# estimate

##### Estimate an ARIMA Model

Estimates an ARIMA model for a univariate time series, including a sparse ARIMA model.

##### Usage

```
estimate(x, p = 0, d = 0, q = 0, PDQ = c(0, 0, 0), S = NA,
method = c("CSS-ML", "ML", "CSS"), intercept = TRUE, output = TRUE, ...)
```

##### Arguments

- x
- a univariate time series.
- p
- the AR order, can be a positive integer or a vector with several positive
integers. The default is
`0`

. - d
- the degree of differencing. The default is
`0`

. - q
- the MA order, can be a positive integer or a vector with several positive
integers. The default is
`0`

. - PDQ
- a vector with three non-negative integers for specification of the seasonal
part of the ARIMA model. The default is
`c(0,0,0)`

. - S
- the period of seasonal ARIMA model. The default is
`NA`

. - method
- fitting method. The default is
`CSS-ML`

. - intercept
- a logical value indicating to include the intercept in ARIMA model. The
default is
`TRUE`

. - output
- a logical value indicating to print the results in R console. The default is
`TRUE`

. - ...
- optional arguments to
`arima`

function.

##### Details

This function is similar to the ESTIMATE statement in ARIMA procedure of SAS,
except that it does not fit a transfer function model for a univariate time series. The
fitting method is inherited from `arima`

in `stats`

package. To be
specific, the pure ARIMA(p,q) is defined as
$$X[t] = \mu + \phi[1]*X[t-1] + ... + \phi[p]*X[p] +
e[t] - \theta[1]*e[t-1] - ... - \theta[q]*e[t-q].$$
The `p`

and `q`

can be a vector for fitting a sparse ARIMA model. For example,
`p = c(1,3),q = c(1,3)`

means the ARMA((1,3),(1,3)) model defined as
$$X[t] = \mu + \phi[1]*X[t-1] + \phi[3]*X[t-3] + e[t]
- \theta[1]*e[t-1] - \theta[3]*e[t-3].$$ The `PDQ`

controls the
order of seasonal ARIMA model, i.e., ARIMA(p,d,q)x(P,D,Q)(S), where S is the seasonal
period. Note that the difference operators `d`

and D = `PDQ`

[2] are different.
The `d`

is equivalent to `diff(x,differences = d)`

and D is
`diff(x,lag = D,differences = S)`

, where the default seasonal period is
`S = frequency(x)`

.

The residual diagnostics plots will be drawn.

##### Value

##### Note

Missing values are removed before the estimate. Sparse seasonal ARIMA(p,d,q)x(P,D,Q)(S) model is not allowed.

##### References

Brockwell, P. J. and Davis, R. A. (1996). Introduction to Time Series and Forecasting. Springer, New York. Sections 3.3 and 8.3.

##### See Also

##### Examples

```
estimate(lh, p = 1) # AR(1) process
estimate(lh, p = 1, q = 1) # ARMA(1,1) process
estimate(lh, p = c(1,3)) # sparse AR((1,3)) process
# seasonal ARIMA(0,1,1)x(0,1,1)(12) model
estimate(USAccDeaths, p = 1, d = 1, PDQ = c(0,1,1))
```

*Documentation reproduced from package aTSA, version 3.1.2, License: GPL-2 | GPL-3*