Matrix containing estimated beta coefficients, (including the intercept when options.intercept=true) standard errors, t-stat and p-values The content of matrix B is as follows: 1st col = beta coefficients The order of the beta coefficients is as follows: 1) trend elements (if present). If the trend is of order two there are r+1 coefficients if the intercept is present otherwise there are just r components;

2) linear part of seasonal component 2, 4, 6, ..., s-2, s-1 coefficients (if present);

3) coefficients associated with the matrix of explanatory variables and/or autoregressive part which have a potential effect on the time series under study (X);

4) non linear part of seasonal component, that is varying amplitude. If varying amplitude is of order k there are k coefficients (if present);

5) level shift component (if present). In this case there are two coefficients, the second (which is also the last element of vector beta) is an integer which specifies the time in which level shift takes place and the first (which is also the penultime element of vector beta) is a real number which identifies the magnitude of the upward (downward) level shift;

2nd col = standard errors;

3rd col = t-statistics;

4th col = p values.

scalar associated with best tentative level shift position. If this field does not exist, forecasts are done assuming no level shift.

$cov(\beta)/\hat \sigma^2$. p-by-p, square matrix.

If the model is linear out.invXX is equal to $(X'X)^{-1}$, else out.invXX is equal to $(A'A)^{-1}$ where $A$ is the matrix of partial derivatives. More precisely:

vector of n fitted values after final (NLS=non linear least squares) step: $ (\hat \eta_1, \hat \eta_2, \ldots, \hat \eta_T)'$

Final scale estimate of the residuals \[ \hat \sigma = cor \times \sum_{i \in S_m} [y_i- \eta(x_i,\hat \beta)]^2/(m-p) \]

where $S_m$ is a set of cardinality $m$ which contains the units not declared as outliers and $p$ is the total number of estimated parameters and cor is a correction factor to make the estimator consistent.

REMARK: structure outEST can be conveniently created by function LTSts.

scalar (order of the trend component).

trend = 1 implies linear trend with intercept, trend = 2 implies quadratic trend, etc.

If this field is empty the simulated time series will not contain a trend. The default value of model.trend is 1.

scalar greater than zero which specifies the length of the seasonal period. For monthly data (default) s=12, for quartely data s=4, ...

The default value of model.s is 12 (that is monthly data are assumed)

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: $(1+\beta_3 t + \beta_4 t^2)\times( \beta_1 \cos( 2 \pi t/s) + \beta_2 \sin ( 2 \pi t/s))$.

If this field is an empty double (default) the simulated time series will not contain a seasonal component.

explanatory variabels. Matrix of size (length(y)+nfore)-by-nexpl.

If model.X is a matrix of size (length(y)+nfore)-by-nexpl, it contains the values of nexpl extra covariates which affect y for periods 1:(length(y)+nfore) where nfore is the requested number of forecasts. If this field is an empty double (default) there is no effect of explanatory variables.

vector with non negative integer numbers specifying the autoregressive components. For example: model.ARp=[1 2] means a AR(2) process;

model.ARp=2 means just the lag 2 component;

model.ARp=[1 2 5 8] means AR(2) + lag 5 + lag 8;

model.ARp=0 (default) means no autoregressive component.

matrix of size r-by-2 containing the list of the units declared as outliers (first column) and corresponding fitted values (second column) or empty scalar. If model.ARtentout is not empty, when the autoregressive component is present, the y values which are used to compute the autoregressive component are replaced by model.tentout(:,2) for the units contained in model.tentout(:,1)