Part A

Interpret what \(\hat{\beta_0}\) means in this context.

Part B

Interpret what \(\hat{\beta_1}\) means in this context.

Part C

The \(R^2=0.023\) for this regression. What are the units of the \(R^2\), and what does this mean?

Part D

The \(SER, \, \hat{\sigma_u}=624.1\) for this regression. What are the units of the SER in this context, and what does it mean? Is the SER large in the context of this regression?

Part E

Suppose Maria is 20 years old. What is her predicted \(\widehat{AWE}\)?

Part F

Suppose the data shows her actual \(AWE\) is $430. What is her residual? Is this a relatively good or a bad prediction?¹

Part G

What does the error term, \(\hat{u_i}\) represent in this case? What might individuals have different values of \(u_i\)?

Part H

Do you think that \(Age\) is exogenous? Why or why not? Would we expect \(\hat{\beta_1}\) to be too large or too small?

Part A

What is the OLS estimate of \(\hat{\beta_1}\)?

Part B

What is the OLS estimate of \(\hat{\beta_0}\)?

Part C

Suppose the OLS estimate of \(\hat{\beta_1}\) has a standard error of \(0.072\). Could we probably reject a null hypothesis of \(H_0: \beta_1=0\) at the 95% level?

Part D

Calculate the \(R^2\) for this model. How much variation in \(Y\) is explained by our model?

Part E

How large is the average residual?

Part A

Clean up the data a bit by mutate()-ing a variable to measure home attendance in millions. This will make it easier to interpret your regression later on.

Part B

Get the correlation between Runs Scored and Home Attendance.

Part C

Plot a scatterplot of Runs Scored (y) on Home Attendance (x). Add a regression line.

Part D

We want to estimate a regression of Runs Scored on Home Attendance:

\[ \widehat{\text{runs scored}_i} = \beta_0 + \beta_1 \text{home attendance}_i \] Run this regression in R.

What are \(\hat{\beta_0}\) and \(\hat{\beta_1}\) for this model? Interpret them in the context of our question. [Hint: make sure to save your regression model as an object, and get a summary() of it. This object will be needed later.]

Part E

Write out the estimated regression equation.

Part F

Make a regression table of the output (using the huxtable package).

Part G

Check the goodness of fit statistics. What is the \(R^2\) and the SER of this model? Interpret them both in the context of our question.

Part H

Now let’s start running some diagnostics of the regression. Make a histogram of the residuals. Do they look roughly normal? [Hint: you will need to use the broom package’s augment() command on your saved regression object to add containing the residuals (.resid), and save this as a new object - to be your data source for the plot in this question and the next question.]

Part I

Make a residual plot.

Part J

Test the regression for heteroskedasticity. Are the errors homoskedastic or heteroskedastic?

[Hint: use the lmtest package’s bptest() command on your saved regression object.]

Run another regression using robust standard errors. [Hint: use the estimatr package’s lm_robust() command and save the output like the following:

reg_robust <-lm_robust(y ~ x, data = the_data, # change y, x, and data names to yours
                              se_type = "stata") # we'll use this method to calculate

Now make another regression output table with huxtable, with one column using regular standard errors (just use your original saved regression object) and another using robust standard errors (use this new saved object).

Part K

Test the data for outliers. If there are any, identify which team(s) and season(s) are outliers. [Hint: use the car package’s outlierTest() command on your saved regression object.]

Part L

Look back at your regression results. What is the marginal effect of home attendance on runs scored? Is this statistically significant? Why or why not?

Part M

Now we’ll try out the infer package to understand the \(p\)-value for our observed slope in our regression model.

First, save the (value of) our sample \(\hat{\beta_1}\) from your regression in Part D as an object, I suggest:

our_slope = 123 # replace "123" with whatever number you found for the slope in part D

Then, install and load the infer package, and then run the following simulation:

# save our simulations as an object (I called it "sims")
sims <- data %>% # "data" here is whatever you named your dataframe!
  specify(y ~ x) %>% # replacing y and x with your variable names
  hypothesize(null = "independence") %>% # H_0 is that slope is 0, x and y are independent
  generate(reps = 1000,
           type = "permute") %>% # make 1000 samples assuming H_0 is true
  calculate(stat = "slope") # estimate slope in each sample

# look at it
sims

# calculate p value
sims %>%
  get_p_value(obs_stat = our_slope,
              direction = "both") # a two-sided H_a: slope =/= 0

Compare to the \(p\)-value in your original regression output in previous parts of this question.

Part N

Make a histogram of the simulated slopes, and plot our sample slope on that histogram, shading the \(p\)-value.

[You can pipe sims into visualize(obs_stat = our_slope), or use ggplot2 to plot a histogram in the normal way, using sims as the data source and add a geom_vline(xintercept = our_slope) to show our finding on the distribution.]