On large market asymptotics for spatial pricecompetition models

Taisuke Otsu and Keita Sunada
January 6, 2022
KEO Discussion Paper No. 165

KEO Discussion Paper No. 165

ON LARGE MARKET ASYMPTOTICS FOR SPATIAL PRICECOMPETITION MODELSTAISUKE OTSU AND KEITA SUNADA
Abstract. We study the problem of weak instruments in a demand estimation of spatial pricecompetition models by Pinkse, Slade, and Brett (2002) (hereafter, PSB). Product characteristics
are included in price instruments and have correlation with prices through the markup. We check
whether product characteristics hold their identification power as the number of product grows
in analogy with random coeﬃcient discrete choice approach investigated by Armstrong (2016).
The conventional weak instruments asymptotics do not work in PSB’s model because a series
estimation is nested in their two-stage least square estimator, and the number of endogenous
regressors (and instruments) also grows as the number of products grows. We provide two
asymptotic results that indicate the lack of inconsistency of PSB’s estimator.

1. Introduction
Economists utilize instruments to solve the simultaneity problem in demand estimation. Since
the influential study by Berry, Levinsohn, and Pakes (1995) (hereafter, BLP) for diﬀerentiatedproduct markets, many papers adopt product characteristics as price instruments, which correlatewith prices through the markup, especially through the market share of each product. However,Armstrong (2016) shows that the market share of each product disappears fast enough as thenumber of products grows in some demand models of BLP. Since the market share is a function of
the product characteristics, these instruments may lose their identifying power and, as a result,lead to inconsistent estimates. Hence the markup formula plays a role of a drifting sequence as
in Staiger and Stock (1997).
This paper studies a weak instrument problem of a demand estimation of spatial price competition models by Pinkse, Slade, and Brett (2002) (hereafter, PSB) in which product characteristics
are included in price instruments. In PSB’s model, consumers’ demands are in a product space,
not in a product characteristic space, and they can consume more than one good. Since BLP
takes a random coeﬃcient discrete choice approach, the demand model of PSB is considerably
diﬀerent from that of BLP. Even under these diﬀerences, however, by rewriting the markup formula induced by the Bertrand equilibrium play, one can see that this formula is a function of
the demand function of each product instead of the market share in BLP. Since the market size
is finite following the existing literature, we expect that the demand function decays to zero as
the number of products grows. Hence the instruments in PSB interact with price in a similar
way to BLP.
1

PSB employ a semiparametric approach, and a series estimation is nested in their estimator.
In the just identified case, the estimation error of their two-stage least squares estimator ✓ˆ ✓ isdenoted byn

1 X
pzi wi0
n
i=1

! 1

n

1 X
p
zi v in
i=1!

=: An 1 bn ,where zi , wi , vi are vectors of instruments, regressors containing series expansion terms and exogenous variables, and regression and approximation errors, respectively. Notice that we cannot
apply conventional weak instruments asymptotics in Staiger and Stock (1997) since An and bn
are growing dimensions. Our first result characterizes the stochastic orders of each element of An
and bn . We find that these are not degenerate, and bn may diverge if the number of expansion
terms grows at a slower rate. Our second result provides an inconsistency result of ✓ˆ given ahigh-level assumption on the maximum eigenvalue of A0n An . For further developments of theseresults, we need to exploit random matrix theory for sample covariance matrix in which elements
are not distributed independently and contain a drifting sequence.

2. Model and estimator
Our model follows that of PSB. There are n sellers of a diﬀerentiated product. For simplicity,
we assume that each firm sells one product. Let qi , pi , and yi be the demand, price, and productcharacteristic for product i. The demand function for product i is given byqi (p, y) = ai +nX(bij pj + cij yj ).

j=1

where p = (p1 , . . . , pn )0 , y = (y1 , . . . , yn )0 , and ({ai }, {bij }, {cij }) are parameters to be estimated.
Suppose firms play the Bertrand pricing game given rival prices, i.e., firm i chooses pi to solvemax(pipi

M Ci )qi (p, y)

Fi ,

(1)

where M Ci and Fi are firm i’s marginal and fixed cost. The best response function of firm i is
0
1
n
X
X
1 @
pi =
ai bii M Ci +bij pj +cij yj A .
2 ii
j6=i

j=1

PSB estimated this best response function by employing a semiparametric approach. Let xi

be a dx -vector of M Ci , finite subset of y, and other exogenous demand and cost variables. Also
let {e` (·)}1`=1 be a sequence of basis functions, {dij } be measures of proximity of firms i and j,P˜
and i` =e` (dij )pj . Based on this notation, the semiparametric model considered by PSB
j6=i

is written as

pi =
=

1
X

↵
˜ ` ˜i` + x0i + ui`=100i ↵ + xi
2

+ vi ,(2)

where vi = ri + ui , ri =P1

˜ ` ˜i` , ↵ = (˜
↵1 , . . . , ↵˜ Ln )0 , and`=Ln +1 ↵

0
10
X
X
X
e1 (dij )pj ,e2 (dij )pj , . . . ,eLn (dij )pj A .
i =@
j6=i

j6=i

j6=i

In our setup, the number of endogenous regressors Ln grows with the number of sellers n. Letting
wi = ( i0 , x0i )0 and ✓ = (↵0 ,0 )0 , this model can be concisely written as p0
i = wi ✓ + vi .

For this model, PSB proposed to estimate ✓ by the (semiparametric) two-stage least squares
based on Kn -dimensional vector of instruments zi . For simplicity, we focus on the just identified
case, i.e., Kn = Ln + dx . As in PSB, we adopt transformed variables of xi as instruments forpi . Let zi = g(xi ) be a Kn -dimensional vector-valued function of xi . Then the semiparametric
instrumental variable estimator for ✓ is written as
! 1 n
n
X
X
0
✓ˆ =
zi w
zi p i .

(3)

i

i=1

i=1

This paper is concerned with the limiting behaviors of the estimator ✓ˆ when the number of
products n increases to infinity under suitable conditions for the price competition models. To
achieve consistency results for ✓ˆ to ✓, it is critical to guarantee suﬃciently strong correlations
between pi ’s contained in the regressors wi and xi generating the instruments zi . To understandthe nature of the problem, observe that the first-order condition of (1) can be written asqi (p, y)+ ui .
bii

pi = M C i(4)Here M Ci is assumed to be an exogenous regressor included in the regression model (2). Thus
we need to guarantee suﬃciently strong correlation between the instruments zi = g(xi ) andmarkup qi (p, y)/bii . However, in the current setup, it is common to assume that the market
Psize is finite, i.e., limn!1 ni=1 qi (p, y) < 1, which implies that qi (p, y) decays to zero as thenumber of products n grows. Therefore, the markup qi (p, y)/bii may not have enough variations
to yield enough correlations with the instruments zi . This phenomenon is thoroughly studied in

Armstrong (2016) for the BLP model on diﬀerentiated product demands. Indeed he conjectured
may emerge in the current model by PSB (see, p. 1964 of Armstrong, 2016). In the next section,
we formally confirm his conjecture.
3. Large market asymptotics
We now study asymptotic properties of the instrumental variables estimator ✓ˆ under thelarge market asymptotics, the number of products n diverges to infinity. Based on the existing
literature, we impose the following assumptions of the demand function qi (p, y) and market size.
Assumption Q. (i) p limn!1

Pn

i=1 qi (y, p) < 1. (ii)Assumption Q (i) says that the market size

Pn

p

p

n max1in qi (y, p)/b1ii ! 0.

i=1 qi (y, p) remains finite as the number of

products n diverges to infinity. This assumption implies that the demand qi (y, p) for eachproduct i decays to 0. Assumption Q (ii) requires that the decay rate of qi (y, p) normalized by3

b1ii should be faster than n 1/2 uniformly over i. An analogous is employed by Armstrong (2016,
Theorem 1) for the BLP model.
We also impose some regularity conditions on the series expansion in (2).
Assumption S. (i) sup1in,`2NP

j6=i |e` (dij )| = O(1). (ii) max1ineach ` 2 N. (iii) sup`2N |↵˜ ` ` | < 1 for some

> 1.

P

j6=i e` (dij )2 = O(1) for

Assumptions S (i) and (iii) are also employed by PSB (their assumptions (vi) and (vii), respectively). Assumptions S (i) and (ii) are on the basis functions {e` (d)}`2N . When the supports of{e` (d)}`2N are finite, these assumptions require that the number of non-zero elements of e` (dij )for i, j = 1, . . . n should be finite. If the supports of {e` (d)}`2N are infinite, Assumptions S (i)and (ii) require that e` (d) should decay fast enough as d ! 1. Assumption S (iii) can be understood as a smoothness condition for the function to be approximated by the series expansion.
Intuitively, larger

is associated with a smoother function.

Based on the above assumptions, we now study the asymptotic properties of the semiparametˆ From (2) and (3), the estimation error of ✓ˆ can be written
ric instrumental variable estimator ✓.
as

n

✓ˆ

✓=

1 X
p
zi wi0
n
i=1

! 1

n

1 X
p
zi v in
i=1!

=: An 1 bn .

(5)

There are two notable features in this expression. First, the matrix An is normalized by n 1/2 ,instead of n 1 for the case of the conventional instrumental variable regression with strong
instruments. This normalization by n 1/2 for An is employed by Staiger and Stock (1997) for
the weak instruments asymptotics. As indicated in the last section, in our setup, the markup
qi (p, y)/bii (and thus wi ) may not have enough correlations with the instruments zi , and hence weadopt the analogous normalization. Second, in contrast to the conventional or weak instruments
asymptotic analysis in Staiger and Stock (1997), An is a Kn ⇥Kn matrix and bn is a Kn ⇥1 vector

so that both components have growing dimensions. In other words, we need to deal with the
problem of weak instruments for semiparametric or series estimators, where not only the numberof instruments Kn but also the number of endogenous regressors Ln grow with the sample sizen. Such an analysis is a substantial challenge in the econometrics literature.1
Although full development of the asymptotic theory for (5) by extending the random matrix
theory is beyond the scope of this paper, we can present two theoretical results to indicate lackˆ The first proposition characterizes the stochastic orders of theof consistency of the estimator ✓.
elements of An and bn .
Proposition 1. Suppose {pi , xi , zi }ni=1 is an i.i.d. triangular array, where each element has thefinite fourth moments, and Assumptions Q and S hold true. Then each element of An is of orderp
Op (1), and each element of bn is of order Op (max{1, nL1n }).
1There are few papers tackle weak instruments in a nonparametric framework despite the problem’s importance.

Han (2020) analyzes this in a nonparametric estimation model of a triangular system. Freyberger (2017) provides positive testability results for the key identification condition in a nonparametric framework, completeness,through the diameter of an identified set.
4

This proposition says that the elements in An and bn do not degenerate, and bn may evendiverge when the Ln (and thus Kn ) grows at a slower rate. Although this result is not enoughto characterize the stochastic order of the whole vector ✓ˆ ✓ = A 1 bn , we can observe analon

gous behaviors of the corresponding terms of An and bn for the case of the weak instruments
asymptotics in Staiger and Stock (1997).
Additionally we provide a lack of consistency result in terms of the Euclidean norm ||✓ˆunder some high level assumption on the matrix An . Let✓||

max (A) be the maximum eigenvalueof a matrix A.
Proposition 2. Suppose {pi , xi , zi }ni=1 is an i.i.d. triangular array, where each element has thefinite fourth moments, and Assumptions Q and S hold true. If

0max (An An )  Cn with probabilitypˆapproaching one (w.p.a.1) and nL2n 2 /Cn ! 0 for some Cn , then ||✓✓|| ! +1.

ˆ TheThis proposition provides suﬃcient conditions to induce inconsistency of the estimator ✓.
additional condition nL2n 2 /Cn ! 0 is analogous to Assumption (viii) in PSB (which requiresPn0nL2n 2 /⇣n ! 0 for a sequence {⇣n } associated with the minimum eigenvalue of
i=1 zi wi ).

In our setup, it is beyond the scope of this paper to characterize the upper bound Cn for the
maximum eigenvalue of the product matrix An A0n with growing dimension, which requires further
developments of the random matrix theory.
To illustrate this point, suppose that An is a Kn ⇥ Kn matrix of independent standard normal

random variables. Then Johnstone (2001, Theorem 1.1) showed that0max (An An )µnnwhere µn = kn2 andn = kn {(Knd! Tracy-Widom law of order 1,1) 1/2 + Kn

1/2 1/3}for kn = (Kn1/2

1)1/2 + Kn . Thus, inthis case, the upper bound Cn can be set as Kn . By Kn = Ln + dx , the additional conditionin Proposition 2 will be nL1n 2 ! 0, which is satisfied when Ln grows fast enough and/orislarge enough.

Finally, we mention how to test the null hypothesis H0 : ✓ = ✓0 . In the conventional weakidentification framework, some asymptotically valid test statistics are proposed. For example,
the S-statistic in Stock and Wright (2000) takes the form
" n
# 1
n
n
X
1 X
1 X
0 1
0
p
SKn (✓) := pgi (✓)
gi (✓)gi (✓)
gi (✓),nnni=1where gi (✓) = zi (piwi0 ✓).

i=1

i=1

If Kn = K is fixed, Stock and Wright (2000, Theorem 2) implies

that SK (✓) converges in distribution to

2 . Then, applying the central limit theorem yields
K

SKn (✓) Kn dp
! N (0, 1).
2Kn
Hence normalized SLn (✓) can be used for constructing an asymptotically valid hypothesis test.
Tn =However, we cannot use this statistic if we are interested in a subset of parameters like finite
dimensional parameterin (2), not in the whole parameter ✓. Developing a framework for thissituation is beyond the scope our paper and left for future research.
5Appendix A. Mathematical appendix
A.1. Proof of Proposition 1. We first consider the matrix An . Without loss of generality,(1,1)

we consider the (1, 1)-element of An , say An. Also we assume E[z1i ] = 0 to simplify thepresentation. By inserting the markup formula in (4), we can decomposenA(1,1)
n

=

1 X X
p
z1i
e1 (dij )pj
n
i=1

=

1
p
n

j6=i

n X
X
i=1 j6=i

n
n
qj
1 XX
1 XX
z1i e1 (dij ) 1 + p
z1i e1 (dij )M Cj + p
z1i e1 (dij )uj
n
n
bjj
i=1
i=1
j6=i

j6=i

=: T1 + T2 + T3 .

For T1 , Assumptions Q (ii) and S (i) and the law of large numbers imply
1
(
)0
n
X
p
qj
1X
|T1 | n max 1 @ max|e1 (dij )|A|z1i | = op (1).
1jn bjj
1in
n
i=1

j6=i

For T2 , observe that

n X
n X X
2 X

E[T22 ] =

n

n X
2 X

=

n

E[z1i z1i1 M Cj M Cj1 ]e1 (dij )e1 (di1 j1 )i=1 i1 =1 j6=i j1 6=i1

i=1 j6=i2{E[z1i M Ci ]E[z1j M Cj ] + E[z1i]E[M Cj2 ]}e1 (dij )2n

X
C1 X Xe1 (dij )2  C1 max
e1 (dij )2 = O(1),1inni=1 j6=ij6=i

for some C1 > 0, where the inequality follows the assumption that z1i and M Ci have thefinite fourth moments, and the last equality follows from Assumption S (ii). Thus, Chebyshev’sPP
inequality implies T2 = Op (1). For T3 , let Ri = z1i j6=i e1 (dij )uj so that T3 = n 1/2 ni=1 Ri .
Note that E[Ri ] = 0,2
E[Ri2 ] = E[z1i]X

2
E[u2j ]e1 (d1j )2 = E[z1i]E[u2i ]j6=i

and
Cov(R1 , R2 ) =X X
X

E[z11 z12 uj1 uj2 ]e1 (d1j1 )e1 (d2j2 )E[z11 z12 u2 uj2 ]e1 (d12 )e1 (d2j2 ) +j2 6=2=

X

X

E[z11 z12 uj1 uj2 ]e1 (d1j1 )e1 (d2j2 )j1 6=1,2 j2 6=1,2

XE[z11 z12 u2 uj2 ]e1 (d12 )e1 (d2j2 ) ++E[z11 z12 u2j1 ]e1 (d1j1 )e1 (d1j2 )j2 6=2= 0.

e1 (d1j )2 ,j6=i

j1 6=1 j2 6=2

=

X

X

X

X

j1 6=1,2 j2 6=1,2,j1

j1 6=1,2

6

E[z11 z12 uj1 uj2 ]e1 (d1j1 )e1 (d2j2 )Thus, we have
n
n
n
X
1X
1X
1X
2
V ar(Ri ) =E[Ri2 ] =E[z1i
]E[u2i ]e1 (d1j )2
n
n
n
i=1
i=1
i=1
j6=i
X
 C2 max
e1 (d1j )2 = O(1),V ar(T3 ) =

1in

j6=i

for some C2 > 0, where the last equality follows from Assumption S (ii). Now Chebyshev’s(1,1)

inequality implies T3 = Op (1). Combining these results, we obtain An= Op (1).

We next consider the vector bn . Without loss of generality, we consider the first element of
bn , say
1
b(1)
n = p

n

n
X
i=1

n

1 X
z1i ui + p
z1i ri =: T4 + T5 .
ni=1

For T4 , the i.i.d. and finite fourth moments assumptions guarantees
1
E[T42 ] =

n

n X
n
X
i=1 j=1n

1X
2 2
E[z1i z1j ui uj ] =E[z1iui ] = O(1).
ni=11 Pn PnThus, T4 is Op (1). For T5 , note that E[T52 ] = n
and thus we haveE[T52 ] =nn
1
1 XX X
n

1
X

i=1

XX

j=1 E[z1i z1j ri rj ] by the i.i.d. assumption,

E[z1i z1j pk pk0 ]˜↵` ↵˜ `0 e` (dik )e`0 (djk0 )i=1 j=1 `=Ln +1 `0 =Ln +1 k6=i k0 6=j

n
n
1
1
X
XX
C3 X X X|↵˜ ` ||↵
˜ `0 ||e` (dik )||e`0 (djk0 )|n
i=1 j=1 `=Ln +1 `0 =Ln +1 k6=i k0 6=j
0
12 0
12
1
X
X
 C3 n @|↵˜ ` |A @ sup|e` (dik )|A1in,`2N k6=i`=Ln +1

0

 C4 nL2n 2 @supX1in,`2N k6=i12|e` (dik )|A = O(nL2n 2 ),(6)for some C3 , C4 > 0, where the first inequality follows from the Cauchy-Schwarz inequalP1
ity and finite fourth moments assumption, the third inequality follows from˜`| `=Ln +1 |↵P1
 C5 L2n 2 for some C5 > 0 by using Assumption S (iii), and the last equal`=Ln +1 C5 `p
ity follows from Assumption S (i). Thus, Chebyshev’s inequality implies T5 = Op ( nL1n ).
p(1)Combining these results, we obtain bn = Op (max{1, nL1n }).
A.2. Proof of Proposition 2. Let

max (A) andmin (A) be the maximum and minimum eigenvalues of a matrix A, respectively. It is suﬃcient to show that Pr{(✓ˆ ✓)0 (✓ˆ ✓)  M } ! 0 foreach M > 0. Take any M > 0. Note that(✓ˆ✓)0 (✓ˆ✓) = b0n (An 1 )0 An 1 bn

min ((An1 0) An 1 )b0n bn =where the last equality follows frommin ((A1 0) A 1) =0 1
min ((AA ) ) =

7

1
,
0
max (AA )b0n bn,0max (An An )for any invertible matrix A. Thus, we havePr{(✓ˆ
⇢ Pr✓) (✓ˆ
0✓)  M }  Pr

b0n bn M,
0max (An An )⇢b0n bnM0max (An An )0max (An An )  Cn

 Pr{b0n bn  Cn M } + o(1) + Pr{ max (An A0n ) > Cn }E[b0n bn ]+ o(1),Cn M

where the third inequality follows from the assumption

w.p.a.1, and thePlast inequality follows from Markov’s inequality. By using the definition bn = p1n ni=1 zi vi =Pnp1
i=1 zi (ri + ui ), we can decomposenn

E[b0n bn ]

=

n

n

0
max (An An )  Cn

n

n

n

1 XX
1 XX
2 XX
E[zi0 zj ri rj ] +E[zi0 zj ui uj ] +E[zi0 zj ri uj ]nnn
i=1 j=1i=1 j=1i=1 j=1=: T1 + T2 + 2T3 .

For T1 , similar arguments to (6) in the proof of Proposition 1 yield
2
0
10
13
n X
n
1
1
X
X
X
X
X
1
T1 =
E 4zi0 zj @↵˜`e` (dih )ph A @↵˜`e` (djk )pk A5n
i=1 j=10

 O(n) @`=Ln +1

1
X`=Ln +1

12 0|↵
˜ ` |A @h6=i`=Ln +1Xsup1in,`2N j6=i

k6=j

12

|e` (dij )|A = O(nL2n 2 ),where the inequality follows from the Cauchy-Schwarz inequality and finite fourth moments
assumption, and the second equality follows from Assumptions S (i) and (iii).
For T2 , the i.i.d. assumption and Cauchy-Schwarz inequality imply
n
q
p
1X
T2 =
E[zi0 zi u2i ]  E[||zi ||4 ] E[u4i ] = O(1).
ni=1For T3 , observe that
T3 =

n
n
1
X
1 XXX
0
E[zi zj uj ph ]↵˜ ` e` (dih )ni=1 j=1 h6=i

=

1
n

n X
X

`=Ln +10E[zi ] E[zj uj pj ]i=1 j6=i

1X↵˜ ` e` (dij )`=Ln +1n
1
C1 X X X|↵˜ ` ||e` (dij )|n
i=1 j6=i `=Ln +1
0
10
1
1
X
X
 C1 @ sup|e` (dij )|A @|↵˜ ` |A = O(L1n1in,`2N j6=i),`=Ln +1

for some C1 > 0, where the second equality follows from E[zj0 zj uj ] = E[zj uj ] = 0, the firstinequality follows from the Cauchy-Schwarz inequality and finite fourth moments assumption,and the last equality follows from Assumptions S (i) and (iii).

8

Combining these results, E[b0n bn ] = O(nL2n 2 ), and thusPr{(✓ˆ

✓)0 (✓ˆ✓)  M }  O(nL2n 2 /Cn ).

Therefore, the conclusion follows by the assumption nL2n 2 /Cn ! 0.
References[1] Armstrong, T. (2016) Large market asymptotics for diﬀerentiated product demand estimators with economic
models of supply, Econometrica, 84, 1961-1980.
[2] Berry, S., Levinsohn, J. and A. Pakes (1995) Automobile prices in market equilibrium, Econometrica, 63,
841-890.
[3] Freyberger, J. (2017) On completeness and consistency in nonparametric instrumental variable models, Econometrica, 85, 1629-1644.
[4] Han, S. (2020) Nonparametric estimation of triangular simultaneous equations models under weak identification, Quantitative Economics, 11, 161-202.
[5] Johnstone, I. M. (2001) On the distribution of the largest eigenvalue in principal components analysis, Annalsof Statistics, 29, 295-327.
[6] Pinkse, J., Slade, M. E. and C. Brett (2002) Spatial price competition: a semiparametric approach, Econometrica, 70, 1111-1153.
[7] Staiger, D. and J. H. Stock (1997) Instrumental variables regression with weak instruments, Econometrica,
65, 557-586.
[8] Stock, J. H. and J. H. Wright (2000) GMM with weak identification, Econometrica, 68, 1055-1096.
Department of Economics, London School of Economics, Houghton Street, London, WC2A
2AE, UK, and Keio Economic Observatory (KEO), 2-15-45 Mita, Minato-ku, Tokyo 108-8345,
Japan.
Email address: t.otsu@lse.ac.uk
Graduate School of Economics, Keio University, 2-15-45 Mita, Minato-ku, Tokyo 108-8345,
Japan.
Email address: sunada@keio.jp

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