Does income growth have a long run effect on the total fertility rate? To answer this question, we look at the GDP per capita and total fertility rate in Kenya between 1990-2015. GDP per capita will be used as a proxy measurement of income. We will use a time series modelling method on the data.

The Autoregressive Distributed Lag Model (ARDL) has been used and cointegration performed to determine whether a long run relationship exist. If confirmed, the long run multipliers will be interpreted.

As ARDL bound cointegration will be used, unit roots on individual variables will not tested. The bound test identify whether cointegration exist at I(0) and I(1) level of integration.

GDP Per Capita and Total Fertility Rate

GDP per capita is a country’s economic output divided by its population. It’s a good indicator of a country’s standard of living. \[\begin{equation} GDP\,Per\,Capita=\frac{Real\,GDP}{Population} \end{equation}\]

Total Fertility Rate is the number of children who would be born per woman if she were to pass through the childbearing years bearing children according to a current schedule of age-specific fertility rates. Specifically, it is the sum of Age Specific Fertility Rates multiplied by class interval for each age group. \[\begin{equation} Total\,Fertility\,Rate= n\times\sum_{\forall i}ASFR_{i} \end{equation}\] Where:
\(ASFR_{i}=\) age-specific fertility rate for women in age group a (expressed as a rate per woman).
\(n\): Class interval for each age group.

With the two concepts defined, we thus we come up with a time series model.

ARDL Model

\[\begin{equation} logTFR_t=\beta_0+\sum_{i=1}^{p}\lambda_{i1}logTFR_{t-1} + \sum_{i=1}^{q}\phi_{1i}logGDP_{t-1}+\epsilon_{t} \end{equation}\] Where:
\(GDP\): Per Capita GDP
\(TFR\): Total Fertility Rate
\(\lambda\) and \(\phi\) are shortrun regression coefficients
\(\epsilon_{t}\) are error terms

If cointegration exist, then we interpret long run coefficients of the Error Correction Model (ECM). The ECM model will be defined as:

\[\begin{equation} \Delta logTFR_t= \beta_0 +\beta_1 logTFR_{t-1}+\beta_2 logGDP_{t-1}+\sum_{i=1}^{p}\lambda_{i1}\Delta logTFR_{t-1} + \sum_{i=1}^{q}\phi_{1i} \Delta logGDP_{t-1}+\epsilon_{t} \end{equation}\]

In this case:
\(\beta_1\) and \(\beta_2\) will be interpreted as the long run parameters.

We now load packages and the data.

library(readxl)
library(tseries)
library(dplyr)
library(readxl)
library(dynlm)
library(ARDL)
library(tidyverse)
library(knitr)
library(dLagM)
library(haven)
library(kableExtra)
#####Open data
data<- read_sav("D:/PARS Folder/fertility.sav")
attach(data)

data<-mutate(data,loggdp=log(gdp),
                          logfert=log(fertility))

The data as viewed for the first 10 cases

attach(data)
## The following objects are masked from data (pos = 3):
## 
##     fertility, gdp, year
head(data,10) %>%
  kable("html") %>%
  kable_styling(font_size=12) %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"))
year gdp fertility loggdp logfert
1990 1523.738 6.066 7.328922 1.802699
1991 1545.982 5.901 7.343414 1.775122
1992 1518.753 5.755 7.325645 1.750069
1993 1511.865 5.633 7.321100 1.728642
1994 1536.920 5.535 7.337535 1.711092
1995 1590.038 5.459 7.371513 1.697266
1996 1637.862 5.400 7.401147 1.686399
1997 1626.998 5.348 7.394492 1.676723
1998 1652.545 5.296 7.410072 1.666952
1999 1668.271 5.239 7.419543 1.656131

We prepare the time as a timeseries.

data<-ts(data,start=1990,end=2017,frequency=1)
logfert=ts(data[,"logfert"],start=1990,end=2017,frequency=1)
loggdp=ts(data[,"loggdp"],start=1990,end=2017,frequency=1)

View the evolving curves for the two series. The first graph show that GDP per capita is raising and the second graph show a general fall in fertility rate.

plot(loggdp,ylab="Log GDP",main="GDP Per Capita")

plot(logfert,ylab="Log Fertility",main="Total Fertility Rate",col=2)

Model is run, but our interest in this case is to determine the optimal number of lags for the dependent and independent variable. In this case ARDL(3,0) is selected. Generally, optimal number of lags are determined through Akaike Information Criterion or Bayesian Information Criterion

model<-auto_ardl(logfert~loggdp,data=data,max_order = 5) 
model2<-model$best_model
model2$order
## [1] 3 0

We then perform ARDL bound test. The test is an F-test with the null hypothesis assuming no cointegration. In our case: \[\begin{equation} H_o: \beta_1=\beta_2=0 \\ H_1: \beta_1\ne \beta_2 \ne 0 \end{equation}\]

From the output, the assumption of no cointegration is rejected as \(p\approx 0.001<0.05\). Thus long-term relationship exist between income growth and fertility rate.

bounds_f_test(model2,case=3)
## 
##  Bounds F-test (Wald) for no cointegration
## 
## data:  d(logfert) ~ L(logfert, 1) + loggdp + d(L(logfert, 1)) + d(L(logfert,     2))
## F = 10.912, p-value = 0.0009601
## alternative hypothesis: Possible cointegration
## null values:
##    k    T 
##    1 1000

The output shows that a 1% growth in income reduces total fertility rate by 0.65%. The relationship between the two is significant.

multipliers(model2)
##          term   estimate std.error t.statistic      p.value
## 1 (Intercept)  6.4519739 0.2832053    22.78196 8.865831e-16
## 2      loggdp -0.6591801 0.0409852   -16.08337 6.606612e-13