T 5 s a t ² u t s y t l n a 2 55 u ² 2 i n c b 1 i

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t ± 5 s   a t ² u t s   y t L N 0 , @ a 2 55 U ² @ / 2 I N " C B   " 1 I N " C B U   " 1

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7 spatial trend: j s   ± longitude of site s k s   ± latitude of site s q t s   ± ) ² * j s   ² + k s   VI. Spatiotemporal models A. Introduction B. Modeling spatial correlation C. Modeling spatiotemporal correlation q t ± Nx 1   vector of unobserved factors for each state q t ± H 1 q t " 1 ² H 2 q t " 2 ² C ² H p q t " p ² 1 t Could have spatial correlation structure to 1 t WBC just take 1 t L N 0 , @ 1 2 I N   and set p ± 1
8 q t ± Hq t " 1 ² 1 t We expect that: h sr ± 0 if r ± R s   and r p s WBC impose this outright. Here we’ll use a Bayesian prior that very strongly moves the data towards this without completely forcing: h sr | s p r , s ± R s   L N 0, A h 3 2   with A h 3 2 very small We also might expect h ss to be very similar (but not forced to be identical) for different s Let h 1 be a random variable that summarizes our prior subjective uncertainty about this common value: h 1 L N m h 1 , A h 1 2   where m h 1 is our prior guess about what h ss would be for a typical state and A h 1 2 , our uncertainty about this guess, might be large

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9 Finally, we may expect a pretty similar coefficient (represented by h 2 L N m h 2 , A h 2 2   ) relating q t s   to the average of neighboring states’ q t " 1 r   with A h 2 2 again large That is, the probability law for our prior on H can be represented as H ± h 1 I N ² h 2 B ² U h h 1 L N m h 1 , A h 1 2   h 2 L N m h 2 , A h 2 2   vec U h   L N 0 , A h 3 2 I N 2   Aside: if we wanted to say there is also a potential aggregate influence of the lagged value of all the other states combined (i.e., of 1 N U q t " 1 ), we could specify H ± h 1 I N ² h 2 B ² h 3 1 N 1 N U ² U h
10 H ± h 1 I N ² h 2 B ² U h i N ± vec I N   b ± vec B   h ± vec H   ± h 1 i N ² h 2 b ² vec U h   prior: h | @ 1 2 L N m h , @ 1 2 M h   m h ± m h 1 i N ² m h 2 b M h ± A h 1 2 i N i N U ² A h 2 2 bb U ² A h 3 2 I N 2 Observed variable to be explained: y t ± N 1   vector of unemployment rates for each state Observed explanatory variables: w t ± k 1   vector of aggregate explanatory variables (common to all states) w t ± 1, t ,...   X t ± N d   matrix of explanatory

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• Winter '09
• JamesHamilton
• Unemployment, Trigraph, Yt, Qt, VI. Spatiotemporal models

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