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* Standard state space models and ARIMA models
*
OPEN DATA bmonth201310.txt
CALENDAR(M) 2004:1
DATA(FORMAT=PRN,NOLABELS,ORG=COLUMNS) 2004:01 2013:10 lgdp
*
* Everything is run on 50*log data
*
set lgdp  = 50.0*lgdp
set d2006 = t>2006:4
*
*
* UC-0 decomposition with AR(2) cycle, broken trend rate.
*
nonlin mu d sn ph1 ph2 se
*
dec frml[rect] af
dec frml[vect] zf
dec frml[symm] swf
*
frml af = ||1.0,0.0,0.0|$
            0.0,ph1,ph2|$
            0.0,1.0,0.0||
frml zf = ||mu+d*d2006,0.0,0.0||
frml swf = %diag(||sn^2,se^2||)
*
compute [vect] c=||1.0,1.0,0.0||
compute [rect] f=%identity(2)~~%zeros(1,2)
*
* Get initial guess values
*
filter(type=hp) lgdp / gdp_hp
set gap_hp = lgdp - gdp_hp
linreg gap_hp
# gap_hp{1 2}
compute ph1=%beta(1),ph2=%beta(2),se=sqrt(%seesq)
set trend = t
linreg gdp_hp
# constant trend
compute mu=%beta(2)
compute sn=sqrt(.1*%seesq)
*
dlm(presample=ergodic,a=af,z=zf,sw=swf,c=c,f=f,y=lgdp,method=bfgs,type=filter) / states0
*
set cycle0 = states0(t)(2)
set trend0 = states0(t)(1)
*
graph(footer="Figure 1 UC-0 Cycle, US Real GDP. Percent Deviation from Trend")
# cycle0
*
* Unrestricted UC model, allowing for correlation between the shocks.
* The maximum occurs at rho=1 (possible since we have only one
* observable, so having just a single effective shock is OK).
*
compute rhone=0.0,sn=.1
nonlin mu d sn ph1 ph2 se rhone=1.0
*
frml swf = ||sn^2|rhone*abs(sn)*abs(se),se^2||
dlm(presample=ergodic,a=af,z=zf,sw=swf,c=c,f=f,y=lgdp,method=bfgs,type=filter) / statesur
set cycleur = statesur(t)(2)
set trendur = statesur(t)(1)
graph(footer="Figure 3 UC-UR Cycle, US Real GDP")
# cycleur
*
* Estimation of unrestricted ARIMA(2,1,2) model. BOXJENK doesn't find
* the boundary solution because it pushes away from unit roots in the
* moving average part.
*
boxjenk(ar=2,diffs=1,ma=2,maxl,noapplydiffs,reg) lgdp
# constant d2006
*
* Estimate of constrained model (AR(2) with broken trend)
*
set trend  = t
set dtrend = %max(0,t-2006:4)
boxjenk(ar=2,diffs=0,reg,maxl) lgdp
# constant trend dtrend