Components Model. Remember that we said that it was useful to think about the components representation

Transcription

1 Componens Model Remember ha we said ha i was useful o hink abou he componens represenaion = T S C Suppose ha C is an AR(p) process Wha model does his impl for?

2 TrendCcle Model For simplici, we sar wih he rend-ccle model = T C And specif he ccle as an AR() C = βc e

3 Inercep onl Suppose he rend is jus an inercep The model is T C = µ = µ = βc C e

4 Parial Differencing Lag he firs equaion, mulipl b β and subrac = µ C Then use β = µ C = ( β ) µ C βc C C = β e o find ( β ) µ β e =

5 Equivalence wih AR() Thus implies wih C = µ C = βc = α β α = ( β )µ e e The model is jus an AR() wih inercep

6 Linear Trend Suppose he rend is a linear ime rend T = µ µ 2 Then C = = µ µ βc 2 e C

7 Parial Differencing Lag he firs equaion, mulipl b β and subrac, and use AR() equaion We find ( ) ( ) ( ) e C C = = = µ β βµ µ β β µ µ µ µ e = 2 β β α

8 Summar : TrendAR() Ccle When he rend is an inercep or a ime rend The componens model = T C C is equivalen wih The componens model is equivalen wih a regression on he rend variables and he lag = βc = T β e e

9 AR(p)Trend A linear rend plus AR(p) is equivalen o A regression on a ime rend plus p lags of p p e C C C C = = β β µ µ 2 p p e = β β γ α

10 Example: Real GDP q 960q 970q 980q 990q 2000q 200q 2020q ime ln_rgdp Fied values Ln(rgdp) and linear rend

11 Residuals from Linear Trend Residuals q 960q 970q 980q 990q 2000q 200q 2020q ime

12 AR(4) on residuals. reg u L(/4).u Source SS df MS Number of obs = 276 F(4, 27) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE = u Coef. Sd. Err. P> [95% Conf. Inerval] u L L L L _cons

13 Fied Values q 960q 970q 980q 990q 2000q 200q 2020q ime Residuals Fied values

14 AR(4) wih rend. reg ln_rgdp ime L(/4).ln_rgdp Source SS df MS Number of obs = 276 F(5, 270) > Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.0087 ln_rgdp Coef. Sd. Err. P> [95% Conf. Inerval] ime ln_rgdp L L L L _cons

15 Fied Values from AR(4) wih Trend q 960q 970q 980q 990q 2000q 200q 2020q ime ln_rgdp Fied values

16 Las 8 ears q 2008q 200q 202q 204q 206q ime ln_rgdp Fied values

17 Residuals Residuals q 960q 970q 980q 990q 2000q 200q 2020q ime

18 Forecass Same as for AR(p) models, bu include ime rend as a regressor h-sep forecas based on α γ β β = h p h p e

19 Forecas for ln(gdp) using AR(4)rend ln(gdp) 2008q 200q3 203q 205q3 208q ime lower forecas inerval forecas upper forecas inerval

20 Forecas for GDP (using exponenial) GDP 2008q 200q3 203q 205q3 208q ime Real Gross Domesic Produc epl ep epu

21 Direc Inerval Forecass sappend, add(4) gen =ln(gdp) reg ime L(/4). predic predic sf,sdf gen L=-.645*sf gen U=.645*sf reg ime L(2/5). predic 2 predic sf2,sdf gen 2L=2-.645*sf2 gen 2U=2.645*sf2 reg ime L(3/6). predic 3 predic sf3,sdf gen 3L=3-.645*sf3 gen 3U=3.645*sf3 reg ime L(4/7). predic 4 predic sf4,sdf gen 4L=4-.645*sf4 gen 4U=4.645*sf4 egen p=rowfirs( 2 3 4) if >=q(207q) egen pl=rowfirs(l 2L 3L 4L) if >=q(207q) egen pu=rowfirs(u 2U 3U 4U) if >=q(207q) label variable p "forecas" label variable pl "lower forecas inerval" label variable pu "upper forecas inerval" sline p pl pu if ime>=q(2008q), ile(ln(gdp)) lpaern (solid longdash shordash shordash) gen ep=exp(p) gen epl=exp(pl) gen epu=exp(pu) sline gdp ep epl epu if ime>=q(2008q), ile(gdp) lpaern (solid longdash shordash shordash)

22 Ieraed Forecas for GDP GDP 2008q 200q3 203q 205q3 208q ime Real Gross Domesic Produc lower forecas inerval ieraed forecas upper forecas inerval

23 Ieraed Inerval Forecass reg ime L(/4). forecas creae ar4 esimae sore model forecas esimaes model forecas solve, simulae(errors,saisic(sddev,prefix(sd_)) reps(000) ) gen f=exp(f_) if ime>=q(207q) gen fl=exp(f_-.645*sd_) gen fu=exp(f_.645*sd_) label variable f "ieraed forecas" label variable fl "lower forecas inerval" label variable fu "upper forecas inerval" sline gdp f fl fu if ime>=q(2008q), ile(gdp) lpaern (solid longdash shordash shordash)

24 Example: Log Sock Volume jan950 0jan960 0jan970 0jan980 0jan990 0jan2000 0jan200 0jan2020 ime index lvolume Fied values Log Volume and Linear Trend

25 Residuals Residuals jan9500jan9600jan9700jan9800jan9900jan20000jan2000jan2020 ime index

26 Model Weekl Daa Fi AR(52)rend

27 Daa and Fied jan950 0jan960 0jan970 0jan980 0jan990 0jan2000 0jan200 0jan2020 ime index lvolume Fied values

28 Las Two Years of Sample jan205 0jul205 0jan206 0jul206 0jan207 ime index lvolume Fied values

29 Residuals Residuals jan9500jan9600jan9700jan9800jan9900jan20000jan2000jan2020 ime index

30 Trend Omission Suppose he ruh is ha he daa have a rend, bu ou fi an AR model wihou a rend. Wha happens? Suppose = µ µ 2 Then = µ 2

31 Example Since If ou esimae an AR(), ou obain You esimae a uni coefficien on he AR lag µ 2 = ˆ ˆ ˆ ˆ ˆ 2 2 = = = = β µ α µ β α e

32 General Effec of Trend Omission If he ruh is = µ µ 2 β Bu ou esimae an AR() wihou a rend ˆ α ˆ β eˆ = Then ou end o find ˆ β This is due o model misspecificaion e

33 Simulaed Example =.0.3 e. gen e=rnormal(0). gen =e. replace =.0*.3*L.e if > (499 real changes made)

34 Simulaed Process

35 Esimae AR() wihou Trend. reg L. Source SS df MS Number of obs = 499 F(, 497) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.676 Coef. Sd. Err. P> [95% Conf. Inerval] L _cons The esimaed AR() coefficien is 0.85, much oo large (rue value was 0.3)

36 Esimae AR() wih Trend. reg L. Source SS df MS Number of obs = 499 F(2, 496) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.962 Coef. Sd. Err. P> [95% Conf. Inerval] L _cons The esimaed AR coef is 0.25, close o he rue 0.3 The esimaed rend coef is 0.0, close o he rue 0.0 The roo MSE decreases from.7 o 0.96

37 Miderm Tuesda March 7, during class Maerial: Lecures hrough las week Problem Ses -6 Diebold: Chapers -7 No on miderm: Chaper 9 & Toda s lecure Review: During class Thursda Bring Quesions!!!!

38 Assignmens Read Diebold Chaper 9 (maerial no on miderm) Read Chaper 6 from The Signal and he Noise Reading Reflecion Due Thursda (3/2) Miderm Tuesda (3/7) Problem Se # 7 Due Tuesda (3/4)