基于MATLAB的spatial panel model 程序阅读与学习。

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发此文的目的是通过研读程序，更深入的了解Elhost在写这个程序时的思想，通过认真研读程序，深入掌握空间面板程序建模的问题。

1. % Demonstration file for Elhorst Panel Data code
   
2. %
   
3. % Dataset downloaded from www.wiley.co.uk/baltagi/
   
4. % Spatial weights matrix constructed by Elhorst
   
5. %
   
6. % written by: J.Paul Elhorst summer 2010
   
7. % University of Groningen
   
8. % Department of Economics
   
9. % 9700AV Groningen
   
10. % the Netherlands
    
11. % j.p.elhorst@rug.nl
    
12. %
    
13. % REFERENCE:
    
14. % Elhorst JP (2010) Matlab Software for Spatial Panels. Under review.
    
15. %
    
16. % Elhorst JP (2010) Spatial Panel Data Models. In Fischer MM, Getis A (Eds.)
    
17. % Handbook of Applied Spatial Analysis, Ch. C.2. Springer: Berlin Heidelberg New York.
    
18. %
    
19. % New:
    
20. 
    
21. % 1) Direct/Indirect effect esimates of the explanatory variables
    
22. % LeSage JP, Pace RK (2009) Introduction to Spatial Econometrics. Boca Raton, Taylor & Francis Group.
    
23. % routine direct_indirect_effects_estimates(results,W,spat_model) is written by J.P. Elhorst
    
24. % routines panel_effects_sar(results,vnames,W) and panel_effects_sar(results,vnames,W)
    
25. % are written and made available by D. Lacombe
    
26. % User may use both routines (note: results are slightly different from each other since they are based on draws from a distrobution
    
27. % or choose one particular routine. If N is large, user should choose
    
28. % Lacombe's routines, since this one is much more efficient computationally
    
29. 
    
30. % 2) Bias correction of coefficient estimates
    
31. % Lee Lf, Yu J. (2010) Estimation of spatial autoregressive models with
    
32. % fixed effects, Journal of Econometrics 154: 165-185.
    
33. 
    
34. % 3) Selection framework to determine which spatial panel data model best
    
35. % describes the data.
    
36. 
    
37. % dimensions of the problem
    
38. load cigarette_mat;
    
39. load US_W;
    
40. T=30; % number of time periods
    
41. N=46; % number of regions
    
42. % row-normalize W
    
43. W=normw(W1); % function of LeSage
    
44. y=A(:,[3]); % column number in the data matrix that corresponds to the dependent variable
    
45. x=A(:,[4,6]); % column numbers in the data matrix that correspond to the independent variables
    
46. for t=1:T
    
47.     t1=(t-1)*N+1;t2=t*N;
    
48.     wx(t1:t2,:)=W*x(t1:t2,:);
    
49. end
    
50. xconstant=ones(N*T,1);
    
51. [nobs K]=size(x);
    
52. % ----------------------------------------------------------------------------------------
    
53. % No fixed effects + spatially lagged dependent variable
    
54. info.lflag=0; % required for exact results
    
55. info.model=0;
    
56. info.fe=0; % Do not print intercept and fixed effects; use info.fe=1 to turn on
    
57. % New routines to calculate effects estimates
    
58. results=sar_panel_FE(y,[xconstant x],W,T,info);
    
59. vnames=char('logcit','intercept','logp','logy');
    
60. % Print out coefficient estimates
    
61. prt_sp(results,vnames,1);
    
62. % Print out effects estimates
    
63. spat_model=0;
    
64. direct_indirect_effects_estimates(results,W,spat_model);
    
65. panel_effects_sar(results,vnames,W);
    
66. % ----------------------------------------------------------------------------------------
    
67. % No fixed effects + spatially lagged dependent variable + spatially
    
68. % independent variables
    
69. info.lflag=0; % required for exact results
    
70. info.model=0;
    
71. info.fe=0; % Do not print intercept and fixed effects; use info.fe=1 to turn on
    
72. % New routines to calculate effects estimates
    
73. results=sar_panel_FE(y,[xconstant x wx],W,T,info);
    
74. vnames=char('logcit','intercept','logp','logy','W*logp','W*logy');
    
75. % Print out coefficient estimates
    
76. prt_sp(results,vnames,1);
    
77. % Print out effects estimates
    
78. spat_model=1;
    
79. direct_indirect_effects_estimates(results,W,spat_model);
    
80. panel_effects_sdm(results,vnames,W);
    
81. % ----------------------------------------------------------------------------------------
    
82. % Spatial fixed effects + spatially lagged dependent variable
    
83. info.lflag=0; % required for exact results
    
84. info.model=1;
    
85. info.fe=0; % Do not print intercept and fixed effects; use info.fe=1 to turn on
    
86. % New routines to calculate effects estimates
    
87. results=sar_panel_FE(y,x,W,T,info);
    
88. vnames=char('logcit','logp','logy');
    
89. % Print out coefficient estimates
    
90. prt_sp(results,vnames,1);
    
91. % Print out effects estimates
    
92. spat_model=0;
    
93. direct_indirect_effects_estimates(results,W,spat_model);
    
94. panel_effects_sar(results,vnames,W);
    
95. % ----------------------------------------------------------------------------------------
    
96. % Spatial fixed effects + spatially lagged dependent variable + spatially
    
97. % independent variables
    
98. info.lflag=0; % required for exact results
    
99. info.model=1;
    
100. info.fe=0; % Do not print intercept and fixed effects; use info.fe=1 to turn on
     
101. % New routines to calculate effects estimates
     
102. results=sar_panel_FE(y,[x wx],W,T,info);
     
103. vnames=char('logcit','logp','logy','W*logp','W*logy');
     
104. % Print out coefficient estimates
     
105. prt_sp(results,vnames,1);
     
106. % Print out effects estimates
     
107. spat_model=1;
     
108. direct_indirect_effects_estimates(results,W,spat_model);
     
109. panel_effects_sdm(results,vnames,W);
     
110. % ----------------------------------------------------------------------------------------
     
111. % Time period fixed effects + spatially lagged dependent variable
     
112. info.lflag=0; % required for exact results
     
113. info.model=2;
     
114. info.fe=0; % Do not print intercept and fixed effects; use info.fe=1 to turn on
     
115. % New routines to calculate effects estimates
     
116. results=sar_panel_FE(y,x,W,T,info);
     
117. vnames=char('logcit','logp','logy');
     
118. % Print out coefficient estimates
     
119. prt_sp(results,vnames,1);
     
120. % Print out effects estimates
     
121. spat_model=0;
     
122. direct_indirect_effects_estimates(results,W,spat_model);
     
123. panel_effects_sar(results,vnames,W);
     
124. % ----------------------------------------------------------------------------------------
     
125. % Time period fixed effects + spatially lagged dependent variable + spatially
     
126. % independent variables
     
127. info.lflag=0; % required for exact results
     
128. info.model=2;
     
129. info.fe=0; % Do not print intercept and fixed effects; use info.fe=1 to turn on
     
130. % New routines to calculate effects estimates
     
131. results=sar_panel_FE(y,[x wx],W,T,info);
     
132. vnames=char('logcit','logp','logy','W*logp','W*logy');
     
133. % Print out coefficient estimates
     
134. prt_sp(results,vnames,1);
     
135. % Print out effects estimates
     
136. spat_model=1;
     
137. direct_indirect_effects_estimates(results,W,spat_model);
     
138. panel_effects_sdm(results,vnames,W);
     
139. % ----------------------------------------------------------------------------------------
     
140. % Spatial and time period fixed effects + spatially lagged dependent variable
     
141. info.lflag=0; % required for exact results
     
142. info.model=3;
     
143. info.fe=0; % Do not print intercept and fixed effects; use info.fe=1 to turn on
     
144. % New routines to calculate effects estimates
     
145. results=sar_panel_FE(y,x,W,T,info);
     
146. vnames=char('logcit','logp','logy');
     
147. % Print out coefficient estimates
     
148. prt_sp(results,vnames,1);
     
149. % Print out effects estimates
     
150. spat_model=0;
     
151. direct_indirect_effects_estimates(results,W,spat_model);
     
152. panel_effects_sar(results,vnames,W);

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关键词：Spatial MATLAB atlab Panel matla 程序 学习

1. % ----------------------------------------------------------------------------------------
   
2. % Spatial and time period fixed effects + spatially lagged dependent variable + spatially
   
3. % independent variables
   
4. % No bias correction
   
5. info.bc=0;
   
6. info.lflag=0; % required for exact results
   
7. info.model=3;
   
8. info.fe=0; % Do not print intercept and fixed effects; use info.fe=1 to turn on
   
9. % New routines to calculate effects estimates
   
10. results=sar_panel_FE(y,[x wx],W,T,info);
    
11. vnames=char('logcit','logp','logy','W*logp','W*logy');
    
12. % Print out coefficient estimates
    
13. prt_sp(results,vnames,1);
    
14. % Print out effects estimates
    
15. spat_model=1;
    
16. direct_indirect_effects_estimates(results,W,spat_model);
    
17. panel_effects_sdm(results,vnames,W);
    
18. % Wald test for spatial Durbin model against spatial lag model
    
19. btemp=results.parm;
    
20. varcov=results.cov;
    
21. Rafg=zeros(K,2*K+2);
    
22. for k=1:K
    
23. Rafg(k,K+k)=1; % R(1,3)=0 and R(2,4)=0;
    
24. end
    
25. Wald_spatial_lag=(Rafg*btemp)'*inv(Rafg*varcov*Rafg')*Rafg*btemp
    
26. prob_spatial_lag=1-chis_cdf (Wald_spatial_lag, K) % probability greater than 0.05 points to insignificance
    
27. % LR test spatial Durbin model against spatial lag model (requires
    
28. % estimation results of the spatial lag model to be available)
    
29. resultssar=sar_panel_FE(y,x,W,T,info);
    
30. LR_spatial_lag=-2*(resultssar.lik-results.lik)
    
31. prob_spatial_lag=1-chis_cdf (LR_spatial_lag,K) % probability greater than 0.05 points to insignificance
    
32. % Wald test spatial Durbin model against spatial error model
    
33. R=zeros(K,1);
    
34. for k=1:K
    
35. R(k)=btemp(2*K+1)*btemp(k)+btemp(K+k); % k changed in 1, 7/12/2010
    
36. % R(1)=btemp(5)*btemp(1)+btemp(3);
    
37. % R(2)=btemp(5)*btemp(2)+btemp(4);
    
38. end
    
39. Rafg=zeros(K,2*K+2);
    
40. for k=1:K
    
41. Rafg(k,k) =btemp(2*K+1); % k changed in 1, 7/12/2010
    
42. Rafg(k,K+k) =1;
    
43. Rafg(k,2*K+1)=btemp(k);
    
44. % Rafg(1,1)=btemp(5);Rafg(1,3)=1;Rafg(1,5)=btemp(1);
    
45. % Rafg(2,2)=btemp(5);Rafg(2,4)=1;Rafg(2,5)=btemp(2);
    
46. end
    
47. Wald_spatial_error=R'*inv(Rafg*varcov*Rafg')*R
    
48. prob_spatial_error=1-chis_cdf (Wald_spatial_error,K) % probability greater than 0.05 points to insignificance
    
49. % LR test spatial Durbin model against spatial error model (requires
    
50. % estimation results of the spatial error model to be available
    
51. resultssem=sem_panel_FE(y,x,W,T,info);
    
52. LR_spatial_error=-2*(resultssem.lik-results.lik)
    
53. prob_spatial_error=1-chis_cdf (LR_spatial_error,K) % probability greater than 0.05 points to insignificance
    
54. % ----------------------------------------------------------------------------------------
    
55. % Spatial and time period fixed effects + spatially lagged dependent variable + spatially
    
56. % independent variables
    
57. info.lflag=0; % required for exact results
    
58. info.model=3;
    
59. info.fe=0; % Do not print intercept and fixed effects; use info.fe=1 to turn on
    
60. info.bc=1;
    
61. % New routines to calculate effects estimates
    
62. results=sar_panel_FE(y,[x wx],W,T,info);
    
63. vnames=char('logcit','logp','logy','W*logp','W*logy');
    
64. % Print out coefficient estimates
    
65. prt_sp(results,vnames,1);
    
66. % Print out effects estimates
    
67. spat_model=1;
    
68. direct_indirect_effects_estimates(results,W,spat_model);
    
69. panel_effects_sdm(results,vnames,W);
    
70. % Wald test for spatial lag model
    
71. btemp=results.parm;
    
72. varcov=results.cov;
    
73. Rafg=zeros(K,2*K+2);
    
74. for k=1:K
    
75. Rafg(k,K+k)=1; % R(1,3)=0 and R(2,4)=0;
    
76. end
    
77. Wald_spatial_lag=(Rafg*btemp)'*inv(Rafg*varcov*Rafg')*Rafg*btemp
    
78. prob_spatial_lag= 1-chis_cdf (Wald_spatial_lag, K) % probability greater than 0.05 points to insignificance
    
79. % LR test spatial Durbin model against spatial lag model (requires
    
80. % estimation results of the spatial lag model to be available)
    
81. resultssar=sar_panel_FE(y,x,W,T,info);
    
82. LR_spatial_lag=-2*(resultssar.lik-results.lik)
    
83. prob_spatial_lag=1-chis_cdf (LR_spatial_lag,K) % probability greater than 0.05 points to insignificance
    
84. % Wald test for spatial error model
    
85. R=zeros(K,1);
    
86. for k=1:K
    
87. R(k)=btemp(2*K+1)*btemp(k)+btemp(K+k); % k changed in 1, 7/12/2010
    
88. % R(1)=btemp(5)*btemp(1)+btemp(3);
    
89. % R(2)=btemp(5)*btemp(2)+btemp(4);
    
90. end
    
91. Rafg=zeros(K,2*K+2);
    
92. for k=1:K
    
93. Rafg(k,k) =btemp(2*K+1); % k changed in 1, 7/12/2010
    
94. Rafg(k,K+k) =1;
    
95. Rafg(k,2*K+1)=btemp(k);
    
96. % Rafg(1,1)=btemp(5);Rafg(1,3)=1;Rafg(1,5)=btemp(1);
    
97. % Rafg(2,2)=btemp(5);Rafg(2,4)=1;Rafg(2,5)=btemp(2);
    
98. end
    
99. Wald_spatial_error=R'*inv(Rafg*varcov*Rafg')*R
    
100. prob_spatial_error= 1-chis_cdf (Wald_spatial_error,K) % probability greater than 0.05 points to insignificance
     
101. % LR test spatial Durbin model against spatial error model (requires
     
102. % estimation results of the spatial error model to be available
     
103. resultssem=sem_panel_FE(y,x,W,T,info);
     
104. LR_spatial_error=-2*(resultssem.lik-results.lik)
     
105. prob_spatial_error=1-chis_cdf (LR_spatial_error,K) % probability greater than 0.05 points to insignificance
     
106. % needed for Hausman test later on
     
107. logliklag=results.lik;
     
108. blagfe=results.parm(1:end-1);
     
109. covblagfe=results.cov(1:end-1,1:end-1);
     
110. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% random effects estimator by ML %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     
111. %
     
112. % Spatial random effects and time period fixed effects + spatially lagged dependent variable + spatially
     
113. % independent variables
     
114. [ywith,xwith,meanny,meannx,meanty,meantx]=demean(y,[x wx],N,T,2); % 2=time dummies
     
115. info.model=1;
     
116. results=sar_panel_RE(ywith,xwith,W,T,info);
     
117. prt_sp(results,vnames,1);
     
118. % Print out effects estimates
     
119. spat_model=1;
     
120. direct_indirect_effects_estimates(results,W,spat_model);
     
121. panel_effects_sdm(results,vnames,W);
     
122. % Wald test for spatial lag model
     
123. btemp=results.parm(1:2*K+2);
     
124. varcov=results.cov(1:2*K+2,1:2*K+2);
     
125. Rafg=zeros(K,2*K+2);
     
126. for k=1:K
     
127. Rafg(k,K+k)=1; % R(1,3)=0 and R(2,4)=0;
     
128. end
     
129. Wald_spatial_lag=(Rafg*btemp)'*inv(Rafg*varcov*Rafg')*Rafg*btemp
     
130. prob_spatial_lag= 1-chis_cdf (Wald_spatial_lag, K) % probability greater than 0.05 points to insignificance
     
131. % Wald test for spatial error model
     
132. R=zeros(K,1);
     
133. for k=1:K
     
134. R(k)=btemp(2*K+1)*btemp(k)+btemp(K+k); % k changed in 1, 7/12/2010
     
135. % R(1)=btemp(5)*btemp(1)+btemp(3);
     
136. % R(2)=btemp(5)*btemp(2)+btemp(4);
     
137. end
     
138. Rafg=zeros(K,2*K+2);
     
139. for k=1:K
     
140. Rafg(k,k) =btemp(2*K+1); % k changed in 1, 7/12/2010
     
141. Rafg(k,K+k) =1;
     
142. Rafg(k,2*K+1)=btemp(k);
     
143. % Rafg(1,1)=btemp(5);Rafg(1,3)=1;Rafg(1,5)=btemp(1);
     
144. % Rafg(2,2)=btemp(5);Rafg(2,4)=1;Rafg(2,5)=btemp(2);
     
145. end
     
146. Wald_spatial_error=R'*inv(Rafg*varcov*Rafg')*R
     
147. prob_spatial_error= 1-chis_cdf (Wald_spatial_error,K) % probability greater than 0.05 points to insignificance
     
148. % needed for Hausman test later on
     
149. logliklagre=results.lik;
     
150. blagre=results.parm(1:end-2);
     
151. covblagre=results.cov(1:end-2,1:end-2);
     
152. % ----------------------------------------------------------------------------------------
     
153. % Hausman test FE versus RE
     
154. hausman=(blagfe-blagre)'*inv(covblagre-covblagfe)*(blagfe-blagre);
     
155. dof=length(blagfe);
     
156. probability=1-chis_prb(abs(hausman),dof);
     
157. % Note: probability < 0.025 implies rejection of random effects model in favor of fixed effects model
     
158. % Use 0.025, since it is a one-sided test
     
159. fprintf(1,'Hausman test-statistic, degrees of freedom and probability = %9.4f,%6d,%9.4f \n',abs(hausman),dof,probability);

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MATLAB R2012b 已经不支持读取 wk1的文件。还好当初已经将数据保存为matlab的数据格式 .mat。
数据的读入是如下两句：
load cigarette_mat;
load US_W;

T=30; % number of time periods
N=46; % number of regions
% row-normalize W
W=normw(W1); % function of LeSage
y=A(:,[3]); % column number in the data matrix that corresponds to the dependent variable
x=A(:,[4,6]); % column numbers in the data matrix that correspond to the independent variables
for t=1:T
    t1=(t-1)*N+1;t2=t*N;
    wx(t1:t2,:)=W*x(t1:t2,:);
end
xconstant=ones(N*T,1);
[nobs K]=size(x);

这部分是空间面板数据的初始处理；为面板模型数据格式做好准备。

第53~65行的代码；是定义了一个具体的空间面板模型，并通过小效应数估计出（直接效应与间接效应），我这里所说的效应也许翻译得不对，记得一些论文上说的是“空间溢出效应”。具体看相关论文与手册。
---------------------
% No fixed effects + spatially lagged dependent variable
info.lflag=0; % required for exact results
info.model=0;
info.fe=0; % Do not print intercept and fixed effects; use info.fe=1 to turn on
% New routines to calculate effects estimates
results=sar_panel_FE(y,[xconstant x],W,T,info);
vnames=char('logcit','intercept','logp','logy');
% Print out coefficient estimates
prt_sp(results,vnames,1);
% Print out effects estimates
spat_model=0;
direct_indirect_effects_estimates(results,W,spat_model);
panel_effects_sar(results,vnames,W);

info是一个数据结构，其中的 info.fe，是info结构里，对 fe 赋值，0与1，在计算机程序里，如在C++里，应该是 boolean 类型，或者其他的如 FOXPRO里，会负值为 0对于 false，1对于 true的逻辑函数。
这里，info.fe=0,就是逻辑否，关闭特定的功能。

results=sar_panel_FE(y,[xconstant x],W,T,info);
具体的估计函数是 sar_panel_FE,是一个 “空间自回归的面板模型”函数。y为因变量，[xconstant x],xconstant 应当是我们所说的常数项 C （ Constant)，x 是各列因变量数据。W 是空间权重矩阵，T是时间时期数，info是一个数据结构，刚才对于模型的各种设定在 info 里给予具体的“设定”， specify。
direct_indirect_effects_estimates(results,W,spat_model);
这句解释了空间的直接效应与空间的间接效应。其实从大师们的论文里可以看到，还有所谓的“空间溢出效应”。

lz能否做个文件，一句一句解释下啊。然后告诉大家如何在matlab上实现程序。我保证该帖必火啊。现在很多人不知道怎么做空间面板。能否将该程序文件发一份给我？非常感谢！598636005@qq.com

gdfoshanluofeng 发表于 2012-10-24 23:38
lz能否做个文件，一句一句解释下啊。然后告诉大家如何在matlab上实现程序。我保证该帖必火啊。现在很多人不 ...

我是业余爱好，工作不稳定，也没心思做这个。其实因为这个爱好，已经严重影响其他方面的正常生活。
暂时不会有一句一句解释这个程序的可能。

只是偶尔心血来潮，随便发两句短文。

至于整个程序，我已经共享在论坛里了，我发帖了的；
是 Elhost 的空间面板，你搜索下我的帖子；

该怎么用，自己探索和与人交流了。
我得做其他事。这个对我来说是一个游戏，有空的时候才玩玩，该忙其他重要的事的时候，会放弃这个的。

哦。原来如此！真是可惜了！

• % Wald test for spatial lag model
• btemp=results.parm;
• varcov=results.cov;
• Rafg=zeros(K,2*K+2);
• for k=1:K
• Rafg(k,K+k)=1; % R(1,3)=0 and R(2,4)=0;
• end
• Wald_spatial_lag=(Rafg*btemp)'*inv(Rafg*varcov*Rafg')*Rafg*btemp
• prob_spatial_lag= 1-chis_cdf (Wald_spatial_lag, K) % probability greater than 0.05 points to insignificance
  

这里是精彩的Wald 检验，通过矩阵运算算出检验统计量。

• % LR test spatial Durbin model against spatial lag model (requires
• % estimation results of the spatial lag model to be available)
• resultssar=sar_panel_FE(y,x,W,T,info);
• LR_spatial_lag=-2*(resultssar.lik-results.lik)
• prob_spatial_lag=1-chis_cdf (LR_spatial_lag,K) % probability greater than 0.05 points to insignificance
• % Wald test for spatial error model
• R=zeros(K,1);
• for k=1:K
• R(k)=btemp(2*K+1)*btemp(k)+btemp(K+k); % k changed in 1, 7/12/2010
• % R(1)=btemp(5)*btemp(1)+btemp(3);
• % R(2)=btemp(5)*btemp(2)+btemp(4);
• end
• Rafg=zeros(K,2*K+2);
• for k=1:K
• Rafg(k,k) =btemp(2*K+1); % k changed in 1, 7/12/2010
• Rafg(k,K+k) =1;
• Rafg(k,2*K+1)=btemp(k);
• % Rafg(1,1)=btemp(5);Rafg(1,3)=1;Rafg(1,5)=btemp(1);
• % Rafg(2,2)=btemp(5);Rafg(2,4)=1;Rafg(2,5)=btemp(2);
• end
• Wald_spatial_error=R'*inv(Rafg*varcov*Rafg')*R
• prob_spatial_error= 1-chis_cdf (Wald_spatial_error,K) % probability greater than 0.05 points to insignificance
  

• % LR test spatial Durbin model against spatial error model (requires
• % estimation results of the spatial error model to be available
• resultssem=sem_panel_FE(y,x,W,T,info);
• LR_spatial_error=-2*(resultssem.lik-results.lik)
• prob_spatial_error=1-chis_cdf (LR_spatial_error,K) % probability greater than 0.05 points to insignificance
  

精彩的LR统计检验量计算，chis_cdf是来自lesage计量经济学工具箱里的函数。   且该模型是对 Spatial Durbin model 与 error model 的对比检验。值得期待。

• % needed for Hausman test later on
• logliklag=results.lik;
• blagfe=results.parm(1:end-1);
• covblagfe=results.cov(1:end-1,1:end-1);
  

tulipsliu 发表于 2012-10-25 12:01
我是业余爱好，工作不稳定，也没心思做这个。其实因为这个爱好，已经严重影响其他方面的正常生活。
暂时 ...

lz你好，为什么我用你发的那个Elhorst的程序包运行里面的demo程序时会报错呢？比如有这些Error: File: demoLMsarsem_panel.m Line: 60 Column: 29
Expression or statement is incorrect--possibly unbalanced (, {, or [.