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.
Raw time series data
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 1947-1961
The best fit for the data 1947-1961 is:
Below is the data on goodness of fit:
Residuals:
Min 1Q Median 3Q Max
-0.38920 -0.27570 -0.01929 0.29579 0.47195
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 393.73307 38.47780 10.23 1.38e-07 ***
years1947to1961 -0.20002 0.01969 -10.16 1.50e-07 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.3295 on 13 degrees of freedom
Multiple R-squared: 0.8881, Adjusted R-squared: 0.8795
F-statistic: 103.2 on 1 and 13 DF, p-value: 1.5e-07
| Year | TFR | TFR residual from best linear fit |
|---|---|---|
| 1947 | 4.566 | 0.27465 |
| 1948 | 4.444 | 0.3526714286 |
| 1949 | 4.357 | 0.4656928571 |
| 1950 | 3.672 | -0.0192857143 |
| 1951 | 3.281 | -0.2102642857 |
| 1952 | 2.995 | -0.2962428571 |
| 1953 | 2.713 | -0.3782214286 |
| 1954 | 2.502 | -0.3892 |
| 1955 | 2.383 | -0.3081785714 |
| 1956 | 2.236 | -0.2551571429 |
| 1957 | 2.058 | -0.2331357143 |
| 1958 | 2.129 | 0.0378857143 |
| 1959 | 2.061 | 0.1699071429 |
| 1960 | 2.008 | 0.3169285714 |
| 1961 | 1.963 | 0.47195 |
Best linear fit: 1961-1973
The best fit for the data 1961-1973 is:
Unlike the other periods, the linear fit here is really bad, largely because of the presence of the anomalous year 1966, but also because minor year-to-year variations dominate the secular trend.
Below is the data on goodness of fit:
Residuals:
Min 1Q Median 3Q Max
-0.45968 0.00582 0.01981 0.04157 0.16769
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -32.61477 22.51756 -1.448 0.175
years1961to1973 0.01763 0.01145 1.540 0.152
Residual standard error: 0.1544 on 11 degrees of freedom
Multiple R-squared: 0.1773, Adjusted R-squared: 0.1025
F-statistic: 2.371 on 1 and 11 DF, p-value: 0.1519
| Year | TFR | TFR residual from best linear fit |
|---|---|---|
| 1961 | 1.963 | 0.0124505495 |
| 1962 | 1.974 | 0.0058241758 |
| 1963 | 2.001 | 0.0151978022 |
| 1964 | 2.045 | 0.0415714286 |
| 1965 | 2.145 | 0.1239450549 |
| 1966 | 1.579 | -0.4596813187 |
| 1967 | 2.224 | 0.1676923077 |
| 1968 | 2.127 | 0.0530659341 |
| 1969 | 2.119 | 0.0274395604 |
| 1970 | 2.129 | 0.0198131868 |
| 1971 | 2.153 | 0.0261868132 |
| 1972 | 2.138 | -0.0064395604 |
| 1973 | 2.135 | -0.0270659341 |
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 1947-1961
Best linear fit 1961-1973
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 |