• Instructions
  • Theory and Concepts
    • Question 1
    • Question 2
    • Question 3
    • Question 4
  • Theory Problems
    • Question 5
      • Part A
      • Part B
      • Part C
      • Part D
    • Question 6
      • Part A
      • Part B
      • Part C
  • R Questions
    • Question 7
      • Part A
      • Part B
      • Part C
      • Part D
      • Part E
    • For every additional inch in height as an adult, we can expect someone’s wages to be $0.11 lower, on average, holding their adolescent height constant.
      • Part F
      • Part G
      • Part H
      • Part I
      • Part J
      • Part K

Instructions

Answers may be longer than I would deem sufficient on an exam. Some might vary slightly based on points of interest, examples, or personal experience. These suggested answers are designed to give you both the answer and a short explanation of why it is the answer.

Theory and Concepts

Question 1

In your own words, describe what omitted variable bias means. What are the two conditions for an omitted variable to cause a bias?

All variables that might influence the dependent variable (Y) that we do not measure and include in our regression are a part of the error term (ϵ). If we omit a variable (Z), it will cause a bias if and only if it meets both of the following conditions: 1. The variable must be a determinant of our dependent variable, corr(Z,Y)≠0, and thus would appear in the error term, ϵ. 2. The variable must be correlated with one of our independent variables of interest in our regression, corr(Z,X)≠0.

If both conditions are met, then if we did not include the omitted variable Z, our estimate of the causal effect of X on Y would be biased, because our estimate (^β1) would pick up some of the effect of Z. If we include Z as another independent variable, then the ^β1 on X will tell us the precise effect of only X→Y, holding Z constant.

Question 2

In your own words, describe what multicollinearity means. What is the cause, and what are the consequences of multicollinearity? How can we measure multicollinearity and its effects? What happens if multicollinearity is perfect?

Multicollinearity just means that two regressors (e.g .X1 and X2) are correlated with each other. This fact does not bias the OLS estimates of these regressors (e.g. ^β1 and ^β2). In fact, the reason X2 is included in the regression is because omitting it would cause omitted variable bias, since corr(X1,X2)≠0 and corr(Y,X2)≠0. However, the variance of these OLS estimators is increased because it is hard to get a precise measure of X1→Y because X2→Y also, and X1 may tend to be certain values (large or small) when X2 is certain values (large or small) so we don’t know counterfactuals (e.g. what if X1 were the opposite of what it tends to be (large or small) when X2 is large or small).

The strength of multicollinearity is simply given by the value of the correlation coefficient between X1 and X2, rX1,X2. We can measure the effect of multicollinearity on the variance of a regressor (Xj)’s coefficient (^βj) with the Variance Inflation Factor:

VIF=11−R2j

where R2j is the R2 from an auxiliary regression of Xj on all of the other regressors.

Multicollinearity is perfect when the correlation between X1 and X2 is 1 or -1. This happens when one regressor (e.g. X1) is an exact linear function of another regressor(s) (e.g.X1=X2100. A regression cannot be run including both variables, as it creates a logical contradiction. In this example, ^β1 would be the marginal effect on Y of changing X1 holding X2 constant – but X2 would naturally change as it is a function of X1!

Question 3

In your own words, describe what a proxy variable is. When or why would we use a proxy variable, and what effects does it have on our model?

If we can think of a candidate variable Z that would cause omitted variable bias (meeting both conditions in question 1), we would need to include this variable in the regression to remove the bias in our OLS estimate for a regressor Xj. If we do not have data on Z, but we do have data on a variable strongly correlated with Z, call it P, then we can use P as a proxy for Z.

corr(Y,Z)≠0corr(Xj,Z)≠0corr(Z,P)≠0

Including the proxy variable removes the bias that variable Z would have on the estimate for Xj by essentially including something strongly resembling Z. It is not important whether or not there is a strong corr(Y,P) or that we have a causal explanation of how P→Y. The only reason we included P in the regression is to get rid of omitted variable bias in Xj caused by Z!

Question 4

Explain how we use Directed Acyclic Graphs (DAGs) to depict a causal model: what are the two criteria that must hold for identifying a causal effect of X on Y? When should we control a variable, and when should we not control a variable?

A Directed Acyclic Graph (DAG) describes a causal model based on making our assumptions about relationships between variables explicit, and in many cases, testable.

Variables are represented as nodes, and causal effects represented as arrows from one node to another (in the direction of the causal effect). We think about the causal effect of X→Y in counterfactual terms: if X had been different, Y would have been different as a response.

When considering the causal effect of X→Y, we must consider all pathways from X to Y (that do not loop, or go through a variable twice), regardless of the direction of the arrows. The paths will be of two types:

  • Causal (front-door) pathways where arrows go from X into Y (including through other mediator variables)
  • Non-causal (back-door) pathways where an arrow leads into X (implying X is partially caused by that variable)

Adjusting or controlling for (in a multivariate regression, this means including the variable in the regression) a variable along a pathway closes that pathway.

Variables should be adjusted (controlled for) such that:

  1. Back-door criterion: no backdoor pathway between X and Y remains open
  2. Front-door criterion: no frontdoor pathway is closed

The one exception is a collider variable, where a variable along a pathway has arrows pointing into it from both directions. This automatically blocks a path (whether front door or back door). Controlling for a collider variable opens the pathway it is on.

See R Practice on Causality and DAGs for examples.

Theory Problems

For the following questions, please show all work and explain answers as necessary. You may lose points if you only write the correct answer. You may use R to verify your answers, but you are expected to reach the answers in this section “manually.”

Question 5

Data were collected from a random sample of 220 home sales from a community in 2017.

^Price=119.2+0.485BDR+23.4Bath+0.156Hsize+0.002Lsize+0.090Age

  • Price: selling price (in $1,000s)
  • BDR: number of bedrooms
  • Bath: number of bathrooms
  • Hsize: size of the house (in ft2)
  • Lsize: lot size (in ft2)
  • Age: age of the house (in years)

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?

From ^β2, $23,400

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?

In this case, ΔBDR=1 and ΔHsize=100. The resulting expected increase in price is 23.4(1)+0.156(100)=39.0, or $39,000.

Part C

Suppose the R2 of this regression is 0.727. Calculate the adjusted ˉR2.

There are n=220 observations and k=6 variables, so:

ˉR2=1−n−1n−k−1(1−R2)=1−220−1220−6−1(1−0.727)=1−219213(0.273)=0.719

Part D

Suppose the following auxiliary regression for BDR has an R2 of 0.841.

^BDR=δ0+δ1Bath+δ2Hsize+δ3Lsize+δ4Age

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

VIF=11−R2j=11−0.841=10.159=6.29

The variance on ^β2 (on Bath) increases by 6.29 times due to multicollinearity between Bath and other X-variables.

Question 6

A researcher wants to investigate the effect of education on average hourly wages. Wage, education, and experience in the dataset have the following correlations:

Wage Education Experience
Wage 1.0000
Education 0.4059
Experience 0.1129 −0.2995 1.0000

She runs a simple regression first, and gets the results:

^Wage=−0.9049+0.5414Education

She runs another regression, and gets the results:

^Experience=35.4615−1.4681Education

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 ^β1 in the first regression overstate or understate the effect of education on wages?

We know that the estimate in the biased regression (first one above) is a function of:

^α1=^β1+^β2^δ1

Where:

  • ^α1: the coefficient on education in the biased regression (0.5414)
  • ^β1: the true effect of education on wages (??)
  • ^β2: the true effect of experience on wages (0.0701)
  • ^δ1: the effect of education on experience (from an auxiliary regression) (-1.4681)

OMV=^β2^δ1=(0.0701)(−1.4681)=−0.1029

Since the bias is negative, it understates the effect of education (due to experience). Note there are other factors that can bias the effect of education, but at least through experience, the bias is negative since the two are negatively related in the data (see the second regression).

Part B

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

We know the biased estimate for education, 0.0611. Plugging in both this and the bias, we get the “true” effect:

α1=β1+β2δ10.5414=β1−0.10290.6443=β1

Part C

The R2 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?

Here we need to calculate the Variance Inflation Factor (VIF) by using the R2 from the auxiliary regression.

VIF=11−R2j=11−(0.0897)=10.9103=1.0985

The variance increases only by 1.0985 times due to fairly weak multicollinearity between education and experience.

R Questions

Answer the following questions using R. When necessary, please write answers in the same document (knitted Rmd to html or pdf, typed .doc(x), or handwritten) as your answers to the above questions. Be sure to include (email or print an .R file, or show in your knitted markdown) your code and the outputs of your code with the rest of your answers.

Question 7

Download the heightwages.csv dataset. This data is a part of a larger dataset from the National Longitudinal Survey of Youth (NLSY) 1979 cohort: a nationally representative sample of 12,686 men and women aged 14-22 years old when they were first surveyed in 1979. They were subsequently interviewed every year through 1994 and then every other year afterwards. There are many included variables, but for now we will just focus on:

  • wage96: Adult hourly wages ($/hr) reported in 1996
  • height85: Adult height (inches) reported in 1985
  • height81: Adolescent height (inches) reported in 1981

We want to figure out what is the effect of height on wages (e.g. do taller people earn more on average than shorter people?)

Part A

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

library(tidyverse)

# load data
heights<-read_csv("https://metricsf19.classes.ryansafner.com/Data/heightwages.csv")

# make scatterplot
ggplot(data=heights, aes(x=height85, y=wage96))+
  geom_jitter(color="blue")+
  geom_smooth(method="lm",color="red")+
  labs(x = "Adult Height in 1985 (inches)",
       y = "Hourly Wage in 1996 ($)")+
  theme_classic(base_family = "Fira Sans Condensed",
           base_size=20)

Part B

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

reg1<-lm(wage96~height85, data=heights)
summary(reg1)
## 
## Call:
## lm(formula = wage96 ~ height85, data = heights)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
##  -17.26   -7.21   -3.38    1.95 1520.48 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -6.97678    5.55895  -1.255 0.209503    
## height85     0.31475    0.08239   3.820 0.000135 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 27.68 on 6711 degrees of freedom
##   (5973 observations deleted due to missingness)
## Multiple R-squared:  0.00217,    Adjusted R-squared:  0.002021 
## F-statistic: 14.59 on 1 and 6711 DF,  p-value: 0.0001346

^Wagesi=−6.98+0.31Heighti$

For every additional inch in height as an adult, we can expect someone’s wages to be $0.31 higher, on average.

Part C

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

For a 5’10’’ person (70 inches), they would earn:

^Wages=−6.98+0.31Height=−6.98+0.31(70)=−6.98+21.70=$14.72

# If you want to calculate it with R
library(broom)
reg1_tidy<-tidy(reg1)

beta_0<-reg1_tidy %>%
  filter(term == "(Intercept)") %>%
  pull(estimate)

beta_1<-reg1_tidy %>%
  filter(term == "height85") %>%
  pull(estimate)

beta_0+beta_1*70 # some rounding error in my calculation above
## [1] 15.05581

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.

heights %>%
  select(wage96, height81, height85) %>%
  cor(use = "pairwise.complete.obs")
##              wage96   height81   height85
## wage96   1.00000000 0.05387358 0.04658124
## height81 0.05387358 1.00000000 0.93429066
## height85 0.04658124 0.93429066 1.00000000

We see that Adolescent height height81 is weakly correlated with wage96 (to be fair, so is adult height), but it is strongly correlated with adult height height85.

Part E

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

reg2<-lm(wage96~height85+height81, data=heights)
summary(reg2)
## 
## Call:
## lm(formula = wage96 ~ height85 + height81, data = heights)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
##  -17.86   -7.23   -3.40    1.99 1520.49 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  -9.2483     5.7749  -1.601   0.1093  
## height85     -0.1068     0.2425  -0.440   0.6598  
## height81      0.4575     0.2459   1.860   0.0629 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 27.89 on 6591 degrees of freedom
##   (6092 observations deleted due to missingness)
## Multiple R-squared:  0.002685,   Adjusted R-squared:  0.002383 
## F-statistic: 8.873 on 2 and 6591 DF,  p-value: 0.0001417

^Wagesi=−9.25−0.11Height85i+0.46Height81i

For every additional inch in height as an adult, we can expect someone’s wages to be $0.11 lower, on average, holding their adolescent height constant.

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?

For that 5’10" person (70 inches) in 1985 and 4’10" (58") in 1981, they would earn:

^Wages=−9.25−0.11Height85+0.46Height81=−9.25−0.11(70)+0.46(58)=−9.25−7.70+25.76=$9.73

# If you want to calculate it with R
library(broom)
reg2_tidy<-tidy(reg2)

multi_beta_0<-reg2_tidy %>%
  filter(term == "(Intercept)") %>%
  pull(estimate)

multi_beta_1<-reg2_tidy %>%
  filter(term == "height85") %>%
  pull(estimate)

multi_beta_2<-reg2_tidy %>%
  filter(term == "height81") %>%
  pull(estimate)

multi_beta_0+multi_beta_1*70+multi_beta_2*58 # some rounding error in my calculation above
## [1] 9.810844

Part G

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

The effect fell by more than a half, and turned negative. The standard error also doubled (likely due to multicollinearity).

Part H

Is there multicollinearity between height85 and height81? Explore with a scatterplot.1

ggplot(data=heights, aes(x=height81, y=height85))+
  geom_jitter(color="blue")+
  geom_smooth(method="lm",color="red")+
  labs(x = "Adult Height in 1985 (inches)",
       y = "Adolescent Height in 1981 (inches)")+
  theme_classic(base_family = "Fira Sans Condensed",
           base_size=20)
## Warning: Removed 2105 rows containing non-finite values (stat_smooth).
## Warning: Removed 2105 rows containing missing values (geom_point).

Part I

Quantify how much multicollinearity affects the variance of the OLS estimates on both heights.2

We simply run vif() (from the car package library) after the recent regression to calculate the Variance Inflation Factor (VIF).

#install.packages("car") # install if you don't have 
library("car") # load the car library for the vif() command
## Loading required package: carData
## 
## Attaching package: 'car'
## The following object is masked from 'package:dplyr':
## 
##     recode
## The following object is masked from 'package:purrr':
## 
##     some
vif(reg2) # run vif 
## height85 height81 
## 8.383416 8.383416

The variance of each OLS estimate is inflated by 8.38 times due to multicollinearity.

Part J

Reach the same number as in part I by running an auxiliary regression.3

aux_reg <- heights %>%
  filter(!is.na(wage96)) %>% # use only for which we have wages
  lm(data = ., height85~height81) # run regression

summary(aux_reg) # look for R-squared
## 
## Call:
## lm(formula = height85 ~ height81, data = .)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -15.3510  -0.3994  -0.1574   0.6006  14.0198 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 3.448524   0.290169   11.88   <2e-16 ***
## height81    0.951601   0.004313  220.62   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.416 on 6592 degrees of freedom
##   (336 observations deleted due to missingness)
## Multiple R-squared:  0.8807, Adjusted R-squared:  0.8807 
## F-statistic: 4.867e+04 on 1 and 6592 DF,  p-value: < 2.2e-16

We find that adult height is positively affected by adolescent height (^β1=0.95, meaning for every inch that someone is when they are an adolescent, their adult height can be predicted to be about the same). The R2 on the auxiliary regression is 0.88.

VIF=11−R21=11−(0.8807)=10.1193=8.38

# in R: 

# extract r.squared using broom
aux_r_sq<-glance(aux_reg) %>%
  pull(r.squared)

# vif formula
1/(1-aux_r_sq)
## [1] 8.383416

Note if you failed to tell R to not drop the data with NA’s you will get a different R2 and hence a different VIF than before. This would use more observations that have data on height81 and height85 but not wage96 which would not be used in the regression…hence, a different n and R2.

Part K

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

library(huxtable)
## 
## Attaching package: 'huxtable'
## The following objects are masked from 'package:ggdag':
## 
##     label, label<-
## The following object is masked from 'package:dplyr':
## 
##     add_rownames
## The following object is masked from 'package:purrr':
## 
##     every
## The following object is masked from 'package:ggplot2':
## 
##     theme_grey
huxreg("Wages (1996)"=reg1,
       "Wages (1996)"=reg2,
       coefs = c("Constant" = "(Intercept)",
                 "Adult Height (1985)" = "height85",
                 "Adolescent Height (1981)" = "height81"),
       statistics = c("N" = "nobs",
                      "R-Squared" = "r.squared",
                      "SER" = "sigma"),
       number_format = 2)
Wages (1996) Wages (1996)
Constant -6.98     -9.25 
(5.56)    (5.77)
Adult Height (1985) 0.31 *** -0.11 
(0.08)    (0.24)
Adolescent Height (1981)         0.46 
        (0.25)
N 6713        6594    
R-Squared 0.00     0.00 
SER 27.68     27.89 
*** p < 0.001; ** p < 0.01; * p < 0.05.

  1. Hint: to avoid overplotting, use geom_jitter() instead of geom_point() to get a better view of the data.↩︎

  2. Hint: You’ll need the car package.↩︎

  3. Hint: There’s some missing wage96 data that may give you a different answer, so filter() your data here by !is.na(wage96) before running this regression - this will include only observations for wage96 that are not NA’s.↩︎