Homework 4

Due: Turn in hard copy at the beginning of class on Wednesday, November 11.

For the coding parts, turn in both your code and its output, and keep your answers concise. You should code the estimators, standard errors, and test statistics yourself rather than using built-in functions, though you are welcome to use built-in functions to check your answers. Use the asymptotic (normal or chi-square) distributions from class to compute critical values and p-values, and treat each dataset as an iid sample.

Data:

• Fertility: RData [description]
• Fertility_subset: RData (a random sample of 1,000 women from Fertility, with the same variables)
• acs: RData [description]

Question 1

For this question, use the Fertility data, which comes from the 1980 U.S. Census and contains married women aged 21-35 with at least two children. We will use the variable boy1st, which is equal to 1 if a woman’s first child is a boy and 0 otherwise. Let \(p = \mathbb{E}[\texttt{boy1st}]\), and consider testing \(\mathbb{H}_0 : p = 0.5\).

1. Propose an estimator for \(p\), which you call \(\hat{p}\). What is your estimate?

2. Calculate the standard error of \(\hat{p}\).

3. Calculate a t-statistic for \(\mathbb{H}_0\). Do you reject \(\mathbb{H}_0\) at the 5% significance level?

4. Calculate the p-value for \(\mathbb{H}_0\). Explain how to interpret it.

5. Calculate a 95% confidence interval for \(p\) and explain how to interpret it.

6. Repeat parts (a)-(e) using the Fertility_subset data.

7. Based on your results from the previous parts, (i) provide an explanation for the differences you see in part (f) relative to the earlier parts, and (ii) comment on the practical and statistical significance of your results.

Question 2

PSE 11.1 (a)-(c). In addition, do the following questions, using the incwage variable in the acs data for \(X\).

4. Compute an estimate of \(cv\).

5. Calculate the standard error of \(\widehat{cv}\).

6. Provide a 95% confidence interval for \(cv\).

7. A natural parametric model for incwage is that it follows an exponential distribution. A random variable that follows an exponential distribution will have its mean equal to its standard deviation, which implies that \(cv = 100\). For this problem, test \(\mathbb{H}_0 : cv = 100\), reporting the t-statistic and p-value. What do you conclude about whether or not wage income follows an exponential distribution on the basis of this test?

Question 3

For this question, return to the Fertility data. In addition to boy1st, we will now use boy2nd, which is equal to 1 if a woman’s second child is a boy and 0 otherwise. Let \(\theta = (\mathbb{E}[\texttt{boy1st}], \mathbb{E}[\texttt{boy2nd}])'\), and consider testing \(\mathbb{H}_0 : \theta = (0.5, 0.5)'\).

1. Let \(\hat{\theta}\) be the vector of sample means of these two variables, and let \(\hat{\mathbf{V}}\) be an estimator of the asymptotic variance of \(\sqrt{n}(\hat{\theta} - \theta)\). Report \(\hat{\theta}\) and \(\hat{\mathbf{V}}\).

2. Calculate the Wald statistic. Report the Wald statistic, its degrees of freedom, the critical value for a test at the 5% significance level, its p-value, and whether you reject \(\mathbb{H}_0\).

3. Repeat parts (a) and (b) using the Fertility_subset data. What differences do you get? Provide an explanation for these differences.