( ) [ ] ( )
Note that pr(i, b) is a quadratic function in bj which helps a lot in calculating the first order conditions.
The four equations (a1) can be solved iteratively using the Fisher Score approach
( )+1 =b( )−H−1
( )
b( ) ∇L( )
b( ),l=0,1,2,... One could use methods to derive values for these data borders, but we do not follow this path here.REFERENCES
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TABLE 1. MEAN ABSOLUTE PERCENTAGE SMOOTHING INDEX (MAPSI) FOR WESTERN GERMANY AND BOLZANO: 1990-1998.
Western Germany Bolzano
MAPSIa,b in % MAPSIa,b in %
Year Raw Data Fitted Data SQb Raw Data Fitted Data SQb 1990 3.26 1.37 0.58
1991 1.75 1.23 0.30
1992 1.56 1.27 0.19 17.34 1.87 0.89
1993 2.13 0.94 0.56 14.59 1.72 0.88
1994 1.94 1.05 0.46 14.02 1.28 0.91
1995 1.95 1.32 0.32 18.99 1.61 0.92
1996 1.98 1.05 0.47 14.57 1.38 0.91
1997 14.21 2.07 0.86
1998 17.62 2.03 0.88
Mean Value over observation period of six Years
2.09 1.18 0.43 15.87 1.71 0.89
a Calculated for the ages 15 to 43.
b See text for detailed information about the indexes.
FIGURE 1a. TOTAL FERTILITY RATE FOR WEST GERMANY; 1954 TO 1999
1 2 3
1954 1959 1964 1969 1974 1979 1984 1989 1994 1999
Total Fertility Rate
FIGURE 1b. MODE OF MOTHERS' AGE FOR WEST GERMANY; 1954 to 1999
20 22 24 26 28 30
1954 1959 1964 1969 1974 1979 1984 1989 1994 1999
Mothers' Age Mode
FIGURE 1c. YOUNG AGE AND OLD AGE FERTILITY INDEX FOR WEST GERMANY; 1954 TO 1999
0,00 0,05 0,10
1954 1959 1964 1969 1974 1979 1984 1989 1994 1999
Cumulative Fertility Rate for Women under Age 18 Cumulative Fertility Rate for Women over Age 39
195 5
1965 197
5 1985
1995 10 15 20 25 30 35 40 45
0 0,2
FIGURE 2. SMOOTH FERTILITY MOUNTAIN OVER WOMEN'S AGE AND TIME;
EXAMPLE BASED ON DATA FOR WEST GERMANY Probability of Childbearing
Observation year
Age of the Mother at Childbearing
FIGURE 3. FERTILITY FUNCTION OVER MOTHERS' AGE FOR WEST GERMANY AND BOLZANO; 1999
0,00 0,04 0,08 0,12
14 19 24 29 34 39 44
Age of the Mother at Childbearing
Relative Childbearing Frequencies
Bolzano West Germany
FIGURE 4a. TOTAL FERTILITY RATE FOR WEST GERMANY AND BOLZANO; 1990 TO 2001
0,8 1 1,2 1,4 1,6
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001
Total Fertility Rate
West German Raw Data Bolzano Raw Data Bolzano Fitted Data
FIGURE 4b. MODE OVER MOTHERS' AGE FOR WEST GERMANY AND BOLZANO; 1990 TO 2001
26 28 30 32
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001
Mothers' Age Mode at Childbearing
West German Raw Data Bolzano Raw Data Bolzano Fitted Data
FIGURE 4c. OLD AGE FERTILITY INDEX FOR WEST GERMANY AND BOLZANO; 1990 TO 2001
0 0,01 0,02 0,03 0,04 0,05
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001
Cumulative Fertility Rate For Women over Age 39
West German Raw Data Bolzano Raw Data Bolzano Fitted Data
FIGURE 4d. YOUNG AGE FERTILITY INDEX FOR WEST GERMANY AND BOLZANO; 1990 TO 2001
0 0,01 0,02 0,03 0,04 0,05
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001
Cumulative Fertility Rate for Women under Age 18
West German Raw Data Bolzano Raw Data Bolzano Fitted Data
1990 1993 199
6 1999
14 19 24 29 34 39
0
FIGURE 5a. FERTILITY MOUNTAIN OVER WOMEN'S AGE AND TIME FOR WEST GERMANY; 1990 TO 1999.
Probability of Childbearing
Observation Year
Age of the Mother at Childbearing
1990 1993
199 6
1999 14 19 24 29 34 39
0
FIGURE 5b. FERTILITY MOUNTAIN OVER WOMEN'S AGE AND TIME FOR BOLZANO; 1990 TO 2001.
Probability of Childbearing
Observation Year
Age of the Mother at Childbearing
FIGURE 6. GOODNESS OF FIT MEASURES FOR WEST GERMANY AND BOLZANO; 1988 TO 1999.
-5%
0%
5%
10%
15%
20%
1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 Mean Absolute/Algebraic Percentage Error over Age Specific Fertility Rate
MAPE West Germany MALPE West Germany MAPE Bolzano MALPE Bolzano
1990 1993
1996 14
19 24 29 34 39
0
FIGURE 7a. SMOOTHED FERTILITY MOUNTAIN OVER WOMEN'S AGE AND TIME FOR WEST GERMANY; 1990 TO 1997.
Probability of Childbearing
Observation Year
Age of the Mother at Childbearing
1991 1994
199
7 14 19 24 29 34 39
0
FIGURE 7b. SMOOTHED FERTILITY MOUNTAIN OVER WOMEN'S AGE AND TIME FOR BOLZANO; 1991 TO 1999.
Probability of Childbearing
Observation Year
Age of the Mother at Childbearing