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### Determination of deterministic regressors (ADF)

Posted: Mon Nov 10, 2014 11:39 am
Hi,

The code below shows a way to perform the Augmented Dickey-Fuller test and to detect deterministic components in a series. It is based on Enders, W. (2008). Applied econometric time series. John Wiley & Sons. As the author mention no procedure can be expected to work well if it is used in a completely mechanical fashion, therefore the procedure should be taken as a suggestion.

Code: Select all

'Nicolas Ronderos Pulido. November of 2014.
'--------------------------Inputs-------------------------------
%series="x" 'Series name
scalar significance=0.1 'Significance levels for the tests 0.1,0.05,0.025 or 0.01
'------------------------------------------------------------------
'Step 1
matrix results
freeze(uroot_ct){%series}.uroot(trend,info=aic,save=results)
if uroot_ct(7,5)<=significance then 'If the null is rejected (there is not unit root)
@uiprompt("The series does not contain a unit root")
delete results significance uroot_ct
stop
endif

'Step 2

!lags=results(2,1)
for !i=1 to !lags
%lag=@str(-!i)
%lags=%lags+"@d("+%series+"("+%lag+"))"
next
equation restricted.ls @d({%series}) c {%lags}
!ssr_r=@ssr
equation unrestricted.ls @d({%series}) {%series}(-1) {%lags} c @trend
freeze(unrestricted_tc) unrestricted
!ssr_ur=@ssr
scalar phi_3=((!ssr_r-!ssr_ur)/2)/(!ssr_ur/(results(1,1)-!lags-3))
scalar obs=@obs({%series})
vector critical_values
call cv(obs,significance,critical_values)
if phi_3>critical_values(1,3) then ' If the null is rejected (the trend and gamma are non-zero)
equation t_test1.ls @d({%series}) c @trend {%lags}
freeze(t_test_1) t_test1
if t_test_1(10,5)<=significance then 'If the null is rejected (the trend is significative)
if unrestricted_tc(9,5)<=significance then ' If the null is rejected (gamma is non-zero)
@uiprompt("The series is trend-stationary")
stop
else
@uiprompt("The series contain a unit root and a quadratic trend") '
delete critical_values obs phi_3 restricted results significance t_test1 t_test_1 unrestricted unrestricted_tc uroot_ct
stop
endif
endif
endif

'Step 3

freeze(uroot_c){%series}.uroot(const,info=aic,save=results)
if uroot_c(7,5)<=significance then 'If the null is rejected (there is not unit root)
@uiprompt("The series does not contain a unit root")
stop
endif

if !lags=0 then
genr first=@d({%series})
!ssr_r=@sumsq(first)
else
equation restricted.ls @d({%series}) {%lags}
!ssr_r=@ssr
endif
equation unrestricted.ls @d({%series}) c {%series}(-1) {%lags}
freeze(unrestricted_c) unrestricted
!ssr_ur=@ssr
scalar phi_1=((!ssr_r-!ssr_ur)/2)/(!ssr_ur/(results(1,1)-!lags-2))
if phi_1>critical_values(1,1) then 'If the null is rejected (the constant and gamma are non-zero)
equation t_test2.ls @d({%series}) c {%lags}
freeze(t_test_2) t_test2
if t_test_2(9,5)<=significance then 'If the null is rejected (the constant is significative)
if unrestricted_c(10,5)<=significance then 'If the null is rejected (gamma is non-zero)
@uiprompt("The series is stationary around a nonzero mean")
stop
else
@uiprompt("The series contain a unit root and a drift") '
delete critical_values obs phi_1 phi_3 restricted significance t_test2 t_test_2 unrestricted unrestricted_c unrestricted_tc uroot_c uroot_ct results
stop
endif
endif
endif

'Step 4

freeze(uroot){%series}.uroot(none,info=aic,save=results)
if uroot(7,5)<=significance then  'If the null is rejected (there is not unit root)
@uiprompt("The series does not contain a unit root")
stop
else
@uiprompt("The series contain a unit root") '
'delete critical_values obs first phi_1 phi_3 restricted results significance unrestricted unrestricted_c unrestricted_tc uroot uroot_c uroot_ct
stop
endif

'Critical values subroutine
subroutine local cv(scalar obs,scalar significance,vector critical_values)
genr n_=NA
n_.fill 25,50,100,250,500,1000
stom(n_,n)
vector(6) nearby=@abs(n-obs)

for !i=1 to 6
if nearby(!i,1)=@min(nearby) then
!position=!i
endif
next

genr criticalphi1=NA
genr criticalphi2=NA
genr criticalphi3=NA

if significance=0.1 then
criticalphi1.fill 4.12,3.94,3.86,3.81,3.79,3.78
criticalphi2.fill 4.67,4.31,4.16,4.07,4.05,4.03
criticalphi3.fill 5.91,5.61,5.47,5.39,5.36,5.34
stom(criticalphi1,critical_phi1)
stom(criticalphi2,critical_phi2)
stom(criticalphi3,critical_phi3)
endif

if significance=0.05 then
criticalphi1.fill 5.18,4.86,4.71,4.63,4.61,4.59
criticalphi2.fill 5.68,5.13,4.88,4.75,4.71,4.68
criticalphi3.fill 7.24,6.73,6.49,6.34,6.30,6.25
stom(criticalphi1,critical_phi1)
stom(criticalphi2,critical_phi2)
stom(criticalphi3,critical_phi3)
endif

if significance=0.025 then
criticalphi1.fill 6.30,5.80,5.57,5.45,5.41,5.38
criticalphi2.fill 6.75,5.94,5.59,5.40,5.35,5.31
criticalphi3.fill 8.65,7.81,7.44,7.25,7.20,7.16
stom(criticalphi1,critical_phi1)
stom(criticalphi2,critical_phi2)
stom(criticalphi3,critical_phi3)
endif

if significance=0.01 then
criticalphi1.fill 7.88,7.06,6.70,6.52,6.47,6.43
criticalphi2.fill 8.21,7.02,6.50,6.22,6.15,6.09
criticalphi3.fill 10.61,9.31,8.73,8.43,8.34,8.27
stom(criticalphi1,critical_phi1)
stom(criticalphi2,critical_phi2)
stom(criticalphi3,critical_phi3)
endif

vector(1,3) critical_values
for !i=1 to 3
critical_values(1,!i)=critical_phi!i(!position)
next

endsub

As an example you can perfom the procedure in simulated processes. You just have to change the series name in quotes (%series).

Code: Select all

wfcreate u 1 1000
genr x=0
genr y=0
genr z=0
smpl 2 1000
x=x(-1)+nrnd
y=0.05+y(-1)+nrnd
z=0.05+0.1*@trend+z(-1)+nrnd
smpl @all

A summary of the algorithm can be found in the attached graph.