invMillsRatio: Inverse Mill's Ratio of probit models
In sampleSelection: Sample Selection Models
Description
Usage
Arguments
Details
Value
Author(s)
References
Examples
View source: R/invMillsRatio.R
Description
Calculates the 'Inverse Mill's Ratios' of univariate and bivariate
probit models.
Usage
1
invMillsRatio( x, all = FALSE )
Arguments
x
probit model estimated by probit, glm or
vglm.
all
a logical value indicating whether the inverse Mill's Ratios
should be calculated for all observations.
Details
The formula to calculate the inverse Mill's ratios for univariate probit
models is taken from Greene (2003, p. 785),
whereas the formulas for bivariate probit models are derived in
Henning and Henningsen (2005).
Value
A data frame that contains the Inverse Mill's Ratios (IMR) and the delta
values (see Greene, 2003, p. 784).
If a univariate probit estimation is provided, the variables
IMR1 and IMR0 are the Inverse Mill's Ratios to correct
for a sample selection bias of y = 1 and y = 0, respectively.
Accordingly, 'delta1' and 'delta0' are the corresponding delta values.
If a bivariate probit estimation is provided, the variables
IMRa1, IMRa0, IMRb1, and IMRb0 are the
Inverse Mills Ratios to correct for a sample selection bias
of y = 1 and y = 0 in equations 'a' and 'b', respectively.
Accordingly, 'deltaa1', 'deltaa0', 'deltab1' and 'deltab0' are the
corresponding delta values.
Author(s)
Arne Henningsen
References
Greene, W. H. (2003)
Econometric Analysis, Fifth Edition, Prentice Hall.
Henning, C.H.C.A and A. Henningsen (2005)
Modeling Price Response of Farm Households in Imperfect
Labor Markets in Poland:
Incorporating Transaction Costs and Heterogeneity into a Farm
Household Approach.
Unpublished, University of Kiel, Germany.
Examples
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## Wooldridge( 2003 ): example 17.5, page 590
data(Mroz87)
myProbit <- glm( lfp ~ nwifeinc + educ + exper + I( exper^2 ) + age +
kids5 + kids618, family = binomial( link = "probit" ), data=Mroz87 )
Mroz87$IMR <- invMillsRatio( myProbit )$IMR1
myHeckit <- lm( log( wage ) ~ educ + exper + I( exper^2 ) + IMR,
data = Mroz87[ Mroz87$lfp == 1, ] )
# using NO labor force participation as endogenous variable
Mroz87$nolfp <- 1 - Mroz87$lfp
myProbit2 <- glm( nolfp ~ nwifeinc + educ + exper + I( exper^2 ) + age +
kids5 + kids618, family = binomial( link = "probit" ), data=Mroz87 )
all.equal( invMillsRatio( myProbit )$IMR1, invMillsRatio( myProbit2 )$IMR0 )
# should be true
# example for bivariate probit
library( "mvtnorm" )
library( "VGAM" )
nObs <- 1000
# error terms (trivariate normal)
sigma <- symMatrix( c( 2, 0.7, 1.2, 1, 0.5, 1 ) )
myData <- as.data.frame( rmvnorm( nObs, c( 0, 0, 0 ), sigma ) )
names( myData ) <- c( "e0", "e1", "e2" )
# exogenous variables (indepently normal)
myData$x0 <- rnorm( nObs )
myData$x1 <- rnorm( nObs )
myData$x2 <- rnorm( nObs )
# endogenous variables
myData$y0 <- -1.5 + 0.8 * myData$x1 + myData$e0
myData$y1 <- ( 0.3 + 0.4 * myData$x1 + 0.3 * myData$x2 + myData$e1 ) > 0
myData$y2 <- ( -0.1 + 0.6 * myData$x1 + 0.7 * myData$x2 + myData$e2 ) > 0
# bivariate probit (using rhobit transformation)
bProbit <- vglm( cbind( y1, y2 ) ~ x1 + x2, family = binom2.rho,
data = myData )
summary( bProbit )
# bivariate probit (NOT using rhobit transformation)
bProbit2 <- vglm( cbind( y1, y2 ) ~ x1 + x2, family = binom2.rho(
lrho = "identitylink" ), data = myData )
summary( bProbit2 )
# inverse Mills Ratios
imr <- invMillsRatio( bProbit )
imr2 <- invMillsRatio( bProbit2 )
all.equal( imr, imr2, tolerance = .Machine$double.eps ^ 0.25)
# tests
# E[ e0 | y1* > 0 & y2* > 0 ]
mean( myData$e0[ myData$y1 & myData$y2 ] )
mean( sigma[1,2] * imr$IMR11a + sigma[1,3] * imr$IMR11b, na.rm = TRUE )
# E[ e0 | y1* > 0 & y2* <= 0 ]
mean( myData$e0[ myData$y1 & !myData$y2 ] )
mean( sigma[1,2] * imr$IMR10a + sigma[1,3] * imr$IMR10b, na.rm = TRUE )
# E[ e0 | y1* <= 0 & y2* > 0 ]
mean( myData$e0[ !myData$y1 & myData$y2 ] )
mean( sigma[1,2] * imr$IMR01a + sigma[1,3] * imr$IMR01b, na.rm = TRUE )
# E[ e0 | y1* <= 0 & y2* <= 0 ]
mean( myData$e0[ !myData$y1 & !myData$y2 ] )
mean( sigma[1,2] * imr$IMR00a + sigma[1,3] * imr$IMR00b, na.rm = TRUE )
# E[ e0 | y1* > 0 ]
mean( myData$e0[ myData$y1 ] )
mean( sigma[1,2] * imr$IMR1X, na.rm = TRUE )
# E[ e0 | y1* <= 0 ]
mean( myData$e0[ !myData$y1 ] )
mean( sigma[1,2] * imr$IMR0X, na.rm = TRUE )
# E[ e0 | y2* > 0 ]
mean( myData$e0[ myData$y2 ] )
mean( sigma[1,3] * imr$IMRX1, na.rm = TRUE )
# E[ e0 | y2* <= 0 ]
mean( myData$e0[ !myData$y2 ] )
mean( sigma[1,3] * imr$IMRX0, na.rm = TRUE )
# estimation for y1* > 0 and y2* > 0
selection <- myData$y1 & myData$y2
# OLS estimation
ols11 <- lm( y0 ~ x1, data = myData, subset = selection )
summary( ols11 )
# heckman type estimation
heckit11 <- lm( y0 ~ x1 + IMR11a + IMR11b, data = cbind( myData, imr ),
subset = selection )
summary( heckit11 )
# estimation for y1* > 0 and y2* <= 0
selection <- myData$y1 & !myData$y2
# OLS estimation
ols10 <- lm( y0 ~ x1, data = myData, subset = selection )
summary( ols10 )
# heckman type estimation
heckit10 <- lm( y0 ~ x1 + IMR10a + IMR10b, data = cbind( myData, imr ),
subset = selection )
summary( heckit10 )
# estimation for y1* <= 0 and y2* > 0
selection <- !myData$y1 & myData$y2
# OLS estimation
ols01 <- lm( y0 ~ x1, data = myData, subset = selection )
summary( ols01 )
# heckman type estimation
heckit01 <- lm( y0 ~ x1 + IMR01a + IMR01b, data = cbind( myData, imr ),
subset = selection )
summary( heckit01 )
# estimation for y1* <= 0 and y2* <= 0
selection <- !myData$y1 & !myData$y2
# OLS estimation
ols00 <- lm( y0 ~ x1, data = myData, subset = selection )
summary( ols00 )
# heckman type estimation
heckit00 <- lm( y0 ~ x1 + IMR00a + IMR00b, data = cbind( myData, imr ),
subset = selection )
summary( heckit00 )
# estimation for y1* > 0
selection <- myData$y1
# OLS estimation
ols1X <- lm( y0 ~ x1, data = myData, subset = selection )
summary( ols1X )
# heckman type estimation
heckit1X <- lm( y0 ~ x1 + IMR1X, data = cbind( myData, imr ),
subset = selection )
summary( heckit1X )
# estimation for y1* <= 0
selection <- !myData$y1
# OLS estimation
ols0X <- lm( y0 ~ x1, data = myData, subset = selection )
summary( ols0X )
# heckman type estimation
heckit0X <- lm( y0 ~ x1 + IMR0X, data = cbind( myData, imr ),
subset = selection )
summary( heckit0X )
# estimation for y2* > 0
selection <- myData$y2
# OLS estimation
olsX1 <- lm( y0 ~ x1, data = myData, subset = selection )
summary( olsX1 )
# heckman type estimation
heckitX1 <- lm( y0 ~ x1 + IMRX1, data = cbind( myData, imr ),
subset = selection )
summary( heckitX1 )
# estimation for y2* <= 0
selection <- !myData$y2
# OLS estimation
olsX0 <- lm( y0 ~ x1, data = myData, subset = selection )
summary( olsX0 )
# heckman type estimation
heckitX0 <- lm( y0 ~ x1 + IMRX0, data = cbind( myData, imr ),
subset = selection )
summary( heckitX0 )
Example output
Loading required package: maxLik
Loading required package: miscTools
Please cite the 'maxLik' package as:
Henningsen, Arne and Toomet, Ott (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics 26(3), 443-458. DOI 10.1007/s00180-010-0217-1.
If you have questions, suggestions, or comments regarding the 'maxLik' package, please use a forum or 'tracker' at maxLik's R-Forge site:
https://r-forge.r-project.org/projects/maxlik/
[1] TRUE
Loading required package: stats4
Loading required package: splines
Attaching package: 'VGAM'
The following object is masked from 'package:sampleSelection':
probit
Call:
vglm(formula = cbind(y1, y2) ~ x1 + x2, family = binom2.rho,
data = myData)
Pearson residuals:
Min 1Q Median 3Q Max
probitlink(mu1) -3.099 -0.9009 0.4622 0.7685 2.346
probitlink(mu2) -3.168 -0.6602 -0.1993 0.6541 5.677
rhobitlink(rho) -7.651 -0.6032 0.2478 0.5139 4.140
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept):1 0.28134 0.04178 6.734 1.66e-11 ***
(Intercept):2 -0.09840 0.04446 -2.213 0.0269 *
(Intercept):3 1.15844 0.12659 9.151 < 2e-16 ***
x1:1 0.35687 0.04369 8.168 3.13e-16 ***
x1:2 0.59222 0.04918 12.043 < 2e-16 ***
x2:1 0.29754 0.04298 6.923 4.43e-12 ***
x2:2 0.67683 0.05130 13.195 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Names of linear predictors: probitlink(mu1), probitlink(mu2), rhobitlink(rho)
Log-likelihood: -1095.843 on 2993 degrees of freedom
Number of Fisher scoring iterations: 3
No Hauck-Donner effect found in any of the estimates
Call:
vglm(formula = cbind(y1, y2) ~ x1 + x2, family = binom2.rho(lrho = "identitylink"),
data = myData)
Pearson residuals:
Min 1Q Median 3Q Max
probitlink(mu1) -3.086 -0.9034 0.4595 0.7653 2.346
probitlink(mu2) -3.273 -0.6617 -0.1966 0.6585 5.436
rho -7.826 -0.5830 0.2583 0.5153 4.036
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept):1 0.28134 0.04178 6.734 1.66e-11 ***
(Intercept):2 -0.09840 0.04446 -2.213 0.0269 *
(Intercept):3 0.52210 0.04604 11.339 < 2e-16 ***
x1:1 0.35687 0.04369 8.168 3.13e-16 ***
x1:2 0.59222 0.04918 12.043 < 2e-16 ***
x2:1 0.29754 0.04298 6.923 4.43e-12 ***
x2:2 0.67683 0.05130 13.195 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Names of linear predictors: probitlink(mu1), probitlink(mu2), rho
Log-likelihood: -1095.843 on 2993 degrees of freedom
Number of Fisher scoring iterations: 3
No Hauck-Donner effect found in any of the estimates
[1] TRUE
[1] 0.7880256
[1] 0.8411649
[1] -0.2699706
[1] -0.3589691
[1] 0.468903
[1] 0.4019226
[1] -0.7789105
[1] -0.8908078
[1] 0.4066549
[1] 0.4080471
[1] -0.5155529
[1] -0.618372
[1] 0.7308693
[1] 0.7616975
[1] -0.5709256
[1] -0.6736764
Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.1136 -0.7843 -0.0275 0.7821 3.4793
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.56270 0.06376 -8.825 < 2e-16 ***
x1 0.46942 0.05975 7.856 4.04e-14 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.134 on 383 degrees of freedom
Multiple R-squared: 0.1388, Adjusted R-squared: 0.1365
F-statistic: 61.72 on 1 and 383 DF, p-value: 4.04e-14
Call:
lm(formula = y0 ~ x1 + IMR11a + IMR11b, data = cbind(myData,
imr), subset = selection)
Residuals:
Min 1Q Median 3Q Max
-2.8721 -0.7544 -0.0240 0.6067 3.5395
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.9969 0.6878 -2.903 0.00391 **
x1 0.8282 0.1296 6.389 4.88e-10 ***
IMR11a 2.1404 2.2446 0.954 0.34090
IMR11b 1.0673 0.3525 3.028 0.00263 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.065 on 381 degrees of freedom
Multiple R-squared: 0.2434, Adjusted R-squared: 0.2374
F-statistic: 40.85 on 3 and 381 DF, p-value: < 2.2e-16
Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.5027 -0.6912 0.0488 0.7793 2.8668
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.83487 0.07539 -24.340 < 2e-16 ***
x1 0.37272 0.07992 4.663 5.46e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.096 on 215 degrees of freedom
Multiple R-squared: 0.09186, Adjusted R-squared: 0.08763
F-statistic: 21.75 on 1 and 215 DF, p-value: 5.461e-06
Call:
lm(formula = y0 ~ x1 + IMR10a + IMR10b, data = cbind(myData,
imr), subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.5825 -0.7011 0.0936 0.7392 2.2169
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -3.2661 1.8446 -1.771 0.078053 .
x1 0.8804 0.1663 5.293 2.99e-07 ***
IMR10a 2.4208 1.6469 1.470 0.143057
IMR10b 1.0472 0.2947 3.554 0.000467 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.9822 on 213 degrees of freedom
Multiple R-squared: 0.2774, Adjusted R-squared: 0.2673
F-statistic: 27.26 on 3 and 213 DF, p-value: 5.822e-15
Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.4618 -0.6188 0.0633 0.7342 2.9974
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.9861 0.1247 -7.906 1.07e-11 ***
x1 0.4616 0.1526 3.025 0.00332 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.128 on 82 degrees of freedom
Multiple R-squared: 0.1004, Adjusted R-squared: 0.0894
F-statistic: 9.148 on 1 and 82 DF, p-value: 0.003323
Call:
lm(formula = y0 ~ x1 + IMR01a + IMR01b, data = cbind(myData,
imr), subset = selection)
Residuals:
Min 1Q Median 3Q Max
-2.21368 -0.61448 0.02581 0.66745 2.18196
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.1408 5.7557 0.198 0.84339
x1 1.0279 0.4067 2.527 0.01346 *
IMR01a 2.1160 3.5098 0.603 0.54829
IMR01b 0.9031 0.3126 2.889 0.00497 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.019 on 80 degrees of freedom
Multiple R-squared: 0.2832, Adjusted R-squared: 0.2563
F-statistic: 10.54 on 3 and 80 DF, p-value: 6.489e-06
Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.3059 -0.7136 -0.0389 0.7179 3.2808
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2.38799 0.06894 -34.639 < 2e-16 ***
x1 0.50504 0.07066 7.148 6.28e-12 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.13 on 312 degrees of freedom
Multiple R-squared: 0.1407, Adjusted R-squared: 0.138
F-statistic: 51.09 on 1 and 312 DF, p-value: 6.28e-12
Call:
lm(formula = y0 ~ x1 + IMR00a + IMR00b, data = cbind(myData,
imr), subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.5236 -0.7402 -0.0121 0.7445 3.0029
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.6417 0.7507 -2.187 0.029509 *
x1 0.7906 0.1267 6.240 1.44e-09 ***
IMR00a 0.1775 1.1859 0.150 0.881131
IMR00b 1.5404 0.4502 3.421 0.000707 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.072 on 310 degrees of freedom
Multiple R-squared: 0.2317, Adjusted R-squared: 0.2243
F-statistic: 31.17 on 3 and 310 DF, p-value: < 2.2e-16
Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)
Residuals:
Min 1Q Median 3Q Max
-4.5384 -0.8422 0.0159 0.8606 3.8369
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.05005 0.05292 -19.84 <2e-16 ***
x1 0.61499 0.05168 11.90 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.264 on 600 degrees of freedom
Multiple R-squared: 0.191, Adjusted R-squared: 0.1896
F-statistic: 141.6 on 1 and 600 DF, p-value: < 2.2e-16
Call:
lm(formula = y0 ~ x1 + IMR1X, data = cbind(myData, imr), subset = selection)
Residuals:
Min 1Q Median 3Q Max
-4.5180 -0.8633 0.0390 0.8578 3.8874
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.62505 0.20925 -7.766 3.53e-14 ***
x1 0.77235 0.07558 10.220 < 2e-16 ***
IMR1X 0.92323 0.32519 2.839 0.00468 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.257 on 599 degrees of freedom
Multiple R-squared: 0.2017, Adjusted R-squared: 0.199
F-statistic: 75.68 on 2 and 599 DF, p-value: < 2.2e-16
Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.5320 -0.8754 -0.0490 0.8899 4.1030
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2.05778 0.06565 -31.344 <2e-16 ***
x1 0.63987 0.06951 9.205 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.258 on 396 degrees of freedom
Multiple R-squared: 0.1763, Adjusted R-squared: 0.1742
F-statistic: 84.74 on 1 and 396 DF, p-value: < 2.2e-16
Call:
lm(formula = y0 ~ x1 + IMR0X, data = cbind(myData, imr), subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.6358 -0.8835 -0.0711 0.9058 3.8633
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.2215 0.3177 -3.844 0.000141 ***
x1 0.8415 0.1019 8.260 2.22e-15 ***
IMR0X 0.8865 0.3296 2.689 0.007463 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.248 on 395 degrees of freedom
Multiple R-squared: 0.1911, Adjusted R-squared: 0.187
F-statistic: 46.65 on 2 and 395 DF, p-value: < 2.2e-16
Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.8057 -0.7642 -0.0296 0.7640 3.5448
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.64705 0.05710 -11.332 <2e-16 ***
x1 0.49052 0.05561 8.821 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.142 on 467 degrees of freedom
Multiple R-squared: 0.1428, Adjusted R-squared: 0.141
F-statistic: 77.82 on 1 and 467 DF, p-value: < 2.2e-16
Call:
lm(formula = y0 ~ x1 + IMRX1, data = cbind(myData, imr), subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.0754 -0.7093 -0.0606 0.6356 3.6354
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.50842 0.11934 -12.640 < 2e-16 ***
x1 0.76903 0.06251 12.303 < 2e-16 ***
IMRX1 1.18395 0.14660 8.076 5.76e-15 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.07 on 466 degrees of freedom
Multiple R-squared: 0.2481, Adjusted R-squared: 0.2448
F-statistic: 76.87 on 2 and 466 DF, p-value: < 2.2e-16
Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.5498 -0.7580 -0.0424 0.8063 3.1909
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2.15926 0.05227 -41.307 <2e-16 ***
x1 0.48537 0.05431 8.938 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.152 on 529 degrees of freedom
Multiple R-squared: 0.1312, Adjusted R-squared: 0.1295
F-statistic: 79.88 on 1 and 529 DF, p-value: < 2.2e-16
Call:
lm(formula = y0 ~ x1 + IMRX0, data = cbind(myData, imr), subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.8148 -0.7703 0.0468 0.7909 2.7010
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.34764 0.11035 -12.212 < 2e-16 ***
x1 0.78002 0.06249 12.483 < 2e-16 ***
IMRX0 1.29836 0.15796 8.219 1.59e-15 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.086 on 528 degrees of freedom
Multiple R-squared: 0.2297, Adjusted R-squared: 0.2268
F-statistic: 78.74 on 2 and 528 DF, p-value: < 2.2e-16
sampleSelection documentation built on Jan. 13, 2021, 7:49 p.m.
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Description Usage Arguments Details Value Author(s) References Examples
View source: R/invMillsRatio.R
Description
Calculates the 'Inverse Mill's Ratios' of univariate and bivariate probit models.
Usage
1 | invMillsRatio( x, all = FALSE )
|
Arguments
x |
probit model estimated by |
all |
a logical value indicating whether the inverse Mill's Ratios should be calculated for all observations. |
Details
The formula to calculate the inverse Mill's ratios for univariate probit models is taken from Greene (2003, p. 785), whereas the formulas for bivariate probit models are derived in Henning and Henningsen (2005).
Value
A data frame that contains the Inverse Mill's Ratios (IMR) and the delta values (see Greene, 2003, p. 784).
If a univariate probit estimation is provided, the variables
IMR1 and IMR0 are the Inverse Mill's Ratios to correct
for a sample selection bias of y = 1 and y = 0, respectively.
Accordingly, 'delta1' and 'delta0' are the corresponding delta values.
If a bivariate probit estimation is provided, the variables
IMRa1, IMRa0, IMRb1, and IMRb0 are the
Inverse Mills Ratios to correct for a sample selection bias
of y = 1 and y = 0 in equations 'a' and 'b', respectively.
Accordingly, 'deltaa1', 'deltaa0', 'deltab1' and 'deltab0' are the
corresponding delta values.
Author(s)
Arne Henningsen
References
Greene, W. H. (2003) Econometric Analysis, Fifth Edition, Prentice Hall.
Henning, C.H.C.A and A. Henningsen (2005) Modeling Price Response of Farm Households in Imperfect Labor Markets in Poland: Incorporating Transaction Costs and Heterogeneity into a Farm Household Approach. Unpublished, University of Kiel, Germany.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 | ## Wooldridge( 2003 ): example 17.5, page 590
data(Mroz87)
myProbit <- glm( lfp ~ nwifeinc + educ + exper + I( exper^2 ) + age +
kids5 + kids618, family = binomial( link = "probit" ), data=Mroz87 )
Mroz87$IMR <- invMillsRatio( myProbit )$IMR1
myHeckit <- lm( log( wage ) ~ educ + exper + I( exper^2 ) + IMR,
data = Mroz87[ Mroz87$lfp == 1, ] )
# using NO labor force participation as endogenous variable
Mroz87$nolfp <- 1 - Mroz87$lfp
myProbit2 <- glm( nolfp ~ nwifeinc + educ + exper + I( exper^2 ) + age +
kids5 + kids618, family = binomial( link = "probit" ), data=Mroz87 )
all.equal( invMillsRatio( myProbit )$IMR1, invMillsRatio( myProbit2 )$IMR0 )
# should be true
# example for bivariate probit
library( "mvtnorm" )
library( "VGAM" )
nObs <- 1000
# error terms (trivariate normal)
sigma <- symMatrix( c( 2, 0.7, 1.2, 1, 0.5, 1 ) )
myData <- as.data.frame( rmvnorm( nObs, c( 0, 0, 0 ), sigma ) )
names( myData ) <- c( "e0", "e1", "e2" )
# exogenous variables (indepently normal)
myData$x0 <- rnorm( nObs )
myData$x1 <- rnorm( nObs )
myData$x2 <- rnorm( nObs )
# endogenous variables
myData$y0 <- -1.5 + 0.8 * myData$x1 + myData$e0
myData$y1 <- ( 0.3 + 0.4 * myData$x1 + 0.3 * myData$x2 + myData$e1 ) > 0
myData$y2 <- ( -0.1 + 0.6 * myData$x1 + 0.7 * myData$x2 + myData$e2 ) > 0
# bivariate probit (using rhobit transformation)
bProbit <- vglm( cbind( y1, y2 ) ~ x1 + x2, family = binom2.rho,
data = myData )
summary( bProbit )
# bivariate probit (NOT using rhobit transformation)
bProbit2 <- vglm( cbind( y1, y2 ) ~ x1 + x2, family = binom2.rho(
lrho = "identitylink" ), data = myData )
summary( bProbit2 )
# inverse Mills Ratios
imr <- invMillsRatio( bProbit )
imr2 <- invMillsRatio( bProbit2 )
all.equal( imr, imr2, tolerance = .Machine$double.eps ^ 0.25)
# tests
# E[ e0 | y1* > 0 & y2* > 0 ]
mean( myData$e0[ myData$y1 & myData$y2 ] )
mean( sigma[1,2] * imr$IMR11a + sigma[1,3] * imr$IMR11b, na.rm = TRUE )
# E[ e0 | y1* > 0 & y2* <= 0 ]
mean( myData$e0[ myData$y1 & !myData$y2 ] )
mean( sigma[1,2] * imr$IMR10a + sigma[1,3] * imr$IMR10b, na.rm = TRUE )
# E[ e0 | y1* <= 0 & y2* > 0 ]
mean( myData$e0[ !myData$y1 & myData$y2 ] )
mean( sigma[1,2] * imr$IMR01a + sigma[1,3] * imr$IMR01b, na.rm = TRUE )
# E[ e0 | y1* <= 0 & y2* <= 0 ]
mean( myData$e0[ !myData$y1 & !myData$y2 ] )
mean( sigma[1,2] * imr$IMR00a + sigma[1,3] * imr$IMR00b, na.rm = TRUE )
# E[ e0 | y1* > 0 ]
mean( myData$e0[ myData$y1 ] )
mean( sigma[1,2] * imr$IMR1X, na.rm = TRUE )
# E[ e0 | y1* <= 0 ]
mean( myData$e0[ !myData$y1 ] )
mean( sigma[1,2] * imr$IMR0X, na.rm = TRUE )
# E[ e0 | y2* > 0 ]
mean( myData$e0[ myData$y2 ] )
mean( sigma[1,3] * imr$IMRX1, na.rm = TRUE )
# E[ e0 | y2* <= 0 ]
mean( myData$e0[ !myData$y2 ] )
mean( sigma[1,3] * imr$IMRX0, na.rm = TRUE )
# estimation for y1* > 0 and y2* > 0
selection <- myData$y1 & myData$y2
# OLS estimation
ols11 <- lm( y0 ~ x1, data = myData, subset = selection )
summary( ols11 )
# heckman type estimation
heckit11 <- lm( y0 ~ x1 + IMR11a + IMR11b, data = cbind( myData, imr ),
subset = selection )
summary( heckit11 )
# estimation for y1* > 0 and y2* <= 0
selection <- myData$y1 & !myData$y2
# OLS estimation
ols10 <- lm( y0 ~ x1, data = myData, subset = selection )
summary( ols10 )
# heckman type estimation
heckit10 <- lm( y0 ~ x1 + IMR10a + IMR10b, data = cbind( myData, imr ),
subset = selection )
summary( heckit10 )
# estimation for y1* <= 0 and y2* > 0
selection <- !myData$y1 & myData$y2
# OLS estimation
ols01 <- lm( y0 ~ x1, data = myData, subset = selection )
summary( ols01 )
# heckman type estimation
heckit01 <- lm( y0 ~ x1 + IMR01a + IMR01b, data = cbind( myData, imr ),
subset = selection )
summary( heckit01 )
# estimation for y1* <= 0 and y2* <= 0
selection <- !myData$y1 & !myData$y2
# OLS estimation
ols00 <- lm( y0 ~ x1, data = myData, subset = selection )
summary( ols00 )
# heckman type estimation
heckit00 <- lm( y0 ~ x1 + IMR00a + IMR00b, data = cbind( myData, imr ),
subset = selection )
summary( heckit00 )
# estimation for y1* > 0
selection <- myData$y1
# OLS estimation
ols1X <- lm( y0 ~ x1, data = myData, subset = selection )
summary( ols1X )
# heckman type estimation
heckit1X <- lm( y0 ~ x1 + IMR1X, data = cbind( myData, imr ),
subset = selection )
summary( heckit1X )
# estimation for y1* <= 0
selection <- !myData$y1
# OLS estimation
ols0X <- lm( y0 ~ x1, data = myData, subset = selection )
summary( ols0X )
# heckman type estimation
heckit0X <- lm( y0 ~ x1 + IMR0X, data = cbind( myData, imr ),
subset = selection )
summary( heckit0X )
# estimation for y2* > 0
selection <- myData$y2
# OLS estimation
olsX1 <- lm( y0 ~ x1, data = myData, subset = selection )
summary( olsX1 )
# heckman type estimation
heckitX1 <- lm( y0 ~ x1 + IMRX1, data = cbind( myData, imr ),
subset = selection )
summary( heckitX1 )
# estimation for y2* <= 0
selection <- !myData$y2
# OLS estimation
olsX0 <- lm( y0 ~ x1, data = myData, subset = selection )
summary( olsX0 )
# heckman type estimation
heckitX0 <- lm( y0 ~ x1 + IMRX0, data = cbind( myData, imr ),
subset = selection )
summary( heckitX0 )
|
Example output
Loading required package: maxLik
Loading required package: miscTools
Please cite the 'maxLik' package as:
Henningsen, Arne and Toomet, Ott (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics 26(3), 443-458. DOI 10.1007/s00180-010-0217-1.
If you have questions, suggestions, or comments regarding the 'maxLik' package, please use a forum or 'tracker' at maxLik's R-Forge site:
https://r-forge.r-project.org/projects/maxlik/
[1] TRUE
Loading required package: stats4
Loading required package: splines
Attaching package: 'VGAM'
The following object is masked from 'package:sampleSelection':
probit
Call:
vglm(formula = cbind(y1, y2) ~ x1 + x2, family = binom2.rho,
data = myData)
Pearson residuals:
Min 1Q Median 3Q Max
probitlink(mu1) -3.099 -0.9009 0.4622 0.7685 2.346
probitlink(mu2) -3.168 -0.6602 -0.1993 0.6541 5.677
rhobitlink(rho) -7.651 -0.6032 0.2478 0.5139 4.140
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept):1 0.28134 0.04178 6.734 1.66e-11 ***
(Intercept):2 -0.09840 0.04446 -2.213 0.0269 *
(Intercept):3 1.15844 0.12659 9.151 < 2e-16 ***
x1:1 0.35687 0.04369 8.168 3.13e-16 ***
x1:2 0.59222 0.04918 12.043 < 2e-16 ***
x2:1 0.29754 0.04298 6.923 4.43e-12 ***
x2:2 0.67683 0.05130 13.195 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Names of linear predictors: probitlink(mu1), probitlink(mu2), rhobitlink(rho)
Log-likelihood: -1095.843 on 2993 degrees of freedom
Number of Fisher scoring iterations: 3
No Hauck-Donner effect found in any of the estimates
Call:
vglm(formula = cbind(y1, y2) ~ x1 + x2, family = binom2.rho(lrho = "identitylink"),
data = myData)
Pearson residuals:
Min 1Q Median 3Q Max
probitlink(mu1) -3.086 -0.9034 0.4595 0.7653 2.346
probitlink(mu2) -3.273 -0.6617 -0.1966 0.6585 5.436
rho -7.826 -0.5830 0.2583 0.5153 4.036
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept):1 0.28134 0.04178 6.734 1.66e-11 ***
(Intercept):2 -0.09840 0.04446 -2.213 0.0269 *
(Intercept):3 0.52210 0.04604 11.339 < 2e-16 ***
x1:1 0.35687 0.04369 8.168 3.13e-16 ***
x1:2 0.59222 0.04918 12.043 < 2e-16 ***
x2:1 0.29754 0.04298 6.923 4.43e-12 ***
x2:2 0.67683 0.05130 13.195 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Names of linear predictors: probitlink(mu1), probitlink(mu2), rho
Log-likelihood: -1095.843 on 2993 degrees of freedom
Number of Fisher scoring iterations: 3
No Hauck-Donner effect found in any of the estimates
[1] TRUE
[1] 0.7880256
[1] 0.8411649
[1] -0.2699706
[1] -0.3589691
[1] 0.468903
[1] 0.4019226
[1] -0.7789105
[1] -0.8908078
[1] 0.4066549
[1] 0.4080471
[1] -0.5155529
[1] -0.618372
[1] 0.7308693
[1] 0.7616975
[1] -0.5709256
[1] -0.6736764
Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.1136 -0.7843 -0.0275 0.7821 3.4793
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.56270 0.06376 -8.825 < 2e-16 ***
x1 0.46942 0.05975 7.856 4.04e-14 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.134 on 383 degrees of freedom
Multiple R-squared: 0.1388, Adjusted R-squared: 0.1365
F-statistic: 61.72 on 1 and 383 DF, p-value: 4.04e-14
Call:
lm(formula = y0 ~ x1 + IMR11a + IMR11b, data = cbind(myData,
imr), subset = selection)
Residuals:
Min 1Q Median 3Q Max
-2.8721 -0.7544 -0.0240 0.6067 3.5395
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.9969 0.6878 -2.903 0.00391 **
x1 0.8282 0.1296 6.389 4.88e-10 ***
IMR11a 2.1404 2.2446 0.954 0.34090
IMR11b 1.0673 0.3525 3.028 0.00263 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.065 on 381 degrees of freedom
Multiple R-squared: 0.2434, Adjusted R-squared: 0.2374
F-statistic: 40.85 on 3 and 381 DF, p-value: < 2.2e-16
Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.5027 -0.6912 0.0488 0.7793 2.8668
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.83487 0.07539 -24.340 < 2e-16 ***
x1 0.37272 0.07992 4.663 5.46e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.096 on 215 degrees of freedom
Multiple R-squared: 0.09186, Adjusted R-squared: 0.08763
F-statistic: 21.75 on 1 and 215 DF, p-value: 5.461e-06
Call:
lm(formula = y0 ~ x1 + IMR10a + IMR10b, data = cbind(myData,
imr), subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.5825 -0.7011 0.0936 0.7392 2.2169
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -3.2661 1.8446 -1.771 0.078053 .
x1 0.8804 0.1663 5.293 2.99e-07 ***
IMR10a 2.4208 1.6469 1.470 0.143057
IMR10b 1.0472 0.2947 3.554 0.000467 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.9822 on 213 degrees of freedom
Multiple R-squared: 0.2774, Adjusted R-squared: 0.2673
F-statistic: 27.26 on 3 and 213 DF, p-value: 5.822e-15
Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.4618 -0.6188 0.0633 0.7342 2.9974
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.9861 0.1247 -7.906 1.07e-11 ***
x1 0.4616 0.1526 3.025 0.00332 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.128 on 82 degrees of freedom
Multiple R-squared: 0.1004, Adjusted R-squared: 0.0894
F-statistic: 9.148 on 1 and 82 DF, p-value: 0.003323
Call:
lm(formula = y0 ~ x1 + IMR01a + IMR01b, data = cbind(myData,
imr), subset = selection)
Residuals:
Min 1Q Median 3Q Max
-2.21368 -0.61448 0.02581 0.66745 2.18196
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.1408 5.7557 0.198 0.84339
x1 1.0279 0.4067 2.527 0.01346 *
IMR01a 2.1160 3.5098 0.603 0.54829
IMR01b 0.9031 0.3126 2.889 0.00497 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.019 on 80 degrees of freedom
Multiple R-squared: 0.2832, Adjusted R-squared: 0.2563
F-statistic: 10.54 on 3 and 80 DF, p-value: 6.489e-06
Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.3059 -0.7136 -0.0389 0.7179 3.2808
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2.38799 0.06894 -34.639 < 2e-16 ***
x1 0.50504 0.07066 7.148 6.28e-12 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.13 on 312 degrees of freedom
Multiple R-squared: 0.1407, Adjusted R-squared: 0.138
F-statistic: 51.09 on 1 and 312 DF, p-value: 6.28e-12
Call:
lm(formula = y0 ~ x1 + IMR00a + IMR00b, data = cbind(myData,
imr), subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.5236 -0.7402 -0.0121 0.7445 3.0029
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.6417 0.7507 -2.187 0.029509 *
x1 0.7906 0.1267 6.240 1.44e-09 ***
IMR00a 0.1775 1.1859 0.150 0.881131
IMR00b 1.5404 0.4502 3.421 0.000707 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.072 on 310 degrees of freedom
Multiple R-squared: 0.2317, Adjusted R-squared: 0.2243
F-statistic: 31.17 on 3 and 310 DF, p-value: < 2.2e-16
Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)
Residuals:
Min 1Q Median 3Q Max
-4.5384 -0.8422 0.0159 0.8606 3.8369
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.05005 0.05292 -19.84 <2e-16 ***
x1 0.61499 0.05168 11.90 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.264 on 600 degrees of freedom
Multiple R-squared: 0.191, Adjusted R-squared: 0.1896
F-statistic: 141.6 on 1 and 600 DF, p-value: < 2.2e-16
Call:
lm(formula = y0 ~ x1 + IMR1X, data = cbind(myData, imr), subset = selection)
Residuals:
Min 1Q Median 3Q Max
-4.5180 -0.8633 0.0390 0.8578 3.8874
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.62505 0.20925 -7.766 3.53e-14 ***
x1 0.77235 0.07558 10.220 < 2e-16 ***
IMR1X 0.92323 0.32519 2.839 0.00468 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.257 on 599 degrees of freedom
Multiple R-squared: 0.2017, Adjusted R-squared: 0.199
F-statistic: 75.68 on 2 and 599 DF, p-value: < 2.2e-16
Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.5320 -0.8754 -0.0490 0.8899 4.1030
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2.05778 0.06565 -31.344 <2e-16 ***
x1 0.63987 0.06951 9.205 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.258 on 396 degrees of freedom
Multiple R-squared: 0.1763, Adjusted R-squared: 0.1742
F-statistic: 84.74 on 1 and 396 DF, p-value: < 2.2e-16
Call:
lm(formula = y0 ~ x1 + IMR0X, data = cbind(myData, imr), subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.6358 -0.8835 -0.0711 0.9058 3.8633
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.2215 0.3177 -3.844 0.000141 ***
x1 0.8415 0.1019 8.260 2.22e-15 ***
IMR0X 0.8865 0.3296 2.689 0.007463 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.248 on 395 degrees of freedom
Multiple R-squared: 0.1911, Adjusted R-squared: 0.187
F-statistic: 46.65 on 2 and 395 DF, p-value: < 2.2e-16
Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.8057 -0.7642 -0.0296 0.7640 3.5448
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.64705 0.05710 -11.332 <2e-16 ***
x1 0.49052 0.05561 8.821 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.142 on 467 degrees of freedom
Multiple R-squared: 0.1428, Adjusted R-squared: 0.141
F-statistic: 77.82 on 1 and 467 DF, p-value: < 2.2e-16
Call:
lm(formula = y0 ~ x1 + IMRX1, data = cbind(myData, imr), subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.0754 -0.7093 -0.0606 0.6356 3.6354
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.50842 0.11934 -12.640 < 2e-16 ***
x1 0.76903 0.06251 12.303 < 2e-16 ***
IMRX1 1.18395 0.14660 8.076 5.76e-15 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.07 on 466 degrees of freedom
Multiple R-squared: 0.2481, Adjusted R-squared: 0.2448
F-statistic: 76.87 on 2 and 466 DF, p-value: < 2.2e-16
Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.5498 -0.7580 -0.0424 0.8063 3.1909
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2.15926 0.05227 -41.307 <2e-16 ***
x1 0.48537 0.05431 8.938 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.152 on 529 degrees of freedom
Multiple R-squared: 0.1312, Adjusted R-squared: 0.1295
F-statistic: 79.88 on 1 and 529 DF, p-value: < 2.2e-16
Call:
lm(formula = y0 ~ x1 + IMRX0, data = cbind(myData, imr), subset = selection)
Residuals:
Min 1Q Median 3Q Max
-3.8148 -0.7703 0.0468 0.7909 2.7010
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.34764 0.11035 -12.212 < 2e-16 ***
x1 0.78002 0.06249 12.483 < 2e-16 ***
IMRX0 1.29836 0.15796 8.219 1.59e-15 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.086 on 528 degrees of freedom
Multiple R-squared: 0.2297, Adjusted R-squared: 0.2268
F-statistic: 78.74 on 2 and 528 DF, p-value: < 2.2e-16
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