The UCM Procedure

Example 35.1 The Airline Series Revisited

The series in this example, the monthly airline passenger series, has already been discussed earlier; see the section A Seasonal Series with Linear Trend. Recall that the series consists of monthly numbers of international airline travelers (from January 1949 to December 1960). Here additional output features of the UCM procedure are illustrated, such as how to use the ESTIMATE and FORECAST statements to limit the span of the data used in parameter estimation and forecasting. The following statements fit a BSM to the logarithm of the airline passenger numbers. The disturbance variance for the slope component is held fixed at value 0; that is, the trend is locally linear with constant slope. In order to evaluate the performance of the fitted model on observed data, some of the observed data are withheld during parameter estimation and forecast computations. The observations in the last two years, years 1959 and 1960, are not used in parameter estimation, while the observations in the last year, year 1960, are not used in the forecasting computations. This is done using the BACK= option in the ESTIMATE and FORECAST statements. In addition, a panel of residual diagnostic plots is obtained using the PLOT=PANEL option in the ESTIMATE statement.

data seriesG;
   set sashelp.air;
   logair = log(air);
proc ucm data = seriesG;
   id date interval = month;
   model logair;
   slope var = 0 noest;
   season length = 12 type=trig;
   estimate back=24 plot=panel;
   forecast back=12 lead=24 print=forecasts;

The following tables display the summary of data used in estimation and forecasting (Output 35.1.1 and Output 35.1.2). These tables provide simple summary statistics for the estimation and forecast spans; they include useful information such as the beginning and ending dates of the span, the number of nonmissing values, etc.

Output 35.1.1: Observation Span Used in Parameter Estimation (partial output)

Variable Type First Last Nobs Mean
logair Dependent JAN1949 DEC1958 120 5.43035

Output 35.1.2: Observation Span Used in Forecasting (partial output)

Variable Type First Last Nobs Mean
logair Dependent JAN1949 DEC1959 132 5.48654

The following tables display the fixed parameters in the model, the preliminary estimates of the free parameters, and the final estimates of the free parameters (Output 35.1.3, Output 35.1.4, and Output 35.1.5).

Output 35.1.3: Fixed Parameters in the Model

The UCM Procedure

Fixed Parameters in the Model
Component Parameter Value
Slope Error Variance 0

Output 35.1.4: Starting Values for the Parameters to Be Estimated

Preliminary Estimates of the Free Parameters
Component Parameter Estimate
Irregular Error Variance 6.64120
Level Error Variance 2.49045
Season Error Variance 1.26676

Output 35.1.5: Maximum Likelihood Estimates of the Free Parameters

Final Estimates of the Free Parameters
Component Parameter Estimate Approx
Std Error
t Value Approx
Pr > |t|
Irregular Error Variance 0.00018686 0.0001212 1.54 0.1233
Level Error Variance 0.00040314 0.0001566 2.57 0.0100
Season Error Variance 0.00000350 1.66319E-6 2.10 0.0354

Two types of goodness-of-fit statistics are reported after a model is fit to the series (see Output 35.1.6 and Output 35.1.7). The first type is the likelihood-based goodness-of-fit statistics, which include the full likelihood of the data, the diffuse portion of the likelihood (see the section Details: UCM Procedure), and the information criteria. The second type of statistics is based on the raw residuals, residual = observed – predicted. If the model is nonstationary, then one-step-ahead predictions are not available for some initial observations, and the number of values used in computing these fit statistics will be different from those used in computing the likelihood-based test statistics.

Output 35.1.6: Likelihood-Based Fit Statistics for the Airline Data

Likelihood Based Fit Statistics
Statistic Value
Full Log Likelihood 180.63
Diffuse Part of Log Likelihood -13.93
Non-Missing Observations Used 120
Estimated Parameters 3
Initialized Diffuse State Elements 13
Normalized Residual Sum of Squares 107
AIC (smaller is better) -355.3
BIC (smaller is better) -347.2
AICC (smaller is better) -355
HQIC (smaller is better) -352
CAIC (smaller is better) -344.2

Output 35.1.7: Residuals-Based Fit Statistics for the Airline Data

Fit Statistics Based on Residuals
Mean Squared Error 0.00156
Root Mean Squared Error 0.03944
Mean Absolute Percentage Error 0.57677
Maximum Percent Error 2.19396
R-Square 0.98705
Adjusted R-Square 0.98680
Random Walk R-Square 0.86370
Amemiya's Adjusted R-Square 0.98630
Number of non-missing residuals used for computing the fit statistics = 107

The diagnostic plots based on the one-step-ahead residuals are shown in Output 35.1.8. The residual histogram and the Q-Q plot show no reasons to question the approximate normality of the residual distribution. The remaining plots check for the whiteness of the residuals. The sample correlation plots, the autocorrelation function (ACF) and the partial autocorrelation function (PACF), also do not show any significant violations of the whiteness of the residuals. Therefore, on the whole, the model seems to fit the data well.

Output 35.1.8: Residual Diagnostics for the Airline Series Using a BSM

Residual Diagnostics for the Airline Series Using a BSM

The forecasts are given in Output 35.1.9. In order to save the space, the upper and lower confidence limit columns are dropped from the output, and only the rows corresponding to the year 1960 are shown. Recall that the actual measurements in the years 1959 and 1960 were withheld during the parameter estimation, and the ones in 1960 were not used in the forecast computations.

Output 35.1.9: Forecasts for the Airline Data

Obs date Forecast StdErr logair Residual
133 JAN60 6.050 0.038 6.033 -0.017
134 FEB60 5.996 0.044 5.969 -0.027
135 MAR60 6.156 0.049 6.038 -0.118
136 APR60 6.124 0.053 6.133 0.010
137 MAY60 6.168 0.058 6.157 -0.011
138 JUN60 6.303 0.061 6.282 -0.021
139 JUL60 6.435 0.065 6.433 -0.002
140 AUG60 6.450 0.068 6.407 -0.043
141 SEP60 6.265 0.071 6.230 -0.035
142 OCT60 6.138 0.073 6.133 -0.005
143 NOV60 6.015 0.075 5.966 -0.049
144 DEC60 6.121 0.077 6.068 -0.053

The figure Output 35.1.10 shows the forecast plot. The forecasts in the year 1960 show that the model predictions were quite good.

Output 35.1.10: Forecast Plot of the Airline Series Using a BSM

Forecast Plot of the Airline Series Using a BSM