10 Ergodic Stationarity

10 Ergodic Stationarity - 241B Lecture Ergodic Stationarity...

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Unformatted text preview: 241B Lecture Ergodic Stationarity The key concept in extending analysis to time series, is a stochastic process. A stochastic process is simply a formal name given to a sequence of random variables. If the index denotes time, then the stochastic process is called a time series. For the time series f Z t g , a realization f z t g is called a sample path. Need for Ergodic Stationarity The fundamental problem in time-series analysis is that we can observe the realiza- tion of the process only once. That is, we have only one sequence of observations corresponding to annual US GDP growth for the post-war period. The sequence is just one of many that could have arisen from the stochastic process, if history had taken a di/erent course we would have a di/erent realization. If we could observe history many times over, then we could assemble a collection of strings for post-war GDP growth. Suppose that 2000 corresponds to the 55th element of the series. Then to estimate the average growth rate for 2000, we would simply av- erage the 55th element across the realizations to generate the ensemble mean. In the language of general equilibrium economics, the ensemble mean is the average across all possible states of nature at any point in calendar time. Of course, it is not possible to observe many di/erent alternative histories. But if the distribution of the growth rate remains unchanged over time (a property referred to as stationarity ) the particular string of 55 numbers that we do observe can be viewed as 55 observations from the same distribution. Further, if the process is not too persistent (what is called ergodicity has this property) each element from the string will contain some information not available from the other elements and, as show below, the time average over the elements of the single string will be consistent for the ensemble mean. Various Classes of Stochastic Processes Stationary Processes A stochastic process f Z t g ( t = 1 ; 2 ; : : : ) is strictly stationary if, for any given &- nite integer r and for any set of subscripts t 1 ; t 2 ; : : : ; t r , the joint distribution of ( Z t ; Z t 1 ; Z t 2 ; : : : ; Z...
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10 Ergodic Stationarity - 241B Lecture Ergodic Stationarity...

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