Spectral Generating Function of Common Structural Time Series Models

Share:

Description

Evaluate the spectral generating function of of common structural models: local level model, local trend model and basic structural model.

Usage

1
stsm.sgf(x, gradient = FALSE, hessian = FALSE, deriv.transPars = FALSE)

Arguments

x

object of class stsm.

gradient

logical. If TRUE, the gradient is returned.

hessian

logical. If TRUE, hessian the gradient is returned.

deriv.transPars

logical. If TRUE, the gradient and the Hessian are scaled by the gradient of the function that transforms the parameters. Ignored if x@transPars is null.

Details

The stationary form of the local level model is (Δ is the differencing operator):

Δ y[t] = v[t] + Δ e[t]

and its spectral generating function at each frequency λ[j] = 2π j/T for j=0,...,T-1 is:

g(λ[j]) = σ^2_2 + 2(1 - cos λ[j]) σ^2_1

The stationary form of the local trend model for a time series of frequency S is:

Δ^2 y[t] = Δ v[t] + w[t-1] + Δ^2 e[t]

and its spectral generating function is:

g(λ[j]) = 2(1 - cos λ[j]) σ^2_2 + σ^2_3 + 4(1 - cos λ[j]) σ^2_1

The stationary form of the basic structural model for a time series of frequency p is:

Δ Δ^p y[t] = Δ^p v[t] + S(L) w[t-1] + Δ^2 s[t] + Δ Δ^p e[t]

and its spectral generating function is:

g(λ[j]) = g_v(λ[j]) σ^2_2 + g_w(λ[j]) σ^2_3 + g_s(λ[j]) σ^2_4 + g_e(λ[j]) σ^2_1

with

g_v(λ[j]) = 2(1 - cos(λ[j] p))

g_w(λ[j]) = (1 - cos(λ[j] p))/(1 - cos(λ[j]))

g_s(λ[j]) = 4 (1 - cos(λ[j]))^2

g_e(λ[j]) = 4 (1 - cos(λ[j])) (1 - cos(λ[j] p))

Value

A list containing the following results:

sgf

spectral generating function of the BSM model at each frequency λ[j] for j=0,…,T-1.

gradient

first order derivatives of the spectral generating function with respect of the parameters of the model.

hessian

second order derivatives of the spsectral generating function with respect of the parameters of the model.

constants

the terms g_v(λ[j]), g_w(λ[j]), g_s(λ[j]) and g_e(λ[j]) that do not depend on the variance parameters.

References

Harvey, A. C. (1989). Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press.

See Also

set.sgfc, stsm-class, stsm.model.

Examples

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
# spectral generating function of the local level plus seasonal model
m <- stsm.model(model = "llm+seas", y = JohnsonJohnson, 
  pars = c("var1" = 2, "var2" = 15), nopars = c("var3" = 30))
res <- stsm.sgf(m)
res$sgf
plot(res$sgf)
res$constants
# the element 'constants' contains the constant variables
# related to each component regardless of whether the 
# variances related to them are in the slot 'pars' or 'nopars'
names(get.pars(m))
colnames(res$constants)

# compare analytical and numerical derivatives
# identical values
m <- stsm.model(model = "llm+seas", y = JohnsonJohnson, 
  pars = c("var1" = 2, "var2" = 15, "var3" = 30))
res <- stsm.sgf(m, gradient = TRUE)

fcn <- function(x, model = m) {
  m <- set.pars(model, x)
  res <- stsm.sgf(m)
  sum(res$sgf)
}

a1 <- numDeriv::grad(func = fcn, x = get.pars(m))
a2 <- colSums(res$grad)
all.equal(a1, a2, check.attributes = FALSE)

# analytical derivatives are evaluated faster than numerically
system.time(a1 <- numDeriv::grad(func = fcn, x = get.pars(m)))
system.time(a2 <- colSums(stsm.sgf(m, gradient = TRUE)$grad))