Greene. Econometric Analysis, 8ed.

CHAPTER 5. Hypothesis Tests and Model Selection

Example 5.1

Load libraries

library(AER)
library(dynlm)
library(dplyr)

Art Appreciation

art <- read.csv("data/TableF4-1.csv", fileEncoding="UTF-8-BOM", header=TRUE)
art$lnP <- log(art$PRICE)
art$Aratio <- art$WIDTH / art$HEIGHT
art$lnArea <- log(art$HEIGHT * art$WIDTH)
summary(OLS1<-lm(lnP ~ lnArea + Aratio, data=art))

## 
## Call:
## lm(formula = lnP ~ lnArea + Aratio, data = art)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.4967 -0.7201 -0.0845  0.7548  3.0073 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -8.34236    0.67820  -12.30   <2e-16 ***
## lnArea       1.31638    0.09205   14.30   <2e-16 ***
## Aratio      -0.09623    0.15784   -0.61    0.542    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.104 on 427 degrees of freedom
## Multiple R-squared:  0.3342, Adjusted R-squared:  0.3311 
## F-statistic: 107.2 on 2 and 427 DF,  p-value: < 2.2e-16

coeftest(OLS1, vcovHC(OLS1, type="HC1"))

## 
## t test of coefficients:
## 
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -8.342361   0.735988 -11.335   <2e-16 ***
## lnArea       1.316384   0.106352  12.378   <2e-16 ***
## Aratio      -0.096233   0.167646  -0.574   0.5663    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

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Example 5.2

Earnings Equation

earning <- read.csv("data/TableF5-1.csv", fileEncoding="UTF-8-BOM", header=TRUE)

earning$KIDS <- with(earning, factor((KL6 + K618) > 0,
                                       levels = c(FALSE, TRUE), labels = c("no", "yes")))
earning$NWINC <- with(earning, (FAMINC - WHRS * WW)/1000)

summary(OLS2 <- lm(log(WHRS * WW) ~ WA + I(WA^2) + WE + KIDS,
            data = earning, subset = LFP == 1))

## 
## Call:
## lm(formula = log(WHRS * WW) ~ WA + I(WA^2) + WE + KIDS, data = earning, 
##     subset = LFP == 1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5305 -0.5266  0.3003  0.8474  1.7568 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  3.2400965  1.7674296   1.833  0.06747 . 
## WA           0.2005573  0.0838603   2.392  0.01721 * 
## I(WA^2)     -0.0023147  0.0009869  -2.345  0.01947 * 
## WE           0.0674727  0.0252486   2.672  0.00782 **
## KIDSyes     -0.3511952  0.1475326  -2.380  0.01773 * 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.19 on 423 degrees of freedom
## Multiple R-squared:  0.041,  Adjusted R-squared:  0.03193 
## F-statistic: 4.521 on 4 and 423 DF,  p-value: 0.001382

vcov(OLS2)

##             (Intercept)            WA       I(WA^2)            WE       KIDSyes
## (Intercept)  3.12380756 -1.440901e-01  1.661740e-03 -9.260920e-03  2.674867e-02
## WA          -0.14409007  7.032544e-03 -8.232369e-05  5.085495e-05 -2.641203e-03
## I(WA^2)      0.00166174 -8.232369e-05  9.739279e-07 -4.976114e-07  3.841018e-05
## WE          -0.00926092  5.085495e-05 -4.976114e-07  6.374903e-04 -5.461931e-05
## KIDSyes      0.02674867 -2.641203e-03  3.841018e-05 -5.461931e-05  2.176587e-02

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Example 5.3

Restricted Investment Equation

invest <- read.csv("data/TableF5-2.csv", fileEncoding="UTF-8-BOM", header=TRUE)

OLS3a<-dynlm(log(REALINVS) ~ TBILRATE + INFL + log(REALGDP) + trend(invest), data=invest)
summary(OLS3 <- update(OLS3a, start = 2))

## 
## Time series regression with "numeric" data:
## Start = 1, End = 203
## 
## Call:
## dynlm(formula = log(REALINVS) ~ TBILRATE + INFL + log(REALGDP) + 
##     trend(invest), data = invest, start = 2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.22313 -0.05540 -0.00312  0.04246  0.31989 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   -9.128432   1.364977  -6.688 2.26e-10 ***
## TBILRATE      -0.008598   0.003195  -2.691  0.00774 ** 
## INFL           0.003306   0.002337   1.415  0.15872    
## log(REALGDP)   1.930156   0.183272  10.532  < 2e-16 ***
## trend(invest) -0.005659   0.001488  -3.803  0.00019 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.08618 on 198 degrees of freedom
## Multiple R-squared:  0.9798, Adjusted R-squared:  0.9793 
## F-statistic:  2395 on 4 and 198 DF,  p-value: < 2.2e-16

OLS4a<-dynlm(log(REALINVS) ~ I(TBILRATE - INFL) + log(REALGDP) + trend(invest), data=invest)
summary(OLS4 <- update(OLS4a, start = 2))

## 
## Time series regression with "numeric" data:
## Start = 1, End = 203
## 
## Call:
## dynlm(formula = log(REALINVS) ~ I(TBILRATE - INFL) + log(REALGDP) + 
##     trend(invest), data = invest, start = 2)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.227897 -0.054542 -0.002435  0.039993  0.313928 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)        -7.902755   1.199306  -6.589 3.87e-10 ***
## I(TBILRATE - INFL) -0.004427   0.002270  -1.950  0.05260 .  
## log(REALGDP)        1.764062   0.160561  10.987  < 2e-16 ***
## trend(invest)      -0.004403   0.001331  -3.308  0.00111 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.0867 on 199 degrees of freedom
## Multiple R-squared:  0.9794, Adjusted R-squared:  0.9791 
## F-statistic:  3154 on 3 and 199 DF,  p-value: < 2.2e-16

deviance(OLS3)

## [1] 1.470565

deviance(OLS4)

## [1] 1.495811

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Example 5.4F

Test for the Earnings Equation

summary(OLS2)$fstatistic

##     value     numdf     dendf 
##   4.52057   4.00000 423.00000

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Example 5.5

Production Functions

productionf <- read.csv("data/TableF5-3.csv", fileEncoding="UTF-8-BOM", header=TRUE)

summary(OLS5 <- lm(log(VALUEADD) ~ log(LABOR) + log(CAPITAL) + I(0.5*log(LABOR)^2) + I(0.5*log(CAPITAL)^2) + log(LABOR):log(CAPITAL), data = productionf))

## 
## Call:
## lm(formula = log(VALUEADD) ~ log(LABOR) + log(CAPITAL) + I(0.5 * 
##     log(LABOR)^2) + I(0.5 * log(CAPITAL)^2) + log(LABOR):log(CAPITAL), 
##     data = productionf)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.33990 -0.10106 -0.01238  0.04605  0.39281 
## 
## Coefficients:
##                         Estimate Std. Error t value Pr(>|t|)  
## (Intercept)              0.94420    2.91075   0.324   0.7489  
## log(LABOR)               3.61364    1.54807   2.334   0.0296 *
## log(CAPITAL)            -1.89311    1.01626  -1.863   0.0765 .
## I(0.5 * log(LABOR)^2)   -0.96405    0.70738  -1.363   0.1874  
## I(0.5 * log(CAPITAL)^2)  0.08529    0.29261   0.291   0.7735  
## log(LABOR):log(CAPITAL)  0.31239    0.43893   0.712   0.4845  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1799 on 21 degrees of freedom
## Multiple R-squared:  0.9549, Adjusted R-squared:  0.9441 
## F-statistic: 88.85 on 5 and 21 DF,  p-value: 2.121e-13

vcov(OLS5)

##                         (Intercept)  log(LABOR) log(CAPITAL)
## (Intercept)              8.47248687 -2.38790338  -0.33129294
## log(LABOR)              -2.38790338  2.39652901  -1.23101576
## log(CAPITAL)            -0.33129294 -1.23101576   1.03278652
## I(0.5 * log(LABOR)^2)   -0.08760011 -0.66580411   0.52305244
## I(0.5 * log(CAPITAL)^2) -0.23317345  0.03476689   0.02636926
## log(LABOR):log(CAPITAL)  0.36354446  0.18311307  -0.22554189
##                         I(0.5 * log(LABOR)^2) I(0.5 * log(CAPITAL)^2)
## (Intercept)                       -0.08760011             -0.23317345
## log(LABOR)                        -0.66580411              0.03476689
## log(CAPITAL)                       0.52305244              0.02636926
## I(0.5 * log(LABOR)^2)              0.50039330              0.14674300
## I(0.5 * log(CAPITAL)^2)            0.14674300              0.08562001
## log(LABOR):log(CAPITAL)           -0.28803386             -0.11604045
##                         log(LABOR):log(CAPITAL)
## (Intercept)                           0.3635445
## log(LABOR)                            0.1831131
## log(CAPITAL)                         -0.2255419
## I(0.5 * log(LABOR)^2)                -0.2880339
## I(0.5 * log(CAPITAL)^2)              -0.1160405
## log(LABOR):log(CAPITAL)               0.1926571

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Example 5.6

A Long-Run Marginal Propensity to Consume

invest <- invest%>%
  mutate(lagC = lag(log(REALCONS), 1),
         lagY = lag(log(REALDPI), 1))
summary(OLS6<-lm(log(REALCONS) ~ log(REALDPI) + lagC, data=invest))

## 
## Call:
## lm(formula = log(REALCONS) ~ log(REALDPI) + lagC, data = invest)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.035243 -0.004606  0.000496  0.005147  0.041754 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.003142   0.010553   0.298  0.76624    
## log(REALDPI) 0.074958   0.028727   2.609  0.00976 ** 
## lagC         0.924625   0.028594  32.337  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.008742 on 200 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.9997, Adjusted R-squared:  0.9997 
## F-statistic: 3.476e+05 on 2 and 200 DF,  p-value: < 2.2e-16

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Example 5.7

J Test for a Consumption Function

summary(OLS7<-lm(log(REALCONS) ~ log(REALDPI) + lagY, data=invest))

## 
## Call:
## lm(formula = log(REALCONS) ~ log(REALDPI) + lagY, data = invest)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.049108 -0.013531 -0.000803  0.011245  0.067266 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.13247    0.02431  -5.448 1.48e-07 ***
## log(REALDPI)  0.86831    0.17601   4.933 1.70e-06 ***
## lagY          0.13454    0.17571   0.766    0.445    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.02179 on 200 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.9982, Adjusted R-squared:  0.9982 
## F-statistic: 5.589e+04 on 2 and 200 DF,  p-value: < 2.2e-16

jtest(OLS7, OLS6)

## J test
## 
## Model 1: log(REALCONS) ~ log(REALDPI) + lagY
## Model 2: log(REALCONS) ~ log(REALDPI) + lagC
##                 Estimate Std. Error t value  Pr(>|t|)    
## M1 + fitted(M2)   1.0432    0.02907 35.8828 < 2.2e-16 ***
## M2 + fitted(M1)  -3.1307    0.49267 -6.3546 1.396e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

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Example 5.8

Size of a RESET Test

set.seed(123)
bstar = NULL 
n = length(art$lnP); B = 5000
for(draw in 1:B)
 {
 Dstar = art[sample(1:n, size=n, replace=T),]
 model = lm(lnP ~ lnArea + Aratio, data=Dstar)
 bstar = rbind( bstar, coef(model) )
}
LS5000 <- cbind(LS5000_b=colMeans(bstar), LS5000_se=sqrt(diag(var(bstar))))
LS5000

##                LS5000_b LS5000_se
## (Intercept) -8.37947283 0.7495613
## lnArea       1.32203910 0.1080326
## Aratio      -0.09665372 0.1702822

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Greene. Econometric Analysis, 8ed.

Solomon Negash

June 11,2020

CHAPTER 5. Hypothesis Tests and Model Selection

Example 5.1

Example 5.2

Example 5.3

Example 5.4F

Example 5.5

Example 5.6

Example 5.7

Example 5.8