Irfan Ahmed, PhD

In many of the problems of interest in finance, the starting point is a time series of prices -- for example, the prices of shares in Ford, taken at 4p.m. each day for 200 days. For a number of statistical reasons, it is preferable not to work directly with the price series, so that raw price series are usually converted into series of returns. Additionally, returns have the added benefit that they are unit-free. So, for example, if an annualized return were 10%, then investors know that they would have got back ￡ 110 for a ￡ 100 investment, or ￡ 1,100 for a ￡ 1,000 investment, and so on. There are two methods used to calculate returns from a series of prices, and these involve the formation of simple returns, and continuously compounded returns, which are achieved as follows: Simple returns Continuously compounded returns Rt =     ( pt − pt − 1) / pt − 1   × 100%      (1.1) rt = 100% × ln( pt/ pt − 1 )  ...

There are broadly three types of data that can be employed in quantitative analysis of financial problems: time series data, cross-sectional data, and panel data. Time series data Time series data, as the name suggests, are data that have been collected over a period of time on one or more variables. Time series data have associated with them a particular frequency of observation or collection of data points. The frequency is simply a measure of the interval over , or the regularity with which , the data are collected or recorded. Following are the examples of time series data. Series Frequency Industrial production Monthly, or quarterly Government budget deficit Annually Money supply Weekly The value of a stock As transactions occur A word on ‘As transactions occur’ is necessary. Much financial data does not start its life as being regularly spaced . For example, the price of common stock for a given comp...

Stationarity and Unit Root Testing l   The stationarity or otherwise of a series can strongly influence its behaviour and properties - e.g. persistence of shocks will be infinite for nonstationary series l   Spurious regressions. If two variables are trending over time, a regression of one on the other could have a high R 2 even if the two are totally unrelated l   If the variables in the regression model are not stationary, then it can be proved that the standard assumptions for asymptotic analysis will not be valid. In other words, the usual “ t -ratios” will not follow a t -distribution, so we cannot validly undertake hypothesis tests about the regression parameters. Stationary and Non-stationary Time Series Stationary Time Series l   A series is said to be stationary if the mean and autocovariances of the series do not depend on time. (A) Strictly Stationary : n   For a strictly stationary time series the distribution of   y(t) is independent...