我来说一个结构方程模型的例子吧——这个例子来自Michael P.Murry的现代计量经济学一书下册。研究人员想对鳕鱼市场的供给与需求建立联立方程模型。供给方程形式为：
PRICE = c+QTY+WINDSPD+RAINY+COLD+STORMY+MIXED+WINDSPD2
需求方程形式为：QTY=c+PRICE+DAY1+DAY2+DAY3+DAY4
其中价格和销量为内生变量，其它和天气和日期有关的为外生变量。所有的外生变量均用来作工具变量。
#首先载入扩展包
library(systemfit)
#读入数据再建立方程
demand <- QTY ~ PRICE+DAY1+DAY2+DAY3+DAY4
supply <- PRICE ~ QTY+WINDSPD+RAINY+COLD+STORMY+MIXED+WINDSPD2
system <- list(demand,supply)
#明确工具变量名称
inst <- ~WINDSPD+WINDSPD2+COLD+RAINY+MIXED+STORMY+DAY1+DAY2+DAY3+DAY4
#利用systemfit命令来获得结果
result2sls <- systemfit(method="2SLS",formula=system,inst=inst)
summary(result2sls)
systemfit results
method: 2SLS
N DF SSR detRCov OLS-R2 McElroy-R2
system 222 208 62.2896 0.057822 0.186339 0.236081
N DF SSR MSE RMSE R2 Adj R2
eq1 111 105 49.3550 0.470047 0.685600 0.184330 0.145489
eq2 111 103 12.9346 0.125579 0.354371 0.193912 0.139129
The covariance matrix of the residuals
eq1 eq2
eq1 0.4700473 0.0347262
eq2 0.0347262 0.1255787
The correlations of the residuals
eq1 eq2
eq1 1.000000 0.142932
eq2 0.142932 1.000000
2SLS estimates for 'eq1' (equation 1)
Model Formula: QTY ~ PRICE + DAY1 + DAY2 + DAY3 + DAY4
Instruments: ~WINDSPD + WINDSPD2 + COLD + RAINY + MIXED + STORMY + DAY1 +
DAY2 + DAY3 + DAY4
Estimate Std. Error t value Pr(>|t|)
(Intercept) 8.5395767 0.1560670 54.71736 < 2.22e-16 ***
PRICE -0.9337678 0.3452685 -2.70447 0.0079832 **
DAY1 -0.0121607 0.2083902 -0.05836 0.9535764
DAY2 -0.5259261 0.2023729 -2.59880 0.0106999 *
DAY3 -0.5626912 0.2070420 -2.71776 0.0076899 **
DAY4 0.1000506 0.2028977 0.49311 0.6229655
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.6856 on 105 degrees of freedom
Number of observations: 111 Degrees of Freedom: 105
SSR: 49.354962 MSE: 0.470047 Root MSE: 0.6856
Multiple R-Squared: 0.18433 Adjusted R-Squared: 0.145489
2SLS estimates for 'eq2' (equation 2)
Model Formula: PRICE ~ QTY + WINDSPD + RAINY + COLD + STORMY + MIXED + WINDSPD2
Instruments: ~WINDSPD + WINDSPD2 + COLD + RAINY + MIXED + STORMY + DAY1 +
DAY2 + DAY3 + DAY4
Estimate Std. Error t value Pr(>|t|)
(Intercept) -3.08018739 5.87202508 -0.52455 0.6010212
QTY 0.05056356 0.12470245 0.40547 0.6859704
WINDSPD 1.44654086 4.06741515 0.35564 0.7228365
RAINY -0.00366122 0.09328092 -0.03925 0.9687676
COLD 0.04688691 0.07791593 0.60176 0.5486544
STORMY 0.38896484 0.14312697 2.71762 0.0077155 **
MIXED 0.21230097 0.09611835 2.20875 0.0294083 *
WINDSPD2 -0.22889804 0.70527628 -0.32455 0.7461792
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.354371 on 103 degrees of freedom
Number of observations: 111 Degrees of Freedom: 103
SSR: 12.934605 MSE: 0.125579 Root MSE: 0.354371
Multiple R-Squared: 0.193912 Adjusted R-Squared: 0.139129
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