计量经济学模型分析方法

1 计量经济学上机模型分析方法总结计量经济学上机模型分析方法总结 一 随机误差项的异方差问题的检验与修正 模型一 Dependent Variable LOG Y Method Least Squares Date 07 29 12 Time 09 03 Sample 1 31 Included observations 31 VariableCoefficientStd Errort StatisticProb C1 0 1 0 0732 LOG X1 0 0 3 0 0040 LOG X2 0 0 10 433850 0000 R squared0 Mean dependent var7 Adjusted R squared0 S D dependent var0 S E of regression0 Akaike info criterion 0 Sum squared resid0 Schwarz criterion 0 Log likelihood12 47249 F statistic54 79806 Durbin Watson stat1 Prob F statistic 0 一 异方差的检验 1 GQ 检验法 模型二 Dependent Variable LOG Y Method Least Squares Date 07 29 12 Time 09 19 Sample 1 12 Included observations 12 VariableCoefficientStd Errort StatisticProb C3 1 3 0 0119 LOG X1 0 0 4 0 0025 LOG X2 0 0 1 0 1890 R squared0 Mean dependent var7 Adjusted R squared0 S D dependent var0 S E of regression0 Akaike info criterion 1 Sum squared resid0 Schwarz criterion 1 Log likelihood14 28638 F statistic9 Durbin Watson stat1 Prob F statistic 0 2 模型三 Dependent Variable LOG Y Method Least Squares Date 07 29 12 Time 09 20 Sample 20 31 Included observations 12 VariableCoefficientStd Errort StatisticProb C 0 1 0 0 8309 LOG X1 0 0 1 0 2153 LOG X2 0 0 7 0 0000 R squared0 Mean dependent var7 Adjusted R squared0 S D dependent var0 S E of regression0 Akaike info criterion 0 Sum squared resid0 Schwarz criterion 0 Log likelihood7 F statistic32 50732 Durbin Watson stat2 Prob F statistic 0 进行模型二和模型三两次回归 目的仅是得到出去中间 7 个样本点以后前后各 12 个样本点的残差平方和 RSS1和 RSS2 然后用较大的 RSS 除以较小的 RSS 即可求出 F 统计量值进行显著性检验 2 怀特检验法 怀特检验法 White 模型一的怀特残差检验结果 White Heteroskedasticity Test F statistic4 Probability0 Obs R squared13 35705 Probability0 Test Equation Dependent Variable RESID 2 Method Least Squares Date 05 29 13 Time 09 04 Sample 1 31 Included observations 31 VariableCoefficientStd Errort StatisticProb C3 2 1 0 1789 LOG X1 0 0 0 0 5327 LOG X1 20 0 0 0 5370 3 LOG X2 0 0 2 0 0101 LOG X2 20 0 2 0 0075 R squared0 Mean dependent var0 Adjusted R squared0 S D dependent var0 S E of regression0 Akaike info criterion 3 Sum squared resid0 Schwarz criterion 3 Log likelihood65 96898 F statistic4 Durbin Watson stat1 Prob F statistic 0 一方面 根据上面的 Obs R2 31 0 13 35705 2 4 说明存在显著的异方差问 题 另一方面 根据下面的辅助回归模型可以看出 LOG X2 与 LOG X2 2 均通过了 t 检验 说明异方差的形式可以用 LOG X2 与 LOG X2 2 的线性组合表示 权变量 可以简单确定为 1 LOG X2 二 加权最小二乘法 WLS 修正 1 方法原理 具体参见教材 2 回归结果分析 模型四 Dependent Variable LOG Y Method Least Squares Date 07 29 12 Time 09 06 Sample 1 31 Included observations 31 Weighting series 1 LOG X2 VariableCoefficientStd Errort StatisticProb C1 0 1 0 0814 LOG X1 0 0 3 0 0006 LOG X2 0 0 9 0 0000 Weighted Statistics R squared0 Mean dependent var7 Adjusted R squared0 S D dependent var0 S E of regression0 Akaike info criterion 0 Sum squared resid0 Schwarz criterion 0 Log likelihood14 15440 F statistic49 27256 Durbin Watson stat2 Prob F statistic 0 Unweighted Statistics 4 R squared0 Mean dependent var7 Adjusted R squared0 S D dependent var0 S E of regression0 Sum squared resid0 Durbin Watson stat2 加权修正以后的模型四怀特检验结果如下 White Heteroskedasticity Test F statistic6 Probability0 Obs R squared15 56541 Probability0 可以看出并没有消除异方差性 加权修正无效 下面采用1 abs e 权变量进行WLS回归 结果如下 模型五 Dependent Variable LOG Y Method Least Squares Date 07 29 12 Time 09 10 Sample 1 31 Included observations 31 Weighting series 1 ABS E VariableCoefficientStd Errort StatisticProb C1 0 4 0 0003 LOG X1 0 0 6 0 0000 LOG X2 0 0 28 688470 0000 Weighted Statistics R squared0 Mean dependent var7 Adjusted R squared0 S D dependent var12 31758 S E of regression0 Akaike info criterion 3 Sum squared resid0 Schwarz criterion 3 Log likelihood56 56339 F statistic1960 131 Durbin Watson stat2 Prob F statistic 0 Unweighted Statistics R squared0 Mean dependent var7 Adjusted R squared0 S D dependent var0 S E of regression0 Sum squared resid0 5 Durbin Watson stat2 对加权以后的模型五进行怀特检验如下 White Heteroskedasticity Test F statistic0 Probability0 Obs R squared0 Probability0 可以看出 模型已经不再存在异方差问题 模型五可以作为修正以后的最终模型 二 随机误差项序列相关性问题的检验与修正 模型一 模型一 Dependent Variable Y Method Least Squares Date 07 29 12 Time 09 48 Sample 1991 2011 Included observations 21 VariableCoefficientStd Errort StatisticProb C178 975555 064213 0 0042 X0 0 17 641570 0000 R squared0 Mean dependent var922 9095 Adjusted R squared0 S D dependent var659 3491 S E of regression162 2653 Akaike info criterion13 10673 Sum squared resid 3 Schwarz criterion13 20621 Log likelihood 135 6207 F statistic311 2248 Durbin Watson stat0 Prob F statistic 0 初始回归模型一经济意义合理 统计指标较为理想 但 DW 值偏低 模型可能存在 序列相关性 一 序列相关性的检验方法 1 自回归模型检验法 Dependent Variable E Method Least Squares Date 07 29 12 Time 09 49 Sample adjusted 1992 2011 Included observations 20 after adjustments 6 VariableCoefficientStd Errort StatisticProb E 1 0 0 3 0 0021 R squared0 Mean dependent var2 Adjusted R squared0 S D dependent var161 7297 S E of regression125 3870 Akaike info criterion12 54939 Sum squared resid 2 Schwarz criterion12 59918 Log likelihood 124 4939 Durbin Watson stat1 说明模型一的随机误差项至少存在一阶正序列相关性 结合该自回归模型的DW值 为1 08 怀疑存在更高阶的序列相关 继续引入e 2 如下 Dependent Variable E Method Least Squares Date 07 29 12 Time 09 49 Sample adjusted 1993 2011 Included observations 19 after adjustments VariableCoefficientStd Errort StatisticProb E 1 1 0 6 0 0000 E 2 0 0 4 0 0008 R squared0 Mean dependent var7 Adjusted R squared0 S D dependent var164 5730 S E of regression93 84710 Akaike info criterion12 02051 Sum squared resid 7 Schwarz criterion12 11993 Log likelihood 112 1949 Durbin Watson stat1 由于e 2 的t检验显著 说明模型一的随机误差项确实存在二阶正序列相关性 结合该二阶自回归模型的DW值为1 95 基本确定不存在更高阶的序列相关 Breusch Godfrey Serial Correlation LM Test F statistic0 Probability0 Obs R squared1 Probability0 可以看出二阶自回归模型的随机误差项不存在序列相关性 论证了原模型仅存在 二阶序列相关 2 DW 检验法 7 0 DW dL 存在正自相关 趋近于 0 DL DW dU 不能确定 DU DW 4 dU 无自相关 趋近于 2 3 LM 检验法 原理 一方面 根据上面的假设检验结果判断是否存在序列相关性 即根据 n p R2统计量值与卡方检验临界值 2 P 进行比较 其中 n 为原模型样本容量 P 为选择 的滞后阶数 R2为下面辅助回归模型的可决系数 若 n p R2 2 P 则拒绝不序列 相关的原假设 说明模型存在显著的序列相关性 另一方面 结合下面的辅助回归模 型中残差滞后变量是否通过 t 检验及 DW 值判断序列相关的具体阶数 方法与上面的 自回归模型检验法相同 选择滞后一阶检验 Breusch Godfrey Serial Correlation LM Test F statistic13 15036 Probability0 Obs R squared8 Probability0 Test Equation Dependent Variable RESID Method Least Squares Date 07 29 12 Time 09 51 Presample missing value lagged residuals set to zero VariableCoefficientStd Errort StatisticProb C 14 2447243 18361 0 0 7453 X0 0 0 0 4417 RESID 1 0 0 3 0 0019 R squared0 Mean dependent var1 30E 13 Adjusted R squared0 S D dependent var158 1566 S E of regression126 7275 Akaike info criterion12 65352 Sum squared resid 4 Schwarz criterion12 80274 Log likelihood 129 8619 F statistic6 Durbin Watson stat1 Prob F statistic 0 说明原模型确实存在一阶序列相关性 结合该辅助回归模型的DW值为1 16 怀疑 存在更高阶的序列相关 引入滞后二阶检验如下 8 Breusch Godfrey Serial Correlation LM Test F statistic20 49152 Probability0 Obs R squared14 84303 Probability0 Test Equation Dependent Variable RESID Method Least Squares Date 07 29 12 Time 09 51 Presample missing value lagged residuals set to zero VariableCoefficientStd Errort StatisticProb C14 0646332 409870 0 6698 X 0 0 0 0 4091 RESID 1 1 0 6 0 0000 RESID 2 0 0 4 0 0008 R squared0 Mean dependent var1 30E 13 Adjusted R squared0 S D dependent var158 1566 S E of regression92 88633 Akaike info criterion12 07027 Sum squared resid 8 Schwarz criterion12 26923 Log likelihood 122 7379 F statistic13 66102 Durbin Watson stat1 Prob F statistic 0 由于e 2 的t检验显著 说明模型一的随机误差项确实存在二阶正序列相关性 结合该二阶自回归模型的DW值为1 95 基本确定不存在更高阶的序列相关 当然可以继续引入滞后三阶检验如下 Breusch Godfrey Serial Correlation LM Test F statistic12 85743 Probability0 Obs R squared14 84303 Probability0 Test Equation Dependent Variable RESID Method Least Squares Date 07 29 12 Time 09 52 Presample missing value lagged residuals set to zero VariableCoefficientStd Errort StatisticProb 9 C14 0646733 407340 0 6794 X 0 0 0 0 4237 RESID 1 1 0 4 0 0009 RESID 2 0 0 1 0 0849 RESID 3 0 0 0 0 9989 R squared0 Mean dependent var1 30E 13 Adjusted R squared0 S D dependent var158 1566 S E of regression95 74504 Akaike info criterion12 16551 Sum squared resid 8 Schwarz criterion12 41421 Log likelihood 122 7379 F statistic9 Durbin Watson stat1 Prob F statistic 0 可以看出并不存在三阶序列相关 二 广义差分法修正 1 方法原理 参考教材自己推导二元线性回归模型存在二阶序列相关时的广义差分模型 2 上机实现结果分析 模型二 模型二 Dependent Variable Y Method Least Squares Date 07 29 12 Time 09 55 Sample adjusted 1992 2011 Included observations 20 after adjustments Convergence achieved after 8 iterations VariableCoefficientStd Errort StatisticProb C160 0892182 89170 0 3936 X0 0 6 0 0000 AR 1 0 0 3 0 0023 R squared0 Mean dependent var958 0450 Adjusted R squared0 S D dependent var655 9980 S E of regression130 5388 Akaike info criterion12 71870 Sum squared resid 3 Schwarz criterion12 86806 Log likelihood 124 1870 F statistic231 4107 Durbin Watson stat1 Prob F statistic 0 Inverted AR Roots 73 由于AR 1 通过t检验 说明模型一确实至少存在一阶序列相关 结合DW值为1 12 10 怀疑存在更高阶序列相关性 LM检验结果如下 Breusch Godfrey Serial Correlation LM Test F statistic6 Probability0 Obs R squared9 Probability0 Test Equation Dependent Variable RESID Method Least Squares Date 07 29 12 Time 09 57 Presample missing value lagged residuals set to zero VariableCoefficientStd Errort StatisticProb C80 86347145 26430 0 5860 X 0 0 1 0 1922 AR 1 0 0 1 0 2099 RESID 1 1 0 3 0 0084 RESID 2 0 0 0 0 7577 R squared0 Mean dependent var 7 24E 11 Adjusted R squared0 S D dependent var123 4773 S E of regression102 1528 Akaike info criterion12 30313 Sum squared resid 8 Schwarz criterion12 55207 Log likelihood 118 0313 F statistic3 Durbin Watson stat2 Prob F statistic 0 说明模型一在一阶广义差分修正后仍然存在序列相关性 继续引入AR 2 进行修正 模型三 模型三 Dependent Variable Y Method Least Squares Date 07 29 12 Time 09 58 Sample adjusted 1993 2011 Included observations 19 after adjustments Convergence achieved after 5 iterations VariableCoefficientStd Errort StatisticProb C210 523342 671174 0 0002 X0 0 19 173600 0000 11 AR 1 1 0 5 0 0000 AR 2 0 0 3 0 0018 R squared0 Mean dependent var998 3158 Adjusted R squared0 S D dependent var648 0772 S E of regression96 86089 Akaike info criterion12 16909 Sum squared resid 5 Schwarz criterion12 36792 Log likelihood 111 6064 F statistic263 6012 Durbin Watson stat2 Prob F statistic 0 Inverted AR Roots 55 80i 55 80i 由于AR 1 和AR 2 都通过t检验 说明模型一确实至少存在二阶序列相关 结合DW 值为2 00 确定不存在更高阶序列相关性 LM检验结果如下 Breusch Godfrey Serial Correlation LM Test F statistic0 Probability0 Obs R squared2 Probability0 可以看出 二阶广义差分修正后的模型三不再存在序列相关性 可以作为最终选 择模型 三 多元线性回归模型分析中解释变量的选取问题 多重共线性的检验与 修正 假设用解释变量x1 x2 x3 x4来解释Y 模型一 Dependent Variable Y Method Least Squares Date 07 29 12 Time 10 35 Sample 1994 2011 Included observations 18 VariableCoefficientStd Errort StatisticProb C 43872 2714512 82 3 0 0086 X14 0 18 466320 0000 X20 0 5 0 0001 R squared0 Mean dependent var44127 11 Adjusted R squared0 S D dependent var4409 100 S E of regression926 6166 Akaike info criterion16 65197 12 Sum squared resid Schwarz criterion16 80036 Log likelihood 146 8677 F statistic184 9504 Durbin Watson stat2 Prob F statistic 0 模型二 Dependent Variable Y Method Least Squares Date 07 29 12 Time 10 36 Sample 1994 2011 Included observations 18 VariableCoefficientStd Errort StatisticProb C 11978 1814072 92 0 0 4090 X15 0 19 568280 0000 X20 0 3 0 0048 X3 0 0 3 0 0031 R squared0 Mean dependent var44127 11 Adjusted R squared0 S D dependent var4409 100 S E of regression694 0715 Akaike info criterion16 11616 Sum squared resid Schwarz criterion16 31402 Log likelihood 141 0454 F statistic224 0086 Durbin Watson stat1 Prob F statistic 0 模型三 Dependent Variable Y Method Least Squares Date 07 29 12 Time 10 37 Sample 1994 2011 Included observations 18 VariableCoefficientStd Errort StatisticProb C 13056 0013437 43 0 0 3490 X16 0 9 0 0000 X20 0 3 0 0034 X3 0 0 3 0 0087 X4 0 0 1 0 1456 R squared0 Mean dependent var44127 11 Adjusted R squared0 S D dependent var4409 100 S E of regression661 8392 Akaike info criterion16 05806 Sum squared resid Schwarz criterion16 30538 13 Log likelihood 139 5225 F statistic185 3683 Durbin Watson stat1 Prob F statistic 0 一 新增解释变量是否应该引入 主要观察引入后相比引入前模型的调整可决系数是否变大 AIC 是否变小 新增解 释变量的 t 检验