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## A test for monotone comparative statics (2013)

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Venue: | – FORTHCOMING IN ADVANCES IN ECONOMETRICS (VOL 31) – |

### Citations

2398 | Microeconomic Theory - Mas-Colell, Whinston, et al. - 1995 |

1092 | Modelling Extremal Events for Insurance and Finance - Embrechts, Klüppelberg, et al. - 1997 |

739 | Extreme Values, Regular Variation, and Point Processes - Resnick - 1987 |

703 | Foundations of Economic Analysis - Samuelson - 1947 |

500 | The Economics of Modern Manufacturing - Milgrom, Roberts - 1990 |

490 | Recursive Macroeconomic Theory. - Ljungqvist, Sargent - 2004 |

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Statistical Inference Under Order Restrictions
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Citation Context ...ndependent Gaussian random variables with known variances.s22 ECHENIQUE AND KOMUNJER A likelihood-ratio (LR) test of H0 against H1 is now available from the existing literature (Bartholomew, 1959a,b; =-=Barlow, Bartholomew, Bremner, and Brunk, 1972-=-; Robertson and Wegman, 1978). We shall review Barholomew’s results, as they are instrumental in showing how the extension of exact results works in the asymptotic case. We introduce the following not... |

281 | Least absolute deviations estimation for the censored regression model
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Citation Context ...the sample size T . It is an extension of the wellknown result for central α-quantiles with α ∈ (0, 1) fixed (Mosteller, 1946; Smirnov, 1952; Siddiqui, 1960; Bahadur, 1966; Bassett and Koenker, 1978; =-=Powell, 1984-=-, 1986), to the case where α increases with the sample size T . Dekkers and de Haan (1989) and Chernozhukov (2005) prove this extension under an additional assumption on the tail behavior of F . While... |

254 |
Tobin’s Marginal q and Average q: A
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(Show Context)
Citation Context ... well-known economic models. Examples are single-person decision models such as models of optimal growth (Barro and Sala-I-Martin, 2003; Ljungqvist and Sargent, 2004) and firms’ investment decisions (=-=Hayashi, 1982-=-; Hayashi and Inoue, 1991), as well as many models in IO (see Vives (1999) for survey). Current econometric literature, on the other hand, has largely remained silent on the issue of formal tests for ... |

220 |
Wald criteria for jointly testing equality and inequality restrictions
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Citation Context ...order restrictions based on the asymptotic distributions of our conditional quantile estimators. This step presents important challenges as the existing results (Gourieroux, Holly, and Monfort, 1982; =-=Kodde and Palm, 1986-=-) apply only to the conditional means; hence, we need to extend them to our extreme conditional quantile framework. Perhaps an even greater difficulty comes from the presence of numerous nuisance para... |

218 |
Estimation of a Model of Entry
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- 1992
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Citation Context ...itions for point identification. A number of recent papers in the (quickly expanding) literature on econometrics of games further carry out estimation in such models (Bresnahan and Reiss, 1990, 1991; =-=Berry, 1992-=-). When the parameters of interest are only set-identified, interesting results on estimation and inference have been derived by (Tamer, 2003; Andrews, Berry, and Jia, 2004; Ciliberto and Tamer, 2004;... |

174 | An Empirical Framework for Testing Theories About Complementarity - Athey, Stern - 1998 |

163 |
Incomplete Simultaneous Discrete Response Model with Multiple Equilibria
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Citation Context ...stimation in such models (Bresnahan and Reiss, 1990, 1991; Berry, 1992). When the parameters of interest are only set-identified, interesting results on estimation and inference have been derived by (=-=Tamer, 2003-=-; Andrews, Berry, and Jia, 2004; Ciliberto and Tamer, 2004; Kim, 2005; Galichon and Henry, 2006). Mostly, the above papers build on discrete-choice methods and are well-suited for models with few choi... |

155 | 2004): “Market Structure and Multiple Equilibria - Ciliberto, Tamer |

139 |
The Relation Between Firm Growth and Q with Multiple Capital Goods: Theory and Evidence from Panel Data on Japanese Firms
- Hayashi, Inoue
- 1991
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Citation Context ...nomic models. Examples are single-person decision models such as models of optimal growth (Barro and Sala-I-Martin, 2003; Ljungqvist and Sargent, 2004) and firms’ investment decisions (Hayashi, 1982; =-=Hayashi and Inoue, 1991-=-), as well as many models in IO (see Vives (1999) for survey). Current econometric literature, on the other hand, has largely remained silent on the issue of formal tests for MCS. The goal of this pap... |

134 | On Regular Variation and its Application to the Weak Convergence of Sample Extremes. Centre Tract 32 - Haan - 1970 |

120 | What Happens When Wal-Mart Comes to Town: An Empirical Analysis of the Discount Retailing Industry - Jia - 2008 |

111 |
Estimation of a model of entry in the airline industry.
- Berry
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Citation Context ...itions for point identification. A number of recent papers in the (quickly expanding) literature on econometrics of games further carry out estimation in such models (Bresnahan and Reiss, 1990, 1991; =-=Berry, 1992-=-). When the parameters of interest are only set-identified, interesting results on estimation and inference have been derived by (Tamer, 2003; Andrews, Berry, and Jia, 2004; Ciliberto and Tamer, 2004;... |

100 | 2007): “Sequential Estimation of Dynamic Discrete - Aguirregabiria, Mira |

96 | Oligopoly Pricing - Vives - 1999 |

85 |
Asymptotic theory of least absolute error regression
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- 1978
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Citation Context ...antiles when αT depends on the sample size T . It is an extension of the wellknown result for central α-quantiles with α ∈ (0, 1) fixed (Mosteller, 1946; Smirnov, 1952; Siddiqui, 1960; Bahadur, 1966; =-=Bassett and Koenker, 1978-=-; Powell, 1984, 1986), to the case where α increases with the sample size T . Dekkers and de Haan (1989) and Chernozhukov (2005) prove this extension under an additional assumption on the tail behavio... |

80 | 2010): “Identification and Estimation of a Discrete Game of Complete Information - Bajari, Hong, et al. |

70 | 2004): “Confidence Regions for Parameters in Discrete Games with Multiple Equilibria, with an Application to Discount Chain Store Location - Andrews, Berry, et al. |

67 | Rationalizability, Learning and Equilibrium
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Citation Context ...odels possess multiple equilibria, and a monotone comparative statics (MCS) prediction holds: There is a smallest and a largest equilibrium, and these change monotonically with explanatory variables (=-=Milgrom and Roberts, 1990-=-; Milgrom and Shannon, 1994; Villas-Boas, 1997). MCS is a feature found in many well-known economic models. Examples are single-person decision models such as models of optimal growth (Barro and Sala-... |

65 |
Simultaenous Equations Models for Dummy Endogenous Variables: A Game Theoretic Formulation with an Application to Labor Force Participation,” Caltech Social Science Working Paper 577
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Citation Context ...ortance of testing economic models that possess multiple equilibria. Proposed solutions to the problem of multiplicity have been to assume the probabilities of various equilibrium realizations known (=-=Bjorn and Vuong, 1984-=-); or finitely parameterized (Bjorn and Vuong, 1985; McKelvey and Palfrey, 1995; Bajari, Hong, and Ryan, 2004; Sweeting, 2005). Without the specification of an equilibrium selection rule, there are al... |

63 | Conditional Quantile Estimation and Inference for ARCH Models,” Econometric Theory - Koenker, Zhao - 1996 |

58 | On the Estimation of the Extreme-Value Index and Large Quantile Estimation - Dekkers, Haan - 1989 |

51 | Sur une mode de croissance réguliere de fonctions - Karamata |

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A Test of Homogeneity for Ordered Alternatives
- Bartholomew
- 1959
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Citation Context ...eans (μ1,...,μk) ofk independent Gaussian random variables with known variances.s22 ECHENIQUE AND KOMUNJER A likelihood-ratio (LR) test of H0 against H1 is now available from the existing literature (=-=Bartholomew, 1959-=-a,b; Barlow, Bartholomew, Bremner, and Brunk, 1972; Robertson and Wegman, 1978). We shall review Barholomew’s results, as they are instrumental in showing how the extension of exact results works in t... |

44 | 2006): “Quantifying equilibrium network externalities in the ACH banking industry,” The RAND - Ackerberg, Gowrisankaran |

42 | Observable Implications of Models with Multiple Equilibria - Jovanovic - 1989 |

41 | Comparative Statics by Adaptive Dynamics and The Correspondence Principle - Echenique - 2002 |

35 |
Limit Distributions for the Terms of a Variational Series
- Smirnov
- 1952
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Citation Context ...ymptotic behavior of intermediate empirical quantiles when αT depends on the sample size T . It is an extension of the wellknown result for central α-quantiles with α ∈ (0, 1) fixed (Mosteller, 1946; =-=Smirnov, 1952-=-; Siddiqui, 1960; Bahadur, 1966; Bassett and Koenker, 1978; Powell, 1984, 1986), to the case where α increases with the sample size T . Dekkers and de Haan (1989) and Chernozhukov (2005) prove this ex... |

33 | 2005), “Extremal quantile regression
- Chernozhukov
(Show Context)
Citation Context ... holds only for quantiles that aresA TEST FOR MONOTONE COMPARATIVE STATICS 5 extreme; hence, we need to use a nonstandard framework to derive their asymptotic distribution (Dekkers and de Haan, 1989; =-=Chernozhukov, 2005-=-). In the second step, we construct a likelihood-ratio test for order restrictions based on the asymptotic distributions of our conditional quantile estimators. This step presents important challenges... |

33 | On the Theory of Order Statistics - Renyi - 1953 |

31 | Coordination Games, Multiple Equilibria and the Timing of Radio Commercials
- Sweeting
- 2005
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Citation Context ...assume the probabilities of various equilibrium realizations known (Bjorn and Vuong, 1984); or finitely parameterized (Bjorn and Vuong, 1985; McKelvey and Palfrey, 1995; Bajari, Hong, and Ryan, 2004; =-=Sweeting, 2005-=-). Without the specification of an equilibrium selection rule, there are alternative approaches to estimation and inference. The first exploits the fact that—despite the multiplicity—some of the model... |

30 | Palfrey (1995): “Quantal Response Equilibria for Normal Form - McKelvey, R |

30 |
Behavior of Regression Quantiles in Non-Stationary, Dependent Cases
- Portnoy
- 1991
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Citation Context ...nt for the true α-quantile under FY |X=x. Consistency of ˆqY |X=x(α) can be extended to cases where (Yx,1,...,Yx,Tx) is a weakly dependent time-series, provided additional assumptions (Pollard, 1991; =-=Portnoy, 1991-=-; Koenker and Zhao, 1996; Komunjer, 2005; Chernozhukov, 2005); for the sake of simplicity, we focus on the independent case. To alleviate the notation, we drop the reference to x when doing so introdu... |

27 | Quasi-Maximum Likelihood Estimation for Conditional Quantiles
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- 2005
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Citation Context ...x. Consistency of ˆqY |X=x(α) can be extended to cases where (Yx,1,...,Yx,Tx) is a weakly dependent time-series, provided additional assumptions (Pollard, 1991; Portnoy, 1991; Koenker and Zhao, 1996; =-=Komunjer, 2005-=-; Chernozhukov, 2005); for the sake of simplicity, we focus on the independent case. To alleviate the notation, we drop the reference to x when doing so introduces no ambiguities. Hence we use the not... |

23 | A Unified Approach to Likelihood Inference on Stochastic Orderings in a Nonparametric Context - Dardanoni, Forcina - 1998 |

23 |
On some Useful ”Inefficient” Statistics
- Mosteller
- 1946
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Citation Context ...emma shows the asymptotic behavior of intermediate empirical quantiles when αT depends on the sample size T . It is an extension of the wellknown result for central α-quantiles with α ∈ (0, 1) fixed (=-=Mosteller, 1946-=-; Smirnov, 1952; Siddiqui, 1960; Bahadur, 1966; Bassett and Koenker, 1978; Powell, 1984, 1986), to the case where α increases with the sample size T . Dekkers and de Haan (1989) and Chernozhukov (2005... |

23 | Foundations of Economic Analysis - A - 1947 |

22 | Sala-I-Martin (2003): Economic Growth - Barro, X |

18 | 2007): “Testing Models with Multiple Equilibria by Quantile Methods - Echenique, Komunjer |

17 |
Likelihood Ratio Tests for Order Restrictions in Exponential Families
- Robertson, Wegman
- 1978
(Show Context)
Citation Context ... variances.s22 ECHENIQUE AND KOMUNJER A likelihood-ratio (LR) test of H0 against H1 is now available from the existing literature (Bartholomew, 1959a,b; Barlow, Bartholomew, Bremner, and Brunk, 1972; =-=Robertson and Wegman, 1978-=-). We shall review Barholomew’s results, as they are instrumental in showing how the extension of exact results works in the asymptotic case. We introduce the following notation: ˆq ≡ (ˆqY |X=x1(αT ),... |

17 |
Comparative Statics of Fixed Points
- Villas-Boas
- 1997
(Show Context)
Citation Context ...arative statics (MCS) prediction holds: There is a smallest and a largest equilibrium, and these change monotonically with explanatory variables (Milgrom and Roberts, 1990; Milgrom and Shannon, 1994; =-=Villas-Boas, 1997-=-). MCS is a feature found in many well-known economic models. Examples are single-person decision models such as models of optimal growth (Barro and Sala-I-Martin, 2003; Ljungqvist and Sargent, 2004) ... |

11 | 2004): “Identification and Estimation of Discrete Games of Complete Information - Bajari, Hong, et al. |

10 |
A Note on Quantiles
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Citation Context ...te empirical quantiles when αT depends on the sample size T . It is an extension of the wellknown result for central α-quantiles with α ∈ (0, 1) fixed (Mosteller, 1946; Smirnov, 1952; Siddiqui, 1960; =-=Bahadur, 1966-=-; Bassett and Koenker, 1978; Powell, 1984, 1986), to the case where α increases with the sample size T . Dekkers and de Haan (1989) and Chernozhukov (2005) prove this extension under an additional ass... |

7 | 2013): “Econometric Analysis of Games with Multiple Equilibria - Paula |

7 | The strategic timing incentives of commercial radio stations: An empirical analysis using multiple equilibria - Sweeting - 2009 |

6 | A Note on Uniform Asymptotic Normality of Intermediate Order Statistics - Falk - 1989 |

6 | 2006): “Dilation Bootstrap: A Natural Approach to Inference - Galichon, Henry |

6 | Measurement Error in Nonlinear Models. Chapmann and - Carroll, Ruppert, et al. - 1995 |

4 | Fernandez-Val (2011): “Inference for Extremal Conditional Quantile Models, with an Application to Market and Birthweight Risks,” The Review of Economic Studies - Chernozhukov, I |

4 | Statistical Inference under Order Restrictions - R, Bremner, et al. - 1972 |

3 | The Almost Sure Representation of Intermediate Order Statistics,” Zeitschrift Fur Wahrscheinlichkeitstheorie und Verwandte Gebiete - Watts - 1980 |

2 | 2005): “Semiparametric Estimation of Signaling Games - Kim |

2 | Levinsohn (1993): “Applications and Limitations of Some Recent - Pakes, Berry, et al. |

2 | Rosen (2008): “The Identification Power of Equilibrium in Games: The Supermodular Case - Molinari, M |

1 | A Test of Homogeneity for Ordered Alternatives - Biometrika, G, et al. - 1959 |

1 | Empirical Models of Discrete Games - J, Ruppert, et al. - 1991 |

1 |
Asymptotics for Least Aboslute Deviation Regression Estimators,” Econometric Theory
- Pollard
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Citation Context ...13) is consistent for the true α-quantile under FY |X=x. Consistency of ˆqY |X=x(α) can be extended to cases where (Yx,1,...,Yx,Tx) is a weakly dependent time-series, provided additional assumptions (=-=Pollard, 1991-=-; Portnoy, 1991; Koenker and Zhao, 1996; Komunjer, 2005; Chernozhukov, 2005); for the sake of simplicity, we focus on the independent case. To alleviate the notation, we drop the reference to x when d... |

1 |
Distribution of Quantiles in Samples Form a Bivariate Population
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(Show Context)
Citation Context ...or of intermediate empirical quantiles when αT depends on the sample size T . It is an extension of the wellknown result for central α-quantiles with α ∈ (0, 1) fixed (Mosteller, 1946; Smirnov, 1952; =-=Siddiqui, 1960-=-; Bahadur, 1966; Bassett and Koenker, 1978; Powell, 1984, 1986), to the case where α increases with the sample size T . Dekkers and de Haan (1989) and Chernozhukov (2005) prove this extension under an... |

1 | 2012): “Identification of Potential Games and Demand Models for Bundles,” unpublished manuscript - Fox, Lazzati |

1 | Censored Regression Quantiles - Renyi - 1986 |

1 | If ¯ H−1 ∈ R0 at 0, then for all sequences {aN} and {a′ N } of positive numbers with limN→∞ aN = limN→∞ a ′ N = 0 and limN→∞ aN/a ′ N lim N→∞ ¯H −1 (aN) ¯H −1 (a′ = 1. N ) = c (with 0 < c < ∞), we have Proof of Lemma 6. Let U(x) ≡ ¯ H −1 (1/x) for all x > - Lemma - 1970 |