Stata 15 help for mf_toeplitzsolve

Title

[M-5] toeplitzsolve() -- Solve linear systems using Toeplitz matrix

Syntax

real matrix toeplitzsolve(real colvector c1, |real matrix Y)

real matrix toeplitzscale(real colvector c1, real matrix Y)

real matrix _toeplitzscale(real colvector c1, real matrix Y, real colvector v, real scalar ldet)

real matrix toeplitzchprod(real colvector c1, real matrix Z)

Description

toeplitzsolve(c1, |Y) solves T*B = Y for B, where T = Toeplitz(c1,c1'). If Y is missing _toeplitzsolve() solves the Yule-Walker equations, where Y = c1[|2\n|].

toeplitzscale(c1, Y) computes solvelower(R,Y) using the Durbin-Levinson algorithm. Here T = Toeplitz(c1,c1') = R*R', R = cholesky(T).

_toeplitzscale(c1, Y, v, ldet) computes solvelower(L,Y) using the Durbin-Levinson algorithm. Here T = Toeplitz(c1,c1') = L*D*L', D = diag(v), and L is lower triangular with 1's on the diagonal.

toeplitzchprod(c1, Z) computes R*Z, where R = cholesky(T) and T = Toeplitz(c1, c1') = R*R'.

Remarks

toeplitzsolve(), toeplitzscale(), and _toeplitzchprod() are designed specifically for time-series applications. The column vector c1 is the autocovariance of the process. Elements c1[h] = cov(Y[t,j],Y[t+h-1,j]), h = 1, ..., n-t+1, so c1[1] = var(y[t]). Here n = length(c1) = rows(Y).

The Yule-Walker estimates of an autoregressive process are found by P = toeplitzsolve(c1), where it is preferred that c1 be the autocorrelation function, that is, c1[1] = 1. toeplitzsolve() is also useful in computing time-series forecasts; see Beran (1994, sec. 8.7).

E = _topelitzscale(c1,Y,v,ldet) is the r x c1 matrix of residuals of the one-step predictions of Y, where columns of Y are generated from the autoregressive process with covariance T = Toeplitz(c1,c1'); see Palma (2007, sec. 4.1.2). Specifically, E[t,j] = Y[t,j] - M[t,1\t,t-1]*Y[|1,j\t-1,j|], where M = cholinv(L), but carried out efficiently using the Durbin-Levinson algorithm. The rows of -M contain the autoregressive parameters of the process. The vector v contains the variances of the residuals and ldet is the log determinate of T. These are all the quantities necessary to compute the log likelihood of Y coming from a zero mean time-series process with covariance T.

Z = topelitzscale(c1,Y) computes Z = E:/sqrt(v), where E and v are from a call to _toeplitzscale().

If Z = rnormal(r,c1,0,1), then Y = toeplitzchprod(c1,Z) is a set of c1 independent zero mean processes of length r with covariances T = Toeplitz(c1,c1'). Typically, c1 = 1. R = cholesky(T) is carried out efficiently using Shur's algorithm (Stewart 1997).

See [M-5] arfimaacf() for generating the autocovariance function, c1, for an autoregressive (fractionally integrated) moving-average process.

Conformability

toeplitzsolve(c1, |Y): c1: r x 1 Y: r x c1 result: r x c1 or r x 1

toeplitzscale(c1, Y): c1: r x 1 Y: r x c1 result: r x c1

_toeplitzscale(c1, Y, v, ldet): c1: r x 1 Y: r x c1 v: r x 1 ldet: 1 x 1 result: r x c1

toeplitzchprod(c1, Z): c1: r x 1 Z: r x c1 result: r x c1

Diagnostics

None.

Source code

toeplitzsolve.mata

References

Beran, J. 1994. Statistics for Long-Memory Processes. Chapman & Hall/CRC.

Palma, W. 2007. Long-Memory Time Series: Theory and Methods. Wiley.

Stewart, M. 1997. Cholesky factorization of semi-definite Toeplitz matrices. Linear Algebra and its Applications 254: 497-525.


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