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;
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