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| The UCM Procedure |
| Parameter Estimation |
The parameter vector in a UCM consists of the variances of the disturbance terms of the unobserved components, the damping coefficients and frequencies in the cycles, the damping coefficient in the autoregression, the lag coefficients of the dependent lags, and the regression coefficients in the regression terms. The regression coefficients are always part of the state vector and are estimated by state smoothing. The remaining parameters are estimated by maximizing either the full diffuse likelihood or the nondiffuse likelihood. The decision to use the full diffuse likelihood or the nondiffuse likelihood depends on the presence or absence of the dependent lag coefficients in the parameter vector. If the parameter vector does not contain any dependent lag coefficients, then the full diffuse likelihood is used. If, on the other hand, the parameter vector does contain some dependent lag coefficients, then the parameters are estimated by maximizing the nondiffuse likelihood. The optimization of the full diffuse likelihood is often unstable when the parameter vector contains dependent lag coefficients. In this sense, when the parameter vector contains dependent lag coefficients, the parameter estimates are not true maximum likelihood estimates.
The optimization of the likelihood, either full or nondiffuse, is carried out using one of several nonlinear optimization algorithms. The user can control many aspects of the optimization process by using the NLOPTIONS statement and by providing the starting values of the parameters while specifying the corresponding components. However, in most cases the default settings work quite well. The optimization process is not guaranteed to converge to a maximum likelihood estimate. In most cases the difficulties in parameter estimation are associated with the specification of a model that is not appropriate for the series being modeled.
Parameter Estimation by Profile Likelihood Optimization
If a disturbance variance, such as the disturbance variance of the irregular component, is a part of the UCM and is a free parameter, then it can be profiled out of the likelihood. This means solving analytically for its optimum and plugging this expression back into the likelihood formula, giving rise to the so-called profile likelihood. The expression of the profile likelihood and the MLE of the profiled variance are given earlier in the section The UCMs as State Space Models, where the computation of the likelihood of the state space model is also discussed.
In some situations the optimization of the profile likelihood can be more efficient because the number of parameters to optimize is reduced by one; however, for a variety of reasons such gains might not always be observed. Moreover, in theory the estimates obtained by optimizing the profile likelihood and the usual likelihood should be the same, but in practice this might not hold because of numerical rounding and other conditions.
In the UCM procedure, by default the usual likelihood is optimized if any of the disturbance variance parameters is held fixed to a nonzero value by using the NOEST option in the corresponding component statement. In other cases the decision whether to optimize the profile likelihood or the usual likelihood is based on several factors that are difficult to document. You can choose which likelihood to optimize during parameter estimation by specifying the PROFILE option for the profile likelihood optimization or the NOPROFILE option for the usual likelihood optimization. In the presence of the PROFILE option, the disturbance variance to profile is checked in a specific order, so that if the irregular component disturbance variance is free then it is always chosen. The situation in other cases is more complicated.
Profiling in the Presence of Fixed Variance Parameters
Note that when the parameter estimation is done by optimizing the profile likelihood, the interpretation of the variance parameters that are held fixed to nonzero values changes. In the presence of the PROFILE option, the disturbance variances that are held at a fixed value by using the NOEST option in their respective component statements are interpreted as being restricted to be that fixed multiple of the profiled variance rather than being fixed at that nominal value. That is, implicitly, the parameter estimation is done under the restriction of holding the disturbance variance ratio fixed at a given value rather than the disturbance variance itself. See Example 31.5 for an example of this type of restriction to obtain a UC model that is equivalent to the famous Hodrick-Prescott filter.
t values
The t values reported in the table of parameter estimates are approximations whose accuracy depends on the validity of the model, the nature of the model, and the length of the observed series. The distributional properties of the maximum likelihood estimates of general unobserved components models have not been explored fully; therefore the probability values that correspond to a t distribution should be interpreted carefully, as they can be misleading. This is particularly true if the parameters in question are close to the boundary of the parameter space. The two sources by Harvey (1989, 2001) are good references for information about this topic. For some parameters, such as, the cycle period, the reported t values are uninformative because comparison of the estimated parameter with zero is never needed. In such cases the t values and the corresponding probability values should be ignored.
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