误差修正模型：
如果用两个变量，人均消费y和人均收入x（从格林的数据获得）来研究误差修正模型。 令z=（y x）’，则模型为：
?zt?A0??zt?1??pi?zt?1??t
i?1k其中，????'
如果令k?1，即滞后项为1，则模型为
?zt?A0??zt?1?p1?zt?1??t
实际上为两个方程的估计：
?yt?ay?b11yt?1?b12xt?1?p11?yt?1?p12?xt?1??1t ?xt?ax?b21yt?1?b22xt?1?p21?yt?1?p22?xt?1??2t
用ols命令做出的结果： gen t=_n tsset t
time variable: t, 1 to 204 gen ly=L.y
(1 missing value generated) gen lx=L.x
(1 missing value generated) reg D.y ly lx D.ly D.lx
Source | SS df MS Number of obs = 202 -------------+------------------------------ F( 4, 197) = 21.07 Model | 37251.2525 4 9312.81313 Prob > F = 0.0000 Residual | 87073.3154 197 441.996525 R-squared = 0.2996 -------------+------------------------------ Adj R-squared = 0.2854 Total | 124324.568 201 618.530189 Root MSE = 21.024
------------------------------------------------------------------------------
D.y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+----------------------------------------------------------------
ly | .0417242 .0187553 2.22 0.027 .0047371 .0787112 lx | -.0318574 .0171217 -1.86 0.064 -.0656228 .001908 ly |
D1. | .1093189 .082368 1.33 0.186 -.0531173 .2717552 lx |
D1. | .0792758 .0566966 1.40 0.164 -.0325344 .1910861 _cons | 2.533504 3.757158 0.67 0.501 -4.875909 9.942916 这是?yt?ay?b11yt?1?b12xt?1?p11?yt?1?p12?xt?1??1t的回归结果，其中ay=2.5335，
b11=0.04172，b12= -0.03186，p11=0.10932，p12=0.07928
同理可得?xt?ax?b21yt?1?b22xt?1?p21?yt?1?p22?xt?1??2t的回归结果，见下 reg D.x ly lx D.ly D.lx
Source | SS df MS Number of obs = 202 -------------+------------------------------ F( 4, 197) = 11.18 Model | 36530.2795 4 9132.56988 Prob > F = 0.0000 Residual | 160879.676 197 816.648101 R-squared = 0.1850 -------------+------------------------------ Adj R-squared = 0.1685 Total | 197409.955 201 982.139082 Root MSE = 28.577
------------------------------------------------------------------------------
D.x | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+----------------------------------------------------------------
ly | .037608 .0254937 1.48 0.142 -.0126676 .0878836 lx | -.0307729 .0232732 -1.32 0.188 -.0766694 .0151237 ly |
D1. | .4149475 .111961 3.71 0.000 .1941517 .6357434 lx |
D1. | -.1812014 .0770664 -2.35 0.020 -.3331825 -.0292203 _cons | 11.20186 5.10702 2.19 0.029 1.130419 21.27331
如果用vec 命令 vec y x, pi
Vector error-correction model
Sample: 3 - 204 No. of obs = 202 AIC = 18.29975 Log likelihood = -1839.275 HQIC = 18.35939 Det(Sigma_ml) = 277863.4 SBIC = 18.44715
Equation Parms RMSE R-sq chi2 P>chi2 ----------------------------------------------------------------
D_y 4 20.9706 0.6671 396.7818 0.0000 D_x 4 28.5233 0.5328 225.8313 0.0000 ----------------------------------------------------------------
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- D_y | _ce1 |
L1. | .0418615 .0069215 6.05 0.000 .0282956 .0554273 y |
LD. | .1091985 .0807314 1.35 0.176 -.0490323 .2674292 x |
LD. | .0793652 .055411 1.43 0.152 -.0292384 .1879687 _cons | -3.602279 3.759537 -0.96 0.338 -10.97084 3.766278 -------------+---------------------------------------------------------------- D_x | _ce1 |
L1. | .0256414 .0094143 2.72 0.006 .0071897 .044093 y |
LD. | .4254495 .1098075 3.87 0.000 .2102308 .6406683 x |
LD. | -.1889879 .0753677 -2.51 0.012 -.3367058 -.04127 _cons | 5.880993 5.113562 1.15 0.250 -4.141405 15.90339 ------------------------------------------------------------------------------
这里_ce1 L1显示的是速度调整参数α的估计值，上述结果没有π的估计，而是在下面的表格中。
Cointegrating equations
Equation Parms chi2 P>chi2 -------------------------------------------
_ce1 1 853.9078 0.0000 -------------------------------------------
Identification: beta is exactly identified
Johansen normalization restriction imposed ------------------------------------------------------------------------------
beta | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- _ce1 |
y | 1 . . . . . x | -.764085 .0261479 -29.22 0.000 -.8153339 -.7128362 _cons | 146.9988 . . . . . ------------------------------------------------------------------------------ 上表中beta显示的β的估计值。
Impact parameters
Equation Parms chi2 P>chi2 -------------------------------------------
D_y 1 36.57896 0.0000 D_x 1 7.418336 0.0065 -------------------------------------------
------------------------------------------------------------------------------
Pi | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- D_y | y |
L1. | .0418615 .0069215 6.05 0.000 .0282956 .0554273 x |
L1. | -.0319857 .0052886 -6.05 0.000 -.0423512 -.0216203 -------------+---------------------------------------------------------------- D_x | y |
L1. | .0256414 .0094143 2.72 0.006 .0071897 .044093 x |
L1. | -.0195922 .0071933 -2.72 0.006 -.0336908 -.0054935
命令pi 显示π的估计值，上表中显示，在第一个方程中协整向量π中，y的L1（滞后一期）的估计值为0.0418615，x的L1（滞后一期）的估计值为-0.0319857，这与ols估计的b11=0.04172，b12= -0.03186很类似；在第二个方程中协整向量π的估计与ols估计的有些差别，可能暗示第二个方程对均衡误差没有反应。
检验协整向量的秩， vecrank y x
Johansen tests for cointegration Trend: constant Number of obs = 202 Sample: 3 - 204 Lags = 2 -------------------------------------------------------------------------------
5% maximum trace critical rank parms LL eigenvalue statistic value 0 6 -1856.3997 . 34.5784 15.41 1 9 -1839.2746 0.15596 0.3282* 3.76 2 10 -1839.1105 0.00162
-------------------------------------------------------------------------------
trace statistic 表明拒绝rank（π）=0的假设，但是不能拒绝rank（π）=1的假设，所以人均消费和人均收入的模型中，协整向量的秩为1。也表明人均消费和人均收入符合误差修正模型。
vec y x, al
al显示α的估计值，即速度调整参数的估计
Adjustment parameters
Equation Parms chi2 P>chi2 -------------------------------------------
D_y 1 36.57896 0.0000 D_x 1 7.418336 0.0065 -------------------------------------------
------------------------------------------------------------------------------
alpha | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- D_y | _ce1 |
L1. | .0418615 .0069215 6.05 0.000 .0282956 .0554273 -------------+---------------------------------------------------------------- D_x | _ce1 |
L1. | .0256414 .0094143 2.72 0.006 .0071897 .044093