Wisconsin Unemployment Rate Forecast Revisited
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1 Wisconsin Unemploymen Rae Forecas Revisied Forecas in Lecure Wisconsin unemploymen November 06 was 4.% Forecass Poin Forecas 50% Inerval 80% Inerval Forecas Forecas December % (4.0%, 4.0%) (3.95%, 4.05%) January % (4.0%, 4.%) (3.9%, 4.%) February % (3.9%, 4.%) (3.8%, 4.%) Realizaions December 06: 4.% January 07: 3.9%
2 Regression wih Correlaed Errors y α + βx + e In some regression models, he errors are correlaed Pure rend Models Pure Seasonaliy Models In hese models he errors can be correlaed Classical and robus sandard errors are no appropriae
3 Example: Sock Volume w 960w 970w 980w 990w 000w 00w Residuals lvolume Fied values Auocorrelaions of e Lag Barle's formula for MA(q) 95% confidence bands 950w 960w 970w 980w 990w 000w 00w
4 Leas-Squares Variance Formula Recall for v x var ( ˆ β ) When he v are uncorrelaed var var v e var a ~ ( ) a ˆ var( v ) β ~ var v [ var( x )] ( v ) var( v ) [ var( x )]
5 General Formula Define f var var v ( v ) When he v are uncorrelaed f, oherwise no. hen var β ( ) a ˆ var( x ) e ~ [ var( x )] f
6 Adjusmen Facor he asympoic variance of leas-squares is he convenional, muliplied by an adjusmen facor for he serial correlaion var β ( ) a ˆ var( x ) e ~ [ var( x )] f
7 Auocovariance of v We wan a useful formula for Since E(v )0, hen E E( v ) ( ) var v ( v v ) cov( v v ) γ ( j) he auocovariance of v j f var var j v ( v )
8 Variance of sum of correlaed v ( ) ( ) j j j j j j v v E v v E v E v var γ
9 Adjusmen Facor Where he (-j) are he auocorrelaions of v ( ) ( ) j j v v f var var
10 his double sum is he sum of all he elemens in he marix here are of he (0) (-) of he () (-) of he () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) j j j +
11 Adjusmen Facor Dividing by If is large ( ) ( ) j j j f j j + ( ) + j f j f
12 Summary: Leas-Squares Variance When he errors are correlaed var β ( ) a ˆ var( x e ) ~ [ var( x )] f f + j ( j) he convenional formula is muliplied by an adjusmen for auocorrelaion
13 HAC Esimaion Esimaion of f For variances and sandard errors under auocorrelaion Called heeroskedasiciy and auocorrelaion consisen (HAC) variance esimaion Muliply convenional variance esimaes by esimaes of f
14 HAC Esimaion he adjusmen is f + j ( j) where (j) are he auocorrelaions of v x e Esimae (j) by sample auocorrelaions using leas-squares residuals Bu in a sample of lengh we canno esimae all auocorrelaions well
15 Unweighed HAC Esimaor For some runcaion parameer m, Original proposal fˆ + j L. Hansen, Hodrick (978) Hal Whie (98) m ˆ ( j)
16 Lars Hansen Professor Lars Hansen, U Chicago Invened Generalized Mehod of Momens, he leading esimaion mehod for applied economerics Inroduced unweighed HAC esimaor for muli-sep regression models 03 Nobel Prize in economics
17 Deficiencies of Unweighed Esimaor fˆ he esimaor is no smooh in he runcaion parameer he sample esimae can be negaive Example of negaive esimae when m fˆ + ˆ Negaive if ()<-/ + ( ) m j ˆ ( j)
18 Example: Liquor Sales ake monhly growh raes Regress on Seasonal Dummies o obain seasonal paern Liquor Sales, Monhly Growh Rae Seasonal y Fied values m 995m 000m 005m 00m 05m ime 06m 06m4 06m7 06m0 07m ime
19 Auocorrelaion of Residual he firs auocorrelaion is less han -/ Auocorrelaions of e Liquor Sales, Monhly Growh Rae, Auocorrelaion Lag Barle's formula for MA(q) 95% confidence bands
20 Weighed HAC Esimaor Called Newey-Wes variance esimaor Whiney Newey, Ken Wes (987) his weighed esimaor is always posiive Smoohly changes in runcaion parameer m ( ) + m j j m j m f ˆ ˆ
21 Whiney Newey and Ken Wes Professor Whiney Newey, MI Leading economeric heoris Professor Ken Wes, Wisconsin Macroeconomis & economerician Forecas evaluaion and comparison Join paper in 987 Weighed HAC esimaor One of he mos referenced papers in economerics
22 Compuaion In SAA, replace regress command wih newey command.newey y x, lag(m) You supply he runcaion parameer m Similar o regression wih robus sandard errors hese are idenical.newey y x, lag(0).reg y x, r
23 Example: Liquor Sales. regress y b.m, r Linear regression Number of obs 99 F(, 87) Prob > F R-squared Roo MSE.0333 Robus y Coef. Sd. Err. P> [95% Conf. Inerval] m _cons
24 Wih Newey-Wes sandard errors. newey y b.m, lag() Regression wih Newey-Wes sandard errors Number of obs 99 maximum lag: F(, 87) Prob > F Newey-Wes y Coef. Sd. Err. P> [95% Conf. Inerval] m _cons
25 runcaion Parameer m should be large when auocorrelaion is large Sophisical daa-dependen mehods o pick m have been developed, bu are no in SAA Sock-Wason defaul (explanaory x s) m 0.75 rend/seasonal defaul m.4 /3 /3
26 Derivaion of Defauls Due o Donald Andrews (99) he opimal m minimizes he mean-squared error of he esimae of f When v is an AR() wih coefficien, Andrews found he opimal m is m C C /3 6 ( ) /3
27 Donald Andrews Leading economeric heoris Conribuions o ime-series Opimal selecion of runcaion parameer ess for srucural change
28 Defaul Values m C C /3 6 ( ) Sock-Wason If boh x and e are AR() wih coef ½, hen v x e has AR() coefficien.5. Plug his in, and C.75 rend-seasonal If x is rend and/or seasonal and e are AR() wih coef ½, hen v x e has AR() coefficien.5. Plug his in, and C.4 /3
29 Liquor Sales again. dis.4*e(n)^(/3) newey y b.m, lag(9) Regression wih Newey-Wes sandard errors Number of obs 99 maximum lag: 9 F(, 87) 8.7 Prob > F Newey-Wes y Coef. Sd. Err. P> [95% Conf. Inerval] m _cons
30 Example: Men s Labor Force Paricipaion Rae, rend Model m 960m 970m 980m 990m 000m 00m 00m ime Civilian Labor Force Paricipaion Rae: 0 years and over, Men Fied values
31 . reg men ime, r Linear regression Number of obs 89 F(, 87) Prob > F R-squared Roo MSE.9307 Robus men Coef. Sd. Err. P> [95% Conf. Inerval] ime _cons dis.4*e(n)^(/3) newey men ime, lag(3) Regression wih Newey-Wes sandard errors Number of obs 89 maximum lag: 3 F(, 87) Prob > F Newey-Wes men Coef. Sd. Err. P> [95% Conf. Inerval] ime _cons
32 Summary In one-sep-ahead forecas regressions If he errors are serially uncorrelaed Use Robus sandard errors reg wih r opion If he errors are correlaed Use Newey-Wes sandard errors newey y x, lag(m) In pure rend or seasonaliy models Se m.4 /3 In dynamic regression Se m.75 /3
33 h-sep-ahead forecass In he AR() Model he opimal h-sep forecasing regression akes he form he error u is a correlaed MA(h-) Unless β0 e y y + + β α h h h h e e e e u u y y β β β β α
34 h-sep-ahead models In any h-sep model he variable v y -h e is generally serially correlaed Generally MA(h-) y + u α + βy h Correc adjusmen erm f + h j ( j)
35 Newey-Wes Sandard Errors Sandard errors can be esimaed using he Newey-Wes mehod runcaion parameer se o forecas horizon mh fˆ + h j h h j ˆ ( j)
36 Example: Unemploymen Rae -monh-ahead forecas wih 4 AR lags Robus sandard errors:. reg ur L (/5).ur, r Linear regression Number of obs 84 F(4, 809) 0.69 Prob > F R-squared 0.58 Roo MSE.785 Robus ur Coef. Sd. Err. P> [95% Conf. Inerval] ur L L L L _cons
37 Example: Unemploymen Rae Newey-Wes sandard errors: Sandard errors on lag 3 and 4 decrease by half Sandard error on consan more han doubles. newey ur L(/5).ur, lag() Regression wih Newey-Wes sandard errors Number of obs 84 maximum lag: F( 4, 809) 3.83 Prob > F Newey-Wes ur Coef. Sd. Err. P> [95% Conf. Inerval] ur L L L L _cons
38 newey and forecasing predic works afer newey command, bu no wih sdf opion newey no appropriae for ieraed forecass Use newey o assess model and examine coefficiens Use reg o compue ou-of-sample forecas inervals
39 Summary In one-sep-ahead forecas regressions If he errors are serially uncorrelaed, use r opion If he errors are correlaed Use newey for sandard errors In pure rend or seasonaliy models se m.4 /3 In dynamic regression se m.75 /3 n Use reg and predic sf, sdf for forecas inervals, or ieraed forecass wih forecas In h-sep-ahead forecas regressions Use newey wih mh for sandard errors Use reg and predic sf, sdf for forecas inervals
40 Join ess y α β β + + y + + p y p How do we assess if a subse of coefficiens are joinly zero? Example: 3 rd +4 h lags. reg gdp L(/4).gdp, r Linear regression Number of obs 75 F(4, 70) 9.86 Prob > F R-squared Roo MSE e Robus gdp Coef. Sd. Err. P> [95% Conf. Inerval] gdp L L L L _cons
41 Join Hypohesis his is a join es of β his can be done wih an F es β In SAA, afer regress (reg) or newey.es L3.gdp L4.gdp Lis variables whose coefficiens are esed for zero.
42 Join ess F es named afer R.A. Fisher (890-96) A founder of modern saisical heory Modern form known as a Wald es, named afer Abraham Wald (90-950) Early conribuor o economerics
43 F es compuaion. es L3.gdp L4.gdp ( ) L3.gdp 0 ( ) L4.gdp 0 F(, 70).09 Prob > F You need o lis each variable separaely SAA describes he hypohesis he value of F is he F-saisic Prob>F is he p-value Small p-values cause rejecion of hypohesis of zero coefficiens Convenionally, rejec hypohesis if p-value < 0.05
44 Example: -sep-ahead GDP AR(4). newey gdp L(/5).gdp, lag() Regression wih Newey-Wes sandard errors Number of obs 74 maximum lag: F( 4, 69) 3.33 Prob > F 0.00 Newey-Wes gdp Coef. Sd. Err. P> [95% Conf. Inerval] gdp L L L L _cons es L3.gdp L4.gdp L5.gdp ( ) L3.gdp 0 ( ) L4.gdp 0 ( 3) L5.gdp 0 F( 3, 69).3 Prob > F
45 esing afer Esimaion he commands predic and es are applied o he mos recenly esimaed model he command es uses he sandard error mehod specified by he esimaion command reg y x : classical F es reg r x, r: heeroskedasiciy-robus F es newey y x, lag(m): correlaion-robus F es (he robus ess are acually Wald saisics)
46 Assignmens Read Wooldridge Chaper. and.5 An elecronic copy is in files a Learn@UW Forecasing Projec Projec Descripion (3/8) Read Chaper 8 from he Signal and he Noise Reading Reflecion hursday (3/30)
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