动态面板简介

Source:
Dynamic Panel Data : IV and GMM Estimation with Stata (Panel)
xtabond cheat sheet

简介

动态面板数据模型，是指通过在静态面板数据模型中引入滞后被解释变量以反映动态滞后效应的模型。这种模型的特殊性在于被解释变量的动态滞后项与随机误差组成部分中的个体效应相关，从而造成估计的内生性。

也就是说在动态面板模型允许使用过去的可观测值，考虑了过去结果对当前结果的影响。而实际上许多经济学问题本质上都是动态的，其中一些经济模型表明，当前的行为依赖于过去的行为，例如就业模型和公司投资问题。

AR(1) 模型

$y_{i,t}=\delta y_{i,t-1}+x'_{i,t} \beta+\alpha_{i}+\varepsilon_{i,t} \quad (1)$ $\ i=1,...,N \qquad t=1,...,T\$

基本设定

$y_{i,t}$ 是一个 $n_{i} \times 1$ 的向量,由时间t上的样本中每个空间单位的被解释变量的一个观测值， $\delta$ 为其在时间上的滞后值 $y_{i,t-1}$ 的响应参数
$x_{i,t}$ 是 $n_{i}\times k$ 的外生解释变量矩阵， $\beta$ 为其响应参数向量
$\alpha_{i}$ 是特定效应，且有 $\alpha_{i} \sim (0,\sigma_{\alpha}^2)$
$\epsilon_{i,t}$ 是服从 $i.i.d$ 分布的干扰项，且有 $\epsilon_{i,t} \sim (0,\sigma_{\epsilon}^2)$

这里就特定效应类型将模型分为两大类

固定效应模型(Fixed effect)
通过消去特定效应项 $\alpha_{i}$ 得到
$y_{i,t}-\overline{y}_{i}=\delta(y_{i,t-1}-\overline{y}_{i.-1})+(x_{i,t}-\overline{x}_{i})\beta+(\epsilon_{i,t}-\overline{\epsilon}_{i})\quad (2)$ $\overline{y}_{i,-1}=\sum_{t=2}^{T}\frac{y_{i,t-1}}{T-1}$ 但是此时 $(y_{i,t-1}-\overline{y}_{i.-1})$ 与 $(\epsilon_{i,t}-\overline{\epsilon}_{i})$ 相关，尽管是在 $\epsilon_{i,t}$ 是序列不相关的的情况下。这是因为 $y_{i,-1}$ 与 $\overline{\epsilon}_{i}$ 相关，后一项的均值包含了 $\epsilon_{i,t-1}$ 这一项，明显与 $y_{i,t-1}$ 是相关的。这可以看出固定效应模型的估计是有偏的。
随机效应模型(Random effect )
若使用广义最小二乘法(GLS)，同理可知， $(y_{i,t-1}-\theta\overline{y}_{i.-1})$ 与 $(\epsilon_{i,t}-\theta\overline{\epsilon}_{i})$ 是相关的。

因此，可以看出，对于一个动态面板数据模型来说，若使用GLS估计方法，不管是固定效应模型还是随机效应模型，它的估计都是有偏且不一致连续的。且偏误 $(bias)$ 的阶数是 $\frac{1}{T}$ ,仅当 $T\rightarrow0$ 时 $bias\rightarrow0$ 。但是当 $T$ 很小， $N\rightarrow \infty$ 时可能会引起很大的偏误。

提出问题：为什么当 $T$ 固定， $N\rightarrow \infty$ 时会出现偏误和不一致连续的情况？
- 首先考虑没有外生向量的模型：
  $y_{i,t}=\delta y_{i,t-1}+\alpha_{i}+\epsilon_{i,t} \quad \left| \delta\right|<1 \quad (3)$ 假设，对于向量 $y_{i,t}$ 我们在 $t=0,1,...,T$ 时有观测值，
  则固定效应模型的估计量为
  $\hat{\delta_{FE}}=\frac{\sum_{i=1}^{N}\sum_{t=1}^{T}(y_{i,t}-\bar{y_{i}})(y_{i,t-1}-\bar{y_{i,-1}})}{\sum_{i=1}^{N}\sum_{t=1}^{T}(y_{i,t-1}-\bar{y_{i,-1}})^2 }\quad (4)$ $\bar{y_{i,t}}=\frac{1}{T}\sum_{t=1}^{T}y_{i,t-1} \ \& \ \bar{y_{i,-1}}=\frac{1}{T}\sum_{t=1}^{T}y_{i,t-1}$ 将等式(3)代入等式(4)中，可以得到估计量的特征 $\hat{\delta_{FE}}=\delta+\frac{\frac{1}{NT}\sum_{i=1}^{N}\sum_{t=1}^{T}(\epsilon_{i,t}-\bar{\epsilon_{i}})(y_{i,t-1}-\bar{y_{i,-1}})}{\frac{1}{NT}\sum_{i=1}^{N}\sum_{t=1}^{T}(y_{i,t-1}-\bar{y_{i,-1}})^2}\quad (5)$
  等式(5)中，当 $T$ 固定， $N\rightarrow \infty$ 的情况下，FE模型的估计值有偏且不一致。因为等式(5)右边最后一项的期望不为零，且在 $N\rightarrow \infty$ 的情况下不收敛到零。Nickell (1981) 和 Hsia (2003) 曾提出过： $plim_{n\rightarrow \infty}\frac{1}{NT}\sum_{i=1}^{N}\sum_{t=1}^{T}(\epsilon_{i,t}-\bar{\epsilon_{i}})(y_{i,t-1}-\bar{y_{i,-1}})=-\frac{\delta_{\epsilon}^2}{T^2}\times\frac{(T-1)-T_{\delta+\delta^T}}{(1-\delta)^2}\neq0 \quad (6)$ 因此，对于固定的 $T$ ，我们有不一致的估计量，不过这与 $\alpha_{i}$ 无关， $\alpha_{i}$ 已经在估计过程过被去掉了。
  这里的问题是被解释变量的动态滞后项与随机误差组成部分中的个体效应相关，如在等式(2)与(5)我们所遇到的问题，即 $y_{i,-1}$ 与 $\overline{\epsilon}_{i}$ 相关。但当 $T\rightarrow \infty$ 时，等式(6)收敛到零，此时当 $T\rightarrow \infty$ 和 $N\rightarrow \infty$ 时， $\delta_{FE}$ 的估计是一致的。

估计方法

IV

为了解决估计量不一致的问题，Anderson 和 Hsio (1981) 提出了工具变量估计(IV)。首先，我们考虑消去个体效应 $\alpha_{i}$ ，对此，做差分得：
$y_{i,t}-y_{i,t-1}=\delta(y_{i,t-1}-y_{i,t-2})+(\epsilon_{i,t}-\epsilon_{i,t-1}) \quad t=2,...,T\quad (7)$

此处若使用最小二乘法(OSL)得到等式(7)的估计量是不一致的，尽管是在 $T\rightarrow \infty$ 的情况下，这是因为 $y_{i,t-1}$ 和 $\epsilon_{i,t-1}$ 相关。

转换规范等式(7)给出一个IV估计的方法，例如， $y_{i,t-2}$ 与 $(y_{i,t-1}-y_{i,t-2})$ 相关，但与 $\epsilon_{i,t-1}$ 不相关，所以选取 $y_{i,t-2}$ 作为工具变量，给出了 $\delta$ 的一个IV估计： $\hat{y_{IV}}=\frac{\sum_{i=1}^{N}\sum_{t=2}^{T}y_{i,t-2}(y_{i,t}-y_{i,t-1})}{\sum_{i=1}^{N}\sum_{t=2}^{T}y_{i,t-2}(y_{i,t-1}-y_{i,t-2})} \quad (8)$
对应的关于等式(8)的一致性条件为 $plim_{N \rightarrow \infty}\frac{1}{N(T-1)}\sum_{i=1}^{N}\sum_{t=2}^{T}(\epsilon_{i,t}-\epsilon_{i,t-1})y_{i,t-2}=0 \quad (9)$ $T\rightarrow \infty \ or \ N\rightarrow \infty \ or \ T\rightarrow \infty \ and \ N \rightarrow \infty$ 注意这里 $(\epsilon_{i,t}-\epsilon_{i,t-1})$ 是MA(1)

Anderson 和 Hsio (1981) 又给出了另一个IV估计的方案，这里选择将 $(y_{i,t-2}-y_{i,t-3})$ 作为工具变量。 $\hat{y_{IV}}^{(2)}=\frac{\sum_{i=1}^{N}\sum_{t=3}^{T}(y_{i,t-2}-y_{i,t-3})(y_{i,t}-y_{i,t-1})}{\sum_{i=1}^{N}\sum_{t=3}^{T}(y_{i,t-2}-y_{i,t-3})(y_{i,t-1}-y_{i,t-2})} \quad (10)$ 同理，得到等式(10)的一致性条件为 $plim_{N \rightarrow \infty}\frac{1}{N(T-2)}\sum_{i=1}^{N}\sum_{t=3}^{T}(\epsilon_{i,t}-\epsilon_{i,t-1})(y_{i,t-2}-y_{i,t-3})=0 \quad (11)$ 且只要 $\epsilon_{i,t}$ 不是自相关的，等式(8)和(11)的一致性都得到了保障。

可以看出，等式(10)构建工具变量时比等式(8)多引入了一个滞后项，这导致缺失了一期的样本数据。这就又提出了一个问题，我们究竟是选择等式(8)还是等式(10)的估计比较好？
而矩估计(MM)法则不需要考虑这个问题，MM估计可以统一所有的估计量且消去样本容量减少的缺点。

GMM

可得等式(9)的矩条件为 $plim_{N \rightarrow \infty}\frac{1}{N(T-1)}\sum_{i=1}^{N}\sum_{t=2}^{T}(\epsilon_{i,t}-\epsilon_{i,t-1})y_{i,t-2}=E[(\epsilon_{i,t}-\epsilon_{i,t-1})y_{i,t-2} ]=0 \quad (12)$
同理，等式(11)的矩条件为 $plim_{N \rightarrow \infty}\frac{1}{N(T-2)}\sum_{i=1}^{N}\sum_{t=3}^{T}(\epsilon_{i,t}-\epsilon_{i,t-1})(y_{i,t-2}-y_{i,t-3})=E[(\epsilon_{i,t}-\epsilon_{i,t-1})(y_{i,t-2}-y_{i,t-3})]=0 \ (13)$ 这两个IV估计值都在估计中附加了一个矩条件。但是，据我们所知，附加更多的矩条件则能增加估计值的效率。

据此，Arellano 和 Bond (1991) 提出可以通过开发附加矩来扩大工具的列表并且是工具的数量随着 t 变化。首先，固定T，这里考虑 T=4。

t = 2 时，矩条件变为： $E[(\epsilon_{i,2}-\epsilon_{i,1})y_{i,0}]=0$
这意味着变量 $y_{i,0}$ 是有效的工具，因为它与 $(y_{i,2}-y_{i,1})$ 高度相关而与 $(\epsilon_{i,2}-\epsilon_{i,1})$ 不相关。
t = 3 时，同理可得，矩条件为： $E[(\epsilon_{i,3}-\epsilon_{i,2})y_{i,1}]=0$ 同时也满足 $E[(\epsilon_{i,2}-\epsilon_{i,1})y_{i,0}]=0$
其中， $y_{i,0}\ y_{i,1}$ 与 $(y_{i,2}-y_{i,1})$ 相关，而与 $(\epsilon_{i,3}-\epsilon_{i,2})$ 不相关。
t = 4 时，我们将得到三组矩条件
$E[(\epsilon_{i,2}-\epsilon_{i,1})y_{i,0}]=0$ $E[(\epsilon_{i,3}-\epsilon_{i,2})y_{i,1}]=0$ $E[(\epsilon_{i,4}-\epsilon_{i, 3})y_{i,2}]=0$

一直继续这么添加矩条件之后，有效的工具集合变为 $(y_{i0},y_{i2}...y_{i,T-2})$
而所有的这些矩条件可以作为广义矩估计(GMM)的一个框架。对于一般的样本大小 T，所有差分后的误差项可排成一个向量：
$\Delta \varepsilon_{i}=\left( \begin{array}{c}{\varepsilon_{i 2}-\varepsilon_{i 1}} \\ {\cdots} \\ {\varepsilon_{i, T}-\varepsilon_{i, T-1}}\end{array}\right) \quad (14)$
由工具变量排成的矩阵为：
$Z_{i} =\left( \begin{array}{ccccccc}{y_{i0}} & {0} & {0} & {\ldots} & {0} & {\ldots} & {0}\\ {0} & {y_{i0}} & {y_{i1}} & {\ldots} & {0} & {\ldots} & {0} \\ {\vdots} & {\vdots} & {\vdots} & {\ldots} & {\vdots} & {\ddots} & {\vdots} \\ {0} & {0} & {0} & {\ldots} & {y_{i0}} & {\ldots} & {y_{i, T-2}} \end{array}\right) \ (15)$ 矩阵 $Z_{i}$ 的每一行都包含了给定时段所有有效工具。因此，所有矩条件的集合可写为： $E\left\{Z_{i}^{\prime} \Delta \varepsilon_{i}\right\}=0 \qquad (16)$ 为了得到GMM估计，将等式(16)改写成 $E\left\{Z_{i}^{\prime}\left(\Delta y_{i}-\gamma \Delta y_{i,-1}\right)\right\}=0 \qquad (17)$

显然，矩条件的数量会增加未知参数的数量。
$\gamma$ 这里我们通过最小化由等式(17)所对应的条件表达的二次型来估计参数 $\gamma$ ：
$\min _{\gamma}\left[\frac{1}{N} \sum_{i=1}^{N} Z_{i}^{\prime}\left(\Delta y_{i}-\gamma \Delta y_{i,-1}\right)\right]^{\prime} W_{N}\left[\frac{1}{N} \sum_{i=1}^{N} Z_{i}^{\prime}\left(\Delta y_{i}-\gamma \Delta y_{i,-1}\right)\right] \quad (18)$

其中 $W_{i}$ 是定义的对称正定的权重矩阵。
然后，通过对等式(18)关于 $\gamma$ 求微分并求解 $\gamma$ 得到GMM估计量： $\hat{\gamma_{G M M}}=\left(\left(\sum_{i=1}^{N} \Delta y_{i,-1}^{\prime} Z_{i}\right) W_{N}\left(\sum_{i=1}^{N} Z_{i}^{\prime} \Delta y_{i,-1}\right)\right)^{-1} \times\left(\sum_{i=1}^{N} \Delta y_{i,-1}^{\prime} Z_{i}\right) W_{N}\left(\sum_{i=1}^{N} Z_{i}^{\prime} \Delta y_{i}\right) \ (19)$

GMM估计并不强制要求 $\varepsilon_{it}$ 关于个体与时间服从独立分布，但是需要注意的是，需要不存在自相关以保证矩条件有效。因此，在一个短面板（即T比较小），建议强制 $\epsilon_{it}$ 不存在自相关，并结合同方差性假设。
Alvarez 和 Arellano (2003) 表示，通常，GMM估计在 $T \rightarrow \infty , N \rightarrow \infty$ 的情况下仍然保持一致性。但是，对于 $T \rightarrow \infty$ 的情况下，GMM估计和FE估计会很接近，这给我们估计方法提供了一个更具有吸引力的选择。

实际应用

Arellano和Bond ( 1991) 提出了一阶差分GMM (FD-GMM)估计方法，主要做法是用变量的水平滞后值作为其一阶差分项的工具变量。具体来说：

由我们上文所考虑的没有外生向量的模型(3)的水平方程 $y_{i,t}=\delta y_{i,t-1}+\alpha_{i}+\varepsilon_{i,t}$ ，考虑它的差分方程
$\Delta y_{it}=\delta \Delta y_{i,t-1} + \Delta \varepsilon_{it} \ (20)$ 因为 $\Delta y_{i,t-1}$ 与 $\Delta \varepsilon_{it}$ 相关联，所以将 $y_{i,t-2}$ 作为 $\Delta y_{i,t-1}$ 的工具变量。这种估计方法就是FD-GMM。

但是, Blundell和Bond (1998)曾指出,一阶差分GMM估计方法容易受到弱工具变量的影响而得到有偏的估计结果。即：

由模型(3)的水平方程可得 $\Delta y_{i,t-1}=(\delta-1)y_{i,t-2}+\alpha_{i}+\varepsilon_{i,t-1}$
当 $\delta$ 接近1的时候，工具变量和外生变量的关系就会变的很弱，这就会产生“弱工具变量问题”。

为了克服弱工具变量的影响, Arellano和Bover (1995) 以及Blundell和Bond (1998)提出了另外一种更加有效的方法,即系统GMM (SYS-GMM)估计方法。其具体做法是将水平方程和差分方程结合起来进行估计,在这种估计方法中,滞后水平作为一阶差分的工具变量,而一阶差分又作为水平变量的工具变量。具体来说：
在上述FD-GMM估计中，可以看作是应用了矩条件 $E(\Delta \varepsilon_{it}y_{i,t-2}=0)$ ，但是却有个矩条件被忽略了，
即： $E(\Delta y_{i,t-1}(\alpha_{i}+\varepsilon_{it}))=E(\Delta y_{i,t-1}(y_{it}-\delta y_{i,t-1}))=0$
而SYS-GMM则是考虑到了这一点，利用了更多的有效信息，使得估计不受 $\delta$ 接近1时的影响。

Stata 范例

我们可以使用xtabond命令来实现FD-GMM估计，这里我们使用数据abdata

 . webuse abdata
 . xtabond n L(0/2).(w k) yr1980-yr1984 year, vce(robust)

Arellano-Bond dynamic panel-data estimation     Number of obs     =        611
Group variable: id                              Number of groups  =        140
Time variable: year
                                                Obs per group:
                                                              min =          4
                                                              avg =   4.364286
                                                              max =          6

Number of instruments =     40                  Wald chi2(13)     =    1318.68
                                                Prob > chi2       =     0.0000
One-step results
                                     (Std. Err. adjusted for clustering on id)
------------------------------------------------------------------------------
             |               Robust
           n |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           n |
         L1. |   .6286618   .1161942     5.41   0.000     .4009254    .8563983
             |
           w |
         --. |  -.5104249   .1904292    -2.68   0.007    -.8836592   -.1371906
         L1. |   .2891446    .140946     2.05   0.040     .0128954    .5653937
         L2. |  -.0443653   .0768135    -0.58   0.564     -.194917    .1061865
             |
           k |
         --. |   .3556923   .0603274     5.90   0.000     .2374528    .4739318
         L1. |  -.0457102   .0699732    -0.65   0.514    -.1828552    .0914348
         L2. |  -.0619721   .0328589    -1.89   0.059    -.1263743    .0024301
             |
      yr1980 |  -.0282422   .0166363    -1.70   0.090    -.0608488    .0043643
      yr1981 |  -.0694052    .028961    -2.40   0.017    -.1261677   -.0126426
      yr1982 |  -.0523678   .0423433    -1.24   0.216    -.1353591    .0306235
      yr1983 |  -.0256599   .0533747    -0.48   0.631    -.1302723    .0789525
      yr1984 |  -.0093229   .0696241    -0.13   0.893    -.1457837    .1271379
        year |   .0019575   .0119481     0.16   0.870    -.0214604    .0253754
       _cons |  -2.543221   23.97919    -0.11   0.916    -49.54158    44.45514
------------------------------------------------------------------------------
Instruments for differenced equation
        GMM-type: L(2/.).n
        Standard: D.w LD.w L2D.w D.k LD.k L2D.k D.yr1980 D.yr1981 D.yr1982
                  D.yr1983 D.yr1984 D.year
Instruments for level equation
        Standard: _cons

下面用xtdpdsys命令来实现SYS-GMM估计

. xtdpdsys n L(0/2).(w k) yr1980-yr1984 year, vce(robust)

System dynamic panel-data estimation            Number of obs     =        751
Group variable: id                              Number of groups  =        140
Time variable: year
                                                Obs per group:
                                                              min =          5
                                                              avg =   5.364286
                                                              max =          7

Number of instruments =     47                  Wald chi2(13)     =    2579.96
                                                Prob > chi2       =     0.0000
One-step results
------------------------------------------------------------------------------
             |               Robust
           n |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           n |
         L1. |   .8221535    .093387     8.80   0.000     .6391184    1.005189
             |
           w |
         --. |  -.5427935   .1881721    -2.88   0.004     -.911604   -.1739831
         L1. |   .3703602   .1656364     2.24   0.025     .0457189    .6950015
         L2. |  -.0726314   .0907148    -0.80   0.423    -.2504292    .1051664
             |
           k |
         --. |   .3638069   .0657524     5.53   0.000     .2349346    .4926792
         L1. |  -.1222996   .0701521    -1.74   0.081    -.2597951     .015196
         L2. |  -.0901355   .0344142    -2.62   0.009    -.1575862   -.0226849
             |
      yr1980 |  -.0308622    .016946    -1.82   0.069    -.0640757    .0023512
      yr1981 |  -.0718417   .0293223    -2.45   0.014    -.1293123    -.014371
      yr1982 |  -.0384806   .0373631    -1.03   0.303    -.1117111    .0347498
      yr1983 |  -.0121768   .0498519    -0.24   0.807    -.1098847    .0855311
      yr1984 |  -.0050903   .0655011    -0.08   0.938    -.1334701    .1232895
        year |   .0058631   .0119867     0.49   0.625    -.0176304    .0293566
       _cons |  -10.59198   23.92087    -0.44   0.658    -57.47602    36.29207
------------------------------------------------------------------------------
Instruments for differenced equation
        GMM-type: L(2/.).n
        Standard: D.w LD.w L2D.w D.k LD.k L2D.k D.yr1980 D.yr1981 D.yr1982
                  D.yr1983 D.yr1984 D.year
Instruments for level equation
        GMM-type: LD.n
        Standard: _cons

可以看到工具变量的数量变多。

xtdpd则是一种更加灵活的替代方法，它可以用比xtabond和xtdpdsys更复杂的结构来拟合在特殊误差和预定变量中具有低阶移动平均关联的模型。

假设检验

序列相关检验

可以看出 Arellano–Bond估计工具变量的设定的关键点在于 $E\left(\Delta y_{i(t-j)} \Delta \varepsilon_{i t}\right)=0 \quad j \geq 2$
我们可以在 Stata 中使用estat abond命令来测试这些条件。
从本质上来说，一般未观测到的的差分项 $\Delta y_{it}$ 应与其因变量的第二期滞后 $y_{i,t-2}$ 和之后的滞后项都无关。如果不是这样，我们又回到了一开始的问题，内生性。所以我们主要关心的是有没有2阶或者更高阶的序列相关性。
下面我们用estat bond命令来测试我们上述的例子：

 . estat abond, artests(4)

Dynamic panel-data estimation                   Number of obs     =        751
Group variable: id                              Number of groups  =        140
Time variable: year
                                                Obs per group:
                                                              min =          5
                                                              avg =   5.364286
                                                              max =          7

Number of instruments =     47                  Wald chi2(13)     =    2579.96
                                                Prob > chi2       =     0.0000
One-step results
                                     (Std. Err. adjusted for clustering on id)
------------------------------------------------------------------------------
             |               Robust
           n |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           n |
         L1. |   .8221535    .093387     8.80   0.000     .6391184    1.005189
             |
           w |
         --. |  -.5427935   .1881721    -2.88   0.004     -.911604   -.1739831
         L1. |   .3703602   .1656364     2.24   0.025     .0457189    .6950015
         L2. |  -.0726314   .0907148    -0.80   0.423    -.2504292    .1051664
             |
           k |
         --. |   .3638069   .0657524     5.53   0.000     .2349346    .4926792
         L1. |  -.1222996   .0701521    -1.74   0.081    -.2597951     .015196
         L2. |  -.0901355   .0344142    -2.62   0.009    -.1575862   -.0226849
             |
      yr1980 |  -.0308622    .016946    -1.82   0.069    -.0640757    .0023512
      yr1981 |  -.0718417   .0293223    -2.45   0.014    -.1293123    -.014371
      yr1982 |  -.0384806   .0373631    -1.03   0.303    -.1117111    .0347498
      yr1983 |  -.0121768   .0498519    -0.24   0.807    -.1098847    .0855311
      yr1984 |  -.0050903   .0655011    -0.08   0.938    -.1334701    .1232895
        year |   .0058631   .0119867     0.49   0.625    -.0176304    .0293566
       _cons |  -10.59198   23.92087    -0.44   0.658    -57.47602    36.29207
------------------------------------------------------------------------------
Instruments for differenced equation
        GMM-type: L(2/.).n
        Standard: D.w LD.w L2D.w D.k LD.k L2D.k D.yr1980 D.yr1981 D.yr1982
                  D.yr1983 D.yr1984 D.year
Instruments for level equation
        GMM-type: LD.n
        Standard: _cons

Arellano-Bond test for zero autocorrelation in first-differenced errors
  +-----------------------+
  |Order |  z     Prob > z|
  |------+----------------|
  |   1  |-4.6414  0.0000 |
  |   2  |-1.0572  0.2904 |
  |   3  |-.19492  0.8455 |
  |   4  | .04472  0.9643 |
  +-----------------------+
   H0: no autocorrelation

可以看到的是，我们可以拒绝一阶序列相关的原假设，但是不能拒绝2、3、4阶序列相关的原假设。所以原模型是合理的。

过度识别检验（estat overid）

检验工具变量是否是与干扰项相关，也就是工具变量是否为外生变量。
原假设是：所有工具变量都是外生。
其中包含 hansen 检验和 sargan 检验，可以用 Stata 中的xtabond2命令实现。

Stata 范例

xtabond命令实现了 Arellano 和 Bond Roodman在1991年所提出的一阶差分矩估计法（FD-GMM），此时它的矩条件是将因变量的滞后和外生变量的一阶差分作为一阶差分方程的工具。
xtdpdsys实现了Arellano、Bover/Blundell 和 Bond在1998年提出的系统 GMM 估计（SYS-GMM），它使用了xtabond的矩条件，同时又将因变量滞后第一差分作为水平方程的工具。
xtdpd则是一种更加灵活的替代方法，它可以用比xtabond和xtdpdsys更复杂的结构来拟合在特殊误差和预定变量中具有低阶移动平均关联的模型。
后估计工具允许您在一阶差分残差中测试序列相关性，并测试过度识别限制的有效性。

例子：
基于Layard和Nickell(1986)的工作，Arellano和Bond(1991)将劳动力需求的动态模型拟合到位于英国的一个具有不平衡面板的公司上。首先，我们根据工资(wages)、资本存量(capital stock)、行业产出(industry output)、年度假人(year dummies) 以及一个时间趋势(a time trend )对就业率(employment) 进行建模，其中包括就业，工资和资本存量的滞后。我们将使用xtabond命令

 . webuse abdata
 . xtabond n L(0/2).(w k) yr1980-yr1984 year, vce(robust)

Arellano-Bond dynamic panel-data estimation     Number of obs     =        611
Group variable: id                              Number of groups  =        140
Time variable: year
                                                Obs per group:
                                                              min =          4
                                                              avg =   4.364286
                                                              max =          6

Number of instruments =     40                  Wald chi2(13)     =    1318.68
                                                Prob > chi2       =     0.0000
One-step results
                                     (Std. Err. adjusted for clustering on id)
------------------------------------------------------------------------------
             |               Robust
           n |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           n |
         L1. |   .6286618   .1161942     5.41   0.000     .4009254    .8563983
             |
           w |
         --. |  -.5104249   .1904292    -2.68   0.007    -.8836592   -.1371906
         L1. |   .2891446    .140946     2.05   0.040     .0128954    .5653937
         L2. |  -.0443653   .0768135    -0.58   0.564     -.194917    .1061865
             |
           k |
         --. |   .3556923   .0603274     5.90   0.000     .2374528    .4739318
         L1. |  -.0457102   .0699732    -0.65   0.514    -.1828552    .0914348
         L2. |  -.0619721   .0328589    -1.89   0.059    -.1263743    .0024301
             |
      yr1980 |  -.0282422   .0166363    -1.70   0.090    -.0608488    .0043643
      yr1981 |  -.0694052    .028961    -2.40   0.017    -.1261677   -.0126426
      yr1982 |  -.0523678   .0423433    -1.24   0.216    -.1353591    .0306235
      yr1983 |  -.0256599   .0533747    -0.48   0.631    -.1302723    .0789525
      yr1984 |  -.0093229   .0696241    -0.13   0.893    -.1457837    .1271379
        year |   .0019575   .0119481     0.16   0.870    -.0214604    .0253754
       _cons |  -2.543221   23.97919    -0.11   0.916    -49.54158    44.45514
------------------------------------------------------------------------------
Instruments for differenced equation
        GMM-type: L(2/.).n
        Standard: D.w LD.w L2D.w D.k LD.k L2D.k D.yr1980 D.yr1981 D.yr1982
                  D.yr1983 D.yr1984 D.year
Instruments for level equation
        Standard: _cons

由于我们的回归模型中包含一个n的滞后，xtabond使用滞后2期和back作为工具。外生变量的差分也可以作为工具。

这里，我们使用xtdpdsys来重新定义模型

. xtdpdsys n L(0/2).(w k) yr1980-yr1984 year, vce(robust)

System dynamic panel-data estimation            Number of obs     =        751
Group variable: id                              Number of groups  =        140
Time variable: year
                                                Obs per group:
                                                              min =          5
                                                              avg =   5.364286
                                                              max =          7

Number of instruments =     47                  Wald chi2(13)     =    2579.96
                                                Prob > chi2       =     0.0000
One-step results
------------------------------------------------------------------------------
             |               Robust
           n |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           n |
         L1. |   .8221535    .093387     8.80   0.000     .6391184    1.005189
             |
           w |
         --. |  -.5427935   .1881721    -2.88   0.004     -.911604   -.1739831
         L1. |   .3703602   .1656364     2.24   0.025     .0457189    .6950015
         L2. |  -.0726314   .0907148    -0.80   0.423    -.2504292    .1051664
             |
           k |
         --. |   .3638069   .0657524     5.53   0.000     .2349346    .4926792
         L1. |  -.1222996   .0701521    -1.74   0.081    -.2597951     .015196
         L2. |  -.0901355   .0344142    -2.62   0.009    -.1575862   -.0226849
             |
      yr1980 |  -.0308622    .016946    -1.82   0.069    -.0640757    .0023512
      yr1981 |  -.0718417   .0293223    -2.45   0.014    -.1293123    -.014371
      yr1982 |  -.0384806   .0373631    -1.03   0.303    -.1117111    .0347498
      yr1983 |  -.0121768   .0498519    -0.24   0.807    -.1098847    .0855311
      yr1984 |  -.0050903   .0655011    -0.08   0.938    -.1334701    .1232895
        year |   .0058631   .0119867     0.49   0.625    -.0176304    .0293566
       _cons |  -10.59198   23.92087    -0.44   0.658    -57.47602    36.29207
------------------------------------------------------------------------------
Instruments for differenced equation
        GMM-type: L(2/.).n
        Standard: D.w LD.w L2D.w D.k LD.k L2D.k D.yr1980 D.yr1981 D.yr1982
                  D.yr1983 D.yr1984 D.year
Instruments for level equation
        GMM-type: LD.n
        Standard: _cons

比较这两个命令输出的页脚说明了这两个估计器之间的关键区别。xtdpdsys将n的滞后差也作为工具纳入水平方程，而xtabond没有。xtdpdsys的工具变量变多了，但是模型的标准误降低了。

这些GMM估计量的矩条件只有在特征误差不存在序列相关性的情况下才有效。由于白噪声的第一个差异必然是自相关的，我们只需要关注第二个和更高的自相关。我们可以使用estat abond测试自相关:

 . estat abond, artests(4)

Dynamic panel-data estimation                   Number of obs     =        751
Group variable: id                              Number of groups  =        140
Time variable: year
                                                Obs per group:
                                                              min =          5
                                                              avg =   5.364286
                                                              max =          7

Number of instruments =     47                  Wald chi2(13)     =    2579.96
                                                Prob > chi2       =     0.0000
One-step results
                                     (Std. Err. adjusted for clustering on id)
------------------------------------------------------------------------------
             |               Robust
           n |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           n |
         L1. |   .8221535    .093387     8.80   0.000     .6391184    1.005189
             |
           w |
         --. |  -.5427935   .1881721    -2.88   0.004     -.911604   -.1739831
         L1. |   .3703602   .1656364     2.24   0.025     .0457189    .6950015
         L2. |  -.0726314   .0907148    -0.80   0.423    -.2504292    .1051664
             |
           k |
         --. |   .3638069   .0657524     5.53   0.000     .2349346    .4926792
         L1. |  -.1222996   .0701521    -1.74   0.081    -.2597951     .015196
         L2. |  -.0901355   .0344142    -2.62   0.009    -.1575862   -.0226849
             |
      yr1980 |  -.0308622    .016946    -1.82   0.069    -.0640757    .0023512
      yr1981 |  -.0718417   .0293223    -2.45   0.014    -.1293123    -.014371
      yr1982 |  -.0384806   .0373631    -1.03   0.303    -.1117111    .0347498
      yr1983 |  -.0121768   .0498519    -0.24   0.807    -.1098847    .0855311
      yr1984 |  -.0050903   .0655011    -0.08   0.938    -.1334701    .1232895
        year |   .0058631   .0119867     0.49   0.625    -.0176304    .0293566
       _cons |  -10.59198   23.92087    -0.44   0.658    -57.47602    36.29207
------------------------------------------------------------------------------
Instruments for differenced equation
        GMM-type: L(2/.).n
        Standard: D.w LD.w L2D.w D.k LD.k L2D.k D.yr1980 D.yr1981 D.yr1982
                  D.yr1983 D.yr1984 D.year
Instruments for level equation
        GMM-type: LD.n
        Standard: _cons

Arellano-Bond test for zero autocorrelation in first-differenced errors
  +-----------------------+
  |Order |  z     Prob > z|
  |------+----------------|
  |   1  |-4.6414  0.0000 |
  |   2  |-1.0572  0.2904 |
  |   3  |-.19492  0.8455 |
  |   4  | .04472  0.9643 |
  +-----------------------+
   H0: no autocorrelation

参考文献：
1、 Dynamic Panel Data : IV and GMM Estimation with Stata (Panel)
2、 xtabond cheat sheet
3、

重现 Arollano and Bond (1991) FD-GMM 结果

clear all
set mem 32m
set matsize 800
use "http://www.stata-press.com/data/r7/abdata.dta"

设置变量，其一阶差分是年度虚拟变量和常数项。
这个步骤一般不是必需的，但需要完全模仿DPD，因为它在FD-GMM中也是需要直接输入时间虚拟和常数项。

forvalues y = 1979/1984 {
    gen yr`y'c = year>=`y'
}
gen cons = year

运行结果

xtabond2 n L(0/1).(l.n w) l(0/2).(k ys) yr198?c cons, gmm(L.n) iv(L(0/1).w l(0/2).(k ys) yr198?c cons) noleveleq noconstant robust

输出结果为：

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
Wald chi2(16) =   1727.45                                      avg =      4.36
Prob > chi2   =     0.000                                      max =         6
------------------------------------------------------------------------------
             |               Robust
           n |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           n |
         L1. |   .6862261   .1445943     4.75   0.000     .4028266    .9696257
         L2. |  -.0853582   .0560155    -1.52   0.128    -.1951467    .0244302
             |
           w |
         --. |  -.6078208   .1782055    -3.41   0.001    -.9570972   -.2585445
         L1. |   .3926237   .1679931     2.34   0.019     .0633632    .7218842
             |
           k |
         --. |   .3568456   .0590203     6.05   0.000      .241168    .4725233
         L1. |  -.0580012   .0731797    -0.79   0.428    -.2014308    .0854284
         L2. |  -.0199475   .0327126    -0.61   0.542    -.0840631    .0441681
             |
          ys |
         --. |   .6085073   .1725313     3.53   0.000     .2703522    .9466624
         L1. |  -.7111651   .2317163    -3.07   0.002    -1.165321   -.2570095
         L2. |   .1057969   .1412021     0.75   0.454    -.1709542     .382548
             |
     yr1980c |   .0029062   .0158028     0.18   0.854    -.0280667    .0338791
     yr1981c |   -.043344   .0169961    -2.55   0.011    -.0766557   -.0100323
     yr1982c |   -.024839   .0202692    -1.23   0.220    -.0645658    .0148878
     yr1983c |  -.0038161   .0219426    -0.17   0.862    -.0468227    .0391905
     yr1984c |   .0040626   .0218975     0.19   0.853    -.0388558     .046981
        cons |   .0095545   .0102896     0.93   0.353    -.0106127    .0297217
------------------------------------------------------------------------------
Instruments for first differences equation
  Standard
    D.(w L.w k L.k L2.k ys L.ys L2.ys yr1980c yr1981c yr1982c yr1983c yr1984c
    cons)
  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(25)   =  67.59  Prob > chi2 =  0.000
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(25)   =  31.38  Prob > chi2 =  0.177
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  iv(w L.w k L.k L2.k ys L.ys L2.ys yr1980c yr1981c yr1982c yr1983c yr1984c cons)
    Hansen test excluding group:     chi2(11)   =  12.01  Prob > chi2 =  0.363
    Difference (null H = exogenous): chi2(14)   =  19.37  Prob > chi2 =  0.151

xtabond2 n L(0/1).(l.n w) l(0/2).(k ys) yr198?c cons, gmm(L.n) iv(L(0/1).w l(0/2).(k ys) yr198?c cons) noleveleq noconstant two

输出结果为：

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 = 41                      Obs per group: min =         4
Wald chi2(16) =   2216.93                                      avg =      4.36
Prob > chi2   =     0.000                                      max =         6
------------------------------------------------------------------------------
           n |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           n |
         L1. |   .6287089   .0904543     6.95   0.000     .4514216    .8059961
         L2. |  -.0651882   .0265009    -2.46   0.014     -.117129   -.0132474
             |
           w |
         --. |  -.5257597   .0537692    -9.78   0.000    -.6311453    -.420374
         L1. |   .3112899   .0940116     3.31   0.001     .1270305    .4955492
             |
           k |
         --. |   .2783619   .0449083     6.20   0.000     .1903432    .3663807
         L1. |   .0140994   .0528046     0.27   0.789    -.0893957    .1175946
         L2. |  -.0402484   .0258038    -1.56   0.119    -.0908229     .010326
             |
          ys |
         --. |   .5919243   .1162114     5.09   0.000     .3641542    .8196943
         L1. |  -.5659863   .1396738    -4.05   0.000    -.8397419   -.2922306
         L2. |   .1005433   .1126749     0.89   0.372    -.1202955     .321382
             |
     yr1980c |   .0006378   .0127959     0.05   0.960    -.0244417    .0257172
     yr1981c |  -.0556422   .0143097    -3.89   0.000    -.0836888   -.0275956
     yr1982c |  -.0209736   .0163224    -1.28   0.199    -.0529648    .0110177
     yr1983c |   .0019072    .014625     0.13   0.896    -.0267574    .0305718
     yr1984c |  -.0165899   .0153035    -1.08   0.278    -.0465842    .0134045
        cons |   .0112155   .0077507     1.45   0.148    -.0039756    .0264066
------------------------------------------------------------------------------
Warning: Uncorrected two-step standard errors are unreliable.

Instruments for first differences equation
  Standard
    D.(w L.w k L.k L2.k ys L.ys L2.ys yr1980c yr1981c yr1982c yr1983c yr1984c
    cons)
  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 =  -3.00  Pr > z =  0.003
Arellano-Bond test for AR(2) in first differences: z =  -0.42  Pr > z =  0.678
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(25)   =  67.59  Prob > chi2 =  0.000
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(25)   =  31.38  Prob > chi2 =  0.177
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  iv(w L.w k L.k L2.k ys L.ys L2.ys yr1980c yr1981c yr1982c yr1983c yr1984c cons)
    Hansen test excluding group:     chi2(11)   =  12.01  Prob > chi2 =  0.363
    Difference (null H = exogenous): chi2(14)   =  19.37  Prob > chi2 =  0.151

xtabond2 n L(0/1).(l.n ys w) k yr198?c cons, gmm(L.n) iv(L(0/1).(ys w) k yr198?c cons) noleveleq noconstant two

输出结果为：

 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 = 38                      Obs per group: min =         4
Wald chi2(13) =   1603.26                                      avg =      4.36
Prob > chi2   =     0.000                                      max =         6
------------------------------------------------------------------------------
           n |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           n |
         L1. |   .4741506   .0853032     5.56   0.000     .3069594    .6413417
         L2. |  -.0529677   .0272843    -1.94   0.052     -.106444    .0005087
             |
          ys |
         --. |    .609776   .1085238     5.62   0.000     .3970732    .8224788
         L1. |  -.4463736   .1248148    -3.58   0.000    -.6910061    -.201741
             |
           w |
         --. |  -.5132049   .0493453   -10.40   0.000      -.60992   -.4164898
         L1. |     .22464   .0800628     2.81   0.005     .0677198    .3815601
             |
           k |   .2927232   .0394626     7.42   0.000      .215378    .3700684
     yr1980c |   .0036333   .0127335     0.29   0.775    -.0213239    .0285905
     yr1981c |   -.050962   .0137101    -3.72   0.000    -.0778334   -.0240907
     yr1982c |  -.0321491   .0139863    -2.30   0.022    -.0595618   -.0047364
     yr1983c |  -.0123558   .0128418    -0.96   0.336    -.0375252    .0128135
     yr1984c |  -.0207296   .0136789    -1.52   0.130    -.0475398    .0060806
        cons |    .010509   .0072515     1.45   0.147    -.0037036    .0247216
------------------------------------------------------------------------------
Warning: Uncorrected two-step standard errors are unreliable.

Instruments for first differences equation
  Standard
    D.(ys L.ys w L.w k yr1980c yr1981c yr1982c yr1983c yr1984c cons)
  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 =  -2.43  Pr > z =  0.015
Arellano-Bond test for AR(2) in first differences: z =  -0.33  Pr > z =  0.739
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(25)   =  75.46  Prob > chi2 =  0.000
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(25)   =  30.11  Prob > chi2 =  0.220
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  iv(ys L.ys w L.w k yr1980c yr1981c yr1982c yr1983c yr1984c cons)
    Hansen test excluding group:     chi2(14)   =  13.33  Prob > chi2 =  0.501
    Difference (null H = exogenous): chi2(11)   =  16.78  Prob > chi2 =  0.115

c
可用数据集缺少销售和库存信息，因此这里必须使用别的工具。该回归完美地复制了 DPD 在 Ox包中 abest3.out 的结果

xtabond2 n L(0/1).(l.n ys w) k yr198?c cons, gmm(L.n) gmm(w k, lag(2 3)) iv(L(0/1).ys yr198?c cons) noleveleq noconstant two

输出结果为：


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 = 59                      Obs per group: min =         4
Wald chi2(13) =   2668.33                                      avg =      4.36
Prob > chi2   =     0.000                                      max =         6
------------------------------------------------------------------------------
           n |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           n |
         L1. |   .8074308   .0491071    16.44   0.000     .7111827    .9036789
         L2. |  -.1134945   .0228508    -4.97   0.000    -.1582813   -.0687076
             |
          ys |
         --. |   .8599923   .1097787     7.83   0.000     .6448299    1.075155
         L1. |  -.8632569   .1101426    -7.84   0.000    -1.079132   -.6473814
             |
           w |
         --. |  -.5686242   .0598841    -9.50   0.000    -.6859949   -.4512536
         L1. |    .640707   .0672002     9.53   0.000      .508997     .772417
             |
           k |   .1833987   .0567488     3.23   0.001     .0721731    .2946243
     yr1980c |   .0095269   .0112312     0.85   0.396    -.0124858    .0315397
     yr1981c |  -.0557186   .0117042    -4.76   0.000    -.0786584   -.0327788
     yr1982c |  -.0558586   .0122055    -4.58   0.000    -.0797809   -.0319362
     yr1983c |  -.0304113   .0133729    -2.27   0.023    -.0566217   -.0042009
     yr1984c |  -.0238496   .0137584    -1.73   0.083    -.0508155    .0031163
        cons |   .0162529    .006453     2.52   0.012     .0036053    .0289005
------------------------------------------------------------------------------
Warning: Uncorrected two-step standard errors are unreliable.

Instruments for first differences equation
  Standard
    D.(ys L.ys yr1980c yr1981c yr1982c yr1983c yr1984c cons)
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(2/3).(w k)
    L(1/8).L.n
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =  -4.07  Pr > z =  0.000
Arellano-Bond test for AR(2) in first differences: z =  -0.68  Pr > z =  0.498
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(46)   =  98.75  Prob > chi2 =  0.000
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(46)   =  58.71  Prob > chi2 =  0.099
  (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(19)   =  19.11  Prob > chi2 =  0.450
    Difference (null H = exogenous): chi2(27)   =  39.60  Prob > chi2 =  0.056
  gmm(w k, lag(2 3))
    Hansen test excluding group:     chi2(22)   =  22.98  Prob > chi2 =  0.403
    Difference (null H = exogenous): chi2(24)   =  35.73  Prob > chi2 =  0.058
  iv(ys L.ys yr1980c yr1981c yr1982c yr1983c yr1984c cons)
    Hansen test excluding group:     chi2(38)   =  40.24  Prob > chi2 =  0.371
    Difference (null H = exogenous): chi2(8)    =  18.47  Prob > chi2 =  0.018

重现 SYS-GMM 结果

clear all
set matsize 800
use "http://www.stata-press.com/data/r7/abdata.dta"

设置变量，其一阶差分是年度虚拟变量和常数项。
这个步骤一般不是必需的，但需要完全模仿DPD，因为它在FD-GMM中也是需要直接输入时间虚拟和常数项。

forvalues y = 1979/1984 {
gen yr`y'c = year>=`y'
}
gen cons = year

运行结果

FD-GMM

xtabond2 n L.n L(0/1).(w k) yr*c cons, gmm(L.(w k n)) iv(yr*c cons) noleveleq noconstant robust

输出结果


Dynamic panel-data estimation, one-step difference GMM
------------------------------------------------------------------------------
Group variable: id                              Number of obs      =       751
Time variable : year                            Number of groups   =       140
Number of instruments = 91                      Obs per group: min =         5
Wald chi2(12) =   1163.33                                      avg =      5.36
Prob > chi2   =     0.000                                      max =         7
------------------------------------------------------------------------------
             |               Robust
           n |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           n |
         L1. |   .7074701   .0841788     8.40   0.000     .5424827    .8724576
             |
           w |
         --. |  -.7087965    .117102    -6.05   0.000    -.9383122   -.4792809
         L1. |   .5000149   .1113282     4.49   0.000     .2818157    .7182141
             |
           k |
         --. |   .4659776    .101044     4.61   0.000      .267935    .6640203
         L1. |  -.2151309   .0858525    -2.51   0.012    -.3833987   -.0468631
             |
     yr1979c |   .0021095   .0177521     0.12   0.905    -.0326839    .0369028
     yr1980c |  -.0265559   .0194641    -1.36   0.172    -.0647048    .0115931
     yr1981c |  -.0326771   .0232915    -1.40   0.161    -.0783275    .0129733
     yr1982c |   .0223882   .0254598     0.88   0.379    -.0275121    .0722885
     yr1983c |   .0188752   .0235881     0.80   0.424    -.0273567    .0651071
     yr1984c |    .010743   .0269194     0.40   0.690    -.0420179     .063504
        cons |   .0057636   .0166077     0.35   0.729    -.0267868     .038314
------------------------------------------------------------------------------
Instruments for first differences equation
  Standard
    D.(yr1979c yr1980c yr1981c yr1982c yr1983c yr1984c cons)
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(1/8).(L.w L.k L.n)
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =  -5.60  Pr > z =  0.000
Arellano-Bond test for AR(2) in first differences: z =  -0.14  Pr > z =  0.891
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(79)   = 125.19  Prob > chi2 =  0.001
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(79)   =  88.80  Prob > chi2 =  0.211
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  iv(yr1979c yr1980c yr1981c yr1982c yr1983c yr1984c cons)
    Hansen test excluding group:     chi2(72)   =  79.36  Prob > chi2 =  0.258
    Difference (null H = exogenous): chi2(7)    =   9.44  Prob > chi2 =  0.223

xtabond2 n L.n L(0/1).(w k) yr*c cons, gmm(L.(w k n)) iv(yr*c cons) noleveleq robust twostep

输出结果为：

Dynamic panel-data estimation, two-step difference GMM
------------------------------------------------------------------------------
Group variable: id                              Number of obs      =       751
Time variable : year                            Number of groups   =       140
Number of instruments = 91                      Obs per group: min =         5
Wald chi2(12) =    909.34                                      avg =      5.36
Prob > chi2   =     0.000                                      max =         7
------------------------------------------------------------------------------
             |              Corrected
           n |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           n |
         L1. |   .6787867   .0890781     7.62   0.000     .5041969    .8533765
             |
           w |
         --. |  -.7198296   .1221408    -5.89   0.000    -.9592211   -.4804381
         L1. |   .4626914   .1134755     4.08   0.000     .2402835    .6850992
             |
           k |
         --. |   .4539046   .1275537     3.56   0.000     .2039038    .7039053
         L1. |  -.1914923   .1044671    -1.83   0.067     -.396244    .0132595
             |
     yr1979c |  -.0023874   .0174565    -0.14   0.891    -.0366015    .0318267
     yr1980c |  -.0258997   .0187008    -1.38   0.166    -.0625525    .0107532
     yr1981c |  -.0317158   .0239383    -1.32   0.185     -.078634    .0152024
     yr1982c |   .0226915   .0268209     0.85   0.398    -.0298764    .0752594
     yr1983c |   .0246047   .0257232     0.96   0.339    -.0258117    .0750211
     yr1984c |   .0105049   .0271836     0.39   0.699    -.0427738    .0637837
        cons |   .0052583   .0156783     0.34   0.737    -.0254706    .0359872
------------------------------------------------------------------------------
Instruments for first differences equation
  Standard
    D.(yr1979c yr1980c yr1981c yr1982c yr1983c yr1984c cons)
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(1/8).(L.w L.k L.n)
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =  -4.46  Pr > z =  0.000
Arellano-Bond test for AR(2) in first differences: z =  -0.17  Pr > z =  0.866
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(79)   = 125.19  Prob > chi2 =  0.001
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(79)   =  88.80  Prob > chi2 =  0.211
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  iv(yr1979c yr1980c yr1981c yr1982c yr1983c yr1984c cons)
    Hansen test excluding group:     chi2(72)   =  79.36  Prob > chi2 =  0.258
    Difference (null H = exogenous): chi2(7)    =   9.44  Prob > chi2 =  0.223

SYS-GMM
eq（level）选项通常也不是必需的，但需要完美的模仿。
同样，dpds2是一个未记录的选项，模拟在DPD中one-step GMM的错误，使误差方差的点估计值加倍（sig2）并影响Sargan和AR（）统计量。
dpds2仅用于演示xtabond2完全匹配DPD的能力。

xtabond2 n L.n L(0/1).(w k) yr1978-yr1984, gmm(L.n, split) gmm(L.(w k)) iv(yr1978-yr1984, eq(level)) h(2) dpds2 robust

输出结果


Dynamic panel-data estimation, one-step system GMM
------------------------------------------------------------------------------
Group variable: id                              Number of obs      =       891
Time variable : year                            Number of groups   =       140
Number of instruments = 113                     Obs per group: min =         6
Wald chi2(12) =   4921.25                                      avg =      6.36
Prob > chi2   =     0.000                                      max =         8
------------------------------------------------------------------------------
             |               Robust
           n |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           n |
         L1. |   .8714137   .0440549    19.78   0.000     .7850676    .9577597
             |
           w |
         --. |  -.7810899   .1159218    -6.74   0.000    -1.008292   -.5538873
         L1. |   .5120738   .1675084     3.06   0.002     .1837634    .8403843
             |
           k |
         --. |   .4688295   .0706695     6.63   0.000     .3303199    .6073391
         L1. |  -.3559805   .0718965    -4.95   0.000     -.496895   -.2150659
             |
      yr1978 |   .0047266     .02076     0.23   0.820    -.0359621    .0454154
      yr1979 |   .0193132   .0245036     0.79   0.431     -.028713    .0673394
      yr1980 |   .0014647   .0247206     0.06   0.953    -.0469868    .0499163
      yr1981 |  -.0211725    .029662    -0.71   0.475    -.0793089    .0369639
      yr1982 |   .0148305   .0274198     0.54   0.589    -.0389113    .0685723
      yr1983 |   .0310377   .0255244     1.22   0.224    -.0189891    .0810646
      yr1984 |   .0201427   .0314874     0.64   0.522    -.0415714    .0818568
       _cons |   .9994287   .3899577     2.56   0.010     .2351257    1.763732
------------------------------------------------------------------------------
Instruments for first differences equation
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(1/8).(L.w L.k)
    L(1/8).L.n
Instruments for levels equation
  Standard
    yr1978 yr1979 yr1980 yr1981 yr1982 yr1983 yr1984
    _cons
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    D.(L.w L.k)
    D.L.n
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =  -5.98  Pr > z =  0.000
Arellano-Bond test for AR(2) in first differences: z =  -0.17  Pr > z =  0.867
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(100)  = 157.41  Prob > chi2 =  0.000
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(100)  = 111.59  Prob > chi2 =  0.201
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  GMM instruments for levels
    Hansen test excluding group:     chi2(79)   =  88.15  Prob > chi2 =  0.225
    Difference (null H = exogenous): chi2(21)   =  23.43  Prob > chi2 =  0.321
  gmm(L.n, eq(diff) lag(1 8))
    Hansen test excluding group:     chi2(72)   =  85.96  Prob > chi2 =  0.125
    Difference (null H = exogenous): chi2(28)   =  25.63  Prob > chi2 =  0.593
  gmm(L.n, eq(diff) lag(1 8)) eq(level) lag(0 0))
    Hansen test excluding group:     chi2(93)   = 106.68  Prob > chi2 =  0.157
    Difference (null H = exogenous): chi2(7)    =   4.91  Prob > chi2 =  0.671
  gmm(L.w L.k, lag(1 .))
    Hansen test excluding group:     chi2(30)   =  43.22  Prob > chi2 =  0.056
    Difference (null H = exogenous): chi2(70)   =  68.37  Prob > chi2 =  0.533
  iv(yr1978 yr1979 yr1980 yr1981 yr1982 yr1983 yr1984, eq(level))
    Hansen test excluding group:     chi2(93)   = 109.91  Prob > chi2 =  0.111
    Difference (null H = exogenous): chi2(7)    =   1.68  Prob > chi2 =  0.976

xtabond2 n L.n L(0/1).(w k) yr1978-yr1984, gmm(L.n, split) gmm(L.(w k)) iv(yr1978-yr1984, eq(level)) h(2) robust twostep

输出结果为：

Dynamic panel-data estimation, two-step system GMM
------------------------------------------------------------------------------
Group variable: id                              Number of obs      =       891
Time variable : year                            Number of groups   =       140
Number of instruments = 113                     Obs per group: min =         6
Wald chi2(12) =   5912.36                                      avg =      6.36
Prob > chi2   =     0.000                                      max =         8
------------------------------------------------------------------------------
             |              Corrected
           n |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           n |
         L1. |    .872881   .0452841    19.28   0.000     .7841259    .9616362
             |
           w |
         --. |  -.7797449   .1165601    -6.69   0.000    -1.008199   -.5512913
         L1. |   .5268032   .1620827     3.25   0.001     .2091269    .8444795
             |
           k |
         --. |   .4700773   .0798591     5.89   0.000     .3135562    .6265983
         L1. |  -.3576081   .0800305    -4.47   0.000     -.514465   -.2007512
             |
      yr1978 |   .0058018   .0197099     0.29   0.768    -.0328288    .0444325
      yr1979 |   .0188977   .0227673     0.83   0.407    -.0257254    .0635207
      yr1980 |   .0028196   .0240708     0.12   0.907    -.0443583    .0499976
      yr1981 |  -.0200226   .0274419    -0.73   0.466    -.0738078    .0337625
      yr1982 |   .0152802   .0233063     0.66   0.512    -.0303992    .0609597
      yr1983 |    .031731   .0234974     1.35   0.177    -.0143231    .0777852
      yr1984 |   .0224206   .0310743     0.72   0.471     -.038484    .0833251
       _cons |   .9484881   .3775501     2.51   0.012     .2085035    1.688473
------------------------------------------------------------------------------
Instruments for first differences equation
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(1/8).(L.w L.k)
    L(1/8).L.n
Instruments for levels equation
  Standard
    yr1978 yr1979 yr1980 yr1981 yr1982 yr1983 yr1984
    _cons
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    D.(L.w L.k)
    D.L.n
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =  -5.81  Pr > z =  0.000
Arellano-Bond test for AR(2) in first differences: z =  -0.15  Pr > z =  0.883
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(100)  = 157.41  Prob > chi2 =  0.000
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(100)  = 111.59  Prob > chi2 =  0.201
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  GMM instruments for levels
    Hansen test excluding group:     chi2(79)   =  88.15  Prob > chi2 =  0.225
    Difference (null H = exogenous): chi2(21)   =  23.43  Prob > chi2 =  0.321
  gmm(L.n, eq(diff) lag(1 8))
    Hansen test excluding group:     chi2(72)   =  85.96  Prob > chi2 =  0.125
    Difference (null H = exogenous): chi2(28)   =  25.63  Prob > chi2 =  0.593
  gmm(L.n, eq(diff) lag(1 8)) eq(level) lag(0 0))
    Hansen test excluding group:     chi2(93)   = 106.68  Prob > chi2 =  0.157
    Difference (null H = exogenous): chi2(7)    =   4.91  Prob > chi2 =  0.671
  gmm(L.w L.k, lag(1 .))
    Hansen test excluding group:     chi2(30)   =  43.22  Prob > chi2 =  0.056
    Difference (null H = exogenous): chi2(70)   =  68.37  Prob > chi2 =  0.533
  iv(yr1978 yr1979 yr1980 yr1981 yr1982 yr1983 yr1984, eq(level))
    Hansen test excluding group:     chi2(93)   = 109.91  Prob > chi2 =  0.111
    Difference (null H = exogenous): chi2(7)    =   1.68  Prob > chi2 =  0.976

重现 Greene 书中结果
重现 Greene, Econometric Analysis, 5th ed. p. 554例子
这里需要加载数据

并使用infile语句提取文件

clear all
set mem 32m
set matsize 800

infile id  year expend revenue grants using h:\macros\T7987.asc, clear
tsset id year

输出：

 panel variable:  id (strongly balanced)
        time variable:  year, 1979 to 1987
                delta:  1 unit

运行结果

无设置年度虚拟变量

xi: xtabond2 expend l(1/3).(expend revenue grants) i.year, gmm(l.expend) iv(i.year) noleveleq twostep h(1)

输出结果为：

Dynamic panel-data estimation, two-step difference GMM
------------------------------------------------------------------------------
Group variable: id                              Number of obs      =      1325
Time variable : year                            Number of groups   =       265
Number of instruments = 30                      Obs per group: min =         5
Wald chi2(17) =    796.67                                      avg =      5.00
Prob > chi2   =     0.000                                      max =         5
------------------------------------------------------------------------------
      expend |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      expend |
         L1. |   1.154936   .3440941     3.36   0.001     .4805244    1.829348
         L2. |  -.0376625   .2267605    -0.17   0.868     -.482105      .40678
         L3. |  -.5644067   .2179554    -2.59   0.010    -.9915915    -.137222
             |
     revenue |
         L1. |  -1.238006   .3617064    -3.42   0.001    -1.946938   -.5290747
         L2. |   .0770075   .2717902     0.28   0.777    -.4556915    .6097065
         L3. |   .6497806   .2693003     2.41   0.016     .1219617    1.177599
             |
      grants |
         L1. |   .0163122    .824194     0.02   0.984    -1.599078    1.631703
         L2. |   1.553793   .7584145     2.05   0.040     .0673281    3.040258
         L3. |   1.789179   .6929651     2.58   0.010     .4309924    3.147366
             |
 _Iyear_1980 |          0  (omitted)
 _Iyear_1981 |          0  (omitted)
 _Iyear_1982 |          0  (omitted)
 _Iyear_1983 |  -.0036579   .0002969   -12.32   0.000    -.0042399   -.0030759
 _Iyear_1984 |  -.0041546   .0005716    -7.27   0.000    -.0052748   -.0030343
 _Iyear_1985 |  -.0037737   .0006463    -5.84   0.000    -.0050404   -.0025071
 _Iyear_1986 |   -.003459   .0007194    -4.81   0.000    -.0048689   -.0020491
 _Iyear_1987 |  -.0025902   .0006666    -3.89   0.000    -.0038968   -.0012837
------------------------------------------------------------------------------
Warning: Uncorrected two-step standard errors are unreliable.

Instruments for first differences equation
  Standard
    D.(_Iyear_1980 _Iyear_1981 _Iyear_1982 _Iyear_1983 _Iyear_1984 _Iyear_1985
    _Iyear_1986 _Iyear_1987)
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(1/8).L.expend
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =  -2.98  Pr > z =  0.003
Arellano-Bond test for AR(2) in first differences: z =  -2.26  Pr > z =  0.024
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(13)   =  39.24  Prob > chi2 =  0.000
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(13)   =  22.83  Prob > chi2 =  0.044
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  iv(_Iyear_1980 _Iyear_1981 _Iyear_1982 _Iyear_1983 _Iyear_1984 _Iyear_1985 _Iyear_1986 _Iyear_1987)
    Hansen test excluding group:     chi2(8)    =   7.82  Prob > chi2 =  0.451
    Difference (null H = exogenous): chi2(5)    =  15.01  Prob > chi2 =  0.010

为了完美匹配，设置一阶差分为年度虚拟变量

forvalues y=1980/1987 {
gen yr`y'c = year>=`y'
}
xtabond2 expend l(1/3).(expend revenue grants) yr*c, gmm(l.expend) iv(yr*c) noleveleq twostep h(1)

输出结果为：

Dynamic panel-data estimation, two-step difference GMM
------------------------------------------------------------------------------
Group variable: id                              Number of obs      =      1325
Time variable : year                            Number of groups   =       265
Number of instruments = 30                      Obs per group: min =         5
Wald chi2(17) =    796.67                                      avg =      5.00
Prob > chi2   =     0.000                                      max =         5
------------------------------------------------------------------------------
      expend |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      expend |
         L1. |   1.154936   .3440941     3.36   0.001     .4805244    1.829348
         L2. |  -.0376625   .2267605    -0.17   0.868     -.482105      .40678
         L3. |  -.5644067   .2179554    -2.59   0.010    -.9915915    -.137222
             |
     revenue |
         L1. |  -1.238006   .3617064    -3.42   0.001    -1.946938   -.5290747
         L2. |   .0770075   .2717902     0.28   0.777    -.4556915    .6097065
         L3. |   .6497806   .2693003     2.41   0.016     .1219617    1.177599
             |
      grants |
         L1. |   .0163122    .824194     0.02   0.984    -1.599078    1.631703
         L2. |   1.553793   .7584145     2.05   0.040     .0673281    3.040258
         L3. |   1.789179   .6929651     2.58   0.010     .4309924    3.147366
             |
     yr1980c |          0  (omitted)
     yr1981c |          0  (omitted)
     yr1982c |          0  (omitted)
     yr1983c |  -.0036579   .0002969   -12.32   0.000    -.0042399   -.0030759
     yr1984c |  -.0004967   .0004128    -1.20   0.229    -.0013057    .0003123
     yr1985c |   .0003809   .0003094     1.23   0.218    -.0002255    .0009872
     yr1986c |   .0003147   .0003282     0.96   0.338    -.0003286     .000958
     yr1987c |   .0008688    .000148     5.87   0.000     .0005788    .0011588
------------------------------------------------------------------------------
Warning: Uncorrected two-step standard errors are unreliable.

Instruments for first differences equation
  Standard
    D.(yr1980c yr1981c yr1982c yr1983c yr1984c yr1985c yr1986c yr1987c)
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(1/8).L.expend
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =  -2.98  Pr > z =  0.003
Arellano-Bond test for AR(2) in first differences: z =  -2.26  Pr > z =  0.024
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(13)   =  39.24  Prob > chi2 =  0.000
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(13)   =  22.83  Prob > chi2 =  0.044
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  iv(yr1980c yr1981c yr1982c yr1983c yr1984c yr1985c yr1986c yr1987c)
    Hansen test excluding group:     chi2(8)    =   7.82  Prob > chi2 =  0.451
    Difference (null H = exogenous): chi2(5)    =  15.01  Prob > chi2 =  0.010

最后编辑于：2019.06.29 08:28:31

人面猴
序言：七十年代末，一起剥皮案震惊了整个滨河市，随后出现的几起案子，更是在滨河造成了极大的恐慌，老刑警刘岩，带你破解...
沈念sama阅读 215,634评论 6赞 497
死咒
序言：滨河连续发生了三起死亡事件，死亡现场离奇诡异，居然都是意外死亡，警方通过查阅死者的电脑和手机，发现死者居然都...
沈念sama阅读 91,951评论 3赞 391
救了他两次的神仙让他今天三更去死
文/潘晓璐我一进店门，熙熙楼的掌柜王于贵愁眉苦脸地迎上来，“玉大人，你说我怎么就摊上这事。” “怎么了？”我有些...
开封第一讲书人阅读 161,427评论 0赞 351
道士缉凶录：失踪的卖姜人
文/不坏的土叔我叫张陵，是天一观的道长。经常有香客问我，道长，这世上最难降的妖魔是什么？我笑而不...
开封第一讲书人阅读 57,770评论 1赞 290
港岛之恋（遗憾婚礼）
正文为了忘掉前任，我火速办了婚礼，结果婚礼上，老公的妹妹穿的比我还像新娘。我一直安慰自己，他们只是感情好，可当我...
茶点故事阅读 66,835评论 6赞 388
恶毒庶女顶嫁案：这布局不是一般人想出来的
文/花漫我一把揭开白布。她就那样静静地躺着，像睡着了一般。火红的嫁衣衬着肌肤如雪。梳的纹丝不乱的头发上，一...
开封第一讲书人阅读 50,799评论 1赞 294
城市分裂传说
那天，我揣着相机与录音，去河边找鬼。笑死，一个胖子当着我的面吹牛，可吹牛的内容都是我干的。我是一名探鬼主播，决...
沈念sama阅读 39,768评论 3赞 416
双鸳鸯连环套：你想象不到人心有多黑
文/苍兰香墨我猛地睁开眼，长吁一口气：“原来是场噩梦啊……” “哼！你这毒妇竟也来了？” 一声冷哼从身侧响起，我...
开封第一讲书人阅读 38,544评论 0赞 271
万荣杀人案实录
序言：老挝万荣一对情侣失踪，失踪者是张志新（化名）和其女友刘颖，没想到半个月后，有当地人在树林里发现了一具尸体，经...
沈念sama阅读 44,979评论 1赞 308
护林员之死
正文独居荒郊野岭守林人离奇死亡，尸身上长有42处带血的脓包…… 初始之章·张勋以下内容为张勋视角年9月15日...
茶点故事阅读 37,271评论 2赞 331
白月光启示录
正文我和宋清朗相恋三年，在试婚纱的时候发现自己被绿了。大学时的朋友给我发了我未婚夫和他白月光在一起吃饭的照片。...
茶点故事阅读 39,427评论 1赞 345
活死人
序言：一个原本活蹦乱跳的男人离奇死亡，死状恐怖，灵堂内的尸体忽然破棺而出，到底是诈尸还是另有隐情，我是刑警宁泽，带...
沈念sama阅读 35,121评论 5赞 340
日本核电站爆炸内幕
正文年R本政府宣布，位于F岛的核电站，受9级特大地震影响，放射性物质发生泄漏。R本人自食恶果不足惜，却给世界环境...
茶点故事阅读 40,756评论 3赞 324
男人毒药：我在死后第九天来索命
文/蒙蒙一、第九天我趴在偏房一处隐蔽的房顶上张望。院中可真热闹，春花似锦、人声如沸。这庄子的主人今日做“春日...
开封第一讲书人阅读 31,375评论 0赞 21
一桩弑父案，背后竟有这般阴谋
文/苍兰香墨我抬头看了看天上的太阳。三九已至，却和暖如春，着一层夹袄步出监牢的瞬间，已是汗流浃背。一阵脚步声响...
开封第一讲书人阅读 32,579评论 1赞 268
情欲美人皮
我被黑心中介骗来泰国打工，没想到刚下飞机就差点儿被人妖公主榨干…… 1. 我叫王不留，地道东北人。一个月前我还...
沈念sama阅读 47,410评论 2赞 368
代替公主和亲
正文我出身青楼，却偏偏与公主长得像，于是被迫代替她去往敌国和亲。传闻我的和亲对象是个残疾皇子，可洞房花烛夜当晚...
茶点故事阅读 44,315评论 2赞 352

动态面板简介

简介

AR(1) 模型

基本设定

估计方法

IV

GMM

实际应用

Stata 范例

假设检验

序列相关检验

过度识别检验（estat overid）

Stata 范例

推荐阅读更多精彩内容