Time Series Analysis

Introduction

Resources

• Time series analysis by James D Hamilton - its considered to be the bible on time series analysis, pretty much covers all the theory , it doesnt have any code or pseudocode

• Introduction to Time Series with R by Metcalfe & Coweperwait

• Analysis of Financial Time Series by Tsay

Aim and Motivation

• Would be exploring time series analysis in the context of finance
• Implementing code from the book by Tsay , would be using the book by Hamilton and Metcalfe as reference
• Aim to understand different time series models in R.
• (Will keep on expanding this section on the go)

Financial Time Series and Linear Time series

Asset Returns

Definition 1 (Simple gross return)
Arithmetic Returns

$$P_t = P_{t-1}(1+R_t) \implies R_t = \frac{P_t - P_{t-1}}{P_t}$$ where $P_t$ and $R_t$ are the price and return at time $t$ respectively
Continuously compounded return or log return

$$r_t = \ln(1+R_t)$$
Dividend Payment

$$r_t = \ln(P_t + D_t) -\ln(P_{t-1})$$ Where $D_t$ is dividend payment between time t-1 and time t .

Definition 2 (Multiperiod gross return)
Similiary we can define multiperiod simple return by $$R_t[k] = exp{[\frac{1}{k}\sum_{j=0}^{k-1}(1+R_{t-j})]} -1$$ where exp is the exponentential function and k is total number periods the asset was held for.

Log return would be $$r_t[k] = r_t +r_{t-1}+.....+r_{t-k+1}$$

Definition 3 (Portfolio Return) $$R_{p,t}= \sum_{i=1}^{N}w_i R_{it}$$

Where $p$ is the portfolio , $i$ the ith asset and $w_i$ the weight of the $i^{th}$ asset in the portfolio

* Load data and header =T give the 1st row of the data file , that is the names of the cloumns of the data set

# da contains the return for 5 five stocks and indexes namely IBM ,value-weighted , equal-weighted and S&P composite index from 1970 to 2008
library(fBasics)
da = read.table("TST/d-ibm3dx7008.txt",header = T) 
dim(da) # dimensions for the data

[1] 9845    5

da[1:5,]# first five rows

ABCDEFGHIJ0123456789

	Date <int>	rtn <dbl>	vwretd <dbl>	ewretd <dbl>	sprtrn <dbl>
1	19700102	0.000686	0.012137	0.033450	0.010211
2	19700105	0.009596	0.006375	0.018947	0.004946
3	19700106	0.000679	-0.007233	-0.005776	-0.006848
4	19700107	0.000678	-0.001272	0.003559	-0.002047
5	19700108	0.002034	0.000564	0.002890	0.000540

5 rows

tail(da)#Last 5 rows

ABCDEFGHIJ0123456789

	Date <int>	rtn <dbl>	vwretd <dbl>	ewretd <dbl>	sprtrn <dbl>
9840	20081223	-0.016953	-0.007618	-0.008332	-0.009717
9841	20081224	-0.000993	0.004463	0.005254	0.005781
9842	20081226	0.010060	0.007170	0.011629	0.005356
9843	20081229	-0.000984	-0.004481	-0.016514	-0.003873
9844	20081230	0.028308	0.024951	0.021692	0.024407
9845	20081231	0.007301	0.017513	0.036731	0.014158

6 rows

ibm = da[,2] # IBM simple returns

sibm = ibm*100 #Percentage simple returns 

b = basicStats(sibm)

s1 = skewness(sibm)
t = s1/sqrt(6/9845) #definition for test statistic here 9845 is N 
pv  = 2*(1-pnorm(t)) # Calculate p-value

cat("skewness =", s1,"\nTest Satistic = ",t,"\np-value = ", pv)

skewness = 0.06139878 
Test Satistic =  2.487093 
p-value =  0.01287919

library(ggplot2)
ibm = as.data.frame(da[,1:2])

ibm["log_return"] = log(1+ibm["rtn"])
colnames(ibm)[2] <- "simple_return"

ggplot(ibm,aes(x = simple_return,colour="Emperical"))+
      ggtitle("Simple return distribution") + 
      xlim (-0.15,0.15)+
      geom_density()+
      stat_function(mapping = aes(colour ="Normal"),
                   fun = dnorm,args = with(ibm,c(mean =mean(simple_return),
                   sd = sd(simple_return))))
ggplot(ibm,aes(x = log_return,colour = "Emperical"))+
      ggtitle("Log return distribution")+ 
      xlim (-0.15,0.15)+geom_density()+
      stat_function(mapping = aes(colour = "Normal"),
             fun = dnorm,args = with(ibm,c(mean = mean(log_return),
             sd = sd(log_return))))+ 
scale_x_continuous("log return distribution",limits = c(-0.15,0.15))

#IBM = ibm[, 2] # IBM simple return values
#IBM = ts(IBM,frequency = 250 ,start = c(1970,1))

#plot.ts(IBM)


#acf(IBM,lag=250,ylim= c(-0.05,0.05))#ACF is covered in detail down below , the x axis label is fraction of unit time 

#libm = ibm[,2]*100
#length(libm)
#mean(libm)
#t.test(libm)
#k1 = kurtosis(libm)
#s1 =skewness(libm)
#t3  = (s1^2)/(6/9845)
#t2 = ((k1)^2)/(24/9845)
#3t2
#a = t2+t3
#a
#normalTest(libm,method = 'jb')

Linear Time Series

Stationarity

Foundation of time series analysis is Stationarity. A time series $\{r_t\}$ is said to be strictly stationary if the joint distribution of $(r_{t_{1}},r_{t_{2}},r_{t_{3}},....,r_{t_{k}})$ is identical to that of $(r_{t_{1+t}},r_{t_{2+t}},r_{t_{3+t}},....,r_{t_{k+t}})$ for all $t$ , where $k$ is an arbitrary positive integer and $(t_1,t_2,t_3,....,t_k)$ is collection of $k$ positive integers. Strict stationarity is very hard to verify emperically
A time series is said to be weakly stationary if the mean of $r_t$ and the covariance between $r_t$ and $r_{t-l}$ are time invariant where $l$ is an arbitrary integer. Weak stationarity enables us to make inference concerning future observations

• $\mathbb{E}(t_t) = \mu$
• $Cov(r_t,r_{t-l}) = \gamma_l$

Weak stationarity is commonly studied and more practical.

The covariance $\gamma_l = Cov(r_t,r_{t-l})$ is called the lag-$l$ autocovariance of $r_t$. From the above defintion it follows that

• $\gamma_0=Var(r_t)$
• $\gamma_{-l} = \gamma_l$

Autocorrelation Function

Consider a weakly stationary return series $r_t$. When the linear dependence between $r_t$ and its past values $r_{t-l}$ is of interest , the concept of correlation is generalised to autocorrelation. The correlation coefficient between $r_t$ and $r_{t-l}$ is called the lag-l autocorrelation of $r_t$ and is commonly denoted by $\rho_l$ , which under the weak stationarity assumption is a function of l only. Its defined by $$\rho_l = \frac{Cov(r_t,r_{t-l})}{\sqrt{Var(r_t)Var(r_{t-l})}} = \frac{Cov(r_t,r_{t-l})}{Var(r_t)} = \frac{\gamma_l}{\gamma_0}$$

Its easy to see that

• $\rho_0 = 1$
• $\rho_l = \rho_{-l}$
• $-1\leq \rho_l \leq 1$

For a given sample of returns $\{r_t\}_{t=1}^T$ , let $\bar r$ be the sample mean. Then the lag-1 autocorrelation of $r_t$ is $$\hat\rho_1 = \frac{\sum_{t=2}^T(r_t - \bar r)(r_{t-1} - \bar r)}{\sum_{t=2}^T(r_t - \bar r )^2}$$

The conditions under which $\hat\rho$ is a consistent estimator of $\rho_1$ are as follows :

• The returns $\{r_t\}$ is an independent and identically distributed sequence
• $\mathbb{E}(r_t^2) < \infty$

Then $\hat\rho_1$ is asymptotically normal with mean zero and variance 1/T

Testing

Let $H_0 : \rho_1 = 0$ be the null hypothesis versus $H_a : \rho_1 \neq 0$ the alternative hypothesis. The test statistic is the usual t ratio (check the statistics section for further description) , which is $\sqrt{T}\hat\rho_1$ and follows asymptotically the standard normal distribution. The null hypothesis $H_0$ is rejected if the t ratio is large in magnitude or equivalently , the p value of the t ratio is small , lets say less than 0.05

set.seed(666)
phi = c(rep(0,11),.9)
sAR = arima.sim(list(order=c(12,0,0), ar=phi), n=37)
sAR = ts(sAR, freq=12)
layout(matrix(c(1,1,2, 1,1,3), nc=2))
par(mar=c(3,3,2,1), mgp=c(1.6,.6,0))
plot(sAR, axes=FALSE, main='seasonal AR(1)', xlab="year", type='c')
Months = c("J","F","M","A","M","J","J","A","S","O","N","D") 
points(sAR, pch=Months, cex=1.25, font=4, col=1:4)
axis(1, 1:4); abline(v=1:4, lty=2, col=gray(.7))
axis(2); box()
ACF = ARMAacf(ar=phi, ma=0, 100)
PACF = ARMAacf(ar=phi, ma=0, 100, pacf=TRUE)
plot(ACF,type="h", xlab="LAG", ylim=c(-.1,1)); abline(h=0) 
plot(PACF, type="h", xlab="LAG", ylim=c(-.1,1)); abline(h=0)