Table Of ContentApplied Time Series Analysis
Applied Time Series
Analysis
A Practical Guide to Modeling and
Forecasting
Terence C. Mills
Loughborough University,Loughborough, UnitedKingdom
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Contents
Introduction ix
1. Time Series and Their Features
Autocorrelation and Periodic Movements 2
Seasonality 4
Stationarity and Nonstationarity 4
Trends 6
Volatility 8
Common Features 9
Time Series Having Natural Constraints 10
Endnotes 12
2. Transforming Time Series 13
Distributional Transformations 13
Stationarity Inducing Transformations 20
Decomposing a Time Series and Smoothing Transformations 23
Endnotes 30
3. ARMA Models for Stationary Time Series 31
Stochastic Processes and Stationarity 31
Wold's Decomposition and Autocorrelation 33
First-Order Autoregressive Processes 35
First-Order Moving Average Processes 36
General AR and MA Processes 37
Autoregressive-Moving Average Models 43
ARMA Model Building and Estimation 46
Endnotes 55
4. ARIMA Models for Nonstationary Time Series 57
Nonstationarity 57
ARIMA Processes 60
ARIMA Modeling 65
Endnotes 68
v
vi Contents
s.
Unit Roots, Difference and Trend Stationarity,
and Fractional Differencing 71
Determining the Order of Integration of a Time Series 71
Testing for a Unit Root 73
Trend Versus Difference Stationarity 77
Testing for More Than One Unit Root 81
Other Approaches to Testing for a Unit Root 83
Estimating Trends Robustly 87
Fractional Differencing and Long Memory 90
Testing for Fractional Differencing 93
Estimating the Fractional Differencing Parameter 96
Endnotes 101
6. Breaking and Nonlinear Trends 103
Breaking Trend Models 103
Breaking Trends and Unit Root Tests 105
Unit Roots Tests When the Break Date Is Unknown 110
Robust Tests for a Breaking Trend 111
Confidence Intervals for the Break Date and Multiple Breaks 112
Nonlinear Trends 112
Endnotes 119
7. An Introduction to Forecasting With Univariate
Models 121
Forecasting With Autoregressive-Integrated-Moving Average
(ARIMA) Models 121
Forecasting a Trend Stationary Process 128
Endnotes 130
8. Unobserved Component Models, Signal Extraction,
and Filters 131
Unobserved Component Models 131
Signal Extraction 136
Filters 139
Endnotes 144
9. Seasonality and Exponential Smoothing 145
Seasonal Patterns in Time Series 145
Modeling Deterministic Seasonality 145
Modeling Stochastic Seasonality 147
Mixed Seasonal Models 152
Seasonal Adjustment 153
Exponential Smoothing 153
Endnotes 159
Contents vii
10. Volatility and Generalized Autoregressive
Conditional Heteroskedastic Processes 161
Volatility 161
Autoregressive Conditional Heteroskedastic Processes 163
Testing for the Presence of ARCH Errors 165
Forecasting From an ARMA-GARCH Model 168
Endnotes 171
11. Nonlinear Stochastic Processes 173
Martingales, Random Walks, and Nonlinearity 173
Nonlinear Stochastic Models 176
Bilinear Models 177
Threshold and Smooth Transition Autoregressions 181
Markov-Switching Models 185
Neural Networks 188
Nonlinear Dynamics and Chaos 189
Testing for Nonlinearity 192
Forecasting With Nonlinear Models 198
Endnotes 199
12. Transfer Functions and Autoregressive Distributed
Lag Modeling 201
Transfer Function-Noise Models 201
Autoregressive Distributed Lag Models 203
Endnotes 210
13. Vector Autoregressions and Granger Causality 211
Multivariate Dynamic Regression Models 211
Vector Autoregressions 212
Granger Causality 213
Determining the Lag Order of a Vector Autoregression 213
Variance Decompositions and Innovation Accounting 216
Structural Vector Autoregressions 222
Endnotes 230
14. Error Correction, Spurious Regressions, and
Cointegration 233
The Error Correction Form of an Autoregressive Distributed
Lag Model 233
Spurious Regressions 234
Error Correction and Cointegration 242
Testing for Cointegration 247
Estimating Cointegrating Regressions 250
Endnotes 253
viii Contents
15. Vector Autoregressions With Integrated Variables,
Vector Error Correction Models, and Common Trends 255
Vector Autoregressions With Integrated Variables 255
Vector Autoregressions With Cointegrated Variables 257
Estimation of Vector Error Correction Models and Tests
of Cointegrating Rank 260
Identification of Vector Error Correction Models 264
Structural Vector Error Correction Models 266
Causality Testing in Vector Error Correction Models 268
Impulse Response Asymptotics in Nonstationary VARs 269
Vector Error Correction Model-X Models 271
Common Trends and Cycles 274
Endnotes 279
16. Compositional and Count Time Series 281
Constrained Time Series 281
Modeling Compositional Data 281
Forecasting Compositional Time Series 283
Time Series Models for Counts: The IN-AR(1) Benchmark Model 288
Other Integer-Valued ARMA Processes 289
Estimation of Integer-Valued ARMA Models 290
Testing for Serial Dependence in Count Time Series 291
Forecasting Counts 293
Intermittent and Nonnegative Time Series 296
Endnotes 296
17. State Space Models 299
Formulating State Space Models 299
The Kalman Filter 303
ML Estimation and the Prediction Error Decomposition 305
Prediction and Smoothing 307
Multivariate State Space Models 308
Endnotes 309
18. Some Concluding Remarks 311
Endnotes 313
References 315
Index 329
Introduction
0.1 Data taking the form of time series, where observations appear sequen-
tially, usually with a fixed time interval between their appearance (every
day, week, month, etc.,), are ubiquitous. Many such series are followed
avidly: for example, the Dow Jones Industrial stock market index opened
2017 with a value of 20,504, closing the year on 24,719; a rise of 20.6%,
one of the largest annual percentage increases on record. By January 26,
2018, the index had reached an intraday high of 26,617 before declining
quicklyto close on February 8 at 23,860; a value approximately equal tothat
of the index at the end of November 2017 and representing a fall of 10.4%
from its peak. Five days later, it closed on February 13 at 24,640; 3.3%
above this “local minimum.” By the end of May 2018, the index stood at
24,415;little changedover the ensuing 3 months.
This “volatility,” which was the subject of great media and, of course,
stock market attention, was surpassed by the behavior of the price of the
crypto-currency bitcoin during a similar period. Bitcoin was priced at $995
at the start of 2017 and $13,850 at the end of the year; an astonishing almost
1300% increase. Yet, during December 17, just a fortnight before, it had
reached an even higher price of $19,871; an almost 1900% increase from the
start of the year. The decline from this high point continued into the new
year, the price falling to $5968 on February 6 (a 70% decline from the peak
price less than 2 months prior), before rebounding again to close at $8545 on
February 13. Since then the price has increased to $11,504 on March 4
before falling back to $6635 on April 6. At the end of May 2018, the price
stood at $7393.
0.2 While financial time series observed at high frequency often display
such wildly fluctuating behavior, there are many other time series, often
from the physical world, which display interesting movements over longer
periods. Fig. I.1 shows the decadal averages of global temperatures from the
1850s onward. The behavior of temperature time series, whether global or
regional, have become the subject of great interest and, in some quarters,
great concern, over the past few decades. Fig. I.1 shows why. Global tem-
peratures were relatively constant from the 1850s to the 1910s before
increasing over the next three decades. There was then a second “hiatus”
between the 1940s and 1970s before temperatures began to increase rapidly
ix
