Compute the (concentrated) conditional log likelihood for ARMA models described by a model template.

This internal helper function computes the (concentrated) conditional log Likelihood of ARMA systems of the form $$a_0 y_t + a_1 y_{t-1} + \cdots + a_p y_{t-p} = b_0 u_t + \cdots + b_q u_{t-q}$$ The conditional likelihood is computed for zero initial values $u_s=y_s=0$ for $s\leq 0$.

Usage

cll_theta_ARMA_cpp(
  th,
  y,
  skip,
  concentrated,
  ib0,
  H_b,
  h_b,
  B1,
  H_B,
  h_B,
  a0,
  A,
  H_A,
  h_A,
  L,
  H_L,
  h_L,
  u,
  dU
)

Arguments

th: $(K)$ dimensional vector of "deep" parameters.
y: $(m,N)$ matrix with the observed outputs: $(y_1,y_2,\ldots,y_N)$.
skip: (integer), omit the first "skip" residuals, when computing the likelihood.
concentrated: (bool), if TRUE then the concentrated, conditional log Likelihood is computed
ib0: $(m, m)$ matrix, is overwritten with the matrix $b_0^{-1}a_0$.
H_b: $(m^2, K)$ matrix.
h_b: $((m^2)$-dimensional vector. Note that vec(b[0]) = H_b*th + h_b.
B1: $(m, mq)$ matrix, is overwritten with $-b_0^{-1}(b_q,...,b_1)$.
H_B: $((m^2)*q, K)$ matrix.
h_B: $((m^2)*q)$-dimensional vector. Note that vec(-(b[q],...,b[1])) = H_B*th + h_B.
a0: $(m, m)$ matrix, is overwritten with $a_0$.
A: $(m, m(q+1))$ matrix, is overwritten with $b_0^{-1}(a_0,...,a_p$.
H_A: $((m^2)*(p+1), K)$ matrix.
h_A: $((m^2)*(p+1))$-dimensional vector. Note that vec((a[0],a[1],...,a[p])) = H_A*th + h_A.
L: $(m,m)$ matrix. If (concentrated==FALSE) then L is overwritten with the left square $L$ of the noise covariance matrix $\Sigma=LL'$ corresponding to the deep parameters th. However, if (concentrated==TRUE) then L is overwritten with sample covariance matrix of the computed residuals!
H_L: $(m^2, K)$ matrix.
h_L: $(m^2)$-dimensional vector. Note that vec(L) = H_L*th + h_L.
u: $(m,N)$ matrix. This matrix is overwritten with (computed) residuals: $(u_1,u_2,\ldots,u_N)$.
dU: $(mN,(m^2)(p+q+2))$ matrix or $(0,0)$ matrix. If non empty this matrix is overwritten with the directional derivatives of the residuals. However, if the matrix is empty then no derivatives are computed.

Value

(double) log Likelihood

Details

This function is mainly used by the function factory ll_FUN. For a more detailed documentation of the (concentrated) conditional log Likelihood, see ll.

The procedure first constructs the ARMA parameter matrices from the given vector th of "deep" parameters.

AR parameters vec((a[0],a[1],...,a[p])) = h_A + H_A * th.
MA parameters vec(b[0]) = h_b + H_b * th and vec(-(b[q],...,b[1])) = h_B + H_B * th
Left square root of noise covariance matrix $\Sigma = LL'$ and vec(L) = h_L + H_L * th.

The residuals (and their directional derivatives) are computed with residuals_ARMA_cpp.

Note

Use this procedure with care!

The procedure does not check the input arguments.
The procedure overwrites some of the input arguments
The data matrices are organized columnwise (to avoid memory shuffling)!
Note also the non standard representation of the coefficient matrices.