Consider the following regression
\[\begin{align*} \text{Score} = 50 + 5 \times \text{Hours} + 10 \times \text{GPA} + U \end{align*}\] where \(\text{Score}\) is a student’s score on a midterm, \(\text{Hours}\) is the number of hours the student studied for the midterm, \(\text{GPA}\) is the student’s current GPA, and \(\mathbb{E}[U \mid \text{Score}, \text{Hours}, \text{GPA}] = 0\). Carefully explain how to interpret the coefficient on \(\text{Hours}\).
Consider the following regression where airq is an indicator of air quality (lower is better) for a particular metropolitan area in California, dens1000 is the number of 1000s of people per square mile, coas indicates whether or not the metro area is on the coast, and medi1000 is the median income in the metro area (in thousands of dollars). The numbers in parentheses are the standard errors of the coefficients.
data("Airq", package = "Ecdat")
library(modelsummary)
Airq$coas <- 1 * (Airq$coas == "yes")
Airq$dens1000 <- Airq$dens / 1000
Airq$medi1000 <- Airq$medi / 1000
reg1 <- lm(airq ~ dens1000 + coas + dens1000 * coas + medi1000, data = Airq)
modelsummary(reg1, fmt = 1, gof_omit = ".")| (1) | |
|---|---|
| (Intercept) | 120.6 |
| (9.5) | |
| dens1000 | -0.3 |
| (2.8) | |
| coas | -31.2 |
| (11.3) | |
| medi1000 | 0.8 |
| (0.4) | |
| dens1000 × coas | -1.2 |
| (3.4) |
Which regressors are statistically significant in this regression? ??
What is the predicted value for the air quality index for a metro area with 1000 people per square mile, that is not located on the coast, and with median income equal to $50,000?
Let \(Y\) denote a person’s age in the United States. Suppose that you have the theory that \(\mathbb{E}[Y] = 35\). You are able to collect a random sample of 100 observations. Using this data, you calculate \(\bar{Y} = 37\) and that \(\widehat{\mathrm{var}}(Y) = 6\).
Calculate a t-statistic for testing the null hypothesis that \(\mathbb{E}[Y]=35\). Do you reject the null hypothesis here? Explain.
What is the standard error of \(\bar{Y}\).
Calculate a p-value for the null hypothesis that \(\mathbb{E}[Y]=35\). How do you interpret it?
Calculate a 95% confidence interval for \(\mathbb{E}[Y]\). How do you interpret it?
Consider the following conditional expectation using country-level data, where \(\text{pcGDP}\) is a country’s per capita GDP (in thousands of dollars), \(\text{Inflation}\) is the country’s current inflation rate, \(\text{Europe}\) is a binary variable indicating whether the country is located in Europe, and where \(\text{Democracy}\) is a binary variable indicating whether a country has democratic institutions.
\[\mathbb{E}[\text{pcGDP} \mid \text{Inflation}, \text{Europe}, \text{Democracy}] = \beta_0 + \beta_1 \text{Inflation} + \beta_2 \text{Inflation} \cdot \text{Europe} + \beta_3 \text{Inflation}^2 + \beta_4 \text{Democracy}\]
Further suppose that \(\beta_0 = 45, \beta_1=-1, \beta_2=-2, \beta_3=-0.1, \beta_4=8\)
What is the expected value of per capita GDP for a European country with democratic institutions whose inflation rate is equal to 4?
What is the marginal contrast for inflation in this model?
What is the average marginal contrast for inflation in this model?
Given the relevant data, how would you estimate the average marginal contrast for inflation?