d_to_coef function

Filter Coefficients of the Fractional Differencing Operator

Filter Coefficients of the Fractional Differencing Operator

Output is with positive signs on the left-hand side of the equation.

d_to_coef(d, max_i = 1000)

Arguments

  • d: the fractional differencing coefficient.
  • max_i: the maximum index up until which to return the coefficient series.

Returns

A numeric vector is returned.

Details

Consider the FARIMA model

(1−B)dYt=ar1Xt−1+...+arpXt−p+ma1et−1+...+maqet−q+et, (1-B)^d Y_t = ar_1 X_{t-1} + ... + ar_p X_{t-p}+ma_1 e_{t-1}+...+ma_q e_{t-q}+e_t,

where ete_t are the innovations and where Xt=(1−B)dYtX_t=(1-B)^d Y_t. dd is the fractional differencing coefficient.

The fractional differencing operator (1−B)d(1-B)^d can alternatively be expressed as an infinite coefficient series, so that

(1−B)d=∑l=0∞blBk, (1-B)^d=\sum_{l=0}^{\infty}b_l B^k,

where BB is the backshift operator and where blb_l, l=0,1,2,...l=0,1,2,..., are the coefficients. Note that b0=1b_0=1 by definition.

The function returns the series of coefficients {bl,l=0,1,2,...}\{b_l, l =0,1,2,...\}.

Examples

d_to_coef(d = 0.3, max_i = 100)

Author(s)

  • Dominik Schulz (Scientific Employee) (Department of Economics, Paderborn University),

    Author

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