Fertility in Japan: curve-fitting the total fertility rate
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.
Contents
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 , we report the correlation between TFR values from 1947 to
and the TFR values from
to 2009.
Lag ![]() |
Autocorrelation between TFR values from 1947 to ![]() ![]() |
Square of autocorrelation (equals ![]() |
---|---|---|
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 ![]() |
Autocorrelation between TFR values from 1973 to ![]() ![]() |
Square of autocorrelation (equals ![]() |
---|---|---|
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 the year. Then, the best linear fit predicting the total fertility rate in year
is:
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:
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:
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:
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 |