g13ds:: Time Series Analysis (NAG Toolbox)

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}

The estimated residual cross-correlation matrix at lag

l

is defined to the

k

k

matrix

{\hat{R}}_{l}

whose

(i, j)

th element is computed as

{\hat{r}}_{i j} (l) = \frac{\sum_{t = l + 1}^{n} ({\hat{ε}}_{i t - l} - {\bar{ε}}_{i}) ({\hat{ε}}_{j t} - {\bar{ε}}_{j})}{\sqrt{\sum_{t = 1}^{n} {({\hat{ε}}_{i t} - {\bar{ε}}_{i})}^{2} \sum_{t = 1}^{n} {({\hat{ε}}_{j t} - {\bar{ε}}_{j})}^{2}}}, l = 0, 1, \dots, i ​ and ​ j = 1, 2, \dots, k,

where

{\hat{ε}}_{i t}

denotes an estimate of the

t

th residual for the

i

th series

ε_{i t}

and

{\bar{ε}}_{i} = \sum_{t = 1}^{n} {\hat{ε}}_{i t} / n

. (Note that

{\hat{R}}_{l}

is an estimate of

E (ε_{t - l} ε_{t}^{T})

, where

E

is the expected value.)

A modified portmanteau statistic,

Q_{(m)}^{*}

, is calculated from the formula (see Li and McLeod (1981))

Q_{(m)}^{*} = \frac{k^{2} m (m + 1)}{2 n} + n \sum_{l = 1}^{m} \hat{r} {(l)}^{T} ({\hat{R}}_{0}^{- 1} \otimes {\hat{R}}_{0}^{- 1}) \hat{r} (l),

where

\otimes

denotes Kronecker product,

{\hat{R}}_{0}

is the estimated residual cross-correlation matrix at lag zero and

\hat{r} (l) = vec ({\hat{R}}_{l}^{T})

, where

vec

of a

k

k

matrix is a vector with the

(i, j)

th element in position

(i - 1) k + j

m

denotes the number of residual cross-correlation matrices computed. (Advice on the choice of

m

is given in Choice of .) Let

l_{C}

denote the total number of ‘free’ parameters in the ARMA model excluding the mean,

μ

, and the residual error covariance matrix

Σ

. Then, under the hypothesis of model adequacy,

Q_{(m)}^{*}

, has an asymptotic

χ^{2}

-distribution on

m k^{2} - l_{C}

degrees of freedom.

Let

\underset{̲}{\hat{r}} = (vec (R_{1}^{T}), vec (R_{2}^{T}), \dots, vec (R_{m}^{T}))

then the covariance matrix of

\underset{̲}{\hat{r}}

is given by

Var (\hat{\underset{̲}{r}}) = [Y - X {(X^{T} G G^{T} X)}^{- 1} X^{T}] / n,

where

Y = I_{m} \otimes (Δ \otimes Δ)

and

G = I_{m} (G G^{T})

Δ

is the dispersion matrix

Σ

in correlation form and

G

a nonsingular

k

k

matrix such that

G G^{T} = Δ^{- 1}

and

G Δ G^{T} = I_{k}

. The construction of the matrix

X

is discussed in Li and McLeod (1981). (Note that the mean,

μ

, plays no part in calculating

Var (\hat{\underset{̲}{r}})

and therefore is not required as input to nag_tsa_multi_varma_diag (g13ds).)

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

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

Accuracy

Further Comments

Timing

Choice of m

The number of residual cross-correlation matrices to be computed,

m

, should be chosen to ensure that when the ARMA model (1) is written as either an infinite order autoregressive process, i.e.,

W_{t} - μ = \sum_{j = 1}^{\infty} π_{j} (W_{t - j} - μ) + ε_{t}

or as an infinite order moving average process, i.e.,

W_{t} - μ = \sum_{j = 1}^{\infty} ψ_{j} ε_{t - j} + ε_{t}

then the two sequences of

k

k

matrices

\{π_{1}, π_{2}, \dots\}

and

\{ψ_{1}, ψ_{2}, \dots\}

are such that

π_{j}

and

ψ_{j}

are approximately zero for

j > m

. An overestimate of

m

is therefore preferable to an under-estimate of

m

. In many instances the choice

m = 10

will suffice. In practice, to be on the safe side, you should try setting

m = 20

Checking a ‘White Noise’ Model

Approximate Standard Errors

Alternative Tests

Example

function g13ds_example


fprintf('g13ds example results\n\n');

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  8.350;
       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];

[k, n] = size(w);
ip = int64(1);
iq = int64(0);
mean_p = true;
m = int64(10);

% Control parameters
iprint = int64(-1);
cgetol =  0.0001;
dishow = int64(2);
ishow  = int64(1);
exact = true;

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

% Initial covariance matrix
qq = zeros(k,k);

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

fprintf('\nOutput from g13ds\n\n');

% Display diagnostics
[qq, r0, r, rcm, chi, idf, siglev, ifail] = ...
    g13ds( ...
           v, ip, iq, m, par, parhld, qq, ishow);

g13ds 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

Output from g13ds


 RESIDUAL CROSS-CORRELATION MATRICES
 -----------------------------------

 LAG     1           :    0.130   0.112
                        ( 0.119)( 0.143)
                          0.094   0.043
                        ( 0.069)( 0.102)

 LAG     2           :   -0.312   0.021
                        ( 0.128)( 0.144)
                         -0.162   0.098
                        ( 0.125)( 0.132)

 LAG     3           :    0.004  -0.176
                        ( 0.134)( 0.144)
                         -0.168  -0.091
                        ( 0.139)( 0.140)

 LAG     4           :   -0.090  -0.120
                        ( 0.137)( 0.144)
                          0.099  -0.232
                        ( 0.142)( 0.143)

 LAG     5           :    0.041   0.093
                        ( 0.140)( 0.144)
                         -0.009  -0.089
                        ( 0.144)( 0.144)

 LAG     6           :    0.234  -0.008
                        ( 0.141)( 0.144)
                          0.069  -0.103
                        ( 0.144)( 0.144)

 LAG     7           :   -0.076   0.007
                        ( 0.142)( 0.144)
                          0.168   0.000
                        ( 0.144)( 0.144)

 LAG     8           :   -0.074   0.559
                        ( 0.143)( 0.144)
                          0.008  -0.101
                        ( 0.144)( 0.144)

 LAG     9           :    0.091   0.193
                        ( 0.144)( 0.144)
                          0.055   0.170
                        ( 0.144)( 0.144)

 LAG    10           :   -0.060   0.061
                        ( 0.144)( 0.144)
                          0.191   0.089
                        ( 0.144)( 0.144)

 SUMMARY TABLE
 -------------

 LAGS   1 -  10

 ***************************
 *            *            *
 * .-........ * .......+.. *
 *            *            *
 ***************************
 *            *            *
 * .......... * .......... *
 *            *            *
 ***************************

 LI-MCLEOD PORTMANTEAU STATISTIC =     49.234
              SIGNIFICANCE LEVEL =      0.086
 (BASED ON  37 DEGREES OF FREEDOM)

On entry,	$k < 1$ ,
or	$kmax < k$ ,
or	$ip < 0$ ,
or	$iq < 0$ ,
or	$ip = iq = 0$ ,
or	$m \leq ip + iq$ ,
or	$m \geq n$ ,
or	$ldrcm < m \times k \times k$ ,
or	liw is too small,
or	lwork is too small.

NAG Toolbox: nag_tsa_multi_varma_diag (g13ds)

▸▿ Contents

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Timing

Choice of m

Checking a ‘White Noise’ Model

Approximate Standard Errors

Alternative Tests

Example