[SOLVED] CS # Lecture 6 Analysis

# Lecture 6 Analysis
# Author: Chris Hansman
# Email: [email protected]
# Date : 14/02/32

#Loading Libraries
library(tidyverse)

##
#Section 1: Loading Data
##
yields0105 <- read_csv(“yields0105.csv”)#————————————————–##Section 2:Plotting Yield Curve from 2001-2005#————————————————–##Gathering Data into a Single Column for Plottingyield_gather <- yields0105 %>%
gather(., value=Yield) %>%
group_by(key) %>%
mutate(number=1)%>%
mutate(Date=cumsum(number))

#Plotting Yields Over Time
ggplot(data=yield_gather, aes(x=Date, y=Yield)) +
geom_line( aes(color = key)) +
theme_minimal()
ggsave(yield_0105.pdf, width=8, height=4)

##
#Section 3: Creating Changes in Yields Over Time
##
yields_diff <- yields0105 %>%
mutate_all(funs(.-lag(.))) %>%
filter(is.na(DGS3MO)==FALSE)

#Gathering Data into a Single Column for Plotting
yield_diff_gather <- yields_diff %>%
gather(., value=Yield_Difference) %>% group_by(key) %>%
mutate(number=1)%>%
mutate(Date=cumsum(number))

#Plotting Yields Differences Over Time
ggplot(data=yield_diff_gather, aes(x=Date, y=Yield_Difference)) +
geom_line( aes(color = key)) +
theme_minimal() + ylab(Yield Difference)
ggsave(yield_diff0105.pdf, width=8, height=4)

##
#Section 4: Performing Principal Components Analysis
##
#Demeaning the Data and converting to matrix form
x_tilde <- yields_diff %>%
mutate_all(funs(.-mean(.)))
x_tilde <- as.matrix(x_tilde)#Storing Meansalpha <- yields_diff %>%
summarize_all(funs(mean(.))) %>%
as.matrix %>%
t()

#Computing Covariance Matrix
Sigma_x <- cov(x_tilde)#Computing corr Matrix rho_x <- cor(x_tilde)#Performing EigendecompositionLambda<-diag(eigen(Sigma_x)$values)Gamma<-eigen(Sigma_x)$vector#Creating Principal Components Variablesp <- t(Gamma)%*%t(x_tilde)p <- as_tibble(t(p))#————————————————–##Section 5:Plotting Loadings on First Three Principal Components#————————————————–##Loading On First Principal Componentp1_loadings <- as_tibble(Gamma[,1]) %>%
mutate(number=1)%>%
mutate(bond=cumsum(number))

#Loading On Second Principal Component
p2_loadings <- as_tibble(Gamma[,2]) %>%
mutate(number=1)%>%
mutate(bond=cumsum(number))

#Loading On Third Principal Component
p3_loadings <- as_tibble(Gamma[,3]) %>%
mutate(number=1)%>%
mutate(bond=cumsum(number))

# Plotting Loadings on First Principal Component
ggplot(p1_loadings, aes(y=value, x=factor(bond))) +
geom_bar(stat=identity) +
scale_x_discrete(breaks = 1:9, labels=c(3 Month,6 Month,1 Year,2 Year,3 Year, 5 Year, 7 Year, 10 Year, 20 Year))+
xlab(Treasury)+
ylab(Loading)+
theme_minimal()
ggsave(p1_loading.pdf, width=6, height=4)

# Plotting Loadings on First Principal Component
ggplot(p2_loadings, aes(y=value, x=factor(bond))) +
geom_bar(stat=identity) +
scale_x_discrete(breaks = 1:9, labels=c(3 Month,6 Month,1 Year,2 Year,3 Year, 5 Year, 7 Year, 10 Year, 20 Year))+
xlab(Treasury)+
ylab(Loading)+
theme_minimal()
ggsave(p2_loading.pdf, width=6, height=4)

# Plotting Loadings on First Principal Component
ggplot(p3_loadings, aes(y=value, x=factor(bond))) +
geom_bar(stat=identity) +
scale_x_discrete(breaks = 1:9, labels=c(3 Month,6 Month,1 Year,2 Year,3 Year, 5 Year, 7 Year, 10 Year, 20 Year))+
xlab(Treasury)+
ylab(Loading)+
theme_minimal()
ggsave(p3_loading.pdf, width=6, height=4)

##
#Section 6:Plotting Cumulative Principal Components
##
#Cumulative Principal Components
p_cumulative <- p %>%
mutate_all(funs(cumsum(.))) %>%
mutate(number=1)%>%
mutate(Date=cumsum(number))

#Plotting First Principal Component
ggplot(data=p_cumulative) +
geom_line(aes(x=Date, y=V1), color=blue)+
ylab(First Principal Component)+
theme_minimal()
ggsave(p1_time.pdf, width=6, height=4)

#Plotting Second Principal Component
ggplot(data=p_cumulative) +
geom_line(aes(x=Date, y=V2), color=red)+
ylab(Second Principal Component)+
theme_minimal()
ggsave(p2_time.pdf, width=6, height=4)

#Plotting First Two Principal Components
ggplot(data=p_cumulative) +
geom_line(aes(x=Date, y=V1), color=blue)+
geom_line(aes(x=Date, y=V2), color=red)+
ylab(First Two Principal Components)+
theme_minimal()
ggsave(p12_time.pdf, width=6, height=4)

##
#Section 7:Predicting Using First Two Principal Components
##
#Extracting First Two Principal Components
p_firsttwo<-p %>%
select(V1, V2) %>%
as.matrix() %>%
t()

#Extrating First Two Eigenvectors
Gamma_firsttwo <- Gamma[,1:2]#Predicted Values of xhat_txhat_t=rep(alpha, ncol(p_firsttwo)) + Gamma_firsttwo%*%p_firsttwoxhat_t<-as.tibble(t(xhat_t))#Cumulative Values of xhat_txhat_c_t <-xhat_t %>%
mutate_all(funs(cumsum(.)))

#Formatting Data For Plotting: Predicted Yield Changes
xhat_t_gather <- xhat_t %>%
gather(., value=Yield) %>%
group_by(key) %>%
mutate(number=1)%>%
mutate(Date=cumsum(number))

#PlottingPredicted Yield Changes
ggplot(data=xhat_t_gather, aes(x=Date, y=Yield)) +
geom_line( aes(color = key)) +
theme_minimal() + ylab(Predicted Yield Changes)
ggsave(predicted_yield_diff.pdf, width=8, height=4)

#Formatting Data For Plotting: Predicted Yields
xhat_c_t_gather <- xhat_c_t %>%
gather(., value=Yield) %>%
group_by(key) %>%
mutate(number=1)%>%
mutate(Date=cumsum(number))

#PlottingPredicted Yields
ggplot(data=xhat_c_t_gather, aes(x=Date, y=Yield)) +
geom_line( aes(color = key)) +
theme_minimal() + ylab(Predicted Yield)
ggsave(predicted_yield.pdf, width=8, height=4)

##
#Section 8:Plotting Fraction of Variance Explained
##
# Plotting Fraction of Variance Explained
ggplot(prop_var, aes(y=value, x=factor(Component))) +
geom_bar(stat=identity, fill=red) +
xlab(Principal Component)+
ylab(Fraction of Variance)+
theme_minimal()
ggsave(frac_Var.pdf, width=6, height=4)