## Exam 1 - 2026

SFX_da <- read.csv("http://www.bauer.uh.edu/rsusmel/4397/Stocks_FX_1973.csv",head=TRUE,sep=",")
names(SFX_da)
x_dat <- SFX_da$Date
x_slb <- SFX_da$SLB
x_cat <- SFX_da$CAT
x_S <- SFX_da$USD_EUR
x_wti <- SFX_da$Crude_WTI
x_gold <- SFX_da$Gold
x_cs <- SFX_da$Cons_sent
x_ip <- SFX_da$IP
x_HS <- SFX_da$House_starts
x_Com <- SFX_da$Commodities
x_Mkt_RF<- SFX_da$Mkt_RF
x_SMB <- SFX_da$SMB
x_HML <- SFX_da$HML
x_CMA <- SFX_da$CMA
x_RMW <- SFX_da$RMW
x_RF <- SFX_da$RF

T <- length(x_slb)
e <- log(x_S[-1]/x_S[-T])
lr_wti <- log(x_wti[-1]/x_wti[-T])
lr_cs <- log(x_cs[-1]/x_cs[-T])
lr_slb <- log(x_slb[-1]/x_slb[-T])
lr_cat <- log(x_cat[-1]/x_cat[-T])
lr_gold <- log(x_gold[-1]/x_gold[-T])
lr_oil <- log(x_wti[-1]/x_wti[-T])
lr_ip <- log(x_ip[-1]/x_ip[-T])
lr_hs <- log(x_HS[-1]/x_HS[-T])
lr_com <- log(x_Com[-1]/x_Com[-T])
x0 <- matrix(1,T-1,1)
Mkt_RF <- x_Mkt_RF[-1]/100
SMB <- x_SMB[-1]/100
HML <- x_HML[-1]/100
CMA <- x_CMA[-1]/100
RMW <- x_RMW[-1]/100
RF <- x_RF[-1]/100

cat_x <- lr_cat - RF			# SLB excess returns 


#### QUESTION 1
x <- lr_hs

## 1.a. Histogram
n_breaks <- 40
h <- hist(x, breaks=n_breaks, col="red", xlab="% Change House Starts",  main = "Distribution - Histogram for HS")

# 1.b. Report moments for changes in HS (lr_oil)
T <- length(x)
m1 <- sum(x)/T         		## Mean
m1
m2 <- sum((x-m1)^2)/T  		## Var
sd <- sqrt(m2)	     		## SD
sd
m3 <- sum((x-m1)^3)/T  		## For numerator of S
m4 <- sum((x-m1)^4)/T  		## For numerator of K
b1 <- m3/m2^(3/2)  		## Sample Skewness
b1
b2 <- (m4/m2^2)        		## Sample Kurtosis
b2

## 1.c Test H0: HS changes are normal
JB <- (b1^2+(b2-3)^2/4)*T/6		# if JB > 5.99, reject H0 
JB

qchisq(.95, df = 2) 	# chi-squared (df=2) value at 5% level

p_val <- 1 - pchisq(JB, df = 2)  	# p-value of LM_test 
p_val

## 1.d Test H0: mean of HS changes = 1%
se_m1 <- sd/sqrt(T) 
t <- (m1-0.01)/se_m1			# if |t| > 1.96, reject H0 
t


## 1.f Empirical Bootstrap 95% C.I. for correlation between lr_hs & cat_x (Need to use boot package)
library(boot)

dat_c <- data.frame(lr_oil, Mkt_RF)
library(boot)
# function to obtain cor from the data
cor_xy <- function(data, i) { 
  d <-data[i,]
  return(cor(d$lr_oil,d$Mkt_RF))
}

boot.samps <- boot(data=dat_c, statistic=cor_xy, R=1000)
boot.ci(boot.samps, type="basic")			# Q: Does CI include 0?



#### QUESTION 2
## Regression 
y <- cat_x

fit_ff4 <- lm(y ~ Mkt_RF + SMB + HML + lr_hs)		# Model 

## 2.a Report Regression
summary(fit_ff4)

## 2.b Drivers (significant variables)

## 2.c Interpret R^2 and report F-goodness of fit test

## 2.d Interpret Beta_1 

## 2.e Is CAT a value stock? (check sign of HML)

## 2.f Check for multicollinearity (using olsrr package)
ols_vif_tol(fit_ibm_ff3)

ols_eigen_cindex(fit_ibm_ff3)

## 2.g Check for outliers (using olsrr package)
x_resid <- residuals(fit_ff4)
x_stand_resid <- x_resid/sd(x_resid) 	# standardized residuals
sum(abs(x_stand_resid) > 2)/T		# Rule of thumg count (5% count is OK)- proportion of "outliers"

library(olsrr)				# need to install package olsrr
cooksd <- cooks.distance(fit_mod)
sum(cooksd > 4/T)/T			# Rule of thumb count (5% count is OK)



## 2.e Compute expected returns according to Model
b_ff4 <- fit_ff4$coefficients
mean_x <- c(mean(Mkt_RF), mean(SMB), mean(HML), mean(lr_cs), mean(lr_gold))
exp_ret <- t(b_ff4[2:5])%*% mean_x	# Expected returns, can be computed using mean of fitted values	

# over/underperfromance of CAT
mean(y)  - exp_ret


#### QUESTION 3
## 3.a beta_1 = 1.
t_beta_1 <- (summary(fit_ff4)$coefficients[2,1] - 1)/summary(fit_ff4)$coefficients[2,2]
t_beta_1 


## 3.b Test H0: SMB=HML=0.
e_ff4 <- fit_ff4$residuals
RSS_U <- sum(e_ff4^2)

#  Restricted Model
fit_r <- lm(y ~ Mkt_RF + lr_hs)	# t-test on US_I: is it significant?
e_r <- fit_r$residuals
RSS_R <- sum(e_r^2)

k <- length(fit_ff4$coefficients)			# k: number of coefficients in Unrestricted Model
k2 <- length(fit_r$coefficients)			# k2: number of coefficients in Restricted Model

J <- k - k2						# J: number of restrictions (= k - k2)
T <- length(e_ff4)
F_test <- ((RSS_R - RSS_U)/J)/(RSS_U/(T-k))
F_test
qf(.95, df1=J, df2=(T-k)) 
p_val <- 1 - pf(F_test, df1=J, df2=(T-k))
p_val

# NOTE: You can also do the test using the car package
library(car)
linearHypothesis(fit_ff4, c("SMB = 0","HML = 0"), test="F") 


## 3.c Test H0: Beta_HML = Beta_hs = 0.1 (using car)
linearHypothesis(fit_ff4, c("HML = -0.1","lr_hs = 0.1"), test="F") 


## 3.d Wald Test to check if GOLD is missing (using lmtest package)
library(lmtest)
fit_ff4_RMW <- lm(y ~ Mkt_RF + SMB + HML + lr_hs + lr_gold)	#Now, U Model
waldtest(fit_ff4_gold, fit_ff4)


## 3.e Wald Test to check if Commodity price changes are missing (using lmtest)
fit_ff4_IP <- lm(y ~ Mkt_RF + SMB + HML + lr_hs + lr_com)	#Now, U Model
waldtest(fit_ff4_IP, fit_ff4)


## 3.f Chow test (with package strucchange)
x_dat[429]
x_break <- 428					# We lost one observation when taking log changes
library(strucchange)
sctest(cat_x ~ Mkt_RF + SMB + HML + lr_hs, type = "Chow", point = x_break)


#### QUESTION 4
## Check answers in solutions