A Partially Linear Kernel Estimator for Categorical Data

Abstract

We extend Robinson's (1988) partially linear estimator to admit the mix of datatypes typically encountered by applied researchers, namely, categorical (nominal and ordinal) and continuous. We also relax the independence assumption that is prevalent in this literature and allow for β-mixing time-series data. We employ Li, Ouyang, and Racine's (2009) categorical and continuous data kernel method, and extend this so that a mix of continuous and/or categorical variables can appear in the nonparametric part of a partially linear time-series model. The estimator appearing in the linear part is shown to be -consistent, which is of course the case for Robinson's (1988) estimator. Asymptotic normality of the nonparametric component is also established. A modest Monte Carlo simulation demonstrates that the proposed estimator can outperform existing nonparametric, semiparametric, and popular parametric specifications that appear in the literature. An application using Survey of Income and Program Participation (SIPP) data to model a dynamic labor supply function is undertaken that provides a robustness check and demonstrates that the proposed method is capable of outperforming popular parametric specifications that have been used to model this dataset.

Keywords:

• Cross-validation
• Kernel Method
• Nonparametric regression
• Partially Linear Model
• Semiparametric regression

JEL Classification:

• C14 - Semiparametric and Nonparametric Methods
• C51 - Model Construction and Estimation

ACKNOWLEDGMENT

We would like to thank (but not implicate) Dan Black for his insightful comments and input.

Notes

See Li and Racine (2007) for a detailed summary of partially linear models.

See Aitchison and Aitken (1976) for details.

One can also use the kernel function for ordered categorical variables and use the kernel function if , and if for unordered categorical variables. Using these complicated kernel functions leads to identical results though the proofs are more involved.

That is, simply replace with in (8), (9), and (10).

Note: Results are based on 1,000 Monte Carlo replications. The independent evaluation sample is always of size n ₂ = 1000. OLS-TRUE and OLS refer to the correctly and the incorrectly specified linear models, respectively; NP refers to the nonparametric categorical and continuous data estimator of Li et al. (2009); PL-Stock refers to Stock's (1989) semiparametric partially linear model having categorical variables in the linear part and continuous variables in the nonparametric part; PL-DM refers to Delgado and Mora's (1995a) semiparametric partially linear model having continuous variables in the linear part and categorical variables in the nonparametric part; PL-MIX refers to the proposed semiparametric partially linear model having categorical and continuous datatypes in both the linear part and the nonparametric part.

As an aside we note that Robinson (1988) refers to misspecified parametric models as -inconsistent.

Note: OLS refers to ordinary least squares. PL-KL refers to the semiparametric partially linear model in Kniesner and Li (2002) that has categorical variables in the linear part. PL-MIX refers to the semiparametric partially linear model with categorical and continuous datatypes in the nonparametric part. The dependent variable is the logarithm of hours worked during the quadramester. Regressors include education (ED), time trend (TM), square of time trend (TM2), a binary indicator of children (KD), a binary indicator of whether a young child was present (YK), a binary indicator for marriage (MR), a lagged dependent variable (LS ₋₁) and the logarithm of the real wage (WG). In the partially linear model PL-MIX, binary indicators KD, YK, and MR are treated as unordered categorical variables and the rest are treated as continuous variables. The number of observations is n ₁ = 3, 570. Standard errors appear in parentheses. Both MSE and report out-of-sample predictive ability on the n ₂ = 510 hold-out observations.