Question

I want to run a two stage probit least square regression in R. Does anyone know how to do this? Is there any package out there? I know it’s possible to do it using Stata, so I imagine it’s possible to do it with R.

Answer 1

You might want to be more specific when you say ‘two-stage-probit-least-squares’. Since you refer to a Stata program that implements this I am guessing you are talking about the CDSIMEQ package, which implements the Amemiya (1978) procedure for the Heckit model (a.k.a Generalized Tobit, a.k.a. Tobit type II model, etc.). As Grant said, systemfit will do a Tobit for you, but not with two equations. The MicEcon package did have a Heckit (but the package has split so many times I don’t know where it is now).

If you want what the CDSIMEQ does, it can easily be implemented in R. I wrote a function that replicates CDSIMEQ:

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)
}

For comparison, we can replicate the sample from the author of CDSIMEQ in their article about the package.

> 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