The UCM Procedure

AUTOREG Statement

  • AUTOREG <options>;

The AUTOREG statement specifies an autoregressive component in the model. An autoregressive component is a special case of cycle that corresponds to the frequency of zero or $\pi $. It is modeled separately for easier interpretation. A stochastic equation for an autoregressive component $r_ t$ can be written as follows:

\[ r_ t = \rho r_{t-1} + \nu _{t} , \; \; \; \; \nu _ t \; \sim \; i.i.d. \; \; N( 0, \sigma _{\nu }^{2} ) \]

The damping factor $\rho $ can take any value in the interval (–1, 1), including –1 but excluding 1. If $\rho = 1$, the autoregressive component cannot be distinguished from the random walk level component. If $\rho = -1$, the autoregressive component corresponds to a seasonal component with a season length of 2, or a nonstationary cycle with period 2. If $| \rho | < 1$, then the autoregressive component is stationary. The following example illustrates the AUTOREG statement. This statement includes an autoregressive component in the model. The damping factor $\rho $ and the disturbance variance $ \sigma _{\nu }^{2}$ are estimated from the data. 

      autoreg;

NOEST=RHO
NOEST=VARIANCE
NOEST=(RHO VARIANCE)

fixes the values of $\rho $ and $\sigma _{\nu }^{2}$ to those specified in the RHO= and VARIANCE= options.

PLOT=FILTER
PLOT=SMOOTH
PLOT=( <FILTER> <SMOOTH> )

requests plotting of the filtered or smoothed estimate of the autoreg component.

PRINT=FILTER
PRINT=SMOOTH
PRINT=(<FILTER> <SMOOTH>)

requests printing of the filtered or smoothed estimate of the autoreg component.

RHO=value

specifies an initial value for the damping factor $\rho $ during the parameter estimation process. The value of $\rho $ must be in the interval (–1, 1), including –1 but excluding 1.

VARIANCE=value

specifies an initial value for the disturbance variance $\sigma _{\nu }^{2}$ during the parameter estimation process. Any nonnegative value, including zero, is an acceptable starting value.