Estimation of random coefficients logit demand models with interactive fixed effects

Estimation of Random Coefficients Logit Demand Models with Interactive Fixed Effects

\abstract{

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{ Keywords:} discrete-choice demand model, interactive fixed effects, factor analysis, panel data, random utility model.\\ { JEL codes:} C23, C25.

Introduction

The Berry, Levinsohn and Pakes BerryLevinsohnPakes1995 (hereafter BLP) demand model, based on the random coefficients logit multinomial choice model, has become the workhorse of demand modeling in empirical industrial organization and antitrust analysis. An important virtue of this model is that it parsimoniously and flexibly captures substitution possibilities between the products in a market. At the same time, the nested simulated GMM procedure proposed by BLP accommodates possible endogeneity of the observed product-specific regressors, notably price. This model and estimation approach has proven very popular (e.g. Nevo nevo1, Petrin petrin; surveyed in Ackerberg et. al. ack_pakes).

Taking a cue from recent developments in panel data econometrics (e.g. Bai and Ng BaiNg2006, Bai Bai2009, and Moon and Weidner MoonWeidner2015,MoonWeidner2015b), we extend the standard BLP demand model by adding interactive fixed effects to the unobserved product characteristic, which is the main “structural error” in the BLP model. This interactive fixed effect specification combines market (or time) specific fixed effects with product specific fixed effects in a multiplicative form, which is often referred to as a factor structure.

Our factor-based approach extends the baseline BLP model in two ways. First, we offer an alternative to the usual moment-based GMM approach. The interactive fixed effects “soak up” some important channels of endogeneity, which may obviate the need for instrumental variables of endogenous regressors such as price. This is important as such instruments may not be easy to identify in practice. Moreover, our analysis of the BLP model with interactive fixed effects illustrates that the problem of finding instruments for price (which arises in any typical demand model) is distinct from the problem of underidentification of some model parameters (such as the variance parameters for the random components), which arises from the specific nonlinearities in the BLP random coefficients demand model. In our setting, the fixed effects may obviate the need for instruments to control for price endogeneity but, as we will point out, we still need to impose additional moment conditions in order to identify these nonlinear parameters. Second, even if endogeneity persists in the presence of the interactive fixed effects, the instruments only need to be exogenous with respect to the residual part of the unobserved product characteristics, which is not explained by the interactive fixed effect. This may expand the set of variables which may be used as instruments.

To our knowledge, the current paper presents the first application of some recent developments in the econometrics of long panels (with product and market fixed effects) to the workhorse demand model in empirical IO. Relative to the existing panel factor literature (for instance, Bai Bai2009, and Moon and Weidner MoonWeidner2015,MoonWeidner2015b) that assume a linear regression with exogenous regressors, the nonlinear model that we consider here poses both identification and estimation challenges. Namely, the usual principal components approach for linear factor models with exogenous regressors is inadequate due to the nonlinearity of the model and the potentially endogenous regressors. At the same time, the conventional GMM approach of BLP cannot be used for identification and estimation due to the presence of the interactive fixed effects.

We propose an alternative identification and estimation scheme which we call the {\em Least Squares-Minimum Distance (LS-MD)} method. It consists of two steps. The first step is a least squares regression of the mean utility on the included product-market specific regressors, factors, and the instrumental variables. The second step minimizes the norm of the least squares coefficient of the instrumental variables in the first step. This estimation approach is similar to the two stage estimation method for a class of instrumental quantile regressions in Chernozhukov and Hansen ChernozhukovHansen2006. We show that under regularity conditions that are comparable to the standard GMM problem, the parameter of interest is point identified and its estimator is consistent. We also derive the limit distribution under asymptotic sequences where both the number of products and the number of markets converge to infinity. In practice, the estimator is simple and straightforward to compute. Monte Carlo simulations demonstrate its good small-sample properties.

Our work complements some recent papers in which alternative estimation approaches and extensions of the standard random coefficients logit model have been proposed, including Villas-Boas and Winer Villasboas1999, Knittel and Metaxoglou KnittelMetaxoglou2014, Dube, Fox and Su DubeFoxSu2012, Harding and Hausman HardingHausman2007, Bajari, Fox, Kim and Ryan bajari_fox_kim_ryan, and Gandhi, Kim and Petrin GandhiKimPetrin2010.

We illustrate our estimator on a dataset of market shares for automobiles, inspired by the exercise in BLP. This application illustrates that our estimator is easy to compute in practice. Significantly, we find that, once factors are included in the specification, the estimation results under the assumption of exogenous and endogenous price are quite similar, suggesting that the factors are indeed capturing much of the unobservable product and time effects leading to price endogeneity.

The paper is organized as follows. Section 2 introduces the model. In Section 3 we discuss how to identify the model when valid instruments are available. In Section 4 we introduce the LS-MD estimation method. Consistency and asymptotic normality are discussed in Section 5. Section 6 contains Monte Carlo simulation results, and Section 7 discusses the empirical example. Section 8 concludes. In the appendix we list the assumptions for the asymptotic analysis and provide technical derivations and proofs of results in the main text.

Notation

We write $A'$ for the transpose of a matrix or vector $A$. For column vectors $v$ the Euclidean norm is defined by $\| v \| = \sqrt{v^{\prime}v}$ . For the $n$-th largest eigenvalues (counting multiple eigenvalues multiple times) of a symmetric matrix $B$ we write $\mu_n(B)$. For an $m\times n$ matrix $A$ the Frobenius norm is $\| A \|_{F} = \sqrt{{\rm Tr}(AA^{\prime})}$, and the spectral norm is $\| A \| = \max_{0 \neq v \in \mathbb{R}^n} \, \frac{ \| A v \|} {\| v\|}$, or equivalently $\| A \| = \sqrt{ \mu_1(A^{\prime}A) }$. Furthermore, we use $P_A = A (A^{\prime}A)^\dagger A'$ and $M_A = \mathbbm{1}_m - A (A^{\prime}A)^\dagger A'$, where $\mathbbm{1}_m$ is the $m\times m$ identity matrix, and $(A^{\prime}A)^\dagger$ denotes a generalized inverse, since $A$ may not have full column rank. The vectorization of an $m\times n$ matrix $A$ is denoted ${\rm vec}(A)$, which is the $mn\times 1$ vector obtained by stacking the columns of $A$. For square matrices $B$, $C$, we use $B>C$ (or $B\geq C$) to indicate that $B-C$ is positive (semi) definite. We use $\nabla$ for the gradient of a function, i.e. $\nabla f(x)$ is the vector of partial derivatives of $f$ with respect to each component of $x$. We use “wpa1” for “with probability approaching one”.

Model

The random coefficients logit demand model is an aggregate market-level model, formulated at the individual consumer-level. Consumer $i$'s utility of product $j$ in market\footnote{ The $t$ subscript can also denote different time periods. } $t$ is given by

where $\epsilon_{ijt}$ is an idiosyncratic product-specific preference shock, and $v_i = (v_{i1}, \ldots, v_{iK})'$ is an idiosyncratic characteristic preference. The mean utility is defined as

where $X_{jt} = \left(X_{1,jt},\ldots,X_{K,jt} \right)'$ is a vector of $K$ observed product characteristics (including price), and $\beta^0=\left(\beta^0_1,\ldots,\beta^0_K\right)'$ is the corresponding vector of coefficients. Following BLP, $\xi^0_{jt}$ denotes unobserved product characteristics of product $j$, which can vary across markets $t$. This is a “structural error”, in that it is observed by all consumers when they make their decisions, but is unobserved by the econometrician. In this paper, we focus on the case where these unobserved product characteristics vary across products and markets according to a factor structure:

where $\lambda^0_j = \left( \lambda^0_{1j},\ldots, \lambda^0_{Rj} \right)'$ is a vector of factor loadings corresponding to the $R$ factors\footnote{ Depending on the specific application one has in mind one may have different interpretations for $\lambda_j$ and $f_t$. For example, in the case of national brands sold in different markets it seems more natural to interpret $\lambda_j$ as the underlying factor (a vector product qualities) and $f_t$ as the corresponding loadings (market specific tastes for these qualities). For convenience, we refer to $f_t$ as factors and $\lambda_j$ as factor loadings throughout the whole paper, which is the typical naming convention in applications where $t$ refers to time.} $f^0_t = \left( f^0_{1t},\ldots, f^0_{Rt} \right)'$, and $e_{jt}$ is a product and market specific error term. Here $\lambda^{0 \prime}_j \, f^0_t$ represent interactive fixed effects, in that both the factors $f^0_t$ and factor loadings $\lambda^0_j$ are unobserved to the econometrician, and can be correlated arbitrarily with the observed product characteristics $X_{jt}$. We assume that the number of factors $R$ is known.\footnote{ Known $R$ is also assumed in Bai (2009) and Moon and Weidner MoonWeidner2015 for the linear regression model with interactive fixed effects. Allowing for $R$ to be unknown presents a substantial technical challenge even for the linear model, and therefore goes beyond the scope of the present paper. } The superscript zero indicates the true parameters, and objects evaluated at the true parameters. Let $\lambda^0 = (\lambda^0_{jr})$ and $f^0 = (\lambda^0_{tr})$ be $J \times R$ and $T \times R$ matrices, respectively.

The factor structure in equation ((ref)) approximates reasonably some unobserved product and market characteristics of interest in an interactive form. For example, television advertising is well-known to be composed of a product-specific component as well as an annual cyclical component (peaking during the winter and summer months).\footnote{ cf. {\em TV Dimensions} tv_dimensions.} The factors and factor loadings can also explain strong correlation of the observed market shares over both products and markets, which is a stylized fact in many industries that has motivated some recent dynamic oligopoly models of industry evolution (e.g. Besanko and Doraszelski BesankoDoraszelski2004). The standard BLP estimation approach, based on moment conditions, allows for weak correlation across markets and products, but does not admit strong correlation due to shocks that affect all products and markets simultaneously, which we model via the factor structure.

To begin with, we assume that the regressors $X_{jt}$ are exogenous with respect to the errors $e_{jt}$, that is, $X_{jt}$ and $e_{jt}$ are uncorrelated for given ($j$, $t$). This assumption, however, is only made for ease of exposition, and in both Section (ref) below and in the empirical illustration, we consider the more general case where regressors (such as price) may be endogenous. Notwithstanding, regressors which are strictly exogenous with respect to $e_{jt}$ can still be endogenous with respect to the $\xi^0_{jt}$, due to correlation of the regressors with the factors and factor loadings. Thus, including the interactive fixed effects may “eliminate” endogeneity problems, so that instruments for endogeneity may no longer be needed. This possibility of estimating a demand model without searching for instruments may be of great practical use in antitrust analysis.

Moreover, when endogeneity persists even given the interactive fixed effects, then our approach may allow for a larger set of IV's. For instance, one criticism of the so-called “Hausman” instruments (cf. Hausman hausmancereal) -- that is, using the price of product $j$ in market $t'$ as an instrument for the price of product $j$ in market $t$ -- is that they may not be independent of “nationwide” demand shocks -- that is, product-specific shocks which are correlated across markets. Our interactive fixed effect $\lambda_j'f_t$ can be interpreted as one type of nationwide demand shock, where the $\lambda_j$ factor loadings capture common (nationwide) components in the shocks across different markets $t$ and $t'$. Since the instruments in our model can be arbitrarily correlated with $\lambda_j$ and $f_t$, the use of Hausman instruments in our model may be (at least partially) immune to the aforementioned criticism.

Next, we introduce the key equations for market shares in the random-coefficient logit demand model. Following Berry, Levinsohn, and Pakes BerryLevinsohnPakes1995, the probability that agent $i$ chooses product $j$ in market $t$ takes the multinomial logit form:

We do not observe individual specific choices, but market shares of the $J$ products in the $T$ markets. The market share of product $j$ in market $t$ is given by

where $G_{\alpha^0}(v)$ is the known distribution of consumer taste $v_i$ over the product characteristic, and $\alpha^0$ is an $L\times 1$ vector of parameters of this distribution.\footnote{The dependence of $\pi_{jt}(\delta_{t},X_t,v_i)$ and $s_{jt}(\alpha^0,\delta_t,X_t)$ on $t$ stems from the arguments $\delta_{t}$ and $X_t$.}

Underlying these derivations are assumptions that (i) the distributions of $\epsilon=(\epsilon_{ijt})$ and $v=(v_i)$ are mutually independent, and are also independent of $X=(X_{jt})$ and $\xi^0=(\xi^0_{jt})$; (ii) $\epsilon_{ijt}$ follows a marginal type I extreme value distribution iid across $i$ and $j$ (but not necessarily independent across $t$).\footnote{When the index $t$ refers to time (or otherwise possesses some natural ordering), then sequential exogeneity is allowed throughout the whole paper, that is, $X_{jt}$ can be correlated with past values of the errors, $e_{js}, s < t$. The errors $e_{jt}$ are assumed to be independent across $j$ and $t$, but heteroscedasticity is allowed. } Moreover, the most often used specification for the distribution of random coefficients in the literature is to assume that they have a multivariate normal distribution, that is, $v \sim {\cal N}(0,\Sigma^0)$, where $\Sigma^0$ is a $K\times K$ matrix of parameters, which can be subject to constraints (e.g. only one or a few regressors may have random coefficients, in which case the components of $\Sigma^0$ are only non-zero for these regressors), and $\alpha^0$ consists of the independent parameters in $\Sigma^0$.\footnote{ We focus in this paper on the case where the functional form of the distribution function $G_{\alpha}$ is known by the researcher. Recent papers have addressed estimation when this is not known; e.g. Bajari, Fox, Kim and Ryan bajari_fox_kim_ryan, bajari_fox_kim_ryan2. }

The observables in this model are the market shares $s_{jt}$ and the regressors $X_{jt}$.\footnote{ In the present paper we assume that the true market shares $s_{jt}=s_{jt}(\delta^0_t)$ are observed. Berry, Linton and Pakes BerryLintonPakes2004 explicitly consider sampling error in the observed market shares in their asymptotic theory. Here, we abstract away from this additional complication and focus on the econometric issues introduced by the factor structure in $\xi^0$. } In addition, we need $M$ instruments $Z_{jt}=( Z_{1,jt}, \ldots, Z_{M,jt} )'$ to construct extra (unconditional) moment conditions, in addition to the unconditional moment conditions constructed by $X_{jt}$, in order to estimate the parameters $\alpha$, with $M \geq L$. These additional instruments are also needed in the usual BLP estimation procedure, even in the absence of the factor structure. Suppose that $X_{jt}$ is exogenous with respect to $\xi^0_{j,t}$. From this, we construct unconditional moment conditions $\mathbb{E}(X_{jt} \xi^0_{j,t}) = 0$. Then, extra moment conditions are still required to identify the covariance parameters in the random coefficients distribution. Notice that those $Z$'s may be non-linear functions of the exogeneous $X$'s, so we do not necessarily need to observe additional exogenous variables.\footnote{ If one is willing to impose the conditional moment condition $\mathbbm{E}(e_{jt} | X_{jt})=0$, then valid $Z_{jt}$ can be constructed as non-linear transformations of $X_{jt}$. }

Let $s = (s_{jt})$, $X_k = (X_{k,jt})$, $Z_{m} = (Z_{m,jt})$ and $e=(e_{jt})$ be $J \times T$ matrices, and also define the tensors $X=(X_{k,jt})$ and $Z= (Z_{m,jt})$, which contain all observed product characteristics and instruments. In the presence of the unobserved factor structure, it is difficult to identify regression parameters of regressors $X_k$ that have a factor structure themselves, which includes product invariant and time invariant regressors. Our assumptions below rule out all those $X_k$ and $Z_m$ that have a low rank when considered as a $J \times T$ matrix.\footnote{ This is exactly analogous to the usual short panel case, in which the presence of fixed effects for each cross-sectional unit precludes identification of the coefficients on time-invariant regressors. If the number of factors $R$ is known accurately, then the coefficients of these low-rank regressors can be identified, but the necessary regularity conditions are relatively cumbersome. For ease of exposition we will therefore rule out both low-rank regressors and low-rank instruments by our assumptions below, and we refer to Bai Bai2009 and Moon and Weidner MoonWeidner2015 for a further discussion of this topic. } The unknown parameters are $\alpha^0$, $\beta^0$, $\lambda^0$, and $f^0$.

The existing literature on demand estimation usually considers asymptotic sequences with either $J$ growing large and $T$ fixed, or $T$ growing large and $J$ fixed. Under these standard asymptotic sequences, the estimation of the nuisance parameters $\lambda^0$ and $f^0$ creates a Neyman and Scott NeymanScott1948 incidental parameter problem: because the number of nuisance parameters grows with the sample size, the estimators for the parameters of interest become inconsistent. Following some recent panel data literature, for example, Hahn and Kuersteiner Hahn:2002p717,Hahn:2004p878 and Hahn and Newey Hahn:2004p882, we handle this problem by considering asymptotic sequences where both $J$ and $T$ become large. Under this alternative asymptotic, the incidental parameter problem is transformed into the issue of asymptotic bias in the limiting distribution of the estimators of the parameters of interest. This asymptotic bias can be characterized and corrected for. Our Monte Carlo simulations suggest that the alternative asymptotic provides a good approximation of the properties of our estimator at finite sample sizes, as long as $J$ and $T$ are moderately large.

Identification

Given the non-linearity of the model, questions regarding the identification of the model parameters of interest are naturally raised. In the following we provide conditions under which the parameters $\alpha$ and $\beta$ as well as the product $\lambda f'$ are identified. We do not consider how to identify $\lambda$ and $f$ separately, because they only enter into the model jointly as $\lambda f'$.\footnote{The transformation $\lambda \rightarrow \lambda S$ and $f \rightarrow f S^{-1}$ gives observationally equivalent parameters for any non-degenerate $R \times R$ matrix $S$. Once the product $\lambda f'$ is identified, one can impose further normalization restrictions to identify $\lambda$ and $f$ separately, if desired.}

Following standard identification arguments (e.g. Matzkin Matzkin2013), our proof demonstrates identification by showing the existence of an injective mapping from the model parameters $(\alpha,\beta,\lambda f')$ and the distribution of the random elements of the model $(e,X,Z)$ to the distribution of the observed data $(s,X,Z)$, where the random elements of the model are comprised of unobserved error terms, product characteristics, and instruments and the observed data are the market shares, product characteristics, and instruments.\footnote{ Injectivity implies that the mapping is one-to-one -- and hence invertible -- along the relevant range. The range of this mapping excludes some distributions of $(s,X,Z)$; for instance, distributions in which some of the market shares take zero values with non-zero probability cannot be generated by our model, due to the multinomial logit structure. See Gandhi, Lu, and Shi GandhiLuShi2013 for additional discussion of estimating discrete-choice demand models when some of the products are observed to have zero market shares.}

Our identification result utilizes a population distribution of a full $J \times T$ panel of observables ($s$, $X$, $Z$), conditional on parameters $\alpha$, $\beta$ and $\lambda f^{\prime}$. The fact that we have nuisance parameters $\lambda_j$ and $f_t$ in both panel dimensions makes the distribution of the full $J \times T$ panel of observables a natural starting point for the identification discussion (where $J$ and $T$ are are finite constants in this section). Normally in the large $N,T$ panel data literature (e.g. in Hahn and Newey Hahn:2004p882, Bai Bai2009, etc.) there is no explicit identification discussion, but consistency as $N,T \rightarrow \infty$ (or in our case $J,T \rightarrow \infty$) is shown directly. The reason is that there is no fixed population distribution that corresponds to the sample as both panel dimensions become large. Thus, when going from identification to estimation there will not be a simple analog principle that allows to treat the sample as multiple draws from the population. This is a general conceptual issue, independent of our paper. The inference results below therefore do not follow immediately from the identification result presented in this section; in particular, the incidental parameter problem (Neyman and Scott NeymanScott1948) related to inference of $\lambda_j$ and $f_t$ needs to be properly addressed.

For our identification result we assume, as in BLP, that there exists a one-to-one relationship between market shares and mean utilities, as summarized by the following assumption. Let ${\cal B}_\alpha \subset \mathbbm{R}^L$ be a given parameter set for $\alpha$.

\newtheorem{INVassumption}{Assumption}

Berry, Gandhi, and Haile BerryGandhiHaile2013 provide general conditions under which this invertibility assumption is satisfied, and Berry and Haile berry_haile_market and Chiappori and Komunjer chiappori_komunjer utilize this inverse mapping in their nonparametric identification results.

Using Assumption (ref) and the specifications (ref) and (ref) we have

In $JT$-vector notation this equation can be written as $\delta^{\rm vec}(\alpha^0) = x \beta^0 + \sum_{r=1}^R f^0_{\cdot r} \otimes \lambda^0_{\cdot r} + e^{\rm vec}$, where $\delta^{\rm vec}(\alpha) = {\rm vec}[ \delta(\alpha,\, s,\, X) ] $ and $e^{\rm vec} = {\rm vec}(e)$ are $JT$-vectors, and $x$ is a $JT \times K$ matrix with columns $x_{.,k} = {\rm vec}\left( X_{k} \right)$. For simplicity we suppress the dependence of $\delta^{\rm vec}(\alpha)$ on $s$ and $X$. It is furthermore convenient to define the $JT \times M$ matrix $z$ with columns $z_{.,m} = {\rm vec}\left( Z_{m} \right)$, the mean utility difference $d(\alpha) = \delta^{\rm vec}(\alpha) - \delta^{\rm vec}(\alpha^0)$, and the unobserved utility difference $\Delta \xi_{\alpha,\beta} = d(\alpha)-x(\beta-\beta^0)$. Both $d(\alpha)$ and $\Delta \xi_{\alpha,\beta} $ are $JT$ vectors. Note that $\Delta \xi_{\alpha,\beta}$ is simply the vectorized difference of the residual unobserved product characteristic at $(\alpha,\beta)$ and $(\alpha^0,\beta^0)$. In the following the indices $j$ and $t$ run from $1$ to $J$ and $1$ to $T$, respectively.

\newtheorem{IDassumption}{Assumption}

In this assumption, and also for the remainder of the paper, we treat the fixed effects $\lambda^0$ and $f^0$ as non-random parameters, that is, all expectations in Assumption ID are implicitly conditional on $\lambda^0$ and $f^0$. The assumptions are discussed in Section (ref) below.

To formulate our identification result we need to introduce some additional notation. We denote the set of joint distributions of $e$, $X$, $Z$ by ${\cal F}_{e,X,Z}$, and the set of joint distributions of $s$, $X$, $Z$ (the observables) by ${\cal F}_{s,X,Z}$. The model described in Section (ref) gives unique market shares $s$ for any given $e$, $X$, $Z$ and parameters $\alpha$, $\beta$, $\lambda f^{\prime}$. The model therefore also uniquely describes the distribution of observables for a given distribution $F_{e,X,Z} \in {\cal F}_{e,X,Z}$ and parameters $\alpha$, $\beta$, $\lambda f^{\prime}$, and we denote this distribution of observables given by the model as $\Gamma(\alpha, \beta, \lambda f^{\prime}, F_{e,X,Z}) \in {\cal F}_{s,X,Z}$. We say that two distributions $F_1,F_2 \in {\cal F}_{s,X,Z}$ are equal if the corresponding joint cdf's are the same, and we write $F_1 = F_2$ in that case. Analogously, we define equality on ${\cal F}_{e,X,Z}$.

The theorem states that if the distribution of observables $F^0_{s,X,Z} =\Gamma(\alpha^0, \beta^0, \lambda^0 f^{0 \prime}, F^0_{e,X,Z})$ is generated from the parameters $(\alpha^0, \beta^0, \lambda^0 f^{0 \prime})$ and $F_{e,X,Z}^0$, satisfying Assumption (ref), then any other $(\alpha, \beta, \lambda f^{\prime})$ and $F_{e,X,Z}$ that generate the same distribution of observables $F^0_{s,X,Z} = \Gamma(\alpha, \beta, \lambda f^{\prime},F_{e,X,Z})$ must be equal to the original $(\alpha^0, \beta^0, \lambda^0 f^{0 \prime})$ and $F_{e,X,Z}^0$. In other words, we can uniquely recover the model parameters from the distribution of observables. Two observationally equivalent model structures $(\alpha^0, \beta^0, \lambda^0 f^{0 \prime}, F^0_{e,X,Z})$ and $(\alpha, \beta, \lambda f^{\prime},F_{e,X,Z})$ need to be identical.

The key tool for the proof of Theorem (ref) is the the expected least squares objective function $$Q\left( \alpha ,\beta ,\gamma ,\lambda ,f ; F^0_{s,X,Z} \right) = \mathbbm{E}_0 \left\{ \sum_{j=1}^J \sum_{t=1}^T \left[ \delta_{jt}(\alpha) - X_{jt}' \beta - Z_{jt}' \gamma - \lambda_j' f_t \right]^2 \right\},$$ where $\gamma \in \mathbbm{R}^L$ is an auxiliary parameter, and $\mathbbm{E}_0$ refers to the expectation under the distribution of observables $F^0_{s,X,Z}$,\footnote{ Normally, we refer to $\mathbbm{E}_0$ simply as $\mathbbm{E}$. We only use different notation here to stress at which point the argument $F^0_{s,X,Z}$ enters into $Q\left( \alpha ,\beta ,\gamma ,\lambda ,f ; F^0_{s,X,Z} \right)$. } which is assumed to be generated from the model, i.e. $F^0_{s,X,Z}= \Gamma(\alpha^0, \beta^0, \lambda^0 f^{0 \prime}, F^0_{e,X,Z})$, with $F^0_{e,X,Z}$ satisfying Assumption (ref).

The true value of the auxiliary parameter $\gamma$ is zero, because of the exclusion restriction on $Z_{jt}$. In the proof of Theorem (ref) we show that under our assumptions the minimizer of $\displaystyle Q\left( \alpha ,\beta ,\gamma ,\lambda ,f ; F^0_{s,X,Z} \right) $ over $(\beta, \lambda, f, \gamma)$, for fixed $\alpha$, only satisfies $\gamma=0$ if and only if $\alpha = \alpha^0$. Thus, by using the expected least squares objective function as a tool we can uniquely identify $\alpha^0$ from the distribution of obervables $F^0_{s,X,Z}$. Having identified $\alpha^0$ we can identify $\beta^0$ and $\lambda^0 f^{0 \prime}$ simply as the unique minimizers of $\displaystyle Q\left(\alpha^0 ,\beta ,\gamma ,\lambda ,f ; F^0_{s,X,Z} \right)$. These findings immediately preclude observational equivalence, {\em viz} two sets of distinct parameters $(\alpha^0, \beta^0, \lambda^0 f^{0 \prime}) \neq (\alpha^1, \beta^1, \lambda^1 f^{1 \prime})$ which are both consistent with the observed distribution $F^0_{s,X,Z}$. For complete details we refer to the proof in the appendix. Furthermore, our identification argument is constructive, as it leads naturally to the LS-MD estimator which we introduce in subsequent sections.

Discussion of the Identification Conditions

In this section we discuss the conditions of the identification theorem. First, we note that when no factors are present ($R=0$), then our identification Assumptions (ref) below essentially require that the unconditional moment conditions $\mathbbm{E}(X_{jt} e_{jt})=0$ and $\mathbbm{E}(Z_{jt} e_{jt})=0$ uniquely identify the model parameters $\alpha$ and $\beta$, thus following the original identification strategy in BLP (1995).\footnote{ As such, our identification results do not add to the literature on non-parametric identification of the BLP model (as in Berry and Haile berry_haile_market, Chiappori and Komunjer chiappori_komunjer, Bajari, Fox, Kim and Ryan bajari_fox_kim_ryan); our concern is, rather, to show that the logit demand model with parametrically-distributed random coefficients can still be identified after the introduction of the interactive fixed effects.}

Assumption $(i)$ demands existence of second moments, assumption $(ii)$ requires the error process to have zero mean, and assumption $(iii)$ imposes exogeneity of the product characteristics $X_{jt}$ and the instruments $Z_{jt}$ with respect to the error $e_{jt}$ (endogenous regressors are discussed in Section (ref)). Apart from the term $M_{(\lambda,\lambda^0)}$, Assumption (ref)$(iv)$ is a standard non-collinearity condition on the product characteristics and the instruments -- which jointly appear as regressors in the first step of (ref). The generalized condition $\mathbbm{E}[(x,z)' (\mathbbm{1}_T \otimes M_{(\lambda,\lambda^0)}) (x,z)] \geq b > 0$ requires non-collinearity of the regressors even after projecting out all directions proportional to the true factor loading $\lambda^0$ and to any other possible factor loadings $\lambda$. A sufficient condition for this assumption is the rank condition ${\rm rank}[\mathbbm{E}(\Xi \, \Xi')] > 2R$ for any non-zero linear combination $\Xi = \beta \cdot X + \gamma \cdot Z$. This rank condition, for example, rules out product-invariant regressors and instruments, as already mentioned above.

Those parts of the conditions $(i)$ to $(iv)$ that do not contain $Z_{jt}$ are used to identify $\beta^0$ and $\lambda^0 f^{0 \prime}$ when $\alpha^0$ is already identified. These conditions are typical regularity conditions for identification of a linear regression model with a modification only required in condition $(iv)$ to accommodate the interactive fixed effects. (See also Moon and Weidner MoonWeidner2015b.)

The key additional assumption that we need for identification of $\alpha^0$ is Assumption (ref)$(v)$. Note that $\Delta \xi_{\alpha^0,\beta^0}=0$, that is, both the left and right side of the inequality in assumption $(v)$ are zero for $(\alpha,\beta)=(\alpha^0,\beta^0)$, which is why this case is explicitly ruled out in the assumption. The left hand side of the inequality in assumption $(v)$ is the sum of squares of that part of $\Delta \xi_{\alpha,\beta}$ that is explained by the regressors $x$ and the instruments $z$. The right hand side is the sum of squares of that part of $\Delta \xi_{\alpha,\beta}$ that is explained by the true factor loading $\lambda^0$ and an arbitrary other factor loading $\lambda$. Thus, the condition is a relevance condition on the instruments, which requires that the explanatory power of the regressors and the instruments needs to be larger than the explanatory power of $\lambda$ and $\lambda^0$ for $\Delta \xi_{\alpha,\beta}$.

A more concrete intuition for Assumption (ref)$(v)$ can be obtained in the case without factors. Without factors, the identification condition simplifies to $\forall (\alpha,\beta) \neq (\alpha^0,\beta^0):$

This can be shown to be equivalent to the statement $\forall \alpha \neq \alpha^0:$

We see that this condition is nothing more than the usual instrument relevance condition (for $z$ in this case) underlying the typical GMM approach in estimating BLP models. It can also be shown to be equivalent to the condition that for all $\alpha \neq \alpha^0$ the matrix $\mathbbm{E}[ (d(\alpha),x)' (x,z)]$ has full rank (equal to $K+1$).

The matrix valued function $\delta(\alpha)=\delta(\alpha,s,X)$ was introduced as the inverse of equation (ref) for the market shares $s_{jt}(\delta_t)$. Thus, once a functional form for $s_{jt}(\delta_t)$ is chosen and some distributional assumptions on the data generating process are made, it is in principle possible to analyze Assumption (ref)$(v)$ further and to discuss validity and optimality of the instruments. Unfortunately, too little is known about the properties of $\delta(\alpha)$ to enable a general analysis.\footnote{ This is a problem not only with our approach, but also with the estimators in BLP, and for Berry, Linton and Pakes BerryLintonPakes2004.} For this reason, in our Monte Carlo simulations in section (ref) below, we provide both analytical and and numerical verifications for Assumption (ref)$(v)$ for the specific setup there.

The final remark is that Assumption (ref)$(v)$ also restricts the family of the distribution of the random coefficient. As a very simple example, suppose that we would specify the distribution $G_{\alpha}$ for the random vector $v$ as $v \sim {\cal N}(\alpha_1,\alpha_2)$, where $\alpha =\left( \alpha _{1},\alpha _{2}\right)$, and we would also include a constant in the vector of regressors $X_{jt}$. Then, the regression coefficient on the constant and $\alpha_1$ cannot be jointly identified (because they both shift mean utility by a constant, but have no other effect), and Assumption (ref)$(v)$ will indeed be violated in this case.

LS-MD Estimator

If $\delta^0_{jt}$ is known, then the above model reduces to the linear panel regression model with interactive fixed effects. Estimation of this model was discussed under fixed $T$ asymptotics in, for example, Holtz-Eakin, Newey and Rosen HoltzEakin-Newey-Rosen1988, and Ahn, Lee, Schmidt AhnLeeSchmidt2001, and for $J,T \rightarrow \infty$ asymptotics in Bai Bai2009, and Moon and Weidner MoonWeidner2015,MoonWeidner2015b.

The computational challenge in estimating the model (ref) lies in accommodating both the model parameters $(\alpha,\beta)$, which in the existing literature has mainly been done in a GMM framework, as well as the nuisance elements $\lambda_j, f_t$, which in the existing literature have been treated using a principal components decomposition in a least-squares context (e.g., Bai Bai2009, and Moon and Weidner MoonWeidner2015,MoonWeidner2015b). Our estimation procedure -- which mimics the identification proof discussed previously -- combines both the GMM approach to demand estimation and the least squares approach to the interactive fixed effect model.

{ Definition:} the {\em least squares-minimum distance (LS-MD)} estimators for $\alpha$ and $\beta$ are defined by

Here, $\beta \in \mathbbm{R}^K$, $\delta(\alpha,\, s,\, X)$, $X_k$ and $Z_m$ are $J \times T$ matrices, $\lambda$ is $J\times R$, $f$ is $T\times R$, $W_{JT}$ is a positive definite $M\times M$ weight matrix, ${\cal B}_\alpha \subset \mathbbm{R}^L$ is an appropriate parameter set for $\alpha$.

Steps 1 and 2 are nested, because $\tilde \gamma_{\alpha}$ defined by step 1 needs to be calculated multiple times while performing the numerical optimization in step 2, but step 3 only needs to be performed once after the calculation of $\widehat \alpha$ in step 2 is finished. Step 1 resembles the linear least-squares estimators with interactive fixed effects considered in Bai Bai2009 and Moon and Weidner MoonWeidner2015, but because our model also includes the nonlinear parameter $\alpha$, this step is nested within step 2, which involves iteration over different candidate values for $\alpha$.

In step 1, we include the IV's $Z_m$ as auxiliary regressors, with coefficients $\gamma \in \mathbbm{R}^M$. Step 2 is based on imposing the exclusion restriction on the IV's, which requires that $\gamma=0$, at the true value of $\alpha$. Thus, we first estimate $\beta$, $\lambda$, $f$, and the instrument coefficients $\gamma$ by least squares for fixed $\alpha$, and subsequently we estimate $\alpha$ by minimizing the norm of $\tilde \gamma_\alpha$ with respect to $\alpha$.

Step 3 in (ref), which defines $\widehat \beta$, is just a repetition of step 1, but with $\alpha=\widehat \alpha$ and $\gamma=0$. One could also use the step 1 estimator $\tilde \beta_{\widehat \alpha}$ to estimate $\beta$. Under the assumptions for consistency of $(\widehat \alpha,\widehat \beta)$ presented below, this alternative estimator is also consistent for $\beta^0$. However, in general $\tilde \beta_{\widehat \alpha}$ has a larger variance than $\widehat \beta$, since irrelevant regressors are included in the estimation of $\tilde \beta_{\widehat \alpha}$.

For given $\alpha$, $\beta$ and $\gamma$ the optimal factors and factor loadings in the least squares problems in step 1 (and step 3) of (ref) turn out to be the principal components estimators for $\lambda$ and $f$. These incidental parameters can therefore be concentrated out easily, and the remaining objective function for $\beta$ and $\gamma$ turns out to be given by an eigenvalue problem (see e.g. Moon and Weidner MoonWeidner2015,MoonWeidner2015b for details), namely

where $\beta \cdot X = \sum_{k=1}^K \, \beta_k \, X_{k}$, $\gamma \cdot Z = \sum_{m=1}^M \, \gamma_m \, Z_m$, and $\mu_r(.)$ refers to the $r$'th largest eigenvalue of the argument matrix. This formulation greatly simplifies the numerical calculation of the estimator, since eigenvalues are easy and fast to compute, and we only need to perform numerical optimization over $\beta$ and $\gamma$, not over $\lambda$ and $f$.

The step 1 optimization problem in (ref) has the same structure as the interactive fixed effect regression model. Thus, for $\alpha = \alpha^0$ it is known from Bai Bai2009, and Moon and Weidner MoonWeidner2015,MoonWeidner2015b that (under their assumptions) $\tilde \beta_{\alpha^0}$ is $\sqrt{JT}$-consistent for $\beta^0$ and asymptotically normal as $J,T \rightarrow \infty$ with $J/T \rightarrow \kappa^2$, $0<\kappa<\infty$.

Step 1 also involves solving for the vector of $\delta$'s which solves the market share equations (2.5), at a given value for $\alpha$. This computational problem is well-studied in the BLP literature.\footnote{We solve it using nonlinear equation solvers, which is a relatively standard procedure from the existing BLP literature. Its validity is ensured by results (in Berry, Levinsohn, Pakes (1995)) showing that, for fixed $\alpha$, these equations constitute a contraction mapping, and the nonlinear equation solver recovers the (unique) fixed point.}

The LS-MD estimator we propose above is distinctive, because of the inclusion of the instruments $Z$ as regressors in the first-step. This can be understood as a generalization of an estimation approach for a linear regression model with endogenous regressors. Consider a simple structural equation $y_{1}=Y_{2}\alpha +e,$ where the endogenous regressors $Y_{2}$ have the reduced form specification $Y_{2}=Z\delta +V$, and $e$ and $V$ are correlated. The two stage least squares estimator of $ \alpha $ is $\widehat{\alpha}_{\rm 2SLS}=\left( Y_{2}^{\prime }P_{Z}Y_{2}\right) ^{-1}Y_{2}^{\prime }P_{Z}y_{1},$ where $P_{Z}=Z\left( Z^{\prime }Z\right) ^{-1}Z^{\prime }$. In this set up, it is possible to show that $\widehat{\alpha} _{\rm 2SLS}$ is also an LS-MD estimator with a suitable choice of the weight matrix. Namely, in the first step the OLS regression of $\left(y_{1}-Y_{2}\alpha \right) $ on $Z$ yields the OLS estimator $\tilde{\gamma}_{\alpha }=\left( Z^{\prime }Z\right) ^{-1}Z^{\prime }\left( y_{1}-Y_{2}\alpha \right)$. Then, in the second step minimizing the distance $\tilde{\gamma}_{\alpha }^{\prime }W\tilde{\gamma} _{\alpha }$ with respect to $\alpha $ gives $\widehat{\alpha}( W) = [ Y_{2}^{\prime }Z( Z^{\prime }Z) ^{-1}W( Z^{\prime }Z) ^{-1}Z^{\prime }Y_{2} ]^{-1} \linebreak[1] [ Y_{2}^{\prime }Z( Z^{\prime }Z) ^{-1}W( Z^{\prime }Z) ^{-1}Z^{\prime }y_{1}]$. Choosing $W=Z^{\prime }Z$ thus results in $\widehat{\alpha}=\widehat{\alpha}\left( Z^{\prime }Z\right) =\widehat{\alpha}_{\rm 2SLS}$. Obviously, for our nonlinear model, strict 2SLS is not applicable; however, our estimation approach can be considered a generalization of this alternative iterative estimator, in which the exogenous instruments $Z$ are included as “extra” regressors in the initial least-squares step.\footnote{ Moreover, the presence of the factors makes it inappropriate to use the moment condition-based GMM approach proposed by BLP, see Appendix (ref). Moment based approaches to factor model estimation like Holtz-Eakin, Newey and Rosen HoltzEakin-Newey-Rosen1988 and Ahn, Lee, Schmidt AhnLeeSchmidt2001,AhnLeeSchmidt2013 would also have to be non-trivially extended to handle the random coefficient parameter $\alpha$ in the presence of two dimensional incidental parameters in a nonlinear framework, but we have not explored this possibility.}

The two-step procedure in the LS-MD estimation is similar to the two stage estimation method in Chernozhukov and Hansen ChernozhukovHansen2006 that investigated endogenous quantile regressions.

Extension: regressor endogeneity with respect to $e_{jt}$

So far, we have assumed that the regressors $X$ could be endogenous only through the factors $\lambda_{j}' f_t$, and they are exogenous wrt $e$. However, this could be restrictive in some applications, for example, when price $p_{jt}$ is determined by $\xi_{jt}$ contemporaneously. Hence, we consider here the possibility that the regressors $X$ could also be correlated with $e$. This is readily accommodated within our framework. Let $X^{\rm end}\subset X$ denote the endogenous regressors, with $\text{dim}(X^{\rm end})=K_2$. (Hence, the number of exogenous regressors equals $K-K_2$.) Similarly, let $\beta^{\rm end}$ denote the coefficients on these regressors, while $\beta$ continues to denote the coefficients on the exogenous regressors. Correspondingly, we assume that $M$, the number of instruments, exceeds $L+K_2$.

{ Definition:} the {\em least-squares minimum distance (LS-MD)} estimators for $\alpha$ and $\beta$ with endogenous regressors $X^{end}$ is defined by:

where ${\cal B}_{\alpha}$ and ${\cal B}^{\rm end}_{\beta}$ are parameter sets for $\alpha$ and $\beta^{\rm end}$.

The difference between this estimator, and the previous one for which all the regressors were assumed exogenous, is that the estimation of $\beta^{\rm end}$, the coefficients on the endogenous regressors $\tilde X$, has been moved to the second step. The estimation procedure in (ref) can me mapped into our original LS-MD procedure in (ref), if we make the following formal replacements:

Thus, by changing the meaning of $\alpha$ and $\delta(\alpha)$ accordingly, the identification result above is still valid, and all results below on the consistency, asymptotic distribution and bias correction of the LS-MD estimator (ref) with only (sequentially) exogenous regressors directly generalize to the estimator (ref) with more general endogenous regressors. Given this discussion, we see that the original BLP (1995) model can be considered a special case of our model in which factors are absent (i.e. $R=0$).

Consistency and Asymptotic Distribution

In this section we present our results on the properties of the LS-MD estimator $\widehat \alpha$ and $\widehat \beta$ defined in (ref) under the asymptotics $J,T \rightarrow \infty$.

The proof of Theorem (ref) is given in the appendix. The similarity between Assumption (ref) and Assumption (ref) is obvious, so that for the most part we can refer to Section (ref) for the interpretation of these assumptions, and in the following we focus on discussing the differences between the consistency and identification assumptions. The one additional assumption is the last one, which requires existence of a positive definite probability limit of the weight matrix $W_{JT}$.

Apart from a rescaling with appropriate powers of $JT$, the Assumptions (ref)$(i)$, $(iii)$, $(iv)$, and $(v)$ are almost exact sample analogs of their identification counterparts in Assumption (ref). The two main differences are that assumption $(i)$ also imposes a Lipschitz-like continuity condition on $\delta(\alpha)$ around $\alpha^0$, and that the right hand-side of the inequality in assumption $(v)$ is not just zero, but a quadratic form in $(\alpha-\alpha^0)$ and $(\beta-\beta^0)$ --- the latter is needed, because expressions which are exactly zero in the identification proof are now only converging to zero asymptotically.

Assumption (ref)$(ii)$ imposes a bound on the the spectral norm of $e$, which is satisfied as long as $e_{jt}$ has mean zero, has a uniformly bounded fourth moment (across $j,t,J,T$) and is weakly correlated across $j$ and $t$.\footnote{ Such a statement on the spectral norm of a random matrix is a typical result in random matrix theory. The difficulty -- and the reason why we prefer such a high-level assumption on the spectral norm of $e$ -- is to specify the meaning of “weakly correlated across $j$ and $t$”. The extreme case is obviously independence across $j$ and $t$, but weaker assumptions are possible. We refer to the discussion in Moon and Weidner MoonWeidner2015 for other examples.} The assumption is therefore the analog of Assumption (ref)$(ii)$.

At finite $J$, $T$, a sufficient condition for existence of $b>0$ such that the inequality in Assumption (ref)$(iv)$ is satisfied, is ${\rm rank}(\Xi) > 2R$ for any non-zero linear combination $\Xi$ of $X_k$ and $Z_m$. This rank condition rules out product-invariant and market-invariant product characteristics $X_k$ and instruments $Z_m$, since those have rank 1 and can be absorbed into the factor structure.\footnote{ Inclusion of product-invariant and market-invariant characteristics (“low-rank regressors”) does not hamper the identification and estimation of the regression coefficients on the other (“high-rank”) regressors. This is because including low-rank regressors is equivalent to increasing the number of factors $R$, and then imposing restrictions on the factors and factors loadings of these new factors. Conditions under which the coefficients of low-rank regressors can be estimated consistently are discussed in Moon and Weidner MoonWeidner2015. } There are many reformulations of this rank condition, but in one formulation or another this rank condition can be found in any of the above cited papers on linear factor regressions, and we refer to Bai Bai2009, and Moon and Weidner MoonWeidner2015 for a further discussion.

Next, we present results on the limiting distribution of $\widehat \alpha$ and $\widehat \beta$. Some further regularity condition are necessary to derive the limiting distribution of our LS-MD estimator, and those are summarized in Assumption (ref) to (ref) in the appendix. These assumptions are straightforward generalization of the assumptions imposed by Moon and Weidner MoonWeidner2015,MoonWeidner2015b for the linear model, except for part $(i)$ of Assumption (ref), which demands that $\delta(\alpha)$ can be linearly approximated around $\alpha^0$ such that the Frobenius norm of the remainder term of the expansion is of order $o_p(\sqrt{JT}\|\alpha-\alpha^0\|)$ in any $\sqrt{J}$ shrinking neighborhood of $\alpha^0$. Notice also that Assumption (ref)(iv) implies $\mathbbm{E}(e_{jt} | X_{jt},Z_{jt} )=0$, while so far we only required $e_{jt}$ to be uncorrelated with $X_{jt}$ and $Z_{jt}$.

The proof of Theorem (ref) is provided in the appendix. Analogous to the least squares estimator in the linear model with interactive fixed effects, there are three bias terms in the limiting distribution of the LS-MD estimator. The bias term $\kappa B_0$ is only present if regressors or instruments are pre-determined, that is, if $X_{jt}$ or $Z_{jt}$ are correlated with $e_{j\tau}$ for $t>\tau$ (but not for $t=\tau$, since this would violate weak exogeneity). A reasonable interpretation of this bias terms thus requires that the index $t$ refers to time, or has some other well-defined ordering. The other two bias terms $\kappa^{-1} B_1$ and $\kappa B_2$ are due to heteroscedasticity of the idiosyncratic error $e_{jt}$ across firms $j$ and markets $t$, respectively. The first and last bias terms are proportional to $\kappa$, and thus are large when $T$ is small compared to $J$, while the second bias terms is proportional to $\kappa^{-1}$, and thus is large when $T$ is large compared to $J$. Note that no asymptotic bias is present if regressors and instruments are strictly exogenous and errors $e_{jt}$ are homoscedastic. There is also no asymptotic bias when $R=0$, since then there are no incidental parameters. For a more detailed discussion of the asymptotic bias, we again refer to Bai Bai2009 and Moon and Weidner MoonWeidner2015.

While the structure of the asymptotic bias terms is analogous to the bias encountered in linear models with interactive fixed effects, we find that the structure of the asymptotic variance matrix for $\widehat \alpha$ and $\widehat \beta$ is analogous to the GMM variance matrix. The LS-MD estimator can be shown to be equivalent to the GMM estimator if no factors are present. In that case the weight matrix ${\cal W}$ that appears in Theorem (ref) can be shown to be the probability limit of the GMM weight matrix that is implicit in our LS-MD approach and, thus, our asymptotic variance matrix exactly coincides with the one for GMM (see also Appendix (ref)). If factors are present, there is no GMM analog of our estimator, but the only change in the structure of the asymptotic variance matrix is the appearance of the projectors $M_{f^0}$ and $M_{\lambda^0}$ in the formulas for $G$, $\Omega$ and ${\cal W}$. The presence of these projectors implies that those components of $X_k$ and $Z_m$ which are proportional to $f^0$ and $\lambda^0$ do not contribute to the asymptotic variance, that is, do not help in the estimation of $\widehat \alpha$ and $\widehat \beta$. This is again analogous the standard fixed effect setup in panel data, where time-invariant components do not contribute to the identification of the regression coefficients.

Using the explicit expressions for the asymptotic bias and variance of the LS-MD estimator, one can provide estimators for this asymptotic bias and variance. By replacing the true parameter values ($\alpha^0$, $\beta^0$, $\lambda^0$, $f^0$) by the estimated parameters ($\widehat \alpha$, $\widehat \beta$, $\widehat \lambda$, $\widehat f$), the error term ($e$) by the residuals ($\widehat e$), and population values by sample values it is easy to define estimators $\widehat B_0$, $\widehat B_1$, $\widehat B_2$, $\widehat G$, $\widehat \Omega$ and $\widehat {\cal W}$ for $B_0$, $B_1$, $B_2$, $G$, $\Omega$ and ${\cal W}$. This is done explicitly in appendix (ref).

The proof is again given in the appendix. Theorem (ref) motivates the introduction of the bias corrected estimator

Under the assumptions of Theorem (ref) the bias corrected estimator is asymptotically unbiased, normally distributed, and has asymptotic variance $\left(G {\cal W} G' \right)^{-1} G {\cal W} \Omega {\cal W} G' \left(G {\cal W} G' \right)^{-1}$, which is consistently estimated by $\left(\widehat G \widehat {\cal W} \widehat G' \right)^{-1} \widehat G \widehat {\cal W} \widehat \Omega \widehat {\cal W} \widehat G' \left(\widehat G \widehat {\cal W} \widehat G' \right)^{-1}$. These results allow inference on $\alpha^0$ and $\beta^0$.

From the standard GMM analysis it is know that the $(K+M) \times (K+M)$ weight matrix ${\cal W}$ which minimizes the asymptotic variance is given by ${\cal W}=c \, \Omega^{-1}$, where $c$ is an arbitrary scalar. If the errors $e_{jt}$ are homoscedastic with variance $\sigma_e^2$ we have $\Omega=\sigma_e^2 \operatorname*{plim}_{J,T \rightarrow \infty} \frac 1 {JT} \left(x^{\lambda f}, z^{\lambda f}\right)' \left(x^{\lambda f}, z^{\lambda f}\right)$, with $x^{\lambda f}$ and $z^{\lambda f}$ defined in Appendix (ref). In this case it is straightforward to show that the optimal ${\cal W} =\sigma_e^2 \, \Omega^{-1}$ is attained by choosing

Under homoscedasticity this choice of weight matrix is optimal in the sense that it minimizes the asymptotic variance of our LS-MD estimator, but nothing is known about the efficiency bound in the presence of interactive fixed effects, that is, a different alternative estimator could theoretically have even lower asymptotic variance.

The unobserved factor loading $\lambda^0$ and factor $f^0$ enter into the definition of $x^{\lambda f}$ and thus also into the optimal $W_{JT}$ in (ref). A consistent estimator for the optimal $W_{JT}$ can be obtained by estimating $\lambda^0$ and $f^0$ in a first stage LS-MD estimation, using an arbitrary positive definite weight matrix.

Under heteroscedasticity of $e_{jt}$ there are in general not enough degrees of freedom in $W_{JT}$ to attain the optimal ${\cal W}$. The reason for this is that we have chosen the first stage of our estimation procedure to be an ordinary least squares step, which is optimal under homoscedasticity but not under heteroscedasticity. By generalizing the first stage optimization to weighted least squares one would obtain the additional degrees of freedom to attain the optimal ${\cal W}$ also under heteroscedasticity, but in the present paper we will not consider this possibility further.

Monte Carlo Simulations

We consider a model with only one regressors $X_{jt}=p_{jt}$, which we refer to as price. The data generating process for mean utility and price is given by

where $\lambda^0_j$, $f^0_t$, $e_{jt}$ and $\tilde p_{jt}$ are mutually independent and are all independent and identically distributed across $j$ and $t$ as ${\cal N}(0,1)$. In the data generating process the number of factors is $R=1$. For the number of factors used in the estimation procedure, $R_{\rm EST}$, we consider the correctly specified case $R_{\rm EST}=R=1$, the misspecified case $R_{\rm EST}=0$, and the case where the number of factors is overestimated $R_{\rm EST}=2$. We have truncated the data generating process for price so that $p_{jt}$ takes no values smaller than 0.2.

The market shares are computed from the mean utilities according to equation (ref) and (ref), where we assume a normally distributed random coefficient on price $p_{jt}$, i.e. $v \sim {\cal N}(0,\alpha^2)$. We chose the parameters of the model to be $\beta^0=-3$ and $\alpha^0=1$. These parameters corresponds to a distribution of consumer tastes where more than $99\%$ of consumers prefer low prices.

Although the regressors are strictly exogenous with respect to $e_{jt}$, we still need an instrument to identify $\alpha$. We choose $Z_{jt} = p_{jt}^2$, the squared price. Thus, the number of instruments is $M=1$. We justify the choice of squared price as an instrument in subsection (ref) by verifying the instrument relevance Assumption (ref)$(v)$ is satisfied for our simulation design.

Simulation results for three different samples sizes $J=T=20$, $50$ and $80$, and three different choices for the number of factors in estimation $R_{\rm EST}=0$, $1$, and $2$ are presented in Table (ref). We find that the estimators for $\widehat \alpha$ and $\widehat \beta$ to be significantly biased when $R_{\rm EST}=0$ factors are chosen in the estimation. This is because the factor and factor loading enter into the distribution of the regressor $p_{jt}$ and the instrument $Z_{jt}$, which makes them endogenous with respect to the total unobserved error $\xi^0_{jt} = \lambda^0_j \, f^0_t \, + \, e_{jt}$, and results in the estimated model with $R_{\rm EST}=0$ to be misspecified. The standard errors of the estimators are also much larger for $R_{\rm EST}=0$ than for $R_{\rm EST}>0$, since the variation of the total unobserved error $\xi^0_{jt}$ is larger than the variation of $e_{jt}$, which is the residual error after accounting for the factor structure.

For the correctly specified case $R_{\rm EST}=R=1$ we find the biases of the estimators $\widehat \alpha$ and $\widehat \beta$ to be negligible relative to the standard errors. For $J=T=20$ the absolute value of the biases is about one tenth the standard errors, and the ratio is even smaller for the larger sample sizes. As the sample size increases from $J=T=20$ to $J=T=50$ and $J=T=80$ one finds the standard error of the estimators to decrease at the rate $1/\sqrt{JT}$, consistent with our asymptotic theory.

The result for the case $R_{\rm EST}=2$ are very similar to those for $R_{\rm EST}=1$, that is, overestimating the number of factors does not affect the estimation quality much in our simulation, at least as long as $R_{\rm EST}$ is small relative to the sample size $J$, $T$.\footnote{ In pure factor models consistent inference procedures on the number of factors are known, e.g. Bai and Ng BaiNg2002, Harding Harding2007, Onatski Onatski2010, and Ahn and Horenstein AhnHorenstein2013. In our model the number of factor can be estimated by applying those pure factor model techniques to the residuals $\widehat \xi = \delta(\widehat \alpha)- \widehat \beta \cdot X$, where $\widehat \alpha$ and $\widehat \beta$ are LS-MD estimator obtained with $R_{\rm EST} \geq R$. Showing consistency of this procedure, however, goes beyond the scope of the current paper. } The biases for the estimators found for $R_{\rm EST}=2$ are still negligible and the standard errors are about $10 \%$ larger for $R_{\rm EST}=2$ than for $R_{\rm EST}=1$ at $J=T=20$, and even less than $10 \%$ larger for the larger sample sizes. The result that choosing $R_{\rm EST}>R$ has only a small effect on the estimator is not covered by the asymptotic theory in this paper, where we assume $R_{\rm EST}=R$, but is consistent with the analytical results found in Moon and Weidner MoonWeidner2015b for the linear model with interactive fixed effects.

We have chosen a data generating process for our simulation where regressors and instruments are strictly exogenous (as opposed to pre-determined) with respect to $e_{jt}$, and where the error distribution $e_{jt}$ is homoscedastic. According to our asymptotic theory there is therefore no asymptotic bias in the estimators $\widehat \alpha$ and $\widehat \beta$, which is consistent with the results in Table (ref). The simulation results for the bias corrected estimators $\widehat \alpha^*$ and $\widehat \beta^*$ are reported in Table (ref) in the appendix, but there is virtually no effect from bias correction here, that is, the results in Table (ref) and Table (ref) are almost identical.

Table (ref) also reports the average estimated standard error based on our asymptotic variance estimator, as well as the empirical size of a nominal $5 \%$ t-test for the hypothesis that the respective parameter equals its true value. Those are not particularly interesting for $R_{\rm EST}<R$, where the model is badly misspecified. For $R_{\rm EST} \geq R$ we find that for small sample sizes ($J=T=20$) our standard errors underestimate the dispersion of the estimator distributions by around $20 \%$, and the t-test is oversized accordingly. For larger sample sizes ($J=T=80$) our standard errors are still a bit too small, but only by around $5 \%$ or less, thus resulting in empirical sizes quite close to the nominal size.

Remarks: Instrument Choice

For the special case where there is only one normally distributed random coefficient attached to the regressor $p_{jt}$, one can write equation (ref) as

For $x\geq 0$ we have the general inequalities $ 1 \geq (1+x)^{-1} \geq 1 -x $. Applying this to (ref) with $x=\sum_{l=1}^J \, \exp\left( \delta_{lt} + p_{lt} v \right)$ one obtains $s^{\rm up}_{jt}(\alpha,\delta_t,X_t) \geq s_{jt}(\alpha,\delta_t,X_t) \geq s^{\rm low}_{jt}(\alpha,\delta_t,X_t)$, where

Here, the integrals over $v$ that appear in the upper and lower bound are solvable analytically, so that we obtain convenient expressions for $s^{\rm up}_{jt}(\alpha,\delta_t,X_t)$ and $s^{\rm low}_{jt}(\alpha,\delta_t,X_t)$.

Consider the specification (ref) for $\beta$ negative and large (in absolute value) relative to $\alpha^2$. Then $\delta_{jt}$ is also negative and large in absolute value, which implies that the $\nu_{jt}=\nu_{jt}(\alpha,\delta_t)$ defined in (ref) is small. For $\nu_{jt} \ll 1$, as here, the lower and upper bounds are almost identical, which implies $s_{jt}(\alpha,\delta_t,X_t) \approx \exp\left( \delta_{jt} + \alpha^2 p_{jt}^2 /2 \right)$, where $\approx$ means almost equal under that approximation. Solving for the mean utility yields $\delta_{jt}(\alpha,s_t,X_t) \approx \log s_{jt}(\alpha,\delta_t,X_t) - \alpha^2 p_{jt}^2 /2$. The difference between $\delta_{jt}(\alpha,s_t,X_t)$ and $\delta^0_{jt} = \delta_{jt}(\alpha^0,s_t,X_t)$ can then be approximated by

This shows that whenever the approximation $\nu_{jt} \ll 1$ is justified, then the squared price $p_{jt}^2$ is a valid instrument to identify $\alpha$. More precisely, equation (ref) implies that the LS-MD estimator with instrument $p_{jt}^2$ is approximately equivalent to the least squares estimator for the linear model with outcome variable $Y_{jt} = \beta p_{jt} + \alpha^2 p_{jt}^2 + \lambda_j' f_t + e_{jt}$. Consistency of this least squared estimator for $\beta$ and $\alpha^2$ in the presence of the parameters $\lambda_j$ and $f_t$ is discussed in Bai Bai2009 and Moon and Weidner MoonWeidner2015.

We have thus shown that $\nu_{jt} \ll 1$ is a sufficient condition for validity of the instrument $p_{jt}^2$. However, for the data-generating process with parameters $\alpha^0=1$ and $\beta^0=-3$ used in the Monte Carlo simulation this is not a good approximation --- when calculating $\nu_{jt}$ in that setup one typically finds values much larger than one. Therefore, we next confirm by numerical methods that $p_{jt}^2$ is also a valid instrument when $\nu_{jt}\ll 1$ does not hold.

The Instrument Relevance Condition: Some Numerical Evidence

We want to verify the instrument relevance Assumption (ref)$(v)$ for the data generating process (ref) in the Monte Carlo Simulations with parameters $\beta^0=-3$, and $\alpha^0=1$. For this purpose we define

$\rho_{\rm IV}(\alpha,\beta)$ is the amount of $\Delta \xi_{\alpha,\beta}$ explained by the instruments and regressors relative to the total variation of $\Delta \xi_{\alpha,\beta}$, i.e. the relative explanatory power of the instruments. $\rho_{\rm F}(\alpha,\beta)$ is the maximum amount of $\Delta \xi_{\alpha,\beta}$ explained by $R$ factor loadings relative the total variation of $\Delta \xi_{\alpha,\beta}$, i.e. the relative explanatory power of the factors. $\rho_{\rm IV}(\alpha,\beta)$ and $\rho_{\rm F}(\alpha,\beta)$ take values betweens 0 and 1.

The difference between the explanatory power of the instruments and regressors and the explanatory power of the factors is given by $\Delta \rho(\alpha,\beta)$. Assumption (ref)$(v)$ requires that $\Delta \rho(\alpha,\beta)>0$ for all $\alpha \in {\cal B}_\alpha$ and $\beta \in \mathbbm{R}^K$.

Figure (ref) contains plots of $\rho_{\rm IV}(\alpha,\beta)$, $\rho_{\rm F}(\alpha,\beta)$ and $ \Delta \rho(\alpha,\beta)$ as a function of $\alpha$ and $\beta$ for one particular draw of the data generating process with $J=T=80$. The sample size is sufficiently large that for different draws the plots in Figure (ref) look essentially identical.\footnote{ The appendix contains additional details on the numerical calculation of $\rho_F(\alpha,\beta)$. } Although the data generating process only contains one factor, we used $R=2$ factors in the calculation of $\rho_{\rm F}(\alpha,\beta)$ and $ \Delta \rho(\alpha,\beta)$ in Figure (ref), in order to verify Assumption (ref)$(v)$ also for the case where the number of factors is overestimated (denoted $R_{\rm EST}$=2 above) --- since $\rho_{\rm F}(\alpha,\beta)$ is an increasing function of $R$, we thus also verify the conditions of $R=1$.

For the given draw and within the examined parameter range one finds that $\rho_{\rm IV}(\alpha,\beta)$ varies between $0.69$ and $1.00$, $\rho_{\rm F}(\alpha,\beta)$ varies between $0.34$ and $0.87$, and $\Delta \rho(\alpha,\beta)$ varies between $0.03$ and $0.49$, in particular $\Delta \rho(\alpha,\beta)>0$, which is what we wanted to verify.

The variation in $\Delta \rho(\alpha,\beta)$ in this example is mostly driven by the variation in $\rho_{\rm F}(\alpha,\beta)$, since $\rho_{\rm IV}(\alpha,\beta)$ for the most part is quite close to one, that is, the explanatory power of the instruments and regressors is very large. The analytical approximation above showed that for $\nu_{jt} \ll 1$ the regressor $p_{jt}$ and the instrument $p_{jt}^2$ perfectly predict $\Delta \xi_{\alpha,\beta}$, that is, we have $\rho_{\rm IV}(\alpha,\beta) \approx 1$ under that approximation. Our numerical result now shows that $p_{jt}^2$ can be a sufficiently powerful instrument also outside the validity range of this approximation.

Empirical illustration: estimation of demand for new automobiles

As an illustration of our procedure, we estimate an aggregate random coefficients logit model of demand for new automobiles, modeled after the analysis in BLP (1995). We compare specifications with and without factors, and with and without price endogeneity. Throughout, we allow for one normally-distributed random coefficient, attached to price.\footnote{ In principle, multiple random coefficients could be accommodated in a straightforward manner; as this application is primarily illustrative, we do not consider this here.}

For this empirical illustration, we use the same data as was used in BLP (1995), which are new automobile sales from 1971-1990.\footnote{ In such a setting, where we have a single national market evolving over time, we can interpret $\lambda_j$ as (unobserved) national advertising for brand $j$, which may be roughly constant across time, and $f_t$ represents the effectiveness or “success” of the advertising, which varies over time. Indeed, for the automobile sector (which is the topic of our empirical example), the dollar amount of national brand-level advertising does not vary much across years, but the success of the ad campaign does vary. } However, our estimation procedure requires a balanced panel for the principal components step. Since there is substantial entry and exit of individual car models, we aggregate up to manufacturer-size level, and assume that consumers choose between aggregate composites of cars.\footnote{ This resembles the treatment in Esteban and Shum's esteban_shum1 empirical study of the new and used car markets, which likewise required a balanced panel. } Furthermore, we also reduce our sample window to the sixteen years 1973-1988. In Table (ref), we list the 23 car aggregates employed in our analysis, along with the across-year averages of the variables.

Except from the aggregation, our variables are the same as in BLP. Market share is given by total sales divided by the number of households in that year. Price is measured in \$1000 of 1983/84 dollars. Our unit for “horse power over weight” (hp/weight) is 100 times horse power over pound. “Miles per dollar” (mpd) is obtained from miles per gallons divided by real price per gallon, and measured in miles over 1983/84 dollars. Size is given by length times width, and measured in $10^{-4} \, {\rm inch}^2$.

We construct instruments using the idea of Berry Berry1994. The instruments for a particular aggregated model and year are given by the averages of hp/weight, mpd and size, over all cars produced by different manufactures in the same year. As the weight matrix in the second step of the LS-MD procedure we use $W_{JT}= \frac 1 {JT} z^{\prime} M_{x} z$, which is the optimal weight matrix under homoscedasticity of $e_{jt}$ and for $R=0$.\footnote{ We do not change the weight matrix when estimating specifications with $R=1$, because we do not want differences in the results for different values of $R$ to be attributed to the change in $W_{JT}$.

We include a constant regressor in the model, although this is a “low-rank” regressor, which is ruled out by our identification and consistency assumptions. However, as discussed in a footnote above the inclusion of a low-rank regressor does not hamper the identification and estimation of the regression coefficients of the other (“high-rank”) regressors. One certainly wants to include a constant regressor when estimating the model with no factors ($R=0$), so to make results easily comparable we include it in all our model specifications.}

\paragraph{Results.} Table (ref) contains estimation results from four specifications of the model. In specification A, prices are considered exogenous (wrt $e_{jt}$), but one factor is present, which captures some degree of price endogeneity (wrt. $\xi_{jt}$). Specification B also contains one factor, but treats prices as endogenous, even conditional on the factor. Specification C corresponds to the BLP (1995) model, where prices are endogenous, but no factor is present. Finally, in specification D, we treat prices as exogenous, and do not allow for a factor. This final specification is clearly unrealistic, but is included for comparison with the other specifications. In table (ref) we report the bias corrected LS-MD estimator (this only makes a difference for specification A and B), which accounts for bias due to heteroscedasticity in the error terms, and due to pre-determined regressors (we choose bandwidth $h=2$ in the construction of $\widehat B_0$). The estimation results without bias correction are reported in table (ref). It turns out, that it makes not much difference, whether the LS-MD estimator, or its bias corrected version are used. The t-values of the bias corrected estimators are somewhat larger, but apart from the constant, which is insignificant anyways, the bias correction changes neither the sign of the coefficients nor the conclusion whether the coefficients are significant at $5\%$ level.

In Specification A, most of the coefficients are precisely estimated. The price coefficient is -4.109, and the characteristics coefficients take the expected signs. The $\alpha$ parameter, corresponding to the standard deviation of the random coefficient on price, is estimated to be 2.092. These point estimates imply that, roughly 97% of the time, the random price coefficient is negative, which is as we should expect.

Compared to this baseline, Specification B allows price to be endogenous (even conditional on the factor). The point estimates for this specifications are virtually unchanged from those in Specification A, except for the constant term. Overall, the estimation results for the specifications A and B are very similar, and show that once factors are taken into account it does not make much difference whether price is treated as exogenous or endogenous. This suggests that the factors indeed capture most of the price endogeneity in this application.

In contrast, the estimation results for specifications C and D, which are the two specifications without any factors, are very different qualitatively. The t-values for specification C are rather small (i.e. standard errors are large), so that the difference in the coefficient estimates in these two specifications are not actually statistically significant. However, the differences in the t-values themselves shows that it makes a substantial difference for the no-factor estimation results whether price is treated as exogenous or endogenous.

Specifically, in Specification C, the key price coefficient and $\alpha$ are substantially smaller in magnitude; furthermore, the standard errors are large, so that none of the estimates are significant at usual significance levels. Moreover, the coefficient on {\em hp/weight} is negative, which is puzzling. In Specification D, which corresponds to a BLP model, but without price endogeneity, we see that the price coefficient is reduced dramatically relative to the other specifications, down to -0.308.

\paragraph{Elasticities.} The sizeable differences in the magnitudes of the price coefficients across the specification with and without factors suggest that these models may imply economically meaningful differences in price elasticities. For this reason, we compute the matrices of own- and cross-price elasticities for Specifications B (see Table (ref)) and C (see Table (ref)). The elasticities were computed using the data in 1988, the final year of our sample. Comparing these two sets of elasticities, the most obvious difference is that the elasticities -- both own- and cross-price -- for Specification C, corresponding to the standard BLP model without factors, are substantially smaller (about one-half in magnitude) than the Specification B elasticities. For instance, reading down the first column of Table (ref), we see that a one-percent increase in the price of a small Chevrolet car would result in a 28% reduction in its market share, but increase the market share for large Chevrolet cars by 1.5%. For the results in Table (ref), however, this same one-percent price increase would reduce the market share for small Chevrolet cars by only 13%, and increase the market share for large Chevrolet cars by less than half a percent.

On the whole, then, this empirical illustration shows that our estimation procedure is feasible even for moderate-sized datasets like the one used here. Including interactive fixed effects delivers results which are strikingly different than those obtained from specifications without these fixed effects.

Conclusion

In this paper, we consider an extension of the popular BLP random coefficients discrete-choice demand model, which underlies much recent empirical work in IO. We add interactive fixed effects in the form of a factor structure on the unobserved product characteristics. The interactive fixed effects can be arbitrarily correlated with the observed product characteristics (including price), which accommodate endogeneity and, at the same time, captures strong persistence in market shares across products and markets. We propose a two-step least squares-minimum distance (LS-MD) procedure to calculate the estimator. Our estimator is easy to compute, and Monte Carlo simulations show that it performs well.

The model in this paper is, to our knowledge, the first application of factor-modeling to a nonlinear setting with endogenous regressors. Since many other models used in applied settings (such as duration models in labor economics, and parametric auction models in IO) have these features, we believe that factor-modeling may prove an effective way of controlling for unobserved heterogeneity in these models. We are exploring these applications in ongoing work.