Econometric Principles and Data Analysis

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Econometric Principles

and Data Analysis

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Econometric Principles and Data Analysis © Centre for Financial and Management Studies SOAS, University of London 1999, revised 2003, 2007, revised 2009, 2010, revised 2013

All rights reserved. No part of this course material may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, including photocopying and recording, or in information storage or retrieval systems, without written permission from the Centre for Financial & Management Studies, SOAS, University of London.

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Econometric Principles & Data

Analysis

Course Introduction and Overview

1 Course

Objectives

This course provides an introduction to econometric methods. In brief, the course examines how we can start from relationships suggested by financial and economic theory, formulate those relationships in mathematical and statistical models, estimate those models using sample data, and make statements based on the parameters of the estimated models. The course examines the assumptions that are necessary for the estimators to have desirable properties, and the assumptions necessary for us to make statistical inference based on the estimated models. In addition, the course explores what happens when these assumptions are not satisfied, and what we can do in these circumstances. The course concludes with an examination of model selection.

2 The Course Authors

The course, and its more advanced sequel, Econometric Analysis and

Applications, were designed and written by Dr Graham Smith, who is Senior Lecturer in the Department of Economics, SOAS, where he teaches

econometrics to MSc students and carries out research on empirical finance. His main research interests focus on emerging stock markets and he has published extensively in international refereed journals. His recent research demonstrates that stock market efficiency is determined by market size, liquidity and the quality of markets.

The course has been revised by Dr Jonathan Simms, who is a tutor for CeFiMS, and has taught at University of Manchester, University of Durham and University of London. He has contributed to development of various CeFiMS courses including Econometric Analysis and Applications;Financial Econometrics, Risk Management: Principles & Applications;Public Financial Management: Reporting and Audit; and Introduction to Law and to Finance.

3 The Course Structure

The paragraphs following the list of topics presented in the units provide brief descriptions of the units’ content. They are intended as an introduction and overview of the course. More complete, detailed explanation, analysis and discussion are provided in the units themselves, and in the course textbook. So don’t worry if you do not understand everything in this short introduction.

Unit 1 Introduction to Econometrics and Regression Analysis

1.1 What is Econometrics? 1.2 How to Use the Course Texts 1.3 Ideas – The Concept of Regression 1.4 Study Guide

1.5 An Example – The Consumption Function 1.6 Summary

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Course Introduction and Overview

Centre for Financial and Management Studies 3

1.8 Exercises

1.9 Answers to Exercises

Unit 2 The Classical Linear Regression Model

2.1 Ideas and Issues 2.2 Study Guide

2.3 Example – the Single Index Model (SIM) 2.4 Summary

2.5 Exercises

Unit 3 Hypothesis Testing

3.3 Example – The Capital Asset Pricing Model 3.4 Summary

3.5 Exercises

Unit 4 The Multiple Regression Model

4.3 Example – A Multi-Index Model 4.4 Summary

4.5 Exercises

Unit 5 Heteroscedasticity

5.3 Example – Price-Earnings Ratio 5.4 Summary

5.5 Exercises

Unit 6 Autocorrelation

6.3 Example – The Single-Index Model 6.4 Summary

6.5 Exercises

Unit 7 Nonnormal Disturbances

7.1 Ideas and Issues 7.2 Study Guide 7.3 Examples 7.4 Summary 7.5 Exercises

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Appendix 1: Small-Sample Critical Values for the Jarque-Bera Test Appendix 2: Stock Market Indices

Unit 8 Model Selection and Course Summary

8.3 Example: the Demand for Money Function 8.4 Summary

8.5 Exercises

8.7 Course Summary: ‘What you do and do not know’

Unit 1 provides an introduction to econometrics and regression analysis. By regression we mean an equation that captures the mathematical relationship between the variables, and also the imperfect nature of that relationship. The unit introduces the stages of an econometric investigation:

• statement of the theory

• collection of data

• mathematical model of the theory (an exact relationship between variables)

• econometric model of the theory (a stochastic model of the relationship between variables)

• parameter estimation

• checking for model adequacy

• tests of hypotheses

• prediction.

Unit 1 also provides guidance on how to use the study materials. In addition, it provides a brief revision of how to calculate financial rates of return. Each unit includes a worked example. (In Unit 1, the example concerns the relation between spot and forward exchange rates.) All of the units also contain exercises for you to do in order to develop your own understanding and confidence, from a wide range of econometric studies. Data for the exercises are provided. The data used in the examples are also provided so that you can replicate the results presented in the unit (replicating the results in the example is presented as an exercise).

The course uses the software package Eviews. Results from Eviews are presented in the units. You are provided with a copy of Eviews to do the unit exercises. Answers for the exercises are provided at the end of each unit, but you look at the answers only after you have done the exercises yourself! Data on the stock price of Delta Airlines Inc. and the New York Stock Exchange Composite Index are introduced in the exercises in Unit 1. This data set is used in a number of units throughout the course, in the worked examples or the exercises. By applying different econometric tools with the same data set, it is hoped you will develop a rounded view of how the

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methods you will learn relate to each other. A variety of other models and data sets are also used.

Unit 2 presents the classical linear regression model. It explains the method of ‘ordinary least squares’ (OLS) and how that can be used to estimate the

unknown parameters of a regression equation using sample data. In this unit we are concerned with models containing two variables; we are trying to discover how one variable – the explanatory variable – explains another variable – the dependent variable – and estimate the parameters in that relationship.

We then need to ask whether we can make statements about the true,

unknown, parameters of the model, based on our estimated values. To do this we need to make a number of assumptions. These assumptions, if satisfied, ensure that the estimators we use have desirable properties (in brief and oversimplified terms: the estimators are accurate and efficient). If the assumptions are satisfied, we can also make predictions about the unknown model parameters, and we can specify, precisely, how confident we are about those predictions. Unit 2 also explains goodness of fit: how closely our estimated model fits our sample data. These ideas are demonstrated using the single-index market model applied to Delta Airlines Inc., and the British retailer Marks & Spencer.

Unit 3 explores how to test hypotheses. Based on our estimated model coefficients, can we answer questions of the form:

• Is the true, unknown coefficient negative, zero, or positive?

• Does it take a particular value?

• Is there actually a relationship between the two variables?

Unit 3 uses the capital asset pricing model (CAPM) for GlaxoSmithKline to demonstrate hypothesis testing. Hypothesis testing is demonstrated further in the exercises with the single-index model. So, for example, we might be concerned with how we can test whether the stock we are interested in is defensive or aggressive; is the company beta less than one or greater than one? The efficiency of foreign exchange markets is also examined. Unit 4 extends the analysis to the multiple regression model; these are regression models in which one variable is explained by two or more

variables. The unit examines the assumptions necessary to estimate and make predictions with such models. The unit asks what happens if, in a multiple regression model, there is a relationship between any of the explanatory variables, in addition to the relationships we hope to discover between the explanatory variables and the dependent variable (this is called multicollin-earity). The techniques of multiple regression are demonstrated with an example of a multi-index model.

Units 5, 6 and 7 are concerned with what happens if a number of the assumptions of the classical linear regression model are not satisfied. What are the consequences for the properties of the ordinary least squares

estimators, and can we still make predictions about the unknown model parameters based on our estimated model?

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Unit 5 is concerned with heteroscedasticity. What is that? Here is a very brief and simplified explanation; a more detailed and precise explanation is

provided in Unit 5. Unit 1 explains how we can specify a mathematical relationship between variables. The actual relationship between variables is not exact, and we attempt to capture this by including an error or disturbance term in the regression equation. One of the assumptions we make is that the variance of the disturbance term – how much it varies about its mean value – is constant for all observations. This is the assumption of homoscedasticity, and is explained in Unit 2.

In some econometric studies this assumption may not be satisfied. Consider a cross-section study of commission rates for different brokerage companies. The disturbance term also attempts to capture those influences on commis-sion rates that we have not included in our model. Is it likely that the variance of this disturbance term will be constant for all brokerage companies? If the variance of the disturbance term is not constant, we say there is heteroscedas-ticity. Unit 5 examines the consequences of heteroscedasticity:

• What are the effects on the properties of OLS estimators, and can we still make predictions based on our estimated model?

The unit examines how heteroscedasticity can be identified, and how we can deal with it, either by transforming the model or by using a different estimation method. If we know what form the heteroscedasticity takes, we can use the method of weighted least squares. Heteroscedasticity is demonstrated with a study of price-earnings ratios estimated for a cross-section of companies. Unit 6 is concerned with autocorrelation. Again, here is a very simple and brief explanation; a more precise and formal explanation is provided in Unit 6. Consider again the disturbance term that we include in our regression equation. The disturbance term reflects the stochastic nature of the relation-ship between variables, and also attempts to capture the elements that we have not included in the model. Another assumption we make about the disturbance term is that the disturbance terms for different observations (e.g. if using annual data, last year and this year, or if using daily data, yesterday and today) are not related.

This is the assumption of noncorrelated disturbances, and is explained in Unit 2. If the disturbances for different observations are related, we say that the disturbance term is serially correlated or ‘autocorrelated’. For example, an economic or financial shock in one month may have persistent effects in following months, and if the model does not explicitly include such

persistence effects, the disturbance terms in different months will be correlated. Unit 6 examines the implications of autocorrelation for the

properties of OLS estimators, and also the consequences for prediction based on OLS estimators. It also shows how to identify autocorrelation using plots and more formal tests, and what can be done to take account of autocorrela-tion, including changing the method of estimation. The effects of

autocorrelated disturbances are demonstrated with the single-index market model for Delta Airlines, and a model of spot and forward exchange rates.

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Unit 7 is concerned with the assumption of normality. In order to make predictions about the true, unknown model parameters, based on our estimated values, we need to assume that the disturbance terms are distri-buted normally – that is, they follow a normal distribution. You are probably already familiar with the normal distribution from your other studies. It is a probability distribution with known properties, which allows us to make statements concerning the unknown model parameters with a certain degree of confidence – for example, we can reject a hypothesis about a parameter with a 5% chance of being wrong, or we can be 95% confident that an unknown parameter takes a value within a certain range of values.

If the disturbance terms are not normally distributed, we are unable to make such predictions, and it also has consequences for the properties of the OLS estimators. Unit 7 explains the effects of having disturbances that are not distributed normally, the tests to detect non-normal disturbances, and what can be done about non-normal disturbances. This includes the use of dummy variables to take account of outliers (data points which are very different from the rest of the sample). These methods are demonstrated with two examples: stock market returns and the single-index model for Marks & Spencer. The exercises include consideration of the SIM for Delta Airlines and for Bank of America.

Unit 8 is concerned with model selection. One of the assumptions we make is that the model we estimate is correctly specified: the regression equation includes all relevant variables, and the functional form of the relationship is specified correctly – variables are included correctly as levels, or their logged values are included, or perhaps squared values of the variables are included. If the model is not correctly specified, this has consequences for the

properties of the OLS estimators and for prediction based on those estima-tors. In particular, Unit 8 examines the consequences of omitting a relevant explanatory variable, including an irrelevant explanatory variable, and using the wrong functional form.

The unit explains methods to identify misspecified equations. These include tests specifically designed to identify misspecified models. In addition, evidence of heteroscedasticity, autocorrelated errors, or non-normal errors, may be a further sign that a model is not correctly specified. Unit 8 also shows how we can decide between different specifications of a particular economic relationship. It demonstrates model selection using the Delta Airlines data set, and also the SIM for IBM stock. Finally, Unit 8 includes a summary of the course, to help with your revision for the final examination. More advanced topics in econometrics are studied in the CeFiMS course

Econometric Analysis & Applications. These include more use of dummy variables, dynamic models: lags and expectations; simultaneous equation models; time series analysis: stationarity and nonstationarity, and forecasting.

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4 Learning

Outcomes

After studying this course you will be able to:

• explain the principles of regression analysis

• outline the assumptions of the classical normal linear regression model, and discuss the significance of these assumptions

• explain the method of ordinary least squares

• produce and interpret plots of data

• use the program Eviews to estimate a regression equation, and interpret the results, for bivariate (two-variable) regression models and multiple regression models

• test hypotheses concerning model parameters

• test joint hypotheses concerning more than one variable

• discuss the consequences of multicollinearity, the methods for identifying multicollinearity, and the techniques for dealing with it

• explain what is meant by heteroscedasticity, and the consequences for OLS estimators and prediction based on those estimators

• assess the methods used to identify heteroscedasticity, including data plots and more formal tests, and the various techniques to deal with heteroscedasticity, including model transformations and estimation by weighted least squares

• explain autocorrelation, and discuss the consequences of autocorrelated disturbances for the properties of OLS estimator and prediction based on those estimators

• outline and discuss the methods used to identify autocorrelated disturbances, and what can done about it, including estimation by generalised least squares

• discuss the consequences of disturbance terms not being normally distributed, tests for nonnormal disturbances, and methods to deal with non-normal disturbances, including the use of dummy variables

• discuss the consequences of specifying equations incorrectly

• discuss the tests used to identify correct model specification, and statistical criteria for choosing between models

• useEviews to conduct tests for heteroscedasticity, correlated disturbances, nonnormal disturbances, functional form, and model selection

• useEviews to estimate models in which the disturbance term is assumed to be heteroscedastic or autocorrelated.

5 Study

Materials

These course units are your central learning resource; they structure your learning unit by unit. Each unit should be studied within a week. The course units are designed in the expectation that studying the unit and the associated readings in the textbook, and completing the exercises, will require 15 to 20 hours during the week.

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Textbook

In addition to the course units you must read the assigned sections from the textbook, which is provided with your course materials:

Damodar N Gujarati and Dawn C Porter (2010) Essentials of Econometrics, New York: McGraw-Hill.

We have specifically used this textbook because it provides an excellent user-friendly introduction to econometric theory and techniques. You will notice that Gujarati and Porter present examples from finance, economics and business, because it is an introduction to econometrics in general. The examples and exercises in the course units are drawn entirely from finance. In each course unit there is a section, called Study Guide, which leads you through the relevant parts of the textbook, and helps you to read and understand the analysis presented there. If, while studying this course, you find you need some revision in basic probability and statistics, you may find it useful to look at parts of Appendices A to D in the textbook, which cover probability, probability distributions, and statistical inference.

Eviews

You have been provided with a copy of Eviews, Student Edition. This is the econometrics software that you will use to do the exercises in the units, and also the data analysis part of your assignments. The results presented in the units are also from Eviews.

Instructions to install Eviews, and to register your copy of the software, are included in the booklet that comes with the Eviews CD. (Your student edition ofEviews will run for two years after installation, and you will be reminded of this every time you open the program.)

You must register your copy of Eviews within 14 days of installing it on your computer. If you do not register your copy within 14 days, the software will stop working.

Eviews is very easy to use. Like any Windows program, you can operate it in a number of ways:

• there are drop-down menus

• selecting an object and then right-clicking provides a menu of available operations

• double-clicking an object opens it

• keyboard shortcuts work.

There is also the option to work with Commands; these are short statements that inform the program what you wish to do, and once you have built up your own vocabulary of useful Commands, this can be a very effective way of working. You can also combine all of these ways of working with Eviews. In each unit there are instructions to help you use Eviews to do the exercises. In addition, Eviews includes help files, which you can read as pdf files, or

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navigate via the Eviews help and search facility. Unit 1 includes a section introducingEviews.

Although easy to use, Eviews is a very powerful program. There are

advanced features that you will not use on this course, and you should not be worried if you see these, either in the menus or the help files. The best advice is to stay focused on the subject that is being studied in each unit, and to do the exercises for the unit; this will reinforce your understanding and also develop your confidence in using data and Eviews.

Exercises

As already noted, there are exercises in every unit. These require you to work withEviews and data files, available from the VLE in the course area for this study session, to do your own econometric analysis. It is very important that you attempt these exercises, and do not just look at the Answers at the end of the units. Your understanding of the material you have studied in the unit will be greatly improved if you do the exercises yourself. You will also develop better understanding and confidence in using Eviews.

The Instructions that accompany the exercises in the first few units are quite detailed, because they are intended to help you to start working with Eviews. As the units progress, it is assumed that you will gradually develop your understanding of the basic Eviews operations, and the Instructions then focus more and more on what new operations are required to do the Exercises in the units. If you find that you have forgotten how to do something, look back at the Instructions in the early units, because the basic operations will be the same.

Podcast

There is a podcast to accompany Econometric Principles & Data Analysis, in which Dr Simms discusses the course with Pasquale Scaramozzino, Professor of Economics at the Centre for Financial and Management Studies. The podcast is 18:26 minutes in length. Timings in the podcast are indicated below in brackets.

The podcast begins by explaining what the course does (from 0:28), and provides advice on how to study econometrics (1:16). Dr Simms then discusses how to get the most out of the materials (2:58), including the examples and exercises, and Eviews, and explains the choice of the textbook,

Essentials of Econometrics. The podcast then addresses the question of how a course in econometrics helps the understanding of financial markets (7:30). The discussion here emphasises the importance of being able to interpret regression results, and assessing the quality of those results; obtaining estimated equations is not enough in itself. Following this, the podcast considers how econometrics bridges the gap between theoretical financial models and financial data (11:56), explaining how econometrics allows us to test whether a particular theoretical model is appropriate or not, and how qualities displayed by the data can be used to improve models.

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In addition to analysing the examples, completing the exercises, and writing your assignments, you are also encouraged to apply the methods you are learning to data sets with which you are familiar from your own working environment (14:37), and to consider how the methods relate to your work or areas of interest – this may enable you to develop a more intuitive under-standing of the econometric techniques. Finally (16:57) there is a summary of the podcast discussion, and a consideration of the general approach to take to your study of econometrics, especially if you are unfamiliar with statistics and maths, or are returning to these subjects after a period of time.

We suggest that you listen to the podcast before you start studying Unit 1, and perhaps again half-way through the course when you have finished Unit 4. It may also provide a helpful revision at the end of the course, reinforcing your understanding of what you have learnt and providing an overall context. We hope that you enjoy this course.

6 Assessment

Your performance on each course is assessed through two written

as-signments and one examination. The asas-signments are written after week four and eight of the course session and the examination is written at a local examination centre in October.

The assignment questions contain fairly detailed guidance about what is required. All assignment answers are limited to 2,500 words and are marked using marking guidelines. When you receive your grade it is accompanied by comments on your paper, including advice about how you might improve, and any clarifications about matters you may not have understood. These comments are designed to help you master the subject and to improve your skills as you progress through your programme.

The written examinations are ‘unseen’ (you will only see the paper in the exam centre) and written by hand, over a three-hour period. We advise that you practise writing exams in these conditions as part of your examination preparation, as it is not something you would normally do.

You are not allowed to take in books or notes to the exam room. This means that you need to revise thoroughly in preparation for each exam. This is especially important if you have completed the course in the early part of the year, or in a previous year.

Preparing for Assignments and Exams

There is good advice on preparing for assignments and exams and writing them in Sections 8.2 and 8.3 of Studying at a Distance by Talbot. We recommend that you follow this advice.

The examinations you will sit are designed to evaluate your knowledge and skills in the subjects you have studied: they are not designed to trick you. If you have studied the course thoroughly, you will pass the exam.

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Understanding assessment questions

Examination and assignment questions are set to test different knowledge and skills. Sometimes a question will contain more than one part, each part

testing a different aspect of your skills and knowledge. You need to spot the key words to know what is being asked of you. Here we categorise the types of things that are asked for in assignments and exams, and the words used. All the examples are from CeFiMS exam papers and assignment questions.

Definitions

Some questions mainly require you to show that you have learned some concepts, by setting out their precise meaning. Such questions are likely to be preliminary and be supplemented by more analytical questions. Generally ‘Pass marks’ are awarded if the answer only contains definitions. They will contain words such as:

Describe Define Examine Distinguish between Compare Contrast Write notes on Outline What is meant by List Reasoning

Other questions are designed to test your reasoning, by explaining cause and effect. Convincing explanations generally carry additional marks to basic definitions. They will include words such as:

Interpret

Explain

What conditions influence

What are the consequences of

What are the implications of

Judgment

Others ask you to make a judgment, perhaps of a policy or a course of action. They will include words like:

Evaluate

Critically examine

Assess

Do you agree that

To what extent does

Calculation

Sometimes, you are asked to make a calculation, using a specified technique, where the question begins:

Use the single index model analysis to

Using any financial model you know

Calculate the standard deviation

Test whether

It is most likely that questions that ask you to make a calculation will also ask for an application of the result, or an interpretation.

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Advice

Other questions ask you to provide advice in a particular situation. This applies to policy papers where advice is asked in relation to a policy problem. Your advice should be based on relevant principles and evidence of what actions are likely to be effective.

Advise

Provide advice on

Explain how you would advise

Critique

In many cases the question will include the word ‘critically’. This means that you are expected to look at the question from at least two points of view, offering a critique of each view and your judgment. You are expected to be critical of what you have read. The questions may begin

Critically analyse Critically consider Critically assess

Critically discuss the argument that

Examine by argument

Questions that begin with ‘discuss’ are similar – they ask you to examine by argument, to debate and give reasons for and against a variety of options, for example

Discuss the advantages and disadvantages of Discuss this statement

Discuss the view that

Discuss the arguments and debates concerning

The grading scheme

Details of the general definitions of what is expected in order to obtain a particular grade are shown below. Remember: examiners will take account of the fact that examination conditions are less conducive to polished work than the conditions in which you write your assignments. These criteria are used in grading all assignments and examinations. Note that as the criteria of each grade rises, it accumulates the elements of the grade below. Assignments awarded better marks will therefore have become comprehensive in both their depth of core skills and advanced skills.

70% and above: Distinction As for the (60-69%) below plus:

• shows clear evidence of wide and relevant reading and an engagement with the conceptual issues

• develops a sophisticated and intelligent argument

• shows a rigorous use and a sophisticated understanding of relevant source materials, balancing appropriately between factual detail and key theoretical issues. Materials are evaluated directly and their assumptions and arguments challenged and/or appraised

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60-69%: Merit As for the (50-59%) below plus:

• shows strong evidence of critical insight and critical thinking

• shows a detailed understanding of the major factual and/or theoretical issues and directly engages with the relevant literature on the topic

• develops a focussed and clear argument and articulates clearly and convincingly a sustained train of logical thought

• shows clear evidence of planning and appropriate choice of sources and methodology

50-59%: Pass below Merit (50% = pass mark)

• shows a reasonable understanding of the major factual and/or theoretical issues involved

• shows evidence of planning and selection from appropriate sources,

• demonstrates some knowledge of the literature

• the text shows, in places, examples of a clear train of thought or argument

• the text is introduced and concludes appropriately 45-49%: Marginal Failure

• shows some awareness and understanding of the factual or theoretical issues, but with little development

• misunderstandings are evident

• shows some evidence of planning, although irrelevant/unrelated material or arguments are included

0-44%: Clear Failure

• fails to answer the question or to develop an argument that relates to the question set

• does not engage with the relevant literature or demonstrate a knowledge of the key issues

• contains clear conceptual or factual errors or misunderstandings

Specimen exam papers

Your final examination will be very similar to the Specimen Exam Paper that you received in your course materials. It will have the same structure and style and the range of question will be comparable. We do not provide past papers or model answers to papers. Our courses are continuously updated and past papers will not be a reliable guide to current and future examinations. The specimen exam paper is designed to be relevant to reflect the exam that will be set on the current edition of the course.

Further information

The OSC will have documentation and information on each year’s examin-ation registrexamin-ation and administrexamin-ation process. If you still have questions, both academics and administrators are available to answer queries. The Regula-tions are also available at , setting out the rules by which exams are governed.

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UNIVERSITY OF LONDON

Centre for Financial and Management Studies

MSc Examination

for External Students 15DFMC230|15DFMC330 Financial Economics

Finance

Econometric Principles and Data Analysis

Specimen Examination

This is a specimen examination paper designed to show you the type of examination you will have at the end of this course. The number of questions and the structure of the examination will be the same, but the wording and requirements of each question will be different.

The examination must be completed in three hours. Answer FOURquestions –

Question One and then THREE other questions.

The examiners give equal weight to each question; therefore, you are advised to distribute your time approximately equally over four questions.

Candidates may use their own electronic calculators in this examination provided they cannot store text. The make and type of calculator MUST BE STATED CLEARLY on the front of the answer book.

Do not remove this Paper from the Examination Room.

It must be attached to your answer book at the end of the examination.

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Page 2 of 5

You must answer Question One and then any other THREE questions. All candidates must attempt Question 1.

1 The Eviews output from estimating a single-index model for Microsoft Corporation using weekly data for the period from 1 September 2009 to 27 August 2012 is provided below. MSFT is the price of Microsoft Corporation stock,Cis an intercept and SP is the Standard & Poor’s 500 index.

Dependent Variable: DLOG(MSFT) Method: Least Squares

Sample (adjusted): 8/09/2009 27/08/2012 Included observations: 156 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

C 8.97E-05 0.001650 0.054359 0.9567

DLOG(SP) 0.867121 0.067407 12.86391 0.0000

R-squared 0.517967 Mean dependent var 0.001896

Adjusted R-squared 0.514837 S.D. dependent var 0.029487

S.E. of regression 0.020539 Akaike info criterion -4.920286

Sum squared resid 0.064962 Schwarz criterion -4.881185

Log likelihood 385.7823 Hannan-Quinn criter. -4.904405

F-statistic 165.4801 Durbin-Watson stat 2.177215

Prob(F-statistic) 0.000000

Breusch-Godfrey Serial Correlation LM Test:

F-statistic 0.821681 Prob. F(2,152) 0.4416

Obs*R-squared 1.668569 Prob. Chi-Square(2) 0.4342

Ramsey RESET Test

Equation: DLOGMSFT_C_DLOGSP Specification: DLOG(MSFT) C DLOG(SP) Omitted Variables: Squares of fitted values

Value df Probability

t-statistic 2.219073 153 0.0280

F-statistic 4.924283 (1, 153) 0.0280

Likelihood ratio 4.941733 1 0.0262

Heteroskedasticity Test: White

F-statistic 0.119699 Prob. F(2,153) 0.8873

Obs*R-squared 0.243711 Prob. Chi-Square(2) 0.8853

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Page 3 of 5

The calculated Jarque-Bera statistic for the least squares estimation of the single-index model is 6.410157 (Prob. = 0.040556).

a Explain the economic rationale underlying the regression equation.

b Interpret the estimated coefficients.

c Discuss the adequacy of the model with respect to

i R

2

ii Serial correlation

iii Functional form

iv Normality

v Heteroscedasticity.

d Predict the value of the return on Microsoft stock if the market return is 2 per cent (or 0.02). Is this forecast likely to be accurate?

2 Explainfourof the following:

a Linear in parameters, and linear in variables

b The method of ordinary least squares (OLS)

c The confidence interval for a slope coefficient

d

e A consistent estimator

f Under the assumptions of the CLRM, OLS estimators are BLUE.

3 Answerallparts of this question. Using daily data for the period 1 March 2010 to 5 April 2012 (532 observations after adjustments), the following multi-index model was estimated by ordinary least squares

ˆ

R_t =0.002+0.902R_M,t +0.103R_O,t+0.001TS_t +0.002RP_t 0.012

(

)

(

0.031

)

(

0.024

)

(

0.002

)

(

0.003

)

(3.1)

( and standard errors are in parentheses)

where is the daily log return on the stock of the American energy multinational ConocoPhillips, is the daily log return on the NYSE

Composite Index, is the daily log return of the Brent crude oil price, is a term structure variable, and is a risk premium variable.

Test the following null hypotheses, explaining carefully in each case the null and alternative hypotheses, the test statistic, degrees of freedom and the critical value of the test statistic.

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Page 4 of 5

a the intercept is zero

b is independent of

c the coefficient on is less than one

d Test the hypothesis that the coefficients on and are both zero. For your information, the following equation was also estimated using the same data and OLS ˆ

R_t =0.0006+0.902R_M,t +0.104R_O,t R2 =0.703 0.0004

(

)

(

0.030

)

(

0.024

)

(3.2)

(Standard errors are in parenthesis.)

4 Answer both parts of this question.

a What is ‘imperfect multicollinearity’ and how might it be detected?

b ‘The theoretical consequences of imperfect

multicollinearity are relatively unimportant but the practical consequences are potentially serious’. Explain and discuss.

5 Answerallparts of this question.

a How might heteroscedasticity arise?

b Explain why heteroscedastic disturbances have consequences for the validity of tandFtests.

c Explain the Park test of heteroscedasticity.

d Given

Y_i =₁+₂X_i+u_i where var

( )

u_i =2X_i2

show how this model can be transformed so that the disturbances have constant variance.

6 Answerallparts of this question.

a What is autocorrelation?

b Why does it matter?

c Explain how the Durbin-Watson test can be used for detecting autocorrelation.

d. For the model

Y_t = +X_t +u_t

u_t =u_t₁+v_t ||<1 v_t ~ IID 0,

( )

2 explain the steps involved in obtaining Cochrane- Orcutt estimates of the unknown parameters.

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Page 5 of 5

7 Answer all parts of this question.

a What is nonnormality?

b What are the consequences for the properties of the OLS estimators if the disturbance terms are not distributed normally?

c How would you examine whether the disturbance terms are distributed normally?

d If there is evidence that the disturbance terms are not distributed normally, what would you do?

8 Answer all parts of this question.

a Explain the characteristics of a ‘good’ econometric model.

b What are the consequences of

i including an irrelevant variable, and

ii using an incorrect functional form?

c How might

i the presence of unnecessary variables, and

ii an incorrect functional form be detected?

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Econometric Principles & Data

Analysis

Unit 1 An Introduction to

Econometrics and Regression

Analysis

Unit Content

This unit provides an introduction to econometrics and regression analysis. It outlines the differences between financial and economic theory and econo-metrics. The unit explains how stochastic relations between variables are different to mathematical relations between variables. It explains how uncer-tainty may be modelled using a disturbance term. The unit introduces the steps involved in an econometric investigation. Unit 1 also introduces you to

Eviews, the econometrics software you will be using for this course.

Learning Outcomes

After studying this unit, the readings, and the exercises, you will be able to discuss and apply the following

• the population regression function

• the sample regression function

• the disturbance (or error) term

• the residual term

• how to use Eviews to open pre-existing text files containing data

• how to create and interpret a scatter plot

• how to obtain summary statistics

• how to create transformations of variables.

Readings for Unit 1

Chapter 1 ‘The Nature and Scope of Econometrics’, from Damodar Gujarati

and Dawn Porter (2010) Essentials of Econometrics,New York:

McGraw-Hill/Irwin.

You will also be asked to read Chapter 1, ‘A Quick Walk Through’, in

Richard Startz (2009) Eviews Illustrated – An Eviews Primer, Irvine

Califor-nia: Quantitative Micro Software. This file will be installed on your computer

when you install your copy of Eviews, and it is accessed via Help in Eviews.

Installation and registration of Eviews

Instructions for installing and registering your copy of the Eviews Student

Edition are in the booklet that comes with the Eviews CD. Instructions to

help you use Eviews to do the Exercises are included in section 1.8 of the

unit.

You must register your copy of Eviews. If you do not, it will stop working 14 days after installation.

Data Files for exercises

You will also be asked to work through exercises, and the data files you need for these are available from the Online Study Centre, in the course area for this study session.

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Unit 1 An Introduction to Econometrics and Regression Analysis

1.1 What is Econometrics?

Welcome to this course. The aim of the course is to give you an introduction to econometric methods or, more specifically, to linear regression, which is the major statistical foundation of econometric work. This course requires that you work with data; we hope you will find this interesting and useful, and that you enjoy the course.

A principal concern of financial and economic theory is relations between variables. In finance, you may have already studied many of these including the capital asset pricing model; arbitrage pricing theory; efficient markets hypothesis; optimal hedging ratios; bid-ask spreads. If you have studied economics, you may be familiar with consumption, investment, and demand for money functions; labour supply and labour demand functions; the expecta-tions-augmented Phillips curve; and many others. You could, in fact, view the whole of economic and finance theory as a set of relations among variables. What is econometrics? Econometrics is concerned with quantifying financial and economic relations. Econometrics is of use in providing numerical estimates of the parameters involved and for testing hypotheses embodied in the theoretical relationships. Broadly defined, econometrics is

… the application of statistical and mathematical methods to the analysis of economic data, with a purpose of giving empirical content to economic theories and verifying them or refuting them.1

This definition is not the only possible one; in fact, in your textbook you will come across a number of definitions, which each puts the emphasis slightly differently. Common to all definitions, however, is the stress on the empirical nature of econometric work: the subject matter of econometrics concerns the interaction of, and confrontation between, theory and data in quantifying economic and financial relationships.

Hence, econometrics is not purely a branch of mathematical economics or mathematical finance. Indeed, mathematical finance or economics need not have any empirical content at all. Econometrics makes use of mathematical methods, but its emphasis is on empirical analysis. However, econometrics is not just a ‘box of tools’ to work with data. It requires, undoubtedly, a good training in statistical techniques, but these techniques need to be situated in an interactive process between theory and the data.

To give empirical content to financial and economic theories and to verify them or refute them, the econometrician is confronted with three types of problems, which are of lesser or no concern to the theorist.

First, in economic or financial theory we develop models out of a priori

reasoning based on relatively simple assumptions. To do this, we abstract from secondary complications by assuming that ‘other things remain equal’

1

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while we investigate the relations between a few key economic or financial variables. In effect, this method reduces to ‘intellectual experimentation’ with causal relations postulated by theory. For example, in demand theory we say that the quantity demanded of a commodity (which is not an inferior good) will fall if its price rises, other things being equal.

This method is fruitful in theory but, unfortunately, in empirical economics and finance the scope for experimentation is severely limited. A researcher cannot alter a commodity’s price (or an asset’s price), holding other things constant, in order to see what happens to demand. In general, financial and economic data are not the outcome of experiments, but rather the product of observational programmes of data gathering and collection in a world where other things are never equal. In econometrics, therefore, we can only resort to careful observation; the basic art of econometric work is more like unravel-ling a complex puzzle than setting up an experiment in a laboratory.

Second, we need to address the difference between deterministic and stochas-tic relationships. This issue arises in a different way in economics and in finance. To make the point, we will explain the distinction between determin-istic and stochastic relationships with an example from economics, and then address it from a financial perspective.

In most economic theory we work with deterministic relationships between

economic variables. Take a simple example: the Keynesian consumption function. In economic theory we assume that if we know the level of aggre-gate real income, consumption will be uniquely determined. That is, for each value of aggregate real income there corresponds a given level of aggregate consumption. This is a convenient device to enable us to work out exact solutions for the interplay between variables within the confines of the assumptions of an economic model.

In reality, however, we do not expect this relationship to be exact: it may be stable perhaps, but it is surely imperfect. Hence, in econometric work we deal with imperfect relationships between variables. It follows that our models cannot be deterministic in nature. We investigate functions between variables that we believe to be reasonably stable, on average, but there will always be a degree of uncertainty about outcomes and conclusions derived from such a model. Econometric modelling requires that we make explicit assumptions

about the character of these imperfections, or disturbances as they are more

commonly labelled. That is, we work with stochastic variables and we need to model their stochastic nature. This is what makes us enter the areas of probability theory and statistical inference and estimation.

How does the distinction between deterministic and stochastic relationships arise in finance? Uncertainty is a fundamental element of risk, and the measurement and management of risk are central aspects of finance. To demonstrate this, consider the single-index model (which you will examine and estimate in Unit 2). In the single-index market model the return on a company stock is considered to be a function of three elements. There is a fixed element which is specific to the company. There is also a deterministic

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relationship between the return on the company stock and the return on a relevant market index: For each value of the return on the market index there corresponds a given value for the return on the company stock. (This part of the model captures the concept of market-determined risk.) In addition, the return on the company stock is explained by a company-specific disturbance or error. (The specific error captures the concept of company-specific risk.) The single-index model includes the company-company-specific disturb-ance not just to make the model more realistic; it is included because we specifically want to understand the stochastic nature of the return on the company stock, and thus get a better understanding of the risk associated with the stock.

Third, in financial and economic models we work with theoretical variables.

Econometrics, in contrast, deals with observed data. Obviously, there is a

certain correspondence between them; data collection is inspired by theoreti-cal frameworks. For example, national income account data were constructed after the ascendancy of Keynesian economics, which concerns the analysis of theoretical aggregates such as output, demand, employment and the price level. However, observed variables do not fully correspond to their theoreti-cal equivalents because of errors in measurement, conceptualisation and coverage. This is usually less of a problem for econometrics applied in a financial context than it is for economics. Financial data on asset prices, for example, is more closely related to the actual transactions taking place, so measurement error is less likely. However, we should be aware that move-ments in financial data may be the result of the particular operating or

reporting features of a market, say, in addition to the desired trading activities of the participants that our theories suggest. In econometrics we need to be aware of the nature of the observed data and its implications for investigating theoretical propositions.

These three elements:

• the fact that we cannot hold other things constant in empirical analysis

• the imperfect nature of relationships between variables and

• the discrepancies between theoretical variables and observed data

give econometrics its distinctive flavour. We cannot move straight from a financial or an economic model (as formulated by theory) to the data before we come to terms with these issues. Econometric methods, therefore, aim to address these issues so as to enable us to engage in meaningful investigation of economic and financial theories.

Note that we talk about methods and, hence, emphasise the need for method-ological groundwork to approach these types of problems. There are no hard and fast rules to deal with them. There is not a box of magic tricks, which always work and give us straight answers. Rather, we are left with the task of studying methodological approaches to issues, which are complex, varied, but challenging.

This course, Econometric Principles and Data Analysis, deals with

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data never behave exactly as our theoretical models would lead us to believe. Theoretical models are useful abstractions, which provide the applied re-searcher with analytical handles to make sense of an often bewildering economic and financial reality. Good theory allows us to search for patterns within the data and to give meaning to such patterns. But we need to disen-tangle these patterns in the middle of a great deal of chance variations and uncertainties of outcomes, which our theories could not possibly aim to explain. Regression analysis provides us with an analytical framework to handle relations between variables, especially between variables whose relation is imperfect.

Indeed, regression analysis seeks to establish statistical regularities among

observed variables. To do this, we need to come to terms with the uncertainty inherent in the behaviour of our data. For this, we need to equip ourselves with statistical theory which allows us to model uncertainty as part of

relations between variables. This is the purpose of this course, Econometric

Principles and Data Analysis, of which this is the first unit. The following are the main points to remember.

• In econometrics we pose the question how to confront theory with data

so as to quantify our financial and economic relationships, to verify them or to refute them.

• In practice, we deal with imperfect relationships between variables

which we can only observe (with errors and, often, through proxies) in

a context which we do not control (we cannot experiment).

• It follows that we can only resort to careful observation of complex

phenomena in order to check our theories against the empirical evidence. This raises questions about econometric methods:

methodological issues about gathering and evaluating such empirical evidence. Whatever conclusions we draw in such a context will always involve a considerable degree of uncertainty, even if our models are correctly specified. For this reason, we resort to probability theory and statistical inference to deal with uncertainty in assessing outcomes and conclusions of empirical analysis.

• Since our concern is primarily with investigating relations between

variables, regression analysis constitutes the major tool of statistical analysis in econometrics.

1.2 How to Use the Course Texts

It is quite possible that you are worried about studying econometrics. After all, it involves working with mathematics and statistics, and you may feel that this is not one of your greatest strengths. Alternatively, you may be one of those who welcome this greater emphasis on mathematics and statistics. Whichever view you hold, it is useful to be aware of a particular problem that invariably arises when studying econometrics.

Teaching and learning econometrics almost inevitably involves a preoccupa-tion with technical details: definipreoccupa-tions of technical terms, mathematical

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derivations, step by step descriptions of statistical procedures etc., all phrased in technical notation. This is normal and, indeed, necessary. But this pre-occupation with technical detail often implies that students lose a perspective on ‘What is it all about?’ or ‘Why are we doing this?’ That is, there is a need to keep a focus on the kind of basic questions, uncluttered by notation and technical detail, which give substance to the subsequent technical exercises. We need to get an overview of a problem before we explore it aided by our technical skills. We need to know the simple questions and intuitive insights which often prompted elaborate technical enquiries.

For this reason, the course texts will always start with a section on ideas or

issues.

The purpose of this is to explain in simple words, with the minimum of technical notation, the basic substance of the unit. The aim is to give you an intuitive feel for the subject matter before going into technical detail. If you feel that mathematics and statistics are not your strongest subjects, this regular section will give you a few ‘analytical handles’ to hold on to when studying relevant techniques.

But, alternatively, if you are confident with mathematics and statistics, it is

important not to skip this section. Technical expertise is not just a question of

one’s ability to work out the steps in a technical procedure or to understand a mathematical derivation. It also involves understanding the type of questions a technique tries to address as well as the assumptions on which it is based. Good technical expertise is more than understanding a set of technical skills (narrowly defined); it also involves analytical insights and judgement of the appropriateness of particular technical procedures in specific conditions. The section on ideas or issues will be self-contained; no references will be made to reading parts of the assigned textbook. Take your time to read it carefully, and to reflect whether you understand the type of questions which will be addressed subsequently in technical detail: ‘get familiar with the forest before you start looking at the trees’. In other words, use this section to provide you with the ‘analytical handles’ to facilitate the study of the relevant techniques.

Next, the course texts will have a reading section, or Study Guide, which

guides your study of the textbook, Gujarati and Porter’s Essentials of

Econometrics. The purpose of these sections is to structure your reading of the textbook as well as to provide brief comments, elaborations and cross-references to exercises and examples, and to suggest short cuts in coping with the material.

The section after that will normally contain one example. This section has two purposes. Firstly, the example highlights a specific aspect of the topic under study in a particular unit of the course. Secondly, the example also tries to give you a bit of the flavour of econometrics in action. Generally, you will be asked to participate in the analysis of the example. The examples aim to highlight the links between economic theory and empirical investigation, and try to illustrate the problems that can arise when we work with real data.

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The next section will provide a brief summary of the main issues raised in the

unit. This will be followed by a section of exercises. It is most important that

you work through all of these exercises. The exercises have three purposes:

• to check your understanding of basic concepts and ideas

• to verify your ability to use technical procedures in practice and

• to develop your skills in interpreting the results of empirical analysis.

The final section of the units will include brief answers to these exercises, which you should not look at until after you’ve worked out the answers for yourself!

You will be using Eviews to do the econometric exercises, and this unit

has an additional section describing this program, which is a widely used

econometrics software package. Instructions to use Eviews will accompany

the exercises, where necessary.

This basic structure of the course texts will be maintained throughout your

study of this course. The section on ideas or issues gives you an overview of

the topic of the unit, using non-technical language. The core of the course

text is the study guide. This guides you through your reading of the textbook

and refers you to the exercises whenever appropriate. The example in each

unit demonstrates a problem dealt with in the course material using real data. By using examples drawn from areas of finance, using real data, this section also aims to provide cross-references to the theory courses.

The summary draws your attention to the main points made in the unit. The

exercises are important and you should always work through them. The exercises will help you to understand the course material. In addition, the knowledge and experience you gain from doing the exercises will help you to write assignments and answer examination questions.

1.3 Ideas – The Concept of Regression

The remainder of this unit will deal with the introduction to regression analysis. As you will see, it is structured along the pattern outlined above.

1.3.1 What is regression?

Regression is the main statistical tool of econometrics. What is regression? Broadly speaking,

… regression methods bring out relations between variables, especially between variables whose relations are imperfect in that we do not have one Y for each X.2

But what do we mean by imperfect relations?

An example may help. Consider the relation between corporate bond spreads

(this is the Y-variable) and the earnings before interest of companies (this is

2

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the X-variable). The spread for a corporate bond is the difference between the

interest rate on the corporate bond and the interest rate on government bonds of equivalent maturity. Interest rates on corporate bonds are higher than those on government bonds to reflect expected default loss, different tax treatments and the riskier return associated with corporate bonds. We would expect that a company with higher earnings before interest would be less likely to default, and hence the bond spread for that company would be lower. Hence, we expect that, on average, the corporate bond spread is inversely related to earnings before interest. But we do not expect this relation to be perfect. That is, if we were to sample 10 companies with identical earnings

before interest (i.e. equal X-values), we would not expect to get 10 identical

corporate spreads (the Y-values). Differences between the markets in which

the firms operate, in management and in other financial variables (e.g. coupon rates, coverage ratios) will account for differences in bond spreads. But, importantly, it is still valid to say that, on average, the bond spread declines as the level of earnings before interest increases. That is what Mosteller and Tukey (quoted above), mean when they say that a relation exists between two variables but that it is imperfect in that we do not have

one Y for each X.

This leads us to the discussion of the concept of regression. Regression

methods aim to bring out this average relation between a dependent variable

on the one hand and one or more independent variables on the other. In our example the average inverse relation between the bond spread and the level

of earnings before interest is the regression of the former variable on the

latter. But, obviously, there will be variation in how markets view the bonds of individual companies that have broadly the same earnings.

In fact, anyone familiar with data analysis knows very well that we can always take an average of one or another aspect of a number of individuals, but we rarely meet the ‘average individual’. So it is also with regression as an average relation: individual observations will rarely conform to the average

relationship between Y and X. Hence, in regression analysis we seek to

establish statistical regularities in the middle of a great deal of chance vari-ation and uncertainty in outcomes. For this reason, regression methods

involve statistical modelling of the chance variation in the data as well as of

the average relationship.

In summary, we hope that our model captures the basic structure of interac-tion between economic and financial variables, and we expect that the

behavioural relations are reasonably stable, but imperfect. At most, we expect these relations to hold ‘on average’. In other words, we seek to discover structure and regularity within data in the middle of a great deal of

uncer-tainty in outcomes. It is similar to separating sound from noise when trying to

listen to a badly tuned radio.

Therefore, a regression model embraces two components:

• a regression line (which defines the basic structure) and

Econometric Principles and Data Analysis

Econometric Principles

and Data Analysis

Econometric Principles & Data

Analysis

Course Introduction and Overview

Contents

1 Course

Objectives

2

The Course Authors

3

The Course Structure

4 Learning

Outcomes

5 Study

Materials

6 Assessment

The grading scheme

Specimen exam papers

Further information

Centre for Financial and Management Studies

MSc Examination

Econometric Principles and Data Analysis

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Econometric Principles & Data

Analysis

Unit 1 An Introduction to

Econometrics and Regression

Analysis

Contents

Unit Content

Learning Outcomes

Readings for Unit 1

1.1 What is Econometrics?

1.2 How to Use the Course Texts

1.3 Ideas – The Concept of Regression

1.3.1

What is regression?