Why State Space estimation changes every time? - r

Question

I don't know if this is the right place for my question (maybe stats exchange?), but I've been stuck some days here
I've been reading this paper from Morley (2002), where he makes an approach to the state space to the Beveridge Nelson decomposition.
So on page 4, Morley uses an example of how to use the state space for an Arima(2,1,2) model.
So I tried to generalize it for any Arima model of order (p,d,q)
So far I have been following the FKF package documentation
So here I put the state space rep for an Arma(3,3) (supposing the series is I(1))
So far I've made a function that computes the Kalman filter for any Arma(p,q), but it changes with every run and I'm not simulating anything because I use the GDP from FRED from 1970 to 2022.
BN_filter <- function(data, order, p_estimates = FALSE){
# Order things up ----
p = order[1]
d = order[2]
q = order[3]
# Set the data
xt <- select_if(us, is.numeric)[[1]]
yt <- xt - lag(xt, n = d)
n <- length(yt)
# Estimate the parameters (guesses) and set the par0 ----
m <- arima(yt, order = c(p, 0 , q), include.mean = FALSE, method = "ML")
coef <- as.matrix(m$coef) # all AR and MA coefficients
rownames(coef) <- NULL
sigma_e <- var(m$residuals, na.rm = T) # Variance of the error
# here is a vector of (p+q)+1 dimension where the last object is the variance
par0 <- as.matrix(c(coef, sigma_e))
# Since series is stationary and arma model doesn't have drift the expected value for the alpha vector is 0
a0 <- as.numeric(rep(0, times = (p+q)))
P0 <- matrix(1e16, nrow = (p+q), ncol = (p+q))
# State - space matrices
dt <- matrix(0, nrow = (p+q))
ct <- matrix(0)
GGt <- matrix(0)
Zt <- matrix(c(1,rep(0, times = (p+q-1))), nrow = 1)
# Set up sample ----
yt_s <- t(yt[(d+1):n])
# Build the Tt matrix and the HHT matrix
Kalman <- function(par, yt, a0, P0, ct, Zt, GGt, dt){
# Build pseudo Tt (this is the Tt matrix from the second row)
Ip = diag(1, nrow = (p-1))
Zpv = matrix(0, nrow = (p-1))
Zm1 = matrix(0, nrow = (p-1), ncol = (q-1))
Zeps = matrix(0, nrow = 1, ncol = (p+q))
Zm2 = matrix(0, nrow = (q-1), ncol = (p-1))
Zqv = matrix(0, nrow = (q-1))
Iq = diag(1, nrow = (q-1))
s_row <- cbind(Ip, Zpv, Zm1, Zpv)
f_row <- cbind(Zm2, Zqv, Iq, Zqv)
pseudo_Tt <- rbind(s_row, Zeps, f_row)
# Tt ----
firstRow <- matrix(par[1:(p+q)], nrow = 1)
Tt <- rbind(firstRow, pseudo_Tt)
#HHt ----
HHt <- matrix(par[(length(par-1):length(par))], ncol = (p+q), nrow = (p+q))
results <- fkf(a0 = a0, P0 = P0, dt = dt, ct = ct,
Tt = Tt, Zt = Zt, HHt = HHt,
GGt = GGt, yt=yt)
return(results)
}
# Optimizing function ----
fn <- function(x0, ...){
-Kalman(par = x0, yt = yt_s,a0 = a0, P0 = P0, ct = ct, Zt = Zt, GGt = GGt, dt= dt)$logLik
}
# Run optimizers -----
## Global optimizer
fit.fkf <- optim(par0, fn, method = "SANN")
## Local optimizer
fit.fkf <- optim(fit.fkf$par, fn)
## Fitted object
fit.obj <- Kalman(fit.fkf$par, yt = yt_s, a0 = a0,
P0 = P0, ct = ct,
Zt = Zt, GGt = GGt, dt = dt)
# Estimates ----
kalman_est <- c(phi = fit.fkf$par[1:p],
theta = fit.fkf$par[(p+1):(p+q)],
sigma_e = tail(fit.fkf$par,1))
colnames(par0) <- "ML Estimates"
if(p_estimates == TRUE){
print(cbind("Kalman Estimates" = kalman_est, "ML Estimates"=par0))
}
return(fit.obj)
}
But I have noticed that when it's an Arma(3,1,3) model the estimated coefficients do not change, but the Arma(4,1,2) model does change:
Here is my output:
> a <- BN_filter(data = us, order = c(3,1,3), p_estimates = TRUE)
Kalman Estimates ML Estimates
phi1 -0.6362326324 -0.5359310692
phi2 0.5288426267 0.6021502455
phi3 1.0348389624 0.9328190861
theta1 0.5993870595 0.5223257304
theta2 -0.4980386193 -0.5754977018
theta3 -0.9717439064 -0.8987930785
sigma_e 0.0001401711 0.0001331292
> a <- BN_filter(data = us, order = c(3,1,3), p_estimates = TRUE)
Kalman Estimates ML Estimates
phi1 -0.6362326324 -0.5359310692
phi2 0.5288426267 0.6021502455
phi3 1.0348389624 0.9328190861
theta1 0.5993870595 0.5223257304
theta2 -0.4980386193 -0.5754977018
theta3 -0.9717439064 -0.8987930785
sigma_e 0.0001401711 0.0001331292
> b <- BN_filter(data = us, order = c(4,1,2), p_estimates = TRUE)
Kalman Estimates ML Estimates
phi1 1.4104289329 -0.0007921442
phi2 -0.3076356643 1.0536717900
phi3 -0.1259039820 0.0330696296
phi4 0.0222118830 -0.0865077506
theta1 -1.4458524512 0.0129934941
theta2 0.4801402654 -0.9869856537
sigma_e 0.0001379343 0.0001306852
> b <- BN_filter(data = us, order = c(4,1,2), p_estimates = TRUE)
Kalman Estimates ML Estimates
phi1 -0.3795696850 -0.0007921442
phi2 0.5461933222 1.0536717900
phi3 0.1572464501 0.0330696296
phi4 0.1603534419 -0.0865077506
theta1 0.3029337586 0.0129934941
theta2 -0.4077502428 -0.9869856537
sigma_e 0.0001557173 0.0001306852
So... can someone explain to me what is happening? Why does it change? What is wrong with this?
Thanks in advance!
Here is the state space model for the Arma(p,q) model, if you find it useful: