Modeling Monthly Inflation Rate Volatility in Kenya Using Arch Type Family Models

64 Pages·2016·1.63 MB·English

UNIVERSITY OF NAIROBI
SCHOOL OF MATHEMATICS

MODELING MONTHLY INFLATION RATE VOLATILITY IN KENYA USING
ARCH TYPE FAMILY MODELS

OKEYO JOHNSON OTIENO
I56/75322/2014

A RESEARCH PROJECT SUBMITTED IN PARTIAL FULFILMENT OF THE
REQUIREMENTS FOR THE DEGREE OF MASTERS OF SCIENCE IN ACTUARIAL
SCIENCE OF THE UNIVERSITY OF NAIROBI

JULY 2016
DECLARATION
I declare that this research project is my own original work and that to the best of my
knowledge has not been submitted before to any other institution for examination:
Signature…………………………. Date…………………………
Okeyo Johnson Otieno
I56/75322/2014
DECLARATION BY THE SUPERVISOR
This research project has been submitted with our approval as the University Supervisor:
Signature …………………………….. Date………………………..
Dr. Philip Ngare
Department of Actuarial Science
School of Mathematics
University of Nairobi

Signature…………………………….. Date………………………..
Dr. Joseph Mwaniki
Department of Actuarial Science
School of Mathematics
University of Nairobi
ii
DEDICATION
This Research Project is dedicated to my wife Faith Moraa and my daughter Joy Adrienne
for the support to the completion of the project.

iii
ACKNOWLEDGEMENT
This research project was made possible through the help and support from a number of
people, including: parents, lecturers, family, and friends. Special dedication of
acknowledgment and gratitude to my supervisors for significant advice and contributions.

iv
LIST OF ABBREVIATIONS
ACF -Autocorrelation Function
AIC -Akaike Information Criterion
AR - Autoregressive
ARCH -Autoregressive Conditional Heteroscedasticity
BIC -Bayesian Information Criterion
CBK -Central Bank of Kenya
CEE - Central and Eastern Europe
GARCH -Generalized Autoregressive Conditional Heteroscedasticity
HQIC -Hannan-Quinn Information Criterion
KNBS -Kenya National Bureau of Statistics
PACF -Partial Autocorrelation Function
SIC -Shwartz Information Criterion
UCL - Upper Class Limit
LCL - Lower Class Limit

v
LIST OF SYMBOLS
 -Error
t
y -Time series observation
t
y2 -Squared observation
t
 -Set of observation
t
r -Return
t
 -Constant part of return
,, -Model parameters
0 1
z -Identically and Independent Standard Normal Random Variable
t
 -Standard deviation
t
2 -Conditional variance
t
 -volatility estimate
b -Volatility coefficient
i
I -Inflation rate
i

vi
TABLE OF CONTENTS

DECLARATION ............................................................................................................................................ ii
DEDICATION ............................................................................................................................................... iii
ACKNOWLEDGEMENT ............................................................................................................................. iv
LIST OF ABBREVIATIONS ......................................................................................................................... v
LIST OF SYMBOLS ..................................................................................................................................... vi
TABLE OF CONTENTS ............................................................................................................................. vii
LIST OF TABLES .......................................................................................................................................... x
LIST OF FIGURES ....................................................................................................................................... xi
ABSTRACT ................................................................................................................................................. xii
CHAPTER ONE ............................................................................................................................................. 1
INTRODUCTION OF THE STUDY ............................................................................................................. 1
1.0 Background of the Study .......................................................................................................................... 1
1.2 Inflation rate volatility .............................................................................................................................. 2
1.3 Statement of the Problem .......................................................................................................................... 3
1.4 Objectives of Study................................................................................................................................... 4
1.4.1 General Objective .................................................................................................................... 4
1.4.2 Specific Objectives .................................................................................................................. 4
1.5 Significance of Study ................................................................................................................................ 4
CHAPTER TWO ............................................................................................................................................ 6
LITERATURE REVIEW ............................................................................................................................... 6
2.0 Introduction .............................................................................................................................................. 6
2.1 Review of Previous Studies ...................................................................................................................... 6
CHAPTER THREE ........................................................................................................................................ 9
RESEARCH METHODOLOGY ................................................................................................................... 9
3.0 Introduction .............................................................................................................................................. 9
vii
3.1 Financial Time series Models ................................................................................................................... 9
3.2: The ARCH Model ................................................................................................................................... 9
3.1.1: Fitting Procedure for ARCH model .................................................................................................... 12
3.3: The GARCH Model............................................................................................................................... 14
3.3 GARCH Model Extensions .................................................................................................................... 17
3.3.1 Symmetric GARCH ............................................................................................................... 17
3.3.2 Asymmetric GARCH Measurement ..................................................................................... 18
3.4 The E-GARCH ....................................................................................................................................... 18
3.5 Glosten-Jagannathan-Runkle (GJR) model: ........................................................................................... 19
3.6 Forecast of Conditional Variance in GARCH model ............................................................................. 20
3.7 Conditional Error distributions ............................................................................................................... 21
3.7.1 Normal distribution................................................................................................................ 21
3.7.2 Student t distribution ............................................................................................................. 21
3.7.3 Generalized error distribution ................................................................................................ 22
3.8 Model selection criteria .......................................................................................................................... 22
CHAPTER FOUR ........................................................................................................................................ 24
DATA ANALYSIS AND RESULTS........................................................................................................... 24
4.0 Introduction ............................................................................................................................................ 24
4.1 Data ......................................................................................................................................................... 24
4.2 Exploratory Data Analysis. ..................................................................................................................... 25
4.3 Inflation Rate Returns Data .................................................................................................................... 26
4.3.1 Simple and log Returns time series plot ................................................................................ 26
4.3.2 Simple and Log returns Descriptive Statistics ....................................................................... 27
4.3.3 THE ACF AND PACF PLOTS ............................................................................................. 28
4.3.4 ARCH Effect Tests ................................................................................................................ 28
4.5 GARCH model. ...................................................................................................................................... 29
4.5.1 GARCH QQ Distribution Plots ............................................................................................. 29
4.6 GJR GARCH model. .............................................................................................................................. 31
4.6.1 GJR GARCH (1,1) distribution QQ plots. ............................................................................ 31
viii
4.6.2 GJR GARCH with Error distribution. .................................................................................. 32
4.7 EGARCH model. .................................................................................................................................... 34
4.7.1 E-GARCH (1,1) with Error distribution QQ Plots. ............................................................... 34
4.7.2 E-GARCH (1,2), E-GARCH (2,1), E-GARCH (2,2) Error distribution QQ plots. ............. 35
4.9 THE SIMULATION. .............................................................................................................................. 37
4.9.1 Conditional SD Simulation Density ...................................................................................... 37
4.9.2 Return Series Simulation Path Density ................................................................................. 39
4.10 Residual Analysis of the Models .......................................................................................................... 40
CHAPTER FIVE .......................................................................................................................................... 44
SUMMARY CONCLUSION AND RECOMMENDATION ...................................................................... 44
5.1 Summary ................................................................................................................................................. 44
5.2 Conclusion .............................................................................................................................................. 44
5.3 Recommendation .................................................................................................................................... 45
REFERENCES ............................................................................................................................................. 46
APPENDIX: R CODES ................................................................................................................................ 48

ix
LIST OF TABLES

Table 4.1: Simple and log Returns Descriptive Statistics ............................................................................. 27
Table 4.2: Ljung box test for Log Returns at different lags. ......................................................................... 28
Table 4.3: The AIC and Log Likelihood distributions of GARCH .............................................................. 30
Table 4.4: The AIC and Log Likelihood for GJR-GARCH with residual distribution. .............................. 33
Table 4.5: E-GARCH with residual distribution. ........................................................................................ 36
Table 4.6: The comparison of GARCH (1,1), GJRGARCH (1,1) & E-GARCH (1,1). ............................... 37
Table 4.7: E-GARCH (1,1) with GED parameter estimates. ........................................................................ 40
Table 4.8: E-GARCH (1,1) with GED distribution Weighted Ljung-Box Test Residuals ........................... 41
Table 4.9: E-GARCH (1,1) with GED conditional distribution Weighted ARCH LM Tests results. .......... 41

x