r - R中的两级最小二乘

Question

我想在R中进行两阶段概率最小二乘回归。有人知道该怎么做吗？那里有什么包裹吗？我知道可以使用Stata来做到这一点，所以我想可以使用R来做到这一点。

Answer 1

当您说“两级概率最小二乘”时，您可能想更具体一些。由于您是指实现该目标的Stata程序，所以我猜您正在谈论CDSIMEQ软件包，该软件包为Heckit模型(也称为通用Tobit，也称为Tobit II型模型等)实现了Amemiya(1978)程序。正如Grant所说，systemfit会为您做一个Tobit，但不会用两个方程式。 MicEcon软件包确实有一个Heckit(但是该软件包 split 了很多次，我不知道它现在在哪里)。

如果您需要CDSIMEQ的功能，则可以在R中轻松实现。我编写了一个复制CDSIMEQ的函数:

tspls <- function(formula1, formula2, data) {
    # The Continous model
    mf1 <- model.frame(formula1, data)
    y1 <- model.response(mf1)
    x1 <- model.matrix(attr(mf1, "terms"), mf1)

    # The dicontionous model
    mf2 <- model.frame(formula2, data)
    y2 <- model.response(mf2)
    x2 <- model.matrix(attr(mf2, "terms"), mf2)

    # The matrix of all the exogenous variables
    X <- cbind(x1, x2)
    X <- X[, unique(colnames(X))]

    J1 <- matrix(0, nrow = ncol(X), ncol = ncol(x1))
    J2 <- matrix(0, nrow = ncol(X), ncol = ncol(x2))
    for (i in 1:ncol(x1)) J1[match(colnames(x1)[i], colnames(X)), i] <- 1
    for (i in 1:ncol(x2)) J2[match(colnames(x2)[i], colnames(X)), i] <- 1

    # Step 1:
    cat("\n\tNOW THE FIRST STAGE REGRESSION")
    m1 <- lm(y1 ~ X - 1)
    m2 <- glm(y2 ~ X - 1, family = binomial(link = "probit"))
    print(summary(m1))
    print(summary(m2))

    yhat1 <- m1$fitted.values
    yhat2 <- X %*% coef(m2)

    PI1 <- m1$coefficients
    PI2 <- m2$coefficients
    V0 <- vcov(m2)
    sigma1sq <- sum(m1$residuals ^ 2) / m1$df.residual
    sigma12 <- 1 / length(y2) * sum(y2 * m1$residuals / dnorm(yhat2))

    # Step 2:
    cat("\n\tNOW THE SECOND STAGE REGRESSION WITH INSTRUMENTS")

    m1 <- lm(y1 ~ yhat2 + x1 - 1)
    m2 <- glm(y2 ~ yhat1 + x2 - 1, family = binomial(link = "probit"))
    sm1 <- summary(m1)
    sm2 <- summary(m2)
    print(sm1)
    print(sm2)

    # Step  3:
    cat("\tNOW THE SECOND STAGE REGRESSION WITH CORRECTED STANDARD ERRORS\n\n")
    gamma1 <- m1$coefficients[1]
    gamma2 <- m2$coefficients[1]

    cc <- sigma1sq - 2 * gamma1 * sigma12
    dd <- gamma2 ^ 2 * sigma1sq - 2 * gamma2 * sigma12
    H <- cbind(PI2, J1)
    G <- cbind(PI1, J2)

    XX <- crossprod(X)                          # X'X
    HXXH <- solve(t(H) %*% XX %*% H)            # (H'X'XH)^(-1)
    HXXVXXH <- t(H) %*% XX %*% V0 %*% XX %*% H  # H'X'V0X'XH
    Valpha1 <- cc * HXXH + gamma1 ^ 2 * HXXH %*% HXXVXXH %*% HXXH

    GV <- t(G) %*% solve(V0)    # G'V0^(-1)
    GVG <- solve(GV %*% G)      # (G'V0^(-1)G)^(-1)
    Valpha2 <- GVG + dd * GVG %*% GV %*% solve(XX) %*% solve(V0) %*% G %*% GVG

    ans1 <- coef(sm1)
    ans2 <- coef(sm2)

    ans1[,2] <- sqrt(diag(Valpha1))
    ans2[,2] <- sqrt(diag(Valpha2))
    ans1[,3] <- ans1[,1] / ans1[,2]
    ans2[,3] <- ans2[,1] / ans2[,2]
    ans1[,4] <- 2 * pt(abs(ans1[,3]), m1$df.residual, lower.tail = FALSE)
    ans2[,4] <- 2 * pnorm(abs(ans2[,3]), lower.tail = FALSE)

    cat("Continuous:\n")
    print(ans1)
    cat("Dichotomous:\n")
    print(ans2)
}

为了进行比较，我们可以将CDSIMEQ作者的样本复制到他们的 article about the package中。

> library(foreign)
> cdsimeq <- read.dta("http://www.stata-journal.com/software/sj3-2/st0038/cdsimeq.dta")
> tspls(continuous ~ exog3 + exog2 + exog1 + exog4,
+     dichotomous ~ exog1 + exog2 + exog5 + exog6 + exog7,
+     data = cdsimeq)

        NOW THE FIRST STAGE REGRESSION
Call:
lm(formula = y1 ~ X - 1)

Residuals:
      Min        1Q    Median        3Q       Max 
-1.885921 -0.438579 -0.006262  0.432156  2.133738 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
X(Intercept)  0.010752   0.020620   0.521 0.602187    
Xexog3        0.158469   0.021862   7.249 8.46e-13 ***
Xexog2       -0.009669   0.021666  -0.446 0.655488    
Xexog1        0.159955   0.021260   7.524 1.19e-13 ***
Xexog4        0.316575   0.022456  14.097  < 2e-16 ***
Xexog5        0.497207   0.021356  23.282  < 2e-16 ***
Xexog6       -0.078017   0.021755  -3.586 0.000352 ***
Xexog7        0.161177   0.022103   7.292 6.23e-13 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 0.6488 on 992 degrees of freedom
Multiple R-squared: 0.5972,     Adjusted R-squared: 0.594 
F-statistic: 183.9 on 8 and 992 DF,  p-value: < 2.2e-16 


Call:
glm(formula = y2 ~ X - 1, family = binomial(link = "probit"))

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-2.49531  -0.59244   0.01983   0.59708   2.41810  

Coefficients:
             Estimate Std. Error z value Pr(>|z|)    
X(Intercept)  0.08352    0.05280   1.582 0.113692    
Xexog3        0.21345    0.05678   3.759 0.000170 ***
Xexog2        0.21131    0.05471   3.862 0.000112 ***
Xexog1        0.45591    0.06023   7.570 3.75e-14 ***
Xexog4        0.39031    0.06173   6.322 2.57e-10 ***
Xexog5        0.75955    0.06427  11.818  < 2e-16 ***
Xexog6        0.85461    0.06831  12.510  < 2e-16 ***
Xexog7       -0.16691    0.05653  -2.953 0.003152 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 1386.29  on 1000  degrees of freedom
Residual deviance:  754.14  on  992  degrees of freedom
AIC: 770.14

Number of Fisher Scoring iterations: 6


        NOW THE SECOND STAGE REGRESSION WITH INSTRUMENTS
Call:
lm(formula = y1 ~ yhat2 + x1 - 1)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.32152 -0.53160  0.04886  0.53502  2.44818 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
yhat2         0.257592   0.021451  12.009   <2e-16 ***
x1(Intercept) 0.012185   0.024809   0.491    0.623    
x1exog3       0.042520   0.026735   1.590    0.112    
x1exog2       0.011854   0.026723   0.444    0.657    
x1exog1       0.007773   0.028217   0.275    0.783    
x1exog4       0.318636   0.028311  11.255   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 0.7803 on 994 degrees of freedom
Multiple R-squared: 0.4163,     Adjusted R-squared: 0.4128 
F-statistic: 118.2 on 6 and 994 DF,  p-value: < 2.2e-16 


Call:
glm(formula = y2 ~ yhat1 + x2 - 1, family = binomial(link = "probit"))

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-2.49610  -0.58595   0.01969   0.59857   2.41281  

Coefficients:
              Estimate Std. Error z value Pr(>|z|)    
yhat1          1.26287    0.16061   7.863 3.75e-15 ***
x2(Intercept)  0.07080    0.05276   1.342 0.179654    
x2exog1        0.25093    0.06466   3.880 0.000104 ***
x2exog2        0.22604    0.05389   4.194 2.74e-05 ***
x2exog5        0.12912    0.09510   1.358 0.174544    
x2exog6        0.95609    0.07172  13.331  < 2e-16 ***
x2exog7       -0.37128    0.06759  -5.493 3.94e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 1386.29  on 1000  degrees of freedom
Residual deviance:  754.21  on  993  degrees of freedom
AIC: 768.21

Number of Fisher Scoring iterations: 6

        NOW THE SECOND STAGE REGRESSION WITH CORRECTED STANDARD ERRORS

Continuous:
                Estimate Std. Error    t value   Pr(>|t|)
yhat2         0.25759209  0.1043073 2.46955009 0.01369540
x1(Intercept) 0.01218500  0.1198713 0.10165068 0.91905445
x1exog3       0.04252006  0.1291588 0.32920764 0.74206810
x1exog2       0.01185438  0.1290754 0.09184073 0.92684309
x1exog1       0.00777347  0.1363643 0.05700519 0.95455252
x1exog4       0.31863627  0.1367881 2.32941597 0.02003661
Dichotomous:
                 Estimate Std. Error    z value     Pr(>|z|)
yhat1          1.26286574  0.7395166  1.7076909 0.0876937093
x2(Intercept)  0.07079775  0.2666447  0.2655134 0.7906139867
x2exog1        0.25092561  0.3126763  0.8025092 0.4222584495
x2exog2        0.22603717  0.2739307  0.8251618 0.4092797527
x2exog5        0.12911922  0.4822986  0.2677163 0.7889176766
x2exog6        0.95609385  0.2823662  3.3860070 0.0007091758
x2exog7       -0.37128221  0.3265478 -1.1369920 0.2555416141