## structural models for time series

Discussion of State Space and Dynamic Stochastic General Equilibrium Models

### structural models for time series

Hello,

I have some problems with the space state models, because it is the first time that I use the instrution dlm. I need to simulate one space state model. Someone can to help me.
The idea is to make a simulation a space state model, with factors no stationary and seasonal, errors have to a distribution noise white gaussian.

And I have a question about structural models, Do I use the same instruction? ¿How do I use the instrution dlm for this kind models?

mcorozcos

Posts: 26
Joined: Wed Feb 08, 2012 11:04 am

### Re: structural models for time series

What's your model? Write it in state-space form. Note, however, that if you're just trying to simulate a model, it's often easier to just do that directly. For instance, if you have a standard local level model:

x(t)=x(t-1)+w(t)
y(t)=x(t)+v(t)

where w(t) and u(t) are the shocks, the easiest way to do that is

Code: Select all
`allocate (# of data points you want)set(first=0.0) x = x{1}+%ran(sigmaw)set y = x+%ran(sigmav)`

To do the same thing with DLM, you would convert that to the state-space model

State equation:

x(t) = [1] x(t-1) + w(t)

Measurement equation:

y(t) = [1] x(t) + v(t)

An example would be:

Code: Select all
`all 100compute sigmav=2.0,sigmaw=1.0dlm(a=1.0,c=1.0,sv=sigmav^2,sw=sigmaw^2,type=simulate,yhat=yhat)set y = %scalar(yhat)`
TomDoan

Posts: 2724
Joined: Wed Nov 01, 2006 5:36 pm

### Re: structural models for time series

Hello Tom,

Thanks for your help. I have two models:

The first one has factors no stationary:

y1t 1 0 f1t e1t
y2t = 0 1 * f2t + e2t
y3t 0 0 e3t

Where:

f1t, f2t: matrix (2x1) of the factors which have an ARIMA model.
e1t, e2t, e3t: matrix (2x1) of noise white Gaussian with distribution normal.
y1t, y2t, y3t: matrix (3X1) of observations
1 0 identity matrix (3x2)
0 1
0 0

The second one has a factor no stationary y other seasonal:

y1t = p11 0 * f1t + e1t
y2t = p21 p22 * f2t + e2t

Where
f1t, f2t: matrix (2x1) of the factors with f1t: is a factor with ARIMA model and f2t: is a factor with SARIMA model
e1t, e2t: matrix (2x1) noise white Gaussian with distribution normal.
y1t, y2t: matrix (2x1) observations
p11 0

I need to simulate two models, I don´t have idea how do I use the instruction dlm for this kind of models?

mcorozcos

Posts: 26
Joined: Wed Feb 08, 2012 11:04 am

### Re: structural models for time series

Again, it's simpler to simulate that in pieces rather than by trying to put together a state-space model with two observables and two separate state vectors. You would need something like this:

Code: Select all
`all 400** This is the ARIMA model (1-L)(1-.4L)y=(1+.3L)u*equation(coeffs=||1.4,-.4,.3||,noconstant) arimaeq y 2 1@armadlm(a=a1,f=f1,c=c1) arimaeq** This is the "airline" SARIMA model (1-L)(1-L^12)y=(1-.7L)(1-.6L^12)u*equation(coeffs=||1.0,1.0,-1.0,-.7,-.6,.42||,noconstant,ar=||1,12,13||,ma=||1,12,13||) sarimaeq y @armadlm(a=a2,f=f2,c=c2) sarimaeqdlm(a=a1,f=f1,c=c1,yhat=fac1,sw=1.0,type=simulate,presample=ergodic)dlm(a=a2,f=f2,c=c2,yhat=fac2,sw=4.0,type=simulate,presample=ergodic)set e1 = %ran(2.0)set e2 = %ran(0.5)*compute p11=1.0,p21=0.5,p22=1.0*set y1 = p11*fac1(t)(1)+e1set y2 = p21*fac1(t)(1)+p22*fac2(t)(1)+e2 `
TomDoan

Posts: 2724
Joined: Wed Nov 01, 2006 5:36 pm

### Re: structural models for time series

mcorozcos

Posts: 26
Joined: Wed Feb 08, 2012 11:04 am

### Re: structural models for time series

Hello Tom,

I have another question. I want to know, how do I remove the common trend for two series in RATS? I know that exist common trend in the series I need to remove in other to analyze the results.

mcorozcos

Posts: 26
Joined: Wed Feb 08, 2012 11:04 am

### Re: structural models for time series

mcorozcos wrote:Hello Tom,

I have another question. I want to know, how do I remove the common trend for two series in RATS? I know that exist common trend in the series I need to remove in other to analyze the results.

This will extract a common H-P trend from a pair of series:

http://www.estima.com/forum/viewtopic.php?f=8&t=1395
TomDoan

Posts: 2724
Joined: Wed Nov 01, 2006 5:36 pm

### Re: structural models for time series

Hello,

I need to create a code in RATS in order to retort the article that I attachment for apply the methodology to colombian data. I hope you can help me.

Thanks
Attachments
Test seasonal integration and cointegration in multivariate unobserved component models 2004.pdf
Article that I need to retort
mcorozcos

Posts: 26
Joined: Wed Feb 08, 2012 11:04 am

### Re: structural models for time series

Hello Tom,

Can you help me with last post (sep 09)?

mcorozcos

Posts: 26
Joined: Wed Feb 08, 2012 11:04 am

### Re: structural models for time series

How far have you gotten on it? This is a paper with no citations in the literature, so it's clearly not in common use. It's not a particularly complicated state-space model, since it just has random walk coefficients on the deterministic seasonals, and the tests appear to be zero restrictions on the variances of those.
TomDoan

Posts: 2724
Joined: Wed Nov 01, 2006 5:36 pm

### Re: structural models for time series

Hello Tom

Can you help to simulate this equation in Rats:

zt(h) = (cos λ(h)t, sin λ(h)t)'

Let s be the number of seasons and λ(h) = 2πh/s, h = 1, ..., [s/2], be the seasonal frequencies. Denote by zt(h), h = 1, ..., [s/2], the spectral indicator variable associated with each of the λ(h), that is zt(h) = (cos λ(h)t, sin λ(h)t)' for h < s/2 and, when s is even, zt(s/2) = cos λ(s/2)t = cos πt.

regards
mcorozcos

Posts: 26
Joined: Wed Feb 08, 2012 11:04 am

### Re: structural models for time series

Is there anything random about those? (I'm puzzled about the use of "simulate", since they appear to be deterministic).
TomDoan

Posts: 2724
Joined: Wed Nov 01, 2006 5:36 pm

### Re: structural models for time series

Hello Tom,

Sorry, I make a mistake in my question, for this reason I make a new one.

I need to simulate some series with base at next structural model for time series:

yt = μt+st + εt,
μt = Xtβ,
st = sum of h = 1 to [s/2] of Zt(h)γt(h),
γt(h) = γt−1(h) + ηt(h), h= 1, ..., [s/2], is a random walk
ηt(h) ∼ IID (0, Σ η(h)) , h= 1, ..., [s/2],
εt ∼ IID (0, Σ ε) ,

where:
Xt = (IN ⊗ xt') ,
β =(β1', ..., βN')' ,
Zt(h) = (IN ⊗ zt(h)') ,
zt(h) = (cos λ(h)t, sin λ(h)t)0 for h < s/2 and, when s is even, zt(s/2) = cos λ(s/2)t = cos πt.
γt(h) =(γ1t(h)', ..., γNt(h)')' , h = 1, ..., [s/2].
ηt(h) is independent of ηs(l)l, i.e. the seasonal components at different frequencies are orthogonal, and also independent of the irregular disturbance εs, for all t, s; ⊗ denotes the Kronecker product.
S=be the number of seasons and
λ(h) = 2πh/s, h = 1, ..., [s/2], be the seasonal frequencies.

mcorozcos

Posts: 26
Joined: Wed Feb 08, 2012 11:04 am

### Re: structural models for time series

The gamma's are the states, so the Z's will be the loadings (C's). Their notation is a bit odd, since they have a combination of 2-vectors and 1-vectors. This sets up the C formula:

Code: Select all
`dec vect[series] zt(s-1)dec frml[vect] cfdo h=1,s/2   set zt(2*h-1) = cos(2*%pi*h*t/s)   if 2*h<=s-1      set zt(2*h) = sin(2*%pi*h*t/s)end do hfrml cf = %xt(zt,t)`
TomDoan

Posts: 2724
Joined: Wed Nov 01, 2006 5:36 pm

### Re: structural models for time series

Hello Tom,

Thanks for your help. I tried to do the last code, but it don´t show a vector of series that I need, for this reason I write you again.

I need to create a vector (100 rows,1 column) with simulated data for this equation:

st = summation of h=1 until h=[s/2] of Zt(h)

where, zt(h), is the spectral indicator variable associated with each of the λ(h), that is:
zt(h) = (cos λ(h)t, sin λ(h)t)' for h < s/2 and,
when s is even, zt(s/2) = cos λ(s/2)t = cos πt.
where s be the number of seasons and λ(h) = 2πh/s, h = 1, ..., [s/2], are the seasonal frequencies.

I hope, you can help me.

Thanks again.

Regards
mcorozcos

Posts: 26
Joined: Wed Feb 08, 2012 11:04 am

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