AR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 )

Transcription

1 AR() Procss Th firs-ordr auorgrssiv procss, AR() is whr is WN(0, σ )

2 Condiional Man and Varianc of AR() Condiional man: Condiional varianc: ) ( ) ( Ω Ω E E ) var( ) ) ( var( ) var( σ Ω Ω Ω Ω E

3 Auocovarianc of AR() Tak h quaion And hn mulipl boh sids b -k Thn ak xpcaions. Sinc is whi nois, i is uncorrlad wih or k k k E( k ) E( k ) E( k ) γ ( k) γ ( k )

4 Auocorrlaion of AR() Dividing b h varianc, his implis W know Thn ρ( k) ρ( k ) ρ( 0) ρ() ρ() ρ( k) ρ(0) ρ() k

5 Auocorrlaion of AR() W hav drivd ρ ( k) k Th auocorrlaion of h saionar AR() is a simpl gomric dca ( < ) If is small, h auocorrlaions dca rapidl o zro wih k If is larg (clos o ) hn h auocorrlaions dca modral Th AR() paramr dscribs h prsisnc in h im sris

6 On-Sp-Ahad Forcas As w showd arlir Thus E E ( Ω Th opimal on-sp-ahad forcas is a linar funcion of h final obsrvd valu ) ( ΩT ) T T

7 -sp-ahad forcas B back-subsiuion Thus and ( ) ) ( ) ( Ω Ω E E T T T E ) ( Ω

8 -sp-ahad forcas This shows ha h opimal -sp-ahad forcas is also a linar funcion of h final obsrvd valu, bu wih h cofficin. E( Ω T ) T T

9 h-sp-ahad forcas Similarl So Opimal forcas: h h h h h h h h h h h E E Ω Ω ) ( ) ( T h T h T E Ω ) (

10 Invrsion of AR() B invring h lag opraor Which is h sam as found b back subsiuion ( ) ( ) i i i i i i L L L 0 0

11 Condiion for Invribili Th opraor (-L) is invribl whn < This is h sam as for h MA() modl is h invrs of h roo of h polnomial -L Th roo of a funcion is h valu whr i crosss h x-axis Th roo of -L is /, h invrs of h roo is Invribili rquirs ha h invrs of h roo b lss han on

12 AR() wih Inrcp An AR() wih inrcp is Taking xpcaions α Thus and E ( ) α E( ) E( µ α µ α µ )

13 Bs Linar Prdicor A linar prdicor of givn - is α Th forcas rror is α Th linar prdicor which minimizs h xpcd squard forcas rror solvs min α, E( α )

14 Las-Squars Th sima of h xpcd squard linar forcas rror is h sum of squard rrors Th las squars sima ˆ ˆ α minimizs h sum of squard rrors, so is h sima of h bs linar prdicor This is a linar rgrssion, raing - as a rgrssor. ˆ

15 Unmplomn Ra. rgrss ur L.ur Sourc SS df MS Numbr of obs 88 F(, 86) Modl Prob > F Rsidual R-squard Adj R-squard Toal Roo MSE ur Cof. Sd. Err. P> [95% Conf. Inrval] ur L _cons

16 Fid AR() To plo a scar and fid rgrssion, wowa scar ur L.ur lfi ur L.ur, il( Unmplomn Ra, 0-4 ar olds ) Unmplomn Ra, 0-4 ar olds ur, L ur Fid valus

17 GDP Growh Ras. rgrss gdp L.gdp Sourc SS df MS Numbr of obs 78 F(, 76) 44.3 Modl Prob > F Rsidual R-squard Adj R-squard 0.35 Toal Roo MSE gdp Cof. Sd. Err. P> [95% Conf. Inrval] gdp L _cons

18 GDP Growh Ras Ral Gross Domsic Produc, L Ral Gross Domsic Produc Fid valus

19 On-Sp-Ahad Forcas Th opimal forcas for T givn T is ˆ T T α T Th forcas using h simas is ˆ T T α ˆ ˆ T

20 Unmplomn Ra, 0-4 ar olds Th simas wr ur Cof. Sd. Err. P> [95% Conf. Inrval] ur L _cons Th valu for Jan 06 is 8.3%, so ˆ : Poin forcas is 8.3% (unchangd from currn) ˆ

21 lis im ur L.ur p if im>m(06m) L. im ur ur p m m m m m m m m m m m m m m

22 Exampl GDP Growh Th simas wr gdp Cof. Sd. Err. P> [95% Conf. Inrval] gdp L _cons Th valu for 4 h quarr 06 is.9%, so ˆ ˆ07 : %

23 GDP Growh L. im gdp gdp p 7. 05q q q q q q q q q

24 On-Sp-Ahad Forcas Error Th forcas rror is ˆ T T T α T T ( α ) T T Th forcas varianc is var( ˆ T T T ) var( T ) σ

25 Forcas varianc simaion Avrag of squard rsiduals whr h las-squars rsiduals ar ˆ ˆ ˆ ˆ σ σ σ T T ˆ ˆ ˆ α

26 Unmplomn Ra, 0-4 ar olds Unmplomn Ra, 0-4 ar olds Rsiduals ur Rsiduals m 960m 970m 980m 990m 000m 00m 00m im 950m 960m 970m 980m 990m 000m 00m 00m im

27 GDP Growh Ral Gross Domsic Produc GDP Growh 950q 960q 970q 980q 990q 000q 00q 00q im Rsiduals Rsiduals 950q 960q 970q 980q 990q 000q 00q 00q im

28 On-Sp-Ahad Inrvals Normal Mhod Assum forcas rror is normall disribud Forcas inrval is poin sima, plus and minus h sandard dviaion of forcas muliplid b a normal prcnil For a 95% inrval: ˆ ˆ ˆ T T ± σ z.05 T For a 90% inrval T ± σ ˆ ˆ ˆ ± σ z.05 T T T T ± σ ˆ ˆ

29 Sandard Dv of Forcas Th forcas varianc is h varianc of h rror, plus h varianc of h sima Th sandard dviaion of h forcas is is squar roo In mos cass h major componn of h forcas varianc is h varianc of h rror Thus a simpl mhod o sima h sandard dviaion of h forcas is o us σ, roo man squard rror, from h rgrssion sima In STATA, for sdf ou can us h command prdic s, sdf Th diffrnc bwn h rms and sdf grows whn h numbr of rgrssors is larg rlaiv o h sampl siz.

30 Exampl: GDP Th Roo MSE is 3.63 Th sdf is 3.64, onl slighl highr!. rgrss gdp L.gdp Sourc SS df MS Numbr of obs 78 F(, 76) 44.3 Modl Prob > F Rsidual R-squard Adj R-squard 0.35 Toal Roo MSE gdp Cof. Sd. Err. P> [95% Conf. Inrval] gdp L _cons

31 Forcas Inrval Consrucion sappnd, add() prdic p prdic s, sdf gn p p.645*s gn p p.645*s lis im p p p s if im>q(06q4) im p p p s q Poin sima.7% Sdf % Inrval [-3.%, 8.7%]

32 Assignmns Rad Dibold hrough Chapr 7 Problm S # 5 Du Tusda (/) Rad Chapr 5 from Th Signal and h Nois Rading Rflcion Du Thursda (/3)