Models, Notation, Goals
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1 Scope Ë ÕÙ Ò Ð Ò ÐÝ Ó ÝÒ Ñ ÑÓ Ð Ü Ô Ö Ñ Ö ² Ñ ¹Ú ÖÝ Ò Ú Ö Ð Ö ÒÙÑ Ö Ð ÔÓ Ö ÓÖ ÔÔÖÓÜ Ñ ÓÒ ß À ÓÖ Ð Ô Ö Ô Ú ß Ë ÑÙÐ ÓÒ Ñ Ó ß ËÑÓÓ Ò ² Ö Ò Ö Ò Ô Ö Ñ Ö ÑÔÐ ß Ã ÖÒ Ð Ñ Ó ÚÓÐÙ ÓÒ Ñ Ó ÓÑ Ò Ô Ö Ð Ð Ö Ò Ð ÓÖ Ñ È µ ÅÙÐ ÔÐ ÓÖ ÑÓ Ð Ò Ò Ò ß º º ¼ Ô Ö Ñ Ö ÇÔ Ò ÕÙ ÓÒ Ö ÓÒ
2 Models, Notation, Goals Ý ½ ¾ µ ÁÒ ÓÖÑ ÓÒ Ñ ½ Ý Ë Ú ÓÖ»Ú Ö Ð Ü Ü ÑÓ Ð Ô Ö Ñ Ö Ç ÖÚ ÓÒ ÑÓ Ð Ô Ý Ü µ Å Ö ÓÚ ÚÓÐÙ ÓÒ ÑÓ Ð Ô Ü Ü ½ µ Ó Ð ÕÙ Ò ÐÐÝ ÙÔ ÔÓ Ö ÓÖ Ô Ü µ Ô Ü ½ ½ µ ÆÙÑ Ö Ð ÔÔÖÓÜ Ñ ÓÒ ÔÓ Ò»Û µ Ü µ µ ½ Æ Ò µ ½ Æ
3 Å ÜÙÖ ÑÓ Ð Some History in Statistics Ö Ö Ó Ô Ö Ñ Ö Å ÜÙÖ Ó ÒÓÖÑ Ð»Ñ ÜÙÖ Ó Ã ÐÑ Ò Ð Ö À ÖÖ ÓÒ ½ µ Ð Ô ² ËÓÖ Ò Ò ½ µ ÅÙÐ ¹ÔÖÓ Ï ² À ÖÖ ÓÒ Ôº ½¾µ Ô Ú ÉÙ Ö ÙÖ Ñ Ó ÈÓÐ Ò Ï ¼µ ÓÑ Ò Ô Ö Ñ Ö Ò Ú Ö Ð È Ö Ñ Ö Ö ÚÓÐÚ» Ô Ò Ñ ÜÔÐÓ Å Ö ÓÚ ÚÓÐÙ ÓÒ ÑÓ Ð Ó Ò Ö Ò Û ÔÓ Ò ² Ò ÖÔÓÐ Æ Ò ÒÙÑ Ö Ð Ò Ö ÓÒ ººº Ù Ð Ñ Ó Ñ ÐÐ Ñ Ò ÓÒ Ë ÑÙÐ ÓÒ Ñ Ó Ò ¼ ÄÓÒ Ö ÓÖÝ Ò Ò Ò Ö Ò Æ Û Ö ÓÒ Ò Ô Ö Ð Ð Ö Ò ÓÓ Ö Ô ËÁÊ È ºººµ
4 (Auxiliary Particle Filter) for States Ì ÓÖ Ð ÙÔ È ² Ë Ô Ö µ Ô Ü ½ ½ µ» Ô Ý ½ Ü ½ µô Ü ½ µ Å ÔÔÖÓÜ Ñ ÓÒ Ó ÔÖ ÓÖ Ô Ü ½ µ Æ ½ µ Ô Ü Ü µ ½ µ È ÙÔ ÖÓÑ ½ ÓÖ ß Ñ µ ½ Ü Ü µ ½ µ ß Ò Û µ ½» µ Ô Ý ½ µ ½µ ÑÔÐ Ùܵ Ò ÓÖ Û ÔÖÓ µ ½ Ñ ½ ÑÔÐ Ü µ ½ Ô Ü ½ Ü µ µ Ò Û µ ½µ ½ Ô Ý ½ Ü µ Ô Ý µ ½ ½µ
5 Filtering for Parameters & States Ì ÓÖ Ð ÙÔ Ô Ü ½ ½ µ» Ô Ý ½ Ü ½ µô Ü ½ µ Û Ô Ü ½ µ Ô µ Ô Ü ½ Ü µô Ü µ Ü Û Ö Ô µô Ü µ Ú Ð Ð ÑÔÐ ÔÓ Ò»Û È Ö Ð Ð Ö Ò ÓÒ Ô Ö Ñ Ö ÒÐÙ Ò Ù Ò Ö Ò Û ÓÒ Ü ÔÓ Ò ÜÔÐ Ò ÓÖ Ò Ý ÙÒ ÓÒ Ô µ ß ÙÒ Ú Ð Ð Ò Ö ÕÙ Ò Ð ÓÒ Ü ß Æ Ê Ò Ö ÓÒ»Ò Û ÔÓ Ò ËÑÓÓ Ò» Ò ÖÔÓÐ ÓÒ Ó ÔÖ ÓÖ ÑÔÐ ² Û
6 Treatment of Parameters: AE Ö Ð ÚÓÐÙ ÓÒ ß ÓÖ ÓÒ Ð ÒÓ Ó Ô Ö Ñ Ö Û Ò Ò ½ ß Ô Ö Ñ Ö ÓÑ Ú Ö Ð ß ½ ½ Û ½ Ò Ô Æ ¼ Ï ½ µ Ö ÙÐ Ò Ò Ô ½ µ Æ ½ Ï ½ µ ÁÒÓÖÔÓÖ Ô Ö Ñ Ö Ò Ô Ö Ð Ð Ö Ò ººº Ù º º Æ Ò È Ý ÓÑÔÙ ÓÒ ÐÐÝ ÙÒ Ö ÒÝ Ó Ô Ö Ñ Ö Ñ ÔÓ Ò ÄÓ Ó Ô Ö Ñ Ö Ò ÓÖÑ ÓÒ Ö Ô Ö ÓÖÑ Ò ÓÚ Ö¹ «Ù ÔÓ Ö ÓÖ Ò Ö Ö ÐÐÝ ÅÓ ÖÒ Ñ Ñ ÒÝ Ô Ö Ñ Ö ß ÐÓ Ó Ò ÓÖÑ ÓÒ ÐÓ
7 Treatment of Parameters: KS à ÖÒ Ð ËÑÓÓ Ò ¹ Ï ¼ º Ö ÔÓÖµ ËÑÓÓ» Ò ÖÔÓÐ Ô Ö Ñ Ö ÑÔÐ ÓÒ ÒÙÓÙ Ö Ù ÓÒ Í Ò Ô Ú ÑÔÓÖ Ò ÑÔÐ Ò Ì Ñ ÑÔÐ»Û µ µ Å Ñ Ò»Ú Ö Ò Ñ Ö Ü Ñ Î Ã ÖÒ Ð Ò Ý Ô µ Æ ½ Ê Ò Ö ÑÔÐ µ Æ Ñ µ ¾ Î µ ß ÑÓÓ Ò Ô Ö Ñ Ö ß Ú ÐÙ Ò Ý Ò ÑÔÓÖ Ò ÑÔÐ Ò µ Ó Ó ÐÓ ÓÒ Ñ Ò ÑÓÓ Ò ÓÖ
8 Treatment of Parameters: KS (cont.) à ÖÒ Ð Û Ë Ö Ò Ï µ Ë Ò Ö ÖÒ Ð ÐÓ ÓÒ Ñ µ µ Ö ÙÐ Ò ÓÚ Ö ÑÓÓ Ò Î µ Î ÓÖÖ Ù Ò ÖÒ Ð ÐÓ ÓÒ Ö Ò Ñ µ µ ½ µñ Û Ö ¾ ½ ¾ Ò Ö ½µ à ÖÒ Ð»Ñ ÜÙÖ ÓÖÖ Å Ñ Ò Ò Ú Ö Ò Ñ Ö Ü Ò µ Ñ Î µ Î
9 Modifying the AE Method ½ ½ Û Ö ½ Æ ¼ Ï ½ µ ÒÓÛ ÓÖÖ Ð Ú ½ µ Ï ½ ¾ Ê ÙÐ ½ µ µ Ñ Ò Î ½ µ Î µ Î ß ÒÓ Ò ÓÖÑ ÓÒ ÐÓ Ò ß Ó» Ô ÓÒ Ó Ï ½ Ë Ò Ö ÓÙÒ ÓÖ Ñ Ó Ï Ò À ÖÖ ÓÒ µ Ï ½ Î ÓÖ Ñ ÐÐ º º ¼º¼½µ Ç Ö ÑÓÖ Ò Ö Ð Ú Ö ÓÒ
10 Modified AE KS ÅÓ ÑÔÐ Û Ô ½ µ Æ ½ ½ µñ ¾ Î µ ½ ¾ ¾ ½ ¾ Ë Ñ ÃË Û Ö Ò ÁÑÔÐ ÔÖ ÓÖ ½ Ô µ Æ ½ µ Æ Ñ µ ¾ Î µ Û Ñ µ µ ½ µñ Ò Ý Ò Ð ÓÙÒ ÓÖ ÖÑ Ò Ò
11 Combined Filtering È Ö Ñ Ö Ü Ò È È µ Ì Ñ Å ÙÑÑ ÖÝ Ü µ µ µ ½ Æ È Ú Ò ÓÖ Ú ÖÝ ÑÔÐ Ò Ñ µ ½ Ü Ü µ ½ Ñ µ Ò Û µ ½» µ µ µ Ó Ü ½ Ó Ô Ý ½ µ ½ Ñ µ µ Ë ÑÔÐ Ùܵ Ò ÓÖ Û ÔÖÓ µ ½ Ò Ò Ñ ½ ÑÔÐ ß µ ½ Æ Ñ µ ¾ Î µ ß Ü µ ½ Ô Ü µ Ò Û µ µ ½µ ½ Ô Ý ½ Ü µ ½ µ ½µ Ô Ý µ ½ ½ Ñ µ µ
12 Simple AR(1) Example ß ÒÓ Ú Ö Ð ÓÒ Ô Ö Ñ Ö ß exact posterior filtering approx ÙÖ ½ ÈÓ Ö ÓÖ ÕÙ Ò Ð ÓÖ Ê ½µ Ô Ö Ñ Ö ¾º ± ¾ ± ¼± ± º ± Ü ¼º ¼º ¼½ ¼º ½ ¼º ¾ ¼º ¾ ÔÔÖÓÜ ¼º ¼º ¼¼ ¼º ½¾ ¼º ¾ ¼º Ì Ð ½ ÉÙ Ò Ð
13 Dynamic Factor Models ß Ö ÓÙ ¼ Ô Ö Ñ Ö Ú Ö Ð ß ÓÖ ÑÓ Ð Ý Õ Ú ÓÖ Ó Ü Ò Ö Ö ÙÖÒ Ðݵ Ú ÓÖ Ó Ð Ò ÓÖ Õ Ñ Ö Ü Ó ÓÖ ÐÓ Ò Ó Æ Ò Ý «Ò Ô Æ ¼ µ Û ½ Õ µ Ä Ò ÓÖ Þ ÖÓ¹Ñ Ò ÒÓÖÑ Ð Û Û Î µ ÜÔ Ü ½ µ ÜÔ Ü µµ Ë ÚÓÐÙ ÓÒ ÑÓ Ð Ë Ú ÓÖ Ü Ü ½ Ü µ ¼ Î ÓÖ Ê ½µ ÑÓ Ð Ü Ü ½ µ Ò Ô Æ ¼ ͵ Û ÓÒ Ð
14 Dynamic Factor Models Ç ÖÚ ÓÒ ÑÓ Ð Ô Ý Ü µ Æ Ý ¼ ÜÔ Ü µ ¼ µ Û Ö ÜÔ Ü µ ÜÔ Ü ½ µ ÜÔ Ü µµ Ë ÚÓÐÙ ÓÒ ÑÓ Ð Ô Ü Ü ½ µ Æ Ü Ü ½ µ ͵ È Ö Ñ Ö Í Õ Ô Ö Ñ Ö
15 Dynamic Factor Model Analysis Ë Ô Ö Ò È ÅÓ Ð Ú Ö ÓÒ Å Å Ò ÐÝ È ÓÒ Ù Ð Ö Ò Ï Å Å ÓÖ Ò ² ÔÓÖ ÓÐ Ó ÓÒ ÖÙ ÓÒ Ù È ÓÒ Ò ÐÝ ÐÝ Ü Ò Ö Ò Ó Õ ÙÖÖ Ò Å Å ÓÚ Ö ¾º Ý Ö Ó ÓÖ Ò ½ ÓÑ Ò Ð Ö Ò Ú È ÓÚ Ö ½ ÓÑÔ Ö ÓÒ Ó È Ö ÙÐ Û ÙÐÐ Å Å Ò ÐÝ ¾ ÏÖ Ò Ð È Ü Í Í ½ µ ¼ ¼ ¼½ ½ ¼ ¼µ Ï Ý ÆÓÖÑ Ð ÖÒ Ð ½ ½ ¼ ¾ µ ¼ ¾ ¼ ¾¼µ ¼ ¼ ¼ ½ µ ¼ ½ ¼ ¾¼µ ½ ¼ ¾ µ ¼ Ô Ö Ñ Ö ººº ½
16 Exchange Rate Time Series.25 DEM.15 JPY 6 x 1 3 CAD time (day) time (day) time (day).25 FRF.15 GBP.2 ESP time (day) time (day) time (day)
17 DFM results at 1 steps «µ 5 x 1 4 α[1] 5 x 1 4 α[2] x α[3] 2 x x α[4] x x α[5] 2 2 x x 1 4 α[6] 5 x x x x 1 3 Ψ[1] x 1 3 Ψ[2].5 x Ψ[3] x 1 3 x 1 3 Ψ[4] x 1 3 x Ψ[5] x x 1 3 Ψ[6] x x x 1 3
18 DFM results at 1 steps µ.75 X[2,1] X[3,1].5 X[3,2] X[4,1] X[4,2].2 X[4,3] X[5,1] X[5,2] X[5,3] X[6,1] X[6,2].4.2 X[6,3]
19 DFM results at 1 steps µ 9 1 µ[1] µ[2] µ[3] logit(φ[1]) logit(φ[2]) logit(φ[3]) 5 1
20 DFM results at 5 steps «µ 5 x 1 4 α[1] 5 x 1 4 α[2] x α[3] 2 x x α[4] x x α[5] 2 2 x x 1 4 α[6] 5 x x x 1 4 x Ψ[1] x 1 3 x 1 3 Ψ[4] x Ψ[2] x 1 3 x Ψ[5] x 1 3 Ψ[3] x x 1 3 Ψ[6] x x x 1 3
21 DFM results at 5 steps µ.75 X[2,1] X[3,1].5 X[3,2] X[4,1] X[4,2].2 X[4,3] X[5,1] X[5,2] X[5,3] X[6,1] X[6,2].4.2 X[6,3]
22 DFM results at 5 steps µ 9 1 µ[1] µ[2] µ[3] logit(φ[1]) logit(φ[2]) logit(φ[3])
23 Comments, issues, future Å Ò ÐÝ ËÑ ÐÐ «Ö Ò Û Ò Å Å Ò È Ð Ö Ö Ñ Ô È Ò ÐÝ ÒÓÖ ÙÒ Ö ÒÝ Ò Í ß ÓÖÖ Ð ÑÓ Û ß «Ö Ò ÓÒ Ô Ö Ñ Ö Ù Ð ¹ÙÔ Ó ÖÖÓÖ Ò ÒÝ ÕÙ Ò Ð Ò ÐÝ Ê Ð Ý ÓÒ Ü Ò Ù ß ÕÙ Ò Ð Ò ÐÝ Ô Ö ÐÐ Ð ÑÓ Ð ß ÐÝ ÙÔ ß Å Å Û Ò È Ñ Ñ ÓÒ Ð ÓÜ
24 Comments, issues, future ÅÓ ÓÒ» Ü Ò ÓÒ È ÁÒ Ö Ò ÔÓ Ö ÓÖ ¹ Ú ÖÝ Ò Ð¹Û ÛÒ ÑÙÐ ÔÐ ÑÓ ººº ÓÑÑÓÒ ÖÒ Ð Ô» Ð Ò Æ Ò ÅÙÐ ÔÐ ÑÓ ÚÓ Ö Ò Ó ÐÓ Ð Ñ Ò Î Ö Ð»ÄÓ Ð ÖÒ Ð Ï Ú Ò ² Ê ÖÝ µ Ù ÓÒ ÓÖÝ µ Ò Ä Ù Ò Ï Ê Ö Ò Ä Ù Ò Ï
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