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Econ226_VI

Econ226_VI - VI Spatiotemporal models A Introduction 1 s...

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1 VI. Spatiotemporal models A. Introduction

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2 s ± location s ± 1,2,..., N t ± date t ± 1,2,..., T y t s   ± variable of interest Example 1 (economics): s ± 1 · Alabama s ± N · Wyoming t ± 1 « 1954:I t ± T · 2004:IV y t s   ± unemployment rate in Colorado in 1972:II Example 2 (Wikle, Berliner, and Cressie): s ± particular location in U.S. midwest at which either temperatures were recorded or are wanted to be inferred y t s   ± average daily maximum temperature at location s in month t
3 y t ± y t 1   y t 2   B y t N   ± unemployment rates for all states observed in quarter t VI. Spatiotemporal models A. Introduction B. Modeling spatial correlation time series: when something is observed at one date, it changes what we expect to see at other dates spatial data: when something is observed at one location, it changes what we expect to see at other locations

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4 time series white noise: / t L N 0, @ / 2   / t independent of / A whenever t p A time series moving average: u t ± / t ² 2 / t " 1 ´ u t is correlated with u t " 1 but not with u t " 2 , u t " 3 ,... time series autoregression: u t ± C u t " 1 ² / t ´ u t is correlated with u t " 1 , u t " 2 ,...
5 spatiotemporal white noise: / t s   L N 0, @ / 2   / t s   independent of / A r   whenever t p A or r p s spatial moving average: Let R s   ± set of all states adjacent to s (note s ± R s   ) n s   ± number of states adjacent to s u t s   ± 2 ¡ n s  ¢ " 1 ! r ² R s   / t r   ² / t s   u t s   is independent of u t r   whenever r and s are more than two states apart Let row s , column r element of B be 1/ n s   if r ² R s   and zero otherwise u t ± I N ² 2 B   / t u t L N 0 , @ 2 I N ² 2 B   I N ² 2 B U

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6 spatial autoregression: u t s   ± C ¡ n s  ¢ " 1 ! r ² R s   u t r   ² / t s   u t ± C Bu t ² / t u t L N 0 , @ / 2 I N " C B   " 1 I N " C B U   " 1   u t s   is correlated with u t r   for all r , s suppose there is a shock a t L N 0, @ a 2   that affects all states equally in addition to u t s   : y t s   ± a t ² u t s   y t ± a t 1 N ² u t (where 1 N is the N 1   vector 1,1,...,1   U ) y t L N 0 , @ a 2 1 N 1 N U ² @ / 2 I N " C B   " 1 I N " C B U   " 1    Or we may have prior information about how state s reacts to this shock 5 s   ± fraction of workers in state s employed in agriculture y
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