Some notes about the program: Since I could not attach the SAS data set elettroload, I have attached a csv file. You will need to read it into a SAS data set and put it in the directory pointed by the libname statement at the start. Before using the data in PROC UCM, a few more steps need to be done: Data prepration for the analysis with PROC UCM. Step 1: Read the hourly load readings from the data set "elettroload" and store them in a new data set, "Test". Test has three columns of interest: Date, Hour, eload (hourly electricity load for that date). The dthour column is not used the in analysis. Step 2: Sort Test by Hour and Date so that Hour can be used as a BY group. In effect, we deal with 24 daily series. Step 3: Add some sinsoidal regressors to the data set. See the explanation in the book. ... View more

Modeling hourly (or even more frequent) electricity load is a well-studied problem and many approaches are suggested.  Many of these approaches can be handled using the UCM procedure.  At this time I am using this forum for illustrating one such approach discussed in Case Study 3 in Chapter 9 of "Time Series Modelling with Unobserved Components," by Pelagatti, M. M. (2015). Boca Raton, FL: CRC Press.  The electricity load data set used in this example is taken from http://www.ucmbook.info/ , which the website associated with the book.   I am also including a link to one of my workshops on UCM and SSM procedures.  This is a link to the slides of a workshop at "The 36th International Symposium on Forecasting Santander, Spain | 19-22 June 2016": https://forecasters.org/wp-content/uploads/gravity_forms/7-621289a708af3e7af65a7cd487aee6eb/2016/07/Selukar_Rajesh_ISF2016.pdf See the example "Hourly Electricity Load in Italy" in the Illustrations section for additional explanations.   Note that each PROC UCM call can take up to 15 minutes to finish (processes 9 years of hourly data).   /* ------------------Program-----------------------------------------------*/ libname wk 'C:\udrive\ucmBook\AX2016\programs'; * directory location of the elettroload data set; data test(drop=h1-h24);    set wk.elettroload;    array hr{24} h1-h24;    do hour=1 to 24;     eload = hr[hour];     dthour = intnx('dthour', dhms(date, 0, 0, 0), hour-1);     output;    end;    format dthour datetime.; run; proc sort data=test;    by hour date; run; data test(drop=j twoPi yf days year);   set test;   array cosf{16} cos1-cos16;   array sinf{16} sin1-sin16;   year = year(date);   yf = mdy(1, 1, year);   days = date - yf; *day within that year;   twoPi = 2*constant('pi')/365.25;   do j=1 to 16;      cosf[j] = cos(twoPi*j*days);      sinf[j] = sin(twoPi*j*days);   end; run; /* Analysis as per Case Study 3 in Chapt 9 of this book. */ proc ucm data=test;    *where hour=1;    id date interval=day;    by hour;    irregular;    level;    cycle period=7 rho=1 noest=(period rho);    cycle period=3.5 rho=1 noest=(period rho);    cycle period=2.3333 rho=1 noest=(period rho);    randomreg cos1-cos16 sin1-sin16;    model eload = dec24 dec25 dec26 jan1 jan6 aug15 easterSat                  easterSun easterMon easterTue holidays holySat                  holySun bridgeDay endYear;    estimate back=14 plot=panel;    forecast back=14 lead=14 outfor=loadfor; run; *An alternate formulation that uses a trigonometric season component with  first 16 harmonics to model the day-of-the-year pattern; title "Model that uses parsimonious trigonometric season"; proc ucm data=test;    *where hour=1;    id date interval=day;    by hour;    irregular;    level;    cycle period=7 rho=1 noest=(period rho);    cycle period=3.5 rho=1 noest=(period rho);    cycle period=2.3333 rho=1 noest=(period rho);    season length=365 type=trig keeph=1 to 16 by 1;    model eload = dec24 dec25 dec26 jan1 jan6 aug15 easterSat                  easterSun easterMon easterTue holidays holySat                  holySun bridgeDay endYear;    estimate back=14 plot=panel;                forecast back=14 lead=14 outfor=loadfor1; run; proc sgplot data=loadfor1;    title "Day of the year pattern for hour=1 in 2006";    where year(date) = 2006 && hour=1;    series x=date y=s_season; run; title; *Yet another formulation that uses spline season to model the day-of-the-year pattern; title "Model that uses spline season"; proc ucm data=test;    *where hour=1;    id date interval=day;    by hour;    irregular;    level;    cycle period=7 rho=1 noest=(period rho);    cycle period=3.5 rho=1 noest=(period rho);    cycle period=2.3333 rho=1 noest=(period rho);    splineseason length=365 degree=3 knots=4 15 30 45 60 75 90 105 120 135 150 165                                           180 195 210 225 240 255 270 285 300 315                                           330 345 360;    model eload = dec24 dec25 dec26 jan1 jan6 aug15 easterSat                  easterSun easterMon easterTue holidays holySat                  holySun bridgeDay endYear;    estimate back=14 plot=panel;                forecast back=14 lead=14 outfor=ucmfor2; run; proc sgplot data=ucmfor2;    title "Day-of-the-year pattern for hour=1 in 2006";    where year(date) = 2006 && hour=1;    series x=date y=s_splseas; run; ... View more

Generally speaking one looks at the significance of the final state of a component if it appears to be time-invariant (decided by looking at the estimate of the disturbance variance and the plot of the component).  Anyway, I will give a better response towards the end of next week.  At the moment I am quite busy because I am traveling tomorrow.  By the way, for an additional example of modeling hourly data using UCM models you can also check out Case Study 3 in Chapt 9 of Pelagatti, M. M. (2015). Time Series Modelling with Unobserved Components. Boca Raton, FL: CRC Press.    ... View more

I ran your program.  The estimates of the disturbance variances associated with the season and block-season components, 0.00380 and 7.88579, respectively, are significant.  This implies that these components are time-varying.  The components significance table you are looking at is based on the state at the final time point of your data set (as is stated in the title of that table).  For time-varying components the information in this table is not very relevant because it is only about that last snapshot of your data.  That is, it does not imply that the block-season component as a whole is insignificant. When I look at the plot of the block-season component, its amplitude varies over time.  In many sections of the data it is quite high, around 50 (whereas your data range is  0 to 121).  So the block-season component is clearly explaining good portion of the data variation in certain sections of your data.   Hope this helps. ... View more

This is a good question.  If your second transfer function spec did not have a denominator polynomial with unit root, you could have easily specified this model by including an extra copy of your I variable, say copy_I, in your input data set.  As it happens, the denominator polynomials in ARIMA cannot be "unstable".  One way to get around this and specify a model that is reasonably close to your model is as follows (this still involves making a copy of the I variable): Note that after multiplying your model equation by (1-B), one gets the following: (1-B) y_t = (omega_01/(1-delta B)) (1-B) X_t + omega_02 X_t + (1-B) N_t. I have renamed your I variable as X.  For simplicity, let us assume that your noise process N_t is a simple white noise.  Then (1-B) N_t is an MA(1) process with MA parameter equal to 1 (which is noninvertible).  In ARIMA you can specify a model close to this rearranged model where the noise is invertible MA(1).  You could do this as follows: Suppose your input data set is TEST. Step1.  Create a copy of variable X:    data test;        set test;        copy_X = X;    run;     Step 2: Specify the model:  proc arima data=test;    i var=y(1) crosscorr=(X copy_X(1)) noprint;    e q=1 input=(/(1) copy_X X) noint method=ml; quit;   This will give you a specification reasonably close to what you want.  If in fact, the true noise process is non-invertible MA(1), your estimated MA parameter will be close to it.   Hope this works OK for you. ... View more