This paper analyzes a procedure called Testing‐Based Forward Model Selection (TBFMS) in linear regression problems. This procedure inductively selects covariates that add predictive power into a working statistical model before estimating a final regression. The criterion for deciding which covariate to include next and when to stop including covariates is derived from a profile of traditional statistical hypothesis tests. This paper proves probabilistic bounds, which depend on the quality of the tests, for prediction error and the number of selected covariates. As an example, the bounds are then specialized to a case with heteroscedastic data, with tests constructed with the help of Huber–Eicker–White standard errors. Under the assumed regularity conditions, these tests lead to estimation convergence rates matching other common high‐dimensional estimators including Lasso.
The correlation between productivity and competition is an oft observed but incompletely understood result. Some suggest that there is a treatment effect of competition on measured productivity, for example, through a reduction of managerial slack. Others argue that greater competition makes unproductive establishments exit by reallocating demand to their productive rivals, raising observed average productivity via selection. I study the ready‐mix concrete industry and offer three perspectives on this ambivalence. First, using a standard decomposition approach, I look for evidence of greater reallocation of demand to productive plants in more competitive markets. Second, I model the establishment exit decision and construct a semiparametric selection correction to quantify the empirical significance of treatment and selection. Finally, I use a grouped instrumental variable quantile regression to test the distributional predictions of the selection hypothesis. I find no evidence for greater selection or reallocation in more competitive markets; instead, all three results suggest that measured productivity responds directly to competition. Potential channels include specialization and managerial inputs.
This paper examines the prices of basic staples in rural Mexico. We document that nonlinear pricing in the form of quantity discounts is common, that quantity discounts are sizable for basic staples, and that the well‐known conditional cash transfer program Progresa has significantly increased quantity discounts, although the program, as documented in previous studies, has not affected unit prices on average. To account for these patterns, we propose a model of price discrimination that nests those of Maskin and Riley (1984) and Jullien (2000), in which consumers differ in their tastes and, because of subsistence constraints, in their ability to pay for a good. We show that under mild conditions, a model in which consumers face heterogeneous subsistence or budget constraints is equivalent to one in which consumers have access to heterogeneous outside options. We rely on known results to characterize the equilibrium price schedule, which is nonlinear in quantity. We analyze the effect of nonlinear pricing on market participation as well as the impact of a market‐wide transfer, analogous to the Progresa one, when consumers are differentially constrained. We show that the model is structurally identified from data on prices and quantities from a single market under common assumptions. We estimate the model using data on three commonly consumed commodities from municipalities and localities in Mexico. Interestingly, we find that relative to linear pricing, nonlinear pricing is beneficial to a large number of households, including those consuming small quantities, mostly because of the higher degree of market participation that nonlinear pricing induces. We also show that the Progresa transfer has affected the slopes of the price schedules of the three commodities we study, which have become steeper as consistent with our model, leading to an increase in the intensity of price discrimination. Finally, we find that a reduced form of our model, in which the size of quantity discounts depends on the hazard rate of the distribution of quantities purchased in a village, accounts for the shift in price schedules induced by the program.
Treating civic preferences as endogenous and government policies and tax capacities as both an influence on and a consequence of their evolution is an important new strand of thinking to which Besley has contributed. I ask: Does his model provide a convincing explanation of the way that civic cultures and the expansion of the state evolved as a matter of historical fact? And I suggest a number of alternative modeling approaches that both would recognize that policy makers take account of the effects of their policy choices on preferences and, consistent with empirical observations, would support equilibria with culturally heterogeneous rather than homogeneous populations.
We propose a way to formalize the relationship between descriptive analysis and structural estimation. A researcher reports an estimate ĉ of a structural quantity of interest c that is exactly or asymptotically unbiased under some base model. The researcher also reports descriptive statistics<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mover accent="true"><a:mi>γ</a:mi><a:mo>ˆ</a:mo></a:mover></a:math>that estimate features γ of the distribution of the data that are related to c under the base model. A reader entertains a less restrictive model that is local to the base model, under which the estimate ĉ may be biased. We study the reduction in worst‐case bias from a restriction that requires the reader's model to respect the relationship between c and γ specified by the base model. Our main result shows that the proportional reduction in worst‐case bias depends only on a quantity we call the informativeness of<d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mover accent="true"><d:mi>γ</d:mi><d:mo>ˆ</d:mo></d:mover></d:math>for ĉ . Informativeness can be easily estimated even for complex models. We recommend that researchers report estimated informativeness alongside their descriptive analyses, and we illustrate with applications to three recent papers.
We study the behavior of the U.S. labor share over the past 90 years. We find that the observed decline of the labor share is entirely explained by the capitalization of intellectual property products in the national income and product accounts.
In a cheap‐talk setting where the conflict of interest between sender and receiver is determined endogenously by the choice of parameters θ i for each agent i , conditions are provided that determine the sign of each agent's inverse demand for θ without assuming that the most informative equilibrium will necessarily be played in the cheap talk game. For two popular functional forms of payoffs, we derive analytically tractable approximations for agent i 's demand for θ . In an application where the θi 's are purchased on a competitive market, we provide conditions for a competitive equilibrium to feature maximal information transmission. In a principal–agent application where the agent's θ is set by the principal, our results show that information transmission will be partial. We consider extensions where: (1) the θ 's are acquired covertly rather than overtly and (2) the θ 's are traded after the sender has received the information.
ANDREWS, GENTZKOW, AND SHAPIRO (2020), henceforth AGS, make important progress on a challenging and fundamental question. The authors consider a setting in which a researcher reports an estimate ĉn that is unbiased under the researcher’s model. The researcher also reports a statistic γ̂n and a measure of informativeness of γ̂n for ĉn, denoted by . AGS show can be related to properties of the distribution of ĉn under models larger than the one employed by the researcher. As a result, can play a role in reassuring a reader who is concerned about some aspects of the researcher’s model. I illustrate the properties of with two examples. The first example is based on the classic selection problem of Heckman (1979) and portrays a setting in which a high is not reassuring. The example shows that if a researcher has assumed random assignment and the reader is concerned about it, then the bias of ĉn under the reader’s model can be arbitrarily large regardless of . The second example is based on the discussion of Attanasio, Meghir, and Santiago (2012) in AGS. This example illustrates that if ĉn and γ̂n are estimators of a common parameter, then can be linked to the properties of a Hausman (1978) test. As a result, √ 1 − can be shown to equal the (normalized) largest bias of ĉn under the reader’s model and, in this sense, a high is reassuring. I show this conclusion holds generally in a classical local asymptotic framework (Bickel, Klassen, Ritov, and Wellner (1993)). The two examples underscore that, as emphasized by AGS, whether a high is reassuring to the reader can crucially depend on the particulars of the researcher’s and reader’s models. I employ the notation in AGS with some minor modifications. The researcher’s model is indexed by η ∈ H. The reader entertains a larger model indexed by (η ζ) ∈ H × Z. I focus on the perspective of the reader, who views the parameter of interest as a function of (η ζ) that we denote by c(η ζ). Under the reader’s model, the distribution of the data is determined by the value of (η ζ). To emphasize this dependence, I write E(η ζ)[ĉn] to denote the expectation of ĉn under (η ζ). The property that ĉn is unbiased under the researcher’s model is equivalent to E(η 0)[ĉn] = c(η 0) for any η ∈H, while the bias under (η ζ) in the reader’s model is given by E(η ζ)[ĉn] − c(η ζ). As in AGS, I consider an analysis local to proper specification, meaning we study (η ζ) that are close to some (η0 0). Since E(η0 0)[ĉn] = c(η0 0) due to ĉn being unbiased under the researcher’s model, the bias of ĉn under any (η ζ) in the reader’s model equals E(η ζ)[ĉn] − c(η ζ)
A patient player privately observes a persistent state and interacts with an infinite sequence of myopic uninformed players. The patient player is either a strategic type who maximizes his payoff or one of several commitment types who mechanically play the same action in every period. I focus on situations in which the uninformed player's best reply to a commitment action depends on the state and where the total probability of commitment types is sufficiently small. I show that the patient player's equilibrium payoff is bounded below his commitment payoff in some equilibria under some of his payoff functions. This is because he faces a trade‐off between building his reputation for commitment and signaling favorable information about the state. When players' stage‐game payoff functions are monotone‐supermodular , the patient player receives high payoffs in all states and in all equilibria. Under an additional condition on the state distribution, my reputation model yields a unique prediction on the patient player's equilibrium payoff and on‐path behavior.
We investigate claims made in Giacomini and White (2006) and Diebold (2015) regarding the asymptotic normality of a test of equal predictive ability. A counterexample is provided in which, instead, the test statistic diverges with probability 1 under the null.