7.6 LR with AR(1) errors and independent errors

We can add some independent error to our model:

\[\begin{equation} \begin{gathered} x_{t} = bx_{t-1} + w_{t}, \text{ } w_t \sim \,\text{N}(0,q) \\ y_{t} = \beta d_t + x_{t} + v_t, \text{ } v_t \sim \,\text{N}(0,r) \end{gathered} \tag{7.10} \end{equation}\]

We’ll generate this data by adding independent error to yt from the previous example.

yt.r <- yt + rnorm(TT, 0, sqrt(r))

We can fit as:

R <- matrix("r")
mod.list <- list(B = B, U = U, R = R, D = D, d = d, A = A)
fit <- MARSS(yt.r, model = mod.list)

Success! abstol and log-log tests passed at 55 iterations.
Alert: conv.test.slope.tol is 0.5.
Test with smaller values (<0.1) to ensure convergence.

MARSS fit is
Estimation method: kem 
Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
Estimation converged in 55 iterations. 
Log-likelihood: -161.9318 
AIC: 333.8635   AICc: 334.5018   
 
       Estimate
R.r       1.034
B.b       0.810
Q.Q       0.255
x0.x0     1.229
D.beta    0.973
Initial states (x0) defined at t=0

Standard errors have not been calculated. 
Use MARSSparamCIs to compute CIs and bias estimates.

This is not a model that can be fit with arima().