MS-ARIMA

[fan]

Hi, Tom. Is there any example about MS-ARIMA model available for people to study? I am trying extend Hamilton's MS-VAR model to an MS-ARMA model. Thanks

[TomDoan]

That would be a special case of a MS State-Space model. It has the path-dependence problem that requires some form of approximation such as the Kim filter.

[fan]

That would be a special case of a MS State-Space model. It has the path-dependence problem that requires some form of approximation such as the Kim filter.

Hi Tom. I am trying to replicate the example of KIMNP111.RPF
5.4.2 from pp 111-115 of Kim and Nelson, "State-space Models with Regime Switching". Lam's model of GNP, estimation with Kim's filter.

I am having following error messages. Could you please help me understand why I am having the message?

Code: Select all

The Error Occurred At Location 170, Line 13 of %MSINIT
Called From Location 127, Line 13 of MSFILTERINIT
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.

The Error Occurred At Location 364, Line 26 of KIMFILTER
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.
The Error Occurred At Location 364, Line 26 of KIMFILTER
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.
The Error Occurred At Location 364, Line 26 of KIMFILTER
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.
The Error Occurred At Location 364, Line 26 of KIMFILTER
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.
The Error Occurred At Location 364, Line 26 of KIMFILTER
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.
The Error Occurred At Location 364, Line 26 of KIMFILTER
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.
The Error Occurred At Location 364, Line 26 of KIMFILTER
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.
The Error Occurred At Location 364, Line 26 of KIMFILTER
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.

In addition, could you please let me know how I can incorporate time-varying transition probability in example?

Thank you

[fan]

That would be a special case of a MS State-Space model. It has the path-dependence problem that requires some form of approximation such as the Kim filter.

Hi Tom, I am trying to estimate a ARMA(2,4) model with mean-switching. I came up my code but there are error messages which i can not figure out what the mistakes are in my code. Could you kindly take a look my code and let me know what mistakes I made?

error message

Code: Select all

The Error Occurred At Location 170, Line 13 of %MSINIT
Called From Location 127, Line 13 of MSFILTERINIT
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.
## MAT2. Matrices with Dimensions 5 x 5 and 2 x 1 Involved in * Operation
The Error Occurred At Location 175, Line 17 of KIMFILTER

My code

Code: Select all

* This is common to any MS state space model analyzed using the
* Kim filter.
*
@MSSetup(states=2)
*
* These are the collapsed SSM mean and variance from the previous time
* period.
*
dec vect[vect] xstar(nstates)
dec vect[symm] sstar(nstates)
*
* This is the filtered best estimate for the state vector
*
dec series[vect] xstates
*
*********************************************************************
*
*
* Fill in the fixed coefficients in the A, C and F matrices
*
compute ndlm=5
dec rect a(ndlm,ndlm)
input a
. . 0 0 0
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0

dec vect phi(2)

dec rect c(1,ndlm)
input c
1 . . . .

dec vect tau(4)

dec rect f(ndlm,1)
compute f=%unitv(ndlm,1)
dec symm sw(ndlm,ndlm)
*
dec vect mu(nstates)
compute sv=0.0
*
dec vect x0(ndlm)
compute x0=%zeros(ndlm,1)
*
nonlin(parmset=initparms) x0
nonlin(parmset=dlmparms) mu sigsq phi tau
nonlin(parmset=msparms) theta
compute theta=%const(0.0)

*********************************************************************
*
* This does a single step of the Kim (approximate) filter
*
function KimFilter time
type integer 		time
*
local integer    	i j
local real       	yerr likely
local symm       	vhat
local rect       	gain
local rect       	phat(nstates,nstates)
local rect    	  	fwork(nstates,nstates)
local rect[vect] 	xwork(nstates,nstates)
local rect[symm] 	swork(nstates,nstates)
*
do i=1,nstates
   do j=1,nstates
      *
      * Do the SSM predictive step
      *
      compute xwork(i,j)=a*xstar(j)
      compute swork(i,j)=a*sstar(j)*tr(a)+sw
      *
      * Do the prediction error for y under state i, and compute the
      * density function for the prediction error.
      *
      compute yerr=indgrowth(time)-(%dot(c,xwork(i,j))+mu(i))
      compute vhat=c*swork(i,j)*tr(c)+sv
      compute gain=swork(i,j)*tr(c)*inv(vhat)
      compute fwork(i,j)=exp(%logdensity(vhat,yerr))
      *
      * Do the SSM update step
      *
      compute xwork(i,j)=xwork(i,j)+gain*yerr
      compute swork(i,j)=swork(i,j)-gain*c*swork(i,j)
   end do j
end do i
*
* Compute the Hamilton filter likelihood
*
compute likely=0.0
do i=1,nstates
   do j=1,nstates
      compute phat(i,j)=p(i,j)*pstar(j)*fwork(i,j)
      compute likely=likely+phat(i,j)
   end do j
end do i
*
* Compute the updated probabilities of the state combinations
*
compute phat=phat/likely
compute pstar=%sumr(phat)
compute pt_t(time)=pstar
*
* Collapse the SSM matrices down to one per state
*
compute xstates(time)=%zeros(ndlm,1)
do i=1,nstates
   compute xstar(i)=%zeros(ndlm,1)
   do j=1,nstates
      compute xstar(i)=xstar(i)+xwork(i,j)*phat(i,j)/pstar(i)
   end do j
   compute sstar(i)=%zeros(ndlm,ndlm)
   do j=1,nstates
      compute sstar(i)=sstar(i)+phat(i,j)/pstar(i)*$
         (swork(i,j)+%outerxx(xstar(i)-xwork(i,j)))
   end do j
   *
   * This is the overall best estimate of the filtered state.
   *
   compute xstates(time)=xstates(time)+xstar(i)*pstar(i)
end do i
compute KimFilter=likely
end
*********************************************************************
*
* This is the start up code for each function evaluation
*
function DLMStart
*
local integer i
*
* Fill in the top row in the A matrix
*
compute %psubmat(a,1,1,tr(phi))
*
* Fill in the C matrix
*
compute %psubmat(c,1,2,tr(tau))
*
* Compute the full matrix for the transition variance
*
compute sw=f*tr(f)*sigsq
*
* Transform the theta to transition probabilities.
*
compute p=%MSLogisticP(theta)
compute pstar=%mcergodic(p)
compute p=%mspexpand(p)
*
* Initialize the KF state and variance. This is set up for estimating
* the pre-sample states.
*
ewise xstar(i)=x0
ewise sstar(i)=%zeros(ndlm,ndlm)
end
*********************************************************************
*
* Get guess values for the means from running an AR(2) on the growth
* rate. However, the guess values for the AR coefficients need to be
* input.
*
boxjenk(ar=2,constant) indgrowth
compute mu(1)=%beta(1)+1.0,mu(2)=%beta(1)-1.0
compute phi=||1.2,-.3||
compute sigsq=%seesq
compute tau=||0.1,0.1,0.1,-0.4||

*
frml kimf = kf=KimFilter(t),log(kf)
compute x0=||5.224,2.699||
*********************************************************************
gset xstates = %zeros(ndlm,1)
@MSFilterInit
maximize(parmset=msparms+dlmparms,start=DLMStart(),method=bfgs,trace) kimf 1945:04 *
maximize(parmset=msparms+dlmparms+initparms,start=DLMStart(),method=bfgs,trace) kimf  1945:04 *

In addition, if I would like to examine the transition probabilities in different periods, such as initial 2 years and last 2 years of a presidency, how i should modify the code here. Many Thanks.

[TomDoan]

That would be a special case of a MS State-Space model. It has the path-dependence problem that requires some form of approximation such as the Kim filter.

Hi Tom. I am trying to replicate the example of KIMNP111.RPF
5.4.2 from pp 111-115 of Kim and Nelson, "State-space Models with Regime Switching". Lam's model of GNP, estimation with Kim's filter.

I am having following error messages. Could you please help me understand why I am having the message?

Code: Select all

The Error Occurred At Location 170, Line 13 of %MSINIT
Called From Location 127, Line 13 of MSFILTERINIT
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.

The Error Occurred At Location 364, Line 26 of KIMFILTER
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.
The Error Occurred At Location 364, Line 26 of KIMFILTER
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.
The Error Occurred At Location 364, Line 26 of KIMFILTER
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.
The Error Occurred At Location 364, Line 26 of KIMFILTER
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.
The Error Occurred At Location 364, Line 26 of KIMFILTER
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.
The Error Occurred At Location 364, Line 26 of KIMFILTER
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.
The Error Occurred At Location 364, Line 26 of KIMFILTER
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.
The Error Occurred At Location 364, Line 26 of KIMFILTER
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.

In addition, could you please let me know how I can incorporate time-varying transition probability in example?

Thank you

You need to add the second line after the initialization of THETA:

compute theta=%const(0.0)
compute p=%MSLogisticP(theta)

You need to recompute P inside the KimFilter function to be dependent upon the entry. You also need to figure out how to handle the pre-sample value of the regime probabilities. Have you looked at the Filardo example?

[TomDoan]

In addition, if I would like to examine the transition probabilities in different periods, such as initial 2 years and last 2 years of a presidency, how i should modify the code here. Many Thanks.

Please post that in PDF. DOCX with formulas isn't portable---it depends upon the software that you have on your computer.

Why are you using such a complicated ARMA model?

[fan]

In addition, if I would like to examine the transition probabilities in different periods, such as initial 2 years and last 2 years of a presidency, how i should modify the code here. Many Thanks.

Please post that in PDF. DOCX with formulas isn't portable---it depends upon the software that you have on your computer.

Why are you using such a complicated ARMA model?

hi tom, thank you for the quick reply. Here is my model. The reason why I am using ARMA model is because the conventional AR model can not well describe the data. Could you please kindly take a look my code and point out my mistakes? Thank you

[TomDoan]

That's not a correct representation for an ARMA(2,4)---I can't even make sense out of what you're trying to do there. The @ARMADLM procedure can create the DLM system matrices for an ARMA model.

The fact that you find that you need an ARMA(2,4) to adequately fit the data with a non-switching model isn't relevant to the choice of the model for a MS version---the MS adds complex dynamic of its own.

[fan]

That's not a correct representation for an ARMA(2,4)---I can't even make sense out of what you're trying to do there. The @ARMADLM procedure can create the DLM system matrices for an ARMA model.

The fact that you find that you need an ARMA(2,4) to adequately fit the data with a non-switching model isn't relevant to the choice of the model for a MS version---the MS adds complex dynamic of its own.

Sorry for the poor explanation of the model. Here is the model I am trying estimate. The final goal is to estimate the probabilities of the economy will be in recession and expansion during election and non-election years, and democratic and republican administrations, respectively.

If non-switching model isn't relevant to the choice of the model for a MS version, how I should select the model for a MS version? I tried the AR(4) with mean-switching model ( the Hamilton model), the estimated state probability misses most of the economic recessions(NBER) for the period before WW2. This is why I am trying to use a different model. Thank you for your great patience

[TomDoan]

It's fairly well known that the Hamilton model doesn't work very well when extended even another 10 years in the postwar period. (That's covered in the Kim-Nelson book). I can't imagine that it would be possible to get the switching model to work in the pre-war era. You probably need a bunch of dummies in transition probabilities, mean shifts and variance shifts to handle the changing dynamics. (Pre-war had much wider variation; post war through early 1980's had milder but rather frequent cycles; after 1980's the expansions are much longer). Whether you end up with much of a "model" if everything has to be dummied out separately is hard to say. At any rate, it's probably not the complexity of the ARMA model that's the problem.

[fan]

It's fairly well known that the Hamilton model doesn't work very well when extended even another 10 years in the postwar period. (That's covered in the Kim-Nelson book). I can't imagine that it would be possible to get the switching model to work in the pre-war era. You probably need a bunch of dummies in transition probabilities, mean shifts and variance shifts to handle the changing dynamics. (Pre-war had much wider variation; post war through early 1980's had milder but rather frequent cycles; after 1980's the expansions are much longer). Whether you end up with much of a "model" if everything has to be dummied out separately is hard to say. At any rate, it's probably not the complexity of the ARMA model that's the problem.

Hi Tom, Thank you so much for your detailed explanation. Just for the purpose of the exercise, could you please kindly take a look the attached model, code and error message in the last post and let me know what the mistake(s) is (are) in the code? Thank you again

[TomDoan]

you forgot the attachments.

[fan]

you forgot the attachments.

Thank you for the quick reply. Here are the attachments


My code

Code: Select all

* This is common to any MS state space model analyzed using the
* Kim filter.
*
@MSSetup(states=2)
*
* These are the collapsed SSM mean and variance from the previous time
* period.
*
dec vect[vect] xstar(nstates)
dec vect[symm] sstar(nstates)
*
* This is the filtered best estimate for the state vector
*
dec series[vect] xstates
*
*********************************************************************
*
*
* Fill in the fixed coefficients in the A, C and F matrices
*
compute ndlm=5
dec rect a(ndlm,ndlm)
input a
. . 0 0 0
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0

dec vect phi(2)

dec rect c(1,ndlm)
input c
1 . . . .

dec vect tau(4)

dec rect f(ndlm,1)
compute f=%unitv(ndlm,1)
dec symm sw(ndlm,ndlm)
*
dec vect mu(nstates)
compute sv=0.0
*
dec vect x0(ndlm)
compute x0=%zeros(ndlm,1)
*
nonlin(parmset=initparms) x0
nonlin(parmset=dlmparms) mu sigsq phi tau
nonlin(parmset=msparms) theta
compute theta=%const(0.0)

*********************************************************************
*
* This does a single step of the Kim (approximate) filter
*
function KimFilter time
type integer       time
*
local integer       i j
local real          yerr likely
local symm          vhat
local rect          gain
local rect          phat(nstates,nstates)
local rect            fwork(nstates,nstates)
local rect[vect]    xwork(nstates,nstates)
local rect[symm]    swork(nstates,nstates)
*
do i=1,nstates
   do j=1,nstates
      *
      * Do the SSM predictive step
      *
      compute xwork(i,j)=a*xstar(j)
      compute swork(i,j)=a*sstar(j)*tr(a)+sw
      *
      * Do the prediction error for y under state i, and compute the
      * density function for the prediction error.
      *
      compute yerr=indgrowth(time)-(%dot(c,xwork(i,j))+mu(i))
      compute vhat=c*swork(i,j)*tr(c)+sv
      compute gain=swork(i,j)*tr(c)*inv(vhat)
      compute fwork(i,j)=exp(%logdensity(vhat,yerr))
      *
      * Do the SSM update step
      *
      compute xwork(i,j)=xwork(i,j)+gain*yerr
      compute swork(i,j)=swork(i,j)-gain*c*swork(i,j)
   end do j
end do i
*
* Compute the Hamilton filter likelihood
*
compute likely=0.0
do i=1,nstates
   do j=1,nstates
      compute phat(i,j)=p(i,j)*pstar(j)*fwork(i,j)
      compute likely=likely+phat(i,j)
   end do j
end do i
*
* Compute the updated probabilities of the state combinations
*
compute phat=phat/likely
compute pstar=%sumr(phat)
compute pt_t(time)=pstar
*
* Collapse the SSM matrices down to one per state
*
compute xstates(time)=%zeros(ndlm,1)
do i=1,nstates
   compute xstar(i)=%zeros(ndlm,1)
   do j=1,nstates
      compute xstar(i)=xstar(i)+xwork(i,j)*phat(i,j)/pstar(i)
   end do j
   compute sstar(i)=%zeros(ndlm,ndlm)
   do j=1,nstates
      compute sstar(i)=sstar(i)+phat(i,j)/pstar(i)*$
         (swork(i,j)+%outerxx(xstar(i)-xwork(i,j)))
   end do j
   *
   * This is the overall best estimate of the filtered state.
   *
   compute xstates(time)=xstates(time)+xstar(i)*pstar(i)
end do i
compute KimFilter=likely
end
*********************************************************************
*
* This is the start up code for each function evaluation
*
function DLMStart
*
local integer i
*
* Fill in the top row in the A matrix
*
compute %psubmat(a,1,1,tr(phi))
*
* Fill in the C matrix
*
compute %psubmat(c,1,2,tr(tau))
*
* Compute the full matrix for the transition variance
*
compute sw=f*tr(f)*sigsq
*
* Transform the theta to transition probabilities.
*
compute p=%MSLogisticP(theta)
compute pstar=%mcergodic(p)
compute p=%mspexpand(p)
*
* Initialize the KF state and variance. This is set up for estimating
* the pre-sample states.
*
ewise xstar(i)=x0
ewise sstar(i)=%zeros(ndlm,ndlm)
end
*********************************************************************
*
* Get guess values for the means from running an AR(2) on the growth
* rate. However, the guess values for the AR coefficients need to be
* input.
*
boxjenk(ar=2,constant) indgrowth
compute mu(1)=%beta(1)+1.0,mu(2)=%beta(1)-1.0
compute phi=||1.2,-.3||
compute sigsq=%seesq
compute tau=||0.1,0.1,0.1,-0.4||

*
frml kimf = kf=KimFilter(t),log(kf)
compute x0=||5.224,2.699||
*********************************************************************
gset xstates = %zeros(ndlm,1)
@MSFilterInit
maximize(parmset=msparms+dlmparms,start=DLMStart(),method=bfgs,trace) kimf 1945:04 *
maximize(parmset=msparms+dlmparms+initparms,start=DLMStart(),method=bfgs,trace) kimf  1945:04 *

error message

Code: Select all

The Error Occurred At Location 170, Line 13 of %MSINIT
Called From Location 127, Line 13 of MSFILTERINIT
## MAT14. Non-invertible Matrix. Using Generalized Inverse for SYMMETRIC.
## MAT2. Matrices with Dimensions 5 x 5 and 2 x 1 Involved in * Operation
The Error Occurred At Location 175, Line 17 of KIMFILTER

[TomDoan]

I'm not seeing where you initialize the transition probabilities.

It doesn't help for you to give me 100 lines out of a program to diagnose a running program error (rather than a syntax error). I need the whole program and data.

[fan]

I'm not seeing where you initialize the transition probabilities.

It doesn't help for you to give me 100 lines out of a program to diagnose a running program error (rather than a syntax error). I need the whole program and data.

Hi Tom. I am sorry to bother you again. I attached my whole program and data sample here. Could you please kindly take a look at my code and help me find out the mistakes? Thank you!!