nag_tsa_multi_varma_estimate (g13dd) fits a vector autoregressive moving average (VARMA) model to an observed vector of time series using the method of Maximum Likelihood (ML). Standard errors of parameter estimates are computed along with their appropriate correlation matrix. The function also calculates estimates of the residual series.

Syntax

[par, qq, niter, rlogl, v, g, cm, ifail] = g13dd(ip, iq, mean_p, par, qq, w, parhld, exact, iprint, cgetol, ishow, 'k', k, 'n', n, 'npar', npar, 'maxcal', maxcal)

[par, qq, niter, rlogl, v, g, cm, ifail] = nag_tsa_multi_varma_estimate(ip, iq, mean_p, par, qq, w, parhld, exact, iprint, cgetol, ishow, 'k', k, 'n', n, 'npar', npar, 'maxcal', maxcal)

Description

Let

W_{t} = {(w_{1 t}, w_{2 t}, \dots, w_{k t})}^{T}

, for

t = 1, 2, \dots, n

, denote a vector of

k

time series which is assumed to follow a multivariate ARMA model of the form

\begin{array}{l} W_{t} - μ = & ϕ_{1} (W_{t - 1} - μ) + ϕ_{2} (W_{t - 2} - μ) + \dots + ϕ_{p} (W_{t - p} - μ) \\ + ε_{t} - θ_{1} ε_{t - 1} - θ_{2} ε_{t - 2} - \dots - θ_{q} ε_{t - q} \end{array}

(1)

where

ε_{t} = {(ε_{1 t}, ε_{2 t}, \dots, ε_{k t})}^{T}

, for

t = 1, 2, \dots, n

, is a vector of

k

residual series assumed to be Normally distributed with zero mean and positive definite covariance matrix

Σ

. The components of

ε_{t}

are assumed to be uncorrelated at non-simultaneous lags. The

ϕ_{i}

and

θ_{j}

are

k

k

matrices of parameters.

\{ϕ_{i}\}

, for

i = 1, 2, \dots, p

, are called the autoregressive (AR) parameter matrices, and

\{θ_{i}\}

, for

i = 1, 2, \dots, q

, the moving average (MA) parameter matrices. The parameters in the model are thus the

p

(

k

k

)

ϕ

-matrices, the

q

(

k

k

)

θ

-matrices, the mean vector,

μ

, and the residual error covariance matrix

Σ

. Let

A (ϕ) = {[\begin{array}{l} ϕ_{1} & I & 0 & . & . & . & 0 \\ ϕ_{2} & 0 & I & 0 & . & . & 0 \\ . & . \\ . & . \\ . & . \\ ϕ_{p - 1} & 0 & . & . & . & 0 & I \\ ϕ_{p} & 0 & . & . & . & 0 & 0 \end{array}]}_{p k \times p k} and B (θ) = {[\begin{array}{l} θ_{1} & I & 0 & . & . & . & 0 \\ θ_{2} & 0 & I & 0 & . & . & 0 \\ . & . \\ . & . \\ . & . \\ θ_{q - 1} & 0 & . & . & . & . & I \\ θ_{q} & 0 & . & . & . & . & 0 \end{array}]}_{q k \times q k}

where

I

denotes the

k

k

identity matrix.

The ARMA model (1) is said to be stationary if the eigenvalues of

A (ϕ)

lie inside the unit circle. Similarly, the ARMA model (1) is said to be invertible if the eigenvalues of

B (θ)

lie inside the unit circle.

The method of computing the exact likelihood function (using a Kalman filter algorithm) is discussed in Shea (1987). A quasi-Newton algorithm (see Gill and Murray (1972)) is then used to search for the maximum of the log-likelihood function. Stationarity and invertibility are enforced on the model using the reparameterisation discussed in Ansley and Kohn (1986). Conditional on the maximum likelihood estimates being equal to their true values the estimates of the residual series are uncorrelated with zero mean and constant variance

Σ

You have the option of setting a argument (exact to false) so that nag_tsa_multi_varma_estimate (g13dd) calculates conditional maximum likelihood estimates (conditional on

W_{0} = W_{- 1} = \dots = W_{1 - p} = ε_{0} = ε_{- 1} = \dots = ε_{1 - q} = 0

). This may be useful if the exact maximum likelihood estimates are close to the boundary of the invertibility region.

You also have the option (see Arguments) of requesting nag_tsa_multi_varma_estimate (g13dd) to constrain elements of the

ϕ

and

θ

matrices and

μ

vector to have pre-specified values.

References

Ansley C F and Kohn R (1986) A note on reparameterising a vector autoregressive moving average model to enforce stationarity J. Statist. Comput. Simulation 24 99–106

Gill P E and Murray W (1972) Quasi-Newton methods for unconstrained optimization J. Inst. Math. Appl. 9 91–108

Shea B L (1987) Estimation of multivariate time series J. Time Ser. Anal. 8 95–110

Parameters

Compulsory Input Parameters

1: $ip$ – int64int32nag_int scalar

p

, the number of AR parameter matrices.

Constraint:

ip \geq 0

2: $iq$ – int64int32nag_int scalar

q

, the number of MA parameter matrices.

Constraint:

iq \geq 0

ip = iq = 0

is not permitted.

3: $mean_p$ – logical scalar

mean_p = true

, if components of

μ

have been estimated and

mean_p = false

, if all elements of

μ

are to be taken as zero.

Constraint:

mean_p = true

false

4: $par (npar)$ – double array

Initial parameter estimates read in row by row in the order

ϕ_{1}, ϕ_{2}, \dots, ϕ_{p}

θ_{1}, θ_{2}, \dots, θ_{q}, μ

Thus,

if $ip > 0$ , $par ((l - 1) \times k \times k + (i - 1) \times k + j)$ must be set equal to an initial estimate of the $(i, j)$ th element of $ϕ_{l}$ , for $l = 1, 2, \dots, p$ , $i = 1, 2, \dots, k$ and $j = 1, 2, \dots, k$ ;
if $iq > 0$ , $par (p \times k \times k + (l - 1) \times k \times k + (i - 1) \times k + j)$ must be set equal to an initial estimate of the $(i, j)$ th element of $θ_{l}$ , $l = 1, 2, \dots, q$ and $i, j = 1, 2, \dots, k$ ;
if $mean_p = true$ , $par ((p + q) \times k \times k + i)$ should be set equal to an initial estimate of the $i$ th component of $μ$ ( $μ (i)$ ). (If you set $par ((p + q) \times k \times k + i)$ to $0.0$ then nag_tsa_multi_varma_estimate (g13dd) will calculate the mean of the $i$ th series and use this as an initial estimate of $μ (i)$ .)

The first

p \times k \times k

elements of par must satisfy the stationarity condition and the next

q \times k \times k

elements of par must satisfy the invertibility condition.

If in doubt set all elements of par to

0.0

5: $qq (kmax, k)$ – double array

kmax, the first dimension of the array, must satisfy the constraint

kmax \geq k

qq (i, j)

must be set equal to an initial estimate of the

(i, j)

th element of

Σ

. The lower triangle only is needed. qq must be positive definite. It is strongly recommended that on entry the elements of qq are of the same order of magnitude as at the solution point. If you set

qq (i, j) = 0.0

, for

i = 1, 2, \dots, k

and

j = 1, 2, \dots, i

, then nag_tsa_multi_varma_estimate (g13dd) will calculate the covariance matrix between the

k

time series and use this as an initial estimate of

Σ

6: $w (kmax, n)$ – double array

kmax, the first dimension of the array, must satisfy the constraint

kmax \geq k

w (i, t)

must be set equal to the

i

th component of

W_{t}

, for

i = 1, 2, \dots, k

and

t = 1, 2, \dots, n

7: $parhld (npar)$ – logical array

parhld (i)

must be set to true if

par (i)

is to be held constant at its input value and false if

par (i)

is a free parameter, for

i = 1, 2, \dots, npar

If in doubt try setting all elements of parhld to false.

8: $exact$ – logical scalar

Must be set equal to true if you wish nag_tsa_multi_varma_estimate (g13dd) to compute exact maximum likelihood estimates. exact must be set equal to false if only conditional likelihood estimates are required.

9: $iprint$ – int64int32nag_int scalar

The frequency with which the automatic monitoring function is to be called.

$iprint > 0$: The ML search procedure is monitored once every iprint iterations and just before exit from the search function.
$iprint = 0$: The search function is monitored once at the final point.
$iprint < 0$: The search function is not monitored at all.

10: $cgetol$ – double scalar

The accuracy to which the solution in par and qq is required.

If cgetol is set to

10^{- l}

and on exit

ifail = 0

ifail \geq 6

, then all the elements in par and qq should be accurate to approximately

l

decimal places. For most practical purposes the value

10^{- 4}

should suffice. You should be wary of setting cgetol too small since the convergence criteria may then have become too strict for the machine to handle.

If cgetol has been set to a value which is less than the machine precision,

ε

, then nag_tsa_multi_varma_estimate (g13dd) will use the value

10.0 \times \sqrt{ε}

instead.

11: $ishow$ – int64int32nag_int scalar

Specifies which of the following two quantities are to be printed.

(i)	table of maximum likelihood estimates and their standard errors (as returned in the output arrays par, qq and cm);
(ii)	table of residual series (as returned in the output array v).

$ishow = 0$: None of the above are printed.
$ishow = 1$: (i) only is printed.
$ishow = 2$: (i) and (ii) are printed.

Constraint:

0 \leq ishow \leq 2

Optional Input Parameters

1: $k$ – int64int32nag_int scalar

Default: the first dimension of the arrays qq, w and the second dimension of the array qq. (An error is raised if these dimensions are not equal.)

k

, the number of observed time series.

Constraint:

k \geq 1

2: $n$ – int64int32nag_int scalar

Default: the second dimension of the array w.

n

, the number of observations in each time series.

3: $npar$ – int64int32nag_int scalar

Default: the dimension of the arrays par, parhld. (An error is raised if these dimensions are not equal.)

The dimension of the arrays par, parhld and g and the second dimension of the array cm.npar is the number of initial parameter estimates.

Constraints:

if $mean_p = false$ , npar must be set equal to $(p + q) \times k \times k$ ;
if $mean_p = true$ , npar must be set equal to $(p + q) \times k \times k + k$ .

The total number of observations

(n \times k)

must exceed the total number of parameters in the model (

npar + k (k + 1) / 2

4: $maxcal$ – int64int32nag_int scalar

Default:

40 \times npar \times (npar + 5)

The maximum number of likelihood evaluations to be permitted by the search procedure.

Constraint:

maxcal \geq 1

Output Parameters

1: $par (npar)$ – double array: If $ifail = 0$ or $ifail \geq 4$ then all the elements of par will be stores the latest estimates of the corresponding ARMA parameters.
2: $qq (kmax, k)$ – double array: If $ifail = 0$ or $ifail \geq 4$ then $qq (i, j)$ will contain the latest estimate of the $(i, j)$ th element of $Σ$ . The lower triangle only is returned.
3: $niter$ – int64int32nag_int scalar: If $ifail = 0$ or $ifail \geq 4$ then niter contains the number of iterations performed by the search function.
4: $rlogl$ – double scalar: If $ifail = 0$ or $ifail \geq 4$ then rlogl contains the value of the log-likelihood function corresponding to the final point held in par and qq.
5: $v (kmax, n)$ – double array: If $ifail = 0$ or $ifail \geq 4$ then $v (i, t)$ will contain an estimate of the $i$ th component of $ε_{t}$ , for $i = 1, 2, \dots, k$ and $t = 1, 2, \dots, n$ , corresponding to the final point held in par and qq.
6: $g (npar)$ – double array: If $ifail = 0$ or $ifail \geq 4$ then $g (i)$ will contain the estimated first derivative of the log-likelihood function with respect to the $i$ th element in the array par. If the gradient cannot be computed then all the elements of g are returned as zero.
7: $cm (ldcm, npar)$ – double array: If $ifail = 0$ or $ifail \geq 4$ then $cm (i, j)$ will contain an estimate of the correlation coefficient between the $i$ th and $j$ th elements in the par array for $1 \leq i \leq npar$ , $1 \leq j \leq npar$ . If $i = j$ , then $cm (i, j)$ will contain the estimated standard error of $par (i)$ . If the $l$ th component of par has been held constant, i.e., $parhld (l)$ was set to true, then the $l$ th row and column of cm will be set to zero. If the second derivative matrix cannot be computed then all the elements of cm are returned as zero.
8: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Note: nag_tsa_multi_varma_estimate (g13dd) may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail = 1$: Constraint: $0 \leq ishow \leq 2$ .

Constraint: $ip \geq 0$ .

Constraint: $iq \geq 0$ .

Constraint: $kmax \geq k$ .

Constraint: $k \geq 1$ .

Constraint: $ldcm \geq npar$ .

Constraint: $maxcal \geq 1$ .

Constraint: $n \times k > npar + k \times (k + 1) / 2$ .

Constraint: $npar =_$ .

Constraint: $npar \geq 0$ .

On entry, $ip = 0$ and $iq = 0$ .

$ifail = 2$: The initial AR parameter estimates are outside the stationarity region.
To proceed, you must try a different starting point.

The initial estimate of $Σ$ is not positive definite. To proceed, you must try a different starting point.

The initial MA parameter estimates are outside the invertibility region. To proceed, you must try a different starting point.

The starting point is too close to the boundary of the admissibility region.

$ifail = 3$: The function cannot compute a sufficiently accurate estimate of the gradient vector at the user-supplied starting point. This usually occurs if either the initial parameter estimates are very close to the ML parameter estimates, or you have supplied a very poor estimate of $Σ$ , or the starting point is very close to the boundary of the stationarity or invertibility region. To proceed, you must try a different starting point.

$ifail = 4$: There have been maxcal log-likelihood evaluations made in the function.

If steady increases in the log-likelihood function were monitored up to the point where this exit occurred, then the exit probably simply occurred because maxcal was set too small, so the calculations should be restarted from the final point held in par and qq. This type of exit may also indicate that there is no maximum to the likelihood surface. Output quantities were computed at the final point held in par and qq, except that if g or cm could not be computed, in which case they are set to zero.

W $ifail = 5$: The conditions for a solution have not all been met, but a point at which the log-likelihood took a larger value could not be found.
Provided that the estimated first derivatives are sufficiently small, and that the estimated condition number of the second derivative (Hessian) matrix, as printed when $iprint \geq 0$ , is not too large, this error exit may simply mean that, although it has not been possible to satisfy the specified requirements, the algorithm has in fact found the solution as far as the accuracy of the machine permits.

Such a condition can arise, for instance, if cgetol has been set so small that rounding error in evaluating the likelihood function makes attainment of the convergence conditions impossible.

If the estimated condition number at the final point is large, it could be that the final point is a solution but that the smallest eigenvalue of the Hessian matrix is so close to zero at the solution that it is not possible to recognize it as a solution. Output quantities were computed at the final point held in par and qq, except that if g or cm could not be computed, in which case they are set to zero.

W $ifail = 6$: The ML solution is so close to the boundary of either the stationarity region or the invertibility region that nag_tsa_multi_varma_estimate (g13dd) cannot evaluate the Hessian matrix. The elements of cm are set to zero, as are the elements of g. All other output quantities are correct.

W $ifail = 7$: An estimate of the second derivative matrix and the gradient vector at the solution point was computed. Either the Hessian matrix was found to be too ill-conditioned to be evaluated accurately or the gradient vector could not be computed to an acceptable degree of accuracy. The elements of cm are set to zero, as are the elements of g. All other output quantities are correct.

W $ifail = 8$: The second-derivative matrix at the solution point is not positive definite. The elements of cm are set to zero. All other output quantities are correct.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

On exit from nag_tsa_multi_varma_estimate (g13dd), if

ifail = 0

ifail \geq 6

and cgetol has been set to

10^{- l}

, then all the parameters should be accurate to approximately

l

decimal places. If cgetol was set equal to a value less than the machine precision,

ε

, then all the parameters should be accurate to approximately

10.0 \times \sqrt{ε}

ifail = 4

on exit (i.e., maxcal likelihood evaluations have been made but the convergence conditions of the search function have not been satisfied), then the elements in par and qq may still be good approximations to the ML estimates. Inspection of the elements of g may help you determine whether this is likely.

Further Comments

Memory Usage

Let

r = \max (ip, iq)

and

s = npar + k \times (k + 1) / 2

. Local workspace arrays of fixed lengths are allocated internally by nag_tsa_multi_varma_estimate (g13dd). The total size of these arrays amounts to

s + k \times r + 52

integer elements and

2 \times s^{2} + s \times (s - 1) / 2 + 15 \times s + k^{2} \times (2 \times ip + iq + {(r + 3)}^{2}) + k \times (2 \times r^{2} + 2 \times r + 3 \times n + 4) + 10

double elements.

Timing

The number of iterations required depends upon the number of parameters in the model and the distance of the user-supplied starting point from the solution.

Constraining for Stationarity and Invertibility

If the solution lies on the boundary of the admissibility region (stationarity and invertibility region) then nag_tsa_multi_varma_estimate (g13dd) may get into difficulty and exit with

ifail = 5

. If this exit occurs you are advised to either try a different starting point or a different setting for exact. If this still continues to occur then you are urged to try fitting a more parsimonious model.

Over-parameterisation

You are advised to try and avoid fitting models with an excessive number of parameters since over-parameterisation can cause the maximization problem to become ill-conditioned.

Standardizing the Residual Series

The standardized estimates of the residual series

ε_{t}

(denoted by

{\hat{e}}_{t}

) can easily be calculated by forming the Cholesky decomposition of

Σ

, e.g.,

G G^{T}

and setting

{\hat{e}}_{t} = G^{- 1} {\hat{ε}}_{t}

. nag_lapack_dpotrf (f07fd) may be used to calculate the array g. The components of

{\hat{e}}_{t}

which are now uncorrelated at all lags can sometimes be more easily interpreted.

Assessing the Fit of the Model

If your time series model provides a good fit to the data then the residual series should be approximately white noise, i.e., exhibit no serial cross-correlation. An examination of the residual cross-correlation matrices should confirm whether this is likely to be so. You are advised to call nag_tsa_multi_varma_diag (g13ds) to provide information for diagnostic checking. nag_tsa_multi_varma_diag (g13ds) returns the residual cross-correlation matrices along with their asymptotic standard errors. nag_tsa_multi_varma_diag (g13ds) also computes a portmanteau statistic and its asymptotic significance level for testing model adequacy. If

ifail = 0

5 \leq ifail \leq 8

on exit from nag_tsa_multi_varma_estimate (g13dd) then the quantities output k, n, v, kmax, ip, iq, par, parhld, and qq will be suitable for input to nag_tsa_multi_varma_diag (g13ds).

Example

This example shows how to fit a bivariate AR(1) model to two series each of length

48

μ

will be estimated and

ϕ_{1} (2, 1)

will be constrained to be zero.

Open in the MATLAB editor: g13dd_example

function g13dd_example


fprintf('g13dd example results\n\n');

ip     = int64(1);
iq     = int64(0);
mean_p = true;

% Set all elements of Q to zero to use the covariance matrix
% between the K time series as the initial estimate of the
% covariance matrix
qq = [0, 0; 0, 0];

% Series
w = [-1.49, -1.62,  5.20,  6.23,  6.21,  5.86,  4.09,  3.18,  2.62,  1.49, ...
      1.17,  0.85, -0.35,  0.24,  2.44,  2.58,  2.04,  0.40,  2.26,  3.34, ...
      5.09,  5.00,  4.78,  4.11,  3.45,  1.65,  1.29,  4.09,  6.32,  7.50, ...
      3.89,  1.58,  5.21,  5.25,  4.93,  7.38,  5.87,  5.81,  9.68,  9.07, ...
      7.29,  7.84,  7.55,  7.32,  7.97,  7.76,  7.00,  8.35;
      7.34,  6.35,  6.96,  8.54,  6.62,  4.97,  4.55,  4.81,  4.75,  4.76, ...
     10.88, 10.01, 11.62, 10.36,  6.40,  6.24,  7.93,  4.04,  3.73,  5.60, ...
      5.35,  6.81,  8.27,  7.68,  6.65,  6.08, 10.25,  9.14, 17.75, 13.30, ...
      9.63,  6.80,  4.08,  5.06,  4.94,  6.65,  7.94, 10.76, 11.89,  5.85, ...
      9.01,  7.50, 10.02, 10.38,  8.15,  8.37, 10.73, 12.14];

% Initial parameter estimates and free parameter flags
par    = zeros(6, 1);
parhld = [false;  false;  true;  false;  false;  false];

% Exact likelihood
exact  = true;
% control parameters
iprint = int64(-1);
cgetol = 0.0001;
ishow  = int64(2);

[par, qq, niter, rlogl, v, g, cm, ifail] = ...
  g13dd( ...
	 ip, iq, mean_p, par, qq, w, parhld, exact, iprint, cgetol, ishow);

g13dd example results


 VALUE OF LOG LIKELIHOOD FUNCTION ON EXIT = -0.20280E+03

 MAXIMUM LIKELIHOOD ESTIMATES OF AR PARAMETER MATRICES
 -----------------------------------------------------

 PHI(1)    ROW-WISE :    0.802   0.065
                       ( 0.091)( 0.102)

                         0.000   0.575
                       ( 0.000)( 0.121)

 MAXIMUM LIKELIHOOD ESTIMATE OF PROCESS MEAN
 -------------------------------------------

                         4.271   7.825
                       ( 1.219)( 0.776)

 MAXIMUM LIKELIHOOD ESTIMATE OF SIGMA MATRIX
 -------------------------------------------

                         2.964

                         0.637   5.380

           RESIDUAL SERIES NUMBER  1
           -------------------------

   T     1      2      3      4      5      6      7      8
 V(T)  -3.33  -1.24   5.75   1.27   0.32   0.11  -1.27  -0.73

   T     9     10     11     12     13     14     15     16
 V(T)  -0.58  -1.26  -0.67  -1.13  -2.02  -0.57   1.24  -0.13

   T    17     18     19     20     21     22     23     24
 V(T)  -0.77  -2.09   1.34   0.95   1.71   0.23  -0.01  -0.60

   T    25     26     27     28     29     30     31     32
 V(T)  -0.68  -1.89  -0.77   2.05   2.11   0.94  -3.32  -2.50

   T    33     34     35     36     37     38     39     40
 V(T)   3.16   0.47   0.05   2.77  -0.82   0.25   3.99   0.20

   T    41     42     43     44     45     46     47     48
 V(T)  -0.70   1.07   0.44   0.28   1.09   0.50  -0.10   1.70

           RESIDUAL SERIES NUMBER  2
           -------------------------

   T     1      2      3      4      5      6      7      8
 V(T)  -0.19  -1.20  -0.02   1.21  -1.62  -2.16  -1.63  -1.13

   T     9     10     11     12     13     14     15     16
 V(T)  -1.34  -1.30   4.82   0.43   2.54   0.35  -2.88  -0.77

   T    17     18     19     20     21     22     23     24
 V(T)   1.02  -3.85  -1.92   0.13  -1.20   0.41   1.03  -0.40

   T    25     26     27     28     29     30     31     32
 V(T)  -1.09  -1.07   3.43  -0.08   9.17  -0.23  -1.34  -2.06

   T    33     34     35     36     37     38     39     40
 V(T)  -3.16  -0.61  -1.30   0.48   0.79   2.87   2.38  -4.31

   T    41     42     43     44     45     46     47     48
 V(T)   2.32  -1.01   2.38   1.29  -1.14   0.36   2.59   2.64

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox