Some Identification Issues in Nonparametric Linear Models with Endogenous Regressors

In applied work economists often seek to relate a given response variable y to some causal parameter mu* associated with it. This parameter usually represents a summarization based on some explanatory variables of the distribution of y, such as a regression function, and treating it as a conditional expectation is central to its identification and estimation. However, the interpretation of mu* as a conditional expectation breaks down if some or all of the explanatory variables are endogenous. This is not a problem when mu* is modelled as a parametric function of explanatory variables because it is well known how instrumental variables techniques can be used to identify and estimate mu*. In contrast, handling endogenous regressors in nonparametric models, where mu* is regarded as fully unknown, presents di±cult theoretical and practical challenges. In this paper we consider an endogenous nonparametric model based on a conditional moment restriction. We investigate identification related properties of this model when the unknown function mu* belongs to a linear space. We also investigate underidentification of mu* along with the identification of its linear functionals. Several examples are provided in order to develop intuition about identification and estimation for endogenous nonparametric regression and related models.

Predator-Induced Plasticity in Spotted Salamanders (Ambystoma maculatum)

The ability to respond plastically to the environment has allowed amphibians to evolve adaptive responses to spatial and temporal variation in predation threat. However, animals exposed to predators may also show costs of plasticity or tradeoffs. This study examines predator-induced plasticity in larval development, behavior, and metamorphosis in the spotted salamander, Ambystoma maculatum. Salamanders were raised in two treatments: with predator cues (a fish predator, genus Lepomis, on the other side of a divided tank), or without predator cues. During the larval stage the predator treatment group experienced higher mortality rates than the no-predator treatment group. Behavioral trials revealed that predator treatment animals ate less than those not exposed, and that this feeding response was immediately inducible and had lasting effects. Animals in the predator treatment group had smaller tail areas during the mid-larval period. Feeding and body size effects may have contributed to increased mortality in the predator-treatment animals. The timing of metamorphic onset was not affected by the presence of predators, but predator-treatment salamanders had shorter snout/vent lengths at metamorphosis. The duration of metamorphosis showed a potentially adaptive plastic response to the presence of predator cues: metamorphosis was longest in the no-predator treatment group, reduced in the predator treatment group, and even further reduced for animals exposed to predator cues only during metamorphosis. Overall, we found a mix of potentially adaptive and costly plastic responses in spotted salamanders.

The Harmonic Oscillator's Frobenius Solution

The Frobenius solution to the differential equations associated with the harmonic oscillator (QM) is carried out in detail.