Part A

Using the group means, write a regression equation for a regression of GDP Growth rate on Common Law. Define

\[\text{Common Law}_i = \begin{cases} 1 & \text{if country } i \text{ has common law} \\ 0 & \text{if country } i \text{ has civil law}\\ \end{cases}\]

Part B

How do we use the regression to find the average GDP Growth rate for common law countries? For civil law countries? For the difference?

Part C

Looking at the coefficients, does there appear to be a statistically significant difference in average GDP Growth Rates between Civil and Common law countries?

Part D

Is the estimate on the difference likely to be unbiased? Why or why not?

Part E

Now using the same table above, reconstruct the regression equation if instead of Common Law, we had used:

\[\text{Civil Law}_i = \begin{cases} 1 & \text{if country } i \text{ has civil law} \\ 0 & \text{if country } i \text{ has common law}\\ \end{cases}\]

Part A

Suppose she runs the following regression: \[\widehat{\text{Price}_h}=119.20+29.76 \, \text{Bedrooms}_h+24.09 \, \text{View}_h+14.06 \, (\text{Bedrooms}_h \times \text{View}_h)\]

What does each coefficient mean?

Part B

Write out two separate regression equations, one for houses with a nice view, and one for homes without a nice view. Explain each coefficient in each regression.

Part C

Suppose she runs the following regression:

\[\widehat{\text{Price}_h}=189.20+42.40 \, \text{Pool}_h+12.10 \, \text{View}_h+12.09 \, (\text{Pool}_h \times \text{View}_h)\]

What does each coefficient mean?

Part D

Find the expected price for:

• a house with no pool and no view
• a house with no pool and a view
• a house with a pool and without a view
• a house with a pool and with a view

Part E

Suppose she obtains the following OLS estimates: \[\widehat{\text{Price}_h}=87.90+53.94 \, \text{Bedrooms}_h+15.29 \, \text{Baths}_h+16.19 \, (\text{Bedrooms}_h \times \text{Baths}_h)\]

What is the marginal effect of adding an additional bedroom if the house has 1 bathroom? 2 bathrooms? 3 bathrooms?

Part F

What is the marginal effect of adding an additional bathroom if the house has 1 bedroom? 2 bedrooms? 3 bedrooms?

Part A

Suppose we estimate the following simple model relating deviation in temperature to year:

\[\widehat{\text{Temperature}_t}=-10.46+0.006 \text{Year}_t\]

Interpret the coefficient on Year (i.e. \(\hat{\beta_1})\)

Part B

Predict the (deviation in) temperature for the year 1900 and for the year 2000.

Part C

Suppose we believe temperature deviations are increasing at an increasing rate, and introduce a quadratic term and estimate the following regression model:

\[\widehat{\text{Temperature}_t}=155.68-0.116 \text{Year}_t+0.000044\text{Year}_t^2\]

What is the marginal effect on (deviation in) global temperature of one additional year elapsing?

Part D

Predict the marginal effect on temperature of one more year elapsing starting in 1900, and in 2000.

Part E

Our quadratic function is a \(U\)-shape. According to the model, at what year was temperature (deviation) at its minimum?

Part A

Suppose we estimate the following simple regression:

\[\widehat{\text{fatalities}_i}=123.98+0.091\text{cell plans}_i\]

Interpret the coefficient on cell plans (i.e. \(\hat{\beta_1})\)

Part B

Now suppose we estimate the regression using a linear-log model:

\[\widehat{\text{fatalities}_i}=-3557.08+515.81\text{ln(cell plans}_i)\]

Interpret the coefficient on ln(cell plans) (i.e. \(\hat{\beta_1})\)

Part C

Now suppose we estimate the regression using a log-linear model:

\[\widehat{\text{ln(fatalities}_i)}=5.43+0.0001\text{cell plans}_i\]

Interpret the coefficient on cell plans (i.e. \(\hat{\beta_1})\)

Part D

Now suppose we estimate the regression using a log-log model:

\[\widehat{\text{ln(fatalities}_i)}=-0.89+0.85\text{ln(cell plans}_i)\]

Interpret the coefficient on cell plans (i.e. \(\hat{\beta_1})\)

Part E

Suppose we include several other variables into our regression and want to determine which variable(s) have the largest effects, a State’s cell plans, population, or amount of miles driven. Suppose we decide to standardize the data to compare units, and we get:

\[\widehat{\text{fatalities}_Z}=4.35+0.002\text{cell plans}_{Z}-0.00007\text{population}_{Z}+0.019\text{miles driven}_{Z}\]

Interpret the coefficients on cell plans, population, and miles driven. Which has the largest effect on fatalities?

Part F

Suppose we wanted to make the claim that it is only miles driven, and neither population nor cell phones determine traffic fatalities. Write (i) the null hypothesis for this claim and (ii) the estimated restricted regression equation.

Part G

Suppose the \(R^2\) on the original regression from (e) was 0.9221, and the \(R^2\) from the restricted regression is 0.9062. With 50 observations, calculate the \(F\)-statistic.

Part A

Using R to examine the data, find the average infant mortality rate for cities with lead pipes and for cities without lead pipes. Calculate the difference, and run a \(t\)-test to determine if this difference is statistically significant.

Part B

Run a regression of infrate on lead, and write down the estimated regression equation. Use the regression coefficients to find:

• the average infant mortality rate for cities with lead pipes
• the average infant mortality rate for cities without lead pipes
• the difference between the averages for cities with or without lead pipes

Part C

Does the pH of the water matter? Include ph in your regression from part B. Write down the estimated regression equation, and interpret each coefficient (note there is no interaction effect here). What happens to the estimate on lead?

Part D

The amount of lead leached from lead pipes normally depends on the chemistry of the water running through the pipes: the more acidic the water (lower pH), the more lead is leached. Create an interaction term between lead and pH, and run a regression of infrate on lead, pH, and your interaction term. Write down the estimated regression equation. Is this interaction significant?

Part E

What we actually have are two different regression lines. Visualize this with a scatterplot between infrate \((Y)\) and ph \((X)\) colored by lead [Hint: make sure lead is a factor variable].

Part F

Do the two regression lines have the same intercept? The same slope? Use the original regression in part D to test these possibilities.

Part G

Take your regression equation from part D and rewrite it as two separate regression equations (one for no lead and one for lead). Interpret the coefficients for each.

Part H

Double check your calculations in Part G are correct by running the regression in D twice, once for cities without lead pipes and once for cities with lead pipes. [Hint: filter() the data first, then use the filtered data for the data = argument in each regression.]

Part I

Use huxtable to make a nice output table of all of your regressions from parts B, C, and D.

Part A

Does economic freedom affect GDP per capita? Create a scatterplot of gdp_pc (y) against econ_freedom (x). Does the effect appear to be linear or nonlinear?

Part B

Run a simple regression of gdp_pc on econ_freedom. Write out the estimated regression equation. What is the marginal effect of econ_freedom on gdp_pc?

Part C

Let’s try a quadratic model. Run a quadratic regression of gdp_pc on econ_freedom. Write out the estimated regression equation.

Part D

Add the quadratic regression to your scatterplot.

Part E

What is the marginal effect of econ_freedom on gdp_pc?

Part F

As a quadratic model, this relationship should predict anecon_freedom score where gdp_pc is at a minimum. What is that minimum Economic Freedom score, and what is the associated GDP per capita?

Part G

Run a cubic model to see if we should keep going up in polynomials. Write out the estimated regression equation. Should we add a cubic term?

Part H

Another way we can test for non-linearity is to run an \(F\)-test on all non-linear variables - i.e. the quadratic term and the cubic term \((\hat{\beta_2}\) and \(\hat{\beta_3}\)) and test against the null hypothesis that: \[H_0: \hat{\beta_2} = \hat{\beta_3} = 0\]

Run this joint hypothesis test, and what can you conclude?

Part I

Instead of a polynomial model, try out a logarithmic model. It is hard to interpret percent changes on an index, but it is easy to understand percent changes in GDP per capita, so run a log-linear regression. Write out the estimated regression equation. What is the marginal effect of econ_freedom?

Part J

Make a scatterplot of your log-linear model with a regression line.

Part K

Put all of your results together in a regression output table with huxtable from your answers in questions B, C, G, and H.