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Econ 399 Chapter11a - (small) ,weneedto properties...

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10 Further Time Series OLS Issues Chapter 10 covered OLS properties for finite  (small) sample time series data -If our Chapter 10 assumptions fail, we need to  derive large sample time series data OLS  properties -for example, if strict exogeneity fails (TS.3) -Unfortunately large sample analysis is more  complicated since observations may be  correlated over time -but some cases still exist where OLS is valid
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11. Further Issues in Using OLS with Time Series Data 11.1 Stationary and Weakly Dependent Time Series 11.2 Asymptotic Properties of OLS 11.3 Using Highly Persistent Time Series in Regression Analysis 11.4 Dynamically Complete Models and the Absence of Serial Correlation 11.5 The Homoskedasticity Assumption for Time Series Models
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11.1 Key Time Series Concepts To derive OLS properties in large time series data  sample, we need to understand two properties: 1)Stationary Process -x distributions are constant over time -a weaker form is Covariance Stationary  Process -x variables differ with distance but not with  time 2) Weakly Dependent Time Series -variables lose connection when separated by  time
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11.1  Stationary Stochastic  Process The stochastic process {x t : t=1,2…}  is  stationary  if for every collection of time indices 1 t 1 <t 2 <…<t m , the joint  distribution of (x t1 , x t2 ,…,x tm ) is the  same as the joint distribution (x t1+h x t2+h ,…,x tm+h ) for all integers h 1
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11.1 Stationary Stochastic Process The above definition has two implications: 1) The sequence {x t : t=1,2…} is  identically  distributed -x 1  has the same distribution as any x t 1) Any joint distribution (ie: the joint distribution  of [x 1, x 2 ]) remains the same over time (ie:  same joint distribution of [x t, x t+1 ])  Basically, any collection of random  variables has the same joint distribution  now as in the future
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11.1 Stationary Stochastic Process It is hard to prove if a data was generated by a  stochastic process -Although certain sequences are obviously not  stationary -Often a weaker form of stationarity suffices -Some texts even call this weaker form  stationarity:
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11.1 Covariance  Stationary  Process The stochastic process {x t : t=1,2…} with a  finite second moment [E(x t 2 )< ]   is  covariance stationary  if: i) E(x t ) is constant ii) Var(x t ) is constant iii) For any t, h 1, Cov(x t ,x t+h ) depends on h  and not on t
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11.1 Covariance Stationary Process Essentially, covariance between variables can 
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