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.

Extractable Soil Phosphorus Concentrations and Creeping Bentgrass Response on Sand Greens

Few studies have directly related turfgrass growth and quality responses to extractable soil P concentrations in sand greens. A 3-yr field experiment was conducted on a sand-based putting green to determine creeping bentgrass (Agrostis stolonifera L.) growth and quality responses to extractable soil P. Extractable soil P concentrations were obtained by using the modified-Morgan, Mehlich-1, and Bray-1 extractants. Critical extractable P concentrations (above which there is a low probability of response to increasing soil P concentrations) for shoot counts, thatch thickness, relative clipping yields, quality ratings, P deficiency ratings, tissue P concentrations, and root weights were determined using Cate-Nelson (CN) and quadratic response and plateau (QRP) models. Both models fit the data relatively well in most cases (R2 values from 0.12 to 0.89), and critical concentrations for the QRP models were always greater than the CN models. Critical extractable P concentrations were lowest for the modified-Morgan extractant (1.4 to 12.0 mg kg(-1)) and greatest for the Mehlich-1 extractant (14.1 to 63.6 mg kg(-1)). Application of estimated critical extractable P concentrations in this study could be used to substantiate observed responses or explain lack of responses in other previously reported creeping bentgrass P studies. We found better model fits with modified-Morgan extractable P for bentgrass quality ratings, deficiency ratings, and tissue P concentrations than with P extracted by the Mehlich or Bray methods. This suggests that the modified-Morgan extractant may have advantages over stronger-acid extractants when used on sand-based media. The results can be used to revise or update existing P fertilization recommendations for bent-grass grown on sand-based media.