Nonparametric analysis of monotone choice Natalia Lazzati John Quah - - PowerPoint PPT Presentation

Nonparametric analysis of monotone choice

Natalia Lazzati John Quah Koji Shirai November, 2018

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 1 / 27

Motivation

Revealed preference theory is often used for single agent models. One of the most well known results is Afriat’s Theorem. We apply a similar approach to the analysis of games: we focus on games with monotone best-replies variation of set constraints and payoff-relevant parameters tight econometric implementation of our theoretical results

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 2 / 27

Main results

What is the empirical content of games with monotone best-replies? We provide a revealed preference test for Nash equilibrium behavior and monotone shape restrictions (RM axiom): we also study Bayesian Nash equilibrium behavior We extend the idea to cross-sectional data. (Manski (2007)) We apply our results to an IO model of entry decisions: method is easy to implement method delivers meaningful restrictions on data

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 3 / 27

Application: IO entry model of airlines

Berry (1992), Ciliberto and Tamer (2009), Kline and Tamer (2016) We observe many markets defined as trips between two airports In each market, two airline firms: i ∈ {1, 2} each firm decides to enter it (yi = E) or not (yi = N)

we observe covariates (x1, x2): Market Presence (P1,P2 ∈ {0, 1}) and Market Size (S ∈ {0, 1})

We test a specific IO model of entry decisions

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 4 / 27

Standard parametric specification

Π1 (y1, y2, x1, ε1) =    α

1x1 + δ11 (y2 = E) + ε1

if y1 = E if y1 = N Π2 (y1, y2, x2, ε2) =    α

2x2 + δ21 (y1 = E) + ε2

if y2 = E if y2 = N

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 5 / 27

Cross-sectional data

Econometrician observes distribution of choices for different (x1, x2). Firm 2 N E Firm 1 N P(N,N |x1, x2) P(N,E | x1, x2) E P(E,N | x1, x2) P(E,E | x1, x2)

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 6 / 27

Standard analysis

Estimate α

1, δ1 and α 2, δ2

Some common assumptions:

firms play Nash equilibrium distribution of (ε1, ε2) belongs to a known family (ε1, ε2) is independent of (x1, x2) interaction effects, δ1 and δ2, are negative

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 7 / 27

Our approach

We offer a joint test of eq behavior and signed effect restrictions. We can find payoffs Π1 (y1, y2, x1, ε1) and Π2 (y1, y2, x2, ε2) such that have single-crossing property in (y1; (−y2, x1)) and (y2; (−y1, x2)) For all

• −y

j , x i

• >
• −y

j , x i

• Πi
• E,y

j , x i , εi

> Πi

• N,y

j , x i , εi

= ⇒ Πi

• E,y

j , x i , εi

> Πi

• N,y

j , x i , εi

• bserved distribution at each covariate arises from a distribution of

pure-strategy Nash eq (We also offer bound estimates for the distribution of firms’ payoffs.)

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 8 / 27

Advantages of our approach

Nonparametric No assumption on eq selection mechanism Very general approach to model heterogeneity

no assumption on the joint distribution of (ε1, ε2) no assumption on group formation

We do assume (ε1, ε2) is independent of (x1, x2)

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 9 / 27

How does the idea work?

Suppose we have the following data (for some fixed x1) x2= (0, 0)

Firm 2 N E Firm 1 N 3/12 3/12 E 4/12 2/12

x2= (0, 1)

Firm 2 N E Firm 1 N 1/12 5/12 E 3/12 3/12

x2= (1, 0)

Firm 2 N E Firm 1 N 2/12 4/12 E 2/12 4/12 Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 10 / 27

How does the idea work?

We test joint hypothesis of Nash eq behavior and signed effect restrictions. Let a (behavioral) type be a sequence of joint choices y1, y2 across covariates x1, x2. We say a (behavioral) type is consistent if it can be generated by eq behavior with payoffs that satisfy the signed restrictions. As possible joint choices and covariate values are finite, we can enumerate all consistent types. The data is consistent with our model if we can decompose the population into a distribution of consistent types that can explain the observations.

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 11 / 27

How does the idea work?

(A bit more on consistent types...) A realization of ε1, ε2 leads to Π1 (y1, y2, x1, ε1) and Π2 (y1, y2, x2, ε2) . Suppose these payoffs satisfy the signed restrictions. Combined with an eq selection rule, these payoffs induce a sequence of Nash eq choices across different covariates x1, x2. This sequence of joint choices across covariates corresponds to one consistent type. Varying ε1, ε2 and the eq selection we generate all consistent types. Notice that many different ε1, ε2 could lead to the same consistent type.

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 12 / 27

How does the idea work?

Data can be rationalized by the following consistent types.

Type Weight

x2= (0, 0) x2= (0, 1) x2= (1, 0)

Action profiles Action profiles Action profiles N,N N,E E,N E,E N,N N,E E,N E,E N,N N,E E,N E,E 1

• 2
• 3
• 4
• 5
• 6
• 7
• ×

⊗ ⊗ ⊗

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 13 / 27

How does the idea work?

Data can be rationalized as follows.

Type Weight

x2= (0, 0) x2= (0, 1) x2= (1, 0)

Action profiles Action profiles Action profiles N,N N,E E,N E,E N,N N,E E,N E,E N,N N,E E,N E,E 1 1/12 1/12 1/12 1/12 2 2/12 2/12 2/12 2/12 3 2/12 2/12 2/12 2/12 4 1/12 1/12 1/12 1/12 5 1/12 1/12 1/12 1/12 6 2/12 2/12 2/12 2/12 7 3/12 3/12 3/12 3/12 Sum 1 3/12 3/12 4/12 2/12 1/12 5/12 3/12 3/12 2/12 4/12 2/12 4/12 Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 14 / 27

Can we explain the data with a linear model?

Notice that Π2 (y1, E, x21, x21, ε2) = α21x21 + α22x22 + δ21 (y1 = E) + ε2 with α21 > 0, α22 > 0, and δ2 < 0. Then ∆Π2 ∆x21 > ∆Π2 ∆x22 ⇐ ⇒ α21 > α22. Whether > or < in the left side, does not depend on realizations of ε2!

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 15 / 27

Can we explain the data with a linear model?

In the data, we have that (I avoid x1 as it is fixed) P (E, E | (1, 0)) − P (E, E | (0, 0)) > P (E, E | (0, 1)) − P (E, E | (0, 0)) = ⇒ α22 < α21 P (N, N | (0, 0)) − P (N, N | (1, 0)) < P (N, N | (0, 0)) − P (N, N | (0, 1)) = ⇒ α22 > α21 This is not possible!

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 16 / 27

In Summary...

Revealed Monotonicity (RM) axiom:

a type is a sequence of joint choices across all covariates in the data not every type is consistent with our model! we offer an axiom that checks consistency of types

Test with cross-sectional data:

we need to find a distribution on consistent types that explains data

Estimation with cross-sectional data:

we can get bounds on different subsets of consistent types

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 17 / 27

Distribution of entry choices in the data

Data are from Kline and Tamer (2016). Covariates = (0, 0, 0) Covariates = (0, 1, 0) N,N N,E E,N E,E N,N N,E E,N E,E 30.37 68.21 0.55 0.87 19 78.51 0.26 2.23 Covariates = (1, 0, 0) Covariates = (1, 1, 0) N,N N,E E,N E,E N,N N,E E,N E,E 19.38 36.71 25.33 18.58 12.15 54.22 4.99 28.64 Covariates = (0, 0, 1) Covariates = (0, 1, 1) N,N N,E E,N E,E N,N N,E E,N E,E 15.88 82.28 0.12 1.73 7.80 88.93 3.27 Covariates = (1, 0, 1) Covariates = (1, 1, 1) N,N N,E E,N E,E N,N N,E E,N E,E 10.64 32.64 30.58 26.14 5.53 50.07 2.14 42.26 Data looks very consistent with our hypothesis!

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 18 / 27

Testing hypothesis

We can find payoffs Π1 (y1, y2, x1, ε1) and Π2 (y1, y2, x2, ε2) such that have single-crossing property in (y1; (−y2, x1)) and (y2; (−y1, x2)) can explain the data as pure-strategy NE

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 19 / 27

Set of consistent types (RM axiom)

There are 48 = 65, 536 possible group types four possible joint choices for each of the eight covariate values Only 482 are consistent with joint Hypothesis (with indif. 1, 809) (0, 0, 0) (0, 1, 0) (1, 0, 0) ... (1, 1, 1) type 1 N,N E,E E,E ... N,N RM axiom type 2 E,E E,E E,E ... N,E RM axiom × ... ... ... ... ... ...

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 20 / 27

Stochastic test

Consistency requires checking whether a system of linear equations Ax = b A is 32 × 482, x is 482 × 1, and b is 32 × 1 has a positive solution in x. A is a matrix of 0’s and 1’s that captures all consistent types

each column captures the behavior of a type under all covariate values

b is the distribution of entry choices x is a possible distribution of group types

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 21 / 27

Empirical results

Data is not consistent with our hypothesis At least one violation is P (N,N|1, 1, 0) + P (E,N|1, 1, 0) = 17.14% < 19% = P (N,N|0, 1, 0)

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 22 / 27

Closest compatible choice distribution

It is quite similar! Covariates = (0, 0, 0) Covariates = (0, 1, 0) N,N N,E E,N E,E N,N N,E E,N E,E 30.06 67.90 0.86 1.18 18.29 78.96 0.71 2.05 Covariates = (1, 0, 0) Covariates = (1, 1, 0) N,N N,E E,N E,E N,N N,E E,N E,E 19.38 36.71 25.33 18.58 12.73 53.64 5.57 28.06 Covariates = (0, 0, 1) Covariates = (0, 1, 1) N,N N,E E,N E,E N,N N,E E,N E,E 15.46 81.86 0.54 2.15 7.86 89.19 0.26 2.69 Covariates = (1, 0, 1) Covariates = (1, 1, 1) N,N N,E E,N E,E N,N N,E E,N E,E 10.64 32.64 30.58 26.14 5.63 49.98 2.24 42.16

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 23 / 27

Small sample considerations

Kitamura and Stoye (2017) The Null-Hypothesis is (H) minx∈R482

+ (b − Ax) (b − Ax) = 0.

The sample counterpart is JN = N minx∈R482

+

• b − Ax
• b − Ax
• = 0.

p-value is about 15%. Thus, we don’t reject the Null-Hypothesis.

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 24 / 27

Application: Bound Estimates

Out of the 482 consistent types, 36 are also consistent with non-strategic interactions

the entry decision of an airline depends on covariates but not on the entry decision of the other airline

We can estimate bounds for the set of 36 non-strategic types Upper bound is 75%

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 25 / 27

Concluding Remarks

We provide a revealed preference test for strategic complementarity

standard time-series data cross-sectional data

We provide out-of-sample predictions of equilibrium points We show our results can be easily implemented

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 26 / 27

Thanks!

Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 27 / 27