scalar (length of seasonal period). For monthly data s=12 (default), for quarterly data s=4, ...

scalar (order of the trend component).

trend = 1 implies linear trend with intercept (default), trend = 2 implies quadratic trend and so on.

Admissible values for trend are, 0, 1, 2 and 3.

scalar (integer specifying number of frequencies, i.e. harmonics, in the seasonal component). Possible values for seasonal are $1, 2, ..., [s/2]$, where $[s/2]=floor(s/2)$.

For example: if seasonal =1 (default) we have: $\beta_1 \cos( 2 \pi t/s) + \beta_2 sin ( 2 \pi t/s)$;

if seasonal =2 we have: $\beta_1 \cos( 2 \pi t/s) + \beta_2 \sin ( 2 \pi t/s) + \beta_3 \cos(4 \pi t/s) + \beta_4 \sin (4 \pi t/s)$.

Note that when $s$ is even the sine term disappears for $j=s/2$ and so the maximum number of trigonometric parameters is $s-1$.

If seasonal is a number greater than 100, then it is possible to specify how the seasonal component grows over time.

For example, seasonal =101 implies a seasonal component which just uses one frequency which grows linearly over time as follows: $(1+\beta_3 t)\times ( \beta_1 cos( 2 \pi t/s) + \beta_2 \sin ( 2 \pi t/s))$.

For example, seasonal =201 implies a seasonal component which just uses one frequency which grows in a quadratic way over time as follows: $(1+\beta_3 t + \beta_4 t^2)\times( \beta_1 \cos( 2 \pi t/s) + \beta_2 \sin ( 2 \pi t/s))$.

seasonal =0 implies a non seasonal model.

matrix of size T-by-nexpl containing the values of nexpl extra covariates which are likely to affect y.

positive integer which specifies the position to include the level shift component.

For example, if model.posLS =13 then the explanatory variable $I(t \geq 13})$ is created.

If this field is not present or if it is empty, the level shift component is not included.

column vector or matrix containing the initial values of parameter estimates which have to be used in the maximization procedure. If model.B is a matrix, then initial estimates are extracted from the first column of this matrix. If this field is empty or if this field is not present, the initial values to be used in the maximization procedure are referred to the OLS parameter estimates of the linear part of the model. The parameters associated to time varying amplitude are initially set to 0.

Remark: the default model is for monthly data with a linear trend (2 parameters) + seasonal component with just one harmonic (2 parameters), no additional explanatory variables and no level shift that is: model=struct;

model.s=12;

model.trend=1;

model.seasonal=1;

model.X='';

model.posLS='';