Dynamic demand for differentiated productswith fixed-effects unobserved heterogeneity∗
Victor Aguirregabiria†
University of Toronto, CEPR
May 8, 2022

Abstract
This paper studies identification and estimation of a dynamic discrete choice modelof demand for differentiated product using consumer-level panel data with few purchase
events per consumer (i.e., short panel). Consumers are forward-looking and their preferences incorporate two sources of dynamics: last choice dependence due to habits andswitching costs, and duration dependence due to inventory, depreciation, or learning. A
key distinguishing feature of the model is that consumer unobserved heterogeneity has aFixed Effects (FE) structure – that is, its probability distribution conditional on the initialvalues of endogenous state variables is unrestricted. I apply and extend recent results toestablish the identification of all the structural parameters as long as the dataset includes
four or more purchase events per household. The parameters can be estimated using asufficient statistic - conditional maximum likelihood (CML) method. An attractive featureof CML in this model is that the sufficient statistic controls for the forward-looking valueof the consumer’s decision problem such that the method does not require solving dynamic
programming problems or calculating expected present values.
Keywords: Structural dynamic discrete choice models; Dynamic demand of differentiatedproducts; Panel data; Fixed effects; Habits; Switching costs; Storable products; Durable
products.
JEL codes: C23, C25, C51, D12.

∗

I have benefited from helpful discussions with Jesús Carro, Jiaying Gu, Francis Guiton, Nail Kashaev, Lance
Lochner, and Matthew Osborne. I would like to thank also comments from the audience in the Applied Economics
and Econometrics seminar at Western University, and in the special session on Econometrics of Dynamic Discrete
Choice at the 2021 conference of the Royal Economic Society.
†Department of Economics, University of Toronto. 150 St. George Street, Toronto, ON, M5S 3G7, Canada,
victor.aguirregabiria@utoronto.ca.

1Introduction

In many markets, consumer demand is dynamic in the sense that consumers’ utility depends
on past decisions. Sources of dynamics in demand include, among others, storable products,1
durable products,2 habit formation and switching costs,3 adoption costs,4 and learning.5 Thesemodels exhibit two main forms of state dependence in consumers’ purchasing decisions: lastchoice dependence and duration dependence. We have last-choice dependence if a consumer’s
previous purchase of a product has a causal effect on her current probability of buying that
product, for instance, as a result of habits, switching costs, or learning. We have duration dependence if the time elapsed since the last purchase has a causal effect on the current buying
decision. Depletion of storable products and depreciation of durable products generate duration
dependence in consumer demand. These forms of state dependence can induce substantial differences between short-run and long-run responses of demand to price changes. This has importanteconomic implications in applications such as evaluating the effects of taxes and subsidies, measurement of firms’ market power, or consumer welfare. The estimation of dynamic structural
demand models using consumer panel data tries to measure these causal effects and use themfor counterfactual analysis and welfare evaluation.
Unobserved heterogeneity plays a fundamental role in dynamic demand models using consumer panel data. Ignoring or incorrectly specifying the correlation between unobserved heterogeneity and pre-determined explanatory variables (e.g., previous purchasing decisions) cangenerate important biases in the estimation of the structural parameters that capture dynamic
causal effects (Heckman, 1981b). Furthermore, the distribution of consumer taste heterogeneity has an important impact on demand price elasticities (Berry, Levinsohn, and Pakes, 1995).
The empirical literature on dynamic demand of differentiated product has considered a Random Effects (RE) approach to model consumer unobserved heterogeneity. RE models impose
parametric restrictions on the distribution of unobserved heterogeneity and on the correlation
1

See Boizot, Robin, and Visser (2001), Pesendorfer (2002), Erdem, Imai, and Keane (2003), and Hendel andNevo (2006).
2
See Esteban and Shum (2007), Goettler and Gordon (2011), and Gowrisankaran and Rysman (2012).
3
See Roy, Chintagunta, and Haldar (1996), Keane (1997), and Osborne (2011).
4
See Ryan and Tucker (2012), and De Groote and Verboven (2019).
5
See Ackerberg (2003), and Ching (2010).

1

between these unobservables and the initial values of the predetermined explanatory variables.
Though this distribution is not identifiable in short panels, its misspecfication can generate
important biases in the estimation of dynamic causal effects. This is the so called initial conditions problem (Heckman, 1981a). Therefore, RE models are not robust to misspecification ofparametric restrictions on unobserved heterogeneity.
Fixed effects (FE) approaches impose no restriction on the distribution of consumers’ unobserved heterogeneity such that the identification of parameters of interest is more robust thanin RE models. However, several identification concerns have inclined researchers to avoid a FE
approach in applications of dynamic discrete choice structural models, and more specifically, indynamic demand models.
A first issue is related to the identification of structural parameters. All the existing (positive) identification results in FE dynamic discrete choice models impose the restriction thatunobserved heterogeneity enters additively in the utility of each choice alternative. However, indynamic programming models, unobserved heterogeneity enters not only in current utility but
also in the continuation value of the forward-looking decision problem, and these continuation
values depend non-additively (and in fact, without a closed-form expression) on both unobserved
heterogeneity and observable state variables. The common wisdom was that FE models cannot
deal with the non-additive unobserved heterogeneity that is inherent to discrete choice dynamic
programming models.
A second important issue is related to the fact that FE methods cannot deliver identificationof the distribution of unobserved heterogeneity with short-panels. Most empirical applications ofdynamic structural models are interested in using the estimated model to obtain Average Partial
Effects (APE) on endogenous variables of changes in explanatory variables or in structuralparameters. Demand price elasticities are examples of these APE s. The common wisdom wasthat APEs are not identified in FE models, as they are expectations over the distribution of theunobserved heterogeneity (e.g., Abrevaya and Hsu, 2021; Honoré and DePaula, 2021).
Two recent studies provide positive identification results for the two issues discussed above.
Aguirregabiria, Gu, and Luo (2021) establish the identification of structural parameters in aclass of FE dynamic panel data logit models where agents are forward-looking. The class of2

models includes two types of endogenous state variables: the lagged choice variable, and thetime duration in the last choice. Aguirregabiria and Carro (2021) prove the point identificationof different AMEs in FE dynamic logit models. For instance, the average causal effect of changes
in the lagged dependent variable or in the duration in last choice are identified.6
In this paper, I apply and extend identification results in Aguirregabiria, Gu, and Luo (2021)
to a FE dynamic panel data model of consumer demand with differentiated product. The model
can incorporate storable or durable products, habit formation, and brand switching costs. I
present a new identification result in FE dynamic discrete choice with forward-looking agents.
I show identification of utility parameters associated to state variables that follow exogenous
stochastic processes. More specifically, in the context of demand models, I establish the identification of parameters that capture the effect of prices. In a FE forward-looking model, thisidentification result relies on a particular structure in the stochastic process of prices. There
are two components in prices: a persistent component (i.e., regular price), and a transitory
component (i.e., temporary promotions).
The structural parameters of the model are estimated using a sufficient statistics - conditional maximum likelihood (CML) method, in the spirit of Chamberlain (1985) and Honoré andKyriazidou (2000). In the context of dynamic structural models, a very helpful implication ofthis CML method is that it is not subject to a curse of dimensionality associated with thecomputational cost of solving a dynamic programming problem, or calculating present valuesof future utilities. This is particularly relevant in applications of dynamic demand of differentiated product as the dimension of the state space increases exponentially with the number of
products, which is typically large. The sufficient statistics - CML method "differences out" thecontinuation (forward-looking) value of the consumer’s decision problem, as this value depends
on the incidental parameters / unobserved heterogeneity. This implies that continuation values
should not be computed to implement the CML estimator. Therefore, the CML estimation ofthis dynamic structural model is computationally as simple as in static or myopic models, andits cost does not depend on the dimension of the state space.7
6

In related work, Chernozhukov, Fernandez-Val, Hahn, and Newey (2013), and more recently Davezies,
D’Haultfoeuille, and Laage (2021), provide partial identification results for a general class of AMEs in binary
choice panel data models.
7
CML estimation implies a different type of curse of dimensionality. The evaluation of the conditional likeli-

3

The rest of the paper is organized as follows. Section 2 describes the model. Section 3 presents
identification results. Section 4 describes the estimation method. In section 5, I summarize anddiscuss some extensions for further research.

2

Model

I present a framework that includes both storable and durable differentiated products. Most of
the features of the model are standard in the literature. The main distinguishing feature is theFixed Effects (FE) nature of consumer unobserved heterogeneity.

2.1

BasicsThere are J different brands of a differentiated product, and we index brands by j ∈ {1, 2, ..., J}.
We index consumers by i. Time is discrete and indexed by t. There is a dichotomy in thedefinition of time in empirical applications of dynamic consumer demand. In most applications,t has the standard interpretation as calendar time: for instance, the unit of time can be a week.
In these applications, the model and data account for time periods where a consumer does notpurchase any variety of the differentiated product. In contrast, a good number of empiricalapplications in this literature consider t as the index for purchase events, such that t = 1 meansa consumer’s first purchase, t = 2 is her second purchase, and so on (see Keane, 1997; Osborne,
2011; among others). The different definition of time t has implications on the interpretationof dynamics in the model, such as duration dependence.8 Here, I follow the standard definitionof t as calendar time. However, all the identification results in this paper apply also to models
where t indexes purchase events.
Every period t, consumer i decides whether to purchase or not one unit of the product, andwhich brand to buy. Variable yit ∈ Y = {0, 1, ..., J} represents the decision of consumer i athood function involves adding up a function over all the possible values of the vector of sufficient statistics. The
number of possible values increases exponentially with the number of time periods in the data, T . However, thisdimensionality problem has an easy solution – at the cost of some efficiency lost in the CML estimator – which
consists of splitting a T -periods history into shorter sub-histories.
8
Also very importantly, the model where t indexes purchase events does not include the decision of "nopurchase" as a choice alternative. That is, there is not an outside alternative.

4

period t, where yit = 0 means "no purchase", and yit = j > 0 means purchase of brand j.
There are two endogenous state variables that capture dynamics in consumer demand: brandchoice in the last purchase, `it ∈ {1, 2, ..., J}; and time duration since the last purchase, dit ∈{1, 2, ..., D}. Last brand choice is related to habit formation and switching costs. Duration sincelast purchase captures the effect of inventory depletion (for storable products) or depreciation(for durable products). By definition, the transition rule of the vector of endogenous statevariables xit ≡ (`it , dit ) is: (` , d + 1) if y = 0
it
itit
xi,t+1 ≡ (`i,t+1 , di,t+1 ) = fx (yit , xit ) ≡(j , 1) if yit = j > 0(1)

In many applications, consumers are located in different geographic markets and face different
prices. This is not a necessary condition for the identification results in this paper, but I allow
for it. Let pit (j) be the price of product j at period t in the market where consumer i is located,
and let pit be the vector with the prices of all products, pit ≡ (pit (j) : j = 1, 2, ..., J).

2.2

Utility

Let Uit be the per-period utility that consumer i obtains at period t. It has four components:
Uit = bit + γi mit − scit + εit(2)

The term bit is the utility from consumption of the branded product; mit represents consumptionof the outside good or numeraire; γi is the marginal utility of the outside good, also referred
as marginal utility of income; scit represents brand switching costs or habits; and εit represents
consumer idiosyncratic taste shocks at the moment of purchase. I describe below each of thesecomponents.
(i) Heterogeneity in consumers’ tastes. The model allows for product differentiation at the
moment of consumption and not only at the moment of purchase. Parameter αi (j) is the flow
utility that consumer i receives from consuming product j when the product has not depreciated (or depleted) over time. It depends on product characteristics and consumer characteristics,
5

some of them unobservable to the researcher. The vector αi ≡ (αi (1), αi (2), ..., αi (J)) represents
the fixed effects for consumer i. The fixed effect αi (j) depends on product and consumer characteristics in an unrestricted way. For instance, it could have the structure αi (j) = x0j β i + ξ 0j ω i ,where xj and ξ j are vectors of observable and unobservable product characteristics, respectively, and β i and ω i are the vectors of marginal utilities of these characteristics for consumer
i. Therefore, the specification of consumer heterogeneity in preferences is very flexible.
(ii) Depreciation / depletion. The flow utility from consumption of the branded producttakes into account depletion or depreciation over time.9 This flow utility has the following form:

bit ≡


 α (j)if yit = j > 0i α (` ) − β dep (` , d ) if y = 0 .
i it
it it
it

(3)

If the consumer purchases a new product (if yit = j > 0), there is no depreciation effect andthe flow utility is αi (j). Otherwise, if the consumer does not purchase a new product (if yit =0), she consumes product `it – her last purchase – and flow utility is αi (`it ) − β dep (`it , dit ).
Parameter β d (j, d) measures the effect on utility of d periods of depreciation in product j. Thisdepreciation/depletion effect can vary across products, and it is a nonparametric function ofduration d.
Arguably, the depletion effect β dep could vary across households. It is important to note thatthe model in this paper allows all the structural parameters, including this depletion effect, todepend on observable household characteristics, such as family size, income, education, or region.
We impose the restriction that β dep does not depend on consumer unobserved heterogeneity.
(iii) Utility from consumption of the composite good. The (dollar) amount of consumption of the composite good, mit , is given by the consumer’s budget constraint:

mit = mi −JX

pit (j) 1{yit = j},

(4)

j=1
9

The datasets used in this literature contain information on consumer purchase histories, but not on consumption and inventories of storable products. To deal with this missing information for storable products, researchers
have imposed different restrictions. Aguirregabiria and Nevo (2013) discuss this issue and different restrictionsin the literature. A common approach is using duration since last purchase, dit , as a proxy of inventory. For
storable products, this is the approach in Erdem, Imai, and Keane (2003).

6

where mi is the consumer’s (weekly) disposable income, andPJ

j=1 pit (j) 1{yit = j} represents

expenditure in the branded product. Since this utility is linear in consumption of the composite
good, income mi does not enter explicitly in the difference between the utilities of two productchoices. However, the parameter γi that represents the marginal utility of the composite good
may depend on consumer income, or on other observable consumer characteristics.
(iv) Product switching costs / habits. Following the standard approach, brand switchingcosts take place at the moment of purchase, and not through the consumption of the product.
The specification of brand switching costs is:scit = β sc (`it , yit ),(5)

Parameter β sc (k, j) represents the cost of switching from brand k to brand j. There is noswitching cost without switching or without a new purchase, such that such that β sc (k, k) = 0
and β sc (k, 0) = 0.
This component of the utility can be also interpreted in terms of habit formation. In thatcase, we have that scit = −β hab (`it , yit ), where parameter β hab (j, j) ≥ 0 for j > 0 is the increasein utility from purchasing the same brand as in last purchase, and for any k 6= j, β hab (k, j)
is restricted to be zero. This habit formation model is equivalent to the brand switching costmodel under the restrictions β sc (k, j) = β sc (k 0 , j) for any k, k 0 6= j. This more parsimonious
specification is quite common in this literature.
Regardless of consumer unobserved heterogeneity or/and forward-looking behavior, it is notpossible to identify all the switching cost parameters β sc (k, j) using data of consumers’ productchoices. In the best case, for any pair of products (k, j), it is possible to identify the following
linear combination of these parameters:
βesc (k, j) = β sc (k, j) + β sc (j, k) − β sc (k, k) − β sc (j, j)(6)

Therefore, we study the identification of these βesc (k, j) parameters.
(v) Extreme value distributed shocks in preferences. Finally, variable εit (j) captures7

other consumer idiosyncratic factors affecting the utility from purchasing product j at period t.
Variables εit ≡ (εit (0), ..., εit (J)) are i.i.d. over (i, t, j) with a type I extreme value distribution.
Both αi and εit are unobservable to the econometrician.
Putting together the different components of the utility function, we have:

Uit = uαi (yit , xit , pit ) + εit (yit ) =α (` ) − β dep (`it , dit ) + εit (0) i itif yit = 0
(7)




 α (j) − γ p (j) − β sc (` , j) + ε (j) if y = j > 0,
i
i it
it
it
it
where uαi (yit , xit , pit ) represents utility excluding unobservable logit shocks.

2.3

Stochastic process for prices

The structure of the stochastic process of prices that I describe here is not necessary for theidentification of the parameters β sc and β dep , but it plays an important role in the identificationof the marginal utility of income parameter, γ, in this forward-looking model. The assumption
is that the price of any product j has two components, one that is persistent over time (zit (j)),
and other that is transitory (eit (j)). That is:pit (j) = ρ (zit (j), eit (j))(8)

where ρ(.) is a known function. Some examples for the ρ(.) function are the linear functionpit (j) = zit (j) + eit (j), or the linear in logs function, pit (j) = exp{zit (j)eit (j)}. The vector of
permanent components zit ≡ (zit (j) : j = 1, 2, ..., J) follows a Markov process of order n. The
vector of transitory components eit ≡ (eit (j) : j = 1, 2, ..., J) is i.i.d. over time, and for any
periods t and s, zit and eis are independently distributed. The stochastic relationship between
the prices of the different products is unrestricted.
This structure can be interpreted in terms of "regular prices", represented by zit , and transitory promotions, represented by eit . For notational simplicty, for the rest of the paper, I consider
that zit follows a first-order Markov process with transition density function fz (zi,t+1 |zit ).
This structure has an important implication on a consumer’s dynamic decision model. The
8whole vector of prices pt (both zt and et ) affects a consumer’s current utility, but the expected
and discounted value of future utilities (the continuation value) depends on zt but not on et .
This exclusion restriction plays an important role in the identification of parameter γ in the
Fixed Effects dynamic forward-looking model in this paper.
It is relevant to note that, given time series data on prices and a specification of the ρ(.)
function, it is possible to identify the parameters in the two stochastic processes that characterize the evolution of prices, and based on these parameters, it is possible to identify the twocomponents zit and eit . For the rest of the paper, I assume that these two vectors, zit and eit ,are observable to the researcher.

2.4

Consumer decision problem

Every period t, the consumer observes the vectors of state variables xit , zit , eit , and εit , andmakes a purchasing decision yit to maximize her expected and discounted intertemporal utilityPs
Et [ ∞
s=0 δi Ui,t+s ], where δi ∈ [0, 1) is consumer i’s time discount factor. This consumer’s
problem is a stationary Markov Decision Process (MDP), and Blackwell’s Theorem establishes
that the value function and the optimal decision rule are time-invariant (Blackwell, 1965).
The decision problem of consumer i at period t is:yit = arg max { uαi (j, xit , pit ) + εit (j) + vαi (fx (j, xit ), zit ) }

(9)

j∈Y

where, as defined in equation (1), fx (j, xit ) represents the value of xi,t+1 given state xit anddecision yit = j; and vαi (fx (j, xit ), zit ) is the continuation value function, i.e., the expected
and discounted value of future utility given current state is (xit , zit ) and current choice j.10 Let
σαi (xit , zit , eit ) be the integrated (or smoothed) value function, that is defined as the expectationof the value function over the distribution of the i.i.d. unobservable state variables εit (Rust,
1994).11 This value function is the unique solution of the integrated Bellman equation. For the
10

In the literature of structural dynamic discrete models, letter v is commonly used to denote the conditional
choice value function. Instead, here I use letter v to denote the continuation value, such that the conditional
choice value function is uαi (j, xit , pit ) + vαi (fx (j, xit ), zit ).
11
That is, if VαRi (xit , zit , eit , εit ) is the value function, then the integrated value function is defined asσαi (xit , zit , eit ) ≡ Vαi (xit , zit , eit , εit ) fε (εit ) dεit , where fε is the density function of εit .

9type I extreme value distribution, the integrated Bellman equation is:σαi (xit , zit , eit ) = log

JX!
exp {uαi (j, xit , pit ) + vαi (fx (j, xit ), zit )}

(10)

j=0

and the continuation value function is:ZZvαi (fx (j, xit ), zit ) ≡ δi

σαi (fx (j, xit ), zi,t+1 , ei,t+1 ) fz (dzi,t+1 |zit ) fe (dei,t+1 )(11)

zi,t+1 ei,t+1

Define the conditional choice probability (CCP) function as:
P (j|xit , zit , eit , αi ) ≡ P r j = arg max {uαi (k, xit , pit ) + εit (k) + vαi (fx (k, xit ), zit )} |xit , zit , eit , αi
k∈Y

(12)
Importantly, this CCP function is conditional on the observable state variables (xit , zit , eit ) andon the unobservable fixed effects αi . The extreme value type I distribution of the unobservables
ε implies that the logarithm of the CCP function has the following form. For any j ∈ Y:
log P (j|xit , zit , eit , αi ) = uαi (j, xit , pit ) + vαi (fx (j, xit ), zit ) − σαi (xit , zit , eit )(13)

Note that the integrated value σαi (xit , zit , eit ) is also the logarithm of the denominator of thelogit CCP function.

3

Identification

The researcher observes a panel dataset of N households over T periods with information on
households’ purchasing decisions and prices. The time length of the panel, T , is short in thesense that it contains only a few purchases per household. The identification results in this paper
consider that T is fixed and – as it is common in proofs of identification – that N is infinite suchthat we have an infinite population of households.
Assumption 1 summarizes restrictions on the model for the identification of structural parameters βesc , β dep , and γ.

10

ASSUMPTION 1. (A) (i.i.d. Logit shocks) εit (j) is i.i.d. over (i, t, j) with type I extreme
value distribution, and is independent of αi . (B) (Strict exogeneity of prices with respect toshocks εit ) For any two periods, t and s, the variables εit (j) and prices pis are independently
distributed. (C) (Sticky prices) The vectors zt − zt−1 and et − et−1 have supports that include aneighborhood around zero.

Assumptions 1(A) and 1(B) rule out aggregate market-level shocks correlated with prices.
This restriction is very common in the literature of structural dynamic demand models, especially in applications using consumer level data.12 This may seem a strong assumption in an
econometric demand model, and at odds with the literature on estimation of static demandmodels using the BLP framework (Berry, Levinsohn, and Pakes, 1995). However, as I explainin the next two paragraphs, the FE approach in this paper controls for two important sourcesof endogeneity in prices.
First, note that the vector of fixed effects αi = (αi (1), αi (2), ..., αi (J)) includes product fixed
effects: in fact, it accounts for any interaction of product effects and consumer effects. Therefore,
these incidental parameters account for time-invariant differences in product quality that canbe correlated with the cross-sectional variation in prices across products. That is, the modelaccounts – in a very general way – for endogeneity of price levels. This means that the potential
concern with Assumption 1(B) is because endogeneity of changes in prices over time.
Second, all the identification results in this paper control for variation over time in thepersistent component of prices, zit and exploit only variation in the transitory component, eit .
Therefore, the method controls for endogeneity of regular prices.
Importantly, we consider a fixed effects (FE) model, in the sense that both the unconditional
distribution Fα (αi ) and the sequence of conditional distributions Fα|x,t (αi |`it , dit , zit , eit ) fort = 1, 2, ..., T are completely unrestricted functions.
12

For instance, Erdem, Imai, and Keane (2003), Hendel and Nevo (2006), Pavlidis and Ellickson, (2017),
Griffith, Nesheim, and O’Connell, (2018), or Mysliwski, Sanches, Junior, and Srisuma, (2020), among many otherpapers in this literature, assume that prices are not correlated with transitory shocks. In contrast, some dynamic
demand models using market level data, such as Gowrisankaran and Rysman (2012), allow for unobserved
market-level demand shocks.

11

3.1

Preliminaries

Let θ be the vector with the structural parameters β sc , β d , and γ. To establish the identificationof θ, I follow Aguirregabiria, Gu, and Luo (2021) who consider a sufficient statistic - conditional
likelihood approach in the spirit of Chamberlain (1985), Magnac (2000), and Honoré and Kyriazidou (2000). I start describing some general features of this approach in the context of the
dynamic demand model.
Let yi ≡ (`i1 , di1 , yi1 , yi2 , ..., yiT ) be the vector with consumer i’s choice history, including theinitial condition (`i1 , di1 ), and let ezi ≡ (zi1 , zi2 , ..., ziT ) and eei ≡ (ei1 , ei2 , ..., eiT ) be the vectors
with the histories of the two components in prices. Taking into account equation (13) for the
log-CCP function, we have that the log-probability of a choice history yi conditional on thehistory of prices and parameters θ and αi is:log P (yi |ezi , eei , αi , θ) = log p1 (xi1 |αi ) +TXuαi (j, xit , pit ) + vαi (fx (j, xit ), zit ) − σαi (xit , zit , eit )

t=1

(14)
where p1 (.) is the probability of the endogenous state variables at period t = 1. In this model,
the log-probability of a choice history has the following structure:log P (yi |ezi , eei , αi , θ) = s(yi , ezi , eei )0 g(αi ) + c(yi , ezi , eei )0 θ(15)

where s(yi , ezi , eei ) and c(yi , ezi , eei ) are vectors of statistics which are functions of (yi , ezi , eei ), suchPTPT
as, for instance, t=1 1{yit = j} or t=1 1{yit = j}pit (j), and g(αi ) is a vector of functions
which include the fixed effects αi .
The structure in equation (15) has two key implications for the identification of θ. First,
equation (15) implies that si ≡ s(yi , ezi , eei ) is a sufficient statistic for αi . That is, the probability
P(yi |ezi , eei , αi , θ, si ) does not depend on αi . To show this, note that:
P(yi |ezi , eei , αi , θ, si ) =P(yi |ezi , eei , αi , θ)
exp {s0i g(αi ) + c0i θ}=P
00P(si |ezi , eei , αi , θ)y: s(y)=si exp {si g(αi ) + c(y) θ}
(16)

= P

exp {c0i θ}
0y: s(y)=si exp {c(y) θ}

12= P(yi |ezi , eei , θ, si )wherePy: s(y)=si is the sum over all the possible choice histories y with s(y) equal to si .

Second, using the expression at the bottom line of equation (16), we have the following
conditional log-likelihood function (at the population level):

E [log P(yi |ezi , eei , θ, si )] = E c0i θ − log X

exp {c(y)0 θ}(17)

y: s(y)=si

The first order conditions for the maximization of this likelihood function with respect to θ
imply the following moment conditions (i.e., likelihood equations):m (θ) ≡ E ci −Xc(y) P(y|ezi , eei , θ, si ) = 0(18)

y: s(y)=si

The Jacobian matrix for this vector of moment conditions – or equivalently, the Hessian of the
log-likelihood function, or the negative of Fisher’s information matrix – is:∂m (θ)= − E [ci − E(ci |si )] [ci − E(ci |si )]0
0
∂θ

(19)

This is the negative of a variance-covariance matrix, and therefore, it is negative semidefinite forany value of θ. In other words, the likelihood function is globally concave in θ. Furthermore,
based on these moment conditions, a necessary and sufficient condition for (point) identificationof θ is that this Jacobian matrix ∂m(θ)/∂θ 0 is nonsingular (i.e., rank identification condition).
The right-hand-side in equation (19) shows that this is the case if and only if the statistics invector ci are linearly independent conditional on si .
A more intuitive description of the identification of the structural parameters consists inshowing that, for every parameter in the vector θ, say θk (the k-th element of vector θ), there
exist two choice histories, say A and B, such that s(A) = s(B) and c(A) − c(B) is a vector
where all the elements are zero except element k that is one. Under these conditions, equation(15) implies that:θk = log P(A) − log P(B),

13

(20)

which shows that parameter θk is identified from the log odds ratio of histories A and B.13 Thisintuitive description of the identification of parameters in FE discrete choice models has beenused by Chamberlain (1985) and Honoré and Kyriazidou (2000), among others. In this paper, I
follow this approach.
It is important to note that for some FE nonlinear panel data models the identification conditions based on this conditional likelihood approach are sufficient but not necessary. Bonhomme(2012) provides a systematic approach (the functional differencing method) to construct moment
restrictions for a general class of FE models. Recently, Honoré and Weidner (2020), Dobronyi,
Gu, and Kim (2021), and Honoré, Muris, and Weidner (2021) have used Bonhomme’s functional
differencing approach to establish new point identification results of parameters in FE models
where the conditional likelihood approach, at least apparently, does not provide identification.
For the rest of this subsection, I present results on the identification of structural parametersin two versions of the demand model: with and without duration dependence. Here, for simplicity, I do not present the expression for the statistics si and ci and instead provide examples ofpairs of histories A and B that identify the different parameters in θ.

3.2

Dynamic demand without duration dependence

Suppose that there is not depreciation or depletion such that β d (`, d) = 0, but there is habit
formation or/and brand switching costs, such that β sc (`, y) 6= 0. This is the demand model in
empirical applications like Roy, Chintagunta, and Haldar (1996), Keane (1997), Osborne (2011),
or Mysliwski, Sanches, Junior, and Srisuma, (2020), among others.
In this model, the only endogenous state variables is `it , and the transition function is:`i,t+1 = fx (yit , `it ) =


 `

it

if yit = 0(21)

 j if y = j > 0
itThen, combining equation (13) with this transition rule, we have the following expression for
13

Note that, in this class of models, every choice history has a strictly positive probability such that log P(A)and log P(B) are finite real numbers.

14

the log-CCP function: α (` ) + v (` , z ) − σ (` , z , e )if yit = 0i it
αi it itαi
it it itlog Pit (yit ) = α (j) + v (j, z ) − σ (` , z , e ) − β sc (` , j) − γ p (j) if y = j > 0
i
αi
it
αi
it it it
it
it
it
(22)
Suppose that T = 4. Let k and j be two different products, and consider the following pairof choice histories:
A = (k, j, k, j);

B = (k, k, j, j)(23)

Taking into account the structure of the log-CCP function in equation (22), we have the following
expression for the log-probabilities of the choice histories:log P(A) = log p1 (k, αi ) + αi (j) + αi (k) + αi (j)+ vαi (j, z2 ) + vαi (k, z3 ) + vαi (j, z4 )(24)

− σαi (k, z2 , e2 ) − σαi (j, z3 , e3 ) − σαi (k, z4 , e4 )− β sc (k, j) − β sc (j, k) − β sc (k, j) − γ (p2 (j) + p3 (k) + p4 (j)) ,andlog P(B) = log p1 (k, αi ) + αi (k) + αi (j) + αi (j)+ vαi (k, z2 ) + vαi (j, z3 ) + vαi (j, z4 )(25)

− σαi (k, z2 , e2 ) − σαi (k, z3 , e3 ) − σαi (j, z4 , e4 )− β sc (k, j) − γ (p2 (k) + p3 (j) + p4 (j)) ,Therefore, the difference between log-probabilities of the two histories is:log P(A) − log P(B) = vαi (j, z2 ) + vαi (k, z3 ) − vαi (k, z2 ) + vαi (j, z3 )− σαi (j, z3 , e3 ) − σαi (k, z4 , e4 ) + σαi (k, z3 , e3 ) + σαi (j, z4 , e4 )(26)

− βesc (k, j) − γ (p2 (j) − p3 (j) − p2 (k) + p3 (k)) ,Given equation (26), we can establish the identification of parameters βesc (k, j) and γ. To
control for the unobserved heterogeneity in the functions v(., αi ) and σ(., αi ), we need to impose

15

the restrictions z2 = z3 = z4 and e3 = e4 . Given these restrictions, we have that:log P(A) − log P(B) = −βesc (k, j) − γ (p2 (j) − p3 (j) − p2 (k) + p3 (k)) ,(27)

Therefore, with p2 (j) − p3 (j) = 0 and p2 (k) − p3 (k) = 0 we have identification of the switching
cost parameter βesc (k, j). And with p2 (j) − p3 (j) − p2 (k) + p3 (k) 6= 0 we have identificationof the price-sensitivity parameter γ. That is, keeping constant the permanent component ofprices between periods 2 and 4, and the transitory component between periods 3 and 4, we canidentify γ using the variation between periods 2 and 3 in the transitory component of the prices
in products j and k.
The pair of histories A and B in equation (23) is only one of the many history pairs withidentification power for the structural parameters. For instance, we can extend this example byincluding periods of no purchase. Let 0n represent a vector of n zeros. For k, j ≥ 1 with k 6= j,
and any two natural numbers n1 and n2 , consider the following choice histories:
A = (k, 0n1 , j, 0n2 , k, 0n2 , j);

B = (k, 0n1 , k, 0n2 , j, 0n2 , j)(28)

It is straightforward to show that the difference between the log-probabilities of these two histories has the following expression:log P(A) − log P(B) =

−

n1 +n
X2 +2

n1 +2nX2 +3

t=n1 +2

t=n1 +n2 +3vαi (j, zt ) − vαi (k, zt ) +vαi (k, zt ) − vαi (j, zt )

n1 +n
X2 +3

n1 +2nX2 +4

t=n1 +3t=n1 +n2 +4σαi (j, zt , et ) − σαi (k, zt , et ) +σαi (k, zt , et ) − σαi (j, zt , et )− βesc (k, j) − γ (pn1 +2 (j) − pn1 +n2 +3 (j) − pn1 +2 (k) + pn1 +n2 +3 (k)) .
(29)
To eliminate the unobserved heterogeneity αi from this difference we need the following conditions on prices: (i) the permanent component zt is constant from period n1 + 2 to n1 + 2n2 + 4;
and (ii) the transitory component et is constant between periods n1 + 3 and n1 + 2n2 + 4. Under

16these conditions, we have that:log P(A) − log P(B) = −βesc (k, j) − γ (pn1 +2 (j) − pn1 +3 (j) − pn1 +2 (k) + pn1 +3 (k)) .

(30)

This equation shows that a change between periods n1 +2 and n1 +3 in the transitory componentof the price of product j or k identifies parameter γ. The switching cost parameter βesc (k, j) is
identified from histories where this transitory component is constant.

3.3

Dynamic demand with duration dependence

Consider now the demand model with duration dependence. The expression for the log-CCP
function is: α (` ) − β dep (` , d ) + v (` , d + 1, z ) − σ (` , d , z , e ) if y = 0
i it
it it
αi it ittαi
it it t t
itlog Pit = α (j) − γ p (j) − β sc (` , j) + v (j, 1, z ) − σ (` , d , z , e ) if y = j > 0
i
it
it
αi
t
αi
it it t tit
(31)
First, it is straightforward to verify that the pair of histories in equation (23) still identifies
scthe switching cost parameters βekj
and the price sensitivity parameter γ. Therefore, I focus

here on the identification of the depretiation parameters β dep (j, d). Furthermore, for notational
simplicity, I consider here that the two price components are constant over the considered choicehistories and omit zt and et as arguments.
Let n be a natural number such that 2 ≤ n ≤ (T − 2)/2. Consider the following choicehistories, both with initial duration d1 = 1:
A = (j, 0n−1 , j, 0n+1 );B = (j, 0n , j, 0n )(32)

Note that these histories do not contain any product switching event such that switching cost
parameters do not appear in the probabilities of these choice histories. Taking into account thestructure of the log-CCP in equation (31), we have the following expressions for log-probabilities17

of these choice histories:log P(A) = log p1 (j, d1 |αi ) + 2(n + 1)αi (j) −

n−1
X

β

dep

(j, d) −

d=1

+

n
X
d=2

vαi (j, d) +

n+2
X

n+1
X

β dep (j, d)

d=1

n
n+1
X
X
vαi (j, d) −σαi (j, d) −σαi (j, d).

d=1

d=1

log P(B) = log p1 (j, d1 |αi ) + 2(n + 1)αi (j) −

d=1

n
n−1
X
X
β dep (j, d) −β dep (j, d)
d=1

d=1

n+1
n+1
n+1
n
X
X
X
X
+
vαi (j, d) +vαi (j, d) −σαi (j, d) −σαi (j, d).
d=2

d=1

d=1

(33)

(34)

d=1

They imply the following expression for the difference between the log-probabilities:
log P(A) − log P(B) = − β dep (j, n + 1) + β dep (j, n) + vαi (j, n + 2) − vαi (j, n + 1)

(35)

The expression in the right-hand-side still depends on the incidental parameters αi such that,without further restrictions, this pair of histories does not identify β dep parameters.
In general, without further restrictions, the duration dependence structural parameters β dep (j, d)
are not identified in the forward-looking model.14 To obtain identification of these parameters,
I follow the same approach as Aguirregabiria, Gu, and Luo (2021) and impose the following
restriction.
ASSUMPTION 2. For any product j, there is a value of duration d∗j – which can vary acrossproducts – such that β dep (j, n) = β dep (j, d∗j ) for any duration n ≥ d∗j .

Importantly, as established in Proposition 6 in Aguirregabiria, Gu, and Luo (2021), the valueof d∗j is identified from the data as long as it is not larger than (T − 1)/2. I reproduce here thisresult from Aguirregabiria, Gu, and Luo (2021).
PROPOSITION. Let d∗j be the value of duration defined in Assumption 2 above. For any
product j and any duration n with 2n+1 ≤ T , define the pair of histories Aj,n = (j, 0n−1 , j, 0n+1 )14Note that in a myopic model (i.e., δi = 0) all the continuation values vαi (j, d) are zero, such that equation(35) implies the identification of β dep (j, n + 1) − β dep (j, n) from log P(A) − log P(B). Here, I do not impose thisrestriction and consider that consumers can be forward-looking.

18

and Bj,n = (j, 0n , j, 0n ). If d∗j ≤ (T −1)/2, then the value d∗j is point identified from the following
expression:d∗j = max{n : log P(Aj,n ) − log P(Bj,n ) 6= 0}



(36)

An important implication of Assumption 2 is that the continuation value function is suchthat vαi (j, n) = vαi (j, d∗j ) for any duration n ≥ d∗j . Combining this property with equation (35),
we have that for n = d∗j − 1:log P(A) − log P(B) = − β dep (j, d∗j ) + β dep (j, d∗j − 1)(37)such that the (local) depreciation rate β dep (j, d∗j ) − β dep (j, d∗j − 1) is identified. If β dep (j, d) isdep

a linear function, i.e., β dep (j, d) = β jd, then equaton (37) implies the identification of thedepproduct-specific depreciation rate β j .

4Estimationdep
sc
Let θ be the vector of parameters (γ, βekj
, β j , k, j ∈ {1, 2, ..., J} with k > j). The dataset is{yit , zit , eit : i = 1, 2, ..., N ; t = 1, 2, ..., T }. Let ezi and eei the vectors with the time series of prices{zit : t = 1, 2, ..., T } and {eit : t = 1, 2, ..., T }, respectively.
In the identification results in section 3, a sufficient statistic for αi was represented as abinary indicator that combines the condition yi ∈ {A ∪ B}, where A and B are two choicehistories, and restrictions on prices, that we can represent as r(zei , eei ) = 0. That is:si ≡ s (yi , ezi , eei ) = 1{yi ∈ A ∪ B and r(ezi , eei ) = 0}(38)

We have shown that:
P(yi |ezi , eei , si = 1) =exp{c(yi , ezi , eei )0 θ}exp{c(A, ezi , eei )0 θ} + exp{c(B, ezi , eei )0 θ}(39)

where c(yi , ezi , eei ) is a vector of known statistics.
There are many pairs of histories A and B that provide sufficient statistics for αi and have19

identification power for θ. Let index these sufficient statistics by m ∈ {1, 2, ..., M }. We also
index by m the different elements that define sufficient statistic sm
i , that is: the corresponding
pair of choice histories, (Am , B m ); the restrictions on prices, rm (ezi , eei ) = 0; and the identifying
statistics, cm (yi , ezi , eei ). Note that we can use m also to index different sub-periods in thepanel dataset.15 Given these M sufficient statistics, we can define the following conditional
log-likelihood function:
L(θ) =

M X
N
X
m
m
m
1{yi ∈ A ∪ B } 1{r (ezi , eei ) = 0} log
m=1 i=1

exp{cm (yi , ezi , eei )0 θ}exp{cm (Am , ezi , eei )0 θ} + exp{cm (B m , ezi , eei )0 θ}(40)

The CML estimator is the value of θ that maximizes L(θ), which is a globally concave function.
In many possible applications, such as those using weekly or daily supermarket scanner data,
price stickiness implies such that the restrictions rm (ezi , eei ) = 0 hold for a non-negligible fractionof observations. Nevertheless, imposing exactly these restrictions typically implies loosing asubstantial amount of observations. This is exactly the issue that motivates the Kernel Weighted
CML in Honoré and Kyriazidou (2000). In the log-likelihood function, we replace the indicator1{rm (ezi , eei ) = 0} with a weight that depends inversely on the magnitude of vector |rm (ezi , eei )|such that we put more weight in the log-likelihood of observations for which |rm (ezi , eei )| is close
to zero. More specifically, we consider the following Kernel Weighted conditional log-likelihood
function:
LKW (θ) =
 m



M X
N
X
r (e
zi , eei )exp{cm (yi , ezi , eei )0 θ}mm
1{yi ∈ A ∪ B } K
logbexp{cm (Am , ezi , eei )0 θ} + exp{cm (B m , ezi , eei )0 θ}N
m=1 i=1(41)
where K(.) is a kernel density function that satisfies the regularity condition K(v) → 0 as||v|| → ∞, and bN is a bandwidth parameter such that bN → 0 and N bN → ∞ and N → ∞.
As shown by Honoré and Kyriazidou (2000), this estimator is consistent and asymptoticallynormal. If prices zit and eit have discrete support, the rate of convergence of the estimator is15
For instance, we can divide a panel dataset with T = 12 periods into three different sub-panels with fourtime periods each, and construct sufficient statistics separately for each sub-panel.

20

N −1/2 . Otherwise, with continuous prices, the rate of convergence is slower than N −1/2 .

5Conclusions

This paper presents a Fixed Effects dynamic panel data model of demand for different products
where consumers are forward looking. I apply and extend recent results from Aguirregabiria,
Gu, and Luo (2021) to establish the identification of all structural parameters in this model.
Several extensions of the results in this paper are interesting topics for further research.
First, an important motivation for the estimation of structural models is using them for counterfactual experiments that consist in evaluating the effects on agents’ behavior of hypotheticalchanges in structural parameters or/and exogenous variables. Recent research by Aguirregabiria
and Carro (2021), Davezies, D’Haultfoeuille, and Laage (2021), and Pakel and Weidner (2021)
present new positive results on the identification of average marginal effects and counterfactuals
in Fixed Effects discrete choice models. It would be interesting applying these results to thisdynamic demand model.
Second, the specification of the sources of demand dynamics in the model of this paper,
though relatively flexible, is restrictive. For instance, the model does not accommodate someforms of consumer learning for experienced goods used in the literature with Random Effects
models (e.g., Ching, 2010).
Third, the model assumes that consumers buy at most one unit of the product per period.
However, it is well-known that forward-looking consumers can buy for inventory (Hendel and
Nevo, 2006; Hendel and Nevo, 2013). It would be interesting and useful to extend the model toincorporate the possibility of consumers purchasing multiple units/
Fourth, in the same spirit as the the recent work by Mysliwski, Sanches, Junior, and Srisuma,(2020), this dynamic demand can be combined with a dynamic game of price competition. The
assumption on the two price components – persistent and transitory – is consistent with anequilibrium of a dynamic pricing game with i.i.d. firms’ private information.
Last but not least, eventually, empirical applications using the model and identification
results in this paper will be key judges of the relative merits and limitations of this framework.

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24