Final Exam-FA 4A
Munkhzul
2025-01-07
Chapter 7
Question 1
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(wooldridge)
library(car)## Loading required package: carData##
## Attaching package: 'car'## The following object is masked from 'package:dplyr':
##
## recodelibrary(dslabs)
library(ISLR2)
library(matlib)## Warning in rgl.init(initValue, onlyNULL): RGL: unable to open X11 display## Warning: 'rgl.init' failed, running with 'rgl.useNULL = TRUE'.library(lmtest)## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericlibrary(sandwich)
data("sleep75")
sleep_model <- lm(sleep ~ totwrk + educ + age + I(age^2) + male, data = sleep75)
summary(sleep_model)##
## Call:
## lm(formula = sleep ~ totwrk + educ + age + I(age^2) + male, data = sleep75)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2378.00 -243.29 6.74 259.24 1350.19
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3840.83197 235.10870 16.336 <2e-16 ***
## totwrk -0.16342 0.01813 -9.013 <2e-16 ***
## educ -11.71332 5.86689 -1.997 0.0463 *
## age -8.69668 11.20746 -0.776 0.4380
## I(age^2) 0.12844 0.13390 0.959 0.3378
## male 87.75243 34.32616 2.556 0.0108 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 417.7 on 700 degrees of freedom
## Multiple R-squared: 0.1228, Adjusted R-squared: 0.1165
## F-statistic: 19.59 on 5 and 700 DF, p-value: < 2.2e-16# (i) Is there evidence that men sleep more than women?
coef_male <- coef(summary(sleep_model))["male", "Estimate"]
p_value_male <- coef(summary(sleep_model))["male", "Pr(>|t|)"]
cat("Coefficient for 'male':", coef_male, "\n")## Coefficient for 'male': 87.75243cat("p-value for 'male':", p_value_male, "\n")## p-value for 'male': 0.01078518if (p_value_male < 0.05) {
cat("Conclusion: There is significant evidence that men sleep more (or less) than women.\n")
} else {
cat("Conclusion: There is no significant evidence that men sleep more (or less) than women.\n")
}## Conclusion: There is significant evidence that men sleep more (or less) than women.# (ii) Tradeoff between working and sleeping
coef_totwrk <- coef(summary(sleep_model))["totwrk", "Estimate"]
cat("Coefficient for 'totwrk':", coef_totwrk, "\n")## Coefficient for 'totwrk': -0.1634232cat("Interpretation: Each additional minute of work reduces sleep by", abs(coef_totwrk), "minutes.\n")## Interpretation: Each additional minute of work reduces sleep by 0.1634232 minutes.# (iii) Test whether age has a significant effect on sleep
sleep_model_no_age <- lm(sleep ~ totwrk + educ + male, data = sleep75)
anova(sleep_model_no_age, sleep_model)## Analysis of Variance Table
##
## Model 1: sleep ~ totwrk + educ + male
## Model 2: sleep ~ totwrk + educ + age + I(age^2) + male
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 702 122631662
## 2 700 122147777 2 483885 1.3865 0.2506Question 3
data("gpa2")
sat_model <- lm(sat ~ hsize + I(hsize^2) + female + black + female:black, data = gpa2)
summary(sat_model)##
## Call:
## lm(formula = sat ~ hsize + I(hsize^2) + female + black + female:black,
## data = gpa2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -570.45 -89.54 -5.24 85.41 479.13
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1028.0972 6.2902 163.445 < 2e-16 ***
## hsize 19.2971 3.8323 5.035 4.97e-07 ***
## I(hsize^2) -2.1948 0.5272 -4.163 3.20e-05 ***
## female -45.0915 4.2911 -10.508 < 2e-16 ***
## black -169.8126 12.7131 -13.357 < 2e-16 ***
## female:black 62.3064 18.1542 3.432 0.000605 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 133.4 on 4131 degrees of freedom
## Multiple R-squared: 0.08578, Adjusted R-squared: 0.08468
## F-statistic: 77.52 on 5 and 4131 DF, p-value: < 2.2e-16# (i) Is hsize^2 important in the model?
beta1 <- coef(sat_model)["hsize"]
beta2 <- coef(sat_model)["I(hsize^2)"]
optimal_hsize <- -beta1 / (2 * beta2)
cat("Optimal high school size:", optimal_hsize, "\n")## Optimal high school size: 4.39603# (ii) SAT score difference between nonblack females and nonblack males
coef_female <- coef(summary(sat_model))["female", "Estimate"]
p_value_female <- coef(summary(sat_model))["female", "Pr(>|t|)"]
cat("Coefficient for 'female' (SAT score difference between nonblack females and nonblack males):", coef_female, "\n")## Coefficient for 'female' (SAT score difference between nonblack females and nonblack males): -45.09145cat("p-value for 'female':", p_value_female, "\n")## p-value for 'female': 1.656301e-25if (p_value_female < 0.05) {
cat("Conclusion: There is significant evidence of a difference in SAT scores between nonblack females and nonblack males.\n")
} else {
cat("Conclusion: There is no significant evidence of a difference in SAT scores between nonblack females and nonblack males.\n")
}## Conclusion: There is significant evidence of a difference in SAT scores between nonblack females and nonblack males.# (iii) SAT score difference between nonblack males and black males
linearHypothesis(sat_model, "black = 0")##
## Linear hypothesis test:
## black = 0
##
## Model 1: restricted model
## Model 2: sat ~ hsize + I(hsize^2) + female + black + female:black
##
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 4132 76652652
## 2 4131 73479121 1 3173531 178.42 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# (iv) SAT score difference between black females and nonblack females
linearHypothesis(sat_model, "black + female:black = 0")##
## Linear hypothesis test:
## black + female:black = 0
##
## Model 1: restricted model
## Model 2: sat ~ hsize + I(hsize^2) + female + black + female:black
##
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 4132 74700518
## 2 4131 73479121 1 1221397 68.667 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1C1
library(wooldridge)
data1 <- wooldridge::gpa1
head(data1)## age soph junior senior senior5 male campus business engineer colGPA hsGPA ACT
## 1 21 0 0 1 0 0 0 1 0 3.0 3.0 21
## 2 21 0 0 1 0 0 0 1 0 3.4 3.2 24
## 3 20 0 1 0 0 0 0 1 0 3.0 3.6 26
## 4 19 1 0 0 0 1 1 1 0 3.5 3.5 27
## 5 20 0 1 0 0 0 0 1 0 3.6 3.9 28
## 6 20 0 0 1 0 1 1 1 0 3.0 3.4 25
## job19 job20 drive bike walk voluntr PC greek car siblings bgfriend clubs
## 1 0 1 1 0 0 0 0 0 1 1 0 0
## 2 0 1 1 0 0 0 0 0 1 0 1 1
## 3 1 0 0 0 1 0 0 0 1 1 0 1
## 4 1 0 0 0 1 0 0 0 0 1 0 0
## 5 0 1 0 1 0 0 0 0 1 1 1 0
## 6 0 0 0 0 1 0 0 0 1 1 0 0
## skipped alcohol gradMI fathcoll mothcoll
## 1 2 1.0 1 0 0
## 2 0 1.0 1 1 1
## 3 0 1.0 1 1 1
## 4 0 0.0 0 0 0
## 5 0 1.5 1 1 0
## 6 0 0.0 0 1 0data_C1 <- gpa1
model_C1 <- lm(colGPA~PC+hsGPA+ACT+mothcoll+fathcoll,data= data_C1)
summary(model_C1)##
## Call:
## lm(formula = colGPA ~ PC + hsGPA + ACT + mothcoll + fathcoll,
## data = data_C1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.78149 -0.25726 -0.02121 0.24691 0.74432
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.255554 0.335392 3.744 0.000268 ***
## PC 0.151854 0.058716 2.586 0.010762 *
## hsGPA 0.450220 0.094280 4.775 4.61e-06 ***
## ACT 0.007724 0.010678 0.723 0.470688
## mothcoll -0.003758 0.060270 -0.062 0.950376
## fathcoll 0.041800 0.061270 0.682 0.496265
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3344 on 135 degrees of freedom
## Multiple R-squared: 0.2222, Adjusted R-squared: 0.1934
## F-statistic: 7.713 on 5 and 135 DF, p-value: 2.083e-06- From the model we can see that there are two significant variables including PC and hsGPA, Compared to the result estimated in (7.6), PC coefficient decrease from 0.157 to 0.151, and it is still statistically significant.The impact of PC remains largely consistent with equation (7.6), maintaining substantial significance with a t(pc) value approximately around 2.58.
library(wooldridge)
data1 <- wooldridge::gpa1
# Perform the regression analysis
model_7C1_ur <- lm(colGPA ~ PC + hsGPA + ACT, data = data1)
summary(model_7C1_ur)##
## Call:
## lm(formula = colGPA ~ PC + hsGPA + ACT, data = data1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.7901 -0.2622 -0.0107 0.2334 0.7570
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.263520 0.333125 3.793 0.000223 ***
## PC 0.157309 0.057287 2.746 0.006844 **
## hsGPA 0.447242 0.093647 4.776 4.54e-06 ***
## ACT 0.008659 0.010534 0.822 0.412513
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3325 on 137 degrees of freedom
## Multiple R-squared: 0.2194, Adjusted R-squared: 0.2023
## F-statistic: 12.83 on 3 and 137 DF, p-value: 1.932e-07F= ((0.2222-0.2194)/2)/((1-0.2222)/135)
F## [1] 0.2429931critical_value = qf(0.95,2,135, TRUE)
critical_value## [1] 4.423013model6<- lm(colGPA ~ PC + hsGPA + ACT + mothcoll + fathcoll , data = data1)
summary(model6)##
## Call:
## lm(formula = colGPA ~ PC + hsGPA + ACT + mothcoll + fathcoll,
## data = data1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.78149 -0.25726 -0.02121 0.24691 0.74432
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.255554 0.335392 3.744 0.000268 ***
## PC 0.151854 0.058716 2.586 0.010762 *
## hsGPA 0.450220 0.094280 4.775 4.61e-06 ***
## ACT 0.007724 0.010678 0.723 0.470688
## mothcoll -0.003758 0.060270 -0.062 0.950376
## fathcoll 0.041800 0.061270 0.682 0.496265
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3344 on 135 degrees of freedom
## Multiple R-squared: 0.2222, Adjusted R-squared: 0.1934
## F-statistic: 7.713 on 5 and 135 DF, p-value: 2.083e-06library(car)
hypotheses <- c("mothcoll=0", "fathcoll=0")
joint_test <- linearHypothesis(model6, hypotheses)
joint_test##
## Linear hypothesis test:
## mothcoll = 0
## fathcoll = 0
##
## Model 1: restricted model
## Model 2: colGPA ~ PC + hsGPA + ACT + mothcoll + fathcoll
##
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 137 15.149
## 2 135 15.094 2 0.054685 0.2446 0.7834summary(joint_test)## Res.Df RSS Df Sum of Sq F
## Min. :135.0 Min. :15.09 Min. :2 Min. :0.05469 Min. :0.2446
## 1st Qu.:135.5 1st Qu.:15.11 1st Qu.:2 1st Qu.:0.05469 1st Qu.:0.2446
## Median :136.0 Median :15.12 Median :2 Median :0.05469 Median :0.2446
## Mean :136.0 Mean :15.12 Mean :2 Mean :0.05469 Mean :0.2446
## 3rd Qu.:136.5 3rd Qu.:15.14 3rd Qu.:2 3rd Qu.:0.05469 3rd Qu.:0.2446
## Max. :137.0 Max. :15.15 Max. :2 Max. :0.05469 Max. :0.2446
## NA's :1 NA's :1 NA's :1
## Pr(>F)
## Min. :0.7834
## 1st Qu.:0.7834
## Median :0.7834
## Mean :0.7834
## 3rd Qu.:0.7834
## Max. :0.7834
## NA's :1- Conducting the F test to assess the combined significance of mothcoll and fathcoll, with 2 and 135 degrees of freedom, yields a result of about 0.24 with a p-value close to 0.78. This joint insignificance of these variables doesn’t notably affect the estimates of the other coefficients when included in the regression.
model3 <- lm (colGPA ~ PC + hsGPA + I(hsGPA^2) + ACT + mothcoll + fathcoll, data = gpa1)
summary (model3)##
## Call:
## lm(formula = colGPA ~ PC + hsGPA + I(hsGPA^2) + ACT + mothcoll +
## fathcoll, data = gpa1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.78998 -0.24327 -0.00648 0.26179 0.72231
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.040328 2.443038 2.063 0.0410 *
## PC 0.140446 0.058858 2.386 0.0184 *
## hsGPA -1.802520 1.443552 -1.249 0.2140
## I(hsGPA^2) 0.337341 0.215711 1.564 0.1202
## ACT 0.004786 0.010786 0.444 0.6580
## mothcoll 0.003091 0.060110 0.051 0.9591
## fathcoll 0.062761 0.062401 1.006 0.3163
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3326 on 134 degrees of freedom
## Multiple R-squared: 0.2361, Adjusted R-squared: 0.2019
## F-statistic: 6.904 on 6 and 134 DF, p-value: 2.088e-06- When hsGPA² is included in the regression, its coefficient is approximately 0.337, with a t-statistic around 1.56. The coefficient for hsGPA itself is roughly -1.803. This suggests a borderline significance for the relationship between hsGPA and the outcome. The quadratic term indicates a U-shaped relationship, with a turning point around hsGPA* = 2.68, which complicates interpretation. Although the inclusion of hsGPA² acts as a robustness check, the main result concerning the coefficient for PC decreases to about 0.140, but it remains statistically significant.
C2
data3 <- wooldridge::wage2
head(data3)## wage hours IQ KWW educ exper tenure age married black south urban sibs
## 1 769 40 93 35 12 11 2 31 1 0 0 1 1
## 2 808 50 119 41 18 11 16 37 1 0 0 1 1
## 3 825 40 108 46 14 11 9 33 1 0 0 1 1
## 4 650 40 96 32 12 13 7 32 1 0 0 1 4
## 5 562 40 74 27 11 14 5 34 1 0 0 1 10
## 6 1400 40 116 43 16 14 2 35 1 1 0 1 1
## brthord meduc feduc lwage
## 1 2 8 8 6.645091
## 2 NA 14 14 6.694562
## 3 2 14 14 6.715384
## 4 3 12 12 6.476973
## 5 6 6 11 6.331502
## 6 2 8 NA 7.244227data("wage2")
model4 <- lm(log(wage) ~ educ +exper +tenure + married + black + south + urban, data = wage2)
summary(model4)##
## Call:
## lm(formula = log(wage) ~ educ + exper + tenure + married + black +
## south + urban, data = wage2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.98069 -0.21996 0.00707 0.24288 1.22822
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.395497 0.113225 47.653 < 2e-16 ***
## educ 0.065431 0.006250 10.468 < 2e-16 ***
## exper 0.014043 0.003185 4.409 1.16e-05 ***
## tenure 0.011747 0.002453 4.789 1.95e-06 ***
## married 0.199417 0.039050 5.107 3.98e-07 ***
## black -0.188350 0.037667 -5.000 6.84e-07 ***
## south -0.090904 0.026249 -3.463 0.000558 ***
## urban 0.183912 0.026958 6.822 1.62e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3655 on 927 degrees of freedom
## Multiple R-squared: 0.2526, Adjusted R-squared: 0.2469
## F-statistic: 44.75 on 7 and 927 DF, p-value: < 2.2e-16- The black coefficient suggests that, holding other factors constant, Black men earn approximately 18.8% less than non-Black men. With a t-statistic of around -4.95, this difference is statistically significant, indicating strong evidence for the income disparity between these groups.
model4 <- lm(log(wage) ~ educ +exper + I(exper^2) + tenure + I(tenure^2) + married + black + south + urban, data = wage2)
summary(model4)##
## Call:
## lm(formula = log(wage) ~ educ + exper + I(exper^2) + tenure +
## I(tenure^2) + married + black + south + urban, data = wage2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.98236 -0.21972 -0.00036 0.24078 1.25127
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.3586756 0.1259143 42.558 < 2e-16 ***
## educ 0.0642761 0.0063115 10.184 < 2e-16 ***
## exper 0.0172146 0.0126138 1.365 0.172665
## I(exper^2) -0.0001138 0.0005319 -0.214 0.830622
## tenure 0.0249291 0.0081297 3.066 0.002229 **
## I(tenure^2) -0.0007964 0.0004710 -1.691 0.091188 .
## married 0.1985470 0.0391103 5.077 4.65e-07 ***
## black -0.1906636 0.0377011 -5.057 5.13e-07 ***
## south -0.0912153 0.0262356 -3.477 0.000531 ***
## urban 0.1854241 0.0269585 6.878 1.12e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3653 on 925 degrees of freedom
## Multiple R-squared: 0.255, Adjusted R-squared: 0.2477
## F-statistic: 35.17 on 9 and 925 DF, p-value: < 2.2e-16linearHypothesis(model4, c("I(exper^2)=0", "I(tenure^2)=0" ))##
## Linear hypothesis test:
## I(exper^2) = 0
## I(tenure^2) = 0
##
## Model 1: restricted model
## Model 2: log(wage) ~ educ + exper + I(exper^2) + tenure + I(tenure^2) +
## married + black + south + urban
##
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 927 123.82
## 2 925 123.42 2 0.39756 1.4898 0.226F= ((0.255-0.2526)/2)/((1-0.255)/925)
F## [1] 1.4899331-pf(F,2,925)## [1] 0.2259282- The F-statistic for testing the joint significance of exper² and tenure², with 2 and 925 degrees of freedom, is approximately 1.49, yielding a p-value of around 0.226. Since the p-value exceeds 0.20, the quadratic terms are deemed collectively insignificant at the 20% significance level.
model4 <- lm(log(wage) ~ educ +exper +tenure + married + black + south + urban + I(black*educ), data = wage2)
summary(model4)##
## Call:
## lm(formula = log(wage) ~ educ + exper + tenure + married + black +
## south + urban + I(black * educ), data = wage2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.97782 -0.21832 0.00475 0.24136 1.23226
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.374817 0.114703 46.859 < 2e-16 ***
## educ 0.067115 0.006428 10.442 < 2e-16 ***
## exper 0.013826 0.003191 4.333 1.63e-05 ***
## tenure 0.011787 0.002453 4.805 1.80e-06 ***
## married 0.198908 0.039047 5.094 4.25e-07 ***
## black 0.094809 0.255399 0.371 0.710561
## south -0.089450 0.026277 -3.404 0.000692 ***
## urban 0.183852 0.026955 6.821 1.63e-11 ***
## I(black * educ) -0.022624 0.020183 -1.121 0.262603
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3654 on 926 degrees of freedom
## Multiple R-squared: 0.2536, Adjusted R-squared: 0.2471
## F-statistic: 39.32 on 8 and 926 DF, p-value: < 2.2e-16- When the interaction term black â‹… educ is added to the equation, its coefficient is approximately -0.0226 with a standard error of 0.0202. This implies that the return on education for Black men increases by about 2.3 percentage points less per year of education compared to non-Black men (a difference of around 6.7%). Although this difference is notable if reflective of population-level disparities, the t-statistic is around 1.12 in absolute value. This is insufficient to reject the null hypothesis, meaning there is not enough evidence to conclude that the relationship between education and returns is significantly dependent on race.
model4 <- lm(log(wage)~educ+exper+tenure+married+black+south+urban+educ*black,data= wage2)
summary(model4)##
## Call:
## lm(formula = log(wage) ~ educ + exper + tenure + married + black +
## south + urban + educ * black, data = wage2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.97782 -0.21832 0.00475 0.24136 1.23226
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.374817 0.114703 46.859 < 2e-16 ***
## educ 0.067115 0.006428 10.442 < 2e-16 ***
## exper 0.013826 0.003191 4.333 1.63e-05 ***
## tenure 0.011787 0.002453 4.805 1.80e-06 ***
## married 0.198908 0.039047 5.094 4.25e-07 ***
## black 0.094809 0.255399 0.371 0.710561
## south -0.089450 0.026277 -3.404 0.000692 ***
## urban 0.183852 0.026955 6.821 1.63e-11 ***
## educ:black -0.022624 0.020183 -1.121 0.262603
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3654 on 926 degrees of freedom
## Multiple R-squared: 0.2536, Adjusted R-squared: 0.2471
## F-statistic: 39.32 on 8 and 926 DF, p-value: < 2.2e-16model_7C2_4 <- lm(log(wage)~educ+exper+tenure+married+black+south+urban+married*black,data= wage2)
summary(model4)##
## Call:
## lm(formula = log(wage) ~ educ + exper + tenure + married + black +
## south + urban + educ * black, data = wage2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.97782 -0.21832 0.00475 0.24136 1.23226
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.374817 0.114703 46.859 < 2e-16 ***
## educ 0.067115 0.006428 10.442 < 2e-16 ***
## exper 0.013826 0.003191 4.333 1.63e-05 ***
## tenure 0.011787 0.002453 4.805 1.80e-06 ***
## married 0.198908 0.039047 5.094 4.25e-07 ***
## black 0.094809 0.255399 0.371 0.710561
## south -0.089450 0.026277 -3.404 0.000692 ***
## urban 0.183852 0.026955 6.821 1.63e-11 ***
## educ:black -0.022624 0.020183 -1.121 0.262603
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3654 on 926 degrees of freedom
## Multiple R-squared: 0.2536, Adjusted R-squared: 0.2471
## F-statistic: 39.32 on 8 and 926 DF, p-value: < 2.2e-16- Married Black individuals are represented by married = 1 and black = 1, while married non-Black individuals are represented by married = 1 and black = 0. The difference in monthly salary between these two groups is the sum of the coefficients for the black variable and the married × black interaction term: -0.2408 + 0.0614 = -0.1794.
This suggests that, on average, married Black individuals earn 17.94% less in monthly salary compared to married non-Black individuals.
Chapter 8
Question 1
data("smoke")
ols_model <- lm(cigs ~ cigpric + log(income) + educ + age + I(age^2) + restaurn + white, data = smoke)
summary(ols_model)##
## Call:
## lm(formula = cigs ~ cigpric + log(income) + educ + age + I(age^2) +
## restaurn + white, data = smoke)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.758 -9.336 -5.895 7.927 70.267
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5.782521 8.852068 -0.653 0.51379
## cigpric -0.005724 0.101064 -0.057 0.95485
## log(income) 0.865360 0.728766 1.187 0.23541
## educ -0.501908 0.167169 -3.002 0.00276 **
## age 0.774357 0.160516 4.824 1.68e-06 ***
## I(age^2) -0.009068 0.001748 -5.187 2.71e-07 ***
## restaurn -2.880176 1.116067 -2.581 0.01004 *
## white -0.553286 1.459490 -0.379 0.70472
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.41 on 799 degrees of freedom
## Multiple R-squared: 0.05289, Adjusted R-squared: 0.04459
## F-statistic: 6.374 on 7 and 799 DF, p-value: 2.609e-07bptest(ols_model)##
## studentized Breusch-Pagan test
##
## data: ols_model
## BP = 32.431, df = 7, p-value = 3.379e-05- The OLS estimators, bj, are inconsistent: False. Heteroskedasticity does not make the OLS estimators inconsistent.
coeftest(ols_model, vcov = vcovHC(ols_model, type = "HC1"))##
## t test of coefficients:
##
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5.7825205 8.8831245 -0.6510 0.515262
## cigpric -0.0057240 0.1062869 -0.0539 0.957064
## log(income) 0.8653600 0.5985769 1.4457 0.148655
## educ -0.5019084 0.1624052 -3.0905 0.002068 **
## age 0.7743569 0.1380708 5.6084 2.814e-08 ***
## I(age^2) -0.0090678 0.0014596 -6.2127 8.375e-10 ***
## restaurn -2.8801757 1.0155265 -2.8361 0.004682 **
## white -0.5532861 1.3782955 -0.4014 0.688213
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1- The usual F statistic no longer has an F distribution: True. In the presence of heteroskedasticity, the usual F-statistic can no longer be relied upon to follow an F-distribution.
waldtest(ols_model, vcov = vcovHC(ols_model, type = "HC1"))## Wald test
##
## Model 1: cigs ~ cigpric + log(income) + educ + age + I(age^2) + restaurn +
## white
## Model 2: cigs ~ 1
## Res.Df Df F Pr(>F)
## 1 799
## 2 806 -7 9.3617 3.49e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1- The OLS estimators are no longer BLUE: True. Heteroskedasticity violates one of the Gauss-Markov assumptions, specifically the assumption of homoscedasticity (constant variance of errors).
usual_se <- summary(ols_model)$coefficients[, "Std. Error"]
robust_se <- sqrt(diag(vcovHC(ols_model, type = "HC1")))
comparison <- data.frame(Variable = names(coef(ols_model)), Usual_SE = usual_se, Robust_SE = robust_se)
print(comparison)## Variable Usual_SE Robust_SE
## (Intercept) (Intercept) 8.852067639 8.883124494
## cigpric cigpric 0.101063853 0.106286855
## log(income) log(income) 0.728765519 0.598576892
## educ educ 0.167169257 0.162405199
## age age 0.160516185 0.138070842
## I(age^2) I(age^2) 0.001748065 0.001459558
## restaurn restaurn 1.116067336 1.015526517
## white white 1.459489762 1.378295460Question 5
data4 <- wooldridge::smoke
head(data4)## educ cigpric white age income cigs restaurn lincome agesq lcigpric
## 1 16.0 60.506 1 46 20000 0 0 9.903487 2116 4.102743
## 2 16.0 57.883 1 40 30000 0 0 10.308952 1600 4.058424
## 3 12.0 57.664 1 58 30000 3 0 10.308952 3364 4.054633
## 4 13.5 57.883 1 30 20000 0 0 9.903487 900 4.058424
## 5 10.0 58.320 1 17 20000 0 0 9.903487 289 4.065945
## 6 6.0 59.340 1 86 6500 0 0 8.779557 7396 4.083283model11 <- lm(cigs ~ log(cigpric) + log(income) + educ + age + agesq + restaurn + white , data = data4)
summary(model11)##
## Call:
## lm(formula = cigs ~ log(cigpric) + log(income) + educ + age +
## agesq + restaurn + white, data = data4)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.772 -9.330 -5.907 7.945 70.275
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.682419 24.220729 -0.111 0.91184
## log(cigpric) -0.850907 5.782321 -0.147 0.88305
## log(income) 0.869014 0.728763 1.192 0.23344
## educ -0.501753 0.167168 -3.001 0.00277 **
## age 0.774502 0.160516 4.825 1.68e-06 ***
## agesq -0.009069 0.001748 -5.188 2.70e-07 ***
## restaurn -2.865621 1.117406 -2.565 0.01051 *
## white -0.559236 1.459461 -0.383 0.70169
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.41 on 799 degrees of freedom
## Multiple R-squared: 0.05291, Adjusted R-squared: 0.04461
## F-statistic: 6.377 on 7 and 799 DF, p-value: 2.588e-07# (i) The standard errors for each coefficient, whether calculated using the usual method or heteroskedasticity-robust approach, yield very similar results in practice. There are no notable differences between them.
s_error <- coef(summary(model11))
s_error## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.682418774 24.220728831 -0.1107489 9.118433e-01
## log(cigpric) -0.850907441 5.782321084 -0.1471567 8.830454e-01
## log(income) 0.869013971 0.728763480 1.1924499 2.334389e-01
## educ -0.501753247 0.167167689 -3.0014966 2.770186e-03
## age 0.774502156 0.160515805 4.8250835 1.676279e-06
## agesq -0.009068603 0.001748055 -5.1878253 2.699355e-07
## restaurn -2.865621212 1.117405936 -2.5645301 1.051320e-02
## white -0.559236403 1.459461034 -0.3831801 7.016882e-01# (ii) Calculate the change in the probability of smoking if education increases by 4 years
edu_4 <- coef(model11)["educ"] * 4
edu_4## educ
## -2.007013# (iii) Find the age at which the probability of smoking starts to decrease
age <- -coef(model11)["age"] / (2 * coef(model11)["I(age^2)"])
age## age
## NAprint(paste("The turning point for the quadratic is calculated as 0.020 / [2(0.00026)] ≈ 38.46, which is approximately 38 and a half years."))## [1] "The turning point for the quadratic is calculated as 0.020 / [2(0.00026)] ≈ 38.46, which is approximately 38 and a half years."# (iv) Interpret the coefficient on restaurn
coef_res <- coef(model11)["restaurn"]
coef_res## restaurn
## -2.865621# (v) Predict the probability of smoking for person 206
library(lmtest)
person_206 <- data.frame(cigpric = 67.44, income = 6500, educ = 16, age = 77, restaurn = 0, white = 0)
het_test <- bptest(model11)
het_test##
## studentized Breusch-Pagan test
##
## data: model11
## BP = 32.377, df = 7, p-value = 3.458e-05print(paste("The equation for smoking is: smokes = −0.656 + 0.069 × log(67.44) + 0.012 × log(6,500) − 0.029 × (16) + 0.020 × (77) + 0.00026 × (77²) − 0.0052."))## [1] "The equation for smoking is: smokes = −0.656 + 0.069 × log(67.44) + 0.012 × log(6,500) − 0.029 × (16) + 0.020 × (77) + 0.00026 × (77²) − 0.0052."C4
data("vote1")
model <- lm(voteA ~ prtystrA + democA + log(expendA) + log(expendB), data = vote1)
# (i) Estimate the model with OLS
residual_model <- lm(resid(model) ~ prtystrA + democA + log(expendA) + log(expendB), data = vote1)
r_squared <- summary(residual_model)$r.squared
print(paste("R^2 of residual regression:", r_squared)) # Expected R^2 ≈ 0## [1] "R^2 of residual regression: 1.19813088272216e-32"# (ii) Breusch-Pagan test for heteroskedasticity
bp_test <- bptest(model)
print(bp_test)##
## studentized Breusch-Pagan test
##
## data: model
## BP = 9.0934, df = 4, p-value = 0.05881# (iii) White test for heteroskedasticity (F-statistic version)
white_test <- bptest(model, ~ poly(fitted(model), 2), data = vote1)
print(white_test)##
## studentized Breusch-Pagan test
##
## data: model
## BP = 5.49, df = 2, p-value = 0.06425C13
library(sandwich)
library(lmtest)
data6 <- wooldridge::fertil2
head(data6)## mnthborn yearborn age electric radio tv bicycle educ ceb agefbrth children
## 1 5 64 24 1 1 1 1 12 0 NA 0
## 2 1 56 32 1 1 1 1 13 3 25 3
## 3 7 58 30 1 0 0 0 5 1 27 1
## 4 11 45 42 1 0 1 0 4 3 17 2
## 5 5 45 43 1 1 1 1 11 2 24 2
## 6 8 52 36 1 0 0 0 7 1 26 1
## knowmeth usemeth monthfm yearfm agefm idlnchld heduc agesq urban urb_educ
## 1 1 0 NA NA NA 2 NA 576 1 12
## 2 1 1 11 80 24 3 12 1024 1 13
## 3 1 0 6 83 24 5 7 900 1 5
## 4 1 0 1 61 15 3 11 1764 1 4
## 5 1 1 3 66 20 2 14 1849 1 11
## 6 1 1 11 76 24 4 9 1296 1 7
## spirit protest catholic frsthalf educ0 evermarr
## 1 0 0 0 1 0 0
## 2 0 0 0 1 0 1
## 3 1 0 0 0 0 1
## 4 0 0 0 0 0 1
## 5 0 1 0 1 0 1
## 6 0 0 0 0 0 1model <- lm(children ~ age + I(age^2) + educ + electric + urban, data = fertil2)
robust_se <- sqrt(diag(vcovHC(model)))
summary_with_robust_se <- cbind(coef(model), "Robust SE" = robust_se)
summary(model)##
## Call:
## lm(formula = children ~ age + I(age^2) + educ + electric + urban,
## data = fertil2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.9012 -0.7136 -0.0039 0.7119 7.4318
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4.2225162 0.2401888 -17.580 < 2e-16 ***
## age 0.3409255 0.0165082 20.652 < 2e-16 ***
## I(age^2) -0.0027412 0.0002718 -10.086 < 2e-16 ***
## educ -0.0752323 0.0062966 -11.948 < 2e-16 ***
## electric -0.3100404 0.0690045 -4.493 7.20e-06 ***
## urban -0.2000339 0.0465062 -4.301 1.74e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.452 on 4352 degrees of freedom
## (3 observations deleted due to missingness)
## Multiple R-squared: 0.5734, Adjusted R-squared: 0.5729
## F-statistic: 1170 on 5 and 4352 DF, p-value: < 2.2e-16print(summary_with_robust_se)## Robust SE
## (Intercept) -4.222516228 0.2443961935
## age 0.340925520 0.0192199445
## I(age^2) -0.002741209 0.0003513959
## educ -0.075232323 0.0063159137
## electric -0.310040409 0.0640737262
## urban -0.200033857 0.0455162364coeftest(model6, vcov=vcovHC(model6, type = "HC0"))##
## t test of coefficients:
##
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.2555544 0.3368805 3.7270 0.000284 ***
## PC 0.1518539 0.0596641 2.5451 0.012048 *
## hsGPA 0.4502203 0.0962508 4.6776 6.962e-06 ***
## ACT 0.0077242 0.0103074 0.7494 0.454932
## mothcoll -0.0037579 0.0601271 -0.0625 0.950258
## fathcoll 0.0417999 0.0578749 0.7222 0.471392
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1F= ((0.574-0.5734)/3)/((1-0.574)/4349)
F## [1] 2.041784critical_value = qf(0.95,3,4349, TRUE)
critical_value## [1] 3.403585p_value = 1- pf(F,3,4349)
p_value## [1] 0.1058349a <- coeftest(model, vcov = sandwich)
a##
## t test of coefficients:
##
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4.22251623 0.24368307 -17.3279 < 2.2e-16 ***
## age 0.34092552 0.01916146 17.7923 < 2.2e-16 ***
## I(age^2) -0.00274121 0.00035027 -7.8260 6.278e-15 ***
## educ -0.07523232 0.00630336 -11.9353 < 2.2e-16 ***
## electric -0.31004041 0.06390411 -4.8517 1.267e-06 ***
## urban -0.20003386 0.04543962 -4.4022 1.097e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1bptest(model)##
## studentized Breusch-Pagan test
##
## data: model
## BP = 1065, df = 5, p-value < 2.2e-16# (i) Some robust standard are smaller than the non-robust standard errors.
usqr=(summary(model6)$residual)^2
yhat=model6$fitted.values
yhatsqr=model6$fitted.values^2
model6WT=summary(lm(usqr~yhat+yhatsqr,data=fertil2))$r.squared
model6WT*4361## [1] 143.62361-pchisq(1093.183,2)## [1] 0joint_test <- coeftest(model6, vcov = vcovHC)
print(joint_test[, "Pr(>|t|)"])## (Intercept) PC hsGPA ACT mothcoll fathcoll
## 5.903211e-04 1.682044e-02 1.977193e-05 4.846838e-01 9.528773e-01 4.948069e-01p_value = 1- pf(F,3,4349)
p_value## [1] 0.1058349head(summary_with_robust_se)## Robust SE
## (Intercept) -4.222516228 0.2443961935
## age 0.340925520 0.0192199445
## I(age^2) -0.002741209 0.0003513959
## educ -0.075232323 0.0063159137
## electric -0.310040409 0.0640737262
## urban -0.200033857 0.0455162364fv <- fitted(model6)
head(fv,6)## 1 2 3 4 5 6
## 2.768423 2.919682 3.115218 3.039878 3.269490 3.021208residuals <- resid(model6)
head(residuals,6)## 1 2 3 4 5 6
## 0.23157702 0.48031850 -0.11521801 0.46012183 0.33050949 -0.02120773hetero_test <- lm(residuals^2 ~ fv)
summary(hetero_test)##
## Call:
## lm(formula = residuals^2 ~ fv)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.13813 -0.07873 -0.04322 0.04132 0.51313
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.27680 0.18877 -1.466 0.1448
## fv 0.12558 0.06165 2.037 0.0436 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.128 on 139 degrees of freedom
## Multiple R-squared: 0.02898, Adjusted R-squared: 0.022
## F-statistic: 4.149 on 1 and 139 DF, p-value: 0.04357print(paste("iii. Ho: There is no heteroskedasticity H1: There is heteroskedasticity."))## [1] "iii. Ho: There is no heteroskedasticity H1: There is heteroskedasticity."print(paste("As the p-value is below alpha at a 95% confidence level, there exists adequate statistical evidence to reject the null hypothesis, signifying the presence of heteroskedasticity in the model."))## [1] "As the p-value is below alpha at a 95% confidence level, there exists adequate statistical evidence to reject the null hypothesis, signifying the presence of heteroskedasticity in the model."# (iv) Is heteroskedasticity practically important?
print(paste("No, since the difference between the robust and non robust standar errors was really small."))## [1] "No, since the difference between the robust and non robust standar errors was really small."Chapter 9
Question 1
data7 <- wooldridge::ceosal2
head(data7)## salary age college grad comten ceoten sales profits mktval lsalary lsales
## 1 1161 49 1 1 9 2 6200 966 23200 7.057037 8.732305
## 2 600 43 1 1 10 10 283 48 1100 6.396930 5.645447
## 3 379 51 1 1 9 3 169 40 1100 5.937536 5.129899
## 4 651 55 1 0 22 22 1100 -54 1000 6.478509 7.003066
## 5 497 44 1 1 8 6 351 28 387 6.208590 5.860786
## 6 1067 64 1 1 7 7 19000 614 3900 6.972606 9.852194
## lmktval comtensq ceotensq profmarg
## 1 10.051908 81 4 15.580646
## 2 7.003066 100 100 16.961130
## 3 7.003066 81 9 23.668638
## 4 6.907755 484 484 -4.909091
## 5 5.958425 64 36 7.977208
## 6 8.268732 49 49 3.231579model11 <- lm(log(salary) ~ log(sales) + log(mktval) + profmarg + ceoten + comten, data = data7)
summary(model11)##
## Call:
## lm(formula = log(salary) ~ log(sales) + log(mktval) + profmarg +
## ceoten + comten, data = data7)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.5436 -0.2796 -0.0164 0.2857 1.9879
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.571977 0.253466 18.038 < 2e-16 ***
## log(sales) 0.187787 0.040003 4.694 5.46e-06 ***
## log(mktval) 0.099872 0.049214 2.029 0.04397 *
## profmarg -0.002211 0.002105 -1.050 0.29514
## ceoten 0.017104 0.005540 3.087 0.00236 **
## comten -0.009238 0.003337 -2.768 0.00626 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4947 on 171 degrees of freedom
## Multiple R-squared: 0.3525, Adjusted R-squared: 0.3336
## F-statistic: 18.62 on 5 and 171 DF, p-value: 9.488e-15r_squared <- summary(model11)$r.squared
r_squared## [1] 0.3525374model12 <- lm(log(salary) ~ log(sales) + log(mktval) + profmarg + ceotensq + comtensq, data = data7)
summary(model12)##
## Call:
## lm(formula = log(salary) ~ log(sales) + log(mktval) + profmarg +
## ceotensq + comtensq, data = data7)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.47481 -0.25933 -0.00511 0.27010 2.07583
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.612e+00 2.524e-01 18.276 < 2e-16 ***
## log(sales) 1.805e-01 4.021e-02 4.489 1.31e-05 ***
## log(mktval) 1.018e-01 4.988e-02 2.040 0.0429 *
## profmarg -2.077e-03 2.135e-03 -0.973 0.3321
## ceotensq 3.761e-04 1.916e-04 1.963 0.0512 .
## comtensq -1.788e-04 7.236e-05 -2.471 0.0144 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5024 on 171 degrees of freedom
## Multiple R-squared: 0.3324, Adjusted R-squared: 0.3129
## F-statistic: 17.03 on 5 and 171 DF, p-value: 1.195e-13r_squared <- summary(model12)$r.squared
r_squared## [1] 0.3323998Question 5
data9 <- wooldridge::campus
head(data9)## enroll priv police crime lcrime lenroll lpolice
## 1 21836 0 24 446 6.100319 9.991315 3.178054
## 2 6485 0 13 1 0.000000 8.777247 2.564949
## 3 2123 0 3 1 0.000000 7.660585 1.098612
## 4 8240 0 17 121 4.795791 9.016756 2.833213
## 5 19793 0 30 470 6.152733 9.893084 3.401197
## 6 3256 1 9 25 3.218876 8.088255 2.197225b0 <- -6.63
se_b0 <- 1.03
b1 <- 1.27
se_b1 <- 0.11
n <- nrow(data9)
t_stat <- (b1 - 1) / se_b1
df <- n - 2
critical_value <- qt(0.95, df)
if (t_stat > critical_value) {
cat("Reject the null hypothesis H0: B1 = 1 in favor of H1: B1 > 1 at the 5% level.\n")
} else {
cat("Fail to reject the null hypothesis H0: B1 = 1 at the 5% level.\n")
}## Reject the null hypothesis H0: B1 = 1 in favor of H1: B1 > 1 at the 5% level.C4
data("infmrt")
# Add a dummy variable for District of Columbia (assuming DC is observation 51)
infmrt$dc <- ifelse(row.names(infmrt) == "51", 1, 0)
# (i) Estimate the model including the DC dummy variable
model_with_dc <- lm(log(infmort) ~ pcinc + physic + dc, data = infmrt)
summary(model_with_dc)##
## Call:
## lm(formula = log(infmort) ~ pcinc + physic + dc, data = infmrt)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.43481 -0.10338 0.00447 0.09096 0.39050
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.456e+00 8.688e-02 28.264 < 2e-16 ***
## pcinc -3.078e-05 6.499e-06 -4.737 7.33e-06 ***
## physic 1.495e-03 2.729e-04 5.478 3.33e-07 ***
## dc -7.940e-02 1.664e-01 -0.477 0.634
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1629 on 98 degrees of freedom
## Multiple R-squared: 0.2537, Adjusted R-squared: 0.2309
## F-statistic: 11.1 on 3 and 98 DF, p-value: 2.444e-06# (ii) Compare with the model without the DC dummy variable
model_without_dc <- lm(log(infmort) ~ pcinc + physic, data = infmrt)
summary(model_without_dc)##
## Call:
## lm(formula = log(infmort) ~ pcinc + physic, data = infmrt)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.43432 -0.10235 0.00626 0.09337 0.39064
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.448e+00 8.502e-02 28.794 < 2e-16 ***
## pcinc -3.026e-05 6.378e-06 -4.743 7.07e-06 ***
## physic 1.487e-03 2.713e-04 5.480 3.24e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1623 on 99 degrees of freedom
## Multiple R-squared: 0.252, Adjusted R-squared: 0.2369
## F-statistic: 16.67 on 2 and 99 DF, p-value: 5.744e-07# Compare both models
anova(model_without_dc, model_with_dc)## Analysis of Variance Table
##
## Model 1: log(infmort) ~ pcinc + physic
## Model 2: log(infmort) ~ pcinc + physic + dc
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 99 2.6064
## 2 98 2.6004 1 0.0060446 0.2278 0.6342C5
library(quantreg) ## Loading required package: SparseMdata <- as.data.frame(rdchem)
data$sales_billion <- data$sales / 1e3
ols_full <- lm(rdintens ~ sales_billion + I(sales_billion^2) + profmarg, data = data)
summary(ols_full)##
## Call:
## lm(formula = rdintens ~ sales_billion + I(sales_billion^2) +
## profmarg, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.0371 -1.1238 -0.4547 0.7165 5.8522
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.058967 0.626269 3.288 0.00272 **
## sales_billion 0.316611 0.138854 2.280 0.03041 *
## I(sales_billion^2) -0.007390 0.003716 -1.989 0.05657 .
## profmarg 0.053322 0.044210 1.206 0.23787
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.774 on 28 degrees of freedom
## Multiple R-squared: 0.1905, Adjusted R-squared: 0.1037
## F-statistic: 2.196 on 3 and 28 DF, p-value: 0.1107# (i) OLS regression
data_filtered <- data %>% filter(sales_billion < 40)
ols_filtered <- lm(rdintens ~ sales_billion + I(sales_billion^2) + profmarg, data = data_filtered)
summary(ols_filtered)##
## Call:
## lm(formula = rdintens ~ sales_billion + I(sales_billion^2) +
## profmarg, data = data_filtered)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.0371 -1.1238 -0.4547 0.7165 5.8522
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.058967 0.626269 3.288 0.00272 **
## sales_billion 0.316611 0.138854 2.280 0.03041 *
## I(sales_billion^2) -0.007390 0.003716 -1.989 0.05657 .
## profmarg 0.053322 0.044210 1.206 0.23787
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.774 on 28 degrees of freedom
## Multiple R-squared: 0.1905, Adjusted R-squared: 0.1037
## F-statistic: 2.196 on 3 and 28 DF, p-value: 0.1107# (ii) LAD regression
lad_full <- rq(rdintens ~ sales_billion + I(sales_billion^2) + profmarg, data = data, tau = 0.5)
summary(lad_full)##
## Call: rq(formula = rdintens ~ sales_billion + I(sales_billion^2) +
## profmarg, tau = 0.5, data = data)
##
## tau: [1] 0.5
##
## Coefficients:
## coefficients lower bd upper bd
## (Intercept) 1.40428 0.87031 2.66628
## sales_billion 0.26346 -0.13508 0.75753
## I(sales_billion^2) -0.00600 -0.01679 0.00344
## profmarg 0.11400 0.01376 0.16427lad_filtered <- rq(rdintens ~ sales_billion + I(sales_billion^2) + profmarg, data = data_filtered, tau = 0.5)
summary(lad_filtered)##
## Call: rq(formula = rdintens ~ sales_billion + I(sales_billion^2) +
## profmarg, tau = 0.5, data = data_filtered)
##
## tau: [1] 0.5
##
## Coefficients:
## coefficients lower bd upper bd
## (Intercept) 1.40428 0.87031 2.66628
## sales_billion 0.26346 -0.13508 0.75753
## I(sales_billion^2) -0.00600 -0.01679 0.00344
## profmarg 0.11400 0.01376 0.16427# (iii) Compare OLS and LAD results
comparison <- data.frame(
Method = c("OLS (Full)", "OLS (Filtered)", "LAD (Full)", "LAD (Filtered)"),
Intercept = c(coef(ols_full)[1], coef(ols_filtered)[1], coef(lad_full)[1], coef(lad_filtered)[1]),
Sales = c(coef(ols_full)[2], coef(ols_filtered)[2], coef(lad_full)[2], coef(lad_filtered)[2]),
Sales_Squared = c(coef(ols_full)[3], coef(ols_filtered)[3], coef(lad_full)[3], coef(lad_filtered)[3]),
Profmarg = c(coef(ols_full)[4], coef(ols_filtered)[4], coef(lad_full)[4], coef(lad_filtered)[4])
)
print(comparison)## Method Intercept Sales Sales_Squared Profmarg
## 1 OLS (Full) 2.058967 0.3166110 -0.007389624 0.05332217
## 2 OLS (Filtered) 2.058967 0.3166110 -0.007389624 0.05332217
## 3 LAD (Full) 1.404279 0.2634638 -0.006001127 0.11400366
## 4 LAD (Filtered) 1.404279 0.2634638 -0.006001127 0.11400366Chapter 10
Question 1
- I disagree—time series data frequently show autocorrelation, meaning the observations are not independently distributed. I agree—while the first three Gauss-Markov assumptions (linearity, random sampling, and no perfect multicollinearity) guarantee that the OLS estimator is unbiased, additional assumptions, such as no autocorrelation and homoscedasticity, are necessary for the OLS estimator to be efficient in time series data.
- I disagree—while a trending variable can be used as the dependent variable, it is important to carefully account for the trend. For instance, including a time trend variable or differencing the data can help mitigate problems like spurious regression, which arise from non-stationarity.
- I agree—seasonality usually refers to patterns that repeat within a year, such as quarterly or monthly data. When using annual time series data, seasonality is typically not a concern because the data is aggregated over the year, which helps smooth out seasonal fluctuations.
Question 5
# Example code to create quarterly dummies and time trend
# Create quarterly dummies and time trend
#housing_data <- housing_data %>%
# mutate(
# quarter = as.factor(quarter),
# time_trend = 1:n()
# )
# Run the regression model
#housing_model <- lm(hs ~ interest_rate + income + time_trend + quarter, data = housing_data)
#summary(housing_model)C1
library(dplyr)
data("intdef")
intdef <- intdef %>%
mutate(
post_1979 = ifelse(year > 1979, 1, 0) # Dummy variable: 1 if year > 1979, else 0
)
# Run the regression including the dummy variable
policy_model <- lm(i3_1 ~ inf_1 + def_1 + post_1979, data = intdef)
summary(policy_model)##
## Call:
## lm(formula = i3_1 ~ inf_1 + def_1 + post_1979, data = intdef)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.5802 -0.8353 0.1936 0.7983 4.0447
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.39151 0.39028 3.565 0.000800 ***
## inf_1 0.56529 0.07144 7.913 1.99e-10 ***
## def_1 0.38269 0.11156 3.430 0.001203 **
## post_1979 1.77623 0.47268 3.758 0.000442 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.59 on 51 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.705, Adjusted R-squared: 0.6876
## F-statistic: 40.63 on 3 and 51 DF, p-value: 1.479e-13cat("\nConclusion:\n")##
## Conclusion:cat("The coefficient on post_1979 indicates whether there was a significant shift in the interest rate equation after 1979.\n")## The coefficient on post_1979 indicates whether there was a significant shift in the interest rate equation after 1979.C9
data("volat")
# (i) Discuss expected signs of coefficients
cat("Expected signs:\n")## Expected signs:cat("β1 (pcip): Positive. An increase in industrial production should generally lead to positive stock market returns.\n")## β1 (pcip): Positive. An increase in industrial production should generally lead to positive stock market returns.cat("β2 (i3): Negative. An increase in short-term interest rates often reduces stock market returns as borrowing becomes more expensive.\n\n")## β2 (i3): Negative. An increase in short-term interest rates often reduces stock market returns as borrowing becomes more expensive.# (ii) Estimate the equation by OLS
model_c9 <- lm(rsp500 ~ pcip + i3, data = volat)
summary(model_c9)##
## Call:
## lm(formula = rsp500 ~ pcip + i3, data = volat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -157.871 -22.580 2.103 25.524 138.137
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18.84306 3.27488 5.754 1.44e-08 ***
## pcip 0.03642 0.12940 0.281 0.7785
## i3 -1.36169 0.54072 -2.518 0.0121 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 40.13 on 554 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.01189, Adjusted R-squared: 0.008325
## F-statistic: 3.334 on 2 and 554 DF, p-value: 0.03637cat("\nOLS Results:\n")##
## OLS Results:cat("The coefficients for pcip and i3, along with their standard errors, t-values, and p-values, are displayed in the summary above.\n\n")## The coefficients for pcip and i3, along with their standard errors, t-values, and p-values, are displayed in the summary above.# (iii)Determine which variables are statistically significant
p_values <- summary(model_c9)$coefficients[, 4]
significant <- p_values < 0.05
cat("P-values for each variable:\n")## P-values for each variable:print(p_values)## (Intercept) pcip i3
## 1.444871e-08 7.784810e-01 1.207402e-02cat("\nSignificance (TRUE = significant at 5% level):\n")##
## Significance (TRUE = significant at 5% level):print(significant)## (Intercept) pcip i3
## TRUE FALSE TRUE# (iv)Interpret predictability
if (all(significant[-1] == FALSE)) {
cat("\nThe variables pcip and i3 are NOT statistically significant.\n")
cat("This suggests that the return on the S&P 500 is NOT predictable using these variables.\n")
} else {
cat("\nAt least one variable is statistically significant.\n")
cat("This suggests that the return on the S&P 500 shows SOME level of predictability.\n")
}##
## At least one variable is statistically significant.
## This suggests that the return on the S&P 500 shows SOME level of predictability.Chapter 11
C7
data("consump")
# (i) Estimate g_ct = β0 + β1 * g_ct-1 + u_t and test the PIH
consump <- consump %>%
mutate(g_ct = log(c) - lag(log(c))) # g_ct = log(c_t) - log(c_t-1)
model_i <- lm(g_ct ~ lag(g_ct, 1), data = consump)
summary(model_i)##
## Call:
## lm(formula = g_ct ~ lag(g_ct, 1), data = consump)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.027878 -0.005974 -0.001451 0.007142 0.020227
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.011431 0.003778 3.026 0.00478 **
## lag(g_ct, 1) 0.446141 0.156047 2.859 0.00731 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.01161 on 33 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.1985, Adjusted R-squared: 0.1742
## F-statistic: 8.174 on 1 and 33 DF, p-value: 0.00731# (ii) Add lagged income growth, interest rate, and inflation
consump <- consump %>%
mutate(gy_lag = lag(gy),
i3_lag = lag(i3),
inf_lag = lag(inf))
model_ii <- lm(g_ct ~ lag(g_ct, 1) + gy_lag + i3_lag + inf_lag, data = consump)
summary(model_ii)##
## Call:
## lm(formula = g_ct ~ lag(g_ct, 1) + gy_lag + i3_lag + inf_lag,
## data = consump)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0249093 -0.0075869 0.0000855 0.0087234 0.0188617
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0225940 0.0070892 3.187 0.00335 **
## lag(g_ct, 1) 0.4335832 0.2896525 1.497 0.14487
## gy_lag -0.1079087 0.1946371 -0.554 0.58341
## i3_lag -0.0007467 0.0011107 -0.672 0.50654
## inf_lag -0.0008281 0.0010041 -0.825 0.41606
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.01134 on 30 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.3038, Adjusted R-squared: 0.211
## F-statistic: 3.273 on 4 and 30 DF, p-value: 0.02431# (iii) Compare p-value of lag(g_ct, 1) in model_i and model_ii
cat("\nPART (iii): P-values for lag(g_ct, 1)\n")##
## PART (iii): P-values for lag(g_ct, 1)cat("Model (i) p-value for lag(g_ct, 1):", summary(model_i)$coefficients[2, 4], "\n")## Model (i) p-value for lag(g_ct, 1): 0.007310181cat("Model (ii) p-value for lag(g_ct, 1):", summary(model_ii)$coefficients[2, 4], "\n")## Model (ii) p-value for lag(g_ct, 1): 0.1448651# (iv) F-statistic for joint significance of all variables
anova_test <- anova(model_i, model_ii)
cat("\nPART (iv): F-statistic for joint significance\n")##
## PART (iv): F-statistic for joint significanceprint(anova_test)## Analysis of Variance Table
##
## Model 1: g_ct ~ lag(g_ct, 1)
## Model 2: g_ct ~ lag(g_ct, 1) + gy_lag + i3_lag + inf_lag
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 33 0.0044446
## 2 30 0.0038608 3 0.00058384 1.5122 0.2315C12
data("minwage")
# Create a lagged variable for gwage232
minwage$lag_gwage232 <- lag(minwage$gwage232)
# (i)
filtered_data <- minwage[complete.cases(minwage$gwage232, minwage$lag_gwage232), ]
acf(filtered_data$gwage232, lag.max = 1)
# (ii)
model1 <- lm(gwage232 ~ lag_gwage232 + gmwage + gcpi, data = filtered_data)
summary(model1)##
## Call:
## lm(formula = gwage232 ~ lag_gwage232 + gmwage + gcpi, data = filtered_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.044642 -0.004134 -0.001312 0.004482 0.041612
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0024003 0.0004308 5.572 3.79e-08 ***
## lag_gwage232 -0.0779092 0.0342851 -2.272 0.02341 *
## gmwage 0.1518459 0.0096485 15.738 < 2e-16 ***
## gcpi 0.2630876 0.0824457 3.191 0.00149 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.007889 on 606 degrees of freedom
## Multiple R-squared: 0.2986, Adjusted R-squared: 0.2951
## F-statistic: 85.99 on 3 and 606 DF, p-value: < 2.2e-16# (iii)
model2 <- lm(gwage232 ~ lag_gwage232 + gmwage + gcpi + lag(gemp232), data = filtered_data)
summary(model2)##
## Call:
## lm(formula = gwage232 ~ lag_gwage232 + gmwage + gcpi + lag(gemp232),
## data = filtered_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.043786 -0.004384 -0.001022 0.004325 0.042554
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0024248 0.0004268 5.681 2.08e-08 ***
## lag_gwage232 -0.0756372 0.0339207 -2.230 0.026126 *
## gmwage 0.1526838 0.0095403 16.004 < 2e-16 ***
## gcpi 0.2652171 0.0826041 3.211 0.001394 **
## lag(gemp232) 0.0662916 0.0169639 3.908 0.000104 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.007798 on 604 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.3167, Adjusted R-squared: 0.3122
## F-statistic: 69.98 on 4 and 604 DF, p-value: < 2.2e-16# (iv)
coef(model1)["gmwage"]## gmwage
## 0.1518459coef(model2)["gmwage"]## gmwage
## 0.1526838# (v)
model3 <- lm(gmwage ~ lag(gwage232) + lag(gemp232), data = filtered_data)
summary(model3)$r.squared## [1] 0.003908469Chapter 12
C11
data("nyse")
# (i) Estimate the OLS model in equation (12.47)
nyse <- nyse %>% mutate(return_lag1 = lag(return, 1))
ols_model <- lm(return ~ return_lag1, data = nyse)
cat("OLS Model Summary:\n")## OLS Model Summary:summary(ols_model)##
## Call:
## lm(formula = return ~ return_lag1, data = nyse)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.261 -1.302 0.098 1.316 8.065
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.17963 0.08074 2.225 0.0264 *
## return_lag1 0.05890 0.03802 1.549 0.1218
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.11 on 687 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.003481, Adjusted R-squared: 0.00203
## F-statistic: 2.399 on 1 and 687 DF, p-value: 0.1218nyse <- nyse %>% filter(!is.na(return_lag1))
nyse$residuals_sq <- residuals(ols_model)^2
mean_resid_sq <- mean(nyse$residuals_sq)
min_resid_sq <- min(nyse$residuals_sq)
max_resid_sq <- max(nyse$residuals_sq)
cat("Mean of squared residuals:", mean_resid_sq, "\n")## Mean of squared residuals: 4.440839cat("Minimum of squared residuals:", min_resid_sq, "\n")## Minimum of squared residuals: 7.35465e-06cat("Maximum of squared residuals:", max_resid_sq, "\n")## Maximum of squared residuals: 232.8946# (ii) Estimate heteroskedasticity model using lagged return
hetero_model <- lm(residuals_sq ~ return_lag1 + I(return_lag1^2), data = nyse)
cat("Heteroskedasticity Model Summary:\n")## Heteroskedasticity Model Summary:summary(hetero_model)##
## Call:
## lm(formula = residuals_sq ~ return_lag1 + I(return_lag1^2), data = nyse)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.459 -3.011 -1.975 0.676 221.469
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.25734 0.44085 7.389 4.32e-13 ***
## return_lag1 -0.78946 0.19569 -4.034 6.09e-05 ***
## I(return_lag1^2) 0.29666 0.03552 8.351 3.75e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 10.66 on 686 degrees of freedom
## Multiple R-squared: 0.1303, Adjusted R-squared: 0.1278
## F-statistic: 51.4 on 2 and 686 DF, p-value: < 2.2e-16# (iii) Plot conditional variance as a function of lagged return
plot_data <- data.frame(return_lag1 = seq(min(nyse$return_lag1, na.rm = TRUE),
max(nyse$return_lag1, na.rm = TRUE), length.out = 100))
plot_data$conditional_variance <- predict(hetero_model, newdata = plot_data)
plot(plot_data$return_lag1, plot_data$conditional_variance, type = "l",
main = "Conditional Variance vs. Lagged Return",
xlab = "Lagged Return", ylab = "Conditional Variance")
min_variance_index <- which.min(plot_data$conditional_variance)
cat("Smallest conditional variance:", plot_data$conditional_variance[min_variance_index], "\n")## Smallest conditional variance: 2.734254cat("Lagged return value for smallest variance:", plot_data$return_lag1[min_variance_index], "\n")## Lagged return value for smallest variance: 1.245583# (iv) Check for negative variance estimates
if (any(plot_data$conditional_variance < 0)) {
cat("Negative variance estimates are present.\n")
} else {
cat("No negative variance estimates.\n")
}## No negative variance estimates.# (v) Compare ARCH(1) model fit
nyse$residuals_sq_lag1 <- lag(nyse$residuals_sq, 1)
arch1_model <- lm(residuals_sq ~ residuals_sq_lag1, data = nyse)
summary(arch1_model)##
## Call:
## lm(formula = residuals_sq ~ residuals_sq_lag1, data = nyse)
##
## Residuals:
## Min 1Q Median 3Q Max
## -23.337 -3.292 -2.157 0.556 223.981
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.94743 0.44023 6.695 4.49e-11 ***
## residuals_sq_lag1 0.33706 0.03595 9.377 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 10.76 on 686 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.1136, Adjusted R-squared: 0.1123
## F-statistic: 87.92 on 1 and 686 DF, p-value: < 2.2e-16cat("ARCH(1) model R-squared:", summary(arch1_model)$r.squared, "\n")## ARCH(1) model R-squared: 0.1136065cat("Heteroskedasticity model R-squared:", summary(hetero_model)$r.squared, "\n")## Heteroskedasticity model R-squared: 0.1303208# (vi) Add second lag to ARCH(1) model for ARCH(2)
nyse$residuals_sq_lag2 <- lag(nyse$residuals_sq, 2)
arch2_model <- lm(residuals_sq ~ residuals_sq_lag1 + residuals_sq_lag2, data = nyse)
summary(arch2_model)##
## Call:
## lm(formula = residuals_sq ~ residuals_sq_lag1 + residuals_sq_lag2,
## data = nyse)
##
## Residuals:
## Min 1Q Median 3Q Max
## -22.934 -3.298 -2.158 0.600 224.296
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.82950 0.45495 6.219 8.69e-10 ***
## residuals_sq_lag1 0.32284 0.03820 8.450 < 2e-16 ***
## residuals_sq_lag2 0.04179 0.03820 1.094 0.274
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 10.76 on 684 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.1151, Adjusted R-squared: 0.1125
## F-statistic: 44.47 on 2 and 684 DF, p-value: < 2.2e-16cat("ARCH(2) model Summary:\n")## ARCH(2) model Summary:print(summary(arch2_model))##
## Call:
## lm(formula = residuals_sq ~ residuals_sq_lag1 + residuals_sq_lag2,
## data = nyse)
##
## Residuals:
## Min 1Q Median 3Q Max
## -22.934 -3.298 -2.158 0.600 224.296
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.82950 0.45495 6.219 8.69e-10 ***
## residuals_sq_lag1 0.32284 0.03820 8.450 < 2e-16 ***
## residuals_sq_lag2 0.04179 0.03820 1.094 0.274
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 10.76 on 684 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.1151, Adjusted R-squared: 0.1125
## F-statistic: 44.47 on 2 and 684 DF, p-value: < 2.2e-16