Class AR
 All Implemented Interfaces:
Serializable
Contrary to the movingaverage (MA) model, the autoregressive model is not always stationary as it may contain a unit root.
The notation AR(p)
indicates an autoregressive
model of order p. For an AR(p)
model to be weakly stationary,
the inverses of the roots of the characteristic polynomial must be less
than 1 in modulus.
Two general approaches are available for determining the order p. The first approach is to use the partial autocorrelation function, and the second approach is to use some information criteria.
Autoregression is a good start point for more complicated models. They often fit quite well (don't need the MA terms). And the fitting process is fast (MLEs require some iterations). In applications, easily fitting autoregressions is important for obtaining initial values of parameters and in getting estimates of the error process. Least squares is a popular choice, as is the YuleWalker procedure. Unlike the YuleWalker procedure, least squares can produce a nonstationary fitted model. Both the YuleWalker and least squares estimators are noniterative and consistent, so they can be used as starting values for iterative methods like MLE.
 See Also:

Nested Class Summary

Constructor Summary

Method Summary
Modifier and TypeMethodDescriptiondouble
Returns adjusted R^{2} statistic.double[]
ar()
Returns the linear coefficients of AR (without intercept).int
df()
Returns the degreeoffreedom of residual standard error.static AR
fit
(double[] x, int p) Fits an autoregressive model with YuleWalker procedure.double[]
Returns the fitted values.double
forecast()
Returns 1step ahead forecast.double[]
forecast
(int l) Returns lstep ahead forecast.double
Returns the intercept.double
mean()
Returns the mean of time series.static AR
ols
(double[] x, int p) Fits an autoregressive model with least squares method.static AR
ols
(double[] x, int p, boolean stderr) Fits an autoregressive model with least squares method.int
p()
Returns the order of AR.double
R2()
Returns R^{2} statistic.double[]
Returns the residuals, that is response minus fitted values.double
RSS()
Returns the residual sum of squares.toString()
double[][]
ttest()
Returns the ttest of the coefficients (including intercept).double
variance()
Returns the residual variance.double[]
x()
Returns the time series.

Constructor Details

AR
Constructor. Parameters:
x
 the time seriesar
 the estimated weight parameters of AR(p).b
 the intercept.method
 the fitting method.


Method Details

x
public double[] x()Returns the time series. Returns:
 the time series.

mean
public double mean()Returns the mean of time series. Returns:
 the mean of time series.

p
public int p()Returns the order of AR. Returns:
 the order of AR.

ttest
public double[][] ttest()Returns the ttest of the coefficients (including intercept). The first column is the coefficients, the second column is the standard error of coefficients, the third column is the tscore of the hypothesis test if the coefficient is zero, the fourth column is the pvalues of test. The last row is of intercept. Returns:
 the ttest of the coefficients.

ar
public double[] ar()Returns the linear coefficients of AR (without intercept). Returns:
 the linear coefficients.

intercept
public double intercept()Returns the intercept. Returns:
 the intercept.

residuals
public double[] residuals()Returns the residuals, that is response minus fitted values. Returns:
 the residuals.

fittedValues
public double[] fittedValues()Returns the fitted values. Returns:
 the fitted values.

RSS
public double RSS()Returns the residual sum of squares. Returns:
 the residual sum of squares.

variance
public double variance()Returns the residual variance. Returns:
 the residual variance.

df
public int df()Returns the degreeoffreedom of residual standard error. Returns:
 the degreeoffreedom of residual standard error.

R2
public double R2()Returns R^{2} statistic. In regression, the R^{2} coefficient of determination is a statistical measure of how well the regression line approximates the real data points. An R^{2} of 1.0 indicates that the regression line perfectly fits the data.In the case of ordinary leastsquares regression, R^{2} increases as we increase the number of variables in the model (R^{2} will not decrease). This illustrates a drawback to one possible use of R^{2}, where one might try to include more variables in the model until "there is no more improvement". This leads to the alternative approach of looking at the adjusted R^{2}.
 Returns:
 R^{2} statistic.

adjustedR2
public double adjustedR2()Returns adjusted R^{2} statistic. The adjusted R^{2} has almost same explanation as R^{2} but it penalizes the statistic as extra variables are included in the model. Returns:
 Adjusted R^{2} statistic.

fit
Fits an autoregressive model with YuleWalker procedure. Parameters:
x
 the time series.p
 the order. Returns:
 the model.

ols
Fits an autoregressive model with least squares method. Parameters:
x
 the time series.p
 the order. Returns:
 the model.

ols
Fits an autoregressive model with least squares method. Parameters:
x
 the time series.p
 the order.stderr
 the flag if estimate the standard errors of parameters. Returns:
 the model.

forecast
public double forecast()Returns 1step ahead forecast. Returns:
 1step ahead forecast.

forecast
public double[] forecast(int l) Returns lstep ahead forecast. Parameters:
l
 the number of steps. Returns:
 lstep ahead forecast.

toString
