Inferences in Binary Dynamic Fixed Models in a Semi-parametric Setup

Abstract

In a longitudinal setup, the so-called generalized estimating equations approach was a popular inference technique to obtain efficient regression estimates until it was discovered that this approach may in fact yield less efficient estimates than an independence assumption-based estimating equation approach. In this paper, we revisit this inference issue in a semi-parametric longitudinal setup for binary data and find that the semi-parametric generalized estimating equations also encounter similar efficiency drawbacks when compared with independence assumption-based approach. This makes the generalized estimating equations approach unacceptable for correlated data analysis. We analyze the repeated binary data by fitting a semi-parametric binary dynamic model. The non-parametric function and the regression parameters involved in the semi-parametric regression function are estimated by using a semi-parametric generalized quasi-likelihood and a semi-parametric quasi-likelihood approach, respectively, whereas the dynamic dependence, that is, the correlation index parameter of the model is estimated by a semi-parametric method of moments. Asymptotic and finite sample properties of the estimators are discussed. The proposed model and the estimation methodology are also illustrated by reanalyzing the well-known respiratory disease data.

Appendix: Asymptotic Properties of the Estimators of the SLDCP Model

The following regularity conditions and/or assumptions (A) are needed to study the asymptotic properties of the estimators of the non-parametric functionψ(⋅), and the parameters β and ρ, for the SLDCP model (2.1).

A.1.

The mean function μ_ij(⋅) in model (2.1) is continuous and

$$\frac{\partial^{r} \mu_{ij}(\cdot)}{\partial {\psi(z_{0})}^{r}} < \infty\;\text{for}\;r = 1,2. $$

A.2.

For i = 1,…,K, the estimating functions

$$f_{i}(\beta)=\frac{\partial [\tilde{\mu}_{i}(\beta, \hat{\psi}(\beta))]'}{\partial \beta} ~~[\tilde{{\Sigma}}_{i}(\beta, \rho,\hat{\psi}(\beta))]^{-1}~ [y_{i} -\tilde{\mu}_{i}(\beta,\hat{\psi}(\beta))]$$

from Eq. 3.12, with $V^{*}_{K}={\sum }^{K}_{i = 1}\text {cov}[f_{i}(\beta )],$ satisfy the Lindeberg’s condition (Amemiya, 1985, Theorem 3.3.6), that is,

$$\lim_{K \rightarrow \infty}{{V}^{*}}^{-1}_{K}{\sum}^{K}_{i = 1}{\sum}_{(f^{\top}_{i} {{V}^{*}}^{-1}_{K} f_{i} ) >\epsilon}f_{i}f^{\top}_{i}g(f_{i})= 0 $$

for all 𝜖 > 0,g(⋅) being the probability distribution of f_i(β). [A proof for the Lindeberg condition in the context of categorical/binary time series data is available in Kaufmann (1987, pp. 89, 93)].

A.3.

Recall from the estimating equation (3.14) for ρ that the standardized residuals are defined as $y_{ij}^{*} = [y_{ij} - \mu _{ij}(\beta ,x_{ij}, \psi (z_{ij}))]/\sqrt {\sigma _{i,jj}(\beta ,x_{ij}, \psi (z_{ij}))}$. For two fixed quantities M₁ and M₂, we now assume that the lag 1 sum of products and sum of squares have bounded variances satisfying

$$\begin{array}{@{}rcl@{}} \mathrm{E}\left[ \left( {\sum}_{j = 1}^{n_{i}-1}\left[ Y_{ij}^{*}Y_{i,j + 1}^{*} - \frac{\sigma_{i,j,j + 1}}{\sqrt{\sigma_{ijj}\sigma_{i,j + 1,j + 1}}} \right]\right)^{2} \right] &<& M_{1},\;\text{and} \end{array} $$

$$\begin{array}{@{}rcl@{}} \mathrm{E}\left[ \left( {\sum}_{j = 1}^{n_{i}}\left[ {Y_{ij}^{*}}^{2} - 1 \right]\right)^{2} \right] &<& M_{2}, \end{array} $$

respectively.

• Consistency of $\hat {\psi }(\cdot )$]

For convenience, in Eq. 3.6, we have shown the estimation for ψ(z₀) for z₀ ≡ z_ℓu for a selected value of ℓ(ℓ = 1,…,K) and u(u = 1,…,n_ℓ). For notational simplicity, here we use μ_ij(z₀) for μ_ij(β,x_ij,ψ(z₀)). Now for known β, and for true binary mean μ_ij ≡ μ_ij(β,x_ij,ψ(z_ij)) given by Eq. ??, a Taylor expansion of $f(\psi (z_{0}),\beta )={\sum }_{i = 1}^{K} {\sum }^{n_{i}}_{j = 1}w_{ij}(z_{0}) [y_{ij}- \mu _{ij}(\beta ,x_{ij}, \psi (z_{0}))]$ (3.6) under the assumption A.1 gives

$$\begin{array}{@{}rcl@{}} &&\!\!\!\!\hat{\psi}(z_{0};\beta) - \psi(z_{0}) \approx \frac{{\sum}_{i = 1}^{K}{\sum}_{j = 1}^{n_{i}}w_{ij}(z_{0})\left[y_{ij} - \mu_{ij}(z_{0})\right]} {{\sum}_{i = 1}^{K}{\sum}_{j = 1}^{n_{i}}w_{ij}(z_{0})\mu_{ij}(z_{0})\left[1-\mu_{ij}(z_{0})\right]} =\frac{f(\psi(z_{0}),\beta)}{f^{\prime}_{\psi(z_{0})}(\psi(z_{0}),\beta)} \\ &{}=&\!\!\!\! \frac{1}{f^{\prime}_{\psi(z_{0})}(\psi(z_{0}),\beta)} \left[{\sum}_{i = 1}^{K}{\sum}_{j = 1}^{n_{i}}w_{ij}(z_{0})\left[y_{ij} - \mu_{ij}\right] + {\sum}_{i = 1}^{K}{\sum}_{j = 1}^{n_{i}}w_{ij}(z_{0})\left[\mu_{ij} - \mu_{ij}(z_{0})\right]\right] \\ &{}=&\!\!\!\! A_{K} + \frac{1}{f^{\prime}_{\psi(z_{0})}(\psi(z_{0}),\beta)}{\sum}_{i = 1}^{K}{\sum}_{j = 1}^{n_{i}}w_{ij}(z_{0}) \left[\mu_{ij} - \mu_{ij}(z_{0})\right] \,, \end{array} $$

(1.1)

where

$$\begin{array}{@{}rcl@{}} A_{K}&=&\frac{1}{B_{K}}\frac{1}{K}{\sum}_{i = 1}^{K}{\sum}_{j = 1}^{n_{i}}p_{ij}(z_{0})\left( y_{ij} - \mu_{ij}\right) \; \text{with} \\ B_{K} &=&\frac{1}{K}{\sum}_{i = 1}^{K}{\sum}_{j = 1}^{n_{i}}p_{ij}(z_{0})\mu_{ij}(z_{0}) \left[1-\mu_{ij}(z_{0})\right], \end{array} $$

with $p_{ij}(z_{0}) \equiv p_{ij}(\frac {z_{0}-z_{ij}}{b})$ as the kernel density defined in Eq. 3.4. Here, b ∝ K^−α for a suitable α(Lin and Carroll 2001; Pagan and Ullah, 1999, p. 28). Notice that E[A_K] = 0 and

$$\text{Var}[A_{K}] = \frac{1}{{B_{K}^{2}}}\frac{1}{K^{2}}{\sum}_{i = 1}^{K} \text{Var}\left[{\sum}_{j = 1}^{n_{i}}p_{ij}(z_{0}) Y_{ij}\right] = \frac{1}{K} Q_{k} $$

with $Q_{K} = \frac {1}{{B_{K}^{2}}}\frac {1}{K}{\sum }_{i = 1}^{K} \text {Var}\left [{\sum }_{j = 1}^{n_{i}}p_{ij}(z_{0}) Y_{ij}\right ] = O(1)$. It then follows, for example, from Amemiya (1985, Theorem 14.4-1) that

$$\begin{array}{@{}rcl@{}} A_{K} &=& o_{p}(1/\sqrt{K}). \end{array} $$

(1.2)

Next by using

$$\mu_{ij} - \mu_{ij}(z_{0}) = \mu_{ij}(z_{0})\left[1-\mu_{ij}(z_{0})\right]\psi^{\prime}(z_{0})(z_{ij}-z_{0}) + O\left( (z_{ij}-z_{0})^{2}\right), $$

after some algebras, the second term in Eq. 6.1 may be expressed as

$$\begin{array}{@{}rcl@{}} &&\frac{1}{f^{\prime}_{\psi(z_{0})}(\psi(z_{0}),\beta)}{\sum}_{i = 1}^{K}{\sum}_{j = 1}^{n_{i}}w_{ij}(z_{0}) \left[\mu_{ij} - \mu_{ij}(z_{0})\right] \\ &=&\frac{\psi^{\prime}(z_{0})\frac{1}{K}\sum\limits_{i = 1}^{K}\sum\limits_{j = 1}^{n_{i}}p_{ij}(z_{0})\, \mu_{ij}(z_{0})\left[1-\mu_{ij}(z_{0})\right](z_{ij}-z_{0})} {\frac{1}{K}\sum\limits_{i = 1}^{K}\sum\limits_{j = 1}^{n_{i}}p_{ij}(z_{0})\,\mu_{ij}(z_{0}) \left[1-\mu_{ij}(z_{0})\right]} + O(b^{2})\\ &=&O(b^{2}), \end{array} $$

(1.3)

because E[z_ij − z₀] for a symmetric such as Gaussian kernel. Thus, by using Eqs. 6.3 and 6.2 in Eq. 6.1, one obtains

$$ \hat{\psi}(z_{0};\beta) - \psi(z_{0}) =o_{p}(1/\sqrt{K})+O(b^{2}), $$

(1.4)

showing that $\hat {\psi }(z_{0};\beta )$ is consistent for ψ(z₀) provided Kb⁴ → 0 as K →∞, that is, $K\frac {1}{K^{4\alpha }}=\frac {1}{K^{4\alpha -1}}\rightarrow 0$ yielding the condition α > 1/4. Note that this convergence result is obtained by minimizing the bias of the estimator(see Eq. 6.1).

• Asymptotic Normality and Consistency of $\hat {\beta }_{SGQL}$

Recall that the SGQL estimator $\hat {\beta }_{SGQL}$ of β is obtained by solving the estimating equation (3.12). Suppose that for true β, the estimating function in Eq. 3.12 is denoted as

$$D_{K}(\beta)= \frac{1}{K}{\sum}_{i = 1}^{K} \frac{\partial [\tilde{\mu}_{i}(\beta, \hat{\psi}(\beta))]'}{\partial \beta} ~~[\tilde{{\Sigma}}_{i}(\beta, \rho,\hat{\psi}(\beta))]^{-1}~ [y_{i} -\tilde{\mu}_{i}(\beta,\hat{\psi}(\beta))]. $$

Thus, $\hat {\beta }_{SGQL}$ must satisfy $D_{K}(\hat {\beta }_{SGQL})= 0$ which by a linear Taylor expansion about true β provides

$$ D_{K}(\beta)+(\hat{\beta}_{SGQL}-\beta)D^{\prime}_{K}(\beta)+o_{p}(1/\sqrt{K})= 0, $$

(1.5)

yielding

$$\begin{array}{@{}rcl@{}} \hat{\beta}_{SGQL}-\beta &=& -\left[D^{\prime}_{K}(\beta)\right]^{-1}[D_{K}(\beta)+o_{p}(1/\sqrt{K})] \\ &=&F^{-1}_{K}(\beta)D_{K}(\beta)+o_{p}(1/\sqrt{K}), \end{array} $$

(1.6)

where

$$F_{K}(\beta) = \frac{1}{K} {\sum}_{i = 1}^{K} \frac{\partial [\tilde{\mu}_{i}(\beta, \hat{\psi}(\beta))]'}{\partial \beta} ~~[\tilde{{\Sigma}}_{i}(\beta, \rho,\hat{\psi}(\beta))]^{-1}~ \frac{\partial \tilde{\mu}_{i}(\beta, \hat{\psi}(\beta))}{\partial \beta}.$$

Notice that in Eq. 6.6, one may write

$$F = \lim_{K\rightarrow\infty}F_{K} = \mathrm{E}_{\hat{\psi}}\left[\frac{\partial \tilde{\mu}_{i}^{\prime}(\beta, \hat{\psi}(z_{i};\beta) )}{\partial\beta}\,{\tilde{{\Sigma}}_{i}}^{-1}(\beta, \hat{\psi}(z_{i}; \beta), {\rho})\frac{\partial \tilde{\mu}_{i}(\beta, \hat{\psi}(z_{i};\beta))}{\partial\beta^{\prime}}\right]. $$

Next by using $Z_{1i} = \frac {\partial \tilde {\mu }_{i}^{\prime }(\beta , \hat {\psi }(z_{i};\beta ) )}{\partial \beta }\,{\tilde {{\Sigma }}_{i}}^{-1}(\beta , \hat {\psi }(z_{i}; \beta ), {\rho }),$ and $v_{1i}^{jk}$ as the (j,k)th element of ${\tilde {{\Sigma }}_{i}}^{-1},$ the estimating function D_K(β) in Eq. 6.6 may be expressed as

$$\begin{array}{@{}rcl@{}} D_{K}(\beta)\!\!\!&=&\!\!\! \frac{1}{K} {\sum}_{i = 1}^{K}Z_{1i} \left( Y_{i} - \mu_{i} \right) \\ &-&\!\!\!\frac{1}{K} {\sum}_{i = 1}^{K}\frac{\partial \tilde{\mu}_{i}^{\prime}(\beta, \hat{\psi}(z_{i};\beta) )}{\partial\beta}\,{\tilde{{\Sigma}}_{i}}^{-1}(\beta, \hat{\psi}(z_{i}; \beta), {\rho})\\&& \times\left[ \tilde{\mu}_{i}(\beta, \hat{\psi}(z_{i};\beta) ) - \mu_{i}(\beta, \psi(z_{i})) \right] \\ &=&\!\!\! \frac{1}{K} {\sum}_{i = 1}^{K}Z_{1i} \left( Y_{i} - \mu_{i} \right) -\frac{1}{K} {\sum}_{i = 1}^{K} {\sum}_{j = 1}^{n_{i}} {\sum}_{k = 1}^{n_{i}}\frac{\partial \tilde{\mu}_{ij}(\beta, \hat{\psi}(z_{ij};\beta))}{\partial\beta}\,v_{1i}^{jk}(\beta, \hat{\psi}, {\rho}) \\ &\times &\!\!\! \mu_{ik}(\beta, \psi(z_{ik})) \left[1-\mu_{ik}(\beta, \psi(z_{ik}))\right] \left[ \hat{\psi}(z_{ik};\beta) - \psi(z_{ik}) \right] + o_{p}(1/\sqrt{K}) \\ &=&\!\!\!\frac{1}{K}{\sum}^{K}_{i = 1}(Z_{1i}-Z_{2i})(Y_{i}-\mu_{i})+O(b^{2})+o_{p}(1/\sqrt{K}), \end{array} $$

(1.7)

by Eq. 6.4, where $Z_{2i} = \left (Z_{2i1},\cdots , Z_{2in_{i}}\right )$ with

$$\begin{array}{@{}rcl@{}} Z_{2ij}&=& \frac{1}{K}{\sum}_{i^{\prime}= 1}^{K} {\sum}_{j^{\prime}= 1}^{n_{i}} {\sum}_{k^{\prime}= 1}^{n_{i}}\frac{1}{B_{K}(z_{i^{\prime}k^{\prime}})}\frac{\partial \tilde{\mu}_{i^{\prime}j^{\prime}}(\beta, \hat{\psi}(z_{i^{\prime}j^{\prime}};\beta))}{\partial\beta}\,v_{1i^{\prime}}^{j^{\prime}k^{\prime}}(\beta, \hat{\psi}, {\rho})\\ &\times& \mu_{i^{\prime}k^{\prime}}(\beta, \psi(z_{i^{\prime}k^{\prime}})) \left[1-\mu_{i^{\prime}k^{\prime}}(\beta, \psi(z_{i^{\prime}k^{\prime}}))\right]\,p_{ij}(z_{i^{\prime}k^{\prime}}), \end{array} $$

where B_K and the kernel density p_ij(z₀) are defined in Eq. 6.1. Hence by using (6.7) in Eq. 6.6, one obtains

$$\begin{array}{@{}rcl@{}} \sqrt{K}\left\{ \hat{\beta}_{SGQL} - \beta \right\} &=& F^{-1} \frac{1}{\sqrt{K}}{\sum}_{i = 1}^{K} \left( Z_{1i} - Z_{2i}\right)(Y_{i} - \mu_{i}) \\ & +& O(\sqrt{K b^{4}}) + o_{p}(1). \end{array} $$

(1.8)

Now because E[Y_i − μ_i] = 0, and cov[Y_i] = Σ_i, by applying the assumption A.2, we can use the Lindeberg-Feller central limit theorem (Amemiya, 1985, Theorem 3.3.6) for independent random variables with non-identical distributions, and obtain

$$\begin{array}{@{}rcl@{}} \sqrt{K}\left\{ \hat{\beta}_{SGQL}- \beta - O(b^{2}) \right\} &\rightarrow& N(0,V_{\beta}), \end{array} $$

(1.9)

where

$$V_{\beta} = F^{-1}\frac{1}{K}\left[{\sum}_{i = 1}^{K} \left( Z_{1i} - Z_{2i}\right){\Sigma}_{i}\left( Z_{1i} - Z_{2i}\right)'\right]F^{-1}.$$

• Consistency of $\hat {\tilde {\rho }}$

We prove the consistency for known β and ψ(z_ij). The result remains valid when β and ψ(z_ij) are replaced by their respective consistent estimates. Notice that for known β and ψ(z_ij), the moment estimator of ρ is given by Eq. 3.14. Now because Y_ij’s are independent for different i, and also because the variances of the lag 1 sum of products and sum of squares are bounded by assumption A.3, for K →∞, we may apply the law of large numbers for independent random variables (Breiman, 1968, Theorem 3.27) and obtain

$$\begin{array}{@{}rcl@{}} && \frac{{\sum}_{i = 1}^{K}{\sum}_{j = 1}^{n_{i}-1}\left( y_{ij}^{*}y_{i,j + 1}^{*} - \frac{\sigma_{i,j,j + 1}}{\sqrt{\sigma_{ijj}\sigma_{i,j + 1,j + 1}}} \right)}{{\sum}_{i = 1}^{K}(n_{i}-1)} \xrightarrow{P} 0\\ &\!\Rightarrow\! & \frac{{\sum}_{i = 1}^{K}{\sum}_{j = 1}^{n_{i}-1}y^{*}_{ij}y^{*}_{i,j + 1}}{{\sum}_{i = 1}^{K}(n_{i}-1)} = \frac{\rho{\sum}_{i = 1}^{K}{\sum}_{j = 1}^{n_{i}-1}\frac{\sqrt{\sigma_{ijj}}}{\sqrt{\sigma_{i,j + 1,j + 1}}}}{{\sum}_{i = 1}^{K}(n_{i}-1)} + o_{p}(1). \end{array} $$

(1.10)

Similarly,

$$\begin{array}{@{}rcl@{}} \frac{{\sum}_{i = 1}^{K}{\sum}_{j = 1}^{n_{i}}\left( {y_{ij}^{*}}^{2}-1\right)}{{\sum}_{i = 1}^{K}n_{i}} &\xrightarrow{P}& 0\hspace{10mm}\Rightarrow\\ \frac{{\sum}_{i = 1}^{K}{\sum}_{j = 1}^{n_{i}}{y_{ij}^{*}}^{2}}{{\sum}_{i = 1}^{K}n_{i}} &=& 1 + o_{p}(1). \end{array} $$

(1.11)

Dividing (6.10) by Eq. 6.11, after some algebras, one obtains

$$\begin{array}{@{}rcl@{}} \hat{\rho} = \frac{\sum\limits_{i = 1}^{K}\sum\limits_{j = 1}^{n_{i}-1}y^{*}_{ij}y^{*}_{i,j + 1}}{ \sum\limits_{i = 1}^{K}\sum\limits_{j = 1}^{n_{i}}{y_{ij}^{*}}^{2}} \frac{\sum\limits_{i = 1}^{K} n_{i}}{\sum\limits_{i = 1}^{K}\sum\limits_{j = 1}^{n_{i}-1}\left[\frac{\sigma_{ijj}}{\sigma_{i,j + 1,j + 1}}\right]^{\frac{1}{2}}} &\xrightarrow{P}& \rho \hspace{3mm}\text{as \(K\rightarrow \infty\)\,.} \end{array} $$

(1.12)

The consistency for $\hat {\tilde {\rho }}$ in Eq. 3.15 follows from Eq. 6.12 because of the fact that $\hat {\tilde {\rho }}$ was constructed by putting consistent estimates forβ and ψ(z_ij) in the formula for $\hat {\rho }$ in Eq. 3.14.