Knowledge that Transforms

We consider the bootstrap unit root tests based on finite order autoregressive integrated models driven by iid innovations, with or without deterministic time trends. A general methodology is developed to approximate asymptotic distributions for the models driven by integrated time series, and used to obtain asymptotic expansions for the Dickey–Fuller unit root tests. The second-order terms in their expansions are of stochastic orders Op(n−1/4) and Op(n−1/2), and involve functionals of Brownian motions and normal random variates. The asymptotic expansions for the bootstrap tests are also derived and compared with those of the Dickey–Fuller tests. We show in particular that the bootstrap offers asymptotic refinements for the Dickey–Fuller tests, i.e., it corrects their second-order errors. More precisely, it is shown that the critical values obtained by the bootstrap resampling are correct up to the second-order terms, and the errors in rejection probabilities are of order o(n−1/2) if the tests are based upon the bootstrap critical values. Through simulations, we investigate how effective is the bootstrap correction in small samples.

ARCH and GARCH models directly address the dependency of conditional second moments, and have proved particularly valuable in modelling processes where a relatively large degree of fluctuation is present. These include financial time series, which can be particularly heavy tailed. However, little is known about properties of ARCH or GARCH models in the heavy–tailed setting, and no methods are available for approximating the distributions of parameter estimators there. In this paper we show that, for heavy–tailed errors, the asymptotic distributions of quasi–maximum likelihood parameter estimators in ARCH and GARCH models are nonnormal, and are particularly difficult to estimate directly using standard parametric methods. Standard bootstrap methods also fail to produce consistent estimators. To overcome these problems we develop percentile–t, subsample bootstrap approximations to estimator distributions. Studentizing is employed to approximate scale, and the subsample bootstrap is used to estimate shape. The good performance of this approach is demonstrated both theoretically and numerically.

High–frequency financial data are not only discretely sampled in time but the time separating successive observations is often random. We analyze the consequences of this dual feature of the data when estimating a continuous–time model. In particular, we measure the additional effects of the randomness of the sampling intervals over and beyond those due to the discreteness of the data. We also examine the effect of simply ignoring the sampling randomness. We find that in many situations the randomness of the sampling has a larger impact than the discreteness of the data.

This note proposes a necessary and sufficient condition on a utility function to guarantee that it generates a demand function satisfying the law of demand. This condition can be interpreted in terms of an agent's attitude towards lotteries in commodity space. As an application, we show that when an agent has an expected utility function, her demand for securities satisfies the law of demand if her coefficient of relative risk aversion does not vary by more than 4.

We provide easy to verify sufficient conditions for the consistency and asymptotic normality of a class of semiparametric optimization estimators where the criterion function does not obey standard smoothness conditions and simultaneously depends on some nonparametric estimators that can themselves depend on the parameters to be estimated. Our results extend existing theories such as those of Pakes and Pollard (1989), Andrews (1994a), and Newey (1994). We also show that bootstrap provides asymptotically correct confidence regions for the finite dimensional parameters. We apply our results to two examples: a ‘hit rate’ and a partially linear median regression with some endogenous regressors.

We consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution.We propose mean-risk models that are solvable by linear programming and generate portfolios whose returns are nondominated in the sense of second-order stochastic dominance. Next, we develop a specialized parametric method for recovering the entire mean-risk efficient frontiers of these models and we illustrate its operation on a large data set involving thousands of assets and realizations.

This paper provides deterministic approximation results for stochastic processes that arise when ¯nite populations recurrently play ¯nite games.The deterministic approximation is de¯ned in continuous time as a system of ordinary di®erential equations of the type studied in evolutionary game theory.We establish precise connections between the long-run behavior of the stochastic process, for large populations, and its deterministic approximation.In particular, we show that if the deterministic solution through the initial state of the stochastic process at some point in time enters a basin of attraction, then the stochastic process will enter any given neighborhood of that attractor within a ¯nite and deterministic time with a probability that exponentially approaches one as the population size goes to in¯nity.The process will remain in this neighborhood for a random time that almost surely exceeds an exponential function of the population size.During this time interval, the process spends almost all time at a certain subset of the attractor, its so-called Birkho® center.We sharpen this result in the special case of ergodic processes.

This paper analyzes the linear regression model y = x + with a conditional median assumption Med( j z) = 0 where z is a vector of exogenous random variables. Added complication arise due to the censoring of the outcome y. We treat the censored model as a model with interval-observed outcomes thus obtaining an incomplete model with inequality restrictions on conditional median regressions. This allows us to use the estimator introduced by Manski and Tamer (2000) to analyze the information contained in these inequality restrictions. We give identication conditions in the absence of censoring and introduce a p N-consistent estimator based on the minimum distance method. We then give suÆcient conditions for global identication of with censored y and endogenous x. In the case of interval data on y and endogenous x, we provide a set-consistent estimator that is based on a modied minimum distance method. In the case where we have point identication, we show that the estimator is p N-normal and derive its asymptotic distribution with a feasible asymptotic variance. A Montecarlo analysis illustrates our estimator. We thank Bo Honore for comments and the Econometrics Research Program at Princeton for support.

We are interested in estimating the average effect of a binary treatment on a scalar outcome. If assignment to the treatment is exogenous or unconfounded, that is, independent of the potential outcomes given covariates, biases associated with simple treatment-control average comparisons can be removed by adjusting for differences in the covariates. Rosenbaum and Rubin (1983) show that adjusting solely for differences between treated and control units in the propensity score removes all biases associated with differences in covariates. Although adjusting for differences in the propensity score removes all the bias, this can come at the expense of efficiency, as shown by Hahn (1998), Heckman, Ichimura, and Todd (1998), and Robins, Mark, and Newey (1992). We show that weighting by the inverse of a nonparametric estimate of the propensity score, rather than the true propensity score, leads to an efficient estimate of the average treatment effect. We provide intuition for this result by showing that this estimator can be interpreted as an empirical likelihood estimator that efficiently incorporates the information about the propensity score.

This paper builds a dynamic industry model with heterogeneous firms that explains why international trade induces reallocations of resources among firms in an industry. The paper shows how the exposure to trade will induce only the more productive firms to enter the export market (while some less productive firms continue to produce only for the domestic market) and will simultaneously force the least productive firms to exit. It then shows how further increases in the industry's exposure to trade lead to additional inter-firm reallocations towards more productive firms. These phenomena have been empirically documented but can not be explained by current general equilibrium trade models, because they rely on a representative firm framework. The paper also shows how the aggregate industry productivity growth generated by the reallocations contributes to a welfare gain, thus highlighting a benefit from trade that has not been examined theoretically before. The paper adapts Hopenhayn's (1992a) dynamic industry model to monopolistic competition in a general equilibrium setting. In so doing, the paper provides an extension of Krugman's (1980) trade model that incorporates firm level productivity differences. Firms with different productivity levels coexist in an industry because each firm faces initial uncertainty concerning its productivity before making an irreversible investment to enter the industry.