PROC SSM

[mawuli_0007]

Please, can anyone help with PROC SSM code for this state space model?

Observation equation:

State equation:

where

Parameters:

[rselukar]

Hello mawuli_0007,
Assuming that the rho*GDP_t term in the right hand side of your
state equation is, in fact, rho*GDP_(t-1), we can formulate your
model as follows:

Latent 2-dimensional state: alpha_t = (GDP_t, mu)
State equation:
alpha_t = T alpha_(t-1) + eta_t
alpha_1 = delta (i.e., both elements diffuse)

Where the 2-dimensional transition matrix is (displayed row-wise)
T = (rho 1-rho; 0 1).
eta_t is two-dimensional white noise with diagonal covariance
(sigma_sq_g, 0).
Note that your mu parameter is treated as a latent state in this
formulation
Even though you don't mention it, I will assume that the parameter
rho is a damping factor, i.e., 0 <= rho <= 1.

Let epsilon_t denote 2-dimensional white noise with full covariance matrix.
Then your observation equations are:
GDP_lt = alpha_t[1] + epsilon_t[1]
GDP_et = alpha_t[1] + epsilon_t[2]

PROC SSM code for this will be something like this:

proc ssm data=test;
id time .. ; *specify the time ID with proper interval;
parms rho / lower=0 upper=1;
parms G_sigma / lower=1.e-8;
zero = 0.0;
one = 1.0;
one_minus_rho = 1 - rho;

state alpha(2) t(g) = (rho one_minus_rho zero one)
cov(d)=(G_sigma zero) a1(2);
comp gdp = alpha[1];
comp mu = alpha[2]; *to output the estimated mu;

state noise(2) type=wn cov(g);
comp ep1 = noise[1];
comp ep2 = noise[2];

model gdp_l = gdp ep1;
model gdp_e = gdp ep2;

output out=for press;
run;

[mawuli_0007]

Hi Secular,

Thank you so much for code sample. You are correct the right hand side is rho*GDP_t-1 and rho < 1. Your code sample run without error, but did not produce any result except input data set information, model summary, and ID variable information. I am trying to replicate a working paper by Aruba et. al. (2013). The paper is entitled Improving GDP Measurement: A Measurement-Error Perspective. They used a slightly different state space representation than your formulation as follows:

State equation:

GDP_t = mu(1-rho) + rho*GDP_(t-1) + eta_Gt

alpha_t = K + T alpha_(t-1) + eta_t

where is K 3 by 1 matrix (displayed row-wise)

K =(mu(1-rho) 0 0)

T is 3 by 3 matrix (displayed row-wise)

T=(rho 0 0, 0 0 0, 0 0 0)

alpha_t is 3 by 1 matrix (displayed row-wise)

alpha_t = (GDP_t eta_It eta_Et)

Observation equation:

GDP_It = GDP_t + eta_It

GDP_Et = GDP_t + eta_Et

Y_t = Z alpha_t

where Y_t is 2 by 1 matrix (displayed row-wise)

Y_t = (GDP_It GDP_Et)

Z is 2 by 3 matrix (displayed row-wise)

Z = (1 1 0, 1 0 1)

alpha_t is 3 by 1 matrix (displayed row-wise)

alpha_t = (GDP_t eta_It eta_Et)

(eta_Gt, eta_It, eta_Et) ~iidN(0,∑) where

∑ is 3 by 3 matrix (displayed row-wise)

∑ = (sigma_sq_GG 0 0, 0 sigma_sq_II sigma_sq_IE, 0 sigma_sq_EI sigma_sq_EE)

My difficulty is with the constant K matrix in the state equation as I've not seen any example in the Proc SSM manual with the K matrix.

[rselukar]

You will have to send me your full code (data and program) and the article for me to comment any further.  My email is rajesh.selukar@sas.com