Question

The regression estimates below are from a research project on the degree of assortative mating among couples in the U.S. Assortative mating is the tendency for couples to be more similar to one another in some ways than would occur at random. In the social sciences, a common way to gauge assortative mating is to use data on educational attainment. The regression estimates here are from a sample of 1.8 million opposite-sex married couples observed in the American Community Survey (ACS) between 2014 and 2018. Wife's years of education is educw and husband's years of education is educh. The standard deviations of these variables are 2.95 for wife's education and 3.01 for husband's education. Note: The questions below build from straightforward to more challenging. Look for opportunities to use results from earlier questions in this assignment. educw =6.06+0.580 educh, R 2 =0.351 4.1. At what level of husband's education would we expect the wife to have 12 years of schooling? 4.2. If a husband has a traditional four-year college degree instead of not attending college at all, how much higher is his wife's expected years of education? 4.3. What is the residual for a couple in which the wife has 12 years of education and the husband has 16 years? 4.4. What is the error term in the regression model, u, for a couple in which the wife has 12 years of education and the husband has 14 years? (a) It must be the same as the residual. (b) It is the magnitude of the residual. (c) It must be zero, because it is uncorrelated with the test scores. (d) It is generally not possible to know the value of the error term. 4.5. The mean of husband's education is 13.82 years. What is the mean of wife's education? 4.6. If you switched the variables in the regression above and ran a simple regression of husband's education (as the dependent variable) on wife's education (as the explanatory variable), what would be the estimated slope coefficient? Hint: Can you solve for Cov (x,y) ? 4.7. If you switched the variables in the regression above and ran a simple regression of husband's education (as the dependent variable) on wife's education (as the explanatory variable), what would be the value of R 2 ?

Answer 1

4.1. Solve for
`\[ 12 = 6.06 + 0.580 * \text{educh} \]`

4.2. Since the regression coefficient for
`\(\text{educh}\) is \(0.580\)`, the difference in expected years of education when the husband has a college degree instead of not attending college is 0.580 years.

4.3. The residual e is given by:

`\[ e = \text{observd value} - \text{predicted value} \]`

4.4. The error term u is the difference between the observed value and the predicted value. It is generally not possible to know the exact value of the error term. So, the answer is (d) It is generally not possible to know the value of the error term.

4.5. The mean of wife's education is given by the intercept of the regression equation. In this case, it is 6.06 years.

4.6. The estimated slope coefficient would be the same, but with roles reversed. So, it would be 0.580.

4.7. The value of
`\(R^2\)` would be the same regardless of which variable is considered dependent or explanatory in a simple linear regression. Therefore, it would be 0.351.

Step-by-step explanation:

Let's go through each question one by one:

4.1. At what level of husband's education would we expect the wife to have 12 years of schooling?

Given the regression equation:

`\[ \text{educw} = 6.06 + 0.580 * \text{educh} \]`

To find when
`\(\text{educw} = 12\)`, substitute 12 for
`\(\text{educw}\)` and solve for
`\(\text{educh}\)`:

`\[ 12 = 6.06 + 0.580 * \text{educh} \]`

Solve for
`\(\text{educh}\)`.

4.2. If a husband has a traditional four-year college degree instead of not attending college at all, how much higher is his wife's expected years of education?

Since the regression coefficient for
`\(\text{educh}\) is \(0.580\)`, the difference in expected years of education when the husband has a college degree instead of not attending college is 0.580 years.

4.3. What is the residual for a couple in which the wife has 12 years of education and the husband has 16 years?

The residual e is given by:

`\[ e = \text{observed value} - \text{predicted value} \]`

Substitute the values into the regression equation to find the predicted value and then calculate the residual.

4.4. What is the error term in the regression model u, for a couple in which the wife has 12 years of education and the husband has 14 years?

The error term u is the difference between the observed value and the predicted value. It is generally not possible to know the exact value of the error term. So, the answer is d It is generally not possible to know the value of the error term.

4.5. The mean of husband's education is 13.82 years. What is the mean of wife's education?

The mean of wife's education is given by the intercept of the regression equation. In this case, it is 6.06 years.

4.6. If you switched the variables in the regression above and ran a simple regression of husband's education (as the dependent variable) on wife's education (as the explanatory variable), what would be the estimated slope coefficient?

The estimated slope coefficient would be the same, but with roles reversed. So, it would be 0.580.

4.7. If you switched the variables in the regression above and ran a simple regression of husband's education (as the dependent variable) on wife's education (as the explanatory variable), what would be the value of
`\(R^2\)`?

The value of
`\(R^2\)` would be the same regardless of which variable is considered dependent or explanatory in a simple linear regression. Therefore, it would be 0.351.