Abstract
The asymptotic distribution is presented for the linear instrumental variables model estimated with a ridge penalty and a prior where the tuning parameter is selected with a holdout sample. The structural parameters and the tuning parameter are estimated jointly by method of moments. A chi-squared statistic permits confidence regions for the structural parameters. The form of the asymptotic distribution provides insights on the optimal way to perform the split between the training and test sample. Results for the linear regression estimated by ridge regression are presented as a special case.
Highlights
Inference for the Linear IV ModelThis paper contributes to the asymptotic distribution theory for ridge parameter estimates for the linear instrumental variables model
When the prior is different from the population parameter value, the asymptotic distribution for the ridge tuning parameter is a mixture with a discrete mass of 1/2 at zero and a truncated normal over the positive reals
Accurate inferences can be performed with the asymptotic distribution of the ridge estimates of the linear IV model when the tuning parameter is empirically selected by a holdout sample
Summary
This paper contributes to the asymptotic distribution theory for ridge parameter estimates for the linear instrumental variables model. In [1], the ridge penalty parameter is estimated jointly with the structural parameters and the asymptotic distribution is characterized as the projection of a stochastic process onto a cone. A related literature concerns the distribution of some empirically selected ridge tuning parameters [14,15,16] These approaches have relied on strong distribution assumptions (e.g., normal error). They are built on tuning parameters as functions of the data, where the functions are determined by assuming that the tuning parameter is fixed. In [1], the asymptotic joint distribution for the parameters in the linear model and the ridge tuning parameter is characterized as the projection of a stochastic process onto a cone This structure occurs because the probability limit of the ridge tuning parameter is on the boundary of the parameters space. A hypothesis for the entire set of structural parameters can be tested using a chi-square test and this statistic can be inverted to give accurate confidence regions
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