Univariate Time Series Models
(M.Sc. Finance - Exercise 4)

Walter Distaso
Imperial College Business School w.distaso@imperial.ac.uk

Question 1
Consider the following three models that a researcher suggests might be reasonable models of stock market prices. yt yt yt = yt−1 + ut = 0.5yt−1 + ut = 0.8yt−1 + ut

(a) What classes of models are these examples of? (b) What would the autocorrelation function for each of these processes look like? (not exactly, just the shape) (c) Which model is more likely to represent stock market prices from a theoretical perspective, and why? If any of the three models truly represented the way stock market prices move, which could potentially be used to make money? (d) Consider the extent of persistence of shocks in the series in each case.

Question 2

You obtain the following estimates for an AR(2) model of some returns data. yt = 0.803yt−1 + 0.682yt−2 + ut , where ut is a white noise error process. By examining the characteristic equation, check the estimated model for stationarity.

Question 3
A researcher is trying to determine the appropriate order of an ARMA model to describe some data, with 200 observations available. She has the following ﬁgures for the log of estimated residual variance (log(ˆ 2 )) for various candidate models. She has σ assumed that an order greater than (3,3) should not be necessary to model the dynamics of the data. What is the “optimal” model order? ARMA(p, q) model order log(ˆ 2 ) σ (0,0) 0.932 (1,0) 0.864 (0,1) 0.902 (1,1) 0.836 (2,1) 0.801 (1,2) 0.821 (2,2) 0.789 (3,2) 0.773 (2,3) 0.782 (3,3) 0.764

Question 4

“Given that the objective of any econometric modeling exercise is to ﬁnd the model that most closely ‘ﬁts’ the data, then adding more lags to an ARMA model will almost invariably lead to a better ﬁt. Therefore, a large model is best because it will ﬁt the data more closely.” Comment on the validity (or otherwise) of this statement.

....2.3 Timeseries models
Timeseries is an ordered sequence of values of a variable at equally spaced time intervals. Timeseries occur frequently when looking at industrial data. The essential difference between modeling data via timeseries methods and the other methods is that Timeseriesanalysis accounts for the fact that data points taken over time may have an internal structure such as autocorrelation, trend or seasonal variation that should be accounted for. A Time-series model explains a variable with regard to its own past and a random disturbance term. Special attention is paid to exploring the historic trends and patterns (such as seasonality) of the timeseries involved, and to predict the future of this series based on the trends and patterns identified in the model. Since timeseries models only require historical observations of a variable, it is less costly in data collection and model estimation.
. Timeseries models can broadly be categorized into linear and nonlinear Models. Linea models depend linearly on previous data points. They include the autoregressive (AR) models, the integrated (I) models, and the moving average (MA) models. The general...

...This paper is a report on the time-seriesanalysis of continuously compounded returns for Ford and GM for the periods January 2002 till April 2007 using monthly stockprices. This analysis is aimed at estimating the ARIMA model that provides the best forecast for the series. This paper will be divided into 2 sections; the first section showing the Ford analysis and the second the GM analysis.
Section 1: Ford
Figure 1: Timeseries plot for raw Ford data.
Figure 1 shows a timeseries plot of the raw Ford stockprices against time. From this plot, a gradual but continuous upward trend can be observed. This trend was disrupted in 2005 when the stockprices experienced a huge rise moving from below 5 to above 25. This rise in stockprice by Ford was not sustained as can be seen from the plot; the prices which reached a peak of above 25 fell to a about 10 by the end of 2005 and fell further in 2006 to a level below 5 fluctuations in the stockprice existed and in 2007 the prices began to level out.
Raw data is likely to be affected by non-stationarity and this can result in bias in the analysis. For the purpose...

...Analysis of Financial TimeSeries
Third Edition
RUEY S. TSAY
The University of Chicago Booth School of Business Chicago, IL
A JOHN WILEY & SONS, INC., PUBLICATION
Analysis of Financial TimeSeries
WILEY SERIES IN PROBABILITY AND STATISTICS Established by WALTER A. SHEWHART and SAMUEL S. WILKS Editors: David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice, Iain M. Johnstone, Geert Molenberghs, David W. Scott, Adrian F. M. Smith, Ruey S. Tsay, Sanford Weisberg Editors Emeriti: Vic Barnett, J. Stuart Hunter, Jozef L. Teugels
A complete list of the titles in this series appears at the end of this volume.
Analysis of Financial TimeSeries
Third Edition
RUEY S. TSAY
The University of Chicago Booth School of Business Chicago, IL
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright 2010 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through...

...listed on the NASDAQ stock exchange under the ticker symbol .
The company's mission statement from the outset was "to organize the world's information and make it universally accessible and useful”, and the company's unofficial slogan is "Don’t be evil”. In 2006, the company moved to its current headquarters in Mountain View, California.
Objectives
1. To fit a multiple regression model to a data set comprising the put, call and strike prices of astock belonging to a company listed on a known index.
2. To use the BSM Model to which provides a mathematical science for the pricing and hedging of European Call and Put options as the American Options market
3. We wanted to analyze the data for Google option prices from the S&P index over the past and present time periods in order to be able to forecast the future.
Literature Review
1. Put call parity
In financial mathematics, put–call parity defines a relationship between the price of a European call option and European put option in a frictionless market —both with the identical strike price and expiry, and the underlying being a liquid asset. In the absence of liquidity, the existence of a forward contract suffices. Put–call parity requires minimal assumptions and thus does not require assumptions such as those of Black–Scholes or other commonly used financial models.
2....

...Secondary Research TimeSeriesAnalysis
VARIABLE FACTOR THAT INCREASING MALAYSIA GDP
Prepared by:
Dina Maya Avinati
Wery Astuti
Faculty of Business
UNIVERSITAS SISWA BANGSA INTERNATIONAL
Mulia Business Park, JL. MT. Haryono Kav. 58-60 Pancoran- South Jakarta
Page | 1
CONTENT
I.
Introduction
1.1
Back Ground of Study
1.2
Problem
1.3
Research Problem
1.4
Research Objective
1.5
Scope and Limitation
1.6
Significant of Study
II.
Literature Review
III.
Methodology
2.1
Time and Place
2.2
Research Framework
2.3
Research Question and Hypothesis
2.4
Data
2.4.1
Type and Source of Data
2.4.2
Data Collection Method
2.4.3
Sampling Method
2.4.4
Data Analysis
2.4.5
Hypothesis Testing
IV.
Research and Discussion
V.
Conclusion and Recommendation
References
Appendix
Letter of Performance
Page | 2
INTRODUCTION
1.1
Back Ground of Study
TimeSeriesAnalysis (TSA) is one name of specific subjects that must be studied
by college students majoring in Management at University of
Siswa Bangsa
International. TimeSeries can define as a sequence of numbers collected at regular
intervals over a period of time. In addition, TimeSeriesAnalysis is method for...

...Timeseries
In statistics, signal processing, econometrics and mathematical finance, a timeseries is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of timeseries are the daily closing value of the Dow Jones index or the annual flow volume of the Nile River at Aswan. Timeseriesanalysis comprises methods for analyzing timeseries data in order to extract meaningful statistics and other characteristics of the data. Timeseries forecasting is the use of a model to predict future values based on previously observed values. Timeseries are very frequently plotted via line charts.
Timeseries data have a natural temporal ordering. This makes timeseriesanalysis distinct from other common data analysis problems, in which there is no natural ordering of the observations (e.g. explaining people's wages by reference to their education level, where the individuals' data could be entered in any order). Timeseriesanalysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. accounting...

...TIMESERIESANALYSIS
Chapter Three
Univariate TimeSeries Models
Chapter Three
Univariate timeseries models c WISE
1
3.1
Preliminaries
We denote the univariate timeseries of interest as yt.
• yt is observed for t = 1, 2, . . . , T ;
• y0, y−1, . . . , y1−p are available;
• Ωt−1 the history or information set at time t − 1.
Call such a sequence of random variables a timeseries.
Chapter Three
Univariate timeseries models c WISE
2
Martingales
Let {yt} denote a sequence of random variables and let It =
{yt, yt−1, . . .} denote a set of conditioning information or information
set based on the past history of yt. The sequence {yt, It} is called a
martingale if
• It−1 ⊂ It (It is a ﬁltration)
• E [|yt|] < ∞
• E [yt|It−1] = yt−1 (martingale property)
Chapter Three
Univariate timeseries models c WISE
3
Random walk model
The most common example of a martingale is the random walk model
yt = yt−1 + εt,
εt ∼ W N (0, σ 2)
where y0 is a ﬁxed initial value.
Letting It = {yt, . . . , y0} implies E [yt|It−1] = yt−1 since E [εt|It−1] = 0.
Chapter Three
Univariate timeseries models c WISE
4
Law of Iterated Expectations
Deﬁnition 1. In general, for...