Description Arguments Details Value References Examples
This function computes the localised and iterated Yule-Walker coefficients
for h-step ahead forecasting of X_{t+h} from X_{t}, ..., X_{t-p+1},
where h = 1, …, H
and p = 1, …, P
.
X |
the data X_1, …, X_T |
P |
the maximum order of coefficients to be computed; has to be a positive integer |
H |
the maximum lead time; has to be a positive integer |
t |
a vector of values t; the elements have to satisfy
|
N |
a vector of values N; the elements have to satisfy
|
For every t \in t
and every N \in N
the (iterated) Yule-Walker
estimates \hat v_{N,T}^{(p,h)}(t) are computed. They are defined as
\hat v_{N,T}^{(p,h)}(t) := e'_1 \big( e_1 \big( \hat a_{N,T}^{(p)}(t) \big)' + H \big)^h, \quad N ≥q 1,
and
\hat v_{0,T}^{(p,h)}(t) := \hat v_{t,T}^{(p,h)}(t),
with
e_1 := ≤ft(\begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \end{array} \right), \quad H := ≤ft( \begin{array}{ccccc} 0 & 0 & \cdots & 0 & 0 \\ 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & \cdots & 0 & 0 \\ \vdots & \ddots & \cdots & 0 & 0 \\ 0 & 0 & \cdots & 1 & 0 \end{array} \right)
and
\hat a_{N,T}^{(p)}(t) := \big( \hatΓ_{N,T}^{(p)}(t) \big)^{-1} \hatγ_{N,T}^{(p)}(t),
where
\hatΓ_{N,T}^{(p)}(t) := \big[ \hat γ_{i-j;N,T}(t) \big]_{i,j = 1, …, p}, \quad \hat γ_{N,T}^{(p)}(t) := \big( \hat γ_{1;N,T}(t), …, \hat γ_{p;N,T}(t) \big)'
and
\hat γ_{k;N,T}(t) := \frac{1}{N} ∑_{\ell=t-N+|k|+1}^{t} X_{\ell-|k|,T} X_{\ell,T}
is the usual lag-k autocovariance estimator (without mean adjustment), computed from the observations X_{t-N+1}, …, X_{t}.
The Durbin-Levinson Algorithm is used to successively compute the solutions to the Yule-Walker equations (cf. Brockwell/Davis (1991), Proposition 5.2.1). To compute the h-step ahead coefficients we use the recursive relationship
\hat v_{i,N,T}^{(p)}(t,h) = \hat a_{i,N,T}^{(p)}(t) \hat v_{1,N,T}^{(p,h-1)}(t) + \hat v_{i+1,N,T}^{(p,h-1)}(t) I\{i ≤q p-1\},
(cf. Section 3.2, Step 3, in Kley et al. (2019)).
Returns a named list with elements coef
, t
, and N
,
where coef
is an array of dimension
P
\times P
\times H
\times
length(t)
\times length(N)
, and
t
, and N
are the parameters provided on the call of the
function. See the example on how to access the vector
\hat v_{N,T}^{(p,h)}(t).
Brockwell, P. J. & Davis, R. A. (1991). Time Series: Theory and Methods. Springer, New York.
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