt = wjt ,
wj 0 ,t ,
wj 0 ,t


0
0
j ∈Fft
j ∈F
/ ft
where wjt = (1, xjt , ωjt ), and Fft is the set of products controlled by
firm f .
This corresponds to the first moment of characteristic differences




X
X
w
zjt0 = wjt ,
djt,k
,
djw0 ,t


0
0
j ∈Fft
j ∈F
/ ft
w = 1, d x , d ω
0
where djt,k
jt,k jt,k , and span(Z ) = span(Z ).
Measuring Substitution Patterns
Illustration: Endogenous Characteristics
37 / 48
Differentiation IVs
Characteristic differences: Let djt,k = p̂kt − p̂jt denote the
exogenous price differences between j and k, where:
p̂jt = π̂0 + π̂1 xjt + π̂2 ωjt .
Two basis functions:
1
Second moment: Euclidian distance




X
X
zjt = wjt ,
dj20 t,j ,
dj20 t,j


0
0
j ∈Fft
2
j ∈F
/ ft
Histogram: Characteristics of local competitors












X
X
wj 0 t ,
wj 0 t
zjt = wjt ,




0
0 / F ,


ft


j ∈ Fft ,
j ∈


dj 0 t,j < sd(d)
dj 0 t,j < sd(d)
Measuring Substitution Patterns
Illustration: Endogenous Characteristics
38 / 48
Weak IV problem is very severe without cost-shifters
0
0
.02
Fraction
.05
.04
.06
.08
Kernel density
.1
.1
Panel structure: J¯ = 50 and T = 10
0
20
40
60
Random coefficient estimate (price)
BLP (1995)
NBH
80
Moment
Note: True value = 20
Measuring Substitution Patterns
Illustration: Endogenous Characteristics
39 / 48
Simulation Results
IV: Sum of charact.
w/o cost w/ cost
IV: Local competitors
w/o cost
w/ cost
βp
Average
RMSE
0.46
2.19
1.08
1.32
1.00
0.22
1.02
0.18
Average
RMSE
13.24
10.84
17.47
7.95
19.10
3.93
19.68
1.51
σp
Measuring Substitution Patterns
Illustration: Endogenous Characteristics
40 / 48
0
.05
Kernel density
.1
.15
.2
The weak IV problem does not go away in large samples...
0
20
40
Random coefficient estimate (price)
BLP (1995)
60
Local competition
Sample size: Solid = 500, Long dash = 1,000, Dash = 2,500.
Measuring Substitution Patterns
Illustration: Endogenous Characteristics
41 / 48
Simulation Results: BLP (1999)’s Approximation
Optimal IV: Jacobian at ξjt = 0:
E
∂σj−1 (st , xt , pt |θ0 ) xt
∂θ
!
∂σj−1 (st , xt , p̂t |θ̂) ≈
∂θ
ξt =0
Where θ̂ is a first-stage estimate using IVs.
IV: Sum of charact.
BLP (1995) Opt. IV
IV: Local competitors
Diff. IV
Opt. IV
βp
Average
RMSE
0.46
2.19
1.29
0.93
1.00
0.22
1.16
0.45
Average
RMSE
13.24
10.84
16.61
28.23
19.10
3.93
17.28
19.07
σp
Measuring Substitution Patterns
Illustration: Endogenous Characteristics
42 / 48
Experiment 4: Natural Experiments
Hotelling example: Exogenous entry of a new product (x 0 = 5)
Three-way panel: product j, market m, and time (t = 0, 1).
Treatment variable:
Djm = 1 (|xjm − 5| < Cutoff)
“Ideal” first-stage: Difference-in-difference regression
∂σj−1 (st , xt , pt |θ0 )
∂θ
= µ̂jm + τ̂t + γ̂Djm × 1(t = 1) + êjmt
GMM Problem:
I
I
I
(1)
Linear characteristics: xjmt = Market/Product FE + After Dummy
Differentiation IV: zjmt = Djm × 1(t = 1)
θ̂gmm is identified from the DiD variation in zjmt .
Measuring Substitution Patterns
Illustration: Endogenous Characteristics
43 / 48
Natural Experiment: Hotelling Example
DGP: δjmt = ξ¯jm + ∆ξjmt , where E (ξ¯jm |x m ) 6= 0
“Diff-in-Diff” specification:
0
2
kdensity theta
4
6
8
z jmt = {Product Dummyjm , 1(t = 1), 1(|xjm − 5| < 1)1(t = 1)}
-2.5
-2
Diff. IV + FE
Measuring Substitution Patterns
-1.5
x
-1
Diff. IV
-.5
BLP (1999)
Illustration: Endogenous Characteristics
44 / 48
Evaluating the Relevance of Instruments
Ex-post: First-stage test
I
First-stage of the non-linear GMM problem at θ̂gmm :
∂σj−1 (st , xt , pt |θ̂gmm )
∂θ
I
= πx xjt + πz zjt + ejt
Standard tests for weak instruments in linear models can be used to
test the relevance of zjt (Wright, 2003).
Ex-ante: IIA test
I
I
A strong instrument for Σ is able to reject the wrong model (Stock
and Wright, 2000)
Under H0 : Σ = 0, the inverse demand equation is independent of x−j :
σjt−1 (st , xt ; θ = 0) = ln sjt /s0t = xjt β + zjt γ + ξjt
I
Standard test statistics for H0 : γ = 0, can be used to test null
hypothesis of IIA preferences
Measuring Substitution Patterns
Testing for Weak IVs
45 / 48
Weak IV Tests: Exogenous characteristics experiment
Weak Identification Test (first-stage)
Kleibergen-Paap Kleibergen-Paap
Dep. var.: ξjt0
Dep. var.: ξˆjt
Diff. IV
Market IV
Distribution
121.068
(0)
0.225
(0.894)
χ2 (L − K + 1)
Measuring Substitution Patterns
IIA Test
Dep. var.:
σjt−1 (θ = 0)
120.722
(0)
0.407
(0.816)
χ2 (L − K + 1)
Testing for Weak IVs
527.239
(0)
6.549
(0.256)
χ2 (L)
46 / 48
Weak IV Tests: Endogenous prices experiment
No cost shifter
IIA test
Weak IV (θ0 )
χ2
p-value
χ2
p-value
Degree of freedom
Measuring Substitution Patterns
Market
(1)
Neighborhood
(2)
Cont. Moments
(3)
2.31
0.47
2.21
0.48
0
70.28
0.00
355.52
0.00
0
99.15
0.00
957.38
0.00
6
Testing for Weak IVs
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Conclusion
What did we do:
I
I
I
Show how that the characteristic model can be used to construct
relevant instruments to identify substitution patterns
And, eliminate the weak IV problem that is present in applied work
Differentiation IV’s: Capture the relative position of each product in
the characteristic space.
What’s next?
I
I
I
Higher-order basis: Lasso
Conduct tests
Non-parametric estimation
Measuring Substitution Patterns
Conclusion
48 / 48