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1 Introduction and Basics........................................................................1
1.1 The Historical Development of Time Series Analysis ...................2
1.2 Graphical Representations of Economic Time Series ....................5
1.3 Ergodicity and Stationarity ...........................................................12
1.4 The Wold Decomposition.............................................................21
References ............................................................................................22
2 Univariate Stationary Processes ........................................................27
2.1 Autoregressive Processes..............................................................27
2.1.1 First Order Autoregressive Processes....................................27
2.1.2 Second Order Autoregressive Processes ...............................40
2.1.3 Higher Order Autoregressive Processes ................................49
2.1.4 The Partial Autocorrelation Function ....................................52
2.1.5 Estimating Autoregressive Processes ....................................56
2.2 Moving Average Processes...........................................................57
2.2.1 First Order Moving Average Processes.................................58
2.2.2 Higher Order Moving Average Processes .............................64
2.3 Mixed Processes ...........................................................................67
2.3.1 ARMA(1,1) Processes ...........................................................67
2.3.2 ARMA(p,q) Processes ...........................................................73
2.4 Forecasting....................................................................................75
2.4.1 Forecasts with Minimal Mean Squared Errors ......................75
2.4.2 Forecasts of ARMA(p,q) Processes.......................................80
2.4.3 Evaluation of Forecasts .........................................................84
2.5 The Relation between Econometric Models and
ARMA Processes..........................................................................87
References ............................................................................................ 88
3 Granger Causality...............................................................................93
3.1 The Definition of Granger Causality ............................................95
3.2 Characterisations of Causal Relations in Bivariate Models..........97
VIII Contents
3.2.1 Characterisations of Causal Relations using the
Autoregressive and Moving Average Representations .........97
3.2.2 Characterising Causal Relations by Using the Residuals
of the Univariate Processes....................................................99
3.3 Causality Tests............................................................................102
3.3.1 The Direct Granger Procedure.............................................102
3.3.2 The Haugh-Pierce Test ........................................................106
3.3.3 The Hsiao Procedure ...........................................................110
3.4 Applying Causality Tests in a Multivariate Setting....................114
3.4.1 The Direct Granger Procedure with More Than Two
Variables .............................................................................114
3.4.2 Interpreting the Results of Bivariate Tests in Systems
With More Than Two Variables .........................................117
3.5 Concluding Remarks ..................................................................118
References ..........................................................................................120
4 Vector Autoregressive Processes .....................................................125
4.1 Representation of the System .....................................................127
4.2 Granger Causality .......................................................................136
4.3 Impulse Response Analysis ........................................................138
4.4 Variance Decomposition ............................................................144
4.5 Concluding Remarks ..................................................................149
References ..........................................................................................150
5 Nonstationary Processes...................................................................153
5.1 Forms of Nonstationarity............................................................153
5.2 Trend Elimination ......................................................................159
5.3 Unit Root Tests...........................................................................163
5.3.1 Dickey-Fuller Tests .............................................................165
5.3.2 The Phillips-Perron Test......................................................171
5.3.3 Unit Root Tests and Structural Breaks ................................176
5.3.4 A Test with the Null Hypothesis of Stationarity .................178
5.4 Decomposition of Time Series ...................................................180
5.5 Further Developments ................................................................187
5.5.1 Fractional Integration ..........................................................187
5.5.2 Seasonal Integration ............................................................189
5.6 Deterministic versus Stochastic Trends in Economic
Time Series .................................................................................191
References ..........................................................................................194
6 Cointegration.....................................................................................199
6.1 Definition and Properties of Cointegrated Processes .................203
Contents IX
6.2 Cointegration in Single Equation Models: Representation,
Estimation and Testing ...............................................................205
6.2.1 Bivariate Cointegration .......................................................205
6.2.2 Cointegration with More Than Two Variables....................208
6.2.3 Testing Cointegration in Static Models ...............................209
6.2.4 Testing Cointegration in Dynamic Models..........................213
6.3 Cointegration in Vector Autoregressive Models ........................218
6.3.1 The Vector Error Correction Representation.......................219
6.3.2 The Johansen Approach.......................................................222
6.3.3 Analysis of Vector Error Correction Models.......................229
6.4 Cointegration and Economic Theory..........................................234
References ..........................................................................................235
7 Autoregressive Conditional Heteroskedasticity .............................241
7.1 ARCH Models ............................................................................245
7.1.1 Definition and Representation.............................................245
7.1.2 Unconditional Moments ......................................................248
7.1.3 Temporal Aggregation.........................................................249
7.2 Generalised ARCH Models ........................................................252
7.2.1 GARCH Models ..................................................................252
7.2.2 The GARCH(1,1) process ...................................................254
7.2.3 Nonlinear Extensions...........................................................257
7.3 Estimation and Testing ...............................................................259
7.4 ARCH/GARCH Models as Instruments of Financial
Market Analysis..........................................................................261
References ..........................................................................................263
Index of Names and Authors ................................................................267
Subject Index..........................................................................................27
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