Nonparametric estimation of additive models with errors-in-variables

Abstract

In the estimation of nonparametric additive models, conventional methods, such as backfitting and series approximation, cannot be applied when measurement error is present in a covariate. This paper proposes a two-stage estimator for such models. In the first stage, to adapt to the additive structure, we use a series approximation together with a ridge approach to deal with the ill-posedness brought by mismeasurement. We derive the uniform convergence rate of this first-stage estimator and characterize how the measurement error slows down the convergence rate for ordinary/super smooth cases. To establish the limiting distribution, we construct a second-stage estimator via one-step backfitting with a deconvolution kernel using the first-stage estimator. The asymptotic normality of the second-stage estimator is established for ordinary/super smooth measurement error cases. Finally, a Monte Carlo study and an empirical application highlight the applicability of the estimator.

Keywords:

• Backfitting
• classical measurement error
• nonparametric additive regression
• ridge regularization
• series estimation

Acknowledgment

The authors would like to thank anonymous referees for helpful comments.

Notes

1 If $\varphi X^{*}$ is treated as the true underlying covariate and $\varphi \ne 0,$ we have $Y-\mu -m_1(Z_1)-\cdots -m_D(Z_D)-U=\tilde{g}(\varphi X^{*}),$ with $\tilde{g}(·)=g(·/\varphi ).$ Then the normalization required in Assumption 1 (6) becomes ${\int }_{\tilde{\mathcal{I}}}\tilde{g}(w)dw=0,$ with $\tilde{\mathcal{I}}=\{\tilde{w}=\varphi w:w\in \mathcal{I}\}.$ Since $\mathcal{I}$ is the range of $X^{*}$ chosen by researchers, instead of $\tilde{\mathcal{I}}$ (which is unknown without a specific value of $\varphi$), we may directly decide the range of interest of $\varphi X^{*}$ based on limited knowledge of $\varphi$ (for example, a set of possible values of $\varphi$).

2 Even though it seems somewhat different, the Sobolev condition imposed here is essentially equivalent to the one used in Meister (2009, eq. (2.30)), which imposes $\int |f^{\text{ft}}(t){|}^2|t{|}^{2\alpha }dt<c.$ First, it is easy to see that $\int |f^{\text{ft}}(t){|}^2{(1+|t{|}^2)}^{\alpha }dt<c$ implies $\int |f^{\text{ft}}(t){|}^2|t{|}^{2\alpha }dt<c.$ For the other direction, we have $\int |f^{\text{ft}}(t){|}^2{(1+|t{|}^2)}^{\alpha }dt\le 2^{\alpha }{\int }_{\left|t\right|\le 1}|f^{\text{ft}}(t){|}^{2}dt+2^{\alpha }\int |f^{\text{ft}}(t){|}^2|t{|}^{2\alpha }dt<c^{\prime },$ where the first inequality follows by $2^{\alpha }|t{|}^{2\alpha }\ge {(1+|t{|}^2)}^{\alpha }\iff |t|\ge 1,$ and the second inequality follows by $f\in L_2(\mathbb{R})$ and Meister (2009, eq. (2.30)).

3 We set α = 2 because Han and Park (2018) assumed that f_j is twice continuously differentiable in their Assumption K4. See Meister (2009, Section A.2) for the relationship between order of differentiability and choice of α.

4 Even though Han and Park (2018) considered a different setup where all covariates are mismeasured, the convergence rate of their smoothed backfitting method would remain the same when only one covariate is mismeasured, as it is independent of their number of covariates d. This is a natural result when the regression function has an additive structure. Therefore, the uniform convergence rate presented in Han and Park (2018, Corollary 3.5) can be directly compared to Theorem 2 when the smoothing parameters α and β are the same.

5 We note that if $\beta >1/2,$ then $f_{ϵ}$ is bounded and continuous.

6 Similarly to the previous case, here we set α = 2 because Horowitz and Mammen (2004) assume that m_j is twice continuously differentiable in their Assumption A2.