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First published online February 1, 2010 2010 年 2 月 1 日首次在线发布

A Control Function Approach to Endogeneity in Consumer Choice Models
消费者选择模型内生性的控制函数方法

Amil Petrin petrin@umn.edu and Kenneth Train train@econ.berkeley.eduView all authors and affiliations
阿米尔·佩特林 petrin@umn.edu 和肯尼斯·特雷恩 train@econ.berkeley.edu 查看所有作者和隶属关系

Volume 47, Issue 1 第 47 卷，第 1 期

https://doi.org/10.1509/jmkr.47.1.3

Abstract 摘要

Endogeneity arises for numerous reasons in models of consumer choice. It leads to inconsistency with standard estimation methods that maintain independence between the model's error and the included variables. The authors describe a control function approach for handling endogeneity in choice models. Observed variables and economic theory are used to derive controls for the dependence between the endogenous variable and the demand error. The theory points to the relationships that contain information on the unobserved demand factor, such as the pricing equation and the advertising equation. The authors’ approach is an alternative to Berry, Levinsohn, and Pakes's (1995) product-market controls for unobserved quality. The authors apply both methods to examine households’ choices among television options, including basic and premium cable packages, in which unobserved attributes, such as quality of programming, are expected to be correlated with price. Without correcting for endogeneity, aggregate demand is estimated to be upward-sloping, suggesting that omitted attributes are positively correlated with demand. Both the control function method and the product-market controls method produce downward-sloping demand estimates that are similar.
在消费者选择模型中，内生性因多种原因产生。它导致标准估计方法的不一致性，这些方法保持模型误差与包含变量之间的独立性。作者描述了一种控制函数方法来处理选择模型中的内生性。通过观察变量和经济理论来推导控制变量，以处理内生变量与需求误差之间的依赖性。理论指出了包含未观察到的需求因素信息的关系，例如定价方程和广告方程。作者的方法是对 Berry、Levinsohn 和 Pakes（1995）未观察到质量的产品市场控制的替代方法。作者应用这两种方法来研究家庭在电视选项（包括基本和高级有线电视套餐）中的选择，其中未观察到的属性（如节目质量）预计与价格相关。如果不纠正内生性，估计的总需求将呈上升趋势，表明遗漏的属性与需求正相关。无论是控制函数法还是产品市场控制法，都会产生相似的向下倾斜的需求估计。

There are several discrete choice demand settings in which researchers have shown that factors not included in the analysis are correlated with the included factors, thus violating the standard independence assumption for consistency (see, e.g., Bass 1969; Berry 1994). In these cases, the estimated impact of the observed factor on demand captures not only that factor's effect but also the effect of the unobserved factors that are correlated with it. For example, products with higher quality usually have higher prices both because the attributes are costly to provide and because they raise demand. When some product attributes are either not observed by the researcher or difficult to measure, such as stylishness of design, estimated price elasticities will be biased in the positive direction.¹
在一些离散选择需求设置中，研究人员已经表明，未包含在分析中的因素与包含的因素相关，从而违反了一致性的标准独立性假设（参见，例如，Bass 1969；Berry 1994）。在这些情况下，观察到的因素对需求的估计影响不仅反映了该因素的作用，还反映了与其相关的未观察到的因素的作用。例如，质量较高的产品通常价格较高，因为这些属性的提供成本较高，并且它们提高了需求。当某些产品属性未被研究人员观察到或难以测量时，例如设计的时尚性，估计的价格弹性将会向正方向偏差。

The problem is often exacerbated by the difficulty of signing this bias. Consider estimated price elasticities with unobserved advertising. Optimizing firms maximize profits with respect to both price and advertising, so in general, they cannot be independent. Firms might raise the price of their products when they advertise if they believe that it stimulates demand. Alternatively, firms may lower price when they advertise (e.g., as a part of a sale). The possibility of either case makes the sign of the bias ambiguous.²
这个问题通常因难以确定这种偏差而加剧。考虑到未观察到的广告的估计价格弹性。优化的公司在价格和广告方面都最大化利润，因此一般来说，它们不能是独立的。如果公司认为广告能刺激需求，它们可能会在广告时提高产品价格。或者，公司在广告时可能会降低价格（例如，作为促销的一部分）。这两种情况的可能性使得偏差的符号变得模糊。

In this article, we propose a control function method for alleviating bias in discrete choice demand settings.³ The approach includes extra variables in the empirical specification to condition out the variation in the unobserved factor that is not independent of the endogenous variable. We derive these controls using economic theory to point to alternative equations that contain information about the unobserved demand factor. Although our empirical application focuses on the pricing equation, any equation that contains information on relevant unobserved factors may be available for use. For example, we anticipate that researchers will explore the advertising equation, which is also affected by the unobserved demand factor.
在本文中，我们提出了一种控制函数方法，以缓解离散选择需求设置中的偏差。 ³ 该方法在经验规范中包括额外变量，以消除与内生变量不独立的未观察因素的变异。我们使用经济理论推导出这些控制变量，以指向包含未观察需求因素信息的替代方程。尽管我们的实证应用侧重于定价方程，但任何包含相关未观察因素信息的方程都可以使用。例如，我们预计研究人员将探索广告方程，因为它也受到未观察需求因素的影响。

The most widely used bias correction method in discrete choice demand settings is the product-market control approach developed by Berry (1994) and Berry, Levinsohn, and Pakes (1995; hereinafter, BLP) for market-level data and then extended to consumer-level data (Berry, Levinsohn, and Pakes 2004; Goolsbee and Petrin 2004). The approach appeals to the aggregate demand equations as a source of information on the unobserved demand factor and has been applied to consumers’ choice among television options (Crawford 2000; Goolsbee and Petrin 2004), minivans (Petrin 2002), and grocery goods (Chintagunta, Dubé, and Goh 2005; Nevo 2001), to name only a few.
在离散选择需求设置中，最广泛使用的偏差校正方法是 Berry（1994）和 Berry、Levinsohn 和 Pakes（1995；以下简称 BLP）为市场级数据开发的产品市场控制方法，然后扩展到消费者级数据（Berry、Levinsohn 和 Pakes 2004；Goolsbee 和 Petrin 2004）。该方法利用总需求方程作为未观察到的需求因素的信息来源，并已应用于消费者在电视选项（Crawford 2000；Goolsbee 和 Petrin 2004）、小型货车（Petrin 2002）和杂货商品（Chintagunta、Dubé和 Goh 2005；Nevo 2001）之间的选择，仅举几例。

Our control function approach provides a useful alternative to the BLP approach. The control function approach is both easier to estimate and available in some situations in which the BLP estimator is not valid. For example, the BLP approach is not consistent in settings in which there are zero, one, or just a small number of purchase observations per product, because it requires that market shares be observed with relatively little sampling error (see Berry, Linton, and Pakes 2004). The BLP approach is also not available for many recently developed empirical demand models, which either maintain assumptions that are not consistent with the BLP setting or are sufficiently complicated to preclude estimating the BLP controls (e.g., Bajari et al. 2007; Fox 2008; Hendel and Nevo 2006). In contrast, our control function approach simply adds new regressors to the demand specification, making it available in all these settings.
我们的控制函数方法为 BLP 方法提供了一种有用的替代方案。控制函数方法不仅更容易估计，而且在某些 BLP 估计器无效的情况下也可用。例如，在每种产品的购买观察数为零、一或仅有少量的情况下，BLP 方法是不一致的，因为它要求市场份额的观察具有相对较小的抽样误差（参见 Berry, Linton, 和 Pakes 2004）。BLP 方法也不适用于许多最近开发的实证需求模型，这些模型要么保持与 BLP 设置不一致的假设，要么复杂到无法估计 BLP 控制（例如，Bajari 等 2007；Fox 2008；Hendel 和 Nevo 2006）。相比之下，我们的控制函数方法只是向需求规范中添加新的回归变量，使其在所有这些情况下都可用。

Either approach is applicable in our empirical application, so we estimate both for comparison. We also provide more discussion relating the approaches in the section “Comparison with Product-Market Control.”
在我们的实证应用中，两种方法都适用，因此我们估计了两者以进行比较。我们还在“与产品市场控制的比较”一节中提供了更多关于这些方法的讨论。

Other methods related to endogeneity in demand settings have been developed. Louviere and colleagues (2005) describe the various manifestations of endogeneity in marketing contexts and the implications for estimation. Building on work by Villas-Boas and Winer (1999), Kuksov and Villas-Boas (2008) describe methods for testing for endogeneity. Villas-Boas and Winer (1999) and Gupta and Park (2009) have developed a maximum likelihood approach, and Yang, Chen, and Allenby (2003) and Jiang, Manchanda, and Rossi (2007) have developed Bayesian methods for handling endogeneity.
在需求设置中与内生性相关的其他方法已经被开发出来。Louviere 及其同事（2005）描述了营销环境中内生性的各种表现及其对估计的影响。基于 Villas-Boas 和 Winer（1999）的工作，Kuksov 和 Villas-Boas（2008）描述了测试内生性的方法。Villas-Boas 和 Winer（1999）以及 Gupta 和 Park（2009）开发了一种最大似然方法，而 Yang、Chen 和 Allenby（2003）以及 Jiang、Manchanda 和 Rossi（2007）则开发了处理内生性的贝叶斯方法。

In the following sections, we describe the control function approach, provide example specifications, discuss the relationship to pricing behavior, and illustrate our approach with an application to households’ choices among television options.
在以下部分中，我们描述了控制函数方法，提供了示例规范，讨论了与定价行为的关系，并通过家庭在电视选项中的选择应用来说明我们的方法。

Model 模型

Consumer n chooses one of the J competing alternatives. The utility that the consumer obtains from alternative j is as follows:
消费者 n 选择 J 个竞争替代品中的一个。消费者从替代品 j 获得的效用如下：

U_{nj} = V (y_{nj} {,x}_{nj} {,β}_{n}) {+ε}_{nj},

(1)

where y_nj is the observed endogenous variable, x_nj is a vector of observed exogenous variables that affect the utility derived from choice j, β_n are parameters that represent the tastes of consumer n, and ε_nj is the unobserved utility.⁴ The endogenous variable might be price, advertising, travel time, or whatever is relevant in the context. The econometric problem arises because ε_nj is not independent of y_nj, as maintained by standard estimation techniques.
其中 y _nj 是观察到的内生变量，x _nj 是影响从选择 j 中获得的效用的观察到的外生变量的向量，β _n 是代表消费者 n 的偏好的参数，ε _nj 是未观察到的效用。 ⁴ 内生变量可能是价格、广告、旅行时间或在上下文中相关的任何内容。经济计量问题的产生是因为 ε _nj 与 y _nj 不是独立的，这与标准估计技术所保持的一致。

The idea behind the control function correction is to derive a proxy variable that conditions on the part of y_nj that depends on ε_nj. If this can be done, the remaining variation in the endogenous variable will be independent of the error, and standard estimation approaches will again be consistent.
控制函数校正的思想是推导出一个代理变量，该变量以 y _nj 中依赖于 ε _nj 的部分为条件。如果能做到这一点，内生变量的剩余变动将与误差无关，标准估计方法将再次一致。

In this discrete choice context, the approach posits that y_nj can be written as a function of all exogenous variables entering utility for any of the choices, denoted as x_n; the variables z_n that do not enter utility directly but affect y_nj (typically the instruments); and a vector of J unobserved terms, μ_n:
在这种离散选择背景下，该方法假设 y _nj 可以写成所有进入任何选择效用的外生变量的函数，记为 x _n ；变量 z _n 不直接进入效用但影响 y _nj （通常是工具变量）；以及一个包含 J 个未观察到项的向量 μ _n ：

y_{nj} = W (x_{n,} z_{n,} μ_{n}) .

(2)

The approach maintains that μ_n and ε_nj are independent of x_n and z_n but are not independent of each other. This equation illustrates the source of the dependence between y_nj and ε_nj—that is, μ_n affects y_nj and is also not independent of ε_nj.
该方法认为 μ _n 和 ε _nj 独立于 x _n 和 z _n ，但彼此之间并不独立。这个方程说明了 y _nj 和 ε _nj 之间依赖关系的来源——即 μ _n 影响 y _nj ，并且也不独立于 ε _nj 。

The key to the control function approach is to note that under the maintained assumptions, conditional on μ_n, ε_nj is independent of y_nj. The feasibility of the control function approach in any setting will be determined by whether the practitioner is able to recover μ_n so it can be conditioned on when the parameters are estimated.⁵
控制函数方法的关键在于注意到，在维持的假设下，条件于 μ _n ，ε _nj 与 y _nj 独立。控制函数方法在任何环境中的可行性将取决于实践者是否能够恢复 μ _n ，以便在估计参数时对其进行条件化。 ⁵

We analyze the control function case when y_nj is additive in its observed and unobserved covariates. A special, illustrative case is when there is a single unobserved factor μ_nj for each choice j:
我们分析了当 y _nj 在其观察到和未观察到的协变量中是加法时的控制函数情况。一个特殊的、说明性的情况是每个选择 j 有一个单一的未观察因素 μ _nj ：

y_{nj} = W (x_{n} {, z}_{n};γ) + μ_{nj},

(3)

where we make explicit γ the parameters of this function. With additivity and the independence assumptions, the controls μ_nj are straightforward to recover using any standard estimator (e.g., ordinary least squares). The question becomes how the new controls are entered into the utility function to condition out the dependence between y_nj and ε_nj.
我们明确指出该函数的参数γ。通过加性和独立性假设，可以使用任何标准估计量（例如，普通最小二乘法）直接恢复控制μ _nj 。问题变成了如何将新的控制项输入效用函数，以消除 y _nj 和ε _nj 之间的依赖性。

One approach enters μ_nj in a flexible manner to condition out any function of it. Decomposing ε_nj into the part that can be explained by a general function of μ_nj and the residual yields the following:
一种方法是以灵活的方式输入 μ _nj 以消除其任何函数。将 ε _nj 分解为可以通过 μ _nj 的一般函数解释的部分和残差，得到以下结果：

ε_{nj} = CF (μ_{nj}; λ) + {\tilde{ε}}_{nj},

(4)

where CF(μ_nj; λ) denotes the control function with parameters λ. The simplest approximation is to specify the control function as linear in μ_nj, in which case the control function is CF(μ_nj; λ) = λμ_nj, λ is a scalar, and utility is as follows:
其中 CF(μ _nj ; λ) 表示具有参数 λ 的控制函数。最简单的近似是将控制函数指定为 μ _nj 的线性函数，在这种情况下，控制函数为 CF(μ _nj ; λ) = λμ _nj ，λ 是一个标量，效用如下：

U_{nj} = V (y_{nj} {,x}_{n} {,β}_{n}) {+λμ}_{nj} + {\tilde{ε}}_{nj} .

(5)

Alternatively, we could allow for a polynomial approximation, adding higher-order terms of μ_nj and the necessary additional parameters.
或者，我们可以允许多项式近似，添加更高阶的μ _nj 项和必要的附加参数。

More generally, it is possible to condition on the entire vector of controls μ_n for any choice j when calculating the control function. In this case, we have the following:
更普遍地，可以在计算控制函数时对任何选择 j 的整个控制向量 μ _n 进行条件处理。在这种情况下，我们有以下内容：

ε_{nj} = CF (μ_{n};λ) + {\tilde{ε}}_{nj},

(6)

which can be approximated to first order with a vector of parameters CF(μ_n; λ) = λ′μ_n.
可以用参数向量 CF(μ _n ; λ) = λ′μ _n 的一阶近似。

Again, higher-order terms are straightforward to add, though parameters increase rapidly in the number of alternative choices. Given the researcher's chosen control function specification, we then have the following:
同样，高阶项的添加也很简单，尽管参数会随着备选项数量的增加而迅速增加。根据研究者选择的控制函数规范，我们有以下内容：

U_{nj} = V (y_{nj} {,x}_{n} {,β}_{n}) +CF (μ_{n},λ) + {\tilde{ε}}_{nj} .

(7)

Conditional on μ_n, the probability that consumer n chooses alternative i is equal to
在 μ _n 条件下，消费者 n 选择替代方案 i 的概率等于

P_{nj} = \int I (U_{nj} {>U}_{nj} \forall j \neq i) f (β_{n}, {\tilde{ε}}_{n}) {dβ}_{n} d {\tilde{ε}}_{n},

(8)

where f(·) is the joint density of β_n and

{\tilde{ε}}_{n}

and I(·) is the indicator function. All that remains to complete the specification is a distributional assumption applied to f(·).
其中 f(·) 是 β _n 和

{\tilde{ε}}_{n}

的联合密度，I(·) 是指示函数。完成规范所需的只是对 f(·) 施加分布假设。

The usual approach is to choose specific functional forms for the distribution of β_n and ε_nj (e.g., normal, logit), though these are typically difficult to motivate with economic theory. They are almost always chosen to be independent of each other, and we maintain that assumption here, which implies in our setting that β_n and

{\tilde{ε}}_{n j}

are independent because conditioning on μ_n cannot induce dependence. In our application, we use normal and logit, though researchers can use whatever assumptions they desire to suit their setup. As in any application, checking the robustness to distributional assumptions is important.
通常的方法是为 β _n 和 ε _nj 的分布选择特定的函数形式（例如，正态分布，逻辑斯蒂分布），尽管这些通常难以用经济理论来解释。它们几乎总是被选择为相互独立的，我们在这里也保持这一假设，这意味着在我们的设置中，β _n 和

{\tilde{ε}}_{n j}

是独立的，因为以 μ _n 为条件不能引起依赖。在我们的应用中，我们使用正态分布和逻辑斯蒂分布，尽管研究人员可以根据他们的设置使用任何他们想要的假设。与任何应用一样，检查对分布假设的稳健性是很重要的。

The model is estimated in two steps. First, the endogenous variable is regressed on observed choice characteristics and the instruments. The residuals of this regression are retained and used to calculate the control function. Second, the choice model is estimated with the control function entering as an extra variable or variables.
该模型分两步估计。首先，将内生变量对观察到的选择特征和工具变量进行回归。保留该回归的残差并用于计算控制函数。其次，带有控制函数作为额外变量的选择模型进行估计。

Because the second step uses an estimate of μ_n from the first step, as opposed to the true μ_n, the asymptotic sampling variance of the second-step estimator needs to take this extra source of variation into account. Either the bootstrap can be implemented, or the standard formulas for two-step estimators can be used (Murphy and Topel 1985; Newey and McFadden 1994). Karaca-Mandic and Train (2003) derive the specific form of these formulas that is applicable to the control function approach. As they note, the bootstrap and asymptotic formulas provide similar standard errors for the application that we describe in our empirical results.
由于第二步使用的是第一步的μ _n 估计值，而不是真实的μ _n ，因此第二步估计量的渐近抽样方差需要考虑这一额外的变异来源。可以实施自助法，或者使用两步估计量的标准公式（Murphy 和 Topel 1985；Newey 和 McFadden 1994）。Karaca-Mandic 和 Train（2003）推导出了适用于控制函数方法的这些公式的具体形式。正如他们所指出的，自助法和渐近公式为我们在实证结果中描述的应用提供了相似的标准误差。

Parametric Functional Forms
参数化函数形式

We consider several parametric forms for the errors in both equations. These parametric forms lead to direct parametric forms for the control function itself and the distribution of the demand residuals conditional on the controls. Although they need not be maintained, they provide an alternative to entering μ_n flexibly in utility and then choosing a distributional assumption for

{\tilde{ε}}_{n j}

.
我们考虑了两个方程中误差的几种参数形式。这些参数形式直接导致了控制函数本身的参数形式以及在控制条件下需求残差的分布。尽管它们不需要被维持，但它们提供了一种替代方法，可以灵活地在效用中输入μ _n ，然后为

{\tilde{ε}}_{n j}

选择一个分布假设。

Example 1: Jointly Normal Errors, Independent over j
示例 1：联合正态误差，j 之间独立

Suppose that μ_nj and ε_nj are jointly normal for each j and i.i.d. over j. Then,
假设 μ _nj 和 ε _nj 对于每个 j 是联合正态分布并且在 j 上是独立同分布的。那么，

CF (μ_{n};λ) = E (ε_{nj} | μ_{n}) = {λμ}_{nj}

(9)

for each j and the deviations

{\tilde{ε}}_{n j} = ε_{n j} - C F (μ_{n}; λ)

are independent of μ_nj and all other regressors. Thus, the control function for each alternative is the residual from the endogenous variable regression interacted with λ, the one coefficient to be estimated. Utility is as follows:
对于每个 j，偏差

{\tilde{ε}}_{n j} = ε_{n j} - C F (μ_{n}; λ)

独立于 μ _nj 和所有其他回归变量。因此，每个备选方案的控制函数是与 λ 交互的内生变量回归的残差，这是要估计的一个系数。效用如下：

U_{n j} = V (y_{n j}, X_{n j}, β_{n}) + λ μ_{n j} + {\tilde{ε}}_{n j},

(10)

where

{\tilde{ε}}_{n j}

is i.i.d. normal with zero mean.
其中

{\tilde{ε}}_{n j}

是独立同分布的均值为零的正态分布。

If β_n is fixed, the model is an independent probit with the residual entering as an extra variable. If β_n is random, the model is a mixed independent probit, mixed over the density of β_n (Train 2003, chaps. 5 and 6 on probit and mixed logit). However, note that the scale of the estimated model differs from that of the original model. In particular,

V a r ({\tilde{ε}}_{n j}) < V a r (ε_{n j})

, such that normalizing by setting

V a r ({\tilde{ε}}_{n j}) = 1

raises the magnitude of coefficients relative to the normalization Var(ε_nj) = 1.
如果 β _n 是固定的，模型是一个独立的 Probit 模型，残差作为一个额外变量进入。如果 β _n 是随机的，模型是一个混合独立 Probit 模型，在 β _n 的密度上混合（Train 2003，第 5 章和第 6 章关于 Probit 和混合 Logit）。然而，请注意，估计模型的尺度与原始模型的尺度不同。特别是，

V a r ({\tilde{ε}}_{n j}) < V a r (ε_{n j})

，因此通过设置

V a r ({\tilde{ε}}_{n j}) = 1

进行归一化会相对于归一化 Var(ε _nj ) = 1 提高系数的大小。

Example 2: Extreme Value and Joint Normal Error Components, Independent over j
示例 2：极值和联合正态误差分量，在 j 之间独立

The previous example can be modified to generate a mixed logit, which has the same normalization for scale in the original and estimated model. This is one of the specifications that Villas-Boas and Winer (1999, Section 2) use. Let

ε_{n j} = ε_{n j}^{1} + ε_{n j}^{2}

, where

ε_{n j}^{1}

and μ_nj are jointly normal and

ε_{n j}^{2}

is i.i.d. extreme value for all j. Then, utility with the control function is
前面的例子可以修改为生成一个混合逻辑模型，该模型在原始模型和估计模型中具有相同的尺度归一化。这是 Villas-Boas 和 Winer（1999，第 2 节）使用的规范之一。设

ε_{n j} = ε_{n j}^{1} + ε_{n j}^{2}

，其中

ε_{n j}^{1}

和μ _nj 是联合正态分布，

ε_{n j}^{2}

对于所有 j 是独立同分布的极值。然后，带有控制函数的效用是

U_{nj} = V (y_{nj}, x_{nj}, β_{n}) + {λμ}_{nj} + {ση}_{nj} + ε_{nj}^{2},

(11)

where η_nj is i.i.d. standard normal. The model is a mixed logit, with mixing over the error components η_nj, whose standard deviation σ is estimated, as well as over the random elements of β_n. The scale in the original utility is normalized by setting the scale of the extreme value distribution for

ε_{n j}^{2}

.
其中 η _nj 是独立同分布的标准正态分布。该模型是一个混合逻辑模型，混合了误差成分 η _nj ，其标准差 σ 被估计，以及 β _n 的随机元素。原始效用中的尺度通过设置

ε_{n j}^{2}

的极值分布的尺度来归一化。

Example 3: Extreme Value and Joint Normal Error Components, Correlation over j
示例 3：极值和联合正态误差分量，j 之间的相关性

The generalization is straightforward conceptually but increases the number of parameters considerably. Let

ε_{n}^{1}

and μ_n be jointly normal with zero mean and covariance Ω. This covariance matrix is 2J × 2J and is composed of submatrices labeled Ω_μμ, Ω_με, Ω_εε. Then,

C F (μ_{n}; λ) = E (ε_{n}^{1} | μ_{n}) = Λ μ_{n}

, where the elements of matrix Λ are related to the elements of

Ω : Λ = Ω_{μ ε} Ω_{μ μ}^{- 1}

. Stacked utilities then become
概念上的推广是直接的，但参数数量显著增加。设

ε_{n}^{1}

和μ _n 为零均值和协方差Ω的联合正态分布。这个协方差矩阵是 2J × 2J，由标记为Ω _μμ 、Ω _με 、Ω _εε 的子矩阵组成。然后，

C F (μ_{n}; λ) = E (ε_{n}^{1} | μ_{n}) = Λ μ_{n}

，其中矩阵Λ的元素与

Ω : Λ = Ω_{μ ε} Ω_{μ μ}^{- 1}

的元素相关。堆叠效用则变为

U_{n} = V (y_{n} {,x}_{n} {,β}_{n}) + Λ μ_{n} + {Γη}_{n} + ε_{n}^{2},

(12)

where η_n is now a vector of J i.i.d. standard normal deviates and Γ is the lower-triangular Choleski matrix of

Ω_{^{ε ε}} - Ω_{μ ε} Ω_{μ μ}^{- 1} Ω_{μ ε}

. In this case, the residuals for each alternative enter the utility of all alternatives, and the mixing is over a set of J normal error components. Villas-Boas and Winer (1999, Section 3) generalize this specification further by allowing

ε_{n}^{2}

to be correlated over alternatives, specifying it to be normally distributed instead of extreme value to accommodate this correlation.
其中 η _n 现在是一个 J 个独立同分布的标准正态偏差的向量，Γ 是

Ω_{^{ε ε}} - Ω_{μ ε} Ω_{μ μ}^{- 1} Ω_{μ ε}

的下三角 Choleski 矩阵。在这种情况下，每个备选方案的残差都会进入所有备选方案的效用，并且混合在一组 J 个正态误差分量中。Villas-Boas 和 Winer（1999，第 3 节）通过允许

ε_{n}^{2}

在备选方案之间相关，进一步推广了这一规范，规定其为正态分布而不是极值分布以适应这种相关性。

Pricing Behavior and the Control Function Approach
定价行为与控制函数方法

Consider consumers’ choice among products, where the endogenous variable y_nj is price p_nj. We investigate the control function approach using some variant of the controls suggested previously with both marginal cost and monopoly pricing. The utility that consumer n obtains from product j is specified as in Example 2:
考虑消费者在产品之间的选择，其中内生变量 y _nj 是价格 p _nj 。我们使用先前建议的控制变量的一些变体，结合边际成本和垄断定价，研究控制函数方法。消费者 n 从产品 j 获得的效用如示例 2 所示：

U_{nj} = V (p_{nj} {,x}_{nj} {,β}_{n}) {+ε}_{nj}^{1} {+ε}_{nj}^{2},

(13)

where

ε_{n j}^{1}

is correlated with price and

ε_{n j}^{2}

is i.i.d. extreme value. Here,

ε_{n j}^{1}

might represent unobserved attributes of the product that are not independent of price. Typically, prices vary over people because different people are in different markets.
其中

ε_{n j}^{1}

与价格相关，

ε_{n j}^{2}

是独立同分布的极值。在这里，

ε_{n j}^{1}

可能代表与价格不独立的产品未观察到的属性。通常，价格因人而异，因为不同的人处于不同的市场。

The marginal cost of product j in consumer n's choice set is denoted as MC(z_nj, ν_nj), where z_nj are exogenous variables observed by the analyst and ν_nj is unobserved. The observed variables z_nj will typically overlap with x_nj insofar as observed attributes of the product affect both demand and cost.
消费者 n 选择集中的产品 j 的边际成本表示为 MC(z _nj , ν _nj )，其中 z _nj 是分析师观察到的外生变量，ν _nj 是未观察到的。观察到的变量 z _nj 通常会与 x _nj 重叠，因为产品的观察属性会同时影响需求和成本。

Marginal Cost Pricing 边际成本定价

Consider marginal cost (MC) pricing, and assume that MC is separable in the unobserved term. The pricing equation becomes
考虑边际成本（MC）定价，并假设 MC 在未观察到的项中是可分离的。定价方程变为

p_{nj} = MC (z_{nj}, v_{nj}) = W (z_{nj},γ) + v_{nj},

(14)

where γ are parameters to be estimated. Following Example 2, assume that

ε_{n j}^{1}

and ν_nj are jointly normal, i.i.d. over j. Correlation may arise, for example, because unobserved attributes affect utility as well as costs, thus entering both

ε_{n j}^{1}

and ν_nj. Given the separability assumption on the unobserved term in the pricing equation, utility becomes
其中 γ 是需要估计的参数。按照示例 2，假设

ε_{n j}^{1}

和ν _nj 是联合正态的，在 j 上是独立同分布的。相关性可能会出现，例如，因为未观察到的属性既影响效用也影响成本，因此同时进入

ε_{n j}^{1}

和ν _nj 。鉴于定价方程中未观察到项的可分性假设，效用变为

U_{nj} = V (p_{nj} {,x}_{nj} {,β}_{n}) + λ v_{nj} + {ση}_{nj} {+ε}_{nj}^{2},

(15)

where η_nj is i.i.d. standard normal. The same specification is appropriate when there is a constant markup over cost in the determination of prices.
其中 η _nj 是独立同分布的标准正态分布。当价格的确定中存在恒定的成本加成时，同样的规格是合适的。

Monopoly Pricing 垄断定价

Consider monopoly pricing, where price depends on the elasticity of demand and marginal cost. The pricing equation for a monopolist is as follows:⁶
考虑垄断定价，其中价格取决于需求弹性和边际成本。垄断者的定价方程如下： ⁶

p_{nj} = [p_{nj} / | e (ε_{n}^{1}) |] +MC (z_{nj}, v_{nj}),

(16)

where

e (ε_{n}^{1})

is the elasticity of demand. This elasticity depends on all factors that affect demand, including attributes that are not observed by the analyst. The elasticity is written as a function of

ε_{n}^{1}

to explicitly denote this dependence.
其中

e (ε_{n}^{1})

是需求弹性。该弹性取决于所有影响需求的因素，包括分析师未观察到的属性。弹性被写成

ε_{n}^{1}

的函数，以明确表示这种依赖性。

Now, suppose that the analyst estimates the following price equation:
现在，假设分析师估计了以下价格方程：

p_{nj} = W (x_{nj} {,z}_{nj},γ) + μ_{nj} .

(17)

Then,

ε_{n}^{1}

enters the pricing equation in a nonseparable manner, suggesting that the additively separable μ_nj will not fully condition out the entire dependence of p_nj on

ε_{n j}^{1}

. Any remaining dependence will bias the estimated price elasticitiy.
然后，

ε_{n}^{1}

以不可分离的方式进入定价方程，这表明可加分离的 μ _nj 将无法完全消除 p _nj 对

ε_{n j}^{1}

的全部依赖。任何剩余的依赖性都会使估计的价格弹性产生偏差。

In this situation, Villas-Boas (2007) suggests working in the reverse direction. Instead of specifying the joint distribution of

〈 v_{n}, ε_{n}^{1} 〉

and then deriving the implications for the distribution of

〈 μ_{n}, ε_{n}^{1} 〉

, Villas-Boas shows that if the price of each product is strictly monotonie in its marginal cost, there is a distribution of

〈 v_{n}, ε_{n}^{1} 〉

and a marginal cost function that is consistent with any given distribution of

〈 μ_{n}, ε_{n}^{1} 〉

. This result implies that the analyst can specify a distribution of

ε_{n}^{1}

conditional on μ_n, as needed for the control function approach, and knows that there is some distribution of ν_n and

ε_{n}^{1}

that gives rise to it.
在这种情况下，Villas-Boas（2007）建议反向操作。与其先指定

〈 v_{n}, ε_{n}^{1} 〉

的联合分布，然后推导出

〈 μ_{n}, ε_{n}^{1} 〉

分布的含义，Villas-Boas 表明，如果每种产品的价格在其边际成本上严格单调，则存在

〈 v_{n}, ε_{n}^{1} 〉

的分布和一个与任何给定

〈 μ_{n}, ε_{n}^{1} 〉

分布一致的边际成本函数。这个结果意味着分析师可以根据μ _n 指定

ε_{n}^{1}

的分布，正如控制函数方法所需，并且知道存在某种 _n 和

ε_{n}^{1}

的分布导致了它。

Empirical Model and Data 经验模型和数据

To illustrate, we apply the control function approach to households’ choice of television reception options. The specification and data are similar to those of Goolsbee and Petrin (2004), who apply the BLP approach. By using a situation in which both approaches can be applied, we are able to compare results.
例如，我们将控制函数方法应用于家庭选择电视接收选项。规格和数据与 Goolsbee 和 Petrin（2004）相似，他们采用了 BLP 方法。通过使用两种方法都可以应用的情况，我们能够比较结果。

In general, households have four alternatives for television: (1) antenna-only, (2) cable with basic or extended service, (3) cable with a premium service added (e.g., HBO), and (4) satellite dish. Basic and extended cable are combined because the data do not differentiate which of these options the households chose. Goolsbee and Petrin (2004) describe the market for cable and satellite television, emphasizing the importance of accounting for endogeneity of price, which arises because unobserved attributes of cable television, like the quality of programming, are not independent of price.
一般来说，家庭有四种电视选择：(1) 仅使用天线，(2) 基本或扩展服务的有线电视，(3) 增加了高级服务（如 HBO）的有线电视，(4) 卫星电视。基本和扩展有线电视被合并在一起，因为数据没有区分家庭选择了哪种选项。Goolsbee 和 Petrin（2004）描述了有线电视和卫星电视的市场，强调了考虑价格内生性的重要性，因为有线电视的未观察到的属性（如节目质量）与价格不是独立的。

Our sample consists of 11,810 households in 172 geographically distinct markets. Each market contains one cable franchise that offers basic, extended, and premium packages. There are several multiple-system operators, such as AT&T and Time Warner, which own many cable franchises throughout the country (thus serving several markets). The price and other attributes of the cable options vary over markets, even for markets served by the same multiple-system operator. Satellite prices do not vary geographically, and the price of antenna-only is assumed to be zero. The price variation that is needed to estimate price impacts arises from the cable alternatives.
我们的样本包括 172 个地理上不同的市场中的 11,810 个家庭。每个市场包含一个提供基本、扩展和高级套餐的有线电视特许经营权。有几家多系统运营商，如 AT&T 和时代华纳，拥有全国许多有线电视特许经营权（因此服务于多个市场）。即使是由同一多系统运营商服务的市场，有线电视选项的价格和其他属性在各个市场之间也有所不同。卫星价格在地理上没有差异，天线仅选项的价格假定为零。估计价格影响所需的价格变化来自有线电视替代方案。

Table 1 provides information about the sampled households and the service options that are available to them. Nearly 85% of the sample lives in single-family dwellings, and average income is approximately $62,000. The most popular television option is basic and extended cable, which is chosen by 45% of the households. Less than a quarter of the households have antenna-only reception. The average price for basic and extended cable is approximately $28 per month, with this price ranging from $16 to $45 (not shown in the table). The fee for premium cable is $40 on average, ranging from $26 to $56. More details of the data appear in the Web Appendix (http://www.marketingpower.com/jmrfeb10).
表 1 提供了抽样家庭及其可用服务选项的信息。样本中近 85%的家庭住在独栋住宅，平均收入约为 62,000 美元。最受欢迎的电视选项是基本和扩展有线电视，45%的家庭选择了这一选项。不到四分之一的家庭仅有天线接收。基本和扩展有线电视的平均价格约为每月 28 美元，价格范围从 16 美元到 45 美元（表中未显示）。高级有线电视的平均费用为 40 美元，价格范围从 26 美元到 56 美元。数据的更多详细信息见网络附录（http://www.marketingpower.com/jmrfeb10）。

Table 1 Demographic Variables and Service Attributes
表 1 人口变量和服务属性

Average income 平均收入	$62,368
Income Groups 收入群体	Share (%) 份额 (%)
<$25,000	19.60
$25,000–$49,999	24.48
$50,000–$74,999	24.39
$75,000–$99,999	17.44
>$ 100,000 >￥100,000	14.09
Unmarried	31.93
Single-family dwelling 独栋住宅	84.58
Rent	16.34
Household Size 家庭规模
1 person 1 人	18.88
2 people 2 人	39.40
3 people 3 人	16.76
4 people 4 人	15.39
5 or more people 5 人或更多	9.56
Chosen Television Option 选择的电视选项
Antenna-only	23.14
Basic and extended cable 基本和扩展电缆	44.79
Premium cable 高级电缆	20.72
Satellite	11.36
Attributes of Service in Household's Area 家庭区域的服务属性	Average
Over-the-air channels 无线电视频道	10.7
Basic/extended cable channels 基本/扩展有线频道	62.9
Additional premium cable channels 额外的高级有线频道	5.8
Price for basic and extended cable 基本和扩展有线电视的价格	27.96
Price for premium cable 优质电缆的价格	39.58

Open in viewer

Because the attributes of the television alternatives are the same for all households in a geographic market, we add a subscript for markets. Let U_njm be the utility that household n that lives in market m obtains from alternative j. The price of alternative j in market m is p_mj, which is not subscripted by n because it is the same for all households in the market m. Price is zero for antenna-only television, and the price of satellite television does not vary over markets or households. The price of the two cable options varies over geographic markets, and unobserved attributes of cable service (e.g., quality of programming) are expected to be correlated with price. The utility of the two cable options (j = 2, 3) is specified as in Example 2:
由于电视替代品的属性在一个地理市场中对所有家庭都是相同的，我们为市场添加一个下标。设 U _njm 为居住在市场 m 的家庭 n 从替代品 j 中获得的效用。市场 m 中替代品 j 的价格为 p _mj ，因为它对市场 m 中的所有家庭都是相同的，所以没有用 n 作为下标。天线电视的价格为零，卫星电视的价格在各个市场或家庭中不变。两种有线电视选项的价格在地理市场中有所不同，有线电视服务的未观察到的属性（例如，节目质量）预计与价格相关。两种有线电视选项（j = 2, 3）的效用如示例 2 中所述：

U_{n j m} = V (p_{m j}, x_{m j}, β_{n}) + ε_{n j}^{1} + ε_{n j}^{2},

(18)

where

ε_{n j}^{1}

is correlated with price,

ε_{n j}^{2}

is i.i.d. extreme value, and x_nj captures exogenous observed attributes. Utility for the two options with constant price (j = 1,4) is the same but without the correlated error component

ε_{n j}^{1}

. Price for the cable options is specified as linear in instruments plus a separable error:
其中

ε_{n j}^{1}

与价格相关，

ε_{n j}^{2}

是独立同分布的极值，x _nj 捕捉外生的观察属性。两个选项的效用在价格恒定（j = 1,4）时相同，但没有相关的误差成分

ε_{n j}^{1}

。电缆选项的价格被指定为线性工具加上可分离误差：

p_{m j} = γ z_{m j} + μ_{m j} .

(19)

We specify μ_mj and

ε_{n j}^{1}

for j = 2, 3 to be jointly normal, independent over j. Then, utility with the control function for alternative j = 2, 3 is as follows:
我们规定 μ _mj 和

ε_{n j}^{1}

对于 j = 2, 3 是联合正态的，在 j 上是独立的。然后，替代 j = 2, 3 的控制函数的效用如下：

U_{njm} = V (p_{mj} {,x}_{mj} {,β}_{n}) + {λμ}_{mj} + σ_{j} η_{nj} {+ε}_{nj}^{2},

(20)

where η_nj is standard normal.
其中 η _nj 是标准正态分布。

To complete the model, we specify V(·) as follows:
为了完成模型，我们将 V(·)指定如下：

\begin{matrix} V (p_{mj} {,x}_{mj} {,β}_{n}) = {αp}_{mj} + \sum_{g = 2 - 5} θ_{g} p_{mj} d_{gn} + {τx}_{mj} \\ + δ_{j} k_{j} + {κk}_{j} s_{n} + {φω}_{n} c_{j} . \end{matrix}

(21)

We specify the price effect to differ by income group. We identified five income groups and take the lowest-income group as the base. The dummy d_gn identifies whether household n is in income group g. The price coefficient for a household in the lowest-income group is α, while that for a household in group g > 1 is α + θ_g. The nonprice attributes x_mj enter with fixed coefficients. The alternative-specific constant for alternative j is k_j. These constants are entered directly and also interacted with demographic variables, s_n.
我们将价格效应按收入组区分。我们确定了五个收入组，并将最低收入组作为基准。虚拟变量 d _gn 用于识别家庭 n 是否属于收入组 g。最低收入组家庭的价格系数为α，而收入组 g > 1 的家庭的价格系数为α + θ _g 。非价格属性 x _mj 以固定系数进入。备选方案 j 的特定常数为 k。这些常数直接输入并与人口变量 s _n 交互。

We include an error component to allow for correlation in unobserved utility over the three nonantenna alternatives. In particular, c_j = 1 if j is one of the three nonantenna alternatives, and c_j = 0 if otherwise, and ω_n is an i.i.d. standard normal deviate. The coefficient φ is the standard deviation of this error component, reflecting the degree of correlation among the nonantenna alternatives.
我们包括一个误差成分，以允许在三个非天线替代方案中未观察到的效用之间的相关性。特别地，如果 j 是三个非天线替代方案之一，则 c = 1，否则 c = 0，ω _n 是一个独立同分布的标准正态偏差。系数 φ 是该误差成分的标准差，反映了非天线替代方案之间的相关程度。

Therefore, the choice probability takes the form of a mixed logit (Brownstone and Train 1999; Train 1998), with the mixing over the distribution of the error components:
因此，选择概率采用混合逻辑模型的形式（Brownstone 和 Train 1999；Train 1998），在误差成分的分布上进行混合：

P_{nj} = \int \frac{e^{v_{ni} (η_{2} {,η}_{3},ω)}}{\sum_{j = 1}^{4} e^{v_{nj (η_{2} {,η}_{3},ω)}}} φ (η_{2}) φ (η_{3}) φ (ω) {dωdη}_{3} {dη}_{2},

(22)

where φ(·) is the standard normal density and
其中φ(·)是标准正态密度函数

\begin{matrix} V (η_{j},ω) = {αp}_{mj} + \sum_{g=2 - 5} θ_{g} p_{mj} d_{gn} + {τx}_{mj} + δ_{j} k_{j} + {κk}_{j} s_{n} \\ + {φωc}_{j} + {λμ}_{mj} + σ_{j} η_{j} . \end{matrix}

(23)

The integral is approximated through simulation: A value of η₂, η₃, and ω is drawn from their standard normal densities, the logit formula is calculated for this draw, the process is repeated for numerous draws, and the results are averaged. To increase accuracy, we use Halton (1960) draws instead of independent random draws. Bhat (2001) finds that 100 Halton draws perform better than 1000 independent random draws, a result that has been confirmed on other data sets (see Hensher 2001; Munizaga and Alvarez-Daziano 2001; Train 2000, 2003).
通过模拟近似积分：从标准正态密度中抽取η ₂ 、η ₃ 和ω的值，计算该抽样的 logit 公式，重复多次抽样，并对结果取平均值。为了提高准确性，我们使用 Halton（1960）抽样代替独立随机抽样。Bhat（2001）发现，100 次 Halton 抽样比 1000 次独立随机抽样表现更好，这一结果在其他数据集上也得到了证实（见 Hensher 2001；Munizaga 和 Alvarez-Daziano 2001；Train 2000, 2003）。

Results 结果

The first step of the approach is to estimate the pricing functions to recover the residuals entering the control functions in the choice model. We regressed the price in each market against the product attributes listed in Table 2 plus Hausman-type price instruments (see Hausman 1997b). We calculated the price instrument for market m as the average price in other markets that are served by the same multiple-system operator as market m.⁷ In our context, these instruments are appropriate if the prices of the same multiple-system operator in other markets reflect common costs of the multiple-system operator but not common demand shocks (e.g., unobserved advertising). We created a separate instrument for the price of extended basic cable and the price of premium cable, and we ran separate regressions for extended basic price and premium price using all instruments in each equation.⁸
该方法的第一步是估计定价函数，以恢复进入选择模型中控制函数的残差。我们将每个市场的价格对表 2 中列出的产品属性以及 Hausman 类型的价格工具变量进行回归（参见 Hausman 1997b）。我们计算市场 m 的价格工具变量为由与市场 m 相同的多系统运营商服务的其他市场的平均价格。 ⁷ 在我们的背景下，如果同一多系统运营商在其他市场的价格反映了多系统运营商的共同成本而不是共同需求冲击（例如，未观察到的广告），那么这些工具变量是合适的。我们为扩展基本有线电视的价格和高级有线电视的价格创建了单独的工具变量，并使用每个方程中的所有工具变量分别对扩展基本价格和高级价格进行回归。 ⁸

Table 2 MIXED LOGIT MODEL OF TELEVISION RECEPTION CHOICE: CONTROL FUNCTION APPROACH
表 2 电视接收选择的混合逻辑模型：控制函数方法

Explanatory Variable 解释变量	Uncorrected	With Control Function 具有控制功能
Variables That Vary over Markets but Are Constant over Consumers in Each Market 在各市场中变化但在每个市场的消费者中保持不变的变量
Price, in dollars per month [1–4] 价格，以每月美元计算 [1–4]	–.0202 (.0047)	–.1003 (.0471)
Number of cable channels [2, 3] 有线频道数量 [2, 3]	–.0023 (.0011)	.0026 (.0039)
Number of premium channels [3] 高级频道数量 [3]	.0375 (.0163)	.0559 (.0382)
Number of over-the-air channels [1] 无线电视频道数量 [1]	.0265 (.0090)	.0232 (.0152)
Whether pay-per-view is offered [2, 3] 是否提供按次付费 [2, 3]	.4315 (.0666)	.5992 (.1792)
Indicator: AT&T is cable company [2] 指标：AT&T 是有线公司 [2]	.1279 (.0946)	–.2072 (.2437)
Indicator: AT&T is cable company [3] 指标：AT&T 是有线公司 [3]	.0993 (.1195)	–.2559 (.2737)
Indicator: Adelphia is cable company [2] 指标：Adelphia 是一家有线电视公司 [2]	.3304 (.1224)	.3443 (.2930)
Indicator: Adelphia is cable company [3] 指标：Adelphia 是一家有线电视公司 [3]	.2817 (.1511)	.2504(.3400)
Indicator: Cablevision is cable company [2] 指标：有线电视公司是有线公司 [2]	.6923 (.2243)	.1031 (.3749)
Indicator: Cablevision is cable company [3] 指标：有线电视公司是有线公司 [3]	1.328 (.2448)	1.015 (.5412)
Indicator: Charter is cable company [2] 指标：Charter 是有线公司 [2]	.0279 (.1010)	–.0587 (.2259)
Indicator: Charter is cable company [3] 指标：Charter 是有线电视公司 [3]	–.0618 (.1310)	–.2171 (.2139)
Indicator: Comcast is cable company [2] 指标：康卡斯特是一家有线电视公司 [2]	.2325 (.1107)	–.1111 (.3694)
Indicator: Comcast is cable company [3] 指标：康卡斯特是一家有线电视公司 [3]	.5010 (.1325)	.2619 (.3210)
Indicator: Cox is cable company [2] 指标：考克斯是有线公司[2]	.2907 (.1386)	–.0720 (.3314)
Indicator: Cox is cable company [3] 指标：考克斯是有线公司[3]	.5258 (.1637)	.1678 (.5065)
Indicator: Time Warner cable company [2] 指标：时代华纳有线公司 [2]	.1393 (.0974)	–.0902 (.2213)
Indicator: Time Warner cable company [3] 指标：时代华纳有线公司 [3]	.2294(.1242)	–.0462 (.2254)
Alternative-specific constant [2] 替代特定常数 [2]	1.119 (.2668)	3.060 (1.054)
Alternative-specific constant [3] 替代特定常数 [3]	.1683 (.3158)	2.439 (1.542)
Alternative-specific constant [4] 替代特定常数 [4]	–.2213 (.4102)	4.386 (2.690)
Variables That Vary over Consumers in Each Market 每个市场中随消费者变化的变量
Price for Income Group 2 [1–4] 收入群体 2 的价格 [1–4]	.0149 (.0024)	.0154 (.0026)
Price for Income Group 3 [1–4] 收入组 3 的价格 [1–4]	.0246 (.0030)	.0253 (.0038)
Price for Income Group 4[1–4] 收入组 4 的价格[1–4]	.0269 (.0034)	.0271 (.0042)
Price for Income Group 5 [1–4] 收入组 5 的价格 [1–4]	.0308 (.0036)	.0311 (.0040)
Education level of household [2] 家庭教育水平 [2]	–.0644 (.0220)	–.0640 (.0254)
Education level of household [3] 家庭教育水平 [3]	–.1137 (.0278)	–.1129 (.0371)
Education level of household [4] 家庭教育水平 [4]	–.1965 (.0369)	–.1987 (.0384)
Household size [2] 家庭规模 [2]	–.0494 (.0240)	–.0556 (.0283)
Household size [3] 家庭规模 [3]	.0160 (.0286)	.0303 (.0421)
Household size [4] 家庭规模 [4]	.0044 (.0357)	.0023 (.0447)
Household rents dwelling [2–3] 家庭租赁住宅 [2–3]	–.2471 (.0867)	–.2719 (.0891)
Household rents dwelling [4] 家庭租赁住宅 [4]	–.2129 (.1562)	–.2008 (.1329)
Single-family dwelling [4] 独栋住宅 [4]	.7622 (.1523)	.7790 (.2071)
Error component for nonantenna alternatives, SD [2–4] 非天线替代方案的误差组件，SD [2–4]	.5087 (.6789)	.4994 (.7344)
Terms to Correct for Endogeneity 纠正内生性的条款
Residual for extended basic cable price [2] 扩展基本有线电视价格的剩余部分 [2]		.0833 (.0481)
Residual for premium cable price [3] 优质电缆价格的剩余部分 [3]		.0929 (.0499)
Error component for basic and extended cable, SD [2] 基本和扩展电缆的错误组件，SD [2]		.0488 (1.423)
Error component for premium cable, SD [3] 高级电缆的错误组件，SD [3]		1.425 (1.142)
Log-likelihood at convergence 收敛时的对数似然值	–14,660.84	–14,645.21
Number of observations 观察次数	11,810	11,810

Notes: Alternatives: (1) antenna-only, (2) basic and extended cable, (3) premium cable, and (4) satellite. Each variable enters the alternatives that are listed in brackets and takes the value of zero in other alternatives. Estimates that are statistically significant (p < .05) are in bold. Standard errors are in parentheses.
注释：替代方案：(1) 仅天线，(2) 基本和扩展有线电视，(3) 高级有线电视，(4) 卫星。每个变量进入括号中列出的替代方案，并在其他替代方案中取零值。统计显著的估计值（p < .05）用粗体表示。标准误差在括号中。

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The residuals from these regressions enter without transformation in the mixed logit model; that is, the control functions are a coefficient times the product-market residual, which is the first and simplest specification proposed in the “Model” section. Specifically, the residual from the extended basic cable price regression enters the extended basic cable alternatives and similarly for the premium cable.
这些回归的残差未经转换直接进入混合逻辑模型；也就是说，控制函数是系数乘以产品市场残差，这是“模型”部分中提出的第一个也是最简单的规范。具体来说，扩展基本有线电视价格回归的残差进入扩展基本有线电视选项，优质有线电视也是如此。

Table 2 gives the estimated parameters. The variables are listed in three groups: (1) those that vary over markets but not over consumers in each market, (2) those that vary over consumers in each market, and (3) the extra variables that are included to correct for endogeneity. The first column gives the model without any correction for the correlation between price and omitted attributes; utility is the same as we specified previously except that the residuals,

{\hat{μ}}_{m j}

_mj, and induced error components, η_j, are not included. The second column applies the control function approach by including the residuals and error components.
表 2 给出了估计参数。变量分为三组：(1)在市场间变化但在每个市场的消费者间不变的变量，(2)在每个市场的消费者间变化的变量，(3)为纠正内生性而包含的额外变量。第一列给出了没有对价格和遗漏属性之间的相关性进行任何修正的模型；效用与我们之前指定的相同，只是残差

{\hat{μ}}_{m j}

_mj 和诱导误差成分η未包含在内。第二列通过包含残差和误差成分应用了控制函数方法。

Without correction, the base price coefficient a is estimated to be –.0202. As we stated previously, the price coefficient is allowed to differ by income group; the estimated price coefficient diffentials by income group, θ_g, are given in the second part of the table because these variables vary over households in each market. For the second income group, the estimated price coefficient is the base of –.0202 plus the differential of .0149, for an estimated price coefficient of –.0053. However, note that for Income Groups 3–5, which comprise the majority of households, the estimated differential exceeds the base in magnitude, such that the estimated price coefficients are positive. This result contradicts the expectation of downward-sloping demand and renders the model implausible for predictive purposes and welfare analysis (because welfare analysis assumes a negative price coefficient).
未经修正，基础价格系数 a 估计为-0.0202。如前所述，价格系数允许因收入组而异；收入组的估计价格系数差异θ _g 在表的第二部分给出，因为这些变量在每个市场的家庭中有所不同。对于第二收入组，估计的价格系数是基础的-0.0202 加上差异的 0.0149，估计的价格系数为-0.0053。然而，请注意，对于收入组 3-5，这些组占大多数家庭，估计的差异在数量上超过了基础，因此估计的价格系数为正。这一结果与需求向下倾斜的预期相矛盾，使得该模型在预测目的和福利分析方面不可信（因为福利分析假设价格系数为负）。

Inclusion of the control functions adjusts the estimated price coefficients in the expected way. We obtained a significant, negative price coefficient for all income groups because the base coefficient estimate increases almost fivefold to –.10. Price elasticities decrease as income rises, with the highest-income group obtaining a price coefficient that is approximately 30% smaller than that of the lowest-income group.
包含控制功能以预期的方式调整了估计的价格系数。我们为所有收入群体获得了显著的负价格系数，因为基准系数估计值几乎增加了五倍，达到-0.10。随着收入的增加，价格弹性下降，最高收入群体的价格系数比最低收入群体的价格系数小约 30%。

The residuals enter significantly and with the expected sign. In particular, a positive residual occurs when the price of the product is higher than can be explained by observed attributes and other observed factors. A positive residual suggests that the product possesses desirable attributes that are not included in the analysis. The residual entering the demand model with a positive coefficient is consistent with this interpretation.
残差显著且符号符合预期。特别是，当产品价格高于可由观察到的属性和其他观察到的因素解释的价格时，会出现正残差。正残差表明产品具有未在分析中包含的理想属性。残差以正系数进入需求模型与这一解释一致。

Neither of the error components is statistically significant, and the hypothesis that both have zero standard deviations cannot be rejected at any meaningful level of confidence. This result might imply that the residuals capture the market-specific unobserved attributed of cable service completely or perhaps reflects the empirical difficulty of estimating alternative-specific normal error components when each alternative also has an i.i.d. extreme value error component.
误差成分均不具有统计显著性，且假设两者的标准差均为零在任何有意义的置信水平下都不能被拒绝。这个结果可能意味着残差完全捕捉了有线服务的市场特定未观察到的属性，或者反映了在每个备选方案也具有独立同分布极值误差成分时估计备选方案特定正态误差成分的经验困难。

Several product attributes are included in the model. In the model without correction, one of these attributes enters with an implausible sign: number of cable channels. With correction, all the product attributes enter with expected signs. In general, the magnitudes are reasonable. An extra premium channel is valued more than an extra cable (nonpremium) channel. An extra over-the-air channel is also valued more than an extra nonpremium cable channel, perhaps because the proliferation of cable channels with low programming content makes the value of extra cable channels relatively low. The option to obtain pay-per-view is valued highly. Note that this attribute, unlike the others, is not on a per-channel basis; its coefficient represents the value of the option to purchase pay-per-view events. The point estimates imply that households are willing to pay $6.00–$8.88 per month for this option, depending on their income.
模型中包含了几个产品属性。在未修正的模型中，其中一个属性的符号不合理：有线频道的数量。经过修正后，所有产品属性的符号都符合预期。总体而言，数值是合理的。一个额外的高级频道比一个额外的普通有线频道更有价值。一个额外的无线频道也比一个额外的普通有线频道更有价值，可能是因为有线频道的泛滥导致其内容质量较低，使得额外有线频道的价值相对较低。按次付费观看的选项被高度重视。请注意，这个属性与其他属性不同，不是按频道计算的；其系数代表了购买按次付费事件的选项的价值。点估计表明，根据家庭收入的不同，家庭愿意为此选项每月支付 6.00 至 8.88 美元。

Several demographic variables enter the model. Their estimated coefficients are fairly similar in the corrected and uncorrected models. The estimates suggest that households with higher education tend to purchase less television reception: The education coefficients are progressively more highly negative for antenna-only (which is zero by normalization), extended basic cable, premium cable, and satellite. Larger households tend not to buy extended basic cable as readily as smaller households. Differences by household size with respect to the other alternatives are highly insignificant. We included a dummy for whether the household rents its dwelling in the two cable alternatives and separately in the satellite alternative. These variables account for the notion that renters are perhaps less able to install a cable hookup and less willing to incur the capital cost of a satellite dish than a household that owns its dwelling. The estimated coefficients are negative, confirming these expectations. Finally, a dummy for whether the household lives in a single-family dwelling enters the satellite alternative to account for the relative difficulty in installing a satellite dish on a multifamily dwelling. As we expected, the estimated coefficient is positive.
几个人口变量进入了模型。它们的估计系数在修正和未修正模型中相当相似。估计结果表明，受教育程度较高的家庭倾向于购买较少的电视接收服务：教育系数对于仅使用天线（通过归一化为零）、扩展基本有线电视、优质有线电视和卫星电视逐渐变得更加负面。较大的家庭不太容易像较小的家庭那样购买扩展基本有线电视。家庭规模对其他选择的差异非常不显著。我们在两个有线电视选项中分别包括了一个家庭是否租房的虚拟变量，并在卫星电视选项中单独包括了一个虚拟变量。这些变量解释了租房者可能不太能够安装有线电视连接，也不太愿意承担卫星天线的资本成本的概念。估计系数为负，证实了这些预期。最后，一个家庭是否居住在单户住宅的虚拟变量进入卫星替代方案，以解释在多户住宅上安装卫星天线的相对困难。正如我们所预期的，估计系数是正的。

Fewer coefficients are significant in the model with correction for endogeneity than in the uncorrected one. We expected this result because the correction for endogeneity attempts to obtain more information from the data (i.e., the relationship of unobserved factors to price and the relationship of observed factors to demand). In other words, the uncorrected model gives a false sense of precision by assuming that price is independent of unobserved factors, when indeed price is related to these factors. Notably, all the coefficients that become insignificant with correction, when they were significant without correction, are for variables that vary over markets but not over consumers in each market. This pattern reflects that unobserved attributes that are correlated with price vary over markets but not over consumers within each market because price itself only varies over markets.
在校正内生性后的模型中，显著的系数比未校正的模型中更少。我们预期到这个结果，因为内生性校正试图从数据中获取更多信息（即未观察因素与价格的关系以及观察因素与需求的关系）。换句话说，未校正的模型通过假设价格与未观察因素独立，给人一种错误的精确感，而实际上价格与这些因素相关。值得注意的是，所有在校正后变得不显著的系数，在未校正时是显著的，都是那些在市场间变化但在每个市场内消费者间不变化的变量。这种模式反映了与价格相关的未观察属性在市场间变化，但在每个市场内的消费者间不变化，因为价格本身只在市场间变化。

Robustness Analysis 稳健性分析

The appropriate control function and distribution for

{\tilde{ε}}_{j n}

is a specification issue. We tried other specifications, including both residuals entering in each cable alternative (to allow for correlation across alternatives as in example 3); a series expansion, both signed and unsigned, of the residuals (to allow for the conditional mean not being exactly as given by a joint normal); correlated rather than independent error components; and exclusion of one or both of the error components (because they are not significant). These alternative specifications all provided similar results.

{\tilde{ε}}_{j n}

的适当控制函数和分布是一个规范问题。我们尝试了其他规范，包括每个电缆替代方案中的残差（如示例 3 中那样允许替代方案之间的相关性）；残差的级数展开，包括有符号和无符号（允许条件均值不完全由联合正态给出）；相关而非独立的误差成分；以及排除一个或两个误差成分（因为它们不显著）。这些替代规范都提供了类似的结果。

As is always the case with endogeneity, the selection of instruments is an issue. As we stated previously, we used the product attributes and Hausman-type prices as instruments, which are widely used but controversial (Bresnahan 1997; Hausman 1997a). With disaggregate demand models, the need for additional instruments is not as stringent as in models with just aggregate data because aggregate demographics do not enter the disaggregate models, but they affect market price. Therefore, they can serve as the extra instruments that are needed for demand estimation.⁹
与内生性问题一样，工具变量的选择也是一个问题。正如我们之前所述，我们使用了产品属性和 Hausman 类型的价格作为工具变量，这些工具变量虽然被广泛使用但也存在争议（Bresnahan 1997；Hausman 1997a）。在非聚合需求模型中，对额外工具变量的需求不像仅有聚合数据的模型那样严格，因为聚合人口统计数据不会进入非聚合模型，但它们会影响市场价格。因此，它们可以作为需求估计所需的额外工具变量。

We reestimated the model without using the prices in other areas as instruments but including the aggregate demographics. With the control function approach, the estimated price coefficient rose when we removed the Hausman-type prices as instruments. This is the direction of change that would be expected if the prices in other markets incorporated the impact of unobserved demand shocks. The other coefficients were not affected under either approach.
我们在不使用其他地区的价格作为工具变量但包括总体人口统计数据的情况下重新估算了模型。使用控制函数方法，当我们去掉 Hausman 类型的价格作为工具变量时，估计的价格系数上升。如果其他市场的价格包含了未观察到的需求冲击的影响，这就是预期的变化方向。其他系数在两种方法下都没有受到影响。

Comparison with Product-Market Control
与产品市场控制的比较

Given the widespread use of the BLP approach, we provide a brief comparison with the control function approach. Then, we discuss BLP model estimation and the results for the same data.
鉴于 BLP 方法的广泛使用，我们简要地与控制函数方法进行比较。然后，我们讨论 BLP 模型估计及相同数据的结果。

The BLP approach uses the aggregate demand equations to recover the unobserved demand factors by matching observed market shares to those predicted by the model. In contrast, our control function approach is based on using different equations for information on the unobserved demand factor, such as the pricing or advertising equation.
BLP 方法使用总需求方程，通过将观察到的市场份额与模型预测的市场份额匹配来恢复未观察到的需求因素。相比之下，我们的控制函数方法是基于使用不同的方程来获取未观察到的需求因素的信息，例如定价或广告方程。

In most applications, the control function approach will be easier to implement than the BLP approach. Often, the first step is just a regression, and the second is maximum likelihood, so the approach can be estimated with standard software packages, such as STATA, SAS (which now has a mixed logit and probit routine), LIMDEP, and Biogeme.¹⁰ The two-step estimator requires us to account for the estimated regressors (as discussed previously), and the sampling covariance can be estimated by bootstrap with these packages.
在大多数应用中，控制函数方法比 BLP 方法更容易实现。通常，第一步只是回归，第二步是最大似然，因此可以使用标准软件包（如 STATA、SAS（现在有混合 logit 和 probit 例程）、LIMDEP 和 Biogeme）来估计该方法。 ¹⁰ 两步估计器要求我们考虑估计的回归变量（如前所述），并且可以通过这些软件包的自举法估计采样协方差。

It is necessary to incorporate a contraction procedure into the estimation routine to implement the BLP estimator. It iteratively calculates the constants that equate predicted and actual shares at each trial value of the parameters. This computation is not trivial, especially when consumer-level data are being used in the estimated specification. Because of this computational burden, the BLP procedure is, to our knowledge, still not available in any of the common statistical packages.
为了实现 BLP 估计器，有必要在估计过程中加入收缩程序。它通过迭代计算常数，使得在每个参数的试验值下预测份额和实际份额相等。这种计算并不简单，特别是在使用消费者级数据进行估计时。由于这种计算负担，据我们所知，BLP 程序在任何常见的统计软件包中仍然不可用。

Because the BLP approach matches observed to predicted shares in a nonlinear setting, it turns out to be sensitive to sampling error in market shares, as Berry, Linton, and Pakes (2004) show. It is not consistent in settings in which there are zero, one, or just a small number of purchase observations per product relative to the number of consumers, as is the case in some data sets.¹¹ It also requires all goods to be strict substitutes, something not required by the control function setup.
由于 BLP 方法在非线性环境中将观察到的份额与预测份额匹配，结果对市场份额中的抽样误差敏感，正如 Berry、Linton 和 Pakes（2004）所示。在每种产品相对于消费者数量只有零、一或少量购买观察的情况下，它是不一致的，正如某些数据集中所存在的情况。 ¹¹ 它还要求所有商品都是严格的替代品，而这是控制函数设置所不需要的。

The BLP approach includes a constant δ_mj for each alternative in each market. All the elements of utility that do not vary within a market are subsumed into these constants. The utility specification given previously becomes the following:
BLP 方法包括每个市场中每个替代方案的常数δ _mj 。在一个市场内不变的所有效用元素都被包含在这些常数中。之前给出的效用规范变成了以下内容：

U_{njm} = δ_{mj} + \sum_{g=2 - 5} θ_{g} p_{mj} d_{gn} + {κk}_{j} s_{n} + {φω}_{n} c_{j} {+ε}_{nj}^{2} .

(24)

The constants are expressed as a function of price and other observed attributes:
常数表示为价格和其他观察属性的函数：

δ_{mj} = {αp}_{mj} + {τx}_{mj} + δ_{j} k_{j} + ε_{mj}^{1} .

(25)

Assuming

ε_{n j}^{2}

and ω_n are i.i.d. extreme value and standard normal, respectively, leads to a mixed logit of the same form as for the control function approach except with constants for each product-market alternative and without the extra error components that are induced by the control function. We estimate the equation for the constants by instrumental variables because utility is assumed to be linear in

ε_{m j}^{1}

and

ε_{m j}^{1}

is correlated with price.
假设

ε_{n j}^{2}

和ω _n 分别是独立同分布的极值和标准正态，导致混合逻辑回归的形式与控制函数方法相同，只是每个产品市场替代项有常数，并且没有由控制函数引起的额外误差成分。我们通过工具变量估计常数的方程，因为效用假设在

ε_{m j}^{1}

中是线性的，并且

ε_{m j}^{1}

与价格相关。

We perform estimation in two stages, with the first stage being the computationally burdensome one. First, we estimate the mixed logit model with constants for each alternative and each market. We recover these constants by solving for the values that match observed to predicted market shares in each market and for each product at every set of parameter values until the minimum is located. Then, these estimated constants are regressed against the product attributes using three-stage least squares.¹² A separate equation is used for the extended basic cable, premium cable, and satellite constants, with the coefficients of the product attributes constrained across equations as in the control function setup (and every characteristics-based setup of which we are aware).
我们分两个阶段进行估计，第一阶段是计算负担较重的阶段。首先，我们估计具有每个替代品和每个市场常数的混合逻辑模型。我们通过求解在每组参数值下匹配观察到的市场份额和预测市场份额的值来恢复这些常数，直到找到最小值。然后，这些估计的常数使用三阶段最小二乘法对产品属性进行回归。 ¹² 对扩展基本有线电视、优质有线电视和卫星常数分别使用单独的方程，产品属性的系数在方程之间受到约束，如控制函数设置（以及我们所知的每个基于特征的设置）中一样。

The results appear in Table 3. The bottom part of the table gives the estimates of the demographic coefficients in the mixed logit model. The top part of the table gives the results of the regression of constants on product attributes. The first column at the top gives the ordinary least squares results, which do not account for omitted attributes, and the second column gives the three-stage least squares results.
结果见表 3。表格的底部给出了混合逻辑模型中人口统计系数的估计值。表格的顶部给出了常数对产品属性回归的结果。顶部的第一列给出了普通最小二乘法的结果，未考虑遗漏的属性，第二列给出了三阶段最小二乘法的结果。

Table 3 MIXED LOGIT MODEL OF TV RECEPTION CHOICE: BLP APPROACH
表 3 电视接收选择的混合逻辑模型：BLP 方法

Explanatory Variable 解释变量	Ordinary Least Squares 普通最小二乘法	Three-Stage Least Squares 三阶段最小二乘法
Variables That Vary over Markets but Are Constant over Consumers in Each Market 在各市场中变化但在每个市场的消费者中保持不变的变量
Price, in dollars per month [1–4] 价格，以每月美元计算 [1–4]	–.0245 (.0091)		–.0922 (.0409)
Number of cable channels [2, 3] 有线频道数量 [2, 3]	–.0024(.0027)		.0017 (.0042)
Number of premium channels [3] 高级频道数量 [3]	.0132 (.0502)		.0463 (.0329)
Number of over-the-air channels (negative) [1] 空中频道数量（负）[1]	.0168 (.0132)		.0196 (.0186)
Whether pay-per-view is offered [2, 3] 是否提供按次付费 [2, 3]	.5872 (.1326)		.7144 (.1814)
Indicator: AT&T is cable company [2] 指标：AT&T 是有线公司 [2]	–.3458 (.2127)		–.2934(.2353)
Indicator: AT&T is cable company [3] 指标：AT&T 是有线公司 [3]	.0158 (.2262)		–.0017 (.2541)
Indicator: Adelphia Comm is cable company [2] 指标：Adelphia Comm 是一家有线电视公司 [2]	.4883 (.2943)		.3837 (.2733)
Indicator: Adelphia Comm is cable company [3] 指标：Adelphia Comm 是一家有线电视公司 [3]	.6111 (.3121)		.5219 (.3065)
Indicator: Cablevision is cable company [2] 指标：有线电视公司是有线公司 [2]	.1905 (.5368)		–.1912 (.5596)
Indicator: Cablevision is cable company [3] 指标：有线电视公司是有线公司 [3]	1.215 (.5829)		.7400 (.6193)
Indicator: Charter Comm is cable company [2] 指标：Charter Comm 是一家有线电视公司 [2]	–.1807 (.2387)		–.1871 (.2196)
Indicator: Charter Comm is cable company [3] 指标：Charter Comm 是一家有线电视公司 [3]	–.0408 (.2539)		–.0685 (.2488)
Indicator: Comcast is cable company [2] 指标：康卡斯特是一家有线电视公司 [2]	–.4097 (.2601)		–.4034 (.2755)
Indicator: Comcast is cable company [3] 指标：康卡斯特是一家有线电视公司 [3]	.1427 (.2755)		.0989 (.3002)
Indicator: Cox Comm is cable company [2] 指标：Cox Comm 是一家有线电视公司 [2]	–.6419 (.4302)		–.6336 (.4225)
Indicator: Cox Comm is cable company [3] 指标：Cox Comm 是一家有线电视公司 [3]	–.0398 (.4564)		–.1563 (.4827)
Indicator: Time Warner is cable company [2] 指标：时代华纳是一家有线电视公司 [2]	–.3756 (.2335)		–.3439 (.2281)
Indicator: Time Warner cable company [3] 指标：时代华纳有线公司 [3]	.0527 (.2503)		–.0009 (.2597)
Alternative-specific constant [2] 替代特定常数 [2]	1.659 (.3486)		3.185 (1.007)
Alternative-specific constant [3] 替代特定常数 [3]	.6462 (.4725)		2.819 (1.480)
Alternative-specific constant [4] 替代特定常数 [4]	.6583 (.1733)		4.635 (.2193)
Variables That Vary over Consumers in Each Market 每个市场中随消费者变化的变量
Price for Income Group 2 [1–4] 收入群体 2 的价格[1–4]		.0156 (.0021)
Price for Income Group 3 [1–4] 收入组 3 的价格 [1–4]		.0273 (.0023)
Price for Income Group 4[1–4] 收入组 4 的价格[1–4]		.0299 (.0027)
Price for Income Group 5 [1–4] 收入组 5 的价格[1–4]		.0353 (.0029)
Education level of household [2] 家庭教育水平 [2]		–.0521 (.0173)
Education level of household [3] 家庭教育水平 [3]		–.1385 (.0203)
Education level of household [4] 家庭教育水平 [4]		–.2525 (.0308)
Household size [2] 家庭规模 [2]		–.0984 (.0240)
Household size [3] 家庭规模 [3]		–.0155 (.0277)
Household size [4] 家庭规模 [4]		–.0235 (.0363)
Household rents dwelling [2, 3] 家庭租赁住宅[2, 3]		–.1494 (.0772)
Household rents dwelling [4] 家庭租赁住宅 [4]		–.5470 (.1349)
Single-family dwelling [4] 独栋住宅 [4]		.1967 (.1023)
Error component for nonantenna alternatives, SD [2–4] 非天线替代方案的误差组件，SD [2–4]		.7775 (.1664)
Log-likelihood at convergence 收敛时的对数似然值		–13,927.40
Number of observations 观测数量		11,810

Notes: Alternatives: (1) antenna-only, (2) basic and extended cable, (3) premium cable, and (4) satellite. Each variable enters the alternatives that are listed in brackets and takes the value zero in other alternatives. Estimates that are statistically significant (p < .05) are in bold. Standard errors are in parentheses.
注释：替代方案：(1) 仅天线，(2) 基本和扩展有线电视，(3) 高级有线电视，(4) 卫星。每个变量进入括号中列出的替代方案，并在其他替代方案中取值为零。统计显著性估计（p < .05）用粗体表示。标准误差在括号中。

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As with the control function approach, the correction for omitted variables raises the estimated price coefficients. Without correction, three of the five income groups received a positive estimated price coefficient. With correction, all groups obtain a significantly negative price coefficient.
与控制函数方法一样，遗漏变量的校正提高了估计的价格系数。未经校正，五个收入组中的三个得到了正的估计价格系数。经过校正，所有组都获得了显著的负价格系数。

The estimated base price coefficient is –.0922, compared with the –.1003 obtained with the control function approach. The estimates of θ_g, the incremental price coefficient for higher-income groups, are similar under the two approaches. As in the control function approach, the number of cable channels obtains a negative coefficient when endogeneity is ignored and, as we expected, becomes positive when the endogeneity is corrected. All the product attributes obtain similar values as with the control function approach.
估计的基础价格系数是-0.0922，而使用控制函数方法得到的是-0.1003。对于高收入群体的增量价格系数θ _g ，两种方法的估计值相似。与控制函数方法一样，当忽略内生性时，有线电视频道的数量获得负系数，而当内生性得到纠正时，正如我们预期的那样，系数变为正值。所有产品属性的估计值与控制函数方法相似。

The demographic coefficients in Table 3 provide similar conclusions as those from the control function approach. Education induces households to buy less television reception. Larger households tend not to buy extended basic cable. Renters tend not to buy cable and satellite as readily as owners. Single-family dwellers tend to purchase satellite reception more readily than households that live in multifamily dwellings. Differences appear not to be statistically significant.
表 3 中的人口系数提供了与控制函数方法类似的结论。教育促使家庭减少购买电视接收服务。较大的家庭往往不购买扩展的基本有线电视。租房者不如房主那样容易购买有线电视和卫星电视。单户家庭比住在多户住宅的家庭更容易购买卫星接收服务。差异似乎在统计上不显著。

Table 4 gives price elasticities from the models for each approach. The two methods give similar elasticities. For example, the same-price elasticity for basic and extended cable is −1.08 with the control function approach and –.97 under the BLP approach.
表 4 给出了每种方法模型的价格弹性。两种方法给出的弹性相似。例如，基本和扩展有线电视的同价弹性在控制函数方法下为-1.08，在 BLP 方法下为-0.97。

Table 4 Estimated Elasticities
表 4 估计弹性

	Control Function 控制功能	BLP
Price of Extended Basic Cable 扩展基本有线电视的价格
Antenna-only share 仅限天线共享	.97	.79
Extended basic cable share 扩展基本有线电视份额	–1.08	–.97
Premium cable share 高级有线电视份额	.76	.88
Satellite share 卫星份额	1.02	.87
Price of Premium Cable 高级电缆价格
Antenna-only share 仅天线共享	.48	.52
Extended basic cable share 扩展基本有线电视份额	.50	.57
Premium cable share 高级有线电视份额	–1.83	–2.04
Satellite share 卫星份额	.48	.58
Price of Satellite 卫星价格
Antenna-only share 仅天线共享	.50	.42
Extended basic cable share 扩展基本有线电视份额	.40	.43
Premium cable share 高级有线电视份额	.37	.45
Satellite share 卫星共享	–3.77	–3.59

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Conclusion 结论

The concern that price, advertising, or other variables are endogenous has proved to be important in many applications. In this article, we propose a control function approach for handling endogeneity in choice models. It uses observed variables and economic theory to derive controls for the part of the unobserved demand factor that is not independent of the endogenous variable. We use the pricing equation in this article to derive controls, but we believe that there are many other possible equations (e.g., the advertising equation) that also contain information on unobserved demand factors.
价格、广告或其他变量是内生的这一担忧在许多应用中已被证明是重要的。在本文中，我们提出了一种控制函数方法来处理选择模型中的内生性问题。它利用观察到的变量和经济理论来推导出对未观察到的需求因素中不独立于内生变量的部分的控制。我们在本文中使用定价方程来推导控制，但我们相信还有许多其他可能的方程（例如，广告方程）也包含关于未观察到的需求因素的信息。

The approach provides an alternative to the commonly used BLP product-market controls for unobserved quality, which is sensitive to sampling error in market shares, more difficult to estimate (especially with consumer-level data), and not applicable for many recently proposed discrete choice demand estimators.
该方法为常用的 BLP 产品市场控制提供了一种替代方案，这些控制对市场份额中的抽样误差敏感，更难估计（尤其是使用消费者级数据时），并且不适用于许多最近提出的离散选择需求估计器。

We apply both methods to examine households’ choices among television options, including basic and premium cable packages, in which unobserved attributes, such as quality of programming, are expected to be correlated with price. Without correcting for endogeneity, aggregate demand for each television option is upward-sloping. The corrected estimates from both the control function method and the product-market controls method produce similar and much more realistic demand elasticities.
我们应用这两种方法来研究家庭在电视选项中的选择，包括基本和高级有线电视套餐，其中未观察到的属性（如节目质量）预计与价格相关。如果不纠正内生性，每个电视选项的总需求是向上倾斜的。通过控制函数方法和产品市场控制方法得到的修正估计产生了相似且更为现实的需求弹性。

Footnotes 脚注

1 Another common bias arises when estimating the elasticity of demand for recreation, shopping, theater, and so on, with respect to travel time. If people with strong tastes for such activities live close to these activities, travel time to the desired activity will be negatively correlated with unobserved taste, leading to a negative bias in the elasticity of demand with respect to travel time.
另一个常见的偏差出现在估计娱乐、购物、剧院等需求的弹性时，涉及到旅行时间。如果对这些活动有强烈兴趣的人住在这些活动附近，前往所需活动的旅行时间将与未观察到的兴趣负相关，从而导致需求弹性对旅行时间的负偏差。

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2 Variation in unobserved variables, including unmeasured attributes and advertising, may itself be caused by changes in demand conditions, such as shifts in tastes. In this case, a model that describes the variation in demand and its relationship to the unobserved variables would more fully represent the situation.
未观察变量的变化，包括未测量的属性和广告，可能本身是由需求条件的变化引起的，例如品味的转变。在这种情况下，描述需求变化及其与未观察变量关系的模型将更全面地代表这种情况。

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3 Heckman and Robb (1985) introduced the term “control function” in the context of selection models, but the concepts date back at least to Heckman (1978) and Hausman (1978). The method has been applied to a Tobit model by Smith and Blundell (1986) and to binary probit models by Rivers and Vuong (1988). Blundell and Powell (2004) include it in their discussion of semiparametric methods for binary choice.
赫克曼和罗布（1985）在选择模型的背景下引入了“控制函数”一词，但这些概念至少可以追溯到赫克曼（1978）和豪斯曼（1978）。该方法已被史密斯和布伦德尔（1986）应用于 Tobit 模型，并被里弗斯和武昂（1988）应用于二元 Probit 模型。布伦德尔和鲍威尔（2004）在他们关于二元选择的半参数方法讨论中也包括了这一方法。

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4 “Observed” and “unobserved” are from the perspective of the practitioner. All terms are observed by the consumer when making the decision.
4 “观察到的”和“未观察到的”是从实践者的角度来看。消费者在做决定时会观察到所有术语。

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5 As Imbens and Newey (2008) show, it is sufficient to condition on any one-to-one function of μ_n.
正如 Imbens 和 Newey（2008）所示，条件于μ _n 的任何一对一函数是足够的。

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6 The more common form of this equation is (p—MC)/p = 1/|e|.
这种方程的更常见形式是 (p—MC)/p = 1/|e|。

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7 To our knowledge, there has been no prior application of the control function approach with cross-sectional market data. Villas-Boas and Winer (1999) use lagged prices as instruments in their time-series model.
据我们所知，控制函数方法尚未在横截面市场数据中应用过。Villas-Boas 和 Winer（1999）在他们的时间序列模型中使用滞后价格作为工具变量。

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8 We discuss the use of alternative instruments in the “Robustness Analysis” section.
我们在“稳健性分析”部分讨论了替代工具的使用。

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9 Consider two households that have the same demographics but live in areas in which the aggregate demographics are different. Part of the price difference between the two areas is presumably attributable to the difference in aggregate demographics. This part of the price difference provides variation in price over households that can be used for estimation of price response.
考虑两个具有相同人口统计特征的家庭，但它们居住在总体人口统计特征不同的地区。两个地区之间的部分价格差异可能归因于总体人口统计特征的差异。这部分价格差异提供了家庭之间的价格变化，可用于估计价格反应。

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10 Matlab and Gauss codes for mixed logit are also available (free) from Train's Web site at http://elsa.berkeley.edu/~train/software.html.
10 个用于混合逻辑回归的 Matlab 和 Gauss 代码也可以从 Train 的网站 http://elsa.berkeley.edu/~train/software.html 免费下载。

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11 For example, in many housing data sets, houses are purchased zero or one time, violating the consistency condition of the BLP estimator. Another example is Martin's (2008) study of customers’ choice between incandescent and compact fluorescent light bulbs (CFLs), where advertising and promotions (e.g., discount coupons) occurred on a weekly basis and varied over stores, and yet it was common for a store not to sell any CFLs in a given week. With the market defined as a store–week, Martin reports that 65% of the market shares were zero for CFLs.
例如，在许多住房数据集中，房屋被购买零次或一次，违反了 BLP 估计量的一致性条件。另一个例子是 Martin（2008）对客户在白炽灯和紧凑型荧光灯（CFL）之间选择的研究，其中广告和促销（例如，折扣券）每周发生并在各个商店之间有所不同，但在某一周内某个商店不卖任何 CFL 是很常见的。在将市场定义为商店-周的情况下，Martin 报告说 65%的 CFL 市场份额为零。

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12 The negative of the number of over-the-air channels enters these equations because this attribute enters the antenna-only alternative in the model of Table 2, whereas it is now entering the constants for the nonantenna alternatives.
12 个无线电视频道的负数进入这些方程，因为在表 2 的模型中，这个属性进入了仅天线的替代方案，而现在它进入了非天线替代方案的常数。

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References 参考文献

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Journal of Marketing Research

Volume 47, Issue 1

Pages: 3 - 13

Article first published online: February 1, 2010

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Authors

Amil Petrin

Kenneth Train

Notes

(e-mail: petrin@umn.edu).

(e-mail: train@econ.berkeley.edu).

The authors thank the two anonymous JMR reviewers for their useful comments and gratefully acknowledge support from the National Bureau of Economic Research. Pradeep Chintagunta served as associate editor for this article.

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Table 1 Demographic Variables and Service Attributes

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Table 2 MIXED LOGIT MODEL OF TELEVISION RECEPTION CHOICE: CONTROL FUNCTION APPROACH

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Table 3 MIXED LOGIT MODEL OF TV RECEPTION CHOICE: BLP APPROACH

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Table 4 Estimated Elasticities

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Abstract 摘要

Model 模型

Parametric Functional Forms参数化函数形式

Example 1: Jointly Normal Errors, Independent over j示例 1：联合正态误差，j 之间独立

Example 2: Extreme Value and Joint Normal Error Components, Independent over j示例 2：极值和联合正态误差分量，在 j 之间独立

Example 3: Extreme Value and Joint Normal Error Components, Correlation over j示例 3：极值和联合正态误差分量，j 之间的相关性

Pricing Behavior and the Control Function Approach定价行为与控制函数方法

Marginal Cost Pricing 边际成本定价

Monopoly Pricing 垄断定价

Empirical Model and Data 经验模型和数据

Results 结果

Robustness Analysis 稳健性分析

Comparison with Product-Market Control与产品市场控制的比较

Conclusion 结论

Footnotes 脚注

References 参考文献

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Authors

Amil Petrin

Kenneth Train

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Parametric Functional Forms
参数化函数形式

Example 1: Jointly Normal Errors, Independent over j
示例 1：联合正态误差，j 之间独立

Example 2: Extreme Value and Joint Normal Error Components, Independent over j
示例 2：极值和联合正态误差分量，在 j 之间独立

Example 3: Extreme Value and Joint Normal Error Components, Correlation over j
示例 3：极值和联合正态误差分量，j 之间的相关性

Pricing Behavior and the Control Function Approach
定价行为与控制函数方法

Comparison with Product-Market Control
与产品市场控制的比较

Article usage^*