Dynamic partially functional linear regression model

Abstract

In this paper, we develop a dynamic partially functional linear regression model in which the functional dependent variable is explained by the first order lagged functional observation and a finite number of real-valued variables. The bivariate slope function is estimated by bivariate tensor-product B-splines. Under some regularity conditions, the large sample properties of the proposed estimators are established. We investigate the finite sample performance of the proposed methods via Monte Carlo simulation studies, and illustrate its usefulness by the analysis of the electricity consumption data.

Appendix. Proofs

The following lemma, which follows easily from Theorem 12.7 of Schumaker (1981), is stated for ease of reference.

Lemma 1

Under Assumption 1, there exists a spline function

$$\begin{aligned} \beta ^*(t,s)=\sum \limits _{i=1}^{M_N}\sum \limits _{j=1}^{M_N}\alpha _{i,j}b_i(t) b_j(s)=\left( {\varvec{B}}^T(t)\otimes {\varvec{B}}^T (s)\right) {{\varvec{\alpha }}}, \end{aligned}$$

which is called spline approximation of $\beta (t,s)$, such that

$$\begin{aligned} \sup _{(t,s)\in [0,1]\times [0,1]} \parallel \beta ^*(t,s)-\beta (t,s) \parallel =O(M_N^{-r}), \end{aligned}$$

where ${\varvec{\alpha }}=(\alpha _{1,1},\alpha _{1,2},\ldots ,\alpha _{1,M_N},\alpha _{2,1},\ldots ,\alpha _{2,M_N},\alpha _{3,1},\ldots ,\alpha _{M_N,M_N})^T.$

Proof of Theorem 1

By Lemma 1, there exists ${\varvec{\alpha }}$ such that

$$\begin{aligned} R(t,s)=\beta (t,s)-{\varvec{B}}^T(t)\otimes {\varvec{B}}^T (s){\varvec{\alpha }}=O(M_N^{-r}) \end{aligned}$$

uniformly in $(t,s)\in [0,1]\times [0,1]$. Then, it has

$$\begin{aligned} \hat{ {\varvec{\alpha }}}- {{\varvec{\alpha }}}= & {} {\varvec{A}}_1^{-1}{\varvec{A}}_2-{ {\varvec{\alpha }}}\\= & {} {\varvec{A}}_1^{-1}({\varvec{A}}_2-{\varvec{A}}_1{ {\varvec{\alpha }}})\\= & {} {\varvec{A}}_1^{-1}\int _0^1\left( \langle Y_1,{\varvec{B}} \rangle ,\ldots ,\langle Y_{N-1},{\varvec{B}} \rangle \right) \otimes {\varvec{B}}(t)({\varvec{I}}_N-{\varvec{P}}_N)[\widetilde{{\varvec{Y}}}(t)-{\varvec{\eta }}^*(t)]dt, \end{aligned}$$

where ${\varvec{\eta }}^*(t)=\left( \int _0^1Y_1(s)\beta ^*(t,s)ds,\ldots ,\int _0^1Y_{N-1}(s)\beta ^*(t,s)ds\right) ^T.$ By the definition of ${\varvec{\eta }}^*(t)$, one has

$$\begin{aligned} \widetilde{{\varvec{Y}}}(t)-{\varvec{\eta }}^*(t)= & {} \left( Y_2(t){-}\int _0^1\beta ^*(t,s)Y_1(s)ds,\ldots ,Y_N(t){-}\int _0^1\beta ^*(t,s)Y_{N-1}(s)ds\right) ^T\\= & {} \left( \varepsilon _2(t)+{\varvec{Z}}_2^T{\varvec{\theta }}(t)+\int _0^1R(t,s)Y_1(s)ds,\ldots ,\varepsilon _N(t)+{\varvec{Z}}_N^T {\varvec{\theta }}(t)\right. \\&\left. +\int _0^1R(t,s)Y_{N-1}(s)ds\right) ^T\\= & {} {{\varvec{\varepsilon }}}(t)+ \widetilde{{\varvec{Z}}}{\varvec{\theta }}(t)+\left( \int _0^1R(t,s)Y_1(s)ds,\ldots ,\int _0^1R(t,s)Y_{N-1}(s)ds\right) ^T\\= & {} {{\varvec{\varepsilon }}}(t)+ \widetilde{{\varvec{Z}}}{\varvec{\theta }}(t)+{\varvec{R}}^*(t), \end{aligned}$$

where

$$\begin{aligned} {{\varvec{\varepsilon }}}(t)=( \varepsilon _2(t), \varepsilon _3(t),\ldots , \varepsilon _N(t))^T \end{aligned}$$

and

$$\begin{aligned} {\varvec{R}}^*(t)=\left( \int _0^1R(t,s)Y_1(s)ds,\ldots ,\int _0^1R(t,s)Y_{N-1}(s)ds\right) ^T. \end{aligned}$$

Invoking the definition of ${\varvec{P}}_N$, it is easy to show that $({\varvec{I}}_N-{\varvec{P}}_N)\widetilde{{\varvec{Z}}}=0.$ Thus, we have

$$\begin{aligned} \hat{ {\varvec{\alpha }}}- { {\varvec{\alpha }}}= & {} {\varvec{A}}_1^{-1}\int _0^1 \left( \langle Y_1,{\varvec{B}} \rangle ,\ldots ,\langle Y_{N-1},{\varvec{B}} \rangle \right) \otimes {\varvec{B}}(t)({\varvec{I}}_N-{\varvec{P}}_N)[{{\varvec{\varepsilon }}}(t)+{\varvec{R}}^*(t)]dt\\= & {} {\varvec{A}}_1^{-1}\int _0^1 \left( \langle Y_1,{\varvec{B}} \rangle ,\langle Y_2,{\varvec{B}} \rangle ,\ldots ,\langle Y_{N-1},{\varvec{B}} \rangle \right) \otimes {\varvec{B}}(t)({\varvec{I}}_N-{\varvec{P}}_N){{\varvec{\varepsilon }}}(t)dt\\&+{\varvec{A}}_1^{-1}\int _0^1 \left( \langle Y_1,{\varvec{B}} \rangle ,\ldots ,\langle Y_{N-1},{\varvec{B}} \rangle \right) \otimes {\varvec{B}}(t)({\varvec{I}}_N-{\varvec{P}}_N) {\varvec{R}}^*(t)dt\\= & {} {\varvec{A}}_1^{-1}({\varvec{I}}_1+{\varvec{II}}_2), \end{aligned}$$

where

$$\begin{aligned} {\varvec{I}}_1=\int _0^1 \left( \langle Y_1,{\varvec{B}} \rangle ,\ldots ,\langle Y_{N-1},{\varvec{B}} \rangle \right) \otimes {\varvec{B}}(t)({\varvec{I}}_N-{\varvec{P}}_N){{\varvec{\varepsilon }}}(t)dt \end{aligned}$$

and

$$\begin{aligned} {\varvec{II}}_2=\int _0^1 \left( \langle Y_1,{\varvec{B}} \rangle ,\ldots ,\langle Y_{N-1},{\varvec{B}} \rangle \right) \otimes {\varvec{B}}(t)({\varvec{I}}_N-{\varvec{P}}_N) {\varvec{R}}^*(t)dt. \end{aligned}$$

Invoking the properties of B-spline and Assumption 1, we can get $\Vert {\varvec{II}}_2\Vert =O_p\left( NM_N^{-r-1}\right)$ and $\Vert {\varvec{I}}_1\Vert =O_p\left( \sqrt{N}M_N^{-1}\right)$ via routine calculation.

First, consider ${\varvec{A}}_1$. By the law of large numbers and Assumptions 1 and 3, one has

$$\begin{aligned} {\varvec{A}}_1= & {} \int _0^1 \left( \langle Y_1,{\varvec{B}}\rangle ,\ldots ,\langle Y_{N-1},{\varvec{B}}\rangle \right) \otimes {\varvec{B}}(t)({\varvec{I}}_N-{\varvec{P}}_N) {{\varvec{B}}}^T(t) \\&\otimes \left( \langle Y_1,{\varvec{B}}\rangle ,\ldots ,\langle Y_{N-1},{\varvec{B}}\rangle \right) ^T dt\\= & {} \int _0^1\int _0^1\int _0^1 {\varvec{B}}(t)\otimes {\varvec{B}}(s) \sum _{n=1}^{N} Y_n(s)({\varvec{I}}_N-{\varvec{P}}_N) Y_n(u) {{\varvec{B}}}^T(t)\otimes {{\varvec{B}}}^T (u)dtdsdu \\= & {} N\int _0^1\int _0^1\int _0^1 {\varvec{B}}(t)\otimes {\varvec{B}}(s) K_Z(s,u) {{\varvec{B}}}^T(t)\otimes {{\varvec{B}}}^T (u)dtduds+o_p(NM_N^{-2}), \end{aligned}$$

where $K_Z(t,s)=E[(Y_n(t)-E(Y_n(t)|Z_n))(Y_{n-1}(s)-E(Y_{n-1}(s)|Z_{n-1}))]$. Combining ${\varvec{I}}_1, {\varvec{II}}_2$ with the properties of B-spline, one has $\Vert \hat{ {\varvec{\alpha }}}- { {\varvec{\alpha }}}\Vert =O_p(M_N^{-r+1})+O_p(M_NN^{-1/2}).$ By triangle inequality and Lemma 1, we have

$$\begin{aligned} \parallel \hat{\beta }(t,s)-\beta (t,s) \parallel ^2\le & {} \parallel \hat{\beta }(t,s)-\beta ^*(t,s) \parallel ^2 + \parallel \beta ^*(t,s)-\beta (t,s) \parallel ^2\\\le & {} M_N^{-2}\Vert \hat{ {\varvec{\alpha }}}- { {\varvec{\alpha }}}\Vert ^2+O(M_N^{-2r})\\= & {} O_p(M_N^{-2r})+O_p(N^{-1})+O(M_N^{-2r})\\= & {} O_p(M_N^{-2r}). \end{aligned}$$

$\square$

Proof of Theorem 2

By the proof of Theorem 1, we have

$$\begin{aligned} \sqrt{\frac{N}{M_N^2}}\left( \hat{\beta }(t,s)-\beta ^*(t,s)\right)= & {} \sqrt{\frac{N}{M_N^2}}{\varvec{B}}^T(t)\otimes {\varvec{B}}^T (s) (\hat{ {\varvec{\alpha }}}- { {\varvec{\alpha }}})\\= & {} \sqrt{\frac{M_N^2}{N}}{\varvec{B}}^T(t)\otimes {\varvec{B}}^T (s) {\varvec{\Sigma }}^{-1}_N({\varvec{I}}_1+{\varvec{II}}_2)+o_p(1)\\= & {} R_1(t,s)+R_2(t,s)+o_p(1), \end{aligned}$$

where

$$\begin{aligned} {\varvec{\Sigma }}_N= & {} M_N^2\int _0^1\int _0^1\int _0^1 {\varvec{B}}(t)\otimes {\varvec{B}}(s) K_Z(s,u) {{\varvec{B}}}^T (t)\otimes {{\varvec{B}}}^T(u)dtduds,\\ R_1(t,s)= & {} \sqrt{\frac{M_N^2}{N}} {\varvec{B}}^T(t)\otimes {\varvec{B}}^T (s) {\varvec{\Sigma }}^{-1}_N\int _0^1 \left( \langle Y_1,{\varvec{B}} \rangle ,\ldots ,\langle Y_{N-1},{\varvec{B}} \rangle \right) \\&\otimes {\varvec{B}}(t)({\varvec{I}}_N-{\varvec{P}}_N){{\varvec{\varepsilon }}}(t)dt, \end{aligned}$$

and

$$\begin{aligned} R_2(t,s)= & {} \sqrt{\frac{M_N^2}{N}} {\varvec{B}}^T(t)\otimes {\varvec{B}}^T (s) {\varvec{\Sigma }}^{-1}_N \int _0^1 \left( \langle Y_1,{\varvec{B}} \rangle ,\ldots ,\langle Y_{N-1},{\varvec{B}} \rangle \right) \\&\otimes {\varvec{B}}(t)({\varvec{I}}_N-{\varvec{P}}_N) {\varvec{R}}^*(t)dt. \end{aligned}$$

First, consider $R_1(t,s).$ By the central limit theorem and Assumption 2, one has $R_1(t,s){\mathop {\longrightarrow }\limits ^{L}}{\varvec{G}}(t,s),$ where ${\varvec{G}}(t,s)$ is a normal distribution with zero mean and the covariance

$$\begin{aligned} \sigma ^2(t, s) =\lim _{N \rightarrow \infty } M_N^2 {\varvec{B}}^T(t)\otimes {\varvec{B}}^T (s) {\varvec{\Sigma }}^{-1}_N {\varvec{H}} {\varvec{\Sigma }}^{-1}_N{\varvec{B}}(t)\otimes {\varvec{B}} (s) \end{aligned}$$

with

$$\begin{aligned} H=\int _0^1\int _0^1{\varvec{B}} (s)C_\varepsilon (t,s){\varvec{B}}^T(s)dtds\otimes \int _0^1\int _0^1{\varvec{B}} (s)K_Z(t,s){\varvec{B}}^T(s)dtds, \end{aligned}$$

where $C_\varepsilon (t,s)=\text {Cov}(\varepsilon _n(t),\varepsilon _n(s)).$

Next, consider $R_2(t,s).$ Invoking Theorem 1 and $N/M_N^{2r+2}=o(1)$, we have

$$\begin{aligned} \Vert R_2(t,s)\Vert= & {} \left\| \sqrt{\frac{M_N^2}{N}} {\varvec{B}}^T(t)\otimes {\varvec{B}}^T (s) {\varvec{\Sigma }}^{-1}_N \right\| \left\| \int _0^1 \left( \langle Y_1,{\varvec{B}} \rangle ,\ldots ,\langle Y_{N-1},{\varvec{B}} \rangle \right) \right. \\&\left. \otimes {\varvec{B}}(t)({\varvec{I}}_N-{\varvec{P}}_N) {\varvec{R}}^*(t)dt\right\| \\= & {} \left\| \sqrt{\frac{M_N^2}{N}} {\varvec{B}}^T(t)\otimes {\varvec{B}}^T (s) {\varvec{\Sigma }}^{-1}_N \right\| \Vert {\varvec{II}}_2\Vert \\= & {} O_p\left( \sqrt{N}M_N^{-r-1}\right) \\= & {} o_p(1). \end{aligned}$$

Combining the asymptotic normality of $R_1(t,s)$ with the convergence rate of $R_2(t,s)$, one has

$$\begin{aligned} \sqrt{\frac{N}{M_N^2}}\left( \hat{\beta }(t,s)-\beta ^*(t,s)\right)&= R_1(t,s)+R_2(t,s)+o_p(1)\\&= R_1(t,s)+o_p(1)\\&{\mathop {\longrightarrow }\limits ^{L}}{\varvec{G}}(t,s). \end{aligned}$$

This completes the proof. $\square$

Proof of Theorem 3

Invoking $\widetilde{{\varvec{Y}}}(t)={\varvec{\eta }}_{\beta }(t)+\widetilde{{\varvec{Z}}}{\varvec{\theta }}(t)+{\varvec{\varepsilon }}(t)$, one has

$$\begin{aligned} \sqrt{N}[{\varvec{\hat{\theta }(t)}}-{\varvec{\theta }}(t)]= & {} \sqrt{N}(\widetilde{{\varvec{Z}}}^T \widetilde{{\varvec{Z}}})^{-1}\widetilde{{\varvec{Z}}}^T[\widetilde{{\varvec{Y}}}(t)- \hat{ {\varvec{\eta }}}_{\hat{\beta }}(t)-\widetilde{{\varvec{Z}}}{\varvec{\theta }}(t)]\\= & {} \sqrt{N}(\widetilde{{\varvec{Z}}}^T \widetilde{{\varvec{Z}}})^{-1}\widetilde{{\varvec{Z}}}^T[{\varvec{\eta }}_{\beta }(t)-{\varvec{\hat{{\varvec{\eta }}}}}_{\hat{\beta }}(t)+{\varvec{\varepsilon }}(t)]\\= & {} R_1(t)+R_2(t), \end{aligned}$$

where $R_1(t)=\sqrt{N}(\widetilde{{\varvec{Z}}}^T \widetilde{{\varvec{Z}}})^{-1}\widetilde{{\varvec{Z}}}^T[{\varvec{\eta }}_{\beta }(t)-{\varvec{\hat{{\varvec{\eta }}}}}_{\hat{\beta }}(t)]$ and $R_2(t)=\sqrt{N}(\widetilde{{\varvec{Z}}}^T \widetilde{{\varvec{Z}}})^{-1}\widetilde{{\varvec{Z}}}^T{\varvec{\varepsilon }}(t)$.

First, consider $R_1(t)$. By Assumption 1 and Theorem 1, we have $\Vert {\varvec{\eta }}_{\beta }(t)-{\varvec{\hat{{\varvec{\eta }}}}}_{\hat{\beta }}(t)\Vert =O_p(NM_N^{-r}).$ Consequently, it has

$$\begin{aligned} \Vert R_1\Vert ^2= & {} N \int _0^1 ({\varvec{\eta }}(t)- \hat{ {\varvec{\eta }}}(t))^T\widetilde{{\varvec{Z}}}(\widetilde{{\varvec{Z}}}^T \widetilde{{\varvec{Z}}})^{-1}(\widetilde{{\varvec{Z}}}^T \widetilde{{\varvec{Z}}})^{-1}\widetilde{{\varvec{Z}}}^T[{\varvec{\eta }}(t)- \hat{ {\varvec{\eta }}}(t)]dt\\= & {} O_p(1)\int _0^1 ({\varvec{\eta }}(t)- \hat{ {\varvec{\eta }}}(t))^T({\varvec{\eta }}(t)- \hat{ {\varvec{\eta }}}(t))dt\\= & {} O_p(M_N^{-r})\\= & {} o_p(1). \end{aligned}$$

Next, consider $R_2(t)$. By the CLT for Hilbert spaces, we have $\frac{1}{\sqrt{N}}\sum _{n=1}^N {\varvec{Z}}_n \varepsilon _n(t) {\mathop {\longrightarrow }\limits ^{L}}{\varvec{B}}_1(t),$ where ${\varvec{B}}_1(t)$ is a Guassian processes with zero mean and the covariance function $C(t, s) = \text {Cov}(\varepsilon _n(t),\varepsilon _n(s))$. Denote ${\varvec{\Sigma }}=\text {E}[{\varvec{Z}}{\varvec{Z}}^T].$ By the strong law of large numbers and the CLT for Hilbert spaces, it has

$$\begin{aligned} R_2(t)= & {} \sqrt{N} (\widetilde{{\varvec{Z}}}^T \widetilde{{\varvec{Z}}})^{-1}\widetilde{{\varvec{Z}}}^T{\varvec{\varepsilon }}(t)\\= & {} {\varvec{\Sigma }}^{-1}\frac{1}{\sqrt{N}}\sum _{n=1}^N {\varvec{Z}}_n \varepsilon _n(t)+o_p(1)\\= & {} {\varvec{\Sigma }}^{-1} {\varvec{B}}_1(t)+o_p(1). \end{aligned}$$

Combining this with $\Vert R_1\Vert =o_p(1),$ one has $\sqrt{N}[{\varvec{\hat{\theta }}}(t)-{\varvec{\theta }}(t)] \longrightarrow {\varvec{G}}(t),$ where ${\varvec{G}}(t)$ is a Guassian processes with zero mean and the covariance function $C_{{\varvec{B}}}(t, s) = {\varvec{\Sigma }}^{-1} \text {Cov}(\varepsilon _n(t),\varepsilon _n(s))$. $\square$