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time series weak dependence

# time series weak dependence - Econ 471 L6 Time Series I...

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Econ 471: L6 Time Series I Sung Y. Park U of Illinois

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Time series I So far we have considered cross-sectional data : data collected at given (flxed) time. For example, years of education, salary and years of work experience of individual i at flxed time, say, 2000. I Now we consider time-series data : data collected over time. For example, salary t , t = 1973 ; 1974 ; ¢ ¢ ¢ ; 2009. I Thus, most importantly, we have to take care of the serial dependence of time series variables, y t , t = 1 ; 2 ; ¢ ¢ ¢ ; n .
Some examples F A Static Model: y t = fl 0 + fl 1 z t + u t ; t = 1 ; 2 ; ¢ ¢ ¢ ; n I Contemporaneous relationship between y and z variables. I A change in z at t has an immediate efiect on y ) ¢ y t = fl 1 ¢ z t when ¢ u t = 0. I Example: Static Phillips curve: inf t = fl 0 + fl 1 unem t + u t ) Contemporaneous trade-ofi efiect in ation and unemployment rates.

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Some examples F (Finite) Distributed Lag Models: gfr t = fi 0 + 1 pe t + 2 pe t ¡ 1 3 pe t ¡ 2 + u t ; t = 1 ; 2 ; ¢ ¢ ¢ ; n where gfr t : the general fertility rate; pe t : real dollar value of the personal tax exemption. I Idea: Decision to have children is related with the tax value of a child. I Tax rate at t may not instantly afiect fertility rate at time t . Tax rates at t ¡ 1 or/and t ¡ 2 may afiect the fertility rate at time t . I Finite distributed model of order 2 : pt t , pe t ¡ 1 and pe t ¡ 2 .
Some examples Consider y t = fi 0 + 0 z t + 1 z t ¡ 1 2 z t ¡ 2 + u t ; t = 1 ; 2 ; ¢ ¢ ¢ ; n What are 1 , 2 and 3 ? I Finite distributed model of order 2. I Suppose there is an unit increase at t and value of z at s 6 = t is a constant, say, c . ) ¢ ¢ ¢ ¢ ¢ ¢ ; z t ¡ 2 = c ; z t ¡ 1 = c ; z t = c + 1 ; z t +1 = c ; ¢ ¢ ¢ ¢ ¢ ¢ y t ¡ 1 = fi 0 + 0 c + 1 c + 2 c y t = fi 0 + 0 ( c + 1) + 1 c + 2 c ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ y t +2 = fi 0 + 0 c + 1 c + 2 ( c + 1) y t +3 = fi 0 + 0 c + 1 c + 2 c

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Some examples I y t ¡ y t ¡ 1 = 0 : immediate change in y due to the one-unit change in z at time t (impact propensity) I y t +1 ¡ y t ¡ 1 = 1 : the change in y one period after the temporary change I y t +2 ¡ y t ¡ 1 = 2 : the change in y two periods after the temporary change I y t +3 ¡ y t ¡ 1 = 0: after two periods y does not change. (why?) ² If we consider the change in y due to a permanent increase in z such that z s = c if s < t and z s = c + 1 otherwise. After one period : 0 + 1 After two periods : 0 + 1 + 2 After three periods : 0 + 1 + 2 0 + 1 + 2 : long-run change (long-run propensity)
Finite sample properties of OLS F Assumptions : A1 (Linear in Parameter) Time series f ( x t 1 ; x t 2 ; ¢ ¢ ¢ ; x tk ; y t ) : t = 1 ; 2 ; ¢ ¢ ¢ ; n g follows the linear model y t = fl 0 + fl 1 x t 1 + fl 2 x t 2 + ¢ ¢ ¢ + fl k x tk + u t . A2 (No perfect collinearity) No independent variable is constant or a perfect linear combination of the others A3 (Zero conditional mean) E ( u t j X ) = 0 ; t = 1 ; 2 ; ¢ ¢ ¢ ; n ; where X is a collection of all independent variables x t = ( x t 1 ; x t 2 ; ¢ ¢ ¢ ; x tk ), i.e., X = ( x 1 ; x 2 ; ¢ ¢ ¢ ; x n )

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Finite sample properties of OLS Remarks: A3
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time series weak dependence - Econ 471 L6 Time Series I...

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