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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_tsa_multi_varma_estimate (g13dd)

Purpose

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, 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, par, qq, w, parhld, exact, iprint, cgetol, ishow, 'k', k, 'n', n, 'npar', npar, 'maxcal', maxcal)

Description

Let Wt = (w1t,w2t,,wkt)T Wt = (w1t,w2t,,wkt)T , for t = 1,2,,nt=1,2,,n, denote a vector of kk time series which is assumed to follow a multivariate ARMA model of the form
Wtμ = φ1(Wt1μ) + φ2(Wt2μ) + + φp(Wtpμ)
+ εtθ1εt1θ2εt2θqεtq
Wt-μ= ϕ1(Wt-1-μ)+ϕ2(Wt-2-μ)++ϕp(Wt-p-μ) +εt-θ1εt-1-θ2εt-2--θqεt-q
(1)
where εt = (ε1t,ε2t,,εkt)T εt = (ε1t,ε2t,,εkt)T , for t = 1,2,,nt=1,2,,n, is a vector of kk residual series assumed to be Normally distributed with zero mean and positive definite covariance matrix ΣΣ. The components of εtεt are assumed to be uncorrelated at non-simultaneous lags. The φiϕi and θjθj are kk by kk matrices of parameters. {φi}{ϕi}, for i = 1,2,,pi=1,2,,p, are called the autoregressive (AR) parameter matrices, and {θi}{θi}, for i = 1,2,,qi=1,2,,q, the moving average (MA) parameter matrices. The parameters in the model are thus the pp (kk by kk) φϕ-matrices, the qq (kk by kk) θθ-matrices, the mean vector, μμ, and the residual error covariance matrix ΣΣ. Let
A(φ) =
[ φ1 I 0 . . . 0 φ2 0 I 0 . . 0 . . . . . . φp − 1 0 . . . 0 I φp 0 . . . 0 0 ]
pk × pk   and B(θ) =
[ θ1 I 0 . . . 0 θ2 0 I 0 . . 0 . . . . . . θq − 1 0 . . . . I θq 0 . . . . 0 ]
qk × qk
A(ϕ)= [ ϕ1 I 0 . . . 0 ϕ2 0 I 0 . . 0 . . . . . . ϕp-1 0 . . . 0 I ϕp 0 . . . 0 0 ] pk×pk   and B(θ)= [ θ1 I 0 . . . 0 θ2 0 I 0 . . 0 . . . . . . θq-1 0 . . . . I θq 0 . . . . 0 ] qk×qk
where II denotes the kk by kk identity matrix.
The ARMA model (1) is said to be stationary if the eigenvalues of A(φ)A(ϕ) lie inside the unit circle. Similarly, the ARMA model (1) is said to be invertible if the eigenvalues of B(θ)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 ΣΣ.
You have the option of setting a parameter (exact to false) so that nag_tsa_multi_varma_estimate (g13dd) calculates conditional maximum likelihood estimates (conditional on W0 = W1 = = W1p = ε0 = ε1 = = ε1q = 0W0=W-1==W1-p=ε0=ε-1== ε1-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 Section [Parameters]) of requesting nag_tsa_multi_varma_estimate (g13dd) to constrain elements of the φϕ and θθ matrices and μμ 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
pp, the number of AR parameter matrices.
Constraint: ip0ip0.
2:     iq – int64int32nag_int scalar
qq, the number of MA parameter matrices.
Constraint: iq0iq0.
ip = iq = 0ip=iq=0 is not permitted.
3:     mean – logical scalar
mean = truemean=true, if components of μμ have been estimated and mean = falsemean=false, if all elements of μμ are to be taken as zero.
Constraint: mean = truemean=true or falsefalse.
4:     par(npar) – double array
npar, the dimension of the array, must satisfy the constraint
  • if mean = falsemean=false, npar must be set equal to (p + q) × k × k(p+q)×k×k;
  • if mean = truemean=true, npar must be set equal to (p + q) × k × k + k(p+q)×k×k+k.
The total number of observations (n × k)(n×k) must exceed the total number of parameters in the model (npar + k(k + 1) / 2npar+k(k+1)/2).
Initial parameter estimates read in row by row in the order φ1,φ2,,φpϕ1,ϕ2,,ϕp, θ1,θ2,,θq,μθ1,θ2,,θq,μ.
Thus,
  • if ip > 0ip>0, par((l1) × k × k + (i1) × k + j)par(l-1)×k×k+(i-1)×k+j must be set equal to an initial estimate of the (i,j)(i,j)th element of φlϕl, for l = 1,2,,pl=1,2,,p, i = 1,2,,ki=1,2,,k and j = 1,2,,kj=1,2,,k;
  • if iq > 0iq>0, par(p × k × k + (l1) × k × k + (i1) × k + j)parp×k×k+(l-1)×k×k+(i-1)×k+j must be set equal to an initial estimate of the (i,j)(i,j)th element of θlθl, l = 1,2,,ql=1,2,,q and i,j = 1,2,,ki,j=1,2,,k;
  • if mean = truemean=true, par((p + q) × k × k + i)par(p+q)×k×k+i should be set equal to an initial estimate of the iith component of μμ (μ(i)μ(i)). (If you set par((p + q) × k × k + i)par(p+q)×k×k+i to 0.00.0 then nag_tsa_multi_varma_estimate (g13dd) will calculate the mean of the iith series and use this as an initial estimate of μ(i)μ(i).)
The first p × k × kp×k×k elements of par must satisfy the stationarity condition and the next q × k × kq×k×k elements of par must satisfy the invertibility condition.
If in doubt set all elements of par to 0.00.0.
5:     qq(kmax,k) – double array
kmax, the first dimension of the array, must satisfy the constraint kmaxkkmaxk.
qq(i,j)qqij must be set equal to an initial estimate of the (i,j)(i,j)th element of ΣΣ. 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) = 0.0qqij=0.0, for i = 1,2,,ki=1,2,,k and j = 1,2,,ij=1,2,,i, then nag_tsa_multi_varma_estimate (g13dd) will calculate the covariance matrix between the kk time series and use this as an initial estimate of ΣΣ.
6:     w(kmax,n) – double array
kmax, the first dimension of the array, must satisfy the constraint kmaxkkmaxk.
w(i,t)wit must be set equal to the iith component of WtWt, for i = 1,2,,ki=1,2,,k and t = 1,2,,nt=1,2,,n.
7:     parhld(npar) – logical array
npar, the dimension of the array, must satisfy the constraint
  • if mean = falsemean=false, npar must be set equal to (p + q) × k × k(p+q)×k×k;
  • if mean = truemean=true, npar must be set equal to (p + q) × k × k + k(p+q)×k×k+k.
The total number of observations (n × k)(n×k) must exceed the total number of parameters in the model (npar + k(k + 1) / 2npar+k(k+1)/2).
parhld(i)parhldi must be set to true if par(i)pari is to be held constant at its input value and false if par(i)pari is a free parameter, for i = 1,2,,npari=1,2,,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 > 0iprint>0
The ML search procedure is monitored once every iprint iterations and just before exit from the search function.
iprint = 0iprint=0
The search function is monitored once at the final point.
iprint < 0iprint<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 10l10-l and on exit ifail = 0ifail=0 or ifail6ifail6, then all the elements in par and qq should be accurate to approximately ll decimal places. For most practical purposes the value 10410-4 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, εε, then nag_tsa_multi_varma_estimate (g13dd) will use the value 10.0 × sqrt(ε)10.0×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 = 0ishow=0
None of the above are printed.
ishow = 1ishow=1
(i) only is printed.
ishow = 2ishow=2
(i) and (ii) are printed.
Constraint: 0ishow20ishow2.

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.)
kk, the number of observed time series.
Constraint: k1k1.
2:     n – int64int32nag_int scalar
Default: The second dimension of the array w.
nn, 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 as declared in the (sub)program from which nag_tsa_multi_varma_estimate (g13dd) is called.npar is the number of initial parameter estimates.
Constraints:
  • if mean = falsemean=false, npar must be set equal to (p + q) × k × k(p+q)×k×k;
  • if mean = truemean=true, npar must be set equal to (p + q) × k × k + k(p+q)×k×k+k.
The total number of observations (n × k)(n×k) must exceed the total number of parameters in the model (npar + k(k + 1) / 2npar+k(k+1)/2).
4:     maxcal – int64int32nag_int scalar
The maximum number of likelihood evaluations to be permitted by the search procedure.
Default: 40 × npar × (npar + 5)40×npar×(npar+5)
Constraint: maxcal1maxcal1.

Input Parameters Omitted from the MATLAB Interface

kmax ldcm

Output Parameters

1:     par(npar) – double array
If ifail = 0ifail=0 or ifail4ifail4 then all the elements of par will be stores the latest estimates of the corresponding ARMA parameters.
2:     qq(kmax,k) – double array
kmaxkkmaxk.
If ifail = 0ifail=0 or ifail4ifail4 then qq(i,j)qqij will contain the latest estimate of the (i,j)(i,j)th element of ΣΣ. The lower triangle only is returned.
3:     niter – int64int32nag_int scalar
If ifail = 0ifail=0 or ifail4ifail4 then niter contains the number of iterations performed by the search function.
4:     rlogl – double scalar
If ifail = 0ifail=0 or ifail4ifail4 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
kmaxkkmaxk.
If ifail = 0ifail=0 or ifail4ifail4 then v(i,t)vit will contain an estimate of the iith component of εtεt, for i = 1,2,,ki=1,2,,k and t = 1,2,,nt=1,2,,n, corresponding to the final point held in par and qq.
6:     g(npar) – double array
If ifail = 0ifail=0 or ifail4ifail4 then g(i)gi will contain the estimated first derivative of the log-likelihood function with respect to the iith 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
ldcmnparldcmnpar.
If ifail = 0ifail=0 or ifail4ifail4 then cm(i,j)cmij will contain an estimate of the correlation coefficient between the iith and jjth elements in the par array for 1inpar1inpar, 1jnpar1jnpar. If i = ji=j, then cm(i,j)cmij will contain the estimated standard error of par(i)pari. If the llth component of par has been held constant, i.e., parhld(l)parhldl was set to true, then the llth 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 = 0ifail=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 = 1ifail=1
On entry,k < 1k<1,
orip < 0ip<0,
oriq < 0iq<0,
orip = iq = 0ip=iq=0,
ornpar(ip + iq) × k × k + Δ × knpar(ip+iq)×k×k+Δ×k, where Δ = 1Δ=1 if mean = truemean=true or Δ = 0Δ=0 if mean = falsemean=false,
orn × knpar + k × (k + 1) / 2n×knpar+k×(k+1)/2,
orkmax < kkmax<k,
ormaxcal < 1maxcal<1,
orishow < 0ishow<0,
orishow > 2ishow>2,
orldcm < nparldcm<npar,
  ifail = 2ifail=2
On entry, either the initial estimate of ΣΣ is not positive definite, or the initial estimates of the AR parameters are such that the model is non-stationary, or the initial estimates of the MA parameters are such that the model is non-invertible. To proceed, you must try a different starting point.
  ifail = 3ifail=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 ΣΣ 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 = 4ifail=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 (as described in Section [Parameters]) are computed at the final point held in par and qq, except that if g or cm cannot be computed, in which case they are set to zero.
W ifail = 5ifail=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 iprint0iprint0, 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 (as described in Section [Parameters]) are computed at the final point held in par and qq, except that if g or cm cannot be computed, in which case they are set to zero.
W ifail = 6ifail=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 will then be set to zero on exit. The elements of g will also be set to zero. All other output quantities will be correct.
W ifail = 7ifail=7
This is an unlikely exit, which could occur in (e04xa), which computes an estimate of the second derivative matrix and the gradient vector at the solution point. 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. In this case the elements of cm will be set to zero on exit as will the elements of g. All other output quantities will be correct.
W ifail = 8ifail=8
The second derivative matrix at the solution point is not positive definite. In this case the elements of cm will be set to zero on exit. All other output quantities will be correct.
  ifail = 999ifail=-999
Internal memory allocation failed.

Accuracy

On exit from nag_tsa_multi_varma_estimate (g13dd), if ifail = 0ifail=0 or ifail6ifail6 and cgetol has been set to 10l10-l, then all the parameters should be accurate to approximately ll decimal places. If cgetol was set equal to a value less than the machine precision, εε, then all the parameters should be accurate to approximately 10.0 × sqrt(ε)10.0×ε.
If ifail = 4ifail=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) r = max(ip,iq)  and s = npar + k × (k + 1) / 2 s = npar+k× (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 × r + 52 s+k×r+52  integer elements and 2 × s2 + s × (s1) / 2 + 15 × s + k2 × (2 × ip + iq + (r + 3)2) + k × (2 × r2 + 2 × r + 3 × n + 4) + 10 2 × s2 +s× (s-1) /2+15× s+ k2 × ( 2×ip+iq+ (r+3) 2 ) +k× ( 2× r2 +2× r+3× 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 = 5ifail=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 εtεt (denoted by te^t) can easily be calculated by forming the Cholesky decomposition of ΣΣ, e.g., GGTGGT and setting t = G1ε̂te^t=G-1ε^t. nag_lapack_dpotrf (f07fd) may be used to calculate the array g. The components of te^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 = 0ifail=0 or 5ifail85ifail8 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

function nag_tsa_multi_varma_estimate_example
ip = int64(1);
iq = int64(0);
mean_p = true;
par = zeros(6, 1);
% 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];
w = [-1.49, -1.62, 5.2, 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, 0.4, 2.26, 3.34, 5.09, 5, 4.78, 4.11, ...
     3.45, 1.65, 1.29, 4.09, 6.32, 7.5, 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, 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, 6.4, 6.24, 7.93, 4.04, 3.73, 5.6, 5.35, 6.81, 8.27, ...
     7.68, 6.65, 6.08, 10.25, 9.14, 17.75, 13.3, 9.63, 6.8, 4.08, 5.06, 4.94, ...
     6.65, 7.94, 10.76, 11.89, 5.85, 9.01, 7.5, 10.02, 10.38, 8.15, 8.37, 10.73, 12.14];
parhld = [false;
     false;
     true;
     false;
     false;
     false];
exact = true;
iprint = int64(-1);
cgetol = 0.0001;
ishow = int64(2);
[parOut, qqOut, niter, rlogl, v, g, cm, ifail] = ...
    nag_tsa_multi_varma_estimate(ip, iq, mean_p, par, qq, w, parhld, exact, iprint, cgetol, ishow)
 

 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

parOut =

    0.8016
    0.0648
         0
    0.5750
    4.2711
    7.8253


qqOut =

    2.9642    0.6373
    0.6373    5.3799


niter =

                    0


rlogl =

 -202.8027


v =

  Columns 1 through 9

   -3.3261   -1.2415    5.7469    1.2704    0.3223    0.1128   -1.2697   -0.7336   -0.5810
   -0.1865   -1.1963   -0.0170    1.2122   -1.6163   -2.1623   -1.6335   -1.1320   -1.3415

  Columns 10 through 18

   -1.2582   -0.6731   -1.1332   -2.0203   -0.5727    1.2360   -0.1309   -0.7728   -2.0894
   -1.2970    4.8173    0.4282    2.5384    0.3527   -2.8828   -0.7657    1.0163   -3.8455

  Columns 19 through 27

    1.3373    0.9464    1.7095    0.2329   -0.0096   -0.5979   -0.6825   -1.8867   -0.7669
   -1.9187    0.1295   -1.1957    0.4080    1.0285   -0.4010   -1.0918   -1.0695    3.4283

  Columns 28 through 36

    2.0514    2.1089    0.9432   -3.3242   -2.5026    3.1626    0.4690    0.0534    2.7677
   -0.0796    9.1687   -0.2322   -1.3434   -2.0630   -3.1558   -0.6117   -1.2952    0.4838

  Columns 37 through 45

   -0.8170    0.2498    3.9851    0.1996   -0.6999    1.0721    0.4391    0.2783    1.0893
    0.7905    2.8687    2.3772   -4.3126    2.3205   -1.0065    2.3817    1.2927   -1.1443

  Columns 46 through 48

    0.5028   -0.1031    1.7031
    0.3580    2.5915    2.6444


g =

   1.0e-03 *

    0.0231
   -0.1487
         0
   -0.0547
    0.0002
   -0.0030


cm =

    0.0906   -0.0814         0   -0.0054   -0.0150    0.0021
   -0.0814    0.1018         0    0.0650    0.0177   -0.0544
         0         0         0         0         0         0
   -0.0054    0.0650         0    0.1206    0.0047   -0.0094
   -0.0150    0.0177         0    0.0047    1.2191    0.3480
    0.0021   -0.0544         0   -0.0094    0.3480    0.7755


ifail =

                    0


function g13dd_example
ip = int64(1);
iq = int64(0);
mean_p = true;
par = zeros(6, 1);
% 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];
w = [-1.49, -1.62, 5.2, 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, 0.4, 2.26, 3.34, 5.09, 5, 4.78, 4.11, ...
     3.45, 1.65, 1.29, 4.09, 6.32, 7.5, 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, 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, 6.4, 6.24, 7.93, 4.04, 3.73, 5.6, 5.35, 6.81, 8.27, ...
     7.68, 6.65, 6.08, 10.25, 9.14, 17.75, 13.3, 9.63, 6.8, 4.08, 5.06, 4.94, ...
     6.65, 7.94, 10.76, 11.89, 5.85, 9.01, 7.5, 10.02, 10.38, 8.15, 8.37, 10.73, 12.14];
parhld = [false;
     false;
     true;
     false;
     false;
     false];
exact = true;
iprint = int64(-1);
cgetol = 0.0001;
ishow = int64(2);
[parOut, qqOut, niter, rlogl, v, g, cm, ifail] = ...
    g13dd(ip, iq, mean_p, par, qq, w, parhld, exact, iprint, cgetol, ishow)
 

 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

parOut =

    0.8016
    0.0648
         0
    0.5750
    4.2711
    7.8253


qqOut =

    2.9642    0.6373
    0.6373    5.3799


niter =

                    0


rlogl =

 -202.8027


v =

  Columns 1 through 9

   -3.3261   -1.2415    5.7469    1.2704    0.3223    0.1128   -1.2697   -0.7336   -0.5810
   -0.1865   -1.1963   -0.0170    1.2122   -1.6163   -2.1623   -1.6335   -1.1320   -1.3415

  Columns 10 through 18

   -1.2582   -0.6731   -1.1332   -2.0203   -0.5727    1.2360   -0.1309   -0.7728   -2.0894
   -1.2970    4.8173    0.4282    2.5384    0.3527   -2.8828   -0.7657    1.0163   -3.8455

  Columns 19 through 27

    1.3373    0.9464    1.7095    0.2329   -0.0096   -0.5979   -0.6825   -1.8867   -0.7669
   -1.9187    0.1295   -1.1957    0.4080    1.0285   -0.4010   -1.0918   -1.0695    3.4283

  Columns 28 through 36

    2.0514    2.1089    0.9432   -3.3242   -2.5026    3.1626    0.4690    0.0534    2.7677
   -0.0796    9.1687   -0.2322   -1.3434   -2.0630   -3.1558   -0.6117   -1.2952    0.4838

  Columns 37 through 45

   -0.8170    0.2498    3.9851    0.1996   -0.6999    1.0721    0.4391    0.2783    1.0893
    0.7905    2.8687    2.3772   -4.3126    2.3205   -1.0065    2.3817    1.2927   -1.1443

  Columns 46 through 48

    0.5028   -0.1031    1.7031
    0.3580    2.5915    2.6444


g =

   1.0e-03 *

    0.0231
   -0.1487
         0
   -0.0547
    0.0002
   -0.0030


cm =

    0.0906   -0.0814         0   -0.0054   -0.0150    0.0021
   -0.0814    0.1018         0    0.0650    0.0177   -0.0544
         0         0         0         0         0         0
   -0.0054    0.0650         0    0.1206    0.0047   -0.0094
   -0.0150    0.0177         0    0.0047    1.2191    0.3480
    0.0021   -0.0544         0   -0.0094    0.3480    0.7755


ifail =

                    0



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