key: cord-0807000-yksmftps
authors: Li, Yinghua; Qin, Yongsong
title: Empirical likelihood for spatial dynamic panel data models
date: 2021-09-29
journal: J Korean Stat Soc
DOI: 10.1007/s42952-021-00150-4
sha: 7f4d4f99867219a9318cbb6728a9484c9bb41fb5
doc_id: 807000
cord_uid: yksmftps

Spatial dynamic panel data (SDPD) models have received great attention in economics in recent 10 years. Existing approaches for the estimation and test of SDPD models are quasi-maximum likelihood (QML) approach and generalized method of moments (GMM). In this article, we introduce the empirical likelihood (EL) method to the statistical inference for SDPD models. The EL ratio statistics are constructed for the parameters of spatial dynamic panel data models. It is shown that the limiting distributions of the empirical likelihood ratio statistics are chi-squared distributions, which are used to construct confidence regions for the parameters of the models. Simulation results show that the EL based confidence regions outperform the normal approximation based confidence regions.

Real data are often observed at different locations and times, which are called as spatial panel data (SPD) . Examples are economic growth rates of major cities in China over last 40 years, monthly unemployment rates of states in USA in the last decade and daily infection rates of COVID-19 in major cites in Hubei province in China over 3 months since December 31, 2019. These data may be modelled by SPD models. The research to various SPD models can be found in Anselin (1988) , Elhorst (2003) , Baltagi et al. (2003) , Baltagi and Li (2006) , Chen and Conley (2001) , Pesaran (2004) , Kapoor et al. (2007) , Baltagi et al. (2007) , Lee and Yu (2010a) , Mutl and Pfaffermayr (2011) , Parent and LeSage (2011) and 

In this article, we suppose that there are n individual units and T time periods and the sampling data satisfy the following SDPD model with spatial error:

where y t = (y 1t , … , y nt ) � is an n-dimensional column vector of observed dependent variables, (| | < 1) characterizes the dynamic effect, x t = (x 1t , … , x nt ) � is an n × p matrix of time-varying exogenous variables, z = (z 1 , … , z n ) � is an n × q matrix of time-invariant exogenous variables, and and are p × 1 and q × 1 regression coefficients, respectively. The disturbance vector t = ( 1t , … , nt ) � is an n × 1 vector of errors. The parameter is a spatial autoregressive coefficient and W n is an n × n spatial weighting matrix of constants, t = ( 1t , … , nt ) � is an n × 1 column vector, and { it } are i.i.d. across t and i with zero mean and variance 2 . The spatial weighting matrix is also called contiguity matrix, which is determined by the spatial dependence of n spatial units. There are many ways to define W n (e.g. pages 17-19 in Anselin 1988) . Let W ij be the (i, j) element of W n . Commonly used W n includes Rook contiguity, Bishop contiguity and Queen contiguity as follows. Rook contiguity: define W ij = 1 if the units i and j share a common side and W ij = 0 , otherwise. Bishop contiguity: define W ij = 1 if the units i and j share a common vertex and W ij = 0 , otherwise. Queen contiguity: define W ij = 1 if the units i and j share a common side or vertex and W ij = 0 , otherwise. The choice of W n is important. Our results hold true for all these commonly used W n .

The models (2.1)-(2.3) in Su and Yang (2015) are as follows:

where = ( 1 , … , n ) � represent the unobservable individual or space-specific effects and other notations are the same as in model (1) and (2). Compared to models (2.1)-(2.3) in Su and Yang (2015) , the only difference is that there is no spacespecific effects in model (1) and (2). Our initial investigation shows that the EL method for SDPD models with space-specific effects may need an adjusted EL method. Further research is needed and left for our future work. We develop the EL method for the SDPD model when y 0 is exogenous. In this case, we can treat y 0 as a fixed constant vector as it contains no information about the model parameters. For convenience, we use k to denote a k × 1 vector of ones, k to denote a k × 1 vector of zeros, and (1) and (2) can be written in a matrix form as:

(2) t = W n t + t , t = 1, 2, … , T,

with or with where ∼ (0, 2 Ω) , with Let = ( � , � , ) � and = ( � , 2 , ) � . We adopt the QML method to derive the estimating equations for the EL method. Under the assumption of normality (which is only used at this moment), based on (3) and (4), the log-likelihood function (ignoring constants) is

Letting above derivatives be 0, we obtain the following estimating equations of the QML method:

− nT 2 + � Ω −1 = 0, Substituting (4) into (7)-(9), we have Noting that X = (X, Z, Y −1 ) and Y −1 contains X, z and , we need to separate out from X . To this end, denote l = (0, Su and Yang (2015) to obtain that Let X 1 = (X, Z) and � X 2 = A x X + (l ⊗ I n )z + Y 0 . Then (10) can be decomposed into For convenience, let e = , i.e. Then (11)-(14) can be rewritten as

Observing that the above estimating equations include the quadratic forms of e, to use the EL method, we need to change the quadratic forms into the linear forms of a well behaved random variables. To this end, we let

. Use h ij,k , a i,1 and a i,2 to denote the (i, j) element of the matrix H k ( k = 1, 2 ), the i-th column of the matrix � X � 1 (I T ⊗ B � ) and the i-th element of the vector � X � 2 (I T ⊗ B � ) , respectively, and adapt the convention that any sum with an upper index of less than one is zero. To deal with the quadratic form in (17) and (19), we follow Kelejian and Prucha (2001) to introduce a martingale difference array. Define the -fields:

Based on (16)-(21), we propose the following EL ratio statistic for ∈ R p+q+3 :

where e i is the ith component of (

. Following Owen (1990) , one can show that where ̃( ) ∈ R p+q+3 is the solution of the following equation:

to denote the vector formed by the diagonal elements of a matrix A, ||a|| to denote the L 2 -norm of a vector a, and min (H) and max (H) to denote the minimum and maximum eigenvalues of a matrix H, respectively. To obtain the asymptotic distribution of ( ) , we need following assumptions.

A1. (1) jt are mutually independent, and they are independent of x ks and z k for all j, k, t, s;

(2) All elements in (x it , z i ) have 4 + 1 moments for some 1 > 0. A2. (1) { it , t = 1, … , T, i = 1, … , n} are independent and identically distributed for all i and t with mean 0, variance 2 > 0 and E| it | 4+ 1 < ∞ for some 1 > 0.

(2) {x it , t = … , −1, 0, 1, …} and {z i } are strictly exogenous and independent across i.

(3) | | < 1. A3. Let W n and {B −1 } be as described above. They satisfy the following conditions:

(1) The row and column sums of W n are uniformly bounded in absolute value;

(2) {B −1 } are uniformly bounded in either row or column sums, uniformly in in a compact parameter space Λ , and

Remark 1 Conditions A1-A3 are common assumptions for spatial models, which are used in Su and Yang (2015) , and the analog of 0 < c 1 ≤ min (nT) −1 Σ p+q+3 is employed in the assumption of Theorem 1 in Kelejian and Prucha (2001) .

We now state the main results.

where 2 p+q+3 is a chi-squared distributed random variable with p + q + 3 degrees of freedom.

Let z (p + q + 3) satisfy P( 2 p+q+3 ≤ z (p + q + 3)) = for 0 < < 1 . It follows from Theorem 1 that an EL based confidence region for with asymptotically correct coverage probability can be constructed as

Recall that = ( � , � , ) � and = ( � , 2 ,

It can be shown that According to Su and Yang (2015) , the QMLE ̂ of satisfies:

where Σ = lim n→∞ 1 nT E[Σ n ( )] and Σ n ( ) = 2 �L ( ). Based on the above asymptotic result, we can obtain the NA based confidence region for . However, we note that the NA method depends on the availability of a consistent estimator of the asymptotic covariance matrix in practical applications, while the EL method does not. This can save the implementation time for the EL method and the EL method outperforms the NA method.

We conducted a small simulation study to compare the finite sample performances of the confidence regions based on EL and NA methods with confidence level = 0.95 , and report the proportion of ( ) ≤ z 0.95 (p + q + 3) and

In the simulations, we used the following two models:

(1) Model 1:

, where x t were generated from N(0, 4), alternatively, x t can be randomly generated in a similar fashion as in Hsiao et al. (2002) , and the elements of z were randomly generated from Bernoulli(0.5). We selected = 1 , = 1 , 2 = 1 and ( , ) were taken as (−0.8, −0.7) , (−0.2, −0.1) , (0.2, 0.1), (0.8, 0.7), (−0.8, 0.7) and (0.2, −0.1) respectively, and ′ it s were i.i.d. from N(0, 1), t(5) and 2 4 − 4 , respectively;

(2) Model 2:

t is an n × 2 matrix, where x (1) t were randomly generated from N(0, 1) and x (2) t were randomly generated from N(0, 4). Moreover, z = z (1) , z (2) is an n × 2 matrix, the elements of z (1) were randomly generated from Bernoulli(0.3) and the elements of z (2) were randomly generated from Bernoulli(0.6). We selected (5) , respectively.

The results of simulations under model 1 are reported in Tables 1, 2 and 3, and the results of simulations under model 2 are reported in Tables 4, 5, 6, 7 and 8.

For the contiguity weight matrix W n = (W ij ) , we took W ij = 1 if spatial units i and j are neighbours by queen contiguity rule (namely, they share common border or vertex), W ij = 0 otherwise (Anselin 1988, P.18) . We considered five ideal cases of spatial units: n = m × m regular grid with m = 7, 10, 13, 16, 20 , denoting W n as grid 49 , grid 100 , grid 169 , grid 256 and grid 400 , respectively. A transformation is often used in applications to convert the matrix W n to the unity of row-sums. We used the standardized version of W n in our simulations, namely W ij was replaced by W ij ∕ ∑ n j=1 W ij . Simulation results under model 1 show that the confidence regions based on NA behave well with coverage probabilities being very close to the nominal level 0.95 when the error term i is normally distributed and n is large, but not well in other cases. The coverage probabilities of the confidence regions based on NA fall to the We can see, from Tables 1, 2, 3, 4, 5, 6, 7 and 8, that the coverage probabilities of confidence regions based on EL method converge to the nominal level 0.95 as the number of spatial units n is large enough, whether the error term i is normally 

In order to illustrate the proposed method in Sect. 2, we conducted a real data analysis. The data come from 288 prefecture-level cities in China, collected from National Bureau of Statistics of China and Anjuke. There were three variables: the logarithm of housing price per square meter ( y t ), the logarithm of income per household ( x t ) and the urbanization rate (z) from the years of 2010 to 2017. In order to ensure the stability and eliminate the influence of dimension, we first did difference and standardization on the above data, and then considered fitting the data via the following model: y t = y t−1 + x t + z + t , t = W n t + t , t = 1, 2, … , 8 , where n = 288 and the spatial weighting matrix W n was selected by the method in Sect. 3. We separately employed the EL method in Sect. 2 and the NA method in Sect. 3 to obtain the confidence intervals for parameters , , , and 2 with confidence level 0.95, which were shown in Table 9 . Table 9 shows that the estimator of the spatial parameter is = 0.3743 , and 0 is not in its confidence interval, which implies that there exists a spatial relationship among the disturbances. The results also show that the lengths of the EL based intervals are uniformly shorter than those of the NA based intervals, which implies that the EL based method performs better than the NA based method for the real data.

In the proof of the main results, we need to use Theorem 1 in Kelejian and Prucha (2001) . We now state this result. Let where ni are real valued random variables, and the a nij and b ni denote the real valued coefficients of the linear-quadratic form. We need the following assumptions in Lemma 1.

(C1) { ni , 1 ≤ i ≤ n} are independent random variables with mean 0 and sup 1≤i≤n,n≥1 E| ni | 4+ 1 < ∞ for some 1 > 0; (C2) For all 1 ≤ i, j ≤ n, n ≥ 1, a nij = a nji , sup 1≤j≤n,n≥1 ∑ n i=1 �a nij � < ∞ , and sup n≥1 n −1 ∑ n i=1 �b ni � 2+ 2 < ∞ for some 2 > 0.

Given above assumptions (C1) and (C2), the mean and variance of Q n are given as (e.g. Kelejian & Prucha, 2001) with 2 ni = E( 2 ni ) and (s) ni = E( s ni ) for s = 3, 4.

Lemma 1 Suppose that Assumptions C1 and C2 hold true and n −1 2 Q ≥ c for some

It is easy to show (e.g. Su & Yang, 2015) that, n −1X� 1 Ω −1X 1 , n −1X� 2 Ω −1X 2 , and n −1X� 1 Ω −1X 2 converge in probability to their expectations. We have and Combine (31)-(36), we have

where Σ p+q+3 is given in (24). From Condition A4, one can see that 

Next we will show that: (1) and (2) To show (1) and (2), it is sufficient to show that (nT) −2 E(S 2 n1 ) → 0 and (nT) −2 E(S 2 n2 ) → 0 , respectively. Obviously,

It follows that

By Conditions A1-A3, we have and (45)

|l 2 + l 3 h ii,1 + l 4 h ii,2 | 4 ≤ Cn −1 → 0,

Similarly, we can prove that From (45)-(48), we have (nT) −2 E(S 2 n1 ) → 0 . Furthermore, Thus,

Note that and (49)

From (53)-(57), we have Further, using (58) and Markov inequality, we obtain ∑ nT i=1 �� i ( )�� 3 = O p (nT 2 ) . Thus (29) is proved.

Proof of Theorem 1 Using Lemma 3 and following the proof of Theorem 1 in Qin (2021) , one can easily show that Theorem 1 holds true.

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This work was partially supported by the National Natural Science Foundation of China (12061017, 12161009). The authors are thankful to the referees for constructive suggestions.

By Conditions A1-A3 and Lemma 2, we have In addition, by Lemma B.2. in Su and Yang (2015) , A � (I T ⊗ B � ) and (I T ⊗ (BAB � )) are uniformly bounded in both row and column sums, it follows that.The conditional expectation and variance given X, Z are denoted as E * and Var * , respectively. Then from (15) and note that E( ) = 0 , we know that the variance of Q n is

Similarly,