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).`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 \(pcGDP\) is a country’s per capita GDP (in thousands of dollars), \(Inflation\) is the country’s current inflation rate, \(Europe\) is a binary variable indicating whether the country is located in Europe, and where \(Democracy\) is a binary variable indicating whether a country has democratic institutions.

\[\mathbb{E}[pcGDP|Inflation, Europe, Democracy] = \beta_0 + \beta_1 Inflation + \beta_2 Inflation \cdot Europe + \beta_3 Inflation^2 + \beta_4 Democracy\]

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 expected value of per capita GDP for a European country with democratic institutions whose inflation rate is equal to 5?

What is the expected value of per capita GDP for a non-European country with democratic institutions whose inflation rate is equal to 4?