LM Test of Neglected Correlated Random E ects and Its Application

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1 LM Test of Neglected Correlated Random E ects and Its Application Jinyong Hahn UCLA Hyungsik Roger Moon USC May, 4 Connan Snider UCLA Introduction Econometric model speci cation is usually tested by exploiting one of the three principles, including (i) the Wald test, which is based on the asymptotic distribution of parameter estimates; (ii) the Likelihood Ratio (LR) test; and (iii) the Lagrange Multiplier (LM) procedure, which is based on the derivative of the log likelihood (score) imposing the hypothesis. The LM procedure seems to dominate the other two in terms of convenience when the objective of the test is to detect neglected heterogeneity in a given econometric model. The LM test statistic is solely based on the parameter estimates under the null hypothesis that there is no neglected heterogeneity and, as such, it eliminates the burden of specifying and estimating the model under the alternative hypothesis that there is some neglected heterogeneity. The possibility of using the LM test as a way of detecting individual heterogeneity was discussed in Breusch and Pagan (98), Chesher (984), Lee and Chesher (986), and Hahn, Newey, and Smith (4), among others. These tests can be viewed as a version of White s (98) Information Matrix test, as discussed in Chesher (984). Alternatively, they can be understood to be a test of overdispersion as in Cox (983), which is our interpretation in this paper. Implicit in these tests is the assumption that the neglected heterogeneity (under the alternative hypothesis) is more or less independent of the randomness that generates the main model, although the assumption is made more explicit in Hahn, Newey, and Smith (4, Lemmas 3. and 3.). In the panel data context, this assumption amounts to the random e ects speci cation, as developed in Balestra and Nerlove (966) or Maddala (97). In other The relationship among the three tests are reviewed in Engle (984), e.g.

2 words, in panel data applications we may roughly say that these tests are designed to detect random e ects, but not xed e ects (or equivalently, correlated random e ects). It is therefore of interest to revisit the LM test and analyze the test of neglected heterogeneity under more general conditions. This paper aims at achieving two distinct goals. The rst is to extend the existing LM test to the situation where the alternative hypothesis is characterized by the xed e ects model, where we adopt convention of equating the xed e ects with the correlated random e ects model. We begin by analyzing a model where the (local) alternative hypothesis is characterized by an arbitrary correlated random e ects, where the mean independence assumption is violated. We obtain a result that the test to detect such an arbitrary correlated random e ects would take the form of the test of conditional moment restriction. We then go on to analyze a more restricted version of correlated random e ects, where the e ects are assumed to be mean independent of the covariates of interest. For the latter case, we obtain an interesting result that the test against the random e ects model, where the conditional variance of the e ects does not depend on the covariates, has certain max-min type optimality property. We will call such test the LM test of overdispersion. In addition, while we show the LM test of overdispersion may have no power in linear models where the mean independence assumption is violated, we show the test may have some robustness properties against such violations in nonlinear models. As an example, we show the test has power against the arbitrary correlated random e ects alternative in the case of the logit model commonly used in applications. These results suggest the most natural applications for the LM test of overdispersion is to nonlinear models where the mean independence assumption is expected to hold. We nd such an application in the context of two-step semiparametric estimation of Bayesian game models commonly applied to the analysis of environments with social or strategic interactions among agents. This leads to the second goal of the paper, which is to draw a connection between panel data analysis and the analysis of multiplicity of equilibrium in games. We argue that multiplicity can be viewed as a particular form of neglected heterogeneity satisfying the mean independence assumption and propose an intuitive speci cation test for a class of two step game estimators. Moreover, the canonical speci cation of these games is a system of logit discrete choice models, to which we show the test is robust against more general forms of neglected heterogeneity that may arise from elsewhere in the model. Two step estimators of Bayesian game models exploit the two requirements of Bayes Nash See Chamberlain (984), e.g.

3 equilibrium. 3 First, equilibrium requires that players strategies are optimal given their beliefs about the distribution of opponent play and, second, that these beliefs are consistent with the true equilibrium distribution, conditional on any common knowledge determinants of player payo s. Under some assumptions about the nature of unobserved heterogeneity, the latter assumption allows the econometrician to, in the rst step, nonparametrically estimate player beliefs directly from the data by estimating conditional equilibrium choice probabilities. Treating these estimates as data, the problem of estimating the model of interdependent actions reduces to one of estimating a single agent choice model in the second step. This two step procedure allows the model to be estimated without ever having to solve the model, an often computationally expensive task. 4 The rst stage estimates conditional choice probabilities generally require pooling across a large number of observed game outcomes, which necessitates restrictive assumptions about two types of unobserved heterogeneity. First, we require there be no persistent unobserved state variables. That is, we rule out, for example, random and xed e ects in player payo functions. Second, while the underlying game is permitted to have multiple equilibria, we have to assume a single equilibrium is played in the data. If the rst stage estimates pool across games where multiple equilibria are played, the resulting choice probabilities with be weighted average of these equilibria, where the weights are given by the equilibrium selection probabilities. Nonlinearity in the second step estimator then leads to inconsistent parameter estimates. Some progress has been made on relaxing these assumptions 5, however, these xes often only apply to relatively restrictive classes of games and outside these classes custom solutions must be developed. Moreover, implementing these solutions often require increasing the computational complexity and burden of estimation. These considerations suggest that speci cation tests of these assumptions may be an attractive toolkit to practitioners. We argue that the LM test of overdispersion naturally applies here, because the meanindependence assumption is natural given the way that the conditional choice probability is computed in the rst step. Even if one were to consider an alternative that does not satisfy the mean-independence assumption, the LM test of overdispersion would have power due to the nonlinearity of the second step. Our proposed LM test can be viewed as a simple triage 3 See, e.g., Aguirregabiria and Mira (), Bajari, Hong, Krainer, and Nekipelov (). 4 This insight has been particularly important in the literature on estimating dynamic games (Aguirregabiria and Mira 7, Bajari, Benkard, and Levin 7, Berry, Pakes, and Ostrovsky 7, Pesendorfer and Schmidt- Dengler 7), where calculating equilibrium is so computationally expensive that full solution approaches to estimation are generally infeasible. 5 See Aguirregabiria and Nevo (3) for a survey. 3

4 tool that allows the researcher to estimate the relatively inexpensive, restrictive model and then use the LM test to decide whether the more complicated model is needed. Again, the main advantage of the LM test over the Wald test and the LR test is that we can construct the test statistics only under the null hypothesis of no heterogeneity (in this case, a single equilibrium is played), and we do not have to identify the model under the alternative that multiple equilibria played. We demonstrate the performance of our test in detecting failure of the single equilibrium assumption in a Monte Carlo exercise based on the application of Sweeting (9), who analyzes the timing of radio commercials among rival stations. Test of Neglected Correlated Random E ects - LM Test Revisited Assume that we observe a random sample (Y i ; X i ), i = ; : : : ; n. The Y and X can be vectors. For example, in the panel data analysis where each individual is observed over T time periods, we will have Y i = (Y i ; : : : ; X it ) and X i = (X i ; : : : ; X it ). We will assume that the conditional density of Y given X is given by the density f (yj x; ; ), where (; ) is the parameter that characterizes the density. Under the null hypothesis, is xed at, but under the alternative hypothesis, the may be a random variable potentially correlated with X. Our purpose is to develop an LM test to detect such neglected heterogeneity in. For simplicity, we will assume that the is a scalar. Following the convention in the literature, e.g., Chesher (984), we will also treat both and as known/ xed, which is the basis of our analysis throughout the rest of this section. Note that we can write f (yj x; ; ) = f (yj x; )without loss of generality. We will assume that under the alternative hypothesis, the is drawn from a distribution whose density is s (j x; ) = g x () where Z wg (wj x) dw b (x) : () Note that the conditional density of given X is allowed to depends on the realization of X. In the existing literature, including Breusch and Pagan (98), Chesher (984), Lee and Chesher (986), and Hahn, Newey, and Smith (4), it is (implicitly or explicitly) assumed that (i) and X are independent of each other; and (ii) the mean of is zero, so that b (x) =. Our speci cation and analysis di er from the existing literature that both of these restrictions are relaxed. In what follows we consider two cases, (i) when b (x) is unknown and not 4

5 necessarily zero and (ii) when b (x) = :. Case : Arbitrary b (x) When the conditional density function of is g observation Y is Z h (yjx; ; ) = x as in (), the density function of the f (yj x; ) g x d: (3) By the change-of-variable = + w, we have Z h (yj x; ; ) = f (yj x; + w) g (wj x) dw: The null hypothesis of interest is that there is no heterogeneity in the parameter. In the model above, this is equivalent to the test of the null H : = against H : 6=. Our LM test is based on s (y; x; ; (yj x; ; h (yj x; ; ) evaluated at =. We can easily see that at =, we have s (y; x; ; ) = f Z (yj x; ) wg (wj x) dw = s (yj x; ) b (x) ; (4) f (yj x; ) where s (yj x; ) f (yj x; ) f (yj x; ) : The analysis leading to (4) implies that the LM test reduces to a test of the moment restriction E [s (yj x; ) b (x)] = : (5) If we are to have power against all possible distribution of, we should have the above conditional moment restriction to hold for all (square integrable) b (x). Obviously it is equivalent to the test of conditional moment restriction E [s (yj x; )j x] = : (6) Tests of conditional moment restrictions in general have been studied by numerous papers in the econometrics literature. 6 Our contribution is the recognition that a particular test of conditional moment restrictions (6) can be viewed as an LM test against arbitrary correlated random e ects. 6 A partial list includes Newey (985), Bierens (99, 9), and Donald, Imbens, and Newey (3). 5

6 Example Consider the model y it = + x it + " it where " it is i.i.d. N (;! ) with! = known. We have so and the test of interest is log f (yj x; ) = (y it x it ) t= s (yj x; ) = (y it x it ) t= " # E (y it x it ) x i; : : : ; x it = : t= Remark In Example, a simple practical version of the test is a J-test for the moment 7 )!3 x i 7 E 6B 4@. C A (y it x it 5 = :. Case : b (x) = x it t= We now consider the special case, where we impose the restriction that b (x) =. The restriction leads to the problem that the LM test statistic in (4) becomes identically equal to. In order to tackle this problem, we follow Chesher (984) and Lee and Chesher (986), e.g., and work instead with p, i.e., Z h (yjx; ; ) = f (yj x; ) p g p x d: (7) We show in Appendix A that the LM test should be based on the p n-standardized sample average of where f (yj x; ) f (yj x; ) (x) ; (8) Z (x) w g (wj x) dw: 7 Here we treat and to be known, although in practice, we would plug in a consistent estimators of and. Therefore, the critical value in practice would have to re ect the sampling variation of the plugged-in estimators as well as the test statistic itself. 6

7 h Because the variance of (8) is E (X i ) ( (X i )) i, where we de ne " # f (Y i j X i ; ) (X i ) E f (Y i j X i ; ) X i ; (9) our test statistic would take the form T n = nx (X i ) (X i ) n i=.. Max-Min Consideration! p n nx i=! f (Y i j X i ; ) f (Y i j X i ; ) (X i ) : () The test statistic in () is inconvenient because it requires that (x) be speci ed. In this section, we provide a justi cation of using (x) = from a (local) max-min perspective. For expositional simplicity, we will suppose that and () is known. Our max-min justi cation is based on the following result: Lemma Under the sequence of distributions characterized by the density Z f (yjx; + p w) g (wj x) dw with = p n and g (wjx) such that R wg (wjx) dw = b (x) = and R w g (wj x) dw = (x) >, the test statistic T n ( ) in () converges to a noncentral distribution with the noncentrality parameter equal to Proof. In Appendix A. () ; () (E [ (X i ) (X i ) (X i )]) E (X i ) ( (X i )) : We will interpret the distribution with g as the true alternative chosen by the nature. The result above describes the asymptotic distribution of the test statistic () under the nature s alternative hypothesis. For a given critical value c, the power of the test is then given by P n [T n ( ) c], where P n denotes the probability measure under the local alternative sequence chosen by the nature. and therefore, the local power is P n [T n ( ) c] is determined by ( () ; ()). Our max-min justi cation is given in the following result: Proposition Consider the problem max min () ; () where (x) and (x) are normalized such that E [ (X i ) (X i )] = E [ (X i ) (X i )] =. The solution is such that () is a constant function. 7

8 Proof. In Appendix A. The decision problem we consider is that for given () ; the nature chooses the alternative () to minimize the lower power, i.e., ( () ; ()), and then, econometrician chooses () by maximizing the local power, i.e., ( () ; ()). The result above establishes that the econometrician s choice is such that () is constant. In other words, a constant choice for (x) is optimal from a max-min decision perspective. Remark Our discussion above suggests that it would be reasonable to take (X i ) = and can be justi ed by the max-min principle. Assuming that (X i ) > almost everywhere, we can see that the (; ()) > for any (X i ) as long as (X i ) > with positive probability. Given that the probability (X i ) > is positive for any reasonable (X i ), we can conclude that the test statistic T n ( ) with (X i ) = has a local power against any reasonable alternatives... Additional Consideration We ask whether the test of overdispersion has any power against the local alternative where the mean independence assumption may be violated. Test of the conditional moment restriction (6) is not expected to have any power if the alternative satis es the mean independence assumption b (x) =. Although the LM test of overdispersion is developed under the assumption that b (x) =, if it has a power even when b (x) 6=, we may argue that it is a more robust than the test of conditional restriction. We provide two examples, and discuss the power of the LM test of overdispersion. Given the max-min consideration, we will take () = without loss of generality, and consider the power of the test statistic () under the local alternative where the mean independence may be violated: Proposition Under the sequence of distributions characterized by the density (3) with = p n, the test statistic () is asymptotically noncentral with the noncentrality parameter equal to (E [ (X i ) b (X i )]) E [ (X i )], where (X i ) E f (Y i j X i ; ) f (Y i j X i ; ) f (Y i j X i ; ) X i : f (Y i j X i ; ) f ( Y i jx i ;) f( Y i jx i is ;) P Proof. Using the same argument as in Appendix A, we can show that p n n i= asymptotically normal with mean equal to E [ (X i ) b (X i )] and variance equal to E [ (X i )], from which the conclusion follows. Below we present two examples, one of which is such that the local power is zero because (X i ) =, and the other one is such that it is not necessarily the case. 8

9 Example (Example - continued) The textbook linear model in Example is such that we have log f (Y i j X i ; ) = C f (Y i j X i ; ) f (Y i j X i ; ) = (y it x it ) t= (y it x it )! T = t= f (Y i j X i ; ) f (Y i j X i ; ) = (y it x it ) = t= t= " it! " it T t= and using the symmetry of the normal distribution, we obtain (X i ) =. It follows that the test of overdispersion does not have any (local) power against the alternative as in (). Example 3 (Panel Logit) Consider the panel logit model where f (Y i j X i ; ) = TY t= exp (x it + ) yit + exp (x it + ) + exp (x it + ) yit We can see that log f (Y i j X i ; ) = y it (x it + ) t= log ( + exp (x it + )) t= f (Y i j X i ; ) f (Y i j X i ; ) = exp (x y it + ) it + exp (x it + ) f (Y i j X i ; ) f (Y i j X i ; ) = t= t=! exp (x y it + ) it + exp (x it + ) t= exp (x it + ) ( + exp (x it + )) and (X i ) = exp (x it + ) ( exp (x it + )) ( + exp (x it + : ))3 t= 3 Application In the previous section, we showed, rst, the test of overdispersion along the lines of Breusch and Pagan (98), Chesher (984), Lee and Chesher (986), and Hahn, Newey, and Smith (4), i.e., the variants of the statistic (8) with () =, have some local max-min optimality properties. Second, we showed that the tests have power against general correlated random 9

10 e ects (b(x) 6= ) in logit models, which is the property that is expected to carry over to other nonlinear models. Therefore, one may argue that the LM test of overdispersion is particularly well-suited for nonlinear models. A potentially important application of our test arises in the context of estimating static binary choice models with social and strategic interactions. These models have been employed to analyze a wide variety of applications including rm entry (Seim 6), the timing of radio commercials (Sweeting 9), labor force participation (Bjorn and Vuong 984), teen sex (Card and Giuliano ), and many more. In these models the payo associated with any particular discrete action, d i f; g, for an agent i depend on the actions of all other agents in the model. The optimal action for agent i then depends on his beliefs about the likely actions of other agents in the model. Under some assumptions about the nature of unobserved heterogeneity in the model, these models can be estimated using a simple two step, semiparametric estimator where, in the rst step, the econometrician estimates the equilibrium beliefs of the agents and in then, in the second step, estimates the parameters of the model that rationalize those estimated beliefs as an equilibrium. We consider a scenario in which the econometrician has access to data fd ig ; X ig g g=;:::;n i=;:::;k g on actions and covariates for K g agents interacting in each of n groups/games. The objective of estimation is to estimate payo s u(d ig ; d ig ; x ig ; ) that rationalize observed choice behavior, where D ig is the vector of choices of all agents except i. Following Bajari, Hong, Krainer, and Nekipelov () we assume the data is generated by a Bayes-Nash equilibrium of a game of incomplete information where players choose actions, d ig f; g to maximize 3 E D i [u(d ig ; D ig ; x ig ; ) + " ig (d ig )jx g ] = 4 X d i Pr(D ig = d i jx g = x g )u(d ig ; d i ; X ig ; ) 5 +" ig (d ig ); where " i (d ig ) is an i.i.d., across agents and games, private information payo shock drawn from a commonly known distribution and we use the convention X g probability that player i chooses d ig = is then = (X g; :::; X Kgg). The f(d ig = jx g ; ) = Pr(" ig () " i () E D i [u(; D ig ; x ig ; ) u(; D ig ; x ig ; )jx g = x g ]) () In a Bayes-Nash equilibrium, player i s beliefs about the distribution of opponent play are required to be consistent with the true distribution of play. This assumption and the observation that the econometrician is able to condition on all the data player i uses in forming his or her beliefs leads to a natural two step estimator for. In the rst step, the econometrician estimates,

11 ideally nonparametrically, conditional choice probabilities (CCP) 8 Pr [D ig = j X g ] = (X g ) () In the second step, the econometrician then formulates an estimator, using a (pseudo) likelihood or moment equation approach, based on the optimality condition (). The basic two step structure of this model was pioneered by Hotz and Miller (993) in the context of dynamic discrete choice models and has subsequently been extended to static discrete games (Agguiregabiria and Mira 3, Bajari, Hong, Krainer, and Nekipelov ) and dynamic games (Agguiregabiria and Mira 7, Bajari, Benkard, and Levin 7, Berry, Pakes, and Ostrovsky 7, Pesendorfer and Schmidt-Dengler 7). The rst stage CCP estimates, by nature of the exercise, require pooling of observations across games and agents, which leads to two sources of potential neglected heterogeneity that may invalidate the approach. First, unobserved individual or game/group level payo heterogeneity and second, multiple equilibria being played across games. Both of these make the CCP inconsistent estimates of equilibrium choice probabilities and, in turn, lead to inconsistency of the second stage estimates. When there are multiple equilibria, E [D i j X g ] can be viewed as a random variable (X g ); the moment equation () de nes a pseudo-parameter (x), which is an average of the (x) weighted by the probability of equilibrium selection. Non-linearity then leads the second stage estimator to be inconsistent for. Consistency thus requires the economist to assume a single equilibrium is played in the data. A similar issue arises in the presence of unobserved heterogeneity in player payo s, often referred to as unobserved state variables in dynamic models. Some progress has been made on identi cation and two step estimation of models with multiple equilibria and unobserved payo heterogeneity, in particular, by Kasahara and Shimotsu (9). However, in contrast to our LM test, the approaches pursued in the literature and both the Wald and LR tests require the econometrician to be able to specify the number of equilibria/types. In addition, even if these numbers are known, the LR test is often complicated due to the technical problem that certain parameters are often not identi ed under the null (Lin and Shao 3, Cho and White 7). Moreover, much of the appeal of the two step estimators is their computational convenience-they can often be programmed with a couple of 8 In the common usage the CCP for this example would be the individual conditional choice probability E[D j jx g ]. Here we implicitly assume players are anonymous and, conditional on X, symmetric so behavior can be described by the number of players that chose d j =. In our usage the CCP is a function of the individual CCPs Y Y Pr[D i = d i jx g ] = E[D j jx g ] ( E[D j jx g ]) j:d i(j)= j:d i(j)=

12 lines of code in commonly used statistical packages like STATA. Both full solution approaches, where the economist actually solves for equilibrium for every parameter guess in the estimation search algorithm, and two step approaches admitting multiple equilibria/payo heterogeneity entail a substantial increase in the computational burden of estimation. In this sense, our LM test can be looked at as a triage tool that allows the researcher to judge whether incurring such costs is necessary. These attractive features of the LM test notwithstanding, our test is limited by the fact that it designed to detect only a local alternative, i.e. cases where the departure from the assumption of a single equilibrium is not too severe. When there are multiple equilibria the very fact that (x) is de ned to a pseudo-parameter makes the test of overdispersion particularly useful. The analysis in the literature including Cox (984), Chesher (984) goes through as long as the heterogeneity is mean-independent of the randomness in Y given X. Given the way that the pseudo-parameter (X g ) is de ned, the mean-independence assumption is satis ed by default. It follows that the test statistic of interest in this application is simply nx f D i j X i ; ^; (X) p n f D i j X i ; ^; : (X) i= Obviously one needs to adjust for the error of estimating as well as (X i ). The (X i ) is often estimated nonparametrically, and as such, analysis along the lines of Newey (994) is necessary in order to characterize the asymptotic variance. From a practical perspective, it can be accommodated either by the bootstrap result as in Chen, Linton, and van Keilegom (3), or by the numerical equivalence result as in Ackerberg, Chen, and Hahn (). 9 4 Monte Carlo Our Monte Carlo exercise is inspired by the application of Sweeting (9), who estimates a strategic model of the timing of radio commercials by radio stations. There may be incentives for radio stations to coordinate the timing of their commercials so listeners do not have incentive to switch stations when commercials come on because, if they do, they will just tune to another commercial. The model is estimated using both a two step approach like those described above and a nested xed point approach that requires repeatedly solving for equilibrium. Multiplicity is endemic to such coordination games, yet identi cation of the two step estimator requires 9 In Appendix B, we present a (straightforward) characterization of how the procedure in Ackerberg, Chen, and Hahn () can be used in our context.

13 assuming, within a geographic market, stations play a single equilibrium strategy. De Paula and Tang (3) analyze this model when the single equilibrium assumption is relaxed and show the model is partially identi ed. In particular, they show the sign of the strategic interaction term can be estimated. Often, however, an analyst is interested in more precise, quantitative answers to counterfactual policy questions and these will generally require point estimates. We assume there are K stations competing in each of n games. For ease of exposition we assume the players are local in the sense that each player only competes in game. Stations compete by choosing a binary action d i f; g, which can be interpreted as a choice to play commercials at :5 or :55 minutes past the hour. The payo associated with each choice are given by: E[u(d i = ; D ig ; x g ; )jx g = x g ] = BX ig + JE[P ig jx g = x g ] + " ig E[u(d i = ; D ig ; x g ; )jx g = x g ] = CX ig + JE[P ig jx g = x g ] + " ig where P j6=i P dig = fd jg = dg K is the proportion of rivals choosing action d and expectations are taken with respect to player i s subjective beliefs about rival play. X ig is a vector of rm speci c covariates. Our setup requires variation in these X s as well as the assumption that X ig are excluded from i s payo. Bajari, Hong Krainer, and Nekipelov () discuss su cient conditions for identi cation of the general model. Sweeting (9) relies on panel arguments in lieu of the this exclusion restriction. We assume X ig f ; ; g and are drawn i.i.d uniform across players and games. Finally we assume the " s are drawn i.i.d from a type extreme value distribution. The probability of observing d i = is given by Pr(d ig = jx g ; ) = Pr(~u(d i = ; X; ) ~u(d i = ; X; ) > )) = exp(x ig + J(E[P ig jx g ] )) + exp(x ig + J(E[P ig jx g ] )) Where we let ~u(d i = d; X; ) = E[u(d i = d; D ig ; x g ; )jx g = x g ]. In the second stage, we formulate a (pseudo) maximum likelihood estimator based on the logit conditional choice probabilities. f(djx; ; ^P (X)) = exp(x ig + J( ^P (X i g) )) + exp(x ig + J( ^P (X i g) ))! dig + exp(x ig + J( ^P (X i g) ))! dig where we have used exchangeability of the X s to rearrange X g as X i g = (X ig ; X ig ) and drop the i index on the CCP. The rst stage estimate ^P is the rst stage CCP estimate: 3

14 p nk P n P K ^P (x i nk g= i= g) = P j6=i fd gj=gfx i g =xi g g K P n P K nk g= i= fxi g = x i gg Letting ^ = ( ^; ^J) denote the maximum likelihood estimates, our test statistic is: nx g= KX i= ^ = p nk nx g= KX i= f (a ig jx g ; ^; ^P ig ) f(a ig jx g ; ^; ^P ig ) = fa i =g 4J ( + exp(x ig + ^J( ^P (X i g) ))( exp( b X ig + ^J( ^P (X i g) )) ( + exp(x ig + J( ^P (X i g) )) 3 +fa i = g 4J ( + exp(b X ig + ^J( ^P (X i g) ))( exp( b X ig + ^J( ^P (X i g) )) ( + exp( b X ig + ^J( ^P (X i g) )) 3 exp( b X ig + ^J( ^P (X i g) )) For our Monte Carlo experiments, we rst solve for all the stable equilibria of our game conditional on X g, we then generate data assuming the X g s are drawn from a discrete uniform distribution in the population. To generate choice data under the single equilibrium assumption we randomly selected an equilibrium from the calculated set of equilibria for each X supp(x) and then generated the pro le of choices by drawing from the distribution induced by this equilibrium for every game in which the realization of X g was X. That is, conditional on X, a single equilibrium is played, though multiple equilibria may be played across the di erent X s. As discussed in Sweeting (9) and Brock and Durlauf () this type of realization of multiplicity actually aids in identi cation of the model, particularly in this type of model where, without realization of multiple equilibria, there is very limited, local variation in conditional choice probabilities. To generate the data under the alternative where the single equilibrium assumption fails, we randomly select an equilibrium from the calculated set of equilibria for each realization of X g in the data and then generate choices by drawing from the distribution induced by this equilibrium. In both the case where the single equilibrium assumption holds and where it does not, we use the same equilibrium selection mechanism. To de ne the mechanism, we start by using iterative updating of strategies to nd an equilibrium associated with an initial starting strategy pro le that has all players choosing an action with probability one. There are 3 (= 5 ) such pro les and all converge to one of the, at most, two stable equilibria of the game. For each X g, in the case of the single equilibrium assumption holding, and for each g, in the case of the single equilibrium assumption failing, the selection mechanism chooses uniformly from the 4!!

15 3 equilibrium realizations of one of the, at most, two equilibria found using our iterative updating algorithm. Intuitively, this causes the mechanism to, for example, tend to select the high equilibrium more often when covariate values tend to be higher. De ning the selection mechanism in this way is a natural way to generate dependence of the equilibrium selection on the covariates. Tables and show the results of our Monte Carlo experiments. Table displays the mean and standard deviation of parameter estimates from simulations assuming di erent sample sizes and true underlying interaction e ects. Table explores the power and size attributes of the test, displaying rejection probabilities from simulations across the di erent sample sizes and true interaction e ects. For each simulation we calculate ~ = ^ p var(^) where we use 5 bootstrap to estimate var(^), and compare it to the standard normal critical values. For both tables the statistics are based on simulations. The last three columns of Table illustrate the impact of the failure of the single equilibrium assumption. Both the impact of fundamentals ( ) and strategic e ects (J) are attenuated. The reason for this is simple. Due to multiplicity, actual choice probabilities are either all high or all low depending on which equilibrium is selected. The CCP estimate lies in between as a selection-probability weighted average of these equilibria. In other words, are CCP estimates suggest that equilibrium choices are more independent than they really are and thus more consistent with a game with relatively low strategic interaction e ects. Table shows our multiplicity test performs quite well in this example. When strategic e ects are weaker, variation in outcomes becomes decreasingly dominated by multiplicity and our test has less power in detecting a violation of the single equilibrium assumption. In our example, this happens both because the direct e ect of X i on a player s payo s is larger relative to the strategic e ects and also indirectly because the equilibrium selection mechanism is relatively more dependent on X g. 5

16 Table : Parameter Estimates Single Eq. True (H) Single Eq. False (H) J n ^ (const.; = ) ^( = ) ^J ^ ^ ^J.(.9).36(.3).88(.3) -.(.7).786(.9).758(.4).75.(.6).(.6).784(.9) -.8(.5).774(.).74(.) 5.(.4).9(.).76(.6) -.33(.3).768(.8).73(.6).3(.9).48(.6) 3.88(.8) -.6(.8).583(.5).7(.6) 3.3(.6).6(.8) 3.49(.) -.65(.5).556(.).676(.) 5 -.3(.4).6(.) 3.4(.7) -.68(.3).549(.7).648(.7).7(.).74(.9) 3.357(.) -.35(.6).365(.).59(.6) 3.5.(.7).4(.) 3.3(.5) -.38(.4).355(.7).53(.) 5.(.4977).8(.) 3.68(.8) -.38(.).349(.5).438(.7) Table : Rejection Frequency Single Eq. True (H) Single Eq. False (H) J n reject (p = :) reject (p = :5) reject (p = :) reject (p = :) reject (p = :5) reject (p = :) Notes:Tab le :Average coe cient estimates over simulated data sets, standard deviation of the simulated estimates in parentheses Tab le Fraction of simulations where the null is rejected at the indicated signi cance level. Test statistic distribution estimated using bootstrap repetitions. 6

17 Appendix A Technical Details for Section A. Some Details Behind (8) By the change-of-variable, we have Z h (yjx; ; ) = f (yjx; + p w) g (wj x) dw: Obviously the score is equal to s (y; ; ) = h (yjx; ; ) yjx; + p g (wj x) dw; and Z p yjx; + w g (wj x) dw = R f f (yjx; + p w w) p g (wj x) dw yjx; + p w wg (wj x) dw p : Using l Hôpital s rule, we obtain lim! R f yjx; + p w wg (w) dw p = lim! R f yjx; + p w w p wg (wj = f (yjx; ) (x) A. Some Details Behind (3) We prove that the test statistic based on (8), i.e., p n nx i= is asymptotically normal with mean equal to f (Y i j X i ; ) f (Y i j X i ; ) (X i ) E (X i ) (X i ) (X i ) (3) Here we use the Lebesgue s dominated convergence theorem under the condition XX of Assumption XX in the appendix. 7

18 and variance equal to. Here E denotes the expectation operator de ned by the null density f (yj x; ) p (x) ; where p (x) is the marginal density of X i. We rst derive the mean and the variance of the statistic p n nx i= f (Y i j X i ; ) f (Y i j X i ; ) (X i ) under the alternative indexed by ; Z k (yj x; ; ) = f (yjx; + w) g (wj x) dw; where g (wjx) is the conditional density of w on x chosen by the nature. In this section, we assume the following regularity conditions: The mixture distribution g satis es. R wg (wjx) dw =. R w g (wjx) dw = (x) and (x) 6= 3. R jwj 3 g (wjx) dw G (x) : Let p (x) be the density function of X i.the conditional density f (yjx; ) is three times continuously di erentiable with respect to the argument and the derivatives satisfy. sup jf (yjx; )j ; sup jf (yjx; )j ; sup jf (yjx; )j F (y; x; ) for some function F (; ; ). R R f ( yjx;) f( yjx;) (x) F (y; x; ) max f; G (x) p (x)g dydx < We denote P ; E ; and V as the probability measure, the expectation operator, and the variance operator, respectively, de ned by the density k (yj x; ; ) p (x) : De ne P = P ; E = E ; and V = V : We rst derive the limits (i) lim! E f (Y i j X i ; ) f (Y i j X i ; ) (X i ) and (ii) lim! V f (Y i j X i ; ) f (Y i j X i ; ) (X i ) 8

19 Part (i): We apply the third order Taylor approximation to k (yj x; ; ) of around = : Then, f (yj x; ) as a function k (yj x; ; ) f (yj x; ) (yj x; ; ) = k (yj x; ; ) k (yj x; ; ~) where ~ lies between and : Here we have (yj x; ; (yjx; + wg (wj x) dw = Z k (yj x; ; f (yjx; + w g (wj x) dw = = f (yjx; ) Z k (yj x; ; = f (yjx; + 3 w 3 g (wj x) dw =~ F (y; x; ) G (x) : Choose = n =4 : Then, since R f (yj x; ) dy =, we have " # nx f E p (Y i j X i ; ) n f (Y i= i j X i ; ) (X i ) = p Z Z f (yj x; ) n f (yj x; ) (x) k (yj x; ; ) p (x) dydx = p Z Z f (yj x; ) n f (yj x; ) (x) [k (yj x; ; ) f (yj x; )] p (x) dydx Z Z f (yj x; ) f (yj x; ) = f (yj x; ) (x) f (yj x; ) (x) f (yj x; ) p (x) dydx + Z Z f (yj x; k (yj x; ; ~) n =4 f (yj x; ) (x) p (x) dydx (5) Because Z Z f (yj x; k (yj x; ; ~) f (yj x; ) (x) p (x) dydx Z Z f (yj x; ) f (yj x; ) (x) F (y; x; ) G (x) p (x) dydx < ; 9

20 we have " # nx f E p (Y i j X i ; ) n f (Y i= i j X i ; ) (X i ) Z Z f (yj x; ) f (yj x; )! f (yj x; ) (x) f (yj x; ) (x) f (yj x; ) p (x) dydx f (Y i j X i ; ) f (Y i j X i ; ) = E f (Y i j X i ; ) (X i ) f (Y i j X i ; ) (X i ) : Part (ii): As for the variance in (ii), we can use the dominated convergence theorem. Note that " # lim V nx f p (Y i j X i ; )! n f (Y i= i j X i ; ) (X i ) f (Y i j X i ; ) = lim V! f (Y i j X i ; ) (X i ) Z Z f (yj x; ) = f (yj x; ) (x) lim k (yj x; ; ) p (x) dydx! Z Z f (yj x; ) f (yj x; ) (x) lim k (yj x; ; ) p (x) dydx :! It follows that lim! V " p n nx i= # f (Y i j X i ; ) f (Y i j X i ; ) (X i ) Z Z f (yj x; ) = f (yj x; ) (x) f (yj x; ) p (x) dydx Z Z f (yj x; ) f (yj x; ) (x) f (yj x; ) p (x) dydx f (Y i j X i ; ) = E f (Y i j X i ; ) (X i ) = : Proof of (3): Next, we derive the limiting distribution under the sequence of local alternative. For this, we let U i = f ( Y i jx i ;) h f( Y i jx i ;) (X i ) E hv f ( Y i jx i ;) i = f ( Y i jx i ;) f( Y i jx i ;) (X i ) i f( Y i jx i ;) (X i ) and consider a triangular sequence fu i : i ng and its underlying probability measure sequence P n =4: Since fu i g i=;:::;n i:i:d: (; ) under P n =4: Then, by the Lindberg-Feller CLT for the triangular array, we have nx p U i ) N (; ) under P n n =4: (6) i=

21 Then, by Parts (i) and (ii) and (6), under P n =4 we have p n = nx i= V n =4 f (Y i j X i ; ) f (Y i j X i ; ) (X i ) f (Y i j X i ; ) f (Y i j X i ; ) (X i ) = pn! nx U i " # nx f + E n =4 p (Y i j X i ; ) n f (Y i= i j X i ; ) (X i ) ) = f (Y i j X i ; ) f (Y i j X i ; ) N (; ) + E f (Y i j X i ; ) (X i ) f (Y i j X i ; ) (X i ) f (Y i j X i ; ) f (Y i j X i ; ) = N E f (Y i j X i ; ) (X i ) f (Y i j X i ; ) (X i ) ; ; i= under P n =4 as required for (3). A.3 Proof of Max-Min The nature moves rst. In order to minimize the power, it tries to minimize the noncentrality parameter ( () ; ()) by choosing (x) given positivity of () and the normalization restriction and (x). This is equivalent to minimizing E [ (X i ) (X i ) (X i )] with respect to () given positivity of () and the normalization restriction. Let X denotes the support of X i. Notice that for any (X i ) with E [ (X i ) (X i )] = ; it follows that E (X i ) (X i ) (X i ) E min xx (x) (X i ) (X i ) = min xx (x). Therefore, min E (X i ) (X i ) (); E[ (X) (X)]= (X i ) min xx (x) : (7) Suppose that x X achieves the minimum of (x) over x X ; i.e. (x ) (x) for all x X : Suppose that the support of X i ; X ; is discrete. We set ( (x) = (x )p(x for x = x ) : for x 6= x Then, we can see that E [ (X) (X)] = and also E (X i ) (X i ) (X i ) = (x ) : (8)

22 This implies that the lower bound of min (); E[ (X) (X)]= E [ (X i ) (X i ) (X i )] is achieved with and that min E (X i ) (X i ) (); E[ (X) (X)]= (X i ) = (x ) = min xx (x) ; (E [ (X i ) (X i ) (X i )]) min (); E[ (X) (X)]= E (X i ) ( (X i )) = ( (x )) E (X i ) ( (X i )) : When X i is a continuous random variable, we consider a sequence of functions ;m (x) such 8 < R x + ;m m (x) p (x) dx if x (x) = x < x < x m + m m : otherwise Then, for all m; we have E (X) ;m (X) = Z x + m x m (x) p (x) dx Also, as m! ; we have as in (8).! Z E (X i ) (X i ) ;m (X) (x) p (x) x m < x < x + dx = : m = m R (x) (x) p (x) x < x < x m + m m R x + m (x) p (x) dx x m! (x ) Now the econometrician moves. He wants to maximize the power by maximizing this ratio ( (x )) E (X i ) ( (X i )) manipulating (). For this purpose, we can see that h E (X i ) (X i ) i " # E (X i ) inf xx (x) and hence, Note that the upper bound of = 4 (x ) E (X i ) ( (x )) E (X i ) ( (X i )) ( 4 (x )) E [ (X i )] ( 4 (x )) = E [ (X i )] ( (x )) E[ (X i )( (X i )) ] ; E[ (X i ; is achievable by choosing )] (x) = (x ) : This justi es a constant choice for (x) from a minimax decision perspective.

23 B An Estimator of the Asymptotic Variance of p n (^ ) If we are to use the results in Ackerberg, Chen, and Hahn (), it su ces to write f (Y i j X i ; (Z i )) E : f (Y i j X i ; (Z i )) We can then view (; ) to be the method of moments estimator log nx f( Y i jx i ; ; b (Z b A n f ( Y i jx i ; ; b (Z b i )) = i= f( Y i jx i ;;(Z i b )) corresponding to the moment equation log f( Yi # jx i ;;(Z i E f ( Y i jx i ;;(Z i = )) f( Y i jx i ;;(Z i )) Note that =. Also note that the asymptotic variance of p nx f Y i j X i ; ; b b (Z i ) n (b ) = pn f Y i j X i ; ; b b (Z i ) i= can be taken care of by the numerical equivalence result as in Ackerberg, Chen, and Hahn (), i.e., we will use the fact that b (Z i ) = Z i ; b in practice for some nite-dimensional b, and apply the usual Murphy-Topel type formula. Let the W i = (Y i ; D i ; X i ; Z i ) : Suppose that one speci es E (D i jz i ) = (Z i ) with a parametric model (Z i ; ) and estimate by solving the exactly identi ed moment condition: n nx S i= D i ; Z i ; ^ = When D i is binary, if we use the logit model with (Z i ; ) = exp(z) i +exp(zi ), the score function is S (D i ; Z i ; ) = (D i (Z i ; )) Z i : If we use the probit model with (Z i ; ) = (Zi) ; then score function is S (D i ; Z i ; ) = (D i )((D i )Zi ) ((D i )Zi ) : Let = (; ) Denote m (W i ; ) = " S (D i ; Z i ; log f( Y i jx i ;;(Z i # = " S (Di ; Z i ; ) f ( Y i jx i ;;(Z i ;)) f( Y i jx i ;;(Z i ;)) # : 3

24 Then, ^ solves Let n nx i= m W i ; ^ = : m (W i ; ; ) = f (Y i j X i ; ; (Z i ; )) f (Y i j X i ; ; (Z i ; )) Then, ^ is estimated by solving the moment condition n P n i= m plugged in for : Denote " ^ ^ # n nx i= m W i ; ^; ^ = : : W i ; ; ^ = after ^ is ^ = 4 Also, denote We have ^ P n n i= m W i ; ^ ^ P m W i ; n n i= m W i ; ^ m W i ; ^; ^ P n n i= m W i ; ^; ^ ^ P m W i ; n n i= m W i ; ^; ^ m W i ; ^; ^ ; = n ; = n ; = n W i ; ^ = n i= W i ; ^; ^ = W i ; ^; ^ = n i= i= i= i ;^) W i ; ^ W i ; ^ W i ; ^; ^ = W i ; ^; ^ f f f h f f i ;^;^) f i f f f f # f @m (W i ; (W i i : 3 5 : 4

25 where f = f Y i j X i ; ^; Z i Y i j X i ; Z i ; ^ : Then, the variance of p N (^ ) is the upper left corner of " ; # " # " ^ ^ ; # : ; ; ^ ^ ; ; Since " ; ; ; " ; ; ; # = " # = 4 ; ; ; ; ; ; ; ; # ; ; 3 5 ; the estimator of the variance of p N (^ h ; ; ; ; i " ; ^ ^ ^ ^ ) is # 4 3 ; ; ; 5 ; = ; ; ; ^ ; ; ; ; ^ ; ; ; ; ; ; ^ ; + ; ^ ; = ; ; ^ ; ; ^ ; ; ; ; ^ + ^ : 5

26 References [] Ackerberg, D., X. Chen, and J. Hahn (): A Practical Asymptotic Variance Estimator for Two-Step Semiparametric Estimators, Review of Economics and Statistics 94, pp [] Aguirregabiria, V. and P. Mira (): Swapping the Nested Fixed Point Algorithm: A Class of Estimators for Discrete Markov Decision Models, Econometrica 7, pp [3] Aguirregabiria, V. and P. Mira (7): Sequential Estimation of Dynamic Discrete Games, Econometrica 75, pp. 53. [4] Aguirregabiria, V. and A. Nevo (3): Recent Developments in Empirical IO: Dynamic Demand and Dynamic Games, in Acemoglu, D., M. Arellano, and E. Deckel, eds. Advances in Economics and Econometrics. Tenth World Congress. Volume 3, Econometrics. [5] Bajari, P., C.L. Benkard, and J. Levin (7): Estimating Dynamic Models of Imperfect Competition, Econometrica 75, pp [6] Bajari, P., H. Hong, J. Krainer and D. Nekipelov (): Estimating Static Models of Strategic Interactions, Journal of Business & Economic Statistics 8, [7] Balestra, P., and M. Nerlove (966): Pooling Cross Section and Time Series Data in the Estimation of a Dynamic Model: The Demand for Natural Gas, Econometrica 34, pp [8] Bierens, H.J. (99): A Consistent Conditional Moment Test of Functional Form, Econometrica, 58, pp [9] Bierens, H.J. (9): Consistent Model Speci cation Tests, Working Paper. [] Breusch, T.S., and A.R. Pagan (98): The Lagrange Multiplier Test and Its Application to Model Speci cation in Econometrics, Review of Economic Studies 47, pp [] Card, D. and L. Giuliano (): Peer E ects and Multiple Equilbria in the Risky Behavior of Friend, UC Berkeley Working Paper. [] Chamberlain, G. (984): Panel Data, in Griliches, Z., and M.D. Intriligator, eds. Handbook of Econometrics, Vol II, pp , North Holland: Amsterdam. 6

27 [3] Chen, X., O. Linton and I. van Keilegom (3): Estimation of Semiparametric Models when the Criterion Function is not Smooth, Econometrica 7, pp [4] Chesher, A. (984): Testing for Neglected Heterogeneity, Econometrica 5, pp [5] Cho, J.S., and H. White (7): Testing for Regime Switching, Econometrica 75, pp [6] Cox, D.R. (983): Some Remarks on Overdispersion, Biometrika 7, pp [7] Engle, R.F. (984): Wald, Likelihood Ratio, and Lagrange Multiplier Tests in Econometrics, in Griliches, Z., and M.D. Intriligator, eds. Handbook of Econometrics, Vol II, pp , North Holland: Amsterdam. [8] Donald, S., G. Imbens, and W.K. Newey (3) "Empirical likelihood estimation and consistent tests with conditional moment restrictions." Journal of Econometrics 7, pp [9] Hahn, J., W.K. Newey, and R.J. Smith (4): Neglected Heterogeneity in Moment Condition Models, Journal of Econometrics 78, pp. 86. [] Hotz, V.J., and R.A. Miller (993): Conditional Choice Probabilities and the Estimation of Dynamic Models, Review of Economic Studies 6, pp [] Kasahara, H., and K. Shimotsu (9): Nonparametric Identi cation of Finite Mixture Models of Dynamic Discrete Choices, Econometrica 77, pp [] Lee, L., and A. Chesher (986): Speci cation Testing when Score Test Statistics Are Identically Zero, Journal of Econometrics 3, pp. 49. [3] Liu, X., and Y. Shao (3): Asymptotics for Likelihood Ratio Tests under Loss of Identi ability, Annals of Statistics 3, pp [4] Maddala, G. S. (97): The Use of Variance Components Models in Pooling Cross Section and Time Series Data, Econometrica 39, pp [5] Newey, W.K. (985): Maximum Likelihood Speci cation Testing and Conditional Moment Tests, Econometrica, 53, pp [6] Newey, W.K. (994): The Asymptotic Variance of Semiparametric Estimators, Econometrica 6,

28 [7] Pakes, A., Ostrovsky, M, and S. Berry (7): Simple Estimators for the Parameters of Discrete Dynamic Games, with Entry/Exit Examples, RAND Journal of Economics 38, [8] Pesendorfer, M. and P. Schmidt-Dengler (8): Asymptotic Least Squares Estimators for Dynamic Games, Review of Economic Studies 75, pp [9] Seim, K. (6): An Empirical Model of Firm Entry with Endogenous Product-Type Choices, RAND Journal of Economics 37, [3] Sweeting, A. (9): The Strategic Timing of Radio Commercials: An Empirical Analysis Using Multiple Equilibria, RAND Journal of Economics 4, pp [3] White, H. (98): Maximum Likelihood Estimation of Misspeci ed Models, Econometrica 5, pp. 5. 8

29 C On-line Appendix: Not for Publication C. Derivation of the Panel Logit Example Note that f (Y i j X i ; ) (X i ) = E f (Y i j X i ; ) = E 4 = t= t= f (Y i j X i ; ) X i f (Y i j X i ; ) y it exp (x it + ) + exp (x it + )!3 exp (x it + ) ( + exp (x it + )) " E y it t=! E " t= exp (x 3 it + ) + exp (x it + ) X i 3 X 5 i exp (x y it + ) it + exp (x it + ) # : # X i The last equality above uses the mathematical induction starting from the following observation: Suppose that A ; A are zero mean independent variables. We then have because E (A + A ) 3 = E A 3 + E A 3 ; (A + A ) 3 = A 3 + 3A A + 3A A + A 3 and recall zero mean assumption and independence, which eliminates all the terms except the two cubic terms. We now claim that (X i ) = t= + = exp (x it + ) ( t= exp(x +) 3 it exp(x +exp(x +) it it +exp(x +) it exp(x it +) +exp(x it +) 3 +exp(x it +) exp (x it + )) ( + exp (x it + ))3 exp(x This is based on the observation that +) it +exp(x +) is the mean of y it, and that if for a Bernouille it variable Z with probability p, we have E (Z p) 3 = ( p) 3 p + ( p) 3 ( p) = p (p ) (p ) which leads the desired result in the panel logit example. C A 9