Modelling Trends and Cycles in Economic Time

Modelling trends and cycles in economic time series has a long history, with the use of linear trends and moving averages forming the basic tool kit of economists until the 1970s. Several developments in econometrics then led to an overhaul of the techniques used to extract trends and cycles from time series. Terence Mills introduces these various approaches to allow students and researchers to appreciate the variety of techniques and the considerations that underpin their choice for modelling trends and cycles.

Terence C. Mills is Professor of Applied Statistics and Econometrics, Department of Economics, Loughborough University, UK.

List of Tables and Figures

ix

Series Editor's Foreword

xi

1 Introduction

1

(12)

1.1 Historical perspective

1

(9)

1.2 Overview of the book

10

(3)

2 'Classical' Techniques of Modelling Trends and Cycles

13

(26)

2.1 The classical trend-cycle decomposition

13

(1)

2.2 Deterministic trend models

14

(4)

2.2.1 Linear trends

14

(1)

2.2.2 Nonlinear trends

15

(1)

2.2.3 Segmented trends and smooth transitions

16

(2)

Example 2.1 A linear trend for US output per capita

18

(1)

Example 2.2 Fitting deterministic trends to UK output

19

(3)

Example 2.3 Fitting a smooth transition to US stock prices

22

(1)

2.3 Estimating trends using moving averages

23

(2)

2.3.1 Simple moving averages

23

(1)

2.3.2 Weighted moving averages

24

(1)

Example 2.4 Weighted moving average trends for US stock prices

25

(1)

2.4 The cyclical component

26

(4)

2.4.1 Autoregressive processes for the cyclical component

26

(4)

2.4.2 Estimating the cyclical component

30

(1)

Example 2.5 The cyclical component of US output per capita

30

(1)

Example 2.6 The cyclical component of UK output

31

(1)

Example 2.7 The cyclical component of US stock prices

32

(1)

2.5 Some problems associated with the 'classical' approach

33

(1)

Example 2.8 Simulating the Slutsky-Yule effect

34

(2)

Further reading and background material

36

(3)

3 Stochastic Trends and Cycles

39

(36)

3.1 An introduction to stochastic trends

39

(9)

3.2 Determining the order of integration of a time series

48

(2)

3.3 Some examples of ARIMA modelling

50

(1)

Example 3.1 Modelling the dollar/sterling exchange rate

50

(1)

Example 3.2 Modelling the FTA All Share index

50

(2)

Example 3.3 Modelling US output

52

(2)

3.4 Trend stationarity versus difference stationarity

54

(4)

Example 3.4 Testing TS versus DS processes for UK equities and US output

58

(1)

3.5 Unobserved component models and signal extraction

59

(5)

3.5.1 Unobserved component models

59

(2)

3.5.2 The Beveridge-Nelson decomposition

61

(3)

Example 3.5 Decomposing UK equities and US output

64

(6)

3.5.3 Signal extraction

66

(1)

3.5.4 Basic structural models

67

(3)

Example 3.6 Basic structural models for UK equities and US output

70

(2)

Further reading and background material

72

(3)

4 Filtering Economic Time Series

75

(28)

4.1 Detrending using linear filters

75

(1)

4.1.1 Symmetric linear filters

75

(1)

Example 4.1 The simple MA filter

76

(5)

4.1.2 Frequency-domain properties of linear filters

77

(4)

Example 4.2 The gain and phase of the simple MA and first differencing filters

81

(1)

Example 4.3 Spurious cyclical behaviour

82

(9)

4.1.3 Designing a low-pass filter

84

(3)

4.1.4 High-pass and band-pass filters

87

(4)

Example 4.4 Band-pass business cycle filters

91

(1)

4.2 The Hodrick-Prescott filter

92

(5)

4.2.1 The Hodrick-Prescott filter in infinite samples

92

(2)

4.2.2 The finite sample H-P filter

94

(1)

4.2.3 Optimising the smoothing parameter

95

(2)

Example 4.5 Band-pass and Hodrick-Prescott cycles for US output

97

(1)

Example 4.6 H-P filtering of UK output

97

(1)

4.3 Filters and structural models

97

(4)

4.3.1 A structural model for the H-P filter

97

(2)

4.3.2 The Butterworth filter

99

(2)

Further reading and background material

101

(2)

5 Nonlinear and Nonparametric Trend Modelling

103

(20)

5.1 Regime shift models

103

(4)

5.1.1 Markov models

103

(4)

Example 5.1 A Markov switching model for US output

107

(5)

5.1.2 STAR models

107

(5)

Example 5.2 A STAR model for UK output

112

(2)

5.2 Nonparametric trends

114

(3)

5.2.1 Smoothing estimators

114

(3)

5.2.2 Kernel regression

117

(1)

5.2.3 Local polynomial regression

Example 5.3 nonparametric trends in US stock prices

117

(2)

5.3 Nonlinear stochastic trends

119

(1)

Further reading and background material

120

(3)

6 Multivariate Modelling of Trends and Cycles

123

(26)

6.1 Common features in time series

123

(4)

6.1.1 Specification and testing for common features

123

(2)

6.1.2 Common cycles and codependence

125

(2)

Example 6.1 Is there a common cycle in US and Canadian output?

127

(1)

6.1.3 Common deterministic trends

127

(1)

Example 6.2 Common trends in the Victorian economy

128

(1)

6.2 Common trends and cycles in a VAR framework

128

(6)

Example 6.3 Common trends and cycles in the UK macroeconomy

134

(3)

6.3 Multivariate basic structural models

137

(2)

6.3.1 Common trends

137

(2)

6.3.2 Common cycles

139

(1)

Example 6.4 A multivariate BSM of the UK macroeconomy

139

(3)

6.4 Multivariate filtering

142

(2)

Example 6.5 Multivariate detrending of UK consumption and output using a common trend restriction

144

(1)

Further reading and background material

145

(4)

7 Conclusions

149

(2)

Computed Examples

151

(10)

References

161

(10)

Author Index

171

(2)

Subject Index

173

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Modelling Trends and Cycles in Economic Time Series

Summary

Author Biography

Table of Contents

Supplemental Materials