# Q1.1 Moments

PPP_da <- read.csv("http://www.bauer.uh.edu/rsusmel/4397/ppp_m.csv",head=TRUE,sep=",")

x_dkk <- PPP_da$DKK_USD
x_sgd <- PPP_da$SGD_USD
summary(x_dkk)
summary(x_sgd)					# You will notice some (191) NA at the start of the sample

T_dkk <- length(x_dkk)
lr_dkk <- log(x_dkk[-1]/x_dkk[-T_dkk])

T_sgd <- length(x_sgd)
lr_sgd <- log(x_sgd[-1]/x_sgd[-T_sgd])		# There will be NA (or NAN = not a number) entries here.

# Different ways to redefine the data, start working with SGD from 1981:Jan (T=97) or use na.omit (it omits NA from data)

lr_sgd_T <- lr_sgd[97:(T_sgd-1)]		# Redefining your data from actual start of sample
lr_sgd_na <- na.omit(lr_sgd)			# Redefining sample by ignoring NA

# For DKK
x <- lr_dkk
T <- length(x)
m1 <- sum(x)/T        		# Mean
m1
m2 <- sum((x-m1)^2)/T 		# Variance
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

# For SGD
x <- lr_sgd_na
T <- length(x)
m1 <- sum(x)/T        		# Mean
m1
m2 <- sum((x-m1)^2)/T 		# Variance
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


n_breaks <- 20
h <- hist(x, breaks=n_breaks, col="red", xlab="% Changes in SGD/USD",  main = "Distribution - Histogram for SGD")


# Q1.2 - Jarque-Bera test
JB <- (b1^2 + (b2-3)^2/4) * T/6
JB						# JB > 5.99 => Reject Normality at 5% level.

## R Note: You can also use the jarque.bera.test(x) form the tseries package.


# Q1.3 - C.I and Tests
# a
monthly_mean <- .01
monthly_SD <- 0.21
mon <-3
quarterly_mean <- monthly_mean * mon 			# annual mean = .01 * 12 = 12%
quarterly_SD =  monthly_SD  * sqrt(mon) 		# annual SD = 0.21 * sqrt(12) = 72.75%

# b
z_98 <- 2.33
ci_98 <- c(monthly_mean - z_98 * monthly_SD,  monthly_mean + z_98 * monthly_SD)
ci_98

# c
N <- 79
s2 <- monthly_SD ^2
var_CI_lb <- (N-1)*s2/qchisq(.99, df=N-1)
var_CI_ub <- (N-1)*s2/qchisq(.01, df=N-1)
sd_CI_lb <- sqrt(var_CI_lb)
sd_CI_lb
sd_CI_ub <- sqrt(var_CI_ub)
sd_CI_ub


# d
t_hat <- (monthly_mean - 0)/(monthly_SD/sqrt(N))
t_hat 							# |t_hat| < 1.96 => Cannot reject monthly mean = 0 at 5% level.



# Q1.4 - Practice Linear Algebra
# a
A <- matrix(runif(9), nrow =3)	
B <- matrix(runif(9), nrow =3)	
det(A)
det(B)
A %*% B
B %*% A

# b
A+B
A-B

# c 
v <-seq(1,9, by=3)
A %*% v

#d
t(v) %*% A

# e
t(v) %*% v

# f 
w <- c(1, 1, 7)
v/w


# Q1.5 - Practice Linear Algebra (continuation)
# a
d1 <- runif(6)
d2 <- rnorm(6)	
D <- cbind(d1,d2)
C <- t(D)%*% D

# b
solve(C)

# c
diag(solve(C))

# d
f <- rnorm(6)
b <- solve(C)%*%t(D)%*%f

# e
f_hat <- D%*%b

# f
e <- f - f_hat

# g
RSS <- as.numeric(t(e)%*% e)			#as.numeric forces R to treat a 1x1 matrix as a scalar, not a matrix (kills the dimensions)


# h
sigma2 <- RSS/(6-2)

# i
Var_b <- sigma2 * solve(C)		
# j
SE_b <- sqrt(diag(Var_b))



# Q1.6  Regression
SFX_da <- read.csv("http://www.bauer.uh.edu/rsusmel/4397/Stocks_FX_1973.csv",head=TRUE,sep=",")
x_pfe <- SFX_da$PFE
x_Mkt_RF<- SFX_da$Mkt_RF
x_RF<- SFX_da$RF
lr_pfe <- log(x_pfe[-1]/x_pfe[-T])
Mkt_RF <- x_Mkt_RF[-1]/100
RF <- x_RF[-1]/100
pfe_x <- lr_pfe - RF

# a (plot)
plot(pfe_x, Mkt_RF, main = "Excess Returns: Pfizer vs Market") 

# b
fit_capm <- lm(pfe_x ~ Mkt_RF)
out_capm <- summary(fit_capm)
out_capm
b_capm <- fit_capm$coefficients			# Extract coefficient from object fit_capm
b_capm_1 <- out_capm$coefficients[,1]		# Extract coefficient from matrix of results in summary(fit_capm)
b_capm_sd1 <- out_capm$coefficients[,2]

# c (t-value of Intercept)
t_alpha <- b_capm[1]/b_capm_sd1[2]

# d (t <- (slope - 1)/SE) has to be greater than 1.645 (if lower, reject H0)
t_beta_1 <- (b_capm[2]-1)/b_capm_sd1[2]

# e Intercept + Slope * 0.005
pred <- b_capm[1] + b_capm[2]*.005


# Q1.7
# a H0: Beta = 0 vs H1: Beta =/= 0

#b 
t_test <- (0.92 - 0)/0.26