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c Copyright (c) James G. MacKinnon, 1996 (corrected 2001-1-8)
c
c urcrouts.f: This is a set of subroutines to estimate critical values
c and P values for unit root and cointegration tests. It is written in
c Fortran 77. Simply call urcval, specifying the first seven arguments.
c The result comes back in the last argument.
c
c A standalone program called urcdist.f is also available.
c
c These routines and the associated data files may be used freely for
c non-commercial purposes, provided that proper attribution is made.
c Please cite the paper
c
c James G. MacKinnon, "Numerical distribution functions for unit root
c and cointegration tests," Journal of Applied Econometrics, 11,
c 1996, 601-618.
c
c The routines and datas may not be incorporated into any book
c or computer program without the express, written consent of the author.
c
c The routines must have access to the files probs.tab and .urc-#.tab
c for # = 1, 2, 3 ..., 12. As currently written, these files must be
c in the current directory or in the directory /usr/local/urcdist. Make
c sure that these files are in the proper format for your computer (i.e.
c that lines are terminated by CR/LF for DOS, Windows, and OS/2 systems
c and by LF alone for Unix systems).
c
c Author authorisation to use under GPL license:
c From - Sat Nov 15 11:56:30 2008
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c for matthieu.stigler@gmail.com; Fri, 14 Nov 2008 21:53:20 -0500
c Date: Fri, 14 Nov 2008 21:53:20 -0500 (EST)
c From: James MacKinnon <jgm@econ.queensu.ca>
c To: Matthieu Stigler <matthieu.stigler@gmail.com>
c Subject: Re: Program for surface response function
c In-Reply-To: <491D8DF3.20907@gmail.com>
c Message-ID: <Pine.LNX.4.62.0811142148490.4293@jgm.econ.queensu.ca>
c References: <491D8DF3.20907@gmail.com>
c MIME-Version: 1.0
c Content-Type: MULTIPART/MIXED; BOUNDARY="260199754-1936399293-1226717600=:4293"
c
c This message is in MIME format. The first part should be readable text,
c while the remaining parts are likely unreadable without MIME-aware tools.
c
c --260199754-1936399293-1226717600=:4293
c Content-Type: TEXT/PLAIN; charset=ISO-8859-1; format=flowed
c Content-Transfer-Encoding: QUOTED-PRINTABLE
c
c On Fri, 14 Nov 2008, Matthieu Stigler wrote:
c
c > mat@cunix:~/Repertoires/urcdist$ f77 -O urcrouts.f -o urcrout
c > /usr/lib/gcc/i486-linux-gnu/3.4.6/../../../../lib/libfrtbegin.a(frtbegin.o):
c > In function `main':
c > (.text+0x35): undefined reference to `MAIN__'
c > collect2: ld a retourn=E9 1 code d'=E9tat d'ex=E9cution
c
c The problem is simply that urcrouts.f contains no main program. Either add
c a main program that calls one or more routines from urcrouts.f (you will
c need this anyway), or compile with the -c switch (instead of -o urcrout).
c This will create urcrouts.o, but it won't be any use without a main
c program.
c
c > Furthermore, I contributed some functions for the R program and would be
c > interested to try to integrate your functions for the package devoted to
c > unit root tests (urca). Do you agree that your code will be used in a R
c > package under the GPL license?
c
c That would be fine with me. But I would want to make sure the R code works
c properly and provides appropriate citations before you make it available.
c
c Cheers,
c
c James G. MacKinnon
c Head, Department of Economics
c Queen's University
c Kingston, Ontario, Canada
c K7L 3N6
c
c Email: jgm@econ.queensu.ca
c Phone: 613 533-2293
c Fax: 613 533-6668
c --260199754-1936399293-1226717600=:4293--
C ******************************************************************************
subroutine fcrit(probs, cnorm, beta, wght, cval, size,
& precrt, nobs, model, nreg, np, nx)
implicit double precision (a-h,o-z)
c
c Copyright (c) James G. MacKinnon, 1995
c Routine to find a critical value for any specified test size.
c Uses GLS to estimate approximating regression.
c
double precision probs(221), cnorm(221), beta(4,221), crits(221),
& wght(221)
double precision yvect(20),xmat(20,4),xomx(4,4),resid(20),gamma(4)
double precision omega(20,20), fits(20)
diffm = 1000.d0
imin = 0
do i=1,221
diff = abs(size - probs(i))
if (diff.lt.diffm) then
diffm = diff
imin = i
if (diffm.lt.1.d-6) go to 100
end if
end do
100 continue
c
nph = np/2
nptop = 221 - nph
if (imin.gt.nph.and.imin.lt.nptop) then
c
c imin is not too close to the end. Use np points around stat.
c
do i=1,np
ic = imin - nph - 1 + i
call eval(beta(1,ic),crits(ic),model,nreg,nobs)
yvect(i) = crits(ic)
xmat(i,1) = 1.d0
xmat(i,2) = cnorm(ic)
xmat(i,3) = xmat(i,2)*cnorm(ic)
xmat(i,4) = xmat(i,3)*cnorm(ic)
end do
c
c form omega matrix
c
do i=1,np
do j=i,np
ic = imin - nph - 1 + i
jc = imin - nph - 1 + j
top = probs(ic)*(1.d0 - probs(jc))
bot = probs(jc)*(1.d0 - probs(ic))
omega(i,j) = wght(ic)*wght(jc)*sqrt(top/bot)
end do
end do
do i=1,np
do j=i,np
omega(j,i) = omega(i,j)
end do
end do
c
nx = 4
call gls(xmat,yvect,omega,gamma,xomx,fits,resid,ssr,ssrt,
& np,nx,20,4,0)
c
c check to see if gamma(4) is needed
c
sd4 = sqrt((ssrt/(np-nx))*xomx(4,4))
ttest = abs(gamma(4))/sd4
if (ttest.gt.precrt) then
call innorz(size,anorm)
cval = gamma(1) + gamma(2)*anorm + gamma(3)*anorm**2
& + gamma(4)*anorm**3
return
else
nx = 3
call gls(xmat,yvect,omega,gamma,xomx,fits,resid,ssr,ssrt,
& np,nx,20,4,1)
call innorz(size,anorm)
cval = gamma(1) + gamma(2)*anorm + gamma(3)*anorm**2
return
end if
c
c imin is close to one of the ends. Use points from imin +/- nph to end.
c
else
if (imin.lt.np) then
np1 = imin + nph
if (np1.lt.5) np1 = 5
do i=1,np1
call eval(beta(1,i),crits(i),model,nreg,nobs)
yvect(i) = crits(i)
xmat(i,1) = 1.d0
xmat(i,2) = cnorm(i)
xmat(i,3) = xmat(i,2)*cnorm(i)
xmat(i,4) = xmat(i,3)*cnorm(i)
end do
else
np1 = 222 - imin + nph
if (np1.lt.5) np1 = 5
do i=1,np1
call eval(beta(1,222-i),crits(222-i),model,nreg,nobs)
ic = 222 - i
yvect(i) = crits(ic)
xmat(i,1) = 1.d0
xmat(i,2) = cnorm(ic)
xmat(i,3) = xmat(i,2)*cnorm(ic)
xmat(i,4) = xmat(i,3)*cnorm(ic)
end do
end if
c
c form omega matrix
c
do i=1,np1
do j=i,np1
if (imin.lt.np) then
top = probs(i)*(1.d0 - probs(j))
bot = probs(j)*(1.d0 - probs(i))
omega(i,j) = wght(i)*wght(j)*sqrt(top/bot)
else
c
c This is to avoid numerical singularities at the upper end
c
omega(i,j) = 0.d0
if (i.eq.j) omega(i,i) = 1.d0
end if
end do
end do
do i=1,np1
do j=i,np1
omega(j,i) = omega(i,j)
end do
end do
c
nx = 4
call gls(xmat,yvect,omega,gamma,xomx,fits,resid,ssr,ssrt,
& np1,nx,20,4,0)
c
c check to see if gamma(4) is needed
c
sd4 = sqrt((ssrt/(np1-nx))*xomx(4,4))
ttest = abs(gamma(4)/sd4)
if (ttest.gt.precrt) then
call innorz(size,anorm)
cval = gamma(1) + gamma(2)*anorm + gamma(3)*anorm**2
& + gamma(4)*anorm**3
return
else
nx = 3
call gls(xmat,yvect,omega,gamma,xomx,fits,resid,ssr,ssrt,
& np1,nx,20,4,1)
call innorz(size,anorm)
cval = gamma(1) + gamma(2)*anorm + gamma(3)*anorm**2
return
end if
c
end if
end
C ******************************************************************************
subroutine fpval(beta, cnorm, wght, probs, pval, stat,
& precrt, nobs, model, nreg, np, nx)
implicit double precision (a-h,o-z)
c
c Copyright (c) James G. MacKinnon, 1995
c Routine to find P value for any specified test statistic.
c
double precision beta(4,221), crits(221), cnorm(221), wght(221),
& probs(221)
double precision yvect(20),xmat(20,4),resid(20),gamma(4)
double precision omega(20,20), fits(20), xomx(4,4)
c
c first, compute all the estimated critical values
c
do i=1,221
call eval(beta(1,i),crits(i),model,nreg,nobs)
end do
c
c find critical value closest to test statistic
c
diffm = 1000.d0
imin = 0
do i=1,221
diff = abs(stat - crits(i))
if (diff.lt.diffm) then
diffm = diff
imin = i
end if
end do
c
nph = np/2
nptop = 221 - nph
if (imin.gt.nph.and.imin.lt.nptop) then
c
c imin is not too close to the end. Use np points around stat.
c
do i=1,np
ic = imin - nph - 1 + i
yvect(i) = cnorm(ic)
xmat(i,1) = 1.d0
xmat(i,2) = crits(ic)
xmat(i,3) = xmat(i,2)*crits(ic)
xmat(i,4) = xmat(i,3)*crits(ic)
end do
c
c form omega matrix
c
do i=1,np
do j=i,np
ic = imin - nph - 1 + i
jc = imin - nph - 1 + j
top = probs(ic)*(1.d0 - probs(jc))
bot = probs(jc)*(1.d0 - probs(ic))
omega(i,j) = wght(ic)*wght(jc)*sqrt(top/bot)
end do
end do
do i=1,np
do j=i,np
omega(j,i) = omega(i,j)
end do
end do
c
nx = 4
call gls(xmat,yvect,omega,gamma,xomx,fits,resid,ssr,ssrt,
& np,nx,20,4,0)
c
c check to see if gamma(4) is needed
c
sd4 = sqrt((ssrt/(np-nx))*xomx(4,4))
ttest = abs(gamma(4))/sd4
if (ttest.gt.precrt) then
crfit = gamma(1) + gamma(2)*stat + gamma(3)*stat**2
& + gamma(4)*stat**3
call ddnor(crfit,pval)
return
else
nx = 3
call gls(xmat,yvect,omega,gamma,xomx,fits,resid,ssr,ssrt,
& np,nx,20,4,1)
crfit = gamma(1) + gamma(2)*stat + gamma(3)*stat**2
call ddnor(crfit,pval)
return
end if
else
c
c imin is close to one of the ends. Use points from imin +/- nph to end.
c
if (imin.lt.np) then
np1 = imin + nph
if (np1.lt.5) np1 = 5
do i=1,np1
yvect(i) = cnorm(i)
xmat(i,1) = 1.d0
xmat(i,2) = crits(i)
xmat(i,3) = xmat(i,2)*crits(i)
xmat(i,4) = xmat(i,3)*crits(i)
end do
else
np1 = 222 - imin + nph
if (np1.lt.5) np1 = 5
do i=1,np1
ic = 222 - i
yvect(i) = cnorm(ic)
xmat(i,1) = 1.d0
xmat(i,2) = crits(ic)
xmat(i,3) = xmat(i,2)*crits(ic)
xmat(i,4) = xmat(i,3)*crits(ic)
end do
end if
c
c form omega matrix
c
do i=1,np1
do j=i,np1
if (imin.lt.np) then
top = probs(i)*(1.d0 - probs(j))
bot = probs(j)*(1.d0 - probs(i))
omega(i,j) = wght(i)*wght(j)*sqrt(top/bot)
else
c
c This is to avoid numerical singularities at the upper end
c
omega(i,j) = 0.d0
if (i.eq.j) omega(i,i) = 1.d0
end if
end do
end do
do i=1,np1
do j=i,np1
omega(j,i) = omega(i,j)
end do
end do
c
nx = 4
call gls(xmat,yvect,omega,gamma,xomx,fits,resid,ssr,ssrt,
& np1,nx,20,4,0)
c
c check to see if gamma(4) is needed
c
sd4 = sqrt((ssrt/(np1-nx))*xomx(4,4))
ttest = abs(gamma(4))/sd4
if (ttest.gt.precrt) then
crfit = gamma(1) + gamma(2)*stat + gamma(3)*stat**2
& + gamma(4)*stat**3
call ddnor(crfit,pval)
else
nx = 3
call gls(xmat,yvect,omega,gamma,xomx,fits,resid,ssr,ssrt,
& np1,nx,20,4,1)
crfit = gamma(1) + gamma(2)*stat + gamma(3)*stat**2
call ddnor(crfit,pval)
end if
c
c check that nothing crazy has happened at the ends
c
if (imin.eq.1.and.pval.gt.probs(1)) pval = probs(1)
if (imin.eq.221.and.pval.lt.probs(221)) pval = probs(221)
return
end if
end
C ******************************************************************************
subroutine eval(beta,cval,model,nreg,nobs)
implicit double precision (a-h,o-z)
c
c Copyright (c) James G. MacKinnon, 1995
c Routine to evaluate response surface for specified betas and sample size.
c
double precision beta(4)
if (nobs.eq.0) then
cval = beta(1)
return
end if
if (model.eq.2) then
onobs = 1.d0/nobs
cval = beta(1) + beta(2)*onobs + beta(3)*onobs**2
return
end if
if (model.eq.3) then
onobs = 1.d0/nobs
cval = beta(1) + beta(2)*onobs + beta(3)*onobs**2
& + beta(4)*onobs**3
return
end if
if (model.eq.4) then
onobs = 1.d0/(nobs - nreg)
cval = beta(1) + beta(2)*onobs + beta(3)*onobs**2
return
end if
if (model.eq.5) then
onobs = 1.d0/(nobs - nreg)
cval = beta(1) + beta(2)*onobs + beta(3)*onobs**2
& + beta(4)*onobs**3
return
end if
C write(6,*) '*** Warning! Error in input file. ***'
return
end
C ******************************************************************************
subroutine gls(xmat,yvect,omega,beta,xomx,fits,resid,ssr,ssrt,
& nobs,nvar,nomax,nvmax,ivrt)
c
c Copyright (c) James G. MacKinnon, 1995
c Subroutine to do GLS estimation the obvious way
c Use only when sample size is small (nobs <= 50)
c 1995-1-3
c
implicit double precision (a-h,o-z)
double precision xmat(nomax,nvmax), yvect(nomax),
& omega(nomax,nomax)
double precision beta(nvmax), xomx(nvmax,nvmax), fits(nomax),
& resid(nomax)
double precision xomy(50)
c
c xomx is covariance matrix of parameter estimates if omega is truly known
c First, invert omega matrix if ivrt=0. Original one gets replaced.
c
if (ivrt.eq.0) call cholx(omega,nomax,nobs,kxx)
c
c form xomx matrix and xomy vector
c
do j=1,nvar
xomy(j) = 0.d0
do l=j,nvar
xomx(j,l) = 0.d0
end do
end do
c
do 21 i=1,nobs
do 21 k=1,nobs
do 24 j=1,nvar
xomy(j) = xomy(j) + xmat(i,j)*omega(k,i)*yvect(k)
do 24 l=j,nvar
xomx(j,l) = xomx(j,l) + xmat(i,j)*omega(k,i)*xmat(k,l)
24 continue
21 continue
c
do j=1,nvar
do l=j,nvar
xomx(l,j) = xomx(j,l)
end do
end do
c
c invert xomx matrix
c
call cholx(xomx,nvmax,nvar,kxx)
c
c now form estimates of beta.
c
do 5 i=1,nvar
beta(i) = 0.d0
do 5 j=1,nvar
beta(i) = beta(i) + xomx(i,j)*xomy(j)
5 continue
c
c find ssr, fitted values, and residuals
c
ssr = 0.d0
do i=1,nobs
fits(i) = 0.d0
do j=1,nvar
fits(i) = fits(i) + xmat(i,j)*beta(j)
end do
resid(i) = yvect(i) - fits(i)
ssr = ssr + resid(i)**2
end do
c
c find ssr from transformed regression
c
ssrt = 0.d0
do i=1,nobs
do k=1,nobs
ssrt = ssrt + resid(i)*omega(k,i)*resid(k)
end do
end do
c
return
end
C ******************************************************************************
subroutine cholx(amat,m,n,kxx)
implicit double precision (a-h,o-z)
c
c Copyright (c) James G. MacKinnon, 1993
c This routine uses the cholesky decomposition to invert a real
c symmetric matrix.
c
double precision amat(m,m)
kxx = 0
do 8 i=1,n
kl = i - 1
do 7 j=i,n
if (i.gt.1) then
do 3 k=1,kl
3 amat(i,j) = amat(i,j) - amat(k,i)*amat(k,j)
else
if (amat(i,i).le.0.d0) then
kxx = i
go to 20
end if
end if
if (i.eq.j) then
amat(i,i) = dsqrt(amat(i,i))
else
if (j.eq.i+1) ooa = 1.d0/amat(i,i)
amat(i,j) = amat(i,j)*ooa
end if
7 continue
8 continue
do 13 i=1,n
do 12 j=i,n
ooa = 1.d0/amat(j,j)
if (i.ge.j) then
t = 1.d0
go to 12
end if
kl = j - 1
t = 0.d0
do 11 k=i,kl
11 t = t - amat(i,k)*amat(k,j)
12 amat(i,j) = t*ooa
13 continue
do 16 i=1,n
do 15 j=i,n
t = 0.d0
do 14 k=j,n
14 t = t + amat(i,k)*amat(j,k)
amat(i,j) = t
19 amat(j,i) = t
15 continue
16 continue
20 return
end
C ******************************************************************************
subroutine ddnor(ystar,gauss)
implicit double precision(a-h,o-z)
c
c Copyright (c) James G. MacKinnon, 1993
c Routine to evaluate cumulative normal distribution
c Written originally in late 1970's
c Modified 1993 to avoid changing the argument
c
c This subroutine uses Cody's method to evaluate the cumulative
c normal distribution. It is probably accurate to 19 or 20
c significant digits. It was written in 1977, based on the Cody
c article referred to in the documentation for IMSL subroutine mdnor.
c
double precision p(6), q(5), a(9), b(8), c(5), d(4)
data p(1)/-6.58749161529837803157d-04/,
1 p(2)/-1.60837851487422766278d-02/,
2 p(3)/-1.25781726111229246204d-01/,
3 p(4)/-3.60344899949804439429d-01/,
4 p(5)/-3.05326634961232344035d-01/,
5 p(6)/-1.63153871373020978498d-02/
data q(1)/2.33520497626869185443d-03/,
1 q(2)/6.05183413124413191178d-02/,
2 q(3)/5.27905102951428412248d-01/,
3 q(4)/1.87295284992346047209d00/,
4 q(5)/2.56852019228982242072d00/
data a(1)/1.23033935479799725272d03/,
1 a(2)/2.05107837782607146532d03/,
2 a(3)/1.71204761263407058314d03/,
3 a(4)/8.81952221241769090411d02/,
4 a(5)/2.98635138197400131132d02/,
5 a(6)/6.61191906371416294775d01/,
6 a(7)/8.88314979438837594118d00/,
7 a(8)/5.64188496988670089180d-01/,
8 a(9)/2.15311535474403846343d-08/
data b(1)/1.23033935480374942043d03/,
1 b(2)/3.43936767414372163696d03/,
2 b(3)/4.36261909014324715820d03/,
3 b(4)/3.29079923573345962678d03/,
4 b(5)/1.62138957456669018874d03/,
5 b(6)/5.37181101862009857509d02/,
6 b(7)/1.17693950891312499305d02/,
7 b(8)/1.57449261107098347253d01/
data c(1)/3.209377589138469472562d03/,
1 c(2)/3.774852376853020208137d02/,
2 c(3)/1.138641541510501556495d02/,
3 c(4)/3.161123743870565596947d00/,
4 c(5)/1.857777061846031526730d-01/
data d(1)/2.844236833439170622273d03/,
1 d(2)/1.282616526077372275645d03/,
2 d(3)/2.440246379344441733056d02/,
3 d(4)/2.360129095234412093499d01/
data orpi/.5641895835477562869483d0/,
1 root2/.70710678118654752440083d0/
c
isw = 1
y = ystar
if (ystar.lt.-16.d0) y = -16.d0
if (ystar.gt.16.d0) y = 16.d0
x = -y*root2
if(x.gt.0.d0) go to 1
if(x.lt.0.d0) go to 2
gauss = .5d0
return
2 continue
x = - x
isw = -1
1 continue
if(x.lt..477d0) go to 10
if(x.le.4.d0) go to 20
c
c evaluate erfc for x.gt.4.0
c
x2 = x*x
xm2 = 1.d0/x2
xm4 = xm2*xm2
xm6 = xm4*xm2
xm8 = xm4*xm4
xm10 = xm6*xm4
top = p(1) + p(2)*xm2 + p(3)*xm4 + p(4)*xm6 + p(5)*xm8 + p(6)*xm10
bot = q(1) + q(2)*xm2 + q(3)*xm4 + q(4)*xm6 + q(5)*xm8 + xm10
crap = orpi + top/(bot*x2)
erfc = dexp(-x2)*crap/x
c
if(isw.eq.-1) erfc = 2.d0 - erfc
gauss = erfc*.5d0
return
20 continue
c
c evaluate erfc for .477.lt.x.le.4.0
c
x2 = x*x
x3 = x2*x
x4 = x2*x2
x5 = x3*x2
x6 = x3*x3
x7 = x3*x4
x8 = x4*x4
top = a(1) + a(2)*x + a(3)*x2 + a(4)*x3 + a(5)*x4 + a(6)*x5 +
& a(7)*x6 + a(8)*x7 + a(9)*x8
bot = b(1) + b(2)*x + b(3)*x2 + b(4)*x3 + b(5)*x4 + b(6)*x5 +
& b(7)*x6 + b(8)*x7 + x8
erfc = dexp(-x2)*top/bot
c
if(isw.eq.-1) erfc = 2.d0 - erfc
gauss = erfc*.5d0
return
10 continue
c
c evaluate erf for x.lt..477
c
x2 = x*x
x4 = x2*x2
x6 = x4*x2
x8 = x4*x4
top = c(1) + c(2)*x2 + c(3)*x4 + c(4)*x6 + c(5)*x8
bot = d(1) + d(2)*x2 + d(3)*x4 + d(4)*x6 + x8
erf = x*top/bot
c
erf = erf*isw
erfc = 1.d0 - erf
gauss = erfc*.5d0
return
end
C ******************************************************************************
subroutine innorz(prob,anorm)
implicit double precision (a-h,o-z)
c
c Copyright (c) James G. MacKinnon, 1995
c Inverse normal routine that adjusts crude result twice.
c It seems to be accurate to about 14 digits.
c Crude result is taken from Abramowitz & Stegun (1968)
c It should have abs. error < 4.5 * 10^-4
c
data c0/2.515517d0/, d1/1.432788d0/, c1/0.802853d0/
data c2/0.010328d0/, d3/0.001308d0/, d2/0.189269d0/
data const/.398942280401432678d0/
c if (prob.lt.0.d0.or.prob.gt.1.d0) then
c write(6,*) 'Attempt to find inverse normal of ', prob
c stop
c end if
pr = prob
if (prob.gt.0.5d0) pr = 1.d0 - prob
arg = 1/pr**2
t = sqrt(log(arg))
anorm = t - (c0 + c1*t + c2*t**2)/
& (1 + d1*t + d2*t**2 + d3*t**3)
c
c now correct crude result by direct method
c
call ddnor(anorm,prob2)
pr2 = 1.d0 - prob2
arg = 1/pr2**2
t = sqrt(log(arg))
anorm2 = t - (c0 + c1*t + c2*t**2)/
& (1 + d1*t + d2*t**2 + d3*t**3)
anorm = anorm + anorm - anorm2
if (prob.lt.0.5d0) anorm = -anorm
c
c now correct better result, using Taylor series approximation
c
call ddnor(anorm,prob2)
error = prob2 - prob
dens = const*dexp(-.5d0*anorm**2)
anorm = anorm - error/dens
return
end
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