Statistical inference for partially linear errors-in-variables panel data models with fixed effects

In this paper, we consider the statistical inference for the partially linear panel data models with fixed effects. We focus on the case where some covariates are measured with additive errors. We propose a modified profile least squares estimator of the regression parameter and the nonparametric components. The asymptotic normality for the parametric component and the rate of convergence for the nonparametric component are established. Consistent estimations of the error variance are also developed. We conduct simulation studies to demonstrate the finite sample performance of our proposed method and we also present an illustrative empirical application.


Introduction
Panel data records information on each individual unit over time, the rich information contained in panel data allows researchers to estimate complex models and answer questions that may not be possible using time series or cross-sectional data alone. Panel data analysis has received a lot of attention during the last two decades due to applications in many disciplines, such as economics, finance, biology, engineering and social sciences. Baltagi (2005) and Hsiao (2003) provided excellent overviews of statistical inference and econometric analysis of parametric panel data models. Semiparametric regression models reduce the high risk of misspecification relative to a fully parametric model and avoid some serious drawbacks of purely nonparametric methods such as the curse of dimensionality, difficulty of interpretation, and lack of extrapolation capability. In the last two decades, various forms of semiparametric regression models have been developed. These include the varying-coefficient model (e.g. Zhao & Lin, 2019), the partially linear regression model (e.g. Engle et al., 1986), the varying-coefficient partially linear model (e.g. J. . The partially linear panel data models with fixed effects is a useful tool for econometric analysis; see, e.g. Henderson et al. (2008), Hu (2014), and Li et al. (2011).
In this paper, we consider the following partially linear panel data models with fixed effects (e.g. Su CONTACT Minxiu Yu yuminxiu01@126.com & Ullah, 2006): where Y it is the response, X it and U it are of dimensions p × 1 and q × 1, respectively, β = (β 1 , . . . , β p ) τ is a vector of p-dimensional unknown parameters, and the superscript τ denotes the transpose of a vector or matrix. m(U it ) is a unknown function and α i is the unobserved individual effects, ε it is the random model error. Here, we assume ε it to be i.i.d. with zero mean and finite variance σ 2 > 0. We allow α i to be correlated with X it , and U it with an unknown correlation structure. Hence, model (1) is a fixed effects model.
Measurement error data are often encountered in many fields, including economics, biomedical sciences and epidemiology. Simply ignoring measurement errors (errors-in-variables), will result in biased estimators. Handing the measurement errors in covariates is generally a challenge for statistical analysis. For the past two decades, regression analysis with measurement errors had much progress. A detailed study can be found in the research of Fuller (1987), J. Fan and Truong (1993), Liang et al. (1999), G. L. Fan et al. (2013) and so on. A statistical analysis of model (1) with additive measurement errors, however, still seems to be missing. The objective of the present paper is to fill this gap.
Specifically, we consider the following partially linear errors-in-variables panel data models with fixed effects where the covariate variable X it is measured with additive error and is not directly observable. Instead, we observe Z it with where the measurement errors η it are independent and identically distributed, independent of (X it , U it , ε it ), we assume that Cov(η it ) = η with η assumed to be known, as in the papers of Zhu and Cui (2003), You and Chen (2006), and other. When η is unknown, we can estimate it by repeatedly measuring Z it ; see Liang et al. (1999) for details.
The individual effects are often viewed as nuisance parameters in the fixed effects panel data regression model. Because of the diverging number of nuisance parameters, the estimation of the parametric and nonparametric components in the partially linear panel data models with fixed effects is no longer straightforward. Great efforts have been made to develop suitable methods for estimations in the last few years. For the model (1), Su and Ullah (2006) adapted a local linear dummy variable approach to remove the unknown fixed effects. In this paper, our aim is to extend the results in Zhou et al. (2010) for fixed effects partially linear regression models with errors in variables to partially linear error-in-variables panel data models with fixed effects. we make statistical inference for the parametric β in partially linear panel data models with fixed effects. Following the estimation procedure proposed by J. , the profile least-squared estimations of β can be obtained. Based on this, a normal-based confidence region for the parametric is constructed. However, because of the attenuation the previously proposed estimators, the profile least squares estimator, are no longer consistent. We propose a modified profile least squares estimator. The resulting estimator is shown to be consistent and asymptotically normal. Consistently estimating the error variance is also considered.
The layout of the remainder of this paper is as follows. In Section 2, we present the estimators of the parametric component, the nonparametric component, as well as the error variance. Assumption conditions and the main result are given in Section 3. The simulated example is provided in Section 4. A real-data example is given in Section 5. All the mathematical proofs of the asymptotic results are given in Appendix.

Model and methodology
To introduce our estimation, we rewrite model (1) in a matrix format yields where B is a nT × n matrix. For our application, the matrix is in a specific form, B = I n ⊗ i T with ⊗ the Kronecker product, I n denotes the n × n identity matrix, and i T denotes the n × 1 vector of ones. There are many approaches to estimate the parameters {β j , j = 1, . . . , p} and the functions m(·). The main idea is from the profile least squares (PLS) approach proposed by J. : suppose that we have a random sam- where m (u) = ∂m(u)/∂u. This leads to the following weighted local least squares problem: find a and b to minimize where a and b are q-dimensional column vectors, K h (·) = K(·/h)/h, K(·) is a kernel function and h is a sequence of positive numbers tending to zero, called bandwidth. For notational convenience, let . Then the solution of problem (4) is given by In (3), m(U it ) is replaced withâ. Then we havê (5) Now we consider a way of removing the unknown fixed effects motivated by a least squares dummy variable model in parametric panel data analysis, for which we solve the following optimization problem: By the least squares method and a slight complex computation, we have where However, X it 's can not be observed in our case and we just have Z it . If we ignore the measurement error and replace X it with Z it in (6), (6) can be used to show that the resulting estimate is inconsistent. It is well known that in linear regression or partially linear regression, inconsistency caused by the measurement error can be overcome by applying the so-called 'correction for attenuation', see Fuller (1987) and Liang et al. (1999) for more details. In the context of partially linear regression, we introduce the modified least squares method to estimate β as follows: and results of J.  suggest that the profile least squares estimator of the nonparametric components m = [m(U 11 ), . . . , m(U 1T ), . . . , m(U nT )] τ can be achieved with the following equation According to the results of Opsomer and Ruppert (1997), the estimate of m has the following form Sometimes, it is also necessary to estimate the error variance σ 2 = E(ε 2 it ) for such tasks as the construction of confidence regions, model-based tests, model selection procedures, single-to-noise ratio determination and so with

Main results
In this section, we will be establishing the asymptotic properties of the proposed estimators from the previous section. Before formulating the main results, we first give the following assumptions: (A1) The random vector U it has a continuous density func- is a symmetric probability density function with a continuous derivative on its compact support U.  (3), if β is the true value of the parameter, then the proposed estimator β of β is asymptotically normal, namely

Theorem 3.2: Suppose that Assumptions
Then the risk of the profile least squares m is bounded as follows:

Theorem 3.3: Suppose that Assumptions
then is a consistent estimator of .

Simulation studies
In this section, we carry out some simulation experiments to study the finite sample performance of the estimators β, m and σ 2 , which are defined in Section 2. Firstly, we consider the following partially linear errors-in-variables panel data models with fixed effects: where (4, 4)), α i = ρX i + w i with ρ = 0.5, 1 and w i ∼N(0, 1) for i = 1, 2, . . . , n, andX i = 1 T T t=1 X it1 . We use ρ to control the correlation between α i andX i . The measurement error η it ∼N(0, η ) where we take η = 0.2 2 I 2 and 0.4 2 I 2 to represent different levels of measurement error.
In our simulations, we took the sample sizes (n, T) = (50, 4), (50, 6), (100, 4) and (100, 6), respectively, and we choose the Epanechnikov kernel K(u) = 3 4 (1 − u 2 )I{|u| ≤ 1}. The 'leave-one-subject-out' cross-validation bandwidth CV(h) is obtained by minimizing whereβ [i] andm [i] (U it ) are estimators of β and m(U it ), respectively, which are computed with all of the measurements but not the ith subject. We consider 500 replications with the nominal level 1 − α = 0.95. We calculate the sample means and standard deviations (SD) of the proposed estimators for the parametric components β 1 , β 2 , and error variance σ 2 in Section 2. In order to compare the proposed modified PLS estimation method for the parametric component and estimator of the error variance with the existing method, we also calculate the sample means and SD of the modified PLS estimators and naive estimators (neglecting the measurement errors). The naive estimators for (β 1 , β 2 ) τ and σ 2 are defined aš Simulation results are summarized in Tables 1 and 2. From  Tables 1 and 2, we make the following observations: (1) There is a clear difference between the SD of the naive estimators and those of the proposed estimators. We think this difference is caused by the measurement errors.
(2) The modified profile least squares estimators for the parametric component and the estimator of the error variance are asymptotically unbiased and have smaller SD than those of the naive estimators.
(3) When ρ increases, the proposed estimator behave better than those of the naive estimators. The SD becomes a little bit bigger.
From Figure 1, we can see that the modified profile least squares estimator of the nonparametric component outperforms the naive profile least squares estimator. The latter is biased.

Real-data analysis
To examine the performance of the proposed method, we analyse a climate data set from the UK met office web site: http://www.metoffice.gov.uk/climate/uk/. The dataset contains mean maximum temperature (in Celsius degrees), mean minimum temperatures, total rainfall (in millimetres), total raindays ≥ 1.0 mm, total sunshine duration (in hours) and days of air frost (in days) from 37 stations covering UK. The objectives of the study are to describe the common trend in the mean maximum temperature series and the relationship of the mean maximum temperatures with total rainfall and total sunshine during the period of January 2006 to December 2015. Data from 16 stations are selected according to data availability. This dataset has been studied by D. G. Li et al. (2011).
Let Y it be the log-transformed mean maximum temperature in tth month in station i, X 1it be the logtransformed total sunshine duration in tth month in station i, because of the measuring mechanism, X 1it can only observe the surrogate variables Z 1it . X 2it be the logtransformed total rainfall in tth month in station i, X 3it be the log-transformed total raindays ≥ 1.0 mm in tth month in station i, respectively. We consider the following model: where U it = t/T, α i = ρX i for i = 1, 2, . . . , n, andX i = 1 T T t=1 X it1 . The measurement error η it ∼N(0, η ) where we take η = 0.2 2 I 2 . A natural question is whether the coefficient of total sunshine duration, the total rainfall and the total raindays ≥ 1.0 mm are statistically significant. To answer this question, the proposed PLR tests are employed. As a result, The PLR tests statistic are λ n (β 1 ) = 80.12, λ n (β 2 ) = 0.8490 and λ n (β 3 ) = 5.1266, respectively. Which provides stark evidence that the variables X 1it and X 3it are significant for the mean maximum temperature at the 0.1 significant level, and which indicates that the coefficient of the variables X 2it is zero. We apply the modified least squares method in Section 2 to models (5.1), and obtain the estimators of parameter β = (β 1 , β 3 ) asβ = (0.6861, 0.2365) with SD = 0.040. which shows that the total raindays ≥ 1.0 mm has no significant effect on the mean maximum temperature, but total sunshine duration is highly positively associated with the mean maximum temperature. In other words, longer sunshine during tends to result in higher maximum temperatures. In addition, the estimators of m(·) are shown in Figure 2.
From Figure 2, we can see that from the beginning of 2006 to the end of 2008, there is an upward trend in the monthly mean maximum temperatures. Thereafter, there is a slight decrease from the beginning of 2009 to the end of 2011. Then from the beginning of 2013 to the end of 2015, there is an increase in the maximum temperatures. From Figure 2, we also can see that the modified profile least squares estimator of the nonparametric component outperforms the naive local profile least squares estimator. The latter has a greater volatility.

Conclusion
In this paper, we have studied the estimation of a partially linear errors-in-variables panel data models with fixed effects. When measurement errors are ignored, the usual profile least squares lead to biased estimators of the parametric and nonparametric components. To deal with this problem, we have proposed a modified profile least squares estimator for the parametric and nonparametric components by correcting the attenuation. We have showed that they were consistent and asymptotically normal.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This research is supported by the National Social Science Fund of China [grant number 18BTJ034].

Appendix. Proofs of the main results
In order to establish this theorem, we need to show some useful lemmas. For the convenience and simplicity, let Each element of the above matrix is in the form of a kernel regression. Similar to the proof of Lemma A.2 in J. , we can derive the desired result.
We now deal with the first term. By Assumption (A6), it is easy to see that Hence, Lemma A.3 holds.

Lemma A.4:
Under the conditions of Theorem 3.1, we have

Proof: By Lemma
Lemma A.5: Under the conditions of Theorem 3.1, we have where λ(A) denotes the eigenvalue of matrix A.
For the term K 2 , we have By Lemma A.1, we have which holds uniformly for i, j = 1, . . . , n. Hence Therefore, . It is obvious that the eigenvalues of K 1 satisfy 0 ≤ λ(K 1 ) ≤ 1 T . Similar to the proof of Lemma A.5 in J. , we have By Lemmas A.2-A.4, we obtain Therefore, applying Lemma A.1 and the fact (A + aB) −1 → A −1 as a → 0, we have Note that Therefore, Invoking the Slutsky theorem and the central limit theorem, we obtain the desired result. The proof of −1 1 −1 being a consistent estimator of −1 1 −1 is straight forward. We here omit the details.

Proof of Theorem 3.2.:
According to the definition of m, we have Similar to the proof of Theorem 3 in J. , we obtain and By the law of large numbers, we have which, together with (A1) and (A2), Theorem 3.2 holds. (9)  = 1 + 2 + 3 .

Proof of Theorem 3.3.: By
Therefore, Because ε it and η it are i.i.d, we have Then, by some simple calculation, we have Therefore, By the same argument as proving Theorem 3 of Zhou et al. (2010), we can show that i = o p (N − 1 2 ), i = 2, 3. Equation (11) holds.
We now show the consistency of . To simplify the notation, we write