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.

Usage

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.

Value

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. doi:10.1093/biomet/69.1.81 .

Author

A. Trapletti

See also

summary.arma for summarizing ARMA model fits; arma-methods for further methods; arima0, ar.

Examples

data(tcm)  
r <- diff(tcm10y)
summary(r.arma <- arma(r, order = c(1, 0)))
#> 
#> Call:
#> arma(x = r, order = c(1, 0))
#> 
#> Model:
#> ARMA(1,0)
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -1.707400 -0.116059  0.006282  0.128118  1.471796 
#> 
#> Coefficient(s):
#>            Estimate  Std. Error  t value Pr(>|t|)    
#> ar1        0.328972    0.039991    8.226 2.22e-16 ***
#> intercept  0.003325    0.011430    0.291    0.771    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Fit:
#> sigma^2 estimated as 0.07287,  Conditional Sum-of-Squares = 40.44,  AIC = 125.89
#> 
summary(r.arma <- arma(r, order = c(2, 0)))
#> 
#> Call:
#> arma(x = r, order = c(2, 0))
#> 
#> Model:
#> ARMA(2,0)
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -1.603801 -0.117037  0.002274  0.129126  1.376261 
#> 
#> Coefficient(s):
#>            Estimate  Std. Error  t value Pr(>|t|)    
#> ar1         0.40790     0.04114    9.915  < 2e-16 ***
#> ar2        -0.23962     0.04113   -5.826 5.67e-09 ***
#> intercept   0.00420     0.01111    0.378    0.705    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Fit:
#> sigma^2 estimated as 0.06881,  Conditional Sum-of-Squares = 38.12,  AIC = 95.94
#> 
summary(r.arma <- arma(r, order = c(0, 1)))
#> 
#> Call:
#> arma(x = r, order = c(0, 1))
#> 
#> Model:
#> ARMA(0,1)
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -1.590634 -0.118433 -0.000165  0.129751  1.339717 
#> 
#> Coefficient(s):
#>            Estimate  Std. Error  t value Pr(>|t|)    
#> ma1        0.506871    0.042113   12.036   <2e-16 ***
#> intercept  0.005287    0.016632    0.318    0.751    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Fit:
#> sigma^2 estimated as 0.06811,  Conditional Sum-of-Squares = 37.8,  AIC = 88.21
#> 
summary(r.arma <- arma(r, order = c(0, 2)))
#> 
#> Call:
#> arma(x = r, order = c(0, 2))
#> 
#> Model:
#> ARMA(0,2)
#> 
#> Residuals:
#>        Min         1Q     Median         3Q        Max 
#> -1.5497077 -0.1224297  0.0009082  0.1343164  1.3639450 
#> 
#> Coefficient(s):
#>            Estimate  Std. Error  t value Pr(>|t|)    
#> ma1        0.449184    0.042387   10.597   <2e-16 ***
#> ma2       -0.117937    0.042662   -2.764   0.0057 ** 
#> intercept  0.004701    0.014613    0.322   0.7477    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Fit:
#> sigma^2 estimated as 0.0673,  Conditional Sum-of-Squares = 37.28,  AIC = 83.58
#> 
summary(r.arma <- arma(r, order = c(1, 1)))
#> 
#> Call:
#> arma(x = r, order = c(1, 1))
#> 
#> Model:
#> ARMA(1,1)
#> 
#> Residuals:
#>        Min         1Q     Median         3Q        Max 
#> -1.5529214 -0.1189386  0.0001515  0.1337559  1.3663775 
#> 
#> Coefficient(s):
#>            Estimate  Std. Error  t value Pr(>|t|)    
#> ar1       -0.202320    0.074961   -2.699  0.00695 ** 
#> ma1        0.658111    0.056927   11.561  < 2e-16 ***
#> intercept  0.006799    0.018195    0.374  0.70866    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Fit:
#> sigma^2 estimated as 0.06731,  Conditional Sum-of-Squares = 37.35,  AIC = 83.62
#> 
plot(r.arma)




data(nino)
s <- nino3.4
summary(s.arma <- arma(s, order=c(20,0)))
#> 
#> Call:
#> arma(x = s, order = c(20, 0))
#> 
#> Model:
#> ARMA(20,0)
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -1.378647 -0.223040  0.007358  0.215996  1.103511 
#> 
#> Coefficient(s):
#>            Estimate  Std. Error  t value Pr(>|t|)    
#> ar1         1.10163     0.04090   26.932  < 2e-16 ***
#> ar2        -0.05461     0.06063   -0.901 0.367693    
#> ar3        -0.17718     0.06060   -2.924 0.003457 ** 
#> ar4         0.07072     0.06056    1.168 0.242900    
#> ar5        -0.05292     0.05990   -0.884 0.376941    
#> ar6         0.06743     0.05932    1.137 0.255661    
#> ar7        -0.17060     0.05877   -2.903 0.003698 ** 
#> ar8        -0.03509     0.05857   -0.599 0.549068    
#> ar9         0.03326     0.05827    0.571 0.568113    
#> ar10        0.14149     0.05829    2.427 0.015206 *  
#> ar11        0.01597     0.05812    0.275 0.783504    
#> ar12        0.16400     0.05791    2.832 0.004629 ** 
#> ar13       -0.22467     0.05833   -3.852 0.000117 ***
#> ar14       -0.01355     0.05869   -0.231 0.817394    
#> ar15       -0.04530     0.05845   -0.775 0.438283    
#> ar16       -0.18815     0.05807   -3.240 0.001196 ** 
#> ar17        0.20923     0.05868    3.566 0.000363 ***
#> ar18        0.07387     0.05868    1.259 0.208074    
#> ar19       -0.13363     0.05792   -2.307 0.021045 *  
#> ar20        0.05234     0.04002    1.308 0.190952    
#> intercept   4.46824     0.79612    5.613 1.99e-08 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Fit:
#> sigma^2 estimated as 0.1079,  Conditional Sum-of-Squares = 62.24,  AIC = 407.41
#> 
summary(s.arma
         <- arma(s, lag=list(ar=c(1,3,7,10,12,13,16,17,19),ma=NULL)))
#> Warning: order is ignored
#> 
#> Call:
#> arma(x = s, lag = list(ar = c(1, 3, 7, 10, 12, 13, 16, 17, 19),     ma = NULL))
#> 
#> Model:
#> ARMA(19,0)
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -1.360875 -0.230887  0.005043  0.226822  1.090989 
#> 
#> Coefficient(s):
#>            Estimate  Std. Error  t value Pr(>|t|)    
#> ar1         1.07772     0.02642   40.793  < 2e-16 ***
#> ar3        -0.16942     0.03016   -5.618 1.93e-08 ***
#> ar7        -0.13913     0.02254   -6.174 6.67e-10 ***
#> ar10        0.16459     0.02931    5.615 1.97e-08 ***
#> ar12        0.20472     0.04754    4.306 1.66e-05 ***
#> ar13       -0.29976     0.04133   -7.253 4.07e-13 ***
#> ar16       -0.21637     0.04349   -4.975 6.54e-07 ***
#> ar17        0.25139     0.04648    5.408 6.37e-08 ***
#> ar19       -0.04510     0.02580   -1.748   0.0804 .  
#> intercept   4.61896     0.77687    5.946 2.75e-09 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Fit:
#> sigma^2 estimated as 0.1094,  Conditional Sum-of-Squares = 63.24,  AIC = 393.87
#> 
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)))
#> Warning: order is ignored
#> 
#> Call:
#> arma(x = s, lag = list(ar = c(1, 3, 7, 10, 12, 13, 16, 17, 19),     ma = 12))
#> 
#> Model:
#> ARMA(19,12)
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -1.339941 -0.209727  0.001678  0.224448  1.065923 
#> 
#> Coefficient(s):
#>            Estimate  Std. Error  t value Pr(>|t|)    
#> ar1         1.06644     0.02807   37.997  < 2e-16 ***
#> ar3        -0.15394     0.03295   -4.673 2.98e-06 ***
#> ar7        -0.12194     0.01971   -6.187 6.14e-10 ***
#> ar10        0.12689     0.02735    4.639 3.50e-06 ***
#> ar12        0.46228     0.10391    4.449 8.63e-06 ***
#> ar13       -0.53619     0.09507   -5.640 1.70e-08 ***
#> ar16       -0.18195     0.05069   -3.589 0.000331 ***
#> ar17        0.24859     0.05087    4.887 1.02e-06 ***
#> ar19       -0.04463     0.02864   -1.558 0.119164    
#> ma12       -0.36376     0.11096   -3.278 0.001044 ** 
#> intercept   3.62293     0.78419    4.620 3.84e-06 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Fit:
#> sigma^2 estimated as 0.103,  Conditional Sum-of-Squares = 59.54,  AIC = 359.84
#> 
summary(s.arma
         <- arma(s, lag=list(ar=c(1,3,7,10,12,13,16,17),ma=12)))
#> Warning: order is ignored
#> 
#> Call:
#> arma(x = s, lag = list(ar = c(1, 3, 7, 10, 12, 13, 16, 17), ma = 12))
#> 
#> Model:
#> ARMA(17,12)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -1.27924 -0.19931  0.01423  0.22825  1.08027 
#> 
#> Coefficient(s):
#>            Estimate  Std. Error  t value Pr(>|t|)    
#> ar1         1.05679     0.02457   43.010  < 2e-16 ***
#> ar3        -0.10723     0.02392   -4.482 7.39e-06 ***
#> ar7        -0.15482     0.01939   -7.984 1.33e-15 ***
#> ar10        0.14458     0.02713    5.330 9.84e-08 ***
#> ar12        0.53544     0.07242    7.394 1.43e-13 ***
#> ar13       -0.64133     0.06248  -10.265  < 2e-16 ***
#> ar16       -0.14278     0.04063   -3.514 0.000441 ***
#> ar17        0.19739     0.03386    5.830 5.53e-09 ***
#> ma12       -0.43009     0.07957   -5.405 6.48e-08 ***
#> intercept   3.01582     0.63664    4.737 2.17e-06 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Fit:
#> sigma^2 estimated as 0.1023,  Conditional Sum-of-Squares = 59.37,  AIC = 353.98
#> 
plot(s.arma)