Fiscal fragmentation and crime control: Is there an efficiency-equity tradeoff?

Abstract

This article investigates the effects of fiscal fragmentation on aggregate crime rates and the spatial disparities in crime rates among counties in a metropolitan area. We begin by developing a model of local government provision of public safety. The model predicts that fiscal fragmentation creates an efficiency-equity tradeoff. To investigate this tradeoff, we estimate a variety of empirical models using county-level panel data drawn from a sample of metropolitan areas in the USA for census years 1990, 2000, and 2010. Our findings suggest that fiscal fragmentation increases efficiency in the provision of public safety; that is, fiscal fragmentation has a negative effect on aggregate crime rates in metropolitan areas. We also find that fiscal fragmentation increases disparities in crime rates among counties in a metropolitan area. In other words, fiscal fragmentation has a negative effect on interpersonal equity in the provision of public safety. We further explore the underlying mechanisms of the efficiency-equity tradeoff in a Spatial-Autoregressive Durbin model with multiplicative spatial interaction terms. Since conventional estimation methods are not suitable for the task at hand, we derive an innovative Maximum Likelihood Estimator for our empirical model. As predicted, we find evidence of both interjurisdictional spillover effects and Tiebout-sorting effects due to fiscal fragmentation.

Appendices

Appendix 1

In this section, we model the households’ choice of the level of public safety, using a median-voter framework. Assume households in jurisdiction j have identical Cobb–Douglas utility functions and differ in household income. Household i in jurisdiction j seeks to maximize utility by choosing the level of public safety, denoted by PS_j, and the amount of a private good x_i. We assume that public safety is a local public good. The household’s maximization problem is given by the following expressions:

$$\mathop {\max }\limits_{x,PS} u\left( {x_{i} ,PS_{j} } \right) = \alpha \log \left( {x_{i} } \right) + \beta \log \left( {PS_{j} } \right)$$

(8)

$${\rm s.t.}\quad P_{0} x_{i} + P_{j} PS_{j} = I_{i}$$

(9)

where P_j is the tax-price of public safety in jurisdiction j; P₀ is the price of a homogeneous private good; and ${I}_{i}$ is the income of household i. The necessary first-order conditions for a local maximum are given as follows:

$$u_{{x_{i} }} = P_{0} \lambda_{i}$$

(10)

$$u_{{PS_{j} }} = P_{j} \lambda_{i}$$

(11)

where λ_i is the Lagrange multiplier associated with the budget constraint of household i in jurisdiction j. Solving for the demand for the private good and public safety, we obtain the following expressions:

$$x_{i}^{D} = \frac{{\alpha I_{i} }}{{P_{0} \left( {\alpha + \beta } \right)}}$$

(12)

$$PS_{ij}^{D} = \frac{{\beta I_{i} }}{{P_{j} \left( {\alpha + \beta } \right)}}$$

(13)

The preference for public safety is increasing with respect to I_i and is single-peaked, which assures us of an equilibrium. Following the median-voter theorem, the demand for public safety in jurisdiction j, $PS_{j}^{D}$ is decided by the median-income household in jurisdiction j. Accordingly, the median-income household’s demand for public safety is given by the following expression:

$$PS_{j}^{D} = D\left( {P_{j} ,P_{0} ,IM_{j} } \right)$$

(14)

where again, P_j is the tax-price of public safety in jurisdiction j; P₀ is the price of a homogeneous private good; and IM_j is the income of the median-income household in jurisdiction j. Every household in jurisdiction j consumes the same amount of public safety. Moreover, public safety obeys the law of downward-sloping demand and is a normal good.

Appendix 2

See Tables 7, 8, 9, 10, 11, 12 and 13.

Table 7 First-stage estimates of the 2SLS regressions

Table 8 OLS estimates of the effect of fiscal fragmentation on crime rates by type of crime (metropolitan areas)

Table 9 2SLS estimates of the effect of fiscal fragmentation on crime rates by type of crime (metropolitan areas)

Table 10 OLS estimates of the effect of fiscal fragmentation on intraregional disparities in crime rates by type of crime (metropolitan areas)

Table 11 2SLS estimates of the effect of fiscal fragmentation on intraregional disparities in crime rates by type of crime (metropolitan areas)

Table 12 2SLS estimates of the effect of fiscal fragmentation on crime rates by type of crime (metropolitan areas, alternative IVs)

Table 13 2SLS estimates of the effect of fiscal fragmentation on intraregional disparities in crime rates by type of crime (metropolitan areas, alternative IVs)

Appendix 3

Without loss of generality, specification (7) can be written in the following matrix form:

$$\begin{aligned} & Y_{nt} = \sum\limits_{j = 1}^{p} {\lambda_{j,0} U_{jn} W_{n} Y_{nt} + X_{nt} \beta_{0} + c_{n} } + \alpha_{t} \iota_{n} + v_{nt} , \\ & v_{nt} = \rho_{0} M_{n} v_{nt} + \varepsilon_{nt} ,\quad t = 1, \ldots T, \\ \end{aligned}$$

(15)

where $Y_{nt}$ is an $n \times 1$ vector of dependent variable, $U_{jn} = {\text{diag}}\left\{ {u_{j,1} , \ldots ,u_{j,n} } \right\}$ $\left( {j = 1, \ldots p} \right)$ is a series of $n \times n$ matrices of non-stochastic exogenous variables, $U_{1n} = I_{n}$ is the identity matrix, $X_{nt}$ is an $n \times k$ matrix of non-stochastic exogenous variables,²⁵$c_{n} = \left( {c_{1,n} , \ldots ,c_{n,n} } \right)^{\prime }$ is an $n \times 1$ vector of fixed effects, $\alpha_{t}$ is the fixed time effect, $\iota_{n}$ is a $n \times 1$ vector of ones, $W_{n}$ and $M_{n}$ are $n \times n$ spatial weights matrices of known constants, and $\varepsilon_{nt}$ is an $n \times 1$ vector of the disturbances where $\varepsilon_{it}$ is i.i.d. across $i$ and $t$ with zero mean and variance $\sigma_{0}^{2}$. Thus, in model (15), we also allow spatial errors. For instance, in specification (7), $Y_{nt} = \left( {CR_{11t} , \ldots ,CR_{{1J_{1} t}} , \ldots ,CR_{I1t} , \ldots ,CR_{{IJ_{I} t}} } \right)^{{\prime }}$, $U_{jn} = {\text{diag}}\left\{ {FF_{1} , \ldots ,FF_{1} , \ldots ,FF_{I} , \ldots ,FF_{I} } \right\}$, $X_{nt} = \left( {X_{11t} , \ldots ,X_{{1J_{1} t}} , \ldots ,X_{I1t} , \ldots ,X_{{IJ_{I} t}} } \right)^{{\prime }}$. As in our empirical study, we only have one multiplicative interactive term in each specification, so $p = 2$. In general, model (15) allows $p$ to be some finite number that satisfies the identification condition, which means model (15) may contain several interactive terms.

Model (15) is related in some sense to the spatial autoregressive model (SAR) model with high-order spatial lags which can characterize spatial interdependence based on different types of relations (e.g., geographic distance, social relation) among cross-sectional units. Such different types of spatial interdependence are captured by the different spatial weight matrices. If we denote $W_{jn} = U_{jn} W_{n}$, then model (15) can be viewed as a $p$-order SAR model with $1$-order SAR disturbance (for short,$SARAR\left( {p,1} \right)$, see Lee and Liu (2010)). However, in essence, model (15) is to capture the varying spatial effects depending on a (possibly exogenous variable $U_{jn}$), while the type of spatial interdependence is assumed to be fixed and exogenously given by the spatial weight matrix. Note that the varying spatial effects are modeled to be linear in $U_{jn}$, one can extend model (15) to a functional coefficients spatial model, where the parameter $\lambda$ that captures the spatial effects is some unspecified smooth function of $U_{jn}$, to specify the possibly nonlinear varying spatial effects.

A central problem in estimating panel models with fixed effect is how to eliminate it, for both the fixed individual effects and fixed time effects. However, from a methodological point of view, the asymptotics are of interest only when both $n$ and $T$ tend to infinity. When $T$ tends to infinity, the fixed-time effects may cause the incidental parameter problem (Neyman and Scott, 1948) in addition to the individual fixed effects if we apply the direct maximum likelihood approach to jointly estimate the common parameters and fixed effects. When $T$ is finite, the fixed-time effects can be regarded as a finite number of additional parameters similar to the role of $\beta$, so the fixed time effects are of no concern. As in our model, $T$ is finite, so we can simply omit the fixed-time effects to avoid the complication of eliminating them. If one is interested in the case when $T$ is large, the transformation approach in Lee and Yu (2010) can also be applied to eliminate the fixed time effects.

With a bit abuse of notation, we denote the time operator $J_{T} = \left( {I_{T} - \frac{1}{T}\iota_{T} \iota_{T}^{{\prime }} } \right)$. To eliminate the fixed individual effects, the transformed model (15) consists of

$$\begin{aligned} & Y_{nt} = \sum\limits_{j = 1}^{p} {\lambda_{j,0} U_{jn} W_{n} Y_{nt} + X_{nt} \beta_{0} + } \tilde{v}_{nt} , \\ & \tilde{v}_{nt} = \rho_{0} M_{n} \tilde{v}_{nt} + \varepsilon_{nt} . \\ \end{aligned}$$

(16)

Note that, the resulting disturbances $\varepsilon_{nt}$ would be linearly dependent over the time dimension. Therefore, without creating linear dependence in the resulting disturbances, a corresponding transformation can be based on the orthonormal eigenvector matrix of $J_{T}$, as in Lee and Yu (2010). The orthogonal transformation in Lee and Yu (2010) includes the Helmert transformation as a special case to eliminate the fixed individual effects. Let $\left[ {F_{T,T - 1} ,\frac{1}{\sqrt T }\iota_{T} } \right]$ be the orthonormal eigenvector matrix of $J_{T}$, where $F_{T,T - 1}$ is the $T \times \left( {T - 1} \right)$ submatrix²⁶ corresponding to the eigenvalues of one. For any $n \times T$ matrix $\left[ {A_{n1} , \ldots A_{nT} } \right]$, define the transformed $n \times \left( {T - 1} \right)$ matrix $\left[ {A_{n1}^{ * } , \ldots A_{{n\left( {T - 1} \right)}}^{*} } \right] = \left[ {A_{n1} , \ldots A_{nT} } \right]F_{T,T - 1}$. Similarly, $X_{nt}^{*} = \left[ {X_{nt,1}^{ * } , \ldots X_{nt,k}^{ * } } \right]$. Note that, $U_{jn}$, which is included in $X_{nt}$, has also been eliminated. Therefore, model (15) can be rewritten as

$$\begin{aligned} & Y_{nt}^{*} = \sum\limits_{j = 1}^{p} {\lambda_{j,0} U_{jn} W_{n} Y_{nt}^{*} + X_{nt}^{*} \beta_{0} } + v_{nt}^{*} , \\ & v_{nt}^{*} = \rho_{0} M_{n} v_{nt}^{*} + \varepsilon_{nt}^{ * } ,\quad t = 1, \ldots ,T - 1, \\ \end{aligned}$$

(17)

Because $\left( {\varepsilon_{n1}^{*\prime} , \ldots ,\varepsilon_{{n\left( {T - 1} \right)}}^{*\prime} } \right)^{\prime } = \left( {F_{T,T - 1}^{{\prime }} \otimes I_{n} } \right)\left( {\varepsilon_{n1}^{{\prime }} , \ldots ,\varepsilon_{nT}^{{\prime }} } \right)^{{\prime }}$ and $\varepsilon_{nt}$ are i.i.d.,

$$E\left( {\varepsilon_{n1}^{*\prime} , \ldots ,\varepsilon_{{n\left( {T - 1} \right)}}^{*\prime} } \right)^{{\prime }} \left( {\varepsilon_{n1}^{*\prime} , \ldots ,\varepsilon_{{n\left( {T - 1} \right)}}^{*\prime} } \right) = \sigma_{0}^{2} \left( {F_{T,T - 1}^{{\prime }} \otimes I_{n} } \right)\left( {F_{T,T - 1} \otimes I_{n} } \right) = \sigma_{0}^{2} I_{{n\left( {T - 1} \right)}} .$$

Hence, $\varepsilon_{it}^{ * }$ is uncorrelated for all $i$ and $t$ (and independent under normality).

If the disturbances were normally distributed, the log-likelihood function of (17) is

$$\begin{aligned} \ln L_{n,T} \left( \theta \right) & = - \frac{{n\left( {T - 1} \right)}}{2}\ln \left( {2\pi \sigma^{2} } \right) + \left( {T - 1} \right)\left[ {\ln \left| {S_{n} \left( \lambda \right)} \right| + \ln \left| {R_{n} \left( \rho \right)} \right|} \right] \\ & \quad - \frac{1}{{2\sigma^{2} }}\sum\limits_{t = 1}^{T - 1} {\varepsilon_{nt}^{ * } \left( \varsigma \right)^{{\prime }} \varepsilon_{nt}^{ * } \left( \varsigma \right)} , \\ \end{aligned}$$

(18)

where $\theta = \left( {\beta^{{\prime }} ,\lambda^{{\prime }} ,\rho ,\sigma^{2} } \right)^{{\prime }}$, $\lambda = \left( {\lambda_{1} , \ldots ,\lambda_{p} } \right)^{\prime }$, $\varepsilon_{nt}^{ * } \left( \varsigma \right) = R_{n} \left( \rho \right)\left[ {S_{n} \left( \lambda \right)Y_{nt}^{ * } - X_{nt}^{ * } \beta } \right]$ and $S_{n} \left( \lambda \right) = I_{n} - \sum\limits_{j = 1}^{p} {\lambda_{j} U_{jn} W_{n} }$. To guarantee that the log-likelihood function is well defined, we only consider the parameter space of $\lambda$ and $\rho$ such that $\left| {S_{n} \left( \lambda \right)} \right| > 0$ and $\left| {R_{n} \left( \rho \right)} \right| > 0$. Let $\left\| \cdot \right\|$ be any matrix norm, one has $\left\| {\sum\limits_{j = 1}^{p} {\lambda_{j} U_{jn} W_{n} } } \right\| \le \left( {\sum\limits_{j = 1}^{p} {\left| {\lambda_{j} } \right|} } \right) \cdot \max_{j = 1, \ldots p} \left\| {U_{jn} W_{n} } \right\|$, and the parameter space may be taken to be $\sum\limits_{j = 1}^{p} {\left| {\lambda_{j} } \right|} \le 1/\max_{j = 1, \ldots p} \left\| {U_{jn} W_{n} } \right\|$. This limits the spatial dependence among the units to a tractable degree (Kelejian and Prucha, 1999) and rules out the unit root case (in time series as a special case). It is also a sufficient condition for $S_{n} \left( \lambda \right)^{ - 1}$ to be uniformly bounded in row and column sums in absolute value, because $S_{n} \left( \lambda \right)^{ - 1} = I_{n} + \sum\limits_{j = 1}^{p} {\lambda_{j} U_{jn} W_{n} } + \left( {\sum\limits_{j = 1}^{p} {\lambda_{j} U_{jn} W_{n} } } \right)^{2} + \cdots$. When $M_{n}$ is row-normalized, a possible parameter space for $\rho$ can be one satisfying $\left| \rho \right| < 1$.

For any $n$-dimensional column vectors $p_{nt}$ and $q_{nt}$, as $\begin{aligned} \sum\limits_{t = 1}^{T - 1} {p_{nt}^{*\prime} q_{nt}^{*} } & = \left( {p_{n1}^{{\prime }} , \ldots ,p_{nT}^{{\prime }} } \right)\left( {F_{T,T - 1} \otimes I_{n} } \right)\left( {F^{\prime}_{T,T - 1} \otimes I_{n} } \right)\left( {q_{n1}^{{\prime }} , \ldots ,q_{nT}^{{\prime }} } \right)^{{\prime }} \\ & = \left( {p_{n1}^{{\prime }} , \ldots ,p_{nT}^{{\prime }} } \right)\left( {J_{T} \otimes I_{n} } \right)\left( {q_{n1}^{{\prime }} , \ldots ,q_{nT}^{{\prime }} } \right)^{{\prime }} = \sum\limits_{t = 1}^{T - 1} {p_{nt}^{{\prime }} q_{nt} } \\ \end{aligned}$.

by using $\left( {p_{n1} , \ldots p_{nT} } \right) = \left( {p_{n1} , \ldots p_{nT} } \right)J_{T}$, the log-likelihood function can be rewritten as

$$\begin{aligned} \ln L_{n,T} \left( \theta \right) & = - \frac{{n\left( {T - 1} \right)}}{2}\ln \left( {2\pi \sigma^{2} } \right) + \left( {T - 1} \right)\left[ {\ln \left| {S_{n} \left( \lambda \right)} \right| + \ln \left| {R_{n} \left( \rho \right)} \right|} \right] \\ &\quad - \frac{1}{{2\sigma^{2} }}\sum\limits_{t = 1}^{T - 1} {\varepsilon_{nt} \left( \varsigma \right)^{{\prime }} \varepsilon_{nt} \left( \varsigma \right)} , \\ \end{aligned}$$

(19)

where $\varepsilon_{nt} \left( \varsigma \right) = R_{n} \left( \rho \right)\left[ {S_{n} \left( \lambda \right)Y_{nt} - X_{nt} \beta } \right]$. For (15), the corresponding estimates are

$$\hat{\beta }_{nT} \left( {\lambda ,\rho } \right) = \left[ {\sum\limits_{t = 1}^{T} {X_{nt}^{{\prime }} R_{n} \left( \rho \right)^{{\prime }} X_{nt} } } \right]^{ - 1} \left[ {\sum\limits_{t = 1}^{T} {X_{nt}^{{\prime }} R_{n} \left( \rho \right)^{{\prime }} R_{n} \left( \rho \right)S_{n} \left( \lambda \right)Y_{nt} } } \right]$$

(20)

$$\begin{aligned} \hat{\sigma }_{nT}^{2} \left( {\lambda ,\rho } \right) & = \frac{1}{{n\left( {T - 1} \right)}}\sum\limits_{t = 1}^{T} {\left[ {S_{n} \left( \lambda \right)Y_{nt} - X_{nt} \hat{\beta }_{nT} \left( {\lambda ,\rho } \right)} \right]}^{{\prime }} R_{n} \left( \rho \right)^{{\prime }} R_{n} \left( \rho \right) \\ & \left[ {S_{n} \left( \lambda \right)Y_{nt} - X_{nt} \hat{\beta }_{nT} \left( {\lambda ,\rho } \right)} \right]. \\ \end{aligned}$$

(21)

and the concentrated log-likelihood function of $\left( {\lambda ,\rho } \right)$ is

$$\begin{aligned} \ln L_{n,T} \left( {\lambda ,\rho } \right) & = - \frac{{n\left( {T - 1} \right)}}{2}\left( {\ln \left( {2\pi } \right) + 1} \right) - \frac{{n\left( {T - 1} \right)}}{2}\ln \hat{\sigma }_{nT}^{2} \left( {\lambda ,\rho } \right) \\ &\quad + \left( {T - 1} \right)\left[ {\ln \left| {S_{n} \left( \lambda \right)} \right| + \ln \left| {R_{n} \left( \rho \right)} \right|} \right], \\ \end{aligned}$$

(22)

Theorem 1

Under the assumptions presented below, $\theta_{0}$ is identified, and the quasi-maximum likelihood estimator $\hat{\theta }_{nT}$ based on (15) is consistent, $\hat{\theta }_{nT} - \theta_{0} \mathop{\longrightarrow}\limits^{p}0$.

The set of assumptions are similar to Assumptions 1–8 in Lee and Yu (2010):

Assumption 1

$W_{n}$ and $M_{n}$ are non-stochastic spatial weights matrices with zero diagonals.

Assumption 2

The disturbances $\left\{ {\varepsilon_{it} } \right\}$; $i = 1, \ldots ,n$ and $t = 1, \ldots ,T$, are i.i.d. across i and t with zero mean, variance $\sigma_{0}^{2}$ and $E|\varepsilon_{it} |^{4 + \eta } < \infty$ for some $\eta > 0$.

Assumption 3

$S_{n} \left( \lambda \right)$ and $R_{n} \left( \rho \right)$ are invertible for all $\lambda \in \Lambda$ and $\rho \in {\rm P}$, where $\Lambda$ and ${\rm P}$ are compact intervals. Furthermore, $\lambda_{0}$ is in the interior of $\Lambda$, and $\rho_{0}$ is in the interior of ${\rm P}$.

Assumption 4

n is large, where T can be finite or large.

Assumption 5

The elements of $X_{nt}$ and $U_{jn}$ are non-stochastic and bounded, uniformly in j, n, and t. Also, under the asymptotic setting in Assumption 4, the limit of $\frac{1}{nT}\sum\limits_{t = 1}^{T} {\tilde{X}_{nt}^{\rm T} R_{n}^{\rm T} R_{n} \tilde{X}_{nt} }$ exists and is nonsingular.

Assumption 6

$W_{n}$ and $M_{n}$ are uniformly bounded in both row and column sums in absolute value (UB). Also $S_{n}^{ - 1} \left( \lambda \right)$ and $R_{n}^{ - 1} \left( \rho \right)$ are UB, uniformly in $\lambda \in \Lambda$ and $\rho \in {\rm P}$.

Assumption 7

Either (a) the limit of $\frac{1}{{n\left( {T - 1} \right)}}\sum\limits_{t = 1}^{T} {\left( {\tilde{X}_{nt} ,G_{n} \tilde{X}_{nt} \beta_{0} } \right)^{\rm T} R_{n}^{\rm T} \left( \rho \right)} R_{n} \left( \rho \right)\left( {\tilde{X}_{nt} ,G_{n} \tilde{X}_{nt} \beta_{0} } \right)$ is nonsingular for each possible $\rho \in {\rm P}$, and the limit of $\left( {\frac{1}{n}\ln |\sigma_{0}^{2} \left( {R_{n}^{ - 1} } \right)^{\rm T} \left( {R_{n}^{ - 1} } \right)| - \frac{1}{n}\ln |\sigma_{n}^{2} \left( \rho \right)\left( {R_{n}^{ - 1} \left( \rho \right)} \right)^{\rm T} \left( {R_{n}^{ - 1} \left( \rho \right)} \right)|} \right)$ is not zero for $\rho \ne \rho_{0}$; or (b) the limit of $\left( {\frac{1}{n}\ln |\sigma_{0}^{2} \left( {R_{n}^{ - 1} } \right)^{\rm T} \left( {S_{n}^{ - 1} } \right)^{\rm T} \left( {S_{n}^{ - 1} } \right)\left( {R_{n}^{ - 1} } \right)| - \frac{1}{n}\ln |\sigma_{n}^{2} \left( {\lambda ,\rho } \right)\left( {R_{n}^{ - 1} \left( \rho \right)} \right)^{\rm T} \left( {S_{n}^{ - 1} \left( \lambda \right)} \right)^{\rm T} \left( {S_{n}^{ - 1} \left( \lambda \right)} \right)\left( {R_{n}^{ - 1} \left( \rho \right)} \right)|} \right)$ is not zero for $\left( {\lambda ,\rho } \right) \ne \left( {\lambda_{0} ,\rho_{0} } \right)$, as n tends to infinity.

Denote $C_{jn} = \ddot{C}_{jn} - \frac{{tr\left( {\ddot{C}_{jn} } \right)}}{n}I_{n}$ and $D_{n} = M_{n} R_{n}^{ - 1} - \frac{{tr\left( {M_{n} R_{n}^{ - 1} } \right)}}{n}I_{n}$.

Assumption 8

The limit of $\frac{1}{{n^{2} }}\left[ {tr\left( {C_{jn}^{s} C_{jn}^{s} } \right)tr\left( {D_{n}^{s} D_{n}^{s} } \right) - tr^{2} \left( {C_{jn}^{s} D_{n}^{s} } \right)} \right]$ is strictly positive for any $j = 1,\ldots ,p$, as n tends to infinity.

Theorem 2

Under the same set of assumptions (the definitions of $\Sigma_{{\theta_{0} ,nT}}$ and $\Omega_{{\theta_{0} ,nT}}$ as well),

$$\sqrt {n(T - 1)} \left( {\hat{\theta }_{nT} - \theta_{0} } \right)\mathop{\longrightarrow}\limits^{d}N\left( {0,\lim \Sigma_{{\theta_{0} ,nT}}^{ - 1} \left( {\Sigma_{{\theta_{0} ,nT}} + \Omega_{{\theta_{0} ,nT}} } \right)\Sigma_{{\theta_{0} ,nT}}^{ - 1} } \right).$$

$$\begin{aligned} \sum_{{\theta_{0} ,nT}} & = - E\left( {\frac{1}{{n\left( {T - 1} \right)}}\frac{{\partial^{2} \ln L_{nT} \left( {\theta_{0} } \right)}}{{\partial \theta \partial \theta^{\rm T} }}} \right) \\ & = \frac{1}{{\sigma_{0}^{2} }}\left( {\begin{array}{*{20}l} {\frac{1}{{n\left( {T - 1} \right)}}\sum\nolimits_{t = 1}^{T} {\left( {\ddot{X}_{nt} ,\ddot{G}_{1n} \ddot{X}_{nt} \beta_{0} , \ldots ,\ddot{G}_{pn} \ddot{X}_{nt} \beta_{0} } \right)^{\rm T} \left( {\ddot{X}_{nt} ,\ddot{G}_{1n} \ddot{X}_{nt} \beta_{0} , \ldots ,\ddot{G}_{pn} \ddot{X}_{nt} \beta_{0} } \right)} } \hfill & * \hfill & * \hfill \\ {0_{{1 \times \left( {k + p} \right)}} } \hfill & 0 \hfill & * \hfill \\ {0_{{1 \times \left( {k + p} \right)}} } \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right) \\ & \quad { + }\left( {\begin{array}{*{20}l} {0_{k \times k} } \hfill & * \hfill & \cdots \hfill & * \hfill & * \hfill & * \hfill \\ {0_{1 \times k} } \hfill & {\frac{1}{n}tr\left( {\ddot{G}_{1n}^{s} \ddot{G}_{1n} } \right)} \hfill & \cdots \hfill & {\frac{1}{n}tr\left( {\ddot{G}_{1n}^{s} \ddot{G}_{pn} } \right)} \hfill & * \hfill & * \hfill \\ \vdots \hfill & \vdots \hfill & \cdots \hfill & \vdots \hfill & * \hfill & * \hfill \\ {0_{1 \times k} } \hfill & {\frac{1}{n}tr\left( {\ddot{G}_{pn}^{s} \ddot{G}_{1n} } \right)} \hfill & \cdots \hfill & {\frac{1}{n}tr\left( {\ddot{G}_{pn}^{s} \ddot{G}_{pn} } \right)} \hfill & * \hfill & * \hfill \\ {0_{1 \times k} } \hfill & {\frac{1}{n}tr\left( {\left( {M_{n} R_{n}^{ - 1} } \right)^{s} \ddot{G}_{1n} } \right)} \hfill & \cdots \hfill & {\frac{1}{n}tr\left( {\left( {M_{n} R_{n}^{ - 1} } \right)^{s} \ddot{G}_{pn} } \right)} \hfill & {\frac{1}{n}tr\left( {\left( {M_{n} R_{n}^{ - 1} } \right)^{s} \left( {M_{n} R_{n}^{ - 1} } \right)} \right)} \hfill & * \hfill \\ {0_{1 \times k} } \hfill & {\frac{1}{{\sigma_{0}^{2} n}}tr\left( {\ddot{G}_{1n} } \right)} \hfill & \cdots \hfill & {\frac{1}{{\sigma_{0}^{2} n}}tr\left( {\ddot{G}_{pn} } \right)} \hfill & {\frac{1}{{\sigma_{0}^{2} n}}tr\left( {M_{n} R_{n}^{ - 1} } \right)} \hfill & {\frac{1}{{2\sigma_{0}^{4} }}} \hfill \\ \end{array} } \right) \\ \end{aligned}$$

$$\begin{aligned} \sum_{{\theta_{0} ,nT}} & = \frac{{\left( {T - 1} \right)}}{T}\frac{{\left( {\mu_{4} - 3\sigma_{0}^{4} } \right)}}{{\sigma_{0}^{4} }} \\ & \times \left( {\begin{array}{*{20}c} {0_{k \times k} } & * & \cdots & * & * & * \\ {0_{1 \times k} } & {\frac{1}{n}\sum\nolimits_{i = 1}^{n} {\left( {\ddot{G}_{1n}^{2} } \right)}_{ii} } & \cdots & {\frac{1}{n}\sum\nolimits_{i = 1}^{n} {\left( {\ddot{G}_{pn} \ddot{G}_{1n} } \right)}_{ii} } & * & * \\ \vdots & \vdots & \cdots & \vdots & * & * \\ {0_{1 \times k} } & {\frac{1}{n}\sum\nolimits_{i = 1}^{n} {\left( {\ddot{G}_{1n} \ddot{G}_{pn} } \right)}_{ii} } & \cdots & {\frac{1}{n}tr\left( {\ddot{G}_{pn}^{2} } \right)_{ii} } & * & * \\ {0_{1 \times k} } & {\frac{1}{n}\sum\nolimits_{i = 1}^{n} {\left( {\ddot{G}_{1n,ii} } \right)} \left( {M_{n} R_{n}^{ - 1} } \right)_{ii} } & \cdots & {\frac{1}{n}\sum\nolimits_{i = 1}^{n} {\left( {\ddot{G}_{pn,ii} } \right)} \left( {M_{n} R_{n}^{ - 1} } \right)_{ii} } & {\frac{1}{n}\sum\nolimits_{i = 1}^{n} {\left( {M_{n} R_{n}^{ - 1} } \right)}_{ii}^{2} } & * \\ {0_{1 \times k} } & {\frac{1}{{2\sigma_{0}^{2} n}}tr\left( {\ddot{G}_{1n} } \right)} & \cdots & {\frac{1}{{2\sigma_{0}^{2} n}}tr\left( {\ddot{G}_{pn} } \right)} & {\frac{1}{{2\sigma_{0}^{2} n}}tr\left( {M_{n} R_{n}^{ - 1} } \right)} & {\frac{1}{{4\sigma_{0}^{4} }}} \\ \end{array} } \right) \\ \end{aligned}$$