Part A

Suppose that a homeowner converts part of an existing living space in her house to a new bathroom. What is the expected increase in the value of the house?

Part B

Suppose a homeowner adds a new bathroom to her house, which also increases the size of the house by 100 square feet. What is the expected increase in the value of the house?

Part C

Suppose the \(R^2\) of this regression is 0.727. Calculate the adjusted \(\bar{R}^2\).

Part D

Suppose the following auxiliary regression for \(BDR\) has an \(R^2\) of 0.841.

\[\widehat{BDR}=\delta_0+\delta_1Bath+\delta_2Hsize+\delta_3Lsize+\delta_4Age\]

Calculate the Variance Inflation Factor for \(BDR\) and explain what it means.

Part A

If the true marginal effect of experience on wages (holding education constant) is 0.0701, calculate the omitted variable bias in the first regression caused by omitting experience. Does the estimate of \(\hat{\beta_1}\) in the first regression overstate or understate the effect of education on wages?

Part B

Knowing this, what would be the true effect of education on wages, holding experience constant?

Part C

The \(R^2\) for the second regression is 0.0897. If she were to run a better regression including both education and experience, how much would the variance of the coefficients on education and experience increase? Why?

Part A

Make a correlation table for our variables listed above.

Hint: select() these variables and then pipe this into cor(., use = "pairwise.complete.obs") to use only observations for which there are data on each variable (to avoid NA’s).

Part B

Consider the following causal model:

library(ggdag)
dagify(bwght ~ cigs + inc,
       cigs ~ price + educ + inc,
       inc ~ educ,
       exposure = "cigs",
       outcome = "bwght") %>%
  tidy_dagitty(seed = 256) %>%
  ggdag_status()+
  theme_dag_blank()+
  theme(legend.position = "none")

Note what we are hypothesizing:

1. bwght is caused by cigs and inc
2. cigs are caused by price, educ, and inc
3. inc is caused by educ

See also how this is written into the notation in R to draw (plot) the DAG.

Create this model on dagitty.net. What does dagitty tell us the testable implications of this causal model?

Part C

Test each implication given to you by dagitty.

• For independencies, e.g. \((x \perp y)\): run a regression of \(y\) on \(x\).
• For conditional independencies, e.g. \((x \perp y | z, a)\): run a regression of \(y\) on \(x, z, a\).

For each, test against the null hypothesis that the relevant coefficient \((\beta_1) =0\) (i.e. \(x\) and \(y\) are indeed independent).

Does this causal model hold up well?

Part D

List all of the possible pathways from cigs to bwght. Which are “front-doors” and which are “back-doors?” Are any blocked by colliders?

Part E

What is the minimal sufficient set of variables we need to control for in order to causally identify the effect of cigs on bwght?

Part F

Estimate the causal effect by running the appropriate regression.

FYI, on dagitty.net, you can change a variable on the diagram to be “adjusted” (controlled for) by clicking it and then hitting the A key.

Part G

We saw some effect between faminc and cigprice. Perhaps there are unobserved factors (such as the economy’s performance) that affect both. Add an unobserved factor u1 to your dagitty model.

FYI, on dagitty.net, you can make a variable be “unobserved” by double-clicking it and then hitting the U key.

Part H

Perhaps our model is poorly specified. Maybe motheduc actually has a causal effect on bwght? Tweak your model from Question 9 on dagitty to add this potential relationship. What testable implications does this new model imply?

Part I

Test these implications appropriately, like you did in Part C. Does this model hold up well?

Part J

Under this new causal model, list all of the possible pathways from cigs to bwght. Which are “front-doors” and which are “back-doors?” Are any blocked by colliders?

Part K

Under this new causal model, what is the minimal sufficient set of variables we need to control in order to causally identify the effect of cigs on bwght?

Part L

Estimate the causal effect in this new model by running the appropriate regression. Compare your answers to those in Part F.

Part M

Try out drawing this model using the ggdag package in R. See my DAG in question 3 for an example.

Part A

Create a quick scatterplot between height85 (as \(X)\) amd wage96 (as \(Y)\).

Part B

Regress wages on adult height. Write the equation of the estimated OLS regression. Interpret the coefficient on height85.

Part C

How much would someone who is 5’10" (70 in) be predicted to earn per hour, according to the model?

Part D

Would adolescent height cause an omitted variable bias if it were left out? Explain using both your intuition, and some statistical evidence with R.

Part E

Now add adolescent height to the regression, and write the new regression equation below, as before. Interpret the coefficient on height85.

Part F

How much would someone who is 5’10" in 1985 and 4’8" in 1981 be predicted to earn, according to the model?

Part G

What happened to the estimate on height85 and its standard error?

Part H

Is there multicollinearity between height85 and height81? Explore with a scatterplot. Hint: to avoid overplotting, use geom_jitter() instead of geom_point() to get a better view of the data.

Part I

Quantify how much multicollinearity affects the variance of the OLS estimates on both heights. Hint: You’ll need the vif command from the car package.

Part J

Reach the same number as in part I by running an auxiliary regression.

Hint: There’s some missing wage96 data that may give you a different answer, so take the data and filter(!is.na(wage96) before running this regression — this will include only observations for wage96 that are not NA’s.

Part K

Make a regression table from part B and D using huxtable.