国际顶级期刊的编辑非常重视内生性问题，一定要处理好内生性问题，03讲了工具变量，本讲中通过动态面板数据能够较好处理内生性问题。
动态面板数据
动态面板数据（Dynamic Panel Data，DPD）：是指在面板模型中，解释变量包含了被假释变量的滞后值。在动态面板数据类型中被解释变量和上一期变量之间存在关系。即， 与 之间是有关系的，上一期的值决定着下一期的值。
动态面板数据模型的设定是在原有的静态面板数据模型的基础上引入被解释变量的滞后期，而其他的都相同。
其中，
为复合误差项，
=
+
，
为随机扰动项，
为不可观测的个体效应。可以很容易的看出，模型中
是一个内生变量，模型存在内生性问题，所以使用传统的最小二乘进行估计，估计结果是有偏且不一致的。
对上述动态面板数据模型进行拟合估计：首先进行一阶差分将原始模型中的不可观测的个体效应
去除，得到差分后的模型为：
由于
与
相关，所以
与
是相关的，所以一阶差分后的动态面板数据模型仍存在内生性问题。Anderson等人在1982年提出了一种为差分变量
-
寻找工具变量的方法。这个工具变量为
。由于差分变量本身包含着
，所以工具变量和内生变量存在高度的相关性，在误差项
不存在自相关的前提下，工具变量
与误差项的差分
-
不相关，因此，
满足工具变量的条件。需要注意的是，
并不是唯一的工具变量，被解释变量滞后三期、四期（即，
，
）都满足工具变量的条件。
同时，他们认为这种相当于两阶段最小二乘估计的结果虽然是一致的，但却并不是有效的，因为他们没有充分利用样本里的所有信息，于是他们提出了使用更多工具变量的**广义矩估计方法（GMM）**来进行动态面板数据模型的估计，工具变量来自更多的滞后期。
广义矩估计GMM
动态面板数据模型的GMM估计方法又可以分为两种，即差分GMM（DIF-GMM）和系统GMM（SYS-GMM）估计方法。
需要注意的是，差分GMM和系统GMM方法主要适用于短动态面板数据。这是因为，虽然基于IV或GMM的估计方法是一致估计量（即当 时，没有偏差），但对于 较小而 较大的长面板则可能存在较严重的偏差。对于长动态面板数据模型的估计可以使用“偏差校正LSDV法”进行估计。
差分GMM的基本思路是：对基本模型进行一阶差分以去除固定效应的影响，然后，用一组滞后的解释变量作为差分方程中相应变量的工具变量。
Blundell和Bond两位作者认为，差分GMM的估计量较易受弱工具变量的影响而产生向下的大的有限样本偏差。为了克服这一问题，Blundell和Bond提出了系统广义矩估计即系统GMM估计方法。
系统GMM估计方法是基于差分GMM之上形成的，结合了差分方程和水平方程，此外，还增加了一组滞后的差分变量作为水平方程相应的工具变量，更具有系统性。
相对来说，系统GMM估计量具有更好的有限样本性质。
系统GMM估计方法的前提假定是：工具变量的一阶差分与固定效应项不相关。然而，到目前为止，并没有方法能够对这一个假定进行检验。
此外，使用系统GMM估计方法的条件是：
（1）大N小T，即短面板数据;
（2）线性函数关系，构造的计量模型要求是线性的；
（3）方程等号左边的变量作为动态变量；
（4）方程等号右边的变量并不是严格外生的；
（5）控制个体固定效应；
（6）默认不存在截面相关问题，并且建议采用双向固定效应。
时间虚拟变量的引入可以使误差项的截面相关变得不相关，所以在模型设定中尽可能地引入时间虚拟变量以减少截面相关的可能。
在理论层面，GMM估计量（差分GMM、系统GMM）的一致性关键取决于各项假设条件是否满足，这需要进行两个假设检验。
（1）通过Hansen过度识别约束检验对所使用的工具变量的有效性进行检验，此检验的原假设是所使用的工具变量与误差项是不相关的。
（2）通过Arellano-Bond的自相关检验方法对差分方程的随机误差项的二阶序列相关进行检验，其原假设是一阶差分方程的随机误差项中不存在二阶序列相关。如果不拒绝原假设则意味着工具变量有效和模型设定正确。
stata操作
数据集
使用英国140家企业1976~1984年的数据来研究就业数据abdata.dta，是非平衡面板，被解释变量为 ,是就业的对数，存在着两期滞后。重要的解释变量有当期和滞后一期的工资水平 ，当期、滞后一期和滞后两期的资本存量 ，以及当期、滞后一期和滞后两期的公司产出 ，所有的变量都取对数形式。
- 描述性统计
des
结果：
obs: 1,031 Layard & Nickell, Unemployment in Britain, Economica 53, 1986 from Ox distvars: 16 21 May 2013 21:52
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------storage display value
variable name type format label variable label
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
ind int %8.0g industry
year int %8.0g
emp float %9.0g employment
wage float %9.0g real wage
cap float %9.0g gross capital stock
indoutpt float %9.0g industry output
n float %9.0g log(employment)
w float %9.0g log(real wage)
k float %9.0g log(gross capital stock)
ys float %9.0g log(industry output)
yr1980 float %9.0g
yr1981 float %9.0g
yr1982 float %9.0g
yr1983 float %9.0g
yr1984 float %9.0g
id float %9.0g firm ID
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Sorted by: id year
sum
结果：
Variable | Obs Mean Std. Dev. Min Max
-------------+---------------------------------------------------------ind | 1,031 5.123181 2.678095 1 9year | 1,031 1979.651 2.21607 1976 1984emp | 1,031 7.891677 15.93492 .104 108.562wage | 1,031 23.9188 5.648418 8.0171 45.2318cap | 1,031 2.507432 6.248712 .0119 47.1079
-------------+---------------------------------------------------------indoutpt | 1,031 103.8012 9.938008 86.9 128.3653n | 1,031 1.056002 1.341506 -2.263364 4.687321w | 1,031 3.142988 .2630081 2.081577 3.8118k | 1,031 -.4415775 1.514132 -4.431217 3.852441ys | 1,031 4.638015 .0939611 4.464758 4.85488
-------------+---------------------------------------------------------yr1980 | 1,031 .1357905 .3427322 0 1yr1981 | 1,031 .1357905 .3427322 0 1yr1982 | 1,031 .1357905 .3427322 0 1yr1983 | 1,031 .0756547 .2645732 0 1yr1984 | 1,031 .0339476 .1811823 0 1
-------------+---------------------------------------------------------id | 1,031 73.20369 41.23333 1 140
- OLS估计
reg n L.n L2.n w L.w k L.k L2.k ys L.ys L2.ys yr*
结果：
Source | SS df MS Number of obs = 751
-------------+---------------------------------- F(15, 735) = 8676.37Model | 1343.3054 15 89.5536936 Prob > F = 0.0000Residual | 7.58634832 735 .010321562 R-squared = 0.9944
-------------+---------------------------------- Adj R-squared = 0.9943Total | 1350.89175 750 1.801189 Root MSE = .1016------------------------------------------------------------------------------n | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------n |L1. | 1.043 0.034 31.01 0.000 0.977 1.109L2. | -0.076 0.033 -2.30 0.022 -0.140 -0.011|w |--. | -0.522 0.049 -10.71 0.000 -0.618 -0.426L1. | 0.474 0.049 9.75 0.000 0.379 0.570|k |--. | 0.342 0.025 13.42 0.000 0.292 0.392L1. | -0.198 0.040 -4.96 0.000 -0.276 -0.119L2. | -0.118 0.028 -4.16 0.000 -0.174 -0.062|ys |--. | 0.429 0.123 3.50 0.001 0.188 0.669L1. | -0.768 0.166 -4.63 0.000 -1.093 -0.442L2. | 0.318 0.111 2.85 0.004 0.099 0.536|yr1980 | 0.011 0.014 0.84 0.401 -0.015 0.038yr1981 | -0.033 0.018 -1.85 0.065 -0.068 0.002yr1982 | -0.026 0.018 -1.39 0.164 -0.062 0.010yr1983 | -0.003 0.018 -0.14 0.885 -0.039 0.033yr1984 | 0.006 0.021 0.26 0.794 -0.036 0.047_cons | 0.284 0.350 0.81 0.418 -0.404 0.972
-----------------------------------------------------------------------------
xi:reg n L.n L2.n w L.w k L.k L2.k ys L.ys L2.ys i.year //LSDV估计
结果：
i.year _Iyear_1976-1984 (naturally coded; _Iyear_1976 omitted)
note: _Iyear_1977 omitted because of collinearity
note: _Iyear_1978 omitted because of collinearitySource | SS df MS Number of obs = 751
-------------+---------------------------------- F(16, 734) = 8136.58Model | 1343.31797 16 83.9573732 Prob > F = 0.0000Residual | 7.57378164 734 .010318504 R-squared = 0.9944
-------------+---------------------------------- Adj R-squared = 0.9943Total | 1350.89175 750 1.801189 Root MSE = .10158------------------------------------------------------------------------------n | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------n |L1. | 1.045 0.034 31.03 0.000 0.979 1.111L2. | -0.077 0.033 -2.33 0.020 -0.141 -0.012|w |--. | -0.524 0.049 -10.74 0.000 -0.619 -0.428L1. | 0.477 0.049 9.79 0.000 0.381 0.572|k |--. | 0.343 0.026 13.46 0.000 0.293 0.393L1. | -0.202 0.040 -5.04 0.000 -0.281 -0.123L2. | -0.116 0.028 -4.06 0.000 -0.172 -0.060|ys |--. | 0.433 0.123 3.53 0.000 0.192 0.674L1. | -0.768 0.166 -4.63 0.000 -1.093 -0.442L2. | 0.312 0.111 2.80 0.005 0.094 0.531|_Iyear_1977 | 0.000 (omitted)_Iyear_1978 | 0.000 (omitted)_Iyear_1979 | 0.016 0.014 1.10 0.270 -0.012 0.044_Iyear_1980 | 0.022 0.017 1.32 0.187 -0.011 0.055_Iyear_1981 | -0.022 0.020 -1.09 0.278 -0.062 0.018_Iyear_1982 | -0.015 0.021 -0.73 0.468 -0.056 0.026_Iyear_1983 | 0.007 0.020 0.36 0.717 -0.033 0.047_Iyear_1984 | 0.015 0.023 0.67 0.504 -0.030 0.061_cons | 0.275 0.351 0.78 0.433 -0.413 0.963
------------------------------------------------------------------------------
- 双向固定效应估计
xtreg n L.n L2.n w L.w k L.k L2.k ys L.ys L2.ys yr*,fe
结果：
Fixed-effects (within) regression Number of obs = 751
Group variable: id Number of groups = 140R-sq: Obs per group:within = 0.7973 min = 5between = 0.9808 avg = 5.4overall = 0.9758 max = 7F(15,596) = 156.25
corr(u_i, Xb) = 0.5474 Prob > F = 0.0000------------------------------------------------------------------------------n | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------n |L1. | 0.732 0.039 18.68 0.000 0.655 0.809L2. | -0.140 0.040 -3.49 0.001 -0.218 -0.061|w |--. | -0.559 0.057 -9.82 0.000 -0.671 -0.448L1. | 0.314 0.061 5.16 0.000 0.195 0.434|k |--. | 0.388 0.031 12.56 0.000 0.327 0.448L1. | -0.079 0.038 -2.07 0.039 -0.154 -0.004L2. | -0.028 0.033 -0.86 0.389 -0.093 0.036|ys |--. | 0.466 0.123 3.80 0.000 0.225 0.708L1. | -0.630 0.158 -3.99 0.000 -0.940 -0.320L2. | 0.061 0.134 0.46 0.648 -0.202 0.325|yr1980 | 0.008 0.013 0.60 0.551 -0.018 0.034yr1981 | -0.029 0.019 -1.53 0.127 -0.066 0.008yr1982 | -0.038 0.020 -1.92 0.055 -0.077 0.001yr1983 | -0.032 0.022 -1.46 0.146 -0.074 0.011yr1984 | -0.015 0.024 -0.62 0.534 -0.063 0.033_cons | 1.797 0.507 3.54 0.000 0.801 2.793
-------------+----------------------------------------------------------------sigma_u | .22630054sigma_e | .09388866rho | .85314812 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(139, 596) = 1.90 Prob > F = 0.0000
xi:xtreg n L.n L2.n w L.w k L.k L2.k ys L.ys L2.ys i.year,fe
结果：
i.year _Iyear_1976-1984 (naturally coded; _Iyear_1976 omitted)
note: _Iyear_1977 omitted because of collinearity
note: _Iyear_1984 omitted because of collinearityFixed-effects (within) regression Number of obs = 751
Group variable: id Number of groups = 140R-sq: Obs per group:within = 0.7973 min = 5between = 0.9809 avg = 5.4overall = 0.9758 max = 7F(16,595) = 146.27
corr(u_i, Xb) = 0.5459 Prob > F = 0.0000------------------------------------------------------------------------------n | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------n |L1. | 0.733 0.039 18.65 0.000 0.656 0.810L2. | -0.139 0.040 -3.48 0.001 -0.218 -0.061|w |--. | -0.560 0.057 -9.81 0.000 -0.672 -0.448L1. | 0.315 0.061 5.17 0.000 0.195 0.435|k |--. | 0.388 0.031 12.55 0.000 0.328 0.449L1. | -0.081 0.038 -2.09 0.037 -0.156 -0.005L2. | -0.028 0.033 -0.85 0.397 -0.092 0.037|ys |--. | 0.469 0.123 3.81 0.000 0.227 0.710L1. | -0.629 0.158 -3.98 0.000 -0.939 -0.318L2. | 0.058 0.135 0.43 0.667 -0.206 0.322|_Iyear_1977 | 0.000 (omitted)_Iyear_1978 | 0.012 0.026 0.46 0.649 -0.039 0.063_Iyear_1979 | 0.017 0.025 0.67 0.503 -0.032 0.065_Iyear_1980 | 0.023 0.025 0.93 0.355 -0.026 0.072_Iyear_1981 | -0.013 0.026 -0.52 0.605 -0.065 0.038_Iyear_1982 | -0.022 0.023 -0.98 0.328 -0.068 0.023_Iyear_1983 | -0.016 0.021 -0.77 0.442 -0.057 0.025_Iyear_1984 | 0.000 (omitted)_cons | 1.780 0.501 3.55 0.000 0.795 2.765
-------------+----------------------------------------------------------------sigma_u | .22568151sigma_e | .09395847rho | .85227336 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(139, 595) = 1.89 Prob > F = 0.0000
- Anderson–Hsiao estimator
*-1.直接估计ivreg D.n (D.L.n=L2.n) D.(L2.n w L.w k L.k L2.k ys L.ys L2.ys yr1980 yr1981 yr1982 yr1983 yr1984)Instrumental variables (2SLS) regressionSource | SS df MS Number of obs = 611
-------------+---------------------------------- F(15, 595) = 5.84Model | -24.6768882 15 -1.64512588 Prob > F = 0.0000Residual | 37.2768667 595 .062650196 R-squared = .
-------------+---------------------------------- Adj R-squared = .Total | 12.5999785 610 .020655702 Root MSE = .2503------------------------------------------------------------------------------D.n | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------n |LD. | 2.308 2.000 1.15 0.249 -1.619 6.235L2D. | -0.224 0.181 -1.23 0.217 -0.580 0.132|w |D1. | -0.810 0.265 -3.05 0.002 -1.331 -0.289LD. | 1.422 1.195 1.19 0.235 -0.925 3.770|k |D1. | 0.253 0.147 1.73 0.085 -0.035 0.541LD. | -0.552 0.624 -0.89 0.376 -1.777 0.672L2D. | -0.213 0.243 -0.88 0.382 -0.690 0.265|ys |D1. | 0.991 0.469 2.11 0.035 0.069 1.912LD. | -1.938 1.457 -1.33 0.184 -4.800 0.924L2D. | 0.487 0.517 0.94 0.346 -0.528 1.502|yr1980 |D1. | -0.017 0.045 -0.39 0.700 -0.105 0.071|yr1981 |D1. | -0.118 0.115 -1.02 0.307 -0.343 0.108|yr1982 |D1. | -0.174 0.158 -1.10 0.270 -0.484 0.136|yr1983 |D1. | -0.224 0.209 -1.07 0.285 -0.634 0.186|yr1984 |D1. | -0.280 0.273 -1.03 0.305 -0.816 0.255|_cons | 0.063 0.064 0.98 0.329 -0.063 0.189
------------------------------------------------------------------------------
Instrumented: LD.n
Instruments: L2D.n D.w LD.w D.k LD.k L2D.k D.ys LD.ys L2D.ys D.yr1980D.yr1981 D.yr1982 D.yr1983 D.yr1984 L2.n
-----------------------------------------------------------------------------
*-2.构建年份虚拟变量后估计tab year,gen(year)ivreg D.n (D.L.n=L2.n) D.(L2.n w L.w k L.k L2.k ys L.ys L2.ys year1 year2 year3 year4 year5 year6 year7 year8 year9)
结果：
. tab year,gen(year)year | Freq. Percent Cum.
------------+-----------------------------------1976 | 80 7.76 7.761977 | 138 13.39 21.141978 | 140 13.58 34.721979 | 140 13.58 48.301980 | 140 13.58 61.881981 | 140 13.58 75.461982 | 140 13.58 89.041983 | 78 7.57 96.611984 | 35 3.39 100.00
------------+-----------------------------------Total | 1,031 100.00. ivreg D.n (D.L.n=L2.n) D.(L2.n w L.w k L.k L2.k ys L.ys L2.ys year1 year2 year3 year4 year5 year6 year7 year8 year9)Instrumental variables (2SLS) regressionSource | SS df MS Number of obs = 611
-------------+---------------------------------- F(15, 595) = 5.84Model | -24.6768882 15 -1.64512588 Prob > F = 0.0000Residual | 37.2768667 595 .062650196 R-squared = .
-------------+---------------------------------- Adj R-squared = .Total | 12.5999785 610 .020655702 Root MSE = .2503------------------------------------------------------------------------------D.n | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------n |LD. | 2.308 2.000 1.15 0.249 -1.619 6.235L2D. | -0.224 0.181 -1.23 0.217 -0.580 0.132|w |D1. | -0.810 0.265 -3.05 0.002 -1.331 -0.289LD. | 1.422 1.195 1.19 0.235 -0.925 3.770|k |D1. | 0.253 0.147 1.73 0.085 -0.035 0.541LD. | -0.552 0.624 -0.89 0.376 -1.777 0.672L2D. | -0.213 0.243 -0.88 0.382 -0.690 0.265|ys |D1. | 0.991 0.469 2.11 0.035 0.069 1.912LD. | -1.938 1.457 -1.33 0.184 -4.800 0.924L2D. | 0.487 0.517 0.94 0.346 -0.528 1.502|year1 |D1. | 0.000 (omitted)|year2 |D1. | 0.000 (omitted)|year3 |D1. | 0.000 (omitted)|year4 |D1. | 0.047 0.045 1.03 0.305 -0.043 0.136|year5 |D1. | 0.076 0.063 1.20 0.230 -0.048 0.201|year6 |D1. | 0.023 0.056 0.40 0.689 -0.088 0.134|year7 |D1. | 0.013 0.056 0.23 0.818 -0.096 0.122|year8 |D1. | 0.010 0.046 0.21 0.830 -0.081 0.101|year9 |D1. | 0.000 (omitted)|_cons | 0.016 0.028 0.58 0.565 -0.038 0.070
------------------------------------------------------------------------------
Instrumented: LD.n
Instruments: L2D.n D.w LD.w D.k LD.k L2D.k D.ys LD.ys L2D.ys D.year1D.year2 D.year3 D.year4 D.year5 D.year6 D.year7 D.year8D.year9 L2.n
-----------------------------------------------------------------------------
- DIF-GMM
. xi:xtabond2 n L.n L2.n w L.w k L.k L2.k ys L.ys L2.ys i.year, gmm(L.n,) iv(L2.n w L.w k L.k L2.k ys L.ys L2.ys i.year) nolevel robust small
i.year _Iyear_1976-1984 (naturally coded; _Iyear_1976 omitted)
Favoring space over speed. To switch, type or click on mata: mata set matafavor speed, perm.
Warning: Two-step estimated covariance matrix of moments is singular.Using a generalized inverse to calculate robust weighting matrix for Hansen test.Difference-in-Sargan/Hansen statistics may be negative.Dynamic panel-data estimation, one-step difference GMM
------------------------------------------------------------------------------
Group variable: id Number of obs = 611
Time variable : year Number of groups = 140
Number of instruments = 41 Obs per group: min = 4
F(18, 140) = 92.63 avg = 4.36
Prob > F = 0.000 max = 6
------------------------------------------------------------------------------| Robustn | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------n |L1. | 0.686 0.147 4.66 0.000 0.395 0.977L2. | -0.085 0.057 -1.50 0.137 -0.198 0.027|w |--. | -0.608 0.181 -3.35 0.001 -0.966 -0.249L1. | 0.393 0.171 2.30 0.023 0.055 0.731|k |--. | 0.357 0.060 5.94 0.000 0.238 0.476L1. | -0.058 0.074 -0.78 0.437 -0.205 0.089L2. | -0.020 0.033 -0.60 0.550 -0.086 0.046|ys |--. | 0.609 0.176 3.47 0.001 0.261 0.956L1. | -0.711 0.236 -3.02 0.003 -1.177 -0.245L2. | 0.106 0.144 0.74 0.463 -0.178 0.390|_Iyear_1977 | 0.000 (omitted)_Iyear_1978 | 0.008 0.032 0.24 0.810 -0.056 0.071_Iyear_1979 | 0.017 0.030 0.58 0.561 -0.041 0.076_Iyear_1980 | 0.030 0.028 1.06 0.293 -0.026 0.085_Iyear_1981 | -0.004 0.030 -0.13 0.894 -0.064 0.056_Iyear_1982 | -0.019 0.023 -0.83 0.407 -0.065 0.027_Iyear_1983 | -0.014 0.019 -0.71 0.479 -0.052 0.024_Iyear_1984 | 0.000 (omitted)
------------------------------------------------------------------------------
Instruments for first differences equationStandardD.(L2.n w L.w k L.k L2.k ys L.ys L2.ys _Iyear_1977 _Iyear_1978 _Iyear_1979_Iyear_1980 _Iyear_1981 _Iyear_1982 _Iyear_1983 _Iyear_1984)GMM-type (missing=0, separate instruments for each period unless collapsed)L(1/8).L.n
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z = -3.60 Pr > z = 0.000
Arellano-Bond test for AR(2) in first differences: z = -0.52 Pr > z = 0.606
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(23) = 67.59 Prob > chi2 = 0.000(Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(23) = 31.38 Prob > chi2 = 0.114(Robust, but weakened by many instruments.)Difference-in-Hansen tests of exogeneity of instrument subsets:iv(L2.n w L.w k L.k L2.k ys L.ys L2.ys _Iyear_1977 _Iyear_1978 _Iyear_1979 _Iyear_1980 _Iyear_1981 _Iyear_1982 _Iyear_1983 _Iyear_1984)Hansen test excluding group: chi2(8) = 12.01 Prob > chi2 = 0.151Difference (null H = exogenous): chi2(15) = 19.37 Prob > chi2 = 0.197
使用lag()选项控制工具变量的滞后期数
. xi:xtabond2 n L.n L2.n w L.w k L.k L2.k ys L.ys L2.ys i.year, gmm(L.n, lag(2 5)) iv(L2.n w L.w k L.k L2.k ys L.ys L2.ys i.year) nolevel robust small nomata
i.year _Iyear_1976-1984 (naturally coded; _Iyear_1976 omitted)
_Iyear_1977 dropped because of collinearity.
_Iyear_1978 dropped because of collinearity.
Building GMM instruments..
2 instrument(s) dropped because of collinearity.
Estimating.
Performing specification tests.Dynamic panel-data estimation, one-step difference GMM
------------------------------------------------------------------------------
Group variable: id Number of obs = 611
Time variable : year Number of groups = 140
Number of instruments = 33 Obs per group: min = 4
F(14, 139) = 117.25 avg = 4.36
Prob > F = 0.000 max = 6
------------------------------------------------------------------------------| Robustn | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------n |L1. | 1.017 0.284 3.58 0.000 0.455 1.578L2. | -0.114 0.051 -2.23 0.027 -0.215 -0.013|w |--. | -0.659 0.204 -3.22 0.002 -1.064 -0.255L1. | 0.634 0.325 1.95 0.053 -0.009 1.276|k |--. | 0.335 0.065 5.12 0.000 0.205 0.464L1. | -0.158 0.117 -1.35 0.179 -0.391 0.074L2. | -0.065 0.051 -1.28 0.204 -0.165 0.036|ys |--. | 0.680 0.198 3.43 0.001 0.289 1.072L1. | -0.993 0.401 -2.48 0.014 -1.785 -0.201L2. | 0.235 0.206 1.14 0.257 -0.173 0.642|_Iyear_1979 | 0.019 0.014 1.41 0.162 -0.008 0.047_Iyear_1980 | 0.038 0.023 1.62 0.107 -0.008 0.084_Iyear_1981 | 0.001 0.033 0.03 0.975 -0.064 0.066_Iyear_1982 | -0.010 0.031 -0.32 0.747 -0.072 0.052_Iyear_1983 | -0.002 0.031 -0.07 0.941 -0.063 0.059_Iyear_1984 | 0.010 0.029 0.34 0.734 -0.047 0.066
------------------------------------------------------------------------------
Instruments for first differences equationStandardD.(L2.n w L.w k L.k L2.k ys L.ys L2.ys _Iyear_1977 _Iyear_1978 _Iyear_1979_Iyear_1980 _Iyear_1981 _Iyear_1982 _Iyear_1983 _Iyear_1984)GMM-type (missing=0, separate instruments for each period unless collapsed)L(2/5).L.n
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z = -2.74 Pr > z = 0.006
Arellano-Bond test for AR(2) in first differences: z = -0.67 Pr > z = 0.504
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(17) = 27.69 Prob > chi2 = 0.049(Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(17) = 21.79 Prob > chi2 = 0.193(Robust, but weakened by many instruments.)
-使用or选项向前正交变换
. xi:xtabond2 n L.n L2.n w L.w k L.k L2.k ys L.ys L2.ys i.year, gmm(L.n, lag(2 5)) iv(L2.n w L.w k L.k L2.k ys L.ys L2.ys i.year) nolevel robust small nomata
i.year _Iyear_1976-1984 (naturally coded; _Iyear_1976 omitted)
_Iyear_1977 dropped because of collinearity.
_Iyear_1978 dropped because of collinearity.
Building GMM instruments..
2 instrument(s) dropped because of collinearity.
Estimating.
Performing specification tests.Dynamic panel-data estimation, one-step difference GMM
------------------------------------------------------------------------------
Group variable: id Number of obs = 611
Time variable : year Number of groups = 140
Number of instruments = 33 Obs per group: min = 4
F(14, 139) = 117.25 avg = 4.36
Prob > F = 0.000 max = 6
------------------------------------------------------------------------------| Robustn | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------n |L1. | 1.017 0.284 3.58 0.000 0.455 1.578L2. | -0.114 0.051 -2.23 0.027 -0.215 -0.013|w |--. | -0.659 0.204 -3.22 0.002 -1.064 -0.255L1. | 0.634 0.325 1.95 0.053 -0.009 1.276|k |--. | 0.335 0.065 5.12 0.000 0.205 0.464L1. | -0.158 0.117 -1.35 0.179 -0.391 0.074L2. | -0.065 0.051 -1.28 0.204 -0.165 0.036|ys |--. | 0.680 0.198 3.43 0.001 0.289 1.072L1. | -0.993 0.401 -2.48 0.014 -1.785 -0.201L2. | 0.235 0.206 1.14 0.257 -0.173 0.642|_Iyear_1979 | 0.019 0.014 1.41 0.162 -0.008 0.047_Iyear_1980 | 0.038 0.023 1.62 0.107 -0.008 0.084_Iyear_1981 | 0.001 0.033 0.03 0.975 -0.064 0.066_Iyear_1982 | -0.010 0.031 -0.32 0.747 -0.072 0.052_Iyear_1983 | -0.002 0.031 -0.07 0.941 -0.063 0.059_Iyear_1984 | 0.010 0.029 0.34 0.734 -0.047 0.066
------------------------------------------------------------------------------
Instruments for first differences equationStandardD.(L2.n w L.w k L.k L2.k ys L.ys L2.ys _Iyear_1977 _Iyear_1978 _Iyear_1979_Iyear_1980 _Iyear_1981 _Iyear_1982 _Iyear_1983 _Iyear_1984)GMM-type (missing=0, separate instruments for each period unless collapsed)L(2/5).L.n
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z = -2.74 Pr > z = 0.006
Arellano-Bond test for AR(2) in first differences: z = -0.67 Pr > z = 0.504
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(17) = 27.69 Prob > chi2 = 0.049(Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(17) = 21.79 Prob > chi2 = 0.193(Robust, but weakened by many instruments.). xi:xtabond2 n L.n L2.n w L.w k L.k L2.k ys L.ys L2.ys i.year, gmm(L.n) iv(L2.n w L.w k L.k L2.k ys L.ys L2.ys i.year) nolevel robus small or
i.year _Iyear_1976-1984 (naturally coded; _Iyear_1976 omitted)
Favoring space over speed. To switch, type or click on mata: mata set matafavor speed, perm.
Warning: Two-step estimated covariance matrix of moments is singular.Using a generalized inverse to calculate robust weighting matrix for Hansen test.Difference-in-Sargan/Hansen statistics may be negative.Dynamic panel-data estimation, one-step difference GMM
------------------------------------------------------------------------------
Group variable: id Number of obs = 611
Time variable : year Number of groups = 140
Number of instruments = 42 Obs per group: min = 4
F(18, 140) = 109.69 avg = 4.36
Prob > F = 0.000 max = 6
------------------------------------------------------------------------------| Robustn | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------n |L1. | 0.653 0.083 7.87 0.000 0.489 0.817L2. | -0.100 0.073 -1.38 0.170 -0.244 0.043|w |--. | -0.558 0.157 -3.56 0.001 -0.867 -0.248L1. | 0.272 0.133 2.04 0.043 0.009 0.535|k |--. | 0.398 0.059 6.78 0.000 0.282 0.514L1. | -0.058 0.055 -1.04 0.300 -0.167 0.052L2. | -0.033 0.042 -0.80 0.427 -0.117 0.050|ys |--. | 0.455 0.171 2.66 0.009 0.116 0.794L1. | -0.579 0.197 -2.93 0.004 -0.969 -0.189L2. | 0.034 0.141 0.24 0.811 -0.245 0.313|_Iyear_1977 | 0.000 (omitted)_Iyear_1978 | 0.012 0.030 0.38 0.703 -0.049 0.072_Iyear_1979 | 0.014 0.030 0.48 0.632 -0.045 0.074_Iyear_1980 | 0.020 0.029 0.71 0.482 -0.037 0.077_Iyear_1981 | -0.015 0.028 -0.54 0.588 -0.071 0.041_Iyear_1982 | -0.025 0.021 -1.21 0.229 -0.067 0.016_Iyear_1983 | -0.018 0.020 -0.90 0.368 -0.058 0.022_Iyear_1984 | 0.000 (omitted)
------------------------------------------------------------------------------
Instruments for orthogonal deviations equationStandardFOD.(L2.n w L.w k L.k L2.k ys L.ys L2.ys _Iyear_1977 _Iyear_1978_Iyear_1979 _Iyear_1980 _Iyear_1981 _Iyear_1982 _Iyear_1983 _Iyear_1984)GMM-type (missing=0, separate instruments for each period unless collapsed)L(1/8).L.n
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z = -4.95 Pr > z = 0.000
Arellano-Bond test for AR(2) in first differences: z = -0.10 Pr > z = 0.918
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(24) = 62.01 Prob > chi2 = 0.000(Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(24) = 31.62 Prob > chi2 = 0.137(Robust, but weakened by many instruments.)Difference-in-Hansen tests of exogeneity of instrument subsets:iv(L2.n w L.w k L.k L2.k ys L.ys L2.ys _Iyear_1977 _Iyear_1978 _Iyear_1979 _Iyear_1980 _Iyear_1981 _Iyear_1982 _Iyear_1983 _Iyear_1984)Hansen test excluding group: chi2(9) = 11.52 Prob > chi2 = 0.242Difference (null H = exogenous): chi2(15) = 20.10 Prob > chi2 = 0.168
-使用更多工具变量
. xi:xtabond2 n L.n L2.n w L.w k L.k L2.k ys L.ys L2.ys i.year, gmm(L.n, lag(1 .)) gmm(w, lag(2 .)) gmm(L.w) gmm(L.k) gmm(k, lag(2 .)) iv(L2.n L2.k ys L.ys L2.ys i.year)
> nolevel robust small
i.year _Iyear_1976-1984 (naturally coded; _Iyear_1976 omitted)
Favoring space over speed. To switch, type or click on mata: mata set matafavor speed, perm.
Warning: Two-step estimated covariance matrix of moments is singular.Using a generalized inverse to calculate robust weighting matrix for Hansen test.Difference-in-Sargan/Hansen statistics may be negative.Dynamic panel-data estimation, one-step difference GMM
------------------------------------------------------------------------------
Group variable: id Number of obs = 611
Time variable : year Number of groups = 140
Number of instruments = 90 Obs per group: min = 4
F(18, 140) = 75.56 avg = 4.36
Prob > F = 0.000 max = 6
------------------------------------------------------------------------------| Robustn | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------n |L1. | 0.818 0.086 9.50 0.000 0.648 0.988L2. | -0.112 0.050 -2.23 0.027 -0.212 -0.013|w |--. | -0.682 0.143 -4.77 0.000 -0.964 -0.399L1. | 0.656 0.203 3.23 0.002 0.255 1.056|k |--. | 0.353 0.122 2.89 0.004 0.111 0.594L1. | -0.154 0.086 -1.78 0.078 -0.325 0.017L2. | -0.030 0.032 -0.95 0.346 -0.094 0.033|ys |--. | 0.651 0.190 3.43 0.001 0.275 1.026L1. | -0.916 0.264 -3.47 0.001 -1.439 -0.394L2. | 0.279 0.186 1.50 0.136 -0.089 0.646|_Iyear_1977 | 0.000 (omitted)_Iyear_1978 | 0.000 (omitted)_Iyear_1979 | 0.011 0.009 1.23 0.221 -0.007 0.030_Iyear_1980 | 0.026 0.017 1.52 0.132 -0.008 0.061_Iyear_1981 | -0.014 0.029 -0.47 0.640 -0.071 0.044_Iyear_1982 | -0.035 0.030 -1.16 0.246 -0.095 0.024_Iyear_1983 | -0.031 0.035 -0.88 0.381 -0.100 0.039_Iyear_1984 | -0.024 0.037 -0.65 0.518 -0.097 0.049
------------------------------------------------------------------------------
Instruments for first differences equationStandardD.(L2.n L2.k ys L.ys L2.ys _Iyear_1977 _Iyear_1978 _Iyear_1979 _Iyear_1980_Iyear_1981 _Iyear_1982 _Iyear_1983 _Iyear_1984)GMM-type (missing=0, separate instruments for each period unless collapsed)L(2/8).kL(1/8).L.kL(1/8).L.wL(2/8).wL(1/8).L.n
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z = -5.39 Pr > z = 0.000
Arellano-Bond test for AR(2) in first differences: z = -0.78 Pr > z = 0.436
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(72) = 120.62 Prob > chi2 = 0.000(Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(72) = 73.72 Prob > chi2 = 0.422(Robust, but weakened by many instruments.)Difference-in-Hansen tests of exogeneity of instrument subsets:gmm(L.n, lag(1 .))Hansen test excluding group: chi2(46) = 43.99 Prob > chi2 = 0.557Difference (null H = exogenous): chi2(26) = 29.72 Prob > chi2 = 0.279gmm(w, lag(2 .))Hansen test excluding group: chi2(65) = 73.72 Prob > chi2 = 0.215Difference (null H = exogenous): chi2(7) = 0.00 Prob > chi2 = 1.000gmm(L.w, lag(1 .))Hansen test excluding group: chi2(52) = 73.72 Prob > chi2 = 0.025Difference (null H = exogenous): chi2(20) = 0.00 Prob > chi2 = 1.000gmm(L.k, lag(1 .))Hansen test excluding group: chi2(67) = 73.72 Prob > chi2 = 0.268Difference (null H = exogenous): chi2(5) = 0.00 Prob > chi2 = 1.000gmm(k, lag(2 .))Hansen test excluding group: chi2(51) = 73.72 Prob > chi2 = 0.020Difference (null H = exogenous): chi2(21) = 0.00 Prob > chi2 = 1.000iv(L2.n L2.k ys L.ys L2.ys _Iyear_1977 _Iyear_1978 _Iyear_1979 _Iyear_1980 _Iyear_1981 _Iyear_1982 _Iyear_1983 _Iyear_1984)Hansen test excluding group: chi2(61) = 56.99 Prob > chi2 = 0.622Difference (null H = exogenous): chi2(11) = 16.72 Prob > chi2 = 0.116
一步法与两步法的比较
. xi:xtabond2 n L.n L2.n w L.w k L.k L2.k ys L.ys L2.ys i.year, gmm(L.n L.w L.k) iv(ys L.ys L2.ys i.year) nolevel robust small nomata //一步法
i.year _Iyear_1976-1984 (naturally coded; _Iyear_1976 omitted)
_Iyear_1977 dropped because of collinearity.
_Iyear_1978 dropped because of collinearity.
Building GMM instruments....
2 instrument(s) dropped because of collinearity.
Estimating.
Performing specification tests.Dynamic panel-data estimation, one-step difference GMM
------------------------------------------------------------------------------
Group variable: id Number of obs = 611
Time variable : year Number of groups = 140
Number of instruments = 90 Obs per group: min = 4
F(14, 139) = 90.85 avg = 4.36
Prob > F = 0.000 max = 6
------------------------------------------------------------------------------| Robustn | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------n |L1. | 0.818 0.086 9.51 0.000 0.648 0.988L2. | -0.112 0.050 -2.23 0.027 -0.212 -0.013|w |--. | -0.682 0.143 -4.78 0.000 -0.964 -0.400L1. | 0.656 0.202 3.24 0.001 0.256 1.056|k |--. | 0.353 0.122 2.89 0.004 0.112 0.593L1. | -0.154 0.086 -1.78 0.077 -0.324 0.017L2. | -0.030 0.032 -0.95 0.345 -0.094 0.033|ys |--. | 0.651 0.190 3.43 0.001 0.276 1.026L1. | -0.916 0.264 -3.47 0.001 -1.438 -0.394L2. | 0.279 0.186 1.50 0.135 -0.088 0.645|_Iyear_1979 | 0.011 0.009 1.23 0.220 -0.007 0.030_Iyear_1980 | 0.026 0.017 1.52 0.131 -0.008 0.061_Iyear_1981 | -0.014 0.029 -0.47 0.639 -0.071 0.044_Iyear_1982 | -0.035 0.030 -1.17 0.245 -0.094 0.024_Iyear_1983 | -0.031 0.035 -0.88 0.380 -0.100 0.038_Iyear_1984 | -0.024 0.037 -0.65 0.517 -0.097 0.049
------------------------------------------------------------------------------
Instruments for first differences equationStandardD.(ys L.ys L2.ys _Iyear_1977 _Iyear_1978 _Iyear_1979 _Iyear_1980_Iyear_1981 _Iyear_1982 _Iyear_1983 _Iyear_1984)GMM-type (missing=0, separate instruments for each period unless collapsed)L(1/.).(L.n L.w L.k)
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z = -5.39 Pr > z = 0.000
Arellano-Bond test for AR(2) in first differences: z = -0.78 Pr > z = 0.436
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(74) = 120.62 Prob > chi2 = 0.001(Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(74) = 73.72 Prob > chi2 = 0.487(Robust, but weakened by many instruments.). xi:xtabond2 n L.n L2.n w L.w k L.k L2.k ys L.ys L2.ys i.year, gmm(L.n L.w L.k) iv(ys L.ys L2.ys i.year) two nolevel robust small nomata //两步法
i.year _Iyear_1976-1984 (naturally coded; _Iyear_1976 omitted)
_Iyear_1977 dropped because of collinearity.
_Iyear_1978 dropped because of collinearity.
Building GMM instruments....
2 instrument(s) dropped because of collinearity.
Estimating.
Computing Windmeijer finite-sample correction...............................................................................................................................
> ..............
Performing specification tests.Dynamic panel-data estimation, two-step difference GMM
------------------------------------------------------------------------------
Group variable: id Number of obs = 611
Time variable : year Number of groups = 140
Number of instruments = 90 Obs per group: min = 4
F(14, 139) = 78.27 avg = 4.36
Prob > F = 0.000 max = 6
------------------------------------------------------------------------------| Correctedn | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------n |L1. | 0.824 0.097 8.51 0.000 0.633 1.016L2. | -0.101 0.053 -1.90 0.059 -0.207 0.004|w |--. | -0.711 0.152 -4.67 0.000 -1.013 -0.410L1. | 0.631 0.178 3.54 0.001 0.279 0.984|k |--. | 0.377 0.135 2.79 0.006 0.110 0.643L1. | -0.169 0.113 -1.49 0.137 -0.392 0.055L2. | -0.058 0.044 -1.32 0.191 -0.145 0.029|ys |--. | 0.662 0.170 3.89 0.000 0.325 0.999L1. | -0.943 0.259 -3.65 0.000 -1.454 -0.432L2. | 0.361 0.196 1.84 0.068 -0.027 0.748|_Iyear_1979 | 0.017 0.010 1.73 0.086 -0.002 0.036_Iyear_1980 | 0.030 0.016 1.83 0.070 -0.002 0.062_Iyear_1981 | -0.012 0.027 -0.44 0.663 -0.066 0.042_Iyear_1982 | -0.022 0.031 -0.71 0.481 -0.084 0.040_Iyear_1983 | -0.005 0.039 -0.12 0.905 -0.082 0.072_Iyear_1984 | -0.002 0.044 -0.03 0.972 -0.088 0.085
------------------------------------------------------------------------------
Instruments for first differences equationStandardD.(ys L.ys L2.ys _Iyear_1977 _Iyear_1978 _Iyear_1979 _Iyear_1980_Iyear_1981 _Iyear_1982 _Iyear_1983 _Iyear_1984)GMM-type (missing=0, separate instruments for each period unless collapsed)L(1/.).(L.n L.w L.k)
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z = -3.92 Pr > z = 0.000
Arellano-Bond test for AR(2) in first differences: z = -0.77 Pr > z = 0.441
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(74) = 120.62 Prob > chi2 = 0.001(Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(74) = 73.72 Prob > chi2 = 0.487(Robust, but weakened by many instruments.)
参考资料
小白学统计|面板数据分析与Stata应用笔记(七)