A Non-Linear Regime Switching Models in Financial Series with Two Regimes

ABSTRACT In this study, two economic series which have changes in regimes were considered. Models considered for the two series are Simple Switching Mixture (SSM) model and Markov Switching Autoregressive (MS-AR) model. Predictions of future transition regime probabilities were performed using the Hamilton filter of m-period transition matrix for MS-AR model, while, the two state ergodic m-step ahead transitions probabilities for SSM model. Subsequently, forecast evaluation measures for the two models were carried out with Mean Absolute Percentage Error (MAPE) and Root Mean Square Error (RMSE). Consumer Price Index (CPI), had a better forecast with SSM model while, Nominal Effective Exchange Rate (NEER) had a better forecast with the MS-AR model.

TABLE OF CONTENT

TITLE PAGE… … …. ….. …. …. …. …. ….… ……. ..i

APPROVAL PAGE… … …. ….. …. …. …. …. ….… ii

CERTIFICATION … …. …. … …. …. …. …. …. iii

DEDICATION …. …. …. …. …. …. …. …. …. iv

ACKNOWLEDGEMENT …. …. …. …. …. …. …. …. v

ABSTRACT …. …. …. …. …. …. …. …. …. ... vi

TABLE OF CONTENTS … …. …. …. …. …. …. …. vii

CHAPTER ONE : INTRODUCTION. ... ... ... ... ... .... …. 1

1.1 Overview ... ... ... ... ... ... .... …. … 1

1.2 Statement of Problem ... ... ... ... ... ... ....... …. 3

1.3 Objectives of the Study ... ... ... ... ... ….. . .... 4

1.4 Significance of the Study ... ... ... ... ... .... …. 4

1.5 Scope of the Study ... ... ... ... ... .... ...... 4

CHAPTER TWO: LITERATURE REVIEW ... ... ... .... ... 5

2.1 Introduction ... ... ... ... ... ... ... .… …. 5

CHAPTER THREE: METHODOLOGY... ... ... ... …... 10

3.1 Introduction... ... ... ... ... ... ... ... .. 10

3.2 The Data ... ... ... ... ... ... ... ….. ….. 10

3.3 Model Identification …… ……. ……. …… …… …… …… ….. …. 10

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3.3.1 Data Stationarity... ... ... ... ... ... .. ... …. 10

3.3.2 CUSUM Test... ... ... ... …………………………. …. 11

3.3.3 Unit Root Test ….. ……..... ... ... ... ... ... …. 12

3.4 Regime Switching Models ……………………………………... …. 12

3.4.1 Assumptions of Regime Switching Models ... ... … 12

3.4.2 Simple Switching Mixture (SSM) Model .... ... … 13

3.4.2.1 Maximum Likelihood Estimation of Mean, Variance and

 Unconditional Probabilities of SSM model... ... ... ..... 14

3.4.2.2 Forecast using matrix of m-period ahead Transition

 Probability for an ergodic two-state Markov Chain…………… 15

3.4.3 Markov Switching Autoregressive (MS-AR)Model... ... .. ... …. 16

3..4.3.1 Properties of Markov Switching Autoregressive Model... …. 16

3.4.3.2 Estimation of Markov Switching Autoregressive Model using

 Hamilton’s Filter Algorithm... .… …………………………….. 18

3.4.3.3 Prediction of Future Regime Probabilities... ... .….. …. … 19

3.4.4 Model Comparison and Forecast Evaluation ... .... ... . … 20

CHAPTER FOUR Application of the Models to Financial Series…………. 21

4.1 Introduction... ... ..... .... .... ....... .. 21

4.2 Numerical Diagnostics Tests ... .... ... ... …. …. 21

4.2.1 Graphical Representation ... .... ... ... ……………. 21

4.2.2 CUSUM Test... ... ... ... ... ... ……………. …. 22

4.2.3 Unit Root Test.... .... ... ... ... ... ……………. …. 23

4.3 Parameter Estimation.... ... ... ... ... ... ... …. 23

4.3.1 Simple Switching Mixture Model... ... ... ... ... …. 23

4.3.1.1 Inference for Consumer Price Index (CPI)……………………. 24

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4.3.1.2 Prediction of future Regimes using m-period ahead

Transition Probabilities matrix for CPI ... ... …. …. … .26

4.3.1.3.Inference for Nominal Effective Exchange Rate (NEER)... …. 27

4.3.1.4 Prediction of future Regimes using m-period ahead

Transition Probabilities matrix for NEER... ... ….... …. … 28

4.3.2 Markov Switching Autoregressive Model ... ... ……….. …. 29

4.3.2.2 Expected Duration in a Particular Regime … … ,,, .. ... ... ………. 30

4.3.2.3The Constant, Variance and the autoregressive coefficient… ……….. 30

4.3.2.4 Inference using Hamilton’s Filter… … ,,, .. ... ... …………….. 32

4.3.2.5 Forecast of future Regimes using Hamilton’s Filter Algorithm… … 34

4.4 Discussion of Results … …. …. …. …. …. 35.

CHAPTER FIVE: Summary, Recommendation and Conclusion... ... ... 38

5.1 Summary... ... ... ... ... ... ... ... ... 38

5.2 Recommendation... ... ... ... ... ... ... 38

5.3 Conclusion... ... ... ... ... ... ... ... ... ... ... 39

References … … … … … … … … … 40

Appendices

Appendices A: Critical Values for Dickey-Fuller Unit Root t-Test … 43

Appendices B: MS-AR Models for CPI and NEER…. …. …. 44

Appendices C: SSM Models for CPI and NEER …. …. …. 46

Appendices D: Summary Table 48