Fertility in Japan: curve-fitting the total fertility rate

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This is a subsidiary page to the fertility in Japan page that discusses different models to explain the total fertility rate data in Japan (1947-2009). We draw on Human Fertility Database data (data downloaded on December 2, 2013, last updated August 8, 2012). The data collected by the Human Fertility Database is based on government data collection by the government of Japan. Details about the modalities of data collection are available here.

Raw time series data

Autocorrelation

Autocorrelation for entire time series

Time series data for fertility is highly autocorrelated: the fertility in a particular year predicts the fertility in the next year more than the fertility in one year predicts the fertility in any other randomly chosen year. The autocorrelation values are given below. For the autocorrelation with lag i, we report the correlation between TFR values from 1947 to 2009 - i and the TFR values from 1947 + i to 2009.

Lag i Autocorrelation between TFR values from 1947 to 2009 - i and the TFR values from 1947 + i to 2009 Square of autocorrelation (equals R^2 for autoregression)
1 0.9789860629 0.9584137114
2 0.9655372551 0.9322621909
3 0.9475415769 0.89783504
4 0.9129172688 0.8334179397
5 0.8653298145 0.7487956879
6 0.810174465 0.6563826637
10 0.6268937442 0.3929957665


Autocorrelation for parts of the time series

1973-2009 (modest decline)

Lag i Autocorrelation between TFR values from 1973 to 2009 - i and the TFR values from 1973 + i to 2009 Square of autocorrelation (equals R^2 for autoregression)
1 0.9869019228 0.9739754053
2 0.9661531286 0.9334518679
3 0.9396322058 0.8829086822
4 0.9158405195 0.8387638571
5 0.8966270252 0.8039400224

Linear model

Single straight line fit

Denote by t the year. Then, the best linear fit predicting the total fertility rate in year t is:

-0.032306t + 65.87150

Quick summary information about the goodness of fit is below (paste from R data, needs to be converted to table):

Residuals:
     Min       1Q   Median       3Q      Max 
-0.77971 -0.19170 -0.02246  0.08087  1.59348 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 65.871500   6.016802   10.95 4.98e-16 ***
yearrange   -0.032306   0.003042  -10.62 1.69e-15 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.439 on 61 degrees of freedom
Multiple R-squared:  0.649,     Adjusted R-squared:  0.6433 
F-statistic: 112.8 on 1 and 61 DF,  p-value: 1.693e-15

The total fertility rate values, along with the residuals from this best linear fit, are below:

Year TFR TFR residual from best linear fit
1947 4.566 1.5934791667
1948 4.444 1.5037847542
1949 4.357 1.4490903418
1950 3.672 0.7963959293
1951 3.281 0.4377015169
1952 2.995 0.1840071045
1953 2.713 -0.065687308
1954 2.502 -0.2443817204
1955 2.383 -0.3310761329
1956 2.236 -0.4457705453
1957 2.058 -0.5914649578
1958 2.129 -0.4881593702
1959 2.061 -0.5238537826
1960 2.008 -0.5445481951
1961 1.963 -0.5572426075
1962 1.974 -0.51393702
1963 2.001 -0.4546314324
1964 2.045 -0.3783258449
1965 2.145 -0.2460202573
1966 1.579 -0.7797146697
1967 2.224 -0.1024090822
1968 2.127 -0.1671034946
1969 2.119 -0.1427979071
1970 2.129 -0.1004923195
1971 2.153 -0.044186732
1972 2.138 -0.0268811444
1973 2.135 0.0024244432
1974 2.04 -0.0602699693
1975 1.9 -0.1679643817
1976 1.84 -0.1956587942
1977 1.786 -0.2173532066
1978 1.774 -0.197047619
1979 1.751 -0.1877420315
1980 1.731 -0.1754364439
1981 1.726 -0.1481308564
1982 1.755 -0.0868252688
1983 1.788 -0.0215196813
1984 1.8 0.0227859063
1985 1.757 0.0120914939
1986 1.719 0.0063970814
1987 1.689 0.008702669
1988 1.657 0.0090082565
1989 1.576 -0.0396861559
1990 1.543 -0.0403805684
1991 1.537 -0.0140749808
1992 1.506 -0.0127693932
1993 1.464 -0.0224638057
1994 1.507 0.0528417819
1995 1.426 0.0041473694
1996 1.43 0.040452957
1997 1.392 0.0347585445
1998 1.388 0.0630641321
1999 1.347 0.0543697197
2000 1.359 0.0986753072
2001 1.336 0.1079808948
2002 1.318 0.1222864823
2003 1.29 0.1265920699
2004 1.288 0.1568976575
2005 1.254 0.155203245
2006 1.313 0.2465088326
2007 1.331 0.2968144201
2008 1.36 0.3581200077
2009 1.361 0.3914255952
Maximum 4.566 1.5934791667
Minimum 1.254 -0.7797146697
Mean 1.971047619 -6.02692499082228E-016
Median 1.786 -0.0224638057
Standard deviation 0.7350500406 0.4354673701

Piecewise linear fit

Eyeballing the data sugests three different subperiods: 1947-1961 (rapid decline), 1961-1973 (slight increase, approximate stability at around replacement), 1973-2009 (decline from replacement to ultra-low, with minor fluctuations).

Best linear fit 1973-2009

The best fit for the data 1973 onward is:

-0.020298t + 41.985133

Below is the data on goodness of fit:

Residuals:
     Min       1Q   Median       3Q      Max 
-0.07044 -0.04925 -0.03329  0.04520  0.19737 

Coefficients:
                    Estimate Std. Error t value Pr(>|t|)    
(Intercept)        41.985133   2.175783   19.30   <2e-16 ***
yearrangesince1973 -0.020298   0.001093  -18.57   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.07097 on 35 degrees of freedom
Multiple R-squared:  0.9079,    Adjusted R-squared:  0.9053 
F-statistic:   345 on 1 and 35 DF,  p-value: < 2.2e-16
Year TFR TFR residual from best linear fit
1973 2.135 0.1973698435
1974 2.04 0.122667615
1975 1.9 0.0029653864
1976 1.84 -0.0367368421
1977 1.786 -0.0704390706
1978 1.774 -0.0621412992
1979 1.751 -0.0648435277
1980 1.731 -0.0645457563
1981 1.726 -0.0492479848
1982 1.755 4.97866287334503E-005
1983 1.788 0.0533475581
1984 1.8 0.0856453295
1985 1.757 0.062943101
1986 1.719 0.0452408725
1987 1.689 0.0355386439
1988 1.657 0.0238364154
1989 1.576 -0.0368658132
1990 1.543 -0.0495680417
1991 1.537 -0.0352702703
1992 1.506 -0.0459724988
1993 1.464 -0.0676747274
1994 1.507 -0.0043769559
1995 1.426 -0.0650791844
1996 1.43 -0.040781413
1997 1.392 -0.0584836415
1998 1.388 -0.0421858701
1999 1.347 -0.0628880986
2000 1.359 -0.0305903272
2001 1.336 -0.0332925557
2002 1.318 -0.0309947843
2003 1.29 -0.0386970128
2004 1.288 -0.0203992413
2005 1.254 -0.0341014699
2006 1.313 0.0451963016
2007 1.331 0.083494073
2008 1.36 0.1327918445
2009 1.361 0.1540896159
Maximum 2.135 0.1973698435
Minimum 1.254 -0.0704390706
Mean 1.5722702703 -3.27065701849033E-015
Median 1.537 -0.0332925557
Standard deviation 0.2305858058 0.0699800163

Linear model for logarithm of total fertility rate

Single straight line fit

The best fit for the logarithm of total fertility rate is:

-0.0149357t + 30.1696837

The data on goodness of fit is below:

Residuals:
     Min       1Q   Median       3Q      Max 
-0.34927 -0.04998 -0.00217  0.03307  0.42880 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) 30.1696837  1.8728392   16.11   <2e-16 ***
yearrange   -0.0149357  0.0009468  -15.78   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1367 on 61 degrees of freedom
Multiple R-squared:  0.8031,    Adjusted R-squared:  0.7999 
F-statistic: 248.9 on 1 and 61 DF,  p-value: < 2.2e-16
Year TFR logTFR Multiplicative residual wrt best logarithmic fit (obtained by dividing actual value by estimated value) Additive residual of log (equals log of preceding column)
1947 4.566 1.5186375481 1.5354171181 0.4288020823
1948 4.444 1.4915548718 1.5168792915 0.4166551267
1949 4.357 1.4717837471 1.509562272 0.4118197227
1950 3.672 1.3007364728 1.2913758095 0.255708169
1951 3.281 1.188148254 1.1712313965 0.1580556709
1952 2.995 1.0969442316 1.0852250331 0.0817873692
1953 2.713 0.9980550337 0.9978362363 -0.002166108
1954 2.502 0.917090412 0.9340783022 -0.0681950089
1955 2.383 0.8683601981 0.9030390316 -0.1019895021
1956 2.236 0.8046885553 0.8600838244 -0.1507254243
1957 2.058 0.7217346374 0.803527699 -0.2187436215
1958 2.129 0.7556523859 0.8437574963 -0.1698901523
1959 2.061 0.7231913019 0.8290991612 -0.1874155156
1960 2.008 0.6971392018 0.8199336186 -0.198531895
1961 1.963 0.6744739153 0.8136203186 -0.2062614608
1962 1.974 0.680061941 0.8304913923 -0.1857377144
1963 2.001 0.6936470556 0.8545187114 -0.1572168791
1964 2.045 0.7153977895 0.8864501171 -0.1205304245
1965 2.145 0.7631395524 0.9437887299 -0.057852941
1966 1.579 0.4567917353 0.7052061995 -0.3492650374
1967 2.224 0.7993073764 1.008219924 0.0081863244
1968 2.127 0.7547125362 0.9787561041 -0.0214727951
1969 2.119 0.7509442793 0.9897475867 -0.0103053313
1970 2.129 0.7556523859 1.0093822358 0.009338496
1971 2.153 0.7668622184 1.036121121 0.0354840492
1972 2.138 0.7598708126 1.0443851761 0.0434283641
1973 2.135 0.7584666467 1.0586133791 0.0569599189
1974 2.04 0.7129498079 1.0267298008 0.0263788007
1975 1.9 0.6418538862 0.9706576959 -0.0297814003
1976 1.84 0.6097655716 0.954150375 -0.0469339941
1977 1.786 0.5799784825 0.9400846425 -0.0617853626
1978 1.774 0.5732368839 0.9478194576 -0.0535912405
1979 1.751 0.5601870533 0.9496086275 -0.0517053504
1980 1.731 0.5486992762 0.9528884755 -0.0482574068
1981 1.726 0.5458065927 0.9644335261 -0.0362143696
1982 1.755 0.5624688569 0.9953942544 -0.0046163847
1983 1.788 0.5810976768 1.0293712262 0.0289481558
1984 1.8 0.5877866649 1.051873505 0.0505728647
1985 1.757 0.5636078092 1.0421956917 0.0413297297
1986 1.719 0.5417427264 1.0349989038 0.0344003676
1987 1.689 0.5241366378 1.0322387629 0.0317299997
1988 1.657 0.5050087384 1.0279204913 0.027537821
1989 1.576 0.4548899914 0.9923839449 -0.0076452053
1990 1.543 0.4337285734 0.986224855 -0.0138709027
1991 1.537 0.4298324646 0.9971727135 -0.0028312908
1992 1.506 0.4094571294 0.9917632045 -0.0082709053
1993 1.464 0.3811724155 0.9786121363 -0.0216198985
1994 1.507 0.4101209196 1.0225140261 0.0222643263
1995 1.426 0.354873322 0.9821143311 -0.0180475506
1996 1.43 0.3576744443 0.9996893406 -0.0003107077
1997 1.392 0.3307415619 0.9877675627 -0.0123078693
1998 1.388 0.3278638621 0.9997501828 -0.0002498485
1999 1.347 0.2978798974 0.984818329 -0.0152980924
2000 1.359 0.3067491352 1.0085431522 0.008506866
2001 1.336 0.2896800751 1.0063938808 0.0063735266
2002 1.318 0.2761154361 1.0077746754 0.0077446083
2003 1.29 0.2546422184 1.0012078401 0.0012071113
2004 1.288 0.2530906277 1.0146982131 0.0145912413
2005 1.254 0.2263384422 1.0027786298 0.0027747765
2006 1.313 0.2723145953 1.0657583914 0.0636866503
2007 1.331 0.2859305394 1.0966261335 0.0922383151
2008 1.36 0.3074846997 1.1373809376 0.1287281961
2009 1.361 0.3082197237 1.1553449308 0.1443989407
Maximum 4.566 1.5186375481 1.5354171181 0.4288020823
Minimum 1.254 0.2263384422 0.7052061995 -0.3492650374
Mean 1.971047619 0.6268281244 1.0095242878 -1.54902545816748E-015
Median 1.786 0.5799784825 0.9978362363 -0.002166108
Standard deviation 0.7350500406 0.3054938373 0.1484170076 0.1355471569

Piecewise linear fit

Best linear fit 1947-1957

Best linear fit 1957-1973

Best linear fit 1973-2009

The best linear fit for the logarithm of total fertility rate is:

-0.0127858t + 25.8988950

The data on goodness of fit is below:

Residuals:
     Min       1Q   Median       3Q      Max 
-0.04215 -0.03424 -0.02049  0.03068  0.09605 

Coefficients:
                     Estimate Std. Error t value Pr(>|t|)    
(Intercept)        25.8988950  1.2431026   20.83   <2e-16 ***
yearrangesince1973 -0.0127858  0.0006244  -20.48   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.04055 on 35 degrees of freedom
Multiple R-squared:  0.923,     Adjusted R-squared:  0.9208 
F-statistic: 419.4 on 1 and 35 DF,  p-value: < 2.2e-16
Year TFR logTFR Multiplicative residual wrt best logarithmic fit (obtained by dividing actual value by estimated value) Additive residual of log (equals log of preceding column)
1973 2.135 0.7584666467 1.0898155098 0.0860084249
1974 2.04 0.7129498079 1.0547221983 0.0532774131
1975 1.9 0.6418538862 0.9949799612 -0.0050326815
1976 1.84 0.6097655716 0.9759585439 -0.024335169
1977 1.786 0.5799784825 0.9595062679 -0.0413364311
1978 1.774 0.5732368839 0.965323305 -0.0352922026
1979 1.751 0.5601870533 0.9650684899 -0.0355562062
1980 1.731 0.5486992762 0.9663220104 -0.0342581562
1981 1.726 0.5458065927 0.9759294181 -0.0243650127
1982 1.755 0.5624688569 1.0050960194 0.0050830786
1983 1.788 0.5810976768 1.037171945 0.0364977255
1984 1.8 0.5877866649 1.0575686433 0.0559725407
1985 1.757 0.5636078092 1.0455881103 0.0445795121
1986 1.719 0.5417427264 1.0361379138 0.0355002564
1987 1.689 0.5241366378 1.031155476 0.0306799948
1988 1.657 0.5050087384 1.0246365071 0.0243379225
1989 1.576 0.4548899914 0.987089073 -0.0129949974
1990 1.543 0.4337285734 0.9788561446 -0.0213705884
1991 1.537 0.4298324646 0.9875966928 -0.0124808702
1992 1.506 0.4094571294 0.980129691 -0.0200703783
1993 1.464 0.3811724155 0.9650558872 -0.0355692651
1994 1.507 0.4101209196 1.0061841092 0.0061650661
1995 1.426 0.354873322 0.9643541228 -0.0362967045
1996 1.43 0.3576744443 0.979503219 -0.0207097552
1997 1.392 0.3307415619 0.9657436907 -0.0348568105
1998 1.388 0.3278638621 0.975359963 -0.0249486833
1999 1.347 0.2978798974 0.9587290088 -0.0421468208
2000 1.359 0.3067491352 0.9797167732 -0.020491756
2001 1.336 0.2896800751 0.9755293921 -0.024774989
2002 1.318 0.2761154361 0.9747699339 -0.025553801
2003 1.29 0.2546422184 0.9663384038 -0.0342411917
2004 1.288 0.2530906277 0.9772556866 -0.0230069553
2005 1.254 0.2263384422 0.9637018526 -0.0369733137
2006 1.313 0.2723145953 1.0220277729 0.0217886665
2007 1.331 0.2859305394 1.0493704759 0.0481904376
2008 1.36 0.3074846997 1.0860317155 0.082530425
2009 1.361 0.3082197237 1.1008155079 0.096051276
Maximum 2.135 0.7584666467 1.1008155079 0.096051276
Minimum 1.254 0.2263384422 0.9587290088 -0.0421468208
Mean 1.5722702703 0.4423133347 1.0007875523 7.78656418622238E-016
Median 1.537 0.4298324646 0.9797167732 -0.020491756
Standard deviation 0.2305858058 0.1440578588 0.0407568627 0.0399820929