# Fit ARMA Models to Time Series Fit an ARMA model to a univariate time series by conditional least squares. For exact maximum likelihood estimation see `arima0`. ```r arma(x, order = c(1, 1), lag = NULL, coef = NULL, include.intercept = TRUE, series = NULL, qr.tol = 1e-07, ...) ``` ## Arguments - `x`: a numeric vector or time series. - `order`: a two dimensional integer vector giving the orders of the model to fit. `order[1]` corresponds to the AR part and `order[2]` to the MA part. - `lag`: a list with components `ar` and `ma`. Each component is an integer vector, specifying the AR and MA lags that are included in the model. If both, `order` and `lag`, are given, only the specification from `lag` is used. - `coef`: If given this numeric vector is used as the initial estimate of the ARMA coefficients. The preliminary estimator suggested in Hannan and Rissanen (1982) is used for the default initialization. - `include.intercept`: Should the model contain an intercept? - `series`: name for the series. Defaults to `deparse(substitute(x))`. - `qr.tol`: the `tol` argument for `qr` when computing the asymptotic standard errors of `coef`. - ``...``: additional arguments for `optim` when fitting the model. ## Details The following parametrization is used for the ARMA(p,q) model: $$ y[t] = a[0] + a[1]y[t-1] + \dots + a[p]y[t-p] + b[1]e[t-1] +\dots + b[q]e[t-q] + e[t], $$ where $a[0]$ is set to zero if no intercept is included. By using the argument `lag`, it is possible to fit a parsimonious submodel by setting arbitrary $a[i]$ and $b[i]$ to zero. `arma` uses `optim` to minimize the conditional sum-of-squared errors. The gradient is computed, if it is needed, by a finite-difference approximation. Default initialization is done by fitting a pure high-order AR model (see `ar.ols`). The estimated residuals are then used for computing a least squares estimator of the full ARMA model. See Hannan and Rissanen (1982) for details. ## Returns A list of class `"arma"` with the following elements: - **lag**: the lag specification of the fitted model. - **coef**: estimated ARMA coefficients for the fitted model. - **css**: the conditional sum-of-squared errors. - **n.used**: the number of observations of `x`. - **residuals**: the series of residuals. - **fitted.values**: the fitted series. - **series**: the name of the series `x`. - **frequency**: the frequency of the series `x`. - **call**: the call of the `arma` function. - **vcov**: estimate of the asymptotic-theory covariance matrix for the coefficient estimates. - **convergence**: The `convergence` integer code from `optim`. - **include.intercept**: Does the model contain an intercept? ## References E. J. Hannan and J. Rissanen (1982): Recursive Estimation of Mixed Autoregressive-Moving Average Order. **Biometrika** 69 , 81--94. ## Author(s) A. Trapletti ## See Also `summary.arma` for summarizing ARMA model fits; `arma-methods` for further methods; `arima0`, `ar`. ## Examples ```r data(tcm) r <- diff(tcm10y) summary(r.arma <- arma(r, order = c(1, 0))) summary(r.arma <- arma(r, order = c(2, 0))) summary(r.arma <- arma(r, order = c(0, 1))) summary(r.arma <- arma(r, order = c(0, 2))) summary(r.arma <- arma(r, order = c(1, 1))) plot(r.arma) data(nino) s <- nino3.4 summary(s.arma <- arma(s, order=c(20,0))) summary(s.arma <- arma(s, lag=list(ar=c(1,3,7,10,12,13,16,17,19),ma=NULL))) acf(residuals(s.arma), na.action=na.remove) pacf(residuals(s.arma), na.action=na.remove) summary(s.arma <- arma(s, lag=list(ar=c(1,3,7,10,12,13,16,17,19),ma=12))) summary(s.arma <- arma(s, lag=list(ar=c(1,3,7,10,12,13,16,17),ma=12))) plot(s.arma) ```

arma function