Kernel Estimation for Panel Data with Heterogeneous Dynamics¹¹1First Version: February, 2018.

Before formally showing the asymptotic properties, we explore the unique features of our asymptotic investigations in an informal manner. By doing so, we clarify the mechanism behind the observation that even very high orders of asymptotic bias matter for our asymptotic analysis.

We here focus on the density estimator for $\hat{\mu}_{i}$, but similar discussions are also relevant for $\hat{\gamma}_{k,i}$ and $\hat{\rho}_{k,i}$. Noting that $\hat{\mu}_{i}-\mu_{i}=\bar{w}_{i}$, we examine the $J$-th order Taylor expansion of $\hat{f}_{\hat{\mu}}(x)$:

where $K^{(j)}$ denotes the $j$-th order derivative and $\tilde{\mu}_{i}$ is between $\mu_{i}$ and $\hat{\mu}_{i}$.

The first term in (2) is the infeasible density estimator based on the true $\mu_{i}$, and its asymptotic behavior is standard and well known in the kernel-smoothing literature. It converges in probability to the density of interest $f_{\mu}(x)$ as $N\to\infty$ and $h\to 0$ with $Nh\to\infty$. In addition, when $Nh^{5}\to C\in[0,\infty)$ also holds, it can hold that:

where $\kappa_{1}\coloneqq\int s^{2}K(s)ds$ and $\kappa_{2}\coloneqq\int K^{2}(s)ds$ and $\mathcal{N}(\mu,\sigma^{2})$ is a normal distribution with mean $\mu$ and variance $\sigma^{2}$.

The unique features in our situation are caused from the second and third terms in (2). For the second term, under regularity conditions, the mean can be evaluated as:

where we used $E((\bar{w}_{i})^{j}|\mu_{i}=x)=O(T^{-j/2})$ (see Assumption 7 below and Lemma 1 in the supplement). Noting that $E(\bar{w}_{i}|\mu_{i})=0$, the bias caused from the second term in (2) can be written as $\sum_{j=2}^{J-1}O(1/\sqrt{T^{j}h^{2j}})$. This bias is negligible when $1/(Th^{2})\to 0$, which is identical to the relative magnitude condition $N^{2}/T^{5}\to 0$ when using the standard bandwidth $h\asymp N^{-1/5}$ in the density estimation with second-order kernels. For the third term in (2), the absolute mean can be evaluated as:

where $0<M<\infty$ denotes a generic positive constant and we use $E|\bar{w}_{i}|^{J}=O(T^{-J/2})$ (see Lemma 1). Hence, the third term in (2) is $O_{p}(1/\sqrt{T^{J}h^{2J+2}})$ by Markov’s inequality. Remarkably, this term does not vanish even when $1/(Th^{2})\to 0$ under which the lower-order terms are negligible. This term can be negligible only if $1/(Th^{2+2/J})\to 0$, which implies $N^{(2+2/J)}/T^{5}\to 0$ when $h\asymp N^{-1/5}$. Note that $N^{(2+2/J)}/T^{5}\to 0$ is “stronger” than $N^{2}/T^{5}$.

The asymptotic investigation above exhibits several unique features. First, it implies that the relative magnitude condition for consistency (and also that for asymptotic normality) varies in the order of the expansion. Specifically, we need $1/(Th^{2+2/J})\to 0$ to achieve the consistency of $\hat{f}_{\hat{\mu}}(x)$ based on the $J$-th order expansion. Second, we can obtain the “weakest” relative magnitude condition $1/(Th^{2})\to 0$ for consistency, only if we execute the infinite-order expansion (that is, as $J\to\infty$). Finally, while we can derive the suitable condition $1/(Th^{2})\to 0$ via the infinite-order expansion, it requires the existence of higher-order moments of $w_{it}$. The evaluation based on the infinite-order expansion demands the existence of $E|w_{it}|^{j}$ for any $j$. Hence, there is a trade-off between the relative magnitude condition and the existence of higher-order moments.

Asymptotic normality requires a further stronger condition. Because the rate of convergence of the kernel estimator is $\sqrt{Nh}$, it requires $N/(T^{J}h^{2J+1})\to 0$. This condition is at best $N^{2}/T^{3}\to 0$, which is obtained under an infinite order expansion with standard bandwidth ($h\asymp N^{-1/5}$). Note that, as in the density estimation above, the highest order of the expansion determines the required condition for asymptotic normality. Such a very high order of bias cannot be corrected in practice, even though methods to correct the first few orders of bias are available in the long-panel literature (e.g., Dhaene and Jochmans, 2015). This result is in stark contrast to the existing studies in which bias correction improves the conditions on the relative magnitudes of $N$ and $T$.

The main reason behind these unique features is that the curvature of the summand (i.e., $K((x-\hat{\mu}_{i})/h)$) depends on the bandwidth $h$. Roughly speaking, as $h\to 0$, the summand function becomes steeper and more “nonlinear.” It exacerbates the bias caused by the nonlinearity and it turns out that even a very high order derivative of $K$ affects the bias. Alternatively, we may also interpret this problem based on the equation $K((x-\hat{\mu}_{i})/h)=K((x-\mu_{i})/h+(\mu_{i}-\hat{\mu}_{i})/h))$. The contribution of the error by using the estimated $\hat{\mu}_{i}$ is $(\mu_{i}-\hat{\mu}_{i})/h$ and it increases as $h\to 0$. Hence, the bias of the density estimator heavily depends on the magnitude of $h$ and the nonlinearity of $K$.

We here formally show the presence of asymptotic biases of the kernel density estimator in (1). We conduct asymptotic investigations based on an infinite-order expansion under which the weakest possible condition on the relative magnitude of $N$ and $T$ is obtained.

We assume the following basic conditions for the data-generating process. These are essentially the same as the assumptions in Okui and Yanagi (2019).

Assumptions 1 and 2 require that the individual time series given $\alpha_{i}$ is strictly stationary across time but i.i.d. across units. The identical distribution across $i$ is essential for our analysis. The independence assumption across $i$ makes our asymptotic investigations tractable, while the consistency result and the same asymptotic biases could be derived even under weak cross-sectional dependence. Note that the i.i.d. assumption does not exclude the presence of heterogeneity in panel data. In our setting, heterogeneity is caused by differences in the realized values of $\{\alpha_{i}\}_{i=1}^{N}$ across units. Assumption 2 also restricts the degree of persistence of the individual time series. The conditions for stationarity and degree of persistence require that the times series for each unit is not a unit root process and that the initial value of each time series is generated from a stationary distribution. Assumption 3 requires the existence of the moments of $w_{it}$, and it allows us to derive the asymptotic biases of the estimators. While we can develop the theoretical properties of the estimators in situations where Assumptions 2 and 3 do not hold for some numbers $r_{m}$ and $r_{d}$, we cannot derive the higher-order biases based on infinite-order expansions in such situations. As a result, in such situations, we demand stronger conditions on the relative magnitudes as discussed in the previous section. Assumption 4 allows us to derive the asymptotic properties of the kernel estimators for $\rho_{k,i}$. All of the assumptions can be satisfied in popular panel data models. For example, they all hold when $y_{it}$ follows a heterogeneous stationary panel autoregressive moving–average model with a Gaussian error term (e.g., $y_{it}=c_{i}+\phi_{i}y_{i,t-1}+u_{it}+\theta_{i}u_{i,t-1}$ with $u_{it}\sim\mathcal{N}(0,\sigma^{2})$).

We also assume the following additional conditions.

Assumption 5 includes the standard conditions for the kernel function, except for infinite differentiability. We require the differentiability in order to expand the kernel estimator for the estimated $\hat{\xi}_{i}$ at the true $\xi_{i}$ based on the infinite-order expansion. Note that the symmetry of $K$ implies that $\int K^{(j)}(s)ds=0$ for any odd $j$.

Assumption 6 requires that $\xi_{i}$ is continuously distributed without probability mass. The continuity of the random variable is essential for implementing kernel-smoothing estimation as it rules out situations where there is no heterogeneity for $\xi_{i}$ (that is, the situation where $\xi_{i}=\xi$ for any $i$ with some constant $\xi$) and where there is finitely grouped heterogeneity (that is, $\xi_{i_{1}}=\xi_{i_{2}}$ for any $i_{1},i_{2}\in\mathbb{I}_{g}$ with some sets $\mathbb{I}_{1},\mathbb{I}_{2},\dots,\mathbb{I}_{G}$ satisfying $\bigoplus_{g=1}^{G}\mathbb{I}_{g}=\{1,2,\dots,N\}$).

Assumption 7 states the existence and smoothness of the conditional expectations. This assumption allows us to derive the exact forms of the asymptotic biases. The convergence rates of the terms are standard and guaranteed by Lemmas 1 and 3 in Appendix B. For example, the assumption requires that $T\cdot E((\bar{w}_{i})^{2}|\mu_{i}=\cdot)=O(1)$ and the convergence rate is consistent with the result in Lemma 1.

The following theorem shows that the kernel density estimators are consistent and asymptotically normal but exhibit asymptotic biases. While the theorem assumes an infinite-order Taylor expansion and the summability of the infinite series of the asymptotic biases directly, we can show their validity under unrestrictive regularity conditions. Because these discussions are highly technical and demand lengthy explanations, they appear in Appendices C and D.

The density estimator can be written as the sum of the infeasible estimator based on the true $\xi_{i}$, say $\hat{f}_{\xi}(x)\coloneqq(Nh)^{-1}\sum_{i=1}^{N}K((x-\xi_{i})/h)$, and the asymptotic biases. The convergence rate of the estimator is the standard order of $O_{p}(1/\sqrt{Nh})$, and the asymptotic distribution is the same as that of the infeasible estimator $\hat{f}_{\xi}(x)$. However, the feasible estimator exhibits asymptotic biases. These results also require the relative magnitude conditions of $N$, $T$, and $h$; that is, $1/(Th^{2})\to 0$ and $N/(T^{3}h^{5})\to 0$ for consistency and asymptotic normality, respectively.

The density estimator for $\mu_{i}$ has two main asymptotic biases given $A_{\mu,1}(x)=0$, but the density estimators for $\gamma_{k,i}$ and $\rho_{k,i}$ have three main asymptotic biases, in addition to the higher-order biases. The first bias of the form $h^{2}\kappa_{1}f_{\xi}^{\prime\prime}(x)/2$ is the standard kernel-smoothing bias. The second bias of the form $A_{\xi,1}(x)/T$ is the incidental parameter bias caused from estimating $\gamma_{k,i}$ and $\rho_{k,i}$ by $\hat{\gamma}_{k,i}$ and $\hat{\rho}_{k,i}$, respectively. The estimation of $\hat{\gamma}_{k,i}$ and $\hat{\rho}_{k,i}$ involves estimating $\mu_{i}$ by $\hat{\mu}_{i}=\bar{y}_{i}$ for each $i$, which becomes a source of the incidental parameter bias. The third bias of the form $A_{\xi,2}(x)/(Th^{2})$ is the second-order nonlinearity bias caused by expanding $K((x-\hat{\xi}_{i})/h)$ for $K((x-\xi_{i})/h)$ by Taylor expansion. Moreover, the $j$-th order nonlinearity bias exhibits the form $A_{\xi,j}(x)/\sqrt{T^{j}h^{2j}}$ for $j\geq 3$.

We need the two conditions, $1/(Th^{2})\to 0$ and $N/(T^{3}h^{5})\to 0$, to ensure the asymptotic negligibility of the higher-order nonlinearity biases. If we use the standard bandwidth $h\asymp N^{-1/5}$ with second-order kernels, the conditions $1/(Th^{2})\to 0$ and $N/(T^{3}h^{5})\to 0$ imply that $N^{2}/T^{5}\to 0$ and that $N^{2}/T^{3}\to 0$, respectively, which are integrated to $N^{2}/T^{3}\to 0$. Note that while the incidental parameter bias and the second-order nonlinearity bias are also asymptotically negligible under these conditions, the practical magnitudes of these biases would be larger than those of the higher-order nonlinearity biases.

We have already discussed the source of the nonlinearity bias in Section 3.1 so here we provide a slightly more detailed discussion of the incidental parameter bias. It does not appear in $\hat{f}_{\hat{\mu}}(x)$ because the estimation error in $\hat{\mu}_{i}$ (that is, $\bar{w}_{i}$) has zero mean. However, errors in $\gamma_{k,i}$ and $\rho_{k,i}$ are not mean-zero. For example, $\hat{\gamma}_{0,i}=\sum_{t=1}^{T}(y_{it}-\bar{y}_{i})^{2}/T=\gamma_{0,i}+\sum_{t=1}^{T}(w_{it}^{2}-\gamma_{0,i})/T-(\bar{w}_{i})^{2}$ and $(\bar{w}_{i})^{2}$ is not mean-zero although it converges to zero at the rate $1/T$. This is the source of the incidental parameter bias $A_{\gamma_{0},1}/T$ and the order $1/T$ comes from the fact that $(\bar{w}_{i})^{2}$ is $O_{p}(1/T)$.

As the incidental parameter bias and the nonlinearity biases in $\hat{f}_{\hat{\xi}}(x)$ may be severe in practice, we propose adoption of the split-panel jackknife to correct them. Among split-panel jackknifes, here we consider half-panel jackknife (HPJ) bias correction. For simplicity, suppose that $T$ is even.⁸⁸8The bias correction with odd $T$ is similar. See Dhaene and Jochmans (2015, page 999) for details. For $\xi_{i}=\mu_{i}$, $\gamma_{k,i}$, or $\rho_{k,i}$, we obtain the estimators $\hat{f}_{\hat{\xi},(1)}(x)$ and $\hat{f}_{\hat{\xi},(2)}(x)$ of $f_{\xi}(x)$ based on two half-panel data $\{\{y_{it}\}_{t=1}^{T/2}\}_{i=1}^{N}$ and $\{\{y_{it}\}_{t=T/2+1}^{T}\}_{i=1}^{N}$, respectively. The HPJ bias-corrected estimator is $\hat{f}_{\hat{\xi}}^{H}(x)\coloneqq\hat{f}_{\hat{\xi}}(x)-(\bar{f}_{\hat{\xi}}(x)-\hat{f}_{\hat{\xi}}(x))$ where $\bar{f}_{\hat{\xi}}(x)\coloneqq[\hat{f}_{\hat{\xi},(1)}(x)+\hat{f}_{\hat{\xi},(2)}(x)]/2$. The term $\bar{f}_{\hat{\xi}}(x)-\hat{f}_{\hat{\xi}}(x)$ estimates the bias in the original estimator $\hat{f}_{\hat{\xi}}(x)$. Importantly, the bandwidths for computing $\hat{f}_{\hat{\xi},(1)}(x)$ and $\hat{f}_{\hat{\xi},(2)}(x)$ must be the same as that for the original estimator $\hat{f}_{\hat{\xi}}(x)$ to reduce the biases.

The next theorem formally shows that the HPJ bias-corrected estimator $\hat{f}_{\hat{\xi}}^{H}(x)$ does not suffer from incidental parameter bias and second-order bias, and does not alter the asymptotic variance of the estimator.

Note that HPJ bias correction does not weaken the relative magnitude condition of $N$, $T$, and $h$ for asymptotic normality in Theorem 1; that is, $N/(T^{3}h^{5})\to 0$. This is because HPJ bias correction cannot eliminate higher-order nonlinearity biases. This result is in stark contrast to the existing literature where bias correction typically weakens the condition on the relative magnitudes of $N$ and $T$ (see, e.g., Dhaene and Jochmans, 2015).

This section considers CI estimation and the selection of optimal bandwidth for density estimation.