Project / Assignment – I DRISHAN SENGUPTA (MB-1122)

PROBLEM

Take a time series data of reasonable length on any financial variable of your interest. Instead of real time series, you may as well consider a time series of artificially generated (i.e., simulated) data such that the DGP of the series incorporates, inter alia, volatility.

i. Plot the data and comment on its nature. Check also if the time series is stationary.

ii. Fit an appropriate conditional mean model to this data. Test if the residuals of the model thus obtained are white noise. Also find empirically whether the squared residuals are autocorrelated or not.

iii. Fit an appropriate volatility model simultaneously with a mean model. Thereafter, test if the standardized residuals as well as the squared standardized residuals are autocorrelated.

iv. Estimate the risk-return type of relationship in the framework of (G)ARCH -In- Mean model, and then comment on the nature of the relationship thus obtained.

SOLUTION

DATA DESCRIPTION :

Monthly returns for IBM stock from 1926 to 1997.

(‘m.ibm2697’ object of class ‘zooreg’, package {FinTS} in R)

Source : http://faculty.chicagogsb.edu/ruey.tsay/teaching/fts2

PART (i)

A time series is said to be strictly stationary if the joint distribution of X(t1),……,X(tk) is the same as the joint distribution of X(t1+α),……….,X(tk+α) for all t1,…., tk,α. This is a quite strong condition to hold in real circumstances. Rather we define a weaker restriction ,called weak stationary , if it’s mean is constant and auto covariance depends on lag; i.e. E(X(t)]=µ Cov[X(t),X(t+α)]=f(α) This weaker definition of stationrity is generally termed as stationariy time series.

INFERENCE:

Monthly returns for IBM stock from 1926 to 1997 showing presence of high volatility (large changes are followed by large changes & small changes are followed by followed by small changes)

R-CODE:

data(m.ibm2697)

data = as.vector(m.ibm2697[ ,1])

y = as.ts(m.ibm2697[ ,1])

y

plot(y, ylab = "Data", main = "Plot showing Monthly returns for IBM stock from 1926 to 1997") There is no simple mathematical method for tracking presence of stationarity in the data however with finite samples the sample acf decays slowly towards zero. One way of tackling this problem in regard to ARIMA models is to ask whether the differencing parameter ‘d’ is exactly equal to 1,if it is then that would mean a unit root is present. Perhaps the most important test for unit root is Augmented Dicky Fuller Test and Phillips-Perron Test, where the null hypothesis is taken to be non-stationarity(i.e. to test for ρ=1)

To check whether the time series under consideration is stationary, we perform the Augmented Dickey-Fuller (ADF) Test and Phillips-Perron (PP) Unit Root Test.

Here, H0 : non – stationarity ag. H1 : stationarity. R-OUTPUT:

Augmented Dickey-Fuller Test(ADF Test)

data: y

Dickey-Fuller = -8.1519, Lag order = 9, p-value = 0.01

alternative hypothesis: stationary

Phillips-Perron Unit Root Test(PP Test)

data: y

Dickey-Fuller Z(alpha) = -784.2656, Truncation lag parameter = 6, p-value = 0.01

alternative hypothesis: stationary

INFERENCE:

Since in both the cases p-value is less than significance level 5% we reject the presence of null hypothesis of non-stationarity; i.e. monthly Pfizer stock data Is stationary.

PART (ii)

From graphical representation of data it is quite evident that high...