## Exam 1 - 2024

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$Date
x_unp<- SFX_da$UNP
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$IP
x_Mkt_RF<- SFX_da$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_ge <- log(x_ge[-1]/x_ge[-T])
lr_slb <- log(x_slb[-1]/x_slb[-T])
lr_unp <- log(x_unp[-1]/x_unp[-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])
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

unp_x <- lr_unp - RF			# SLB excess returns 


#### QUESTION 1
x <- lr_oil

# compute moments for changes in oil (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.a Test H0: mean of oil changes = 0
se_m1 <- sd/sqrt(T) 
t <- m1/se_m1
t


## 1.b Test H0: oil changes are normal
JB <- (b1^2+(b2-3)^2/4)*T/6
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.c 95% C.I. for correlation between lr_oil & Mkt_RF (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="perc")

# Note: You can also build a 95% CI assuming a Normal approximation for SD
m_cor <- boot.samps$t0
sd_cor <- sqrt(1-m_cor^2)/sqrt(T-2)

cor_lb <- m_cor - 1.96 * sd_cor 
cor_lb
cor_ub <- m_cor + 1.96 * sd_cor  
cor_ub


#### QUESTION 2
## Regression 
y <- unp_x

fit_ff5 <- lm(y ~ Mkt_RF + SMB + HML + lr_cs + lr_oil)		# Model 

## 2.a Report Regression
summary(fit_ff5)

## 2.b Drivers (significant variables)

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

## 2.d Interpret Beta_1

## 2.e Check for outliers (using olsrr package)
x_resid <- residuals(fit_ff5)
x_stand_resid <- x_resid/sd(x_resid) 	# standardized residuals
sum(x_stand_resid > 2)/T		# Rule of thumg count (5% count is OK)- proportion of "outliers"

library(olsrr)				# need to install package olsrr
x_lev <- ols_leverage(fit_ibm_ff3)	# leverage residuals
sum(x_lev > (2*k+2)/T) 			# Rule of thumb count (5% count is OK) - proportion of "levarage observations"	

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

ols_eigen_cindex(fit_ibm_ff3)

## 2.e Compute expected returns according to Model
b_ff5 <- fit_ff5$coefficients
mean_x <- c(mean(Mkt_RF), mean(SMB), mean(HML), mean(lr_cs), mean(lr_gold))
exp_ret <- t(b_ff5[2:6])%*% mean_x		

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


#### QUESTION 3
## 3.a beta_1 = 1.
t_beta_1 <- (summary(fit_ff5)$coefficients[2,1] - 1)/summary(fit_ff5)$coefficients[2,2]
t_beta_1 


## 3.b Test H0: CAPM true.
e_ff5 <- fit_ff5$residuals
RSS_U <- sum(e_ff5^2)

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

k <- length(fit_ff5$coefficients)			# k: number of coefficients in Unrestricted Model
k2 <- length(fit_capm$coefficients)			# k2: number of coefficients in Restricted Model

J <- k - k2						# J: number of restrictions (= k - k2)
T <- length(e_ff5)
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_ff5, c("SMB = 0","HML = 0", "lr_cs = 0", "lr_oil = 0"), test="F") 


## 3.c Test H0: Beta_CS = Beta_oil = -0.3 (using car)
linearHypothesis(fit_ff5, c("lr_cs = -0.1","lr_oil = 0.3"), test="F") 


## 3.d Wald Test to check if RMW is missing (using lmtest package)
library(lmtest)
fit_ff5_RMW <- lm(y ~ Mkt_RF + SMB + HML + lr_cs + lr_oil + RMW)	#Now, U Model
waldtest(fit_ff5_RMW, fit_ff5)

## 3.e Wald Test to check if IP changes are missing (using lmtest)
fit_ff5_IP <- lm(y ~ Mkt_RF + SMB + HML + lr_cs + lr_oil + RMW)	#Now, U Model
waldtest(fit_ff5_IP, fit_ff5)

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



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