Assignment Task

The general fertility rate ( gfr ) is the number of children born to every 1,000 women of child- bearing age. In this study, we explore the determinants of gfr using three datasets.

The first dataset comprises a time series of gfr in the United States from 1920 to 1984. Drawing on the methodology outlined in Whittington, Alm, and Peters (1990), we apply the following regression equation for our analysis:

The explanatory variables include the average real dollar value of the personal tax exemption ( pe ); a technology shock variable ( pill ), which is assigned a value of 1 starting from 1963 to reflect the availability of the birth control pill for contraception; and a macroeconomic environment shock variable ( ww2 ), set to 1 for the years 1941 to 1945, corresponding to the period of the United States` involvement in World War II.

1. Using OLS to estimate equation (1). Report the results in equation or tabular form. Interpret the estimated coefficients of β₁ , β₂ ,β ₃ .

2. An econometrician argues that, for both biological and behavioral reasons, decisions to have children would not immediately result from changes in the personal exemption ( pe ). Therefore he proposes to use a finite distributed lag (FDL) model of order 2, as the following Equation (2):

Using OLS to estimate equation (2). What is the long-run propensity (LRP) of the tax exemption on the fertility rate? Propose a regression equation to perform a t-test on the significance of the LRP and perform the test. What do you conclude from the result of this test (use 5% significance level)?

3. Another econometrician analyzing a time series presented in Figure 1 argues that Model (1) is misspecified due to an observable linear time trend in the variable of interest post-1957. The concern is that omitting a control for this time trend could introduce bias into the estimates. Do you agree with this statement. If you agree with the econometrician`s critique, identify the likely direction of bias in the coefficient for the “pill” variable and provide a rationale for your conclusion.

The second dataset comprises cross-sectional data for a sample of 4,361 women born between the years 1934 and 1961. This dataset is analyzed to shed light on individual women’s birth decisions. The dependent variable in this analysis is the number of children each woman has had (nchld), with the major explanatory variables being the years of education (educ) and the age of the women at the time of the interview. Additionally, to account for varying perspectives on childbirth across different age groups, we segment the sample into four cohorts ( − ) based on the women ’ s birth years, with each cohort encompassing a span of six birth years (for instance, dummy corresponds to those born in 1934-1940).

We consider a regression of the number of children with respect to females’ education level, age, and the three cohort dummies as follows:

We run OLS regression on Equation (3), and obtain the following regression results:

4. We analyze the residuals obtained from the regression and use Figure 2 to illustrate the relationship between the residuals and the education level. Based on this visual analysis, identify any potential issues with regression (3). Conduct appropriate statistical tests to verify if Model (3) exhibits this problem, using a 5% significance level. What conclusions can be drawn from the outcomes of these tests?

5. Model (3) includes dummy variables for different cohorts. Why has the associated dummy variable for the first cohort ¹ been excluded? Using the regression results of Model (3), interpret the coefficients of each cohort dummy.

6. Perform statistical tests to determine whether cohort dummies jointly statistically significantly affect the number of children at the 5% significance level. What conclusions can be drawn from this analysis?

To obtain the causal impact of birth control pill’s introduction on the fertility choice, we use a third data set, which is a retrospective longitudinal data set for women born in 1934-1947. These women were asked to recall, for each year from 1958 to 1967, whether they gave birth to a child ( newborn ). Additionally, we include a variable indicating access to birth control pills by age 16 ( buypill ). Therefore, the birth control pill was only effective for those who had access to it during the child-bearing ages. We consider the following setting:

7. Write down the population equation for those buypill = 1 and buypill = 0

separately. Interpret the coefficient of the interaction term.

8. In this retrospective data, the dependent variable (nebrn ) variable may have been measured with some errors as respondents could inaccurately recall specific years of childbirth. However, those errors from bad m