BAY_SEA

Bayesian seasonal adjustment modeling.  The model allows for a decomposition of a time series into trend, seasonal, and an error component. 

Required Arguments

W — Vector containing the stationary time series. (Input)

Optional Arguments

IORDER — Order of trend differencing. (Input)
Default: IORDER = 2.

ISORDER — Order of seasonal differencing. (Input)
Default: ISORDER = 1.

NFOCAST — Number of forecasted values.   (Input)
Default: NFOCAST = 0.

NPERIOD — Number of seasonals within a period. (Input)
Default: NPERIOD = 12.

RIGID — Controls rigidity of the seasonal pattern. (Input)
Default: RIGID = 1.0.

LOGT — Model option. (Input)

0

Non-additive model

1

Log additive model

             Default: LOGT = 0.

IPRINT — Printing option. (Input) 

0

No printing

1

Prints final results only

2

Prints intermediate and final results

Default: IPRINT = 0.

ABIC — The Akaike Bayesian Information Criterion for the estimated model.   (Output)

TREND — Vector of length size(W)+ NFOCAST containing the estimated trend component for each data value followed by the trend estimates for the NFOCAST forecasted values. (Output)

SEASONAL — Vector of length size(W)+ NFOCAST containing the estimated seasonal components for each data value followed by the estimates for the NFOCAST forecasted seasonal values. (Output)

COMP — Vector of length size(W)containing the estimated irregular components.   (Output)

SMOOTHED — Vector of length size(W)+ NFOCAST containing the estimated smoothed estimates for each of the time series values followed by the NFOCAST forecast values. (Output)

FORTRAN 90 Interface

Generic:                              CALL BAY_SEA (W [,…])

Specific:                             The specific interface names are S_BAY_SEA and D_BAY_SEA.

Description

Routine BAY_SEA is based upon the algorithm published by Akaike (1980).  This algorithm uses a Bayesian approach to the problem of fitting the following autoregressive model for a time series Wt  decomposed into a trend and a seasonal component. 

Adopting the notation described earlier in the Usage Notes section of this chapter, if

 

then a seasonal autoregressive model can be represented by the following relationship:

where Wt is the stationary time series with mean μ , Tt denotes an underlying trend, St
seasonal component and At a noise or irregular component.

A non-Bayesian approach to this problem would be to estimate the trend and seasonal components by minimizing

where p is the period of the seasonal component, and d, r , and z are properly chosen constants.

In BAY_SEA, the approach is to select the parameter d, which controls the smoothness of the trend and seasonality estimates, using Bayesian methods.  The prior distribution controls the smoothness of the trend and seasonal components by assuming low order Gaussian autoregressive models for some differences of these components.  The choice of the variance of the Gaussian distribution is realized by maximizing the log likelihood of the Bayesian model.

The other smoothing parameters, r and z, are determined by the value of RIGID.  The default value for RIGID is 1.  Larger values of RIGID produce a more rigid seasonal pattern.  Normally, a series is first fit using the default value for RIGID.  The smoothness of the trend and seasonality estimates are examined and then RIGID is either increased or decreased depending upon whether more or less seasonal smoothing is needed.

Additionally, BAY_SEA selects the optimum autoregressive model as the model that minimizes ABIC.

,

where the likelihood in this case is the mixed Bayesian maximum likelihood.  Smaller values of represent a better fit.

Example

This example uses unadjusted unemployment for women over 20 years of age in the U.S. for 1991-2001, as reported by the U.S. Bureau of Labor Statistics (www.bls.gov). Figure 1 displays this data together with the smoothed preditions from BAY_SEA. Figure 2 displays the same data with trend predictions from BAY_SEA.

 

       use bay_sea_int

       use wrrrl_int

 

       IMPLICIT     NONE

!                                  SPECIFICATIONS FOR PARAMETERS

       integer, parameter :: focast=12, nobs=132, nyears=11

       real(kind(1e0))           :: b(nobs)

!                                  U.S. Labor Statistics

!                                  unemployment for women

!                                  over 20 years of age

       data b/ 2968D0,3009D0,2962D0,2774D0,3040D0,3165D0,& !1991

               3104D0,3313D0,3178D0,3142D0,3129D0,3107D0,&

               3397D0,3447D0,3328D0,3229D0,3286D0,3577D0,& !1992

               3799D0,3867D0,3655D0,3360D0,3310D0,3369D0,&

               3643D0,3419D0,3108D0,3118D0,3146D0,3385D0,& !1993

               3458D0,3468D0,3330D0,3244D0,3135D0,3005D0,&

               3462D0,3272D0,3275D0,2938D0,2894D0,3106D0,& !1994

               3150D0,3289D0,3136D0,2829D0,2776D0,2467D0,&

               2944D0,2787D0,2749D0,2762D0,2578D0,2900D0,& !1995

               3100D0,3102D0,2934D0,2864D0,2652D0,2456D0,&

               3088D0,2774D0,2701D0,2555D0,2677D0,2741D0,& !1996

               3052D0,2966D0,2772D0,2723D0,2705D0,2640D0,&

               2898D0,2788D0,2718D0,2406D0,2520D0,2645D0,& !1997

               2708D0,2811D0,2666D0,2380D0,2292D0,2187D0,&

               2750D0,2595D0,2554D0,2213D0,2218D0,2449D0,& !1998

               2532D0,2639D0,2449D0,2326D0,2302D0,2065D0,&

               2447D0,2398D0,2381D0,2250D0,2086D0,2397D0,& !1999

               2573D0,2475D0,2299D0,2054D0,2127D0,1935D0,&

               2425D0,2245D0,2298D0,2005D0,2208D0,2379D0,& !2000

               2459D0,2539D0,2182D0,1959D0,2012D0,1834D0,&

               2404D0,2329D0,2285D0,2175D0,2245D0,2492D0,& !2001

               2636D0,2892D0,2784D0,2771D0,2878D0,2856D0/

 

       integer  :: i

       real(kind(1e0)) :: abicm

       real(kind(1e0)) :: trend(nobs+focast), season(nobs+focast)

       real(kind(1e0)) :: irreg(nobs), smooth(nobs+focast)

       character       :: months(focast)*3, years(nyears+2)*4

       data months/'Jan', 'Feb', 'Mar', 'Apr', 'May', 'Jun', &

                   'Jul', 'Aug', 'Sep', 'Oct', 'Nov', 'Dec'/

       data years/'    ', '1991', '1992', '1993', '1994', '1995', &

                  '1996', '1997','1998','1999','2000','2001','2002'/

 

       call bay_sea(b, trend=trend, seasonal=season, &

                    comp=irreg, abic=abicm, iorder=2, isorder=1,&

                    nfocast=focast, smoothed=smooth)

 

       write(*,*) 'ABIC = ', abicm

       call wrrrl('TREND with last 12 values forecasted',trend,months, &

                   years, nra=12,nca=nyears+1,lda=12)

 

         call wrrrl('SEASONAL with last 12 values forecasted',season,&

                      months,years, nra=12,nca=nyears+1, lda=12)

         call wrrrl('IRREGULAR=Original data-TREND-SEASONAL',irreg, &

                     months,years, nra=12,nca=nyears,lda=12)

 

       end

Output

 

 ABIC =  1297.6403

 

                    TREND with last 12 values forecasted

         1991     1992     1993     1994     1995     1996     1997     1998

 Jan   2879.8   3318.9   3422.6   3228.7   2827.3   2795.7   2743.1   2481.3

 Feb   2918.2   3359.9   3387.8   3206.0   2815.8   2785.6   2720.6   2469.7

 Mar   2955.1   3399.1   3355.5   3177.0   2812.4   2777.9   2694.2   2455.4

 Apr   2990.0   3436.1   3329.5   3142.1   2814.7   2773.2   2665.2   2439.0

 May   3022.7   3469.0   3309.6   3103.9   2819.3   2771.3   2636.0   2422.8

 Jun   3052.8   3496.1   3294.8   3064.8   2825.5   2771.0   2607.4   2408.5

 Jul   3082.8   3514.5   3283.8   3025.7   2830.9   2772.4   2580.8   2396.8

 Aug   3116.2   3521.9   3276.0   2987.3   2833.2   2773.6   2557.2   2387.7

 Sep   3153.4   3517.8   3270.3   2949.0   2831.7   2774.7   2536.3   2380.3

 Oct   3193.8   3503.7   3264.8   2911.3   2826.1   2774.5   2518.0   2373.6

 Nov   3235.6   3482.4   3256.8   2876.6   2816.6   2770.4   2503.0   2366.6

 Dec   3277.6   3455.2   3245.3   2847.6   2805.9   2760.2   2491.5   2359.1

 

         1999     2000     2001     2002

 Jan   2352.1   2235.9   2206.0   3166.1

 Feb   2345.6   2237.6   2239.0   3275.1

 Mar   2338.1   2239.3   2281.4   3384.1

 Apr   2328.3   2238.9   2333.1   3493.0

 May   2315.5   2235.2   2393.9   3602.0

 Jun   2301.4   2226.3   2464.7   3710.9

 Jul   2286.0   2212.8   2545.7   3819.9

 Aug   2269.8   2196.6   2636.7   3928.9

 Sep   2255.5   2181.3   2735.7   4037.8

 Oct   2244.6   2171.9   2840.4   4146.8

 Nov   2238.3   2172.1   2948.2   4255.8

 Dec   2235.6   2183.3   3057.2   4364.7

 

                   SEASONAL with last 12 values forecasted

        1991    1992    1993    1994    1995    1996    1997    1998    1999

 Jan   162.9   165.6   169.3   172.0   173.8   176.3   176.4   177.3   176.4

 Feb    51.4    51.5    50.5    49.5    48.6    48.8    50.0    51.1    51.0

 Mar   -24.0   -23.9   -23.4   -18.8   -16.3   -13.0    -8.7    -4.9    -3.0

 Apr  -191.0  -190.1  -189.1  -188.0  -186.6  -187.8  -188.5  -187.9  -186.6

 May  -140.6  -143.3  -145.4  -147.4  -148.1  -147.1  -147.1  -147.9  -147.7

 Jun    67.2    66.6    65.8    64.4    63.3    62.2    62.7    63.6    64.9

 Jul   176.9   180.1   181.6   183.0   185.5   186.6   185.8   186.1   187.2

 Aug   251.9   253.0   252.6   253.3   253.1   252.8   253.7   254.4   255.2

 Sep    76.4    77.2    77.1    77.3    75.5    73.3    72.6    70.7    68.9

 Oct   -79.5   -80.5   -80.1   -80.8   -81.5   -84.3   -87.6   -90.1   -93.4

 Nov  -119.8  -120.7  -120.5  -120.2  -120.4  -119.7  -119.7  -118.1  -117.6

 Dec  -235.7  -237.9  -243.0  -247.9  -250.5  -251.2  -254.2  -256.2  -257.5

 

        2000    2001    2002

 Jan   177.2   177.6   177.5

 Feb    50.8    51.6    51.5

 Mar    -1.9    -1.7    -1.8

 Apr  -187.2  -186.5  -186.6

 May  -145.7  -145.6  -145.7

 Jun    65.6    65.1    65.0

 Jul   186.4   184.7   184.7

 Aug   256.8   256.9   256.9

 Sep    67.3    67.0    67.0

 Oct   -95.1   -94.6   -94.6

 Nov  -117.3  -116.6  -116.6

 Dec  -258.1  -257.1  -257.1

 

                   IRREGULAR=Original data-TREND-SEASONAL

        1991    1992    1993    1994    1995    1996    1997    1998    1999

 Jan   -74.7   -87.5    51.1    61.2   -57.1   116.0   -21.5    91.4   -81.5

 Feb    39.4    35.6   -19.3    16.5   -77.4   -60.4    17.3    74.2     1.5

 Mar    30.9   -47.2  -224.1   116.8   -47.1   -63.9    32.5   103.5    45.9

 Apr   -25.0   -17.0   -22.5   -16.1   133.9   -30.5   -70.6   -38.0   108.3

 May   157.8   -39.7   -18.2   -62.6   -93.1    52.8    31.2   -56.9   -81.9

 Jun    45.0    14.3    24.4   -23.1    11.2   -92.1   -25.1   -23.2    30.7

 Jul  -155.6   104.4    -7.4   -58.6    83.6    93.0   -58.6   -50.9    99.9

 Aug   -55.0    92.1   -60.6    48.4    15.7   -60.4     0.1    -3.1   -49.9

 Sep   -51.8    60.0   -17.4   109.6    26.8   -76.0    57.1    -2.1   -25.3

 Oct    27.7   -63.3    59.3    -1.5   119.4    32.8   -50.3    42.5   -97.1

 Nov    13.2   -51.7    -1.3    19.5   -44.3    54.3   -91.3    53.5     6.3

 Dec    65.1   151.7     2.7  -132.8   -99.4   131.0   -50.3   -37.9   -43.1

 

        2000    2001

 Jan    11.9    20.4

 Feb   -43.5    38.4

 Mar    60.5     5.3

 Apr   -46.7    28.5

 May   118.5    -3.3

 Jun    87.1   -37.7

 Jul    59.9   -94.4

 Aug    85.6    -1.6

 Sep   -66.6   -18.7

 Oct  -117.8    25.3

 Nov   -42.7    46.4

 Dec   -91.2    56.0

Figure 8-4  Sample Smoothed Predictions from BAY_SEA

Figure 8- 5 Sample Trend Predictions from BAY_SEA

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