Status

Returns To Schooling

Returns To Schooling

Ability Bias

Mincer with Unobserved Ability

Mincer with Unobserved Ability

Angrist and Krueger (1991): Birthdate is as good as Random

AK91 IV setup

AK91 Estimation: Two Stage Least Squares (2SLS)

AK91 allow us to introduce a widely used variation of our simple IV estimator: 2SLS

1. We estimate a first stage model which uses only exogenous variables (like $z$) to explain our endgenous regressor $s$.

2. We then use the first stage model to predict values of $s$ in what is called the second stage or the reduced form model. Performing this procedure is supposed to take out any impact of $A$ in the correlation we observe in our data between $s$ and $y$.

$$\begin{array}{cc}\text{1. Stage:}{s}_i& ={\alpha }_0+{\alpha }_1{z}_i+{\eta }_i\\ \text{2. Stage:}{y}_i& ={\beta }_0+{\beta }_1{\hat{s}}_i+u_i\end{array}$$

Conditions:

1. Relevance of the IV: ${\alpha }_1\ne 0$
2. Independence (IV assignment as good as random): $E\left[\eta |z\right]=0$
3. Exogeneity (our exclusion restriction): $E\left[u|z\right]=0$

Data on birth quarter and wages

Let's load the data and look at a quick summary

data("ak91", package = "masteringmetrics")
# from the modelsummary package
datasummary_skim(data.frame(ak91),histogram = TRUE)

AK91 Data Transformations

• We want to create the q4 dummy which is TRUE if you are born in the 4th quarter.

• create factor versions of quarter and year of birth.

ak91 <- mutate(ak91,
               qob_fct = factor(qob),
               q4 = as.integer(qob == "4"),
               yob_fct = factor(yob))
# get mean wage by year/quarter
ak91_age <- ak91 %>%
  group_by(qob, yob) %>%
  summarise(lnw = mean(lnw), s = mean(s)) %>%
  mutate(q4 = (qob == 4))

AK91 Figure 1: First Stage!

Let's reproduce AK91's first figure now on education as a function of quarter of birth!

ggplot(ak91_age, aes(x = yob + (qob - 1) / 4, y = s )) +
  geom_line() + 
  geom_label(mapping = aes(label = qob, color = q4)) +
  guides(label = FALSE, color = FALSE) +
  scale_x_continuous("Year of birth", breaks = 1930:1940) +
  scale_y_continuous("Years of Education", breaks = seq(12.2, 13.2, by = 0.2),
                     limits = c(12.2, 13.2)) +
  theme_bw()

AK91 Figure 1: First Stage!

AK91 Figure 2: Impact of IV on outcome

What about earnings for those groups?

ggplot(ak91_age, aes(x = yob + (qob - 1) / 4, y = lnw)) +
  geom_line() +
  geom_label(mapping = aes(label = qob, color = q4)) +
  scale_x_continuous("Year of birth", breaks = 1930:1940) +
  scale_y_continuous("Log weekly wages") +
  guides(label = FALSE, color = FALSE) +  
  theme_bw()

AK91 Figure 2: Impact of IV on outcome

Running IV estimation in R

library(estimatr)
# create a list of models
mod <- list()
# standard (biased!) OLS
mod$ols <- lm(lnw ~ s, data = ak91)
# IV: born in q4 is TRUE?
# doing IV manually in 2 stages.
mod[["1. stage"]] <- lm(s ~ q4, data = ak91)
ak91$shat         <- predict(mod[["1. stage"]])
mod[["2. stage"]] <- lm(lnw ~ shat, data = ak91)
# run 2SLS
# doing IV all in one go
# notice the formula!
# formula = y ~ x | z
mod$`2SLS`  <- iv_robust(lnw ~ s | q4,
                         data = ak91,
                         diagnostics = TRUE)

AK91 Results Table

Remember the F-Statistic?

• We encountered this before: it's useful to test restricted vs unrestricted models against each other.

• Here, we are interested whether our instruments are jointly significant. Of course, with only one IV, that's not more informative than the t-stat of that IV.

• This F-Stat compares the predictive power of the first stage with and without the IVs. If they have very similar predictive power, the F-stat will be low, and we will not be able to reject the H0 that our IVs are jointly insignificant in the first stage model. 😞

Additional Control Variables

• We saw a clear time trend in education earlier.

• There are also business-cycle fluctuations in earnings

• We should somehow control for different time periods.

• Also, we can use more than one IV! Here is how:

Additional Control Variables

# we keep adding to our `mod` list:
mod$ols_yr  <- update(mod$ols, . ~ . + yob_fct)  #  previous OLS model
# add exogenous vars on both sides of the `|` !
mod[["2SLS_yr"]] <- estimatr::iv_robust(lnw ~ s  + yob_fct | q4 + yob_fct, data = ak91, diagnostics = TRUE )  
# use all quarters as IVs
mod[["2SLS_all"]] <- estimatr::iv_robust(lnw ~ s  + yob_fct | qob_fct + yob_fct, data = ak91, diagnostics = TRUE  )

Additional Control Variables

AK91: Taking Stock - The Quarter of Birth (QOB) IV

IV Identification

Let's go back to our simple linear model:

$$y={\beta }_0+{\beta }_{1}x+u$$

where we fear that $Cov(x,u)\ne 0$, $x$ is endogenous.

Conditions for IV

1. first stage or relevance: $Cov(z,x)\ne 0$
2. IV exogeneity: $Cov(z,u)=0$: the IV is exogenous in the outcome equation.

Valid Model (A) vs Invalid Model (B) for IV z

IV Identification

IV Identification

$$\begin{array}{cc}Cov(z,y)& =Cov(z,{\beta }_0+{\beta }_{1}x+u)\\ & ={\beta }_{1}Cov(z,x)+Cov(z,u)\end{array}$$

IV Estimator

Just plugging in for the population moments:

$${\hat{\beta }}_1=\frac{{\sum }_{i=1}^n(z_i-\overline{z})(y_i-\overline{y})}{{\sum }_{i=1}^n(z_i-\overline{z})(x_i-\overline{x})}$$

• The intercept estimate is ${\hat{\beta }}_0=\overline{y}-{\hat{\beta }}_1\overline{x}$

• Given both assumptions 1. and 2. are satisfied, we say that the IV estimator is consistent for ${\beta }_1$. We write

$$\text{plim}\left({\hat{\beta }}_1\right)={\beta }_1$$

in words: the probability limit of ${\hat{\beta }}_1$ is the true ${\beta }_1$.

• If this is true, we say that this estimator is consistent.

IV Inference

Assuming $E\left(u^2|z\right)={\sigma }^2$ the variance of the IV slope estimator is

$$Var\left({\hat{\beta }}_{1,IV}\right)=\frac{{\sigma }^2}{n{\sigma }_x^2{\rho }_{x,z}^2}$$

• ${\sigma }_x^2$ is the population variance of $x$,

• ${\sigma }^2$ the one of $u$, and

• ${\rho }_{x,z}$ is the population correlation between $x$ and $z$.

You can see 2 important things here:

1. Without the term ${\rho }_{x,z}^2$, this is like OLS variance.
2. As sample size $n$ increases, the variance decreases.

IV Variance is Always Larger than OLS Variance

• Replace ${\rho }_{x,z}^2$ with $R_{x,z}^2$, i.e. the R-squared of a regression of $x$ on $z$:

$$Var\left({\hat{\beta }}_{1,IV}\right)=\frac{{\sigma }^2}{n{\sigma }_x^2{R}_{x,z}^2}$$

1. Given $R_{x,z}^2<1$ in most real life situations, we have that $Var\left({\hat{\beta }}_{1,IV}\right)>Var\left({\hat{\beta }}_{1,OLS}\right)$ almost certainly.

2. The higher the correlation between $z$ and $x$, the closer their $R_{x,z}^2$ is to 1. With $R_{x,z}^2=1$ we get back to the OLS variance. This is no surprise, because that implies that in fact $z=x$.

So, if you have a valid, exogenous regressor $x$, you should not perform IV estimation using $z$ to obtain $\hat{\beta }$, since your variance will be unnecessarily large.

Returns to Education for Married Women

Consider the following model for married women's wages:

$$\log wage={\beta }_0+{\beta }_{1}educ+u$$ Let's run an OLS on this, and then compare it to an IV estimate using father's education. Keep in mind that this is a valid IV $z$ if

1. fatheduc and educ are correlated
2. fatheduc and $u$ are not correlated.

Returns to Education for Married Women

data(mroz,package = "wooldridge")
mods = list()
mods$OLS <- lm(lwage ~ educ, data = mroz)
mods[['First Stage']] <- lm(educ ~ fatheduc, data = subset(mroz, inlf == 1))
mods$IV  <- estimatr::iv_robust(lwage ~ educ | fatheduc, data = mroz)

IV Standard Errors

IV with a Weak Instrument

• IV is consistent under given assumptions.

• However, even if we have only very small $Cor(z,u)$, we can get wrong-footed

• Small corrleation between $x$ and $z$ can produce inconsistent estimates.

Weak Stuff

To illustrate this point, let's assume we want to look at the impact of number of packs of cigarettes smoked per day by pregnant women (packs) on the birthweight of their child (bwght):

$$\log \left(bwght\right)={\beta }_0+{\beta }_{1}packs+u$$

We are worried that smoking behavior is correlated with a range of other health-related variables which are in $u$ and which could impact the birthweight of the child. So we look for an IV. Suppose we use the price of cigarettes (cigprice), assuming that the price of cigarettes is uncorrelated with factors in $u$. Let's run the first stage of cigprice on packs and then let's show the 2SLS estimates:

Weak Stuff

data(bwght, package = "wooldridge")
mods <- list()
mods[["First Stage"]] <- lm(packs ~ cigprice, data = bwght)
mods[["IV"]] <- estimatr::iv_robust(log(bwght) ~  packs | cigprice, data = bwght, diagnostics = TRUE)

Weak Stuff