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1、 CHAPTER11373Cointegration and State Space Modelsn this chapter, we introduce the concepts of cointegrated processes and state space models, as well as the relative estimation methods. State space models were introduced in the engineering literature in the 1960s especially through the work of Rudolf
2、 E. Kalman. Cointegration analy-sis is a more recent econometric tool. The first articles to introduce coin- tegrated models were penned by Engle and Granger in the second half of the 1980s. Though vector autoregressive (VAR) processes and state space mod- els are equivalent representations of the s
3、ame processes, deeper insight into the relationship between state space models and cointegration was gained more recently when it was understood that cointegration implies a reduced number of common stochastic trends. The idea behind cointe- gration that there are feedback mechanisms that force proc
4、esses to stay close together is therefore intimately related to the idea that the behav- ior of large sets of processes is driven by the dynamics of a smaller num- ber of variables. COINTEGRATIONCointegration is one of the key concepts of modern econometrics. Lets start by giving an intuitive explan
5、ation of cointegration and its proper- ties. Two or more processes are said to be cointegratedif they stay close to each other even if they “drift about” as individual processes. A color- ful illustration is that of the drunken man and his dog: Both stumble about aimlessly but never drift too far ap
6、art. Cointegration is an impor-Ic11-Cointegration Page 373 Thursday, October 26, 2006 2:08 PM374FINANCIAL ECONOMETRICStant concept both for economics and financial modeling. It implements the notion that there are feedbacks that keep variables mutually aligned. To introduce the notion of cointegrati
7、on, recall the concepts of station- ary processes and integrated processes. Key Features of CointegrationLets first give an intuitive characterization to the concept of cointegra- tion in the case of two stochastic processes. Cointegration can be under- stood in terms of its three key features: Redu
8、ction of order of integration Regression Common trendsFirst, considerreduction of order of integration. Two or more stochastic processes that are integrated of order one or higher are said to be coin- tegrated if there are linear combinations of the processes with a lowerorder of integration. In fin
9、ancial econometrics, cointegration is usually a property of processes integrated of order one that admit linear combina- tions integrated of order zero (stationary). As we will see, it is also pos-sible to define fractional cointegration between fractionally integrated processes. Second, the concept
10、 of cointegration can be also stated in terms oflinear regression. Two or more processes integrated of order one are said to be cointegrated if it is possible to make a meaningful linear regression of one process on the other(s). In general, it is not possible to make a meaningful linear regression
11、of one integrated process over another. However, regression is possible if the two processes are cointe- grated. Cointegration is that property that allows one to meaningfully regress one integrated process on other integrated processes. Finally, a property of cointegrated processes is the presence
12、of inte- grated common trends. Given nprocesses with rcointegrating relation- ships, it is possible to determine n-rcommon trends. Common trends are integrated processes such that any of the noriginal processes can be expressed as a linear regression on the common trends. Cointegration entails dimen
13、sionality reduction insofar as common trends are the com- mon drivers of a set of processes.Long-Run EquilibriumGiven nprocesses integrated of order one, the processes are said to be cointegrated if there is a linear combination of the processes that is sta- tionary. If the processes are stock price
14、s, cointegration means that evenc11-Cointegration Page 374 Thursday, October 26, 2006 2:08 PMCointegration and State Space Models375if the stock prices are individually integrated of order onefor example arithmetic random walksthere are portfolios that are stationary. The linear relationships that p
15、roduce stationary processes are called cointe- gratingrelationships. Cointegrated processes are characterized by a short-term dynamics and a long-run equilibrium. Note that this latter property does not mean that cointegrated processes tendto a long-term equilibrium. On the contrary, the relative be
16、havior is stationary. Long-run equilibrium is the static regression function, that is, the relationship between the pro- cesses after eliminating the short-term dynamics. In general, there can be many linearly independent cointegrating relationships. Given nprocesses integrated of order one, there can be a maximum of n 1 cointegrating relationships. Cointegrating relation-ships are not uniquely defined: In fact, any linear combination of cointe- grating relationships is another co
