# varmaFit¶

## Purpose¶

Computes exact maximum likelihood parameter estimates for a VARMA model.

## Format¶

vmo = varmaFit(y, p[, d, q, vmc])
vmo = varmaFit(dataset, formula, p[, d, q, vmc])
Parameters
• y (Nx1 vector) – data.

• x (Nxk vector) – independent data.

• dataset (string) – name of data set or null string.

• formula (string) – formula string of the model. E.g. “y ~ X1 + X2” ‘y’ is the name of dependent variable, ‘X1’ and ‘X2’ are names of independent variables; E.g. “y ~ .” , ‘.’ means including all variables except dependent variable ‘y’;

• p (scalar) – order of AR process. Default = 0.

• d (scalar) – Optional input, the order of differencing to achieve stationarity. Default = 0.

• q (scalar) – Optional input, number of MA matrices to be estimated. Default = 0.

• vmc (struct) –

Optional input, an instance of a varmamtControl structure. The following members of vmc are referenced within this routine:

scalar, number of AR lags in the ADF test statistic. Default = 2.

vmc.critl

scalar, the significance levels defining p-values. Default = .95.

vmc.ctl

instance of an sqpsolvemtControl structure.

 vmc.ctl.covType scalar, if 2, QML standard errors are computed, if 0, none; otherwise Wald-type. vmc.ctl.printIters scalar, iteration information printed every vmc.ctl.printIters-th iteration.

See documentation for sqpsolvemtControl for further information regarding members of this structure.

string, specifies the format for the output header. vmc.header can contain zero or more of the following characters:

 t title is to be printed. l lines are to bracket the title. d a date and time is to be printed. v version number of program is to be printed. f file name being analyzed is to be printed

Example:

vmc.header = "tld";


If vmc.header = " ", no header is printed. Default = “tldvf”.

vmc.indEquations

KxL matrix of zeros and ones. Used to set zero restrictions on the variables to be estimated. Only used if the number of equations, vmc.L is greater than one. Elements set to indicate the coefficients to be estimated. If vmc.L = 1, all coefficients will be estimated. If vmc.L > 1 and vmc.indEquations is set to a missing value (the default), all coefficients will be estimated.

vmc.lags

scalar, number of lags over which ACF and Diagnostics are calculated. Default = 12.

vmc.nodet

scalar. Set vmc.nodet = 1 to suppress the constant term from the fitted regression and include it in the co-integrating regression; otherwise, set vmc.nodet = 0. Default = 0.

vmc.nwtrunc

scalar, the number of autocorrelations to use in calculating the Newey-West correction. If vmc.nwtrunc = 0, GAUSS will use a truncation lag given by Newey and West, vmc.nwtrunc = $$4(T / 100)^{2/9}$$.

vmc.output

scalar. Set to 0 to suppress all printing from varmaFit. Set vmc.output > 0 to print results. Default = 1.

vmc.scale

scalar or an Lx1 vector, scales for the time series. If scalar, all series are multiplied by the value. If an Lx1 vector, each series is multiplied by the corresponding element of vmc.scale. Default = 4/standard deviation (found to be best by experimentation).

vmc.setConstraints

scalar, set to a nonzero value to impose stationarity and invertibility by constraining roots of the AR and MA characteristic equations to be outside the unit circle. Set to zero to estimate an unconstrained model. Default = 1.

vmc.start

Instance of a PV structure containing starting values. See VES-Starting Values for discussion of setting starting values. By default, varmaFit calculates starting values.

vmc.title

string, a title to be printed at the top of the output header (see vmc.header). By default, no title is printed (vmc.title = " ").

Returns

vmo (struct) –

An instance of a varmamtOut structure containing the following members:

vmo.acfm

Lx(p*L) matrix, the autocorrelation function. The first L columns are the lag 1 ACF; the last L columns are the lag p ACF.

vmo.aic

Lx1 vector, the Akaike Information Criterion.

vmo.arroots

px1 vector of AR roots, possibly complex.

vmo.bic

Lx1 vector, the Schwarz Bayesian Information Criterion.

vmo.covpar

QxQ matrix of estimated parameters. The parameters are in the row-major order: AR(1) to AR(p), MA(1) to MA(q), beta (if x variables were present in the estimation), and the constants.

vmo.fct

Lx1 vector, the likelihood value.

vmo.lagr

An instance of an sqpsolvemtLagrange structure containing the following members:

 vmo.lagr.lineq linear equality constraints. vmo.lagr.nlineq nonlinear equality constraints. vmo.lagr.linineq linear inequality constraints. vmo.lagr.nlinineq nonlinear inequality constraints. vmo.lagr.bounds bounds. When an inequality or bounds constraint is active, its associated Lagrangean is nonzero. The linear Lagrangeans precede the nonlinear Lagrangeans in the covariance matrices.

vmo.lrs

Lx1 vector, the Likelihood Ratio Statistic.

vmo.maroots

qx1 vector of MA roots, possibly complex.

vmo.pacfm

Lx(p*L) matrix, the partial autocorrelation function, computed only if a univariate model is estimated. The first L columns are the lag 1 ACF; the last L columns are the lag p ACF.

vmo.par

An instance of a PV structure containing the parameter estimates, which can be retrieved using pvUnpack.

For example,

struct varmamtOut vout;
vout = varmaFit(vmc, y, 0);
ph = pvUnpack(vout.par, "phi");
th = pvUnpack(vout.par, "theta");
vc = pvUnpack(vout.par, "vc");


The complete set of parameter matrices and arrays that can be unpacked depending on the model is:

 phi Lxpxp array, autoregression coefficients. theta Lxqxq array, moving average coefficients. vc LxL residual covariance matrix. beta LxK regression coefficient matrix. beta0 Lx1 constant vector. zeta Lxpxar array of ecm coefficients. pi LxL matrix. Note that pi is a reserved word in GAUSS. Users will need to assign this to a different variable name.

vmo.portman

vmc.lags-(p+q)x3 matrix of portmanteau statistics for the multivariate model and Ljung-Box statistics for the univariate model. The time period is in column one, the Qs (portmanteau) statistic in column two and the p-value in column three.

vmo.residuals

TxL matrix, residuals.

vmo.retcode

2x1 vector, return code.

First element:

0

normal convergence.

1

forced exit.

2

maximum number of iterations exceeded.

3

function calculation failed.

4

5

Hessian calculation failed.

6

line search failed.

7

error with constraints.

Second element:

0

covariance matrix of parameters failed.

1

ML covariance matrix.

2

QML covariance matrix.

3

Cross-Product covariance matrix.

vmo.ss

Lx2 matrix, the sum of squares for Y in column one and the sum of squared error in column 2.

## Examples¶

### Data matrices¶

new;
cls;
library tsmt;

// Create file name with full path


## Remarks¶

Errors are assumed to be distributed $$N(0, Q)$$. The estimation procedure assumes that all series are stationary. Setting vmc.SetConstraints to a nonzero value enforces stationarity, by constraining the roots of the characteristic equation

$1 - \Phi_1z - \Phi_2z^2 - ... - \Phi_pz^p$

to be outside the unit circle (where $$\Phi_i, i = 1, ..., p$$ are the AR coefficient matrices).

If any estimated parameters in the coefficient matrices are on a constraint boundary, the Lagrangeans associated with these parameters will be nonzero. These Lagrangeans are stored in vmo.lagr. Standard errors are generally not available for parameters on constraint boundaries.

tsmt

varmamt.src