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空间计量的多面检验: 以 cigar 为例

 计量经济圈 2023-05-05 发布于浙江

谁把极地铺进太虚, 把大地挂在太空, 又时时护理与查验?

Elhorst J P (2012) Matlab software for spatial panels. International Regional Science Review. doi:10.1177/0160017612452429 一文, 以 cigar 数据集演示估计空间面板模型的程序, 包括模型选择与检验以及偏误修正方法, 以提供一个选择性框架来决定哪种模型能更好地描述数据.

4种空间交互效应和4类检验

近年来空间计量经济学文献越来越热衷于基于空间面板的模型设置以及计量经济学关系的估计. 这种兴趣可以解释如下, 和截面数据单一方程相比, 面板数据为研究者提供了扩展模型的可能性, 虽然截面数据一直以来是空间计量经济学文献最初的关注点.

当今的一个 (空间) 计量经济学研究者对模型有更多的选择.

  • 首先, 他应该问自己, 应该考虑下列哪种空间交互效应:

    (1) 空间滞后的因变量;

    (2) 空间滞后的自变量;

    (3) 空间自相关的误差项;

    (4) 以上情况的组合.

  • 其次, 他应该问自己, 是否考虑空间特定效应和/或时间特定效应, 是固定效应还是随机效应. 已经开发并可以获得不同统计检验的两个程序可以帮助研究者在不同的选择中进行挑选.

    (A) 第一个程序提供 (稳健) LM 检验, 是古典 LM 检验 (Burridge, 1980; Anselin, 1988) 和稳健 LM 检验 (Anselin et al., 1996) 从截面数据到空间面板数据的一般化. 这个一般化是基于 Elhorst (2010a) .

    (B) 第二个程序包含一个框架来检验空间滞后模型 SLM 、空间误差模型 SEM 和空间杜宾模型 (spatial Durbin model, SDM), 以及选择固定效应模型、随机效应模型, 或者固定/随机效应均不存在的模型.

3个模型

空间滞后模型 SLM

  • 是因变量, 表示截面单元 在时间 上的数据.
  • 变量 表示相邻区域上因变量 和 的交互效应,
  • 是 和 个元素预先设置的非负 维的空间权重矩阵 , 用来描述样本空间单元的分布.
  • 是内生交互效应的响应参数, 假定 是权重矩阵行标准化以后的特征根.
  • 是常数项的参数. 是外生变量的 维向量,
  • 是末知参数的 维向量.
  • 是独立同分布的误差项, 服从 分布,
  • 表示空间固定效应,
  • 表示时间固定效应.

空间误差模型 SEM

被称为自相关系数. 单元的误差项 取决于空间权重矩阵 相邻单元 的误差项, 以及 ,

LM 检验

为检验空间滞后模型 SLM 和空间误差模型 SEM 哪个比不存在空间交互效应模型更适合描述数据, 需用 LM (Lagrange Multiplier) 统计量检验空间滞后因变量和空间误差自相关, 同时运用稳健 LM 统计量检验存在局部空间误差自相关情况下的空间滞后因变量, 以及存在局部空间滞后因变量时的空间误差自相关.

由于稳健 LM 检验的结果依赖于包含的效应, 因此建议给出不同面板设置模型的 LM 统计量. Matlab 程序 (LMsarsem_panel.m), 和一个演示文件 (demoLMsarsem_panel.m), 计算 LM 统计量.

空间杜宾模型 SDM

如果基于 LM 检验拒绝非空间模型而接受空间滞后模型 SLM 或者空间误差模型 SEM, 那么要谨慎选择两个模型中的一个. LeSage & Pace (2009, Ch.6) 建议同时考虑空间杜宾模型 SDM (If the nonspatial model on the basis of these LM tests is rejected in favor of the spatial lag model or the spatial error model, one should be careful to endorse one of these two models. LeSage and Pace (2009, chap. 6) recommend to also consider the spatial Durbin model). 该模型用空间滞后的自变量拓展了空间滞后模型 SLM, 形式如下:

LR 及 Wald 检验

和 都是 维参数向量. 该模型可以用来检验假设 和 .

第一个假设检验是否空间杜宾模型 SDM 会被简化 (simplified) 空间滞后模型 SLM,

第二个假设检验是否空间杜宾模型 SDM 会简化 (simplified) 为空间误差模型 SEM (burridge, 1981). 两个检验均服从自由度为 的卡方分布.

如果同时估计空间滞后模型 SLM 和空间误差模型 SEM, 那么这些检验可以采取似然比检验 (likelihood ratio, LR) 的形式.

如果不估计这些模型, 仅采取 Wald 检验.

如果两个假设 都被拒绝了, 那么空间杜宾模型 SDM 更好地描述数据.

相反, 若第一个假设不能被拒绝, 则选择空间滞后模型 SLM, 并且稳健的 LM 检验也说明要选择空间滞后模型 SLM.

类似地, 如果第二个假设不能被拒绝, 则选择空间误差模型 SEM, 并且稳健的 LM 检验也说明要选择空间误差模型 SEM .

若这些条件不满足, 也就是说, 稳健的 LM 检验指向其他的模型, 那么应该采用空间杜宾模型 SDM. 这是因为这个模型是空间滞后和空间误差模型 SEM 的一般化形式.

空间计量经济学文献被划分为, 是否应用特殊到一般的方法, 或者一般到特殊的方法 (Florax et al.,2003; Mur & Angula,2009). 检验的程序概括了以上两种方法的混合.

首先, 非空间模型用来检验与空间滞后模型 SLM 以及空间误差模型 SEM 的不同 (特殊到一般的方法--Stge: Specific-to-General, Mur&Angulo2009) .

Image


如果非空间模型被拒绝了, 那么空间杜宾模型 SDM 用来检验是否会简化为空间滞后或者空间误差模型 SEM (一般到特殊的方法--Gets: General-to-Specific, Mur&Angulo2009).

Image


如果两种检验确定了选择空间滞后模型 SLM 或者空间误差模型 SEM, 那么可以认为该模型就是最好的.

相反, 如果非空间模型被拒绝了, 而接受空间滞后或者空间误差模型 SEM 而不是空间杜宾模型 SDM, 那么最好采用更一般的模型.

Image

                    (Yang et al. 2023)

cigar 的例子

Baltagi & Li (2004) 基于美国 46 个州的面板数据估计了一个香烟需求模型

  • 表示 14 岁以上吸烟者的人均香烟需求数量, 单位: 包.
  • 指每包香烟实际零售均价.
  • 指人均实际可支配收入.

方程 (13) 中包含每一年的地区虚拟变量和时间虚拟变量. 本文将研究是否这些固定效应是联合显著的, 以及是否用随机效应代替.

表 1 给出了非空间面板模型的回归结果, 并检验是否空间滞后模型 SLM 或空间误差模型 SEM 更合适. 通过运行演示文件 “demoLMsarsem_panel.m” 可以得到该结果并复制.

使用古典 LM 检验时, 在不考虑空间或者时间固定效应, 不存在空间滞后因变量的假设和不存在空间自相关误差项的假设在 和 显著水平上均被拒绝.

使用稳健检验时, 不存在空间自相关误差项的假设在 5%和 显著水平上均被拒绝, 但考虑空间或者时间固定效应情况下, 不存在空间滞后因变量的假设在 和 显著水平上不能被拒绝. 显然, 选择空间还是时间固定效应是一个重要的问题.

表1  用没有空间交互效应的面板数据模型对香烟的需求进行估计的结果


(1)
Pooled OLS		(2)
Spatial fixed
effects		(3)
Time-period
fixed effects		(4)
Spatial and
time-period
fixed effects
log()-0.859
(-25.16)		-0.702
(-38.88)		-1.205
(-22.66)		-1.035
(-25.63)
log()0.268
(10.85)		-0.011
(-0.66)		0.565
(18.66)		0.529
(11.67)
Intercept3.485
(30.75)



0.0340.0070.0280.005

0.3210.8530.4400.896
Log L370.31425.2503.91661.7
LM spatial lag66.47136.4344.0446.90
LM spatial error153.04255.7262.8654.65
Robust LM spatial lag58.2629.510.331.16
Robust LM spatial error144.84148.8019.158.91

为检验 (虚无) 假设: 空间固定效应是联合不显著, 需要进行似然比检验. 结果 (2315.7, 自由度为 46, ) 表明拒绝零假设. 同样地, 也应该拒绝时间固定效应不是联合显著的零假设 (473.1, 自由度为 30, . 这些检验证明, 要选择空间和时间固定效应模型, 这就是双向固定效应模型 (TWFE, Baltagi, 2005).

到目前为止, 检验结果指向双向固定效应模型中的空间误差模型 SEM. 从我们第二部分给出的检验程序来看, 应该考虑空间杜宾形式的香烟需求模型. 模型结果见表 2 第 (1) (2) 列, 通过程序 “demopanelscompare.m” 获得.

表2  用具有空间和时间特定效应的空间杜宾模型 SDM 对香烟的需求进行估计的结果

Determinants(1)
Spatial and time-period
fixed effects
(2)
Spatial and time-period
fixed effects bias-corrected
(3)
Random spatial effects,
fixed time-period effects

0.219(6.67)
0.264(8.25)
0.224(6.82)

-1.003
(-25.02)
-1.001
(-24.36)
-1.007
(-24.91)

0.601
(10.51)
0.603
(10.27)
0.593
(10.71)

0.045
(0.55)
0.093
(1.13)
0.066
(0.81)

-0.292
(-3.73)
-0.314
(-3.93)
-0.271
(-3.55)
phi



0.087
(6.81)

0.005
0.005
0.005
(Pseudo) 0.901
0.902
0.880
(Pseudo) Corrected 0.400
0.400
0.317
log L1691.4
1691.4
1555.5
Wald test spatial lag14.83(=.001)
17.96(=.000)
13.90(=.001)
LR test satial lag15.75(=.000)
17.96(=.000)
14.48(=.000)
Wald test satial error8.98(=.011)
8.18(=.017)
7.38(=.025)
LR test satial error8.23(=.016)
8.28(=.016)
7.27(=.026)
Direct effect -1.015(-24.34)-1.014(-25.44)-1.013(-24.73)-1.012(-23.93)-1.018(-24.64)-1.018(-25.03)
Indirect effect -0.210(-2.40)-0.211(-2.37)-0.220(-2.26)-0.215(-2.12)-0.199(-2.28)-0.195(-2.19)
Total effect -1.225(-12.56)-1.225(-12.37)-1.232(-11.31)-1.228(-11.26)-1.217(-12.43)-1.213(-12.21)
Direct effect 0.591 (10.62)0.594(10.44)0.594(10.45)0.594(10.67)0.586(10.68)0.583(10.53)
Indirect effect -0.194(-2.29)-0.194(-2.27)-0.197(-2.15)-0.196(-2.18)-0.169(-2.03)-0.171(-2.06)
Total effect 0.397(5.05)0.400(5.19)0.397(4.61)0.398(4.62)0.417(5.45)0.412(5.37)

注: 括符中为  值. 直接和间接效应估计: 左栏  每次抽样时计算, 右栏  通过式  计算.  Corrected R 为不含固定效应的 R.

  • 空间滞后因变量 () 和自变量 () 的系数对偏误修正程序十分敏感. 这就是为什么要在 Matlab 程序中建立偏误修正程序来处理固定效应空间滞后和固定效应空间误差模型 SEM 的主要原因 (程序 “sar_panel_FE.m' “sem_panel_FE.m').

  • 利用 Wald 或 LR 检验来检验是否空间杜宾模型 SDM 会简化成空间误差模型 SEM, 即 . 表 2 中第 (2) 列结果说明应该拒绝原假设.

  • 利用 Wald 或 LR 检验来检验是否空间杜宾模型 SDM 会简化成空间滞后模型 SLM, 即 . 结果显示该假设被拒绝了. 因此应选择空间杜宾模型 SDM.

  • Hausman 检验假设: 选择随机效应而不是固定效应. 结果说明拒绝随机效应模型.

  • 另一种检验方式是估计参数 “phi” (Baltagi, 2005 中是 ), 该参数度量附着数据截面成分的权重, 取值范围 . 如果参数为 0 , 说明随机效应模型向固定效应靠拢; 如果参数为 1 , 说明它向不受任何限制的空间特定效应靠拢. 结果和 Hausman 检验相同, 固定效应和随机效应模型显著不同.

  • 进一步, 检验 (SDM等) 直接效应和间接效应的符号和显著性, 因显著性水平重要且灵活, 故应审慎选择模型.

Stata

(读入数据参阅 前篇推文)

clear all
use Wct_bin.dta
spmat dta Wst m1-m46, norm(row) replace

* Panel data set up
use cigar
xtset state year

*参 Carlos Mendez
**Non-spatial panel

*Pooled OLS
reg logc logp logy
estimate store pool

*Region FE
xtreg logc logp logy, fe
estimate store rfe

*Time FE
reg logc logp logy i.year
estimate store tfe

*Two-way FE
xtreg logc logp logy i.year, fe
estimate store rtfe

*Hausman test...

*Comparison
estimates table pool rfe tfe rtfe, b(%7.2f) star(0.1 0.05 0.01) stf(%9.0f)

使用 xsmle 但少 LM 检验

**Spatial panel

*SDM with two-way FE
xsmle logc logp logy, fe type(both) wmat(Wst) mod(sdm) effects nsim(999) nolog
estimate store sdm1

*Lee and Yu correction
xsmle logc logp logy, fe type(both) leeyu wmat(Wst) mod(sdm) effects nsim(999) nolog
estimate store sdm2

*Comparison
estimates table sdm1 sdm2, b(%7.3f) star(0.1 0.05 0.01) stf(%9.0f)

*Wald tests
quietly xsmle logc logp logy, fe type(both) leeyu wmat(Wst) mod(sdm) effects nsim(999) nolog

* Wald test: Reduce to SAR? (NO if p < 0.05)
test ([Wx]logp = 0) ([Wx]logy = 0)

* Wald test: Reduce to SLX? (NO if p < 0.05)
test ([Spatial]rho = 0)

* Wald test: Reduce to SEM? (NO if p < 0.05)
testnl ([Wx]logp = -[Spatial]rho*[Main]logp) ([Wx]logy = -[Spatial]rho*[Main]logy)

*lrtest
lrtest sdm1 sdm2

使用 spregxt 提供多面检验

spregxt logc logp logy, nc(46) wmfile(Wct_bin) model(sar) mfx(log) pmfx tests predict(Yh) resid(Ue)


==============================================================================
*** Binary (0/1) Weight Matrix: (1380x1380) : NC=46 NT=30 (Non Normalized)
------------------------------------------------------------------------------
==============================================================================
* MLE Spatial Panel Lag Normal Model (SAR)
==============================================================================
logc = logp logy
------------------------------------------------------------------------------
Sample Size = 1380 | Cross Sections Number = 46
Wald Test = 402.6298 | P-Value > Chi2(2) = 0.0000
F-Test = 201.3149 | P-Value > F(2 , 1332) = 0.0000
R2 (R-Squared) = 0.2262 | Raw Moments R2 = 0.9985
R2a (Adjusted R2) = 0.1989 | Raw Moments R2 Adj = 0.9985
Root MSE (Sigma) = 0.2007 | Log Likelihood Function = 348.1546
------------------------------------------------------------------------------
- R2h= 0.2878 R2h Adj= 0.2627 F-Test = 278.23 P-Value > F(2 , 1332)0.0000
- R2r= 0.9985 R2r Adj= 0.9985 F-Test = 3.1e+05 P-Value > F(3 , 1332)0.0000
------------------------------------------------------------------------------
logc | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
logc |
logp | -.8302382 .0349615 -23.75 0.000 -.8987615 -.7617148
logy | .0440118 .0253011 1.74 0.082 -.0055775 .0936011
_cons | 4.871404 .1186427 41.06 0.000 4.638869 5.103939
-------------+----------------------------------------------------------------
/Rho | -.002663 .0006449 -4.13 0.000 -.003927 -.001399
/Sigma | .188014 .0035788 52.54 0.000 .1809997 .1950283
------------------------------------------------------------------------------
LR Test SAR vs. OLS (Rho=0): 17.0515 P-Value > Chi2(1) 0.0000
Acceptable Range for Rho: -0.3690 < Rho < 0.1970
------------------------------------------------------------------------------

==============================================================================
* Panel Model Selection Diagnostic Criteria - Model= (sar)
==============================================================================
- Log Likelihood Function LLF = 348.1546
---------------------------------------------------------------------------
- Akaike Information Criterion (1974) AIC = 0.0391
- Akaike Information Criterion (1973) Log AIC = -3.2426
---------------------------------------------------------------------------
- Schwarz Criterion (1978) SC = 0.0395
- Schwarz Criterion (1978) Log SC = -3.2313
---------------------------------------------------------------------------
- Amemiya Prediction Criterion (1969) FPE = 0.0404
- Hannan-Quinn Criterion (1979) HQ = 0.0392
- Rice Criterion (1984) Rice = 0.0391
- Shibata Criterion (1981) Shibata = 0.0391
- Craven-Wahba Generalized Cross Validation (1979) GCV = 0.0391
------------------------------------------------------------------------------

==============================================================================
*** Spatial Panel Aautocorrelation Tests - Model= (sar)
*** Binary (0/1) Weight Matrix (W): (Non Normalized)
==============================================================================
Ho: Error has No Spatial AutoCorrelation
Ha: Error has Spatial AutoCorrelation

- GLOBAL Moran MI = 0.2234 P-Value > Z(11.941) 0.0000
- GLOBAL Geary GC = 0.6853 P-Value > Z(-10.410) 0.0000
- GLOBAL Getis-Ords GO = -0.9132 P-Value > Z(-11.941) 0.0000
------------------------------------------------------------------------------
- Moran MI Error Test = 2.9432 P-Value > Z(156.817) 0.0032
------------------------------------------------------------------------------
- LM Error (Burridge) = 140.1836 P-Value > Chi2(1) 0.0000
- LM Error (Robust) = 150.8628 P-Value > Chi2(1) 0.0000
------------------------------------------------------------------------------
Ho: Spatial Lagged Dependent Variable has No Spatial AutoCorrelation
Ha: Spatial Lagged Dependent Variable has Spatial AutoCorrelation

- LM Lag (Anselin) = 23.8390 P-Value > Chi2(1) 0.0000
- LM Lag (Robust) = 34.5182 P-Value > Chi2(1) 0.0000
------------------------------------------------------------------------------
Ho: No General Spatial AutoCorrelation
Ha: General Spatial AutoCorrelation

- LM SAC (LMErr+LMLag_R) = 174.7018 P-Value > Chi2(2) 0.0000
- LM SAC (LMLag+LMErr_R) = 174.7018 P-Value > Chi2(2) 0.0000
------------------------------------------------------------------------------

==============================================================================
*** Panel Unit Roots Tests - Model= (sar)
==============================================================================
Ho: All Panels are Stationary - Ha: Some Panels Have Unit Roots

- Hadri Z Test (No Trend - No Robust) = 87.2984 P-Value > Z(0,1) 0.0000
- Hadri Z Test (No Trend - Robust) = 72.1427 P-Value > Z(0,1) 0.0000
- Hadri Z Test ( Trend - No Robust) = 74.7931 P-Value > Z(0,1) 0.0000
- Hadri Z Test ( Trend - Robust) = 64.7472 P-Value > Z(0,1) 0.0000
------------------------------------------------------------------------------
==============================================================================
* (1) (DF): Dickey-Fuller Test
* (2) (ADF): Augmented Dickey-Fuller Test
* (3) (APP): Augmented Phillips-Perron Test
--------------------------------------------------
Ho: All Panels Have Unit Roots (Non stationary)
Ha: At Least One Panel is Stationary
------------------------------------------------------------------------------
Ho: Non Stationary [0.05, 0.01 < P-Value]
Ha: Stationary [0.05, 0.01 > P-Value]
------------------------------------------------------------------------------
*** (1) Dickey-Fuller (DF) Test:
--------------------------------------------------
- DF Test: [Lag = 0] (No Trend) = 10.2853 P-Value > Z(0,1) 1.0000
* Since [.05 < 1.0000]: Variable (logc) has Non Stationary (Unit Roots)
------------------------------------------------------------------------------
- DF Test: [Lag = 0] ( Trend) = 8.6732 P-Value > Z(0,1) 1.0000
* Since [.05 < 1.0000]: Variable (logc) has Non Stationary (Unit Roots)
------------------------------------------------------------------------------

------------------------------------------------------------------------------
*** (2) Augmented Dickey-Fuller (ADF) Test:
--------------------------------------------------
- ADF Test: [Lag = 1] (No Trend) = 11.5440 P-Value > Z(0,1) 1.0000
* Since [.05 < 1.0000]: Variable (logc) has Non Stationary (Unit Roots)
------------------------------------------------------------------------------
- ADF Test: [Lag = 1] ( Trend) = 8.8444 P-Value > Z(0,1) 1.0000
* Since [.05 < 1.0000]: Variable (logc) has Non Stationary (Unit Roots)
------------------------------------------------------------------------------

------------------------------------------------------------------------------
*** (3) Augmented Phillips-Perron (APP) Test:
--------------------------------------------------
- APP Test: [Lag = 1] (No Trend) = 10.8299 P-Value > Z(0,1) 1.0000
* Since [.05 < 1.0000]: Variable (logc) has Non Stationary (Unit Roots)
------------------------------------------------------------------------------
- APP Test: [Lag = 1] ( Trend) = 8.6843 P-Value > Z(0,1) 1.0000
* Since [.05 < 1.0000]: Variable (logc) has Non Stationary (Unit Roots)
------------------------------------------------------------------------------

==============================================================================
*** Panel Error Component Tests - Model= (sar)
==============================================================================
* Panel Random Effects Tests
Ho: No AR(1) Autocorrelation - Ha: AR(1) Autocorrelation
Ho: Pooled OLS - No Significance Difference among Panels
Ha: Random Effect - Significance Difference among Panels

- Breusch-Pagan LM Test -Two Side =8415.2978 P-Value > Chi2(1) 0.0000
- Breusch-Pagan ALM Test -Two Side =7242.5917 P-Value > Chi2(1) 0.0000
------------------------------------------------------------------------------
- Sosa-Escudero-Yoon LM Test -One Side = 91.7349 P-Value > Chi2(1) 0.0000
- Sosa-Escudero-Yoon ALM Test -One Side = 85.1034 P-Value > Chi2(1) 0.0000
------------------------------------------------------------------------------
- Baltagi-Li LM Autocorrelation Test =1358.6621 P-Value > Chi2(1) 0.0000
- Baltagi-Li ALM Autocorrelation Test = 185.9560 P-Value > Chi2(1) 0.0000
------------------------------------------------------------------------------
- Baltagi-Li LM AR(1) Joint Test =8601.2538 P-Value > Chi2(2) 0.0000
------------------------------------------------------------------------------



spregxt logc logp logy, nc(46) wmfile(Wct_bin) model(sdm) mfx(log) pmfx tests predict(Yh) resid(Ue)


【以下报告跟上文误重】==============================================================================
*** Binary (0/1) Weight Matrix: (1380x1380) : NC=46 NT=30 (Non Normalized)
------------------------------------------------------------------------------
==============================================================================
* MLE Spatial Panel Lag Normal Model (SAR)
==============================================================================
logc = logp logy
------------------------------------------------------------------------------
Sample Size = 1380 | Cross Sections Number = 46
Wald Test = 402.6298 | P-Value > Chi2(2) = 0.0000
F-Test = 201.3149 | P-Value > F(2 , 1332) = 0.0000
R2 (R-Squared) = 0.2262 | Raw Moments R2 = 0.9985
R2a (Adjusted R2) = 0.1989 | Raw Moments R2 Adj = 0.9985
Root MSE (Sigma) = 0.2007 | Log Likelihood Function = 348.1546
------------------------------------------------------------------------------
- R2h= 0.2878 R2h Adj= 0.2627 F-Test = 278.23 P-Value > F(2 , 1332)0.0000
- R2r= 0.9985 R2r Adj= 0.9985 F-Test = 3.1e+05 P-Value > F(3 , 1332)0.0000
------------------------------------------------------------------------------
logc | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
logc |
logp | -.8302382 .0349615 -23.75 0.000 -.8987615 -.7617148
logy | .0440118 .0253011 1.74 0.082 -.0055775 .0936011
_cons | 4.871404 .1186427 41.06 0.000 4.638869 5.103939
-------------+----------------------------------------------------------------
/Rho | -.002663 .0006449 -4.13 0.000 -.003927 -.001399
/Sigma | .188014 .0035788 52.54 0.000 .1809997 .1950283
------------------------------------------------------------------------------
LR Test SAR vs. OLS (Rho=0): 17.0515 P-Value > Chi2(1) 0.0000
Acceptable Range for Rho: -0.3690 < Rho < 0.1970
------------------------------------------------------------------------------

==============================================================================
* Panel Model Selection Diagnostic Criteria - Model= (sar)
==============================================================================
- Log Likelihood Function LLF = 348.1546
---------------------------------------------------------------------------
- Akaike Information Criterion (1974) AIC = 0.0391
- Akaike Information Criterion (1973) Log AIC = -3.2426
---------------------------------------------------------------------------
- Schwarz Criterion (1978) SC = 0.0395
- Schwarz Criterion (1978) Log SC = -3.2313
---------------------------------------------------------------------------
- Amemiya Prediction Criterion (1969) FPE = 0.0404
- Hannan-Quinn Criterion (1979) HQ = 0.0392
- Rice Criterion (1984) Rice = 0.0391
- Shibata Criterion (1981) Shibata = 0.0391
- Craven-Wahba Generalized Cross Validation (1979) GCV = 0.0391
------------------------------------------------------------------------------

==============================================================================
*** Spatial Panel Aautocorrelation Tests - Model= (sar)
*** Binary (0/1) Weight Matrix (W): (Non Normalized)
==============================================================================
Ho: Error has No Spatial AutoCorrelation
Ha: Error has Spatial AutoCorrelation

- GLOBAL Moran MI = 0.2234 P-Value > Z(11.941) 0.0000
- GLOBAL Geary GC = 0.6853 P-Value > Z(-10.410) 0.0000
- GLOBAL Getis-Ords GO = -0.9132 P-Value > Z(-11.941) 0.0000
------------------------------------------------------------------------------
- Moran MI Error Test = 2.9432 P-Value > Z(156.817) 0.0032
------------------------------------------------------------------------------
- LM Error (Burridge) = 140.1836 P-Value > Chi2(1) 0.0000
- LM Error (Robust) = 150.8628 P-Value > Chi2(1) 0.0000
------------------------------------------------------------------------------
Ho: Spatial Lagged Dependent Variable has No Spatial AutoCorrelation
Ha: Spatial Lagged Dependent Variable has Spatial AutoCorrelation

- LM Lag (Anselin) = 23.8390 P-Value > Chi2(1) 0.0000
- LM Lag (Robust) = 34.5182 P-Value > Chi2(1) 0.0000
------------------------------------------------------------------------------
Ho: No General Spatial AutoCorrelation
Ha: General Spatial AutoCorrelation

- LM SAC (LMErr+LMLag_R) = 174.7018 P-Value > Chi2(2) 0.0000
- LM SAC (LMLag+LMErr_R) = 174.7018 P-Value > Chi2(2) 0.0000
------------------------------------------------------------------------------

==============================================================================
*** Panel Unit Roots Tests - Model= (sar)
==============================================================================
Ho: All Panels are Stationary - Ha: Some Panels Have Unit Roots

- Hadri Z Test (No Trend - No Robust) = 87.2984 P-Value > Z(0,1) 0.0000
- Hadri Z Test (No Trend - Robust) = 72.1427 P-Value > Z(0,1) 0.0000
- Hadri Z Test ( Trend - No Robust) = 74.7931 P-Value > Z(0,1) 0.0000
- Hadri Z Test ( Trend - Robust) = 64.7472 P-Value > Z(0,1) 0.0000
------------------------------------------------------------------------------
==============================================================================
* (1) (DF): Dickey-Fuller Test
* (2) (ADF): Augmented Dickey-Fuller Test
* (3) (APP): Augmented Phillips-Perron Test
--------------------------------------------------
Ho: All Panels Have Unit Roots (Non stationary)
Ha: At Least One Panel is Stationary
------------------------------------------------------------------------------
Ho: Non Stationary [0.05, 0.01 < P-Value]
Ha: Stationary [0.05, 0.01 > P-Value]
------------------------------------------------------------------------------
*** (1) Dickey-Fuller (DF) Test:
--------------------------------------------------
- DF Test: [Lag = 0] (No Trend) = 10.2853 P-Value > Z(0,1) 1.0000
* Since [.05 < 1.0000]: Variable (logc) has Non Stationary (Unit Roots)
------------------------------------------------------------------------------
- DF Test: [Lag = 0] ( Trend) = 8.6732 P-Value > Z(0,1) 1.0000
* Since [.05 < 1.0000]: Variable (logc) has Non Stationary (Unit Roots)
------------------------------------------------------------------------------

------------------------------------------------------------------------------
*** (2) Augmented Dickey-Fuller (ADF) Test:
--------------------------------------------------
- ADF Test: [Lag = 1] (No Trend) = 11.5440 P-Value > Z(0,1) 1.0000
* Since [.05 < 1.0000]: Variable (logc) has Non Stationary (Unit Roots)
------------------------------------------------------------------------------
- ADF Test: [Lag = 1] ( Trend) = 8.8444 P-Value > Z(0,1) 1.0000
* Since [.05 < 1.0000]: Variable (logc) has Non Stationary (Unit Roots)
------------------------------------------------------------------------------

------------------------------------------------------------------------------
*** (3) Augmented Phillips-Perron (APP) Test:
--------------------------------------------------
- APP Test: [Lag = 1] (No Trend) = 10.8299 P-Value > Z(0,1) 1.0000
* Since [.05 < 1.0000]: Variable (logc) has Non Stationary (Unit Roots)
------------------------------------------------------------------------------
- APP Test: [Lag = 1] ( Trend) = 8.6843 P-Value > Z(0,1) 1.0000
* Since [.05 < 1.0000]: Variable (logc) has Non Stationary (Unit Roots)
------------------------------------------------------------------------------

==============================================================================
*** Panel Error Component Tests - Model= (sar)
==============================================================================
* Panel Random Effects Tests
Ho: No AR(1) Autocorrelation - Ha: AR(1) Autocorrelation
Ho: Pooled OLS - No Significance Difference among Panels
Ha: Random Effect - Significance Difference among Panels

- Breusch-Pagan LM Test -Two Side =8415.2978 P-Value > Chi2(1) 0.0000
- Breusch-Pagan ALM Test -Two Side =7242.5917 P-Value > Chi2(1) 0.0000
------------------------------------------------------------------------------
- Sosa-Escudero-Yoon LM Test -One Side = 91.7349 P-Value > Chi2(1) 0.0000
- Sosa-Escudero-Yoon ALM Test -One Side = 85.1034 P-Value > Chi2(1) 0.0000
------------------------------------------------------------------------------
- Baltagi-Li LM Autocorrelation Test =1358.6621 P-Value > Chi2(1) 0.0000
- Baltagi-Li ALM Autocorrelation Test = 185.9560 P-Value > Chi2(1) 0.0000
------------------------------------------------------------------------------
- Baltagi-Li LM AR(1) Joint Test =8601.2538 P-Value > Chi2(2) 0.0000
------------------------------------------------------------------------------

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