Filardo (1994) maximum likelihood, problem with maximize

[Jules89]

Hello Tom,
again I have go a question about the replication of the Filardo (1994) Paper using maximum likelihood.
The code is:

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****************************************************************************************
************************* Reading in and Transforming the Data *************************
****************************************************************************************


cal(m) 1948
open data filardo.dat
data(format=prn,org=cols) 1948:1 1992:8 year month ip cli dfi xli sp fed tspread


set ipgr = 100.0*log(ip/ip{1})

stats(noprint) ipgr * 1959:12
compute stdearly=sqrt(%variance)
stats(noprint) ipgr 1960:1 *
compute stdlate=sqrt(%variance)

set ipgradjust = %if(t<=1959:12,ipgr*stdlate/stdearly,ipgr)
set g = ipgradjust
set cligr   = 100.0*log(cli/cli{1})
set spgr    = 100.0*log(sp/sp{1})
set xlidiff = xli-xli{1}
set ffdiff  = fed-fed{1}
set tspdiff = tspread-tspread{1}
set dfismooth = dfi+2*dfi{1}+2*dfi{2}+dfi{3}


* Center variabes to zero means
dofor s = cligr spgr xlidiff ffdiff tspdiff dfismooth
   diff(center) s
end dofor s


*****************************************************************************************
**************************** Setting up the Markov Switching ****************************
*****************************************************************************************

source msvarsetup.src

compute nlags=4

compute gstart=1948:6,gend=1992:8
@MSVARSetup(lags=nlags, Switch=M)
# g

nonlin(parmset=common) mu phi sigma
nonlin(parmset=fixed)  p

frml loglftp = log(%MSVarProb(t))

@MSVARInitial gstart gend

maximize(parmset=fixed+common,start=(pstar=%MSVARInit()), reject=%MSVARInitTransition()==0.0,$
 method=bfgs) loglftp gstart gend

**********************************************************************
******************** Time varying probabilities **********************
**********************************************************************

dec equation p1eq p2eq
dec vector   v1   v2
nonlin(parmset=tv) v1 v2

**********************************************************************
********** Overwriting the standard fixed transition matrix **********
**********************************************************************

function  %MSVARPmat  time
type rect %MSVARPmat
type int  time
*
local integer i j
local rect    pexpand
local real    z
*
compute p(1,1)=%(z=exp(%dot(%eqnxvector(p1eq,time),v1)),z/(1+z))
compute p(1,2)=%(z=exp(%dot(%eqnxvector(p2eq,time),v2)),1/(1+z))
compute %MSVARInitTransition()
*
if nexpand==nstates {
   dim pexpand(nstates,nstates-1)
   ewise pexpand(i,j)=%if(i==nstates,-1.0,i==j)
   compute %MSVARPmat=pexpand*p
   ewise %MSVARPmat(i,j)=%if(i==nstates,1.0+%MSVARPmat(i,j),%MSVARPmat(i,j))
}
else {
   dim %MSVARPmat(nexpand,nexpand)
   ewise %MSVARPmat(i,j)=MSVARTransProbs(MSVARTransLookup(i,j))
}
end

**********************************************************************
******* Initialize the probabilities using only the intercepts *******
**********************************************************************

function TVPInit
type vector TVPInit
*
local rect a
local integer i j
*
compute p=||%logistic(v1(1),1.0),1-%logistic(v2(1),1.0)||
compute TVPInit=%MSVARInit()
end

**********************************************************************
********************* Calculating the Likelihood *********************
**********************************************************************

frml logltvtp = %MSVARPMat(t),log(%MSVarProb(t))

**********************************************************************
****************** Initialize the other Parameters *******************
**********************************************************************

*** Define the logistic index for the transitions
equation p1eq *
# constant cligr{1}
equation p2eq *
# constant cligr{1}


***Starting Values Except Sigma (initialized in @MSVARInitial)
compute v1=log(.7/.3)~~0.0
compute v2=log(.95/.05)~~0.0
compute mu(1)=-1.7,mu(2)=.3


*********************************************************************
**************** Maximization of the Log-Likelihood *****************
*********************************************************************

maximize(parmset=tv+common,start=(pstar=TVPInit()),$
 method=bfgs) logltvtp gstart gend

The code produces a slightly different output, whenever I rerun the following part:

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**********************************************************************
****************** Initialize the other Parameters *******************
**********************************************************************

*** Define the logistic index for the transitions
equation p1eq *
# constant cligr{1}
equation p2eq *
# constant cligr{1}


***Starting Values Except Sigma (initialized in @MSVARInitial)
compute v1=log(.7/.3)~~0.0
compute v2=log(.95/.05)~~0.0
compute mu(1)=-1.7,mu(2)=.3


*********************************************************************
**************** Maximization of the Log-Likelihood *****************
*********************************************************************

maximize(parmset=tv+common,start=(pstar=TVPInit()),$
 method=bfgs) logltvtp gstart gend

For the first time it produces:

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Convergence in    21 Iterations. Final criterion was  0.0000062 <=  0.0000100
Monthly Data From 1948:06 To 1992:08
Usable Observations                       531
Function Value                      -595.8451

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  V1(1)                         1.330191142  0.498559139      2.66807  0.00762882
2.  V1(2)                        -1.191674820  0.575598988     -2.07032  0.03842227
3.  V2(1)                         4.285987836  0.743873498      5.76172  0.00000001
4.  V2(2)                         1.649581410  0.532863384      3.09569  0.00196354
5.  MU(1)(1)                     -0.881213302  0.184496277     -4.77632  0.00000179
6.  MU(2)(1)                      0.472025129  0.073745579      6.40072  0.00000000
7.  PHI(1)(1,1)                   0.210996431  0.046587036      4.52908  0.00000592
8.  PHI(2)(1,1)                   0.072441700  0.047977295      1.50992  0.13106478
9.  PHI(3)(1,1)                   0.103730162  0.048833808      2.12415  0.03365789
10. PHI(4)(1,1)                   0.102676182  0.049932939      2.05628  0.03975539
11. SIGMA(1,1)                    0.472889311  0.034647662     13.64852  0.00000000

For the second one:

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MAXIMIZE - Estimation by BFGS
Convergence in    21 Iterations. Final criterion was  0.0000062 <=  0.0000100
Monthly Data From 1948:06 To 1992:08
Usable Observations                       531
Function Value                      -595.8451

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  V1(1)                         1.330191138  0.498557616      2.66808  0.00762863
2.  V1(2)                        -1.191674820  0.575600618     -2.07032  0.03842282
3.  V2(1)                         4.285987824  0.743866100      5.76177  0.00000001
4.  V2(2)                         1.649581403  0.532856659      3.09573  0.00196328
5.  MU(1)(1)                     -0.881213302  0.184495564     -4.77634  0.00000179
6.  MU(2)(1)                      0.472025129  0.073745368      6.40074  0.00000000
7.  PHI(1)(1,1)                   0.210996431  0.046587035      4.52908  0.00000592
8.  PHI(2)(1,1)                   0.072441700  0.047977564      1.50991  0.13106694
9.  PHI(3)(1,1)                   0.103730162  0.048833803      2.12415  0.03365788
10. PHI(4)(1,1)                   0.102676182  0.049932854      2.05629  0.03975505
11. SIGMA(1,1)                    0.472889311  0.034647353     13.64864  0.00000000

For the third one:

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MAXIMIZE - Estimation by BFGS
Convergence in    21 Iterations. Final criterion was  0.0000062 <=  0.0000100
Monthly Data From 1948:06 To 1992:08
Usable Observations                       531
Function Value                      -595.8451

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  V1(1)                         1.330191125  0.498560770      2.66806  0.00762901
2.  V1(2)                        -1.191674824  0.575612285     -2.07027  0.03842675
3.  V2(1)                         4.285987810  0.743827047      5.76208  0.00000001
4.  V2(2)                         1.649581394  0.532831667      3.09588  0.00196232
5.  MU(1)(1)                     -0.881213303  0.184494672     -4.77636  0.00000178
6.  MU(2)(1)                      0.472025128  0.073746027      6.40069  0.00000000
7.  PHI(1)(1,1)                   0.210996431  0.046587067      4.52908  0.00000592
8.  PHI(2)(1,1)                   0.072441700  0.047978115      1.50989  0.13107137
9.  PHI(3)(1,1)                   0.103730162  0.048833952      2.12414  0.03365842
10. PHI(4)(1,1)                   0.102676182  0.049933545      2.05626  0.03975779
11. SIGMA(1,1)                    0.472889311  0.034647396     13.64862  0.00000000

So some of the estimated parmeters are slightly different, but that should not be when I Use the same code with the same data, shouldn't it?


Thank you in advance and best regards

Jules

[TomDoan]

What you're doing resets the v's and the mu's but not the phi's and sigma's. Different phi's and sigma's means (slightly) different optimization paths which means slightly different estimates.

[Jules89]

So would adding a @MSVARINITIAL before the maximize instruction solve the problem?

[TomDoan]

Probably. But why is having estimates agreeing to eight significant digits a "problem"?

[Jules89]

Just wanted to make sure that my results are completely correct. When I use this code for other applications, there might be bigger differences when I rerun it again and again

[TomDoan]

MS models are notorious for multiple modes, often very different modes with similar likelihoods. However, if you're getting convergence to the same mode, the coefficients will agree to at least six significant digits. You really can't ask for more than that.