Say we estimate \[ price_i = 10 + 1.5\times bedroom_i + 2\times bathroom_i + 12\times area_i \]
beta_bedroom = 1.5
beta_bathroom = 2
beta_area = 12
alpha = 10
# all else constant, add one bedroom
# change equal to:
beta_bedroom[1] 1.5# all else constant, add one bedroom and add 3 to area
# change equal to:
beta_bedroom + 3*beta_area[1] 37.5Dor - remember, discuss ‘average’ change, and not ‘individual’ change.
Is this causal?
Load data, create variables
# load data
path = "/Users/dorleventer/Dropbox/teaching/undergrad_econometrics_spring_2023"
insert_data_path <- file.path(path, "data")
df = read.csv(glue::glue("{insert_data_path}/wage2.csv"))
# create pexp variable
df = df %>%
mutate(
log_wage = log(wage),
pexp = age - educ - 6,
pexp2 = pexp^2
) |> tibble()
df# A tibble: 935 × 19
obs age black brthord educ feduc hours iq kww lwage married meduc
<int> <int> <int> <int> <int> <int> <int> <int> <int> <dbl> <int> <int>
1 1 31 0 2 12 8 40 93 35 6.65 1 8
2 2 37 0 NA 18 14 50 119 41 6.69 1 14
3 3 33 0 2 14 14 40 108 46 6.72 1 14
4 4 32 0 3 12 12 40 96 32 6.48 1 12
5 5 34 0 6 11 11 40 74 27 6.33 1 6
6 6 35 1 2 16 NA 40 116 43 7.24 1 8
7 7 30 0 2 10 8 40 91 24 6.40 0 8
8 8 38 0 3 18 NA 40 114 50 6.99 1 8
9 9 36 0 3 15 5 45 111 37 7.05 1 14
10 10 36 0 1 12 11 40 95 44 6.91 1 12
# … with 925 more rows, and 7 more variables: sibs <int>, south <int>,
# urban <int>, wage <int>, log_wage <dbl>, pexp <dbl>, pexp2 <dbl>Note that experience is fully determined by age and educ.
summary(lm(log_wage ~ educ + pexp + pexp2 + age, data=df))
Call:
lm(formula = log_wage ~ educ + pexp + pexp2 + age, data = df)
Residuals:
Min 1Q Median 3Q Max
-1.86059 -0.22867 0.03587 0.26465 1.37655
Coefficients: (1 not defined because of singularities)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.0529170 0.2133277 23.686 < 0.0000000000000002 ***
educ 0.0838286 0.0072517 11.560 < 0.0000000000000002 ***
pexp 0.0671036 0.0239406 2.803 0.00517 **
pexp2 -0.0015824 0.0008356 -1.894 0.05856 .
age NA NA NA NA
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.3939 on 931 degrees of freedom
Multiple R-squared: 0.1282, Adjusted R-squared: 0.1254
F-statistic: 45.64 on 3 and 931 DF, p-value: < 0.00000000000000022age was omitted!
Why? Due to co-linearity with pexp and educ.
How did it choose to omit age? Last variable, if we change order:
summary(lm(log_wage ~ educ + age + pexp + pexp2, data=df))
Call:
lm(formula = log_wage ~ educ + age + pexp + pexp2, data = df)
Residuals:
Min 1Q Median 3Q Max
-1.86059 -0.22867 0.03587 0.26465 1.37655
Coefficients: (1 not defined because of singularities)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.6502953 0.3429839 13.558 < 0.0000000000000002 ***
educ 0.0167250 0.0237091 0.705 0.48072
age 0.0671036 0.0239406 2.803 0.00517 **
pexp NA NA NA NA
pexp2 -0.0015824 0.0008356 -1.894 0.05856 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.3939 on 931 degrees of freedom
Multiple R-squared: 0.1282, Adjusted R-squared: 0.1254
F-statistic: 45.64 on 3 and 931 DF, p-value: < 0.00000000000000022New order, omitted last variable (this time, pexp).
What can we do? Choose which variable to omit, or define pexp differently.
We begin with omitting age.
summary(lm(log_wage ~ educ + pexp + pexp2, data=df))$coefficients[,1:2] Estimate Std. Error
(Intercept) 5.052917009 0.2133277308
educ 0.083828588 0.0072516770
pexp 0.067103612 0.0239406453
pexp2 -0.001582433 0.0008355912We continue with adding IQ.
summary(lm(log_wage ~ educ + pexp + pexp2 + iq, data=df))$coefficients[,1:2] Estimate Std. Error
(Intercept) 4.697127993 0.2167882552
educ 0.063366078 0.0078313824
pexp 0.067876010 0.0234694198
pexp2 -0.001570575 0.0008191349
iq 0.006106719 0.0009805248We added IQ, and the effect of education on wages decreased.
How could we have seen that coming?
If we note two things:
That the correlation between IQ and educ is positive.
cor(df$iq, df$educ)[1] 0.515697IQ and log_wage is positive.cor(df$iq, df$lwage)[1] 0.3147877educ (\(\hat{\beta}_{educ}\)), is biased upwards when we estiamte a model without IQ.It is constructive to think of the two variables case:
Putting it all together:
\[\begin{align} \hat{\beta}_1 &= \frac{\hat{cov}(educ,y)}{\hat{var}(educ)} - \hat{\beta}_2\frac{\hat{cov}(educ,IQ)}{\hat{var}(educ)} \\ \leftrightarrow \hat{\beta}_1 &= \hat{\tilde{\beta}}_1 - \hat{\beta}_2\hat{\gamma}_1 \\ \leftrightarrow (\text{Short Model Estimator:})\quad\hat{\tilde{\beta}}_1 &= \hat{\beta}_1 + \hat{\beta}_2\hat{\gamma}_1 \\ &= Truth + Bias \end{align}\]Translating what we wrote above to math,
we argued that \(\hat{\gamma}_1>0\) and \(\hat{\beta}_2>0\),
and so the estimator in the short model is upward biased.
Bottom line: adding IQ to estimation will decrease coefficient on educ.
We continue with adding fathers education.
summary(lm(log_wage ~ educ + pexp + pexp2 + iq + feduc, data=df))$coefficients[,1:2] Estimate Std. Error
(Intercept) 4.769617360 0.2352397408
educ 0.054716879 0.0088027999
pexp 0.058818673 0.0255789941
pexp2 -0.001155708 0.0009027647
iq 0.005783902 0.0011411345
feduc 0.013049787 0.0047694996Compare to previous estimators. What changed?
What does this for mean feduc as an OV?
Now add mothers education.
summary(lm(log_wage ~ educ + pexp + pexp2 + iq + feduc + meduc, data=df))$coefficients[,1:2] Estimate Std. Error
(Intercept) 4.718291292 0.2420661714
educ 0.054332388 0.0089634235
pexp 0.064914445 0.0265187182
pexp2 -0.001421293 0.0009426224
iq 0.005469130 0.0011657645
feduc 0.009249090 0.0055402097
meduc 0.008844003 0.0063002757What happened to the slope of educ?
What happened to stat. significance of feduc?
If we want a un-biased model, which would you choose?
Dor: Remember, small discussion on trade-off of adding more variables (bad controls, variance).
True model is long model \(prod = \beta_0 + \beta_1training + \beta_2 ability + u\)
Assumption: \(cor(training,ability)<0\)
What happens if we estimate small model \(prod = \alpha_0 + \alpha_1training + v\)?
Can you guess the sign of the bias? What if:
ability positively correlated with productivity, but
ability is negatively correlated with training?
Then \(\hat{\alpha}_1\) is downward-biased, w.r.t \(\beta_1\).
Lets show this explicitly.
From our assumption: \[cor(training,ability)<0\rightarrow cov(training,ability)<0\rightarrow\frac{cov(training,ability)}{var(training)}<0\]
Hence, when estimating \(ability = \gamma_0 + \gamma_1 training + e\), the estimator \(\hat{\gamma}_1 < 0\).
From OLS formula, \[\hat{\alpha}_1 = \frac{cov(training,prod)}{var(training)}\]
From OLS formula, \[\hat{\beta}_1 = \frac{cov(training,prod)}{var(training)} - \hat{\beta}_2\frac{cov(training,ability)}{var(training)}\]
Combining, we get \[\hat{\beta}_1 = \hat{\alpha}_1 - \hat{\beta}_2\hat{\gamma}_1\leftrightarrow \hat{\alpha}_1 = \hat{\beta}_1 + \hat{\beta}_2\hat{\gamma}_1\]
To sign the bias, all we need are the signs of \(\hat{\beta}_2\) and \(\hat{\gamma}_1\). Since argued that \(\hat{\gamma}_1<0\), and \(\hat{\beta}_2>0\), then \(\hat{\alpha}_1 = \hat{\beta}_1 + \text{Something negative}\). We say that this estimator is hence downward-biased.
Lets look at \(\alpha_0\) directly, by inputing the formulas.
OLS formula \[\hat{\alpha}_0 = \overline{prod} - \hat{\alpha}_1 \overline{training}\] \[ \hat{\beta}_0 = \overline{prod} - \hat{\beta}_1 \overline{training} - \hat{\beta}_2\overline{ability}\] \[\hat{\gamma}_0 = \overline{ability} - \hat{\gamma}_1 \overline{training}\]
Input mean of explained variable into single covariate regression
\[\hat{\alpha}_0 = \hat{\beta}_0 + \hat{\beta}_1\overline{training} + \hat{\beta}_2\overline{ability} - \hat{\alpha}_1 \overline{training}\]
Input the OVB formula for the slope we found above \[\begin{align*} \hat{\alpha}_{0} & =\hat{\beta}_{0}+\hat{\beta}_{1}\overline{training}+\hat{\beta}_{2}\overline{ability}-\hat{\alpha}_{1}\overline{training}\\ & =\hat{\beta}_{0}+\hat{\beta}_{2}\overline{ability}-\left(\hat{\alpha}_{1}-\hat{\beta}_{1}\right)\overline{training}\\ & =\hat{\beta}_{0}+\hat{\beta}_{2}\overline{ability}-\left(\hat{\beta}_{2}\hat{\gamma}_{1}\right)\overline{training}\\ & =\hat{\beta}_{0}+\hat{\beta}_{2}\left(\overline{ability}-\hat{\gamma}_{1}\overline{training}\right) \end{align*}\]
Input estimate for the intercept from the covariate regression \[\begin{align*} \hat{\alpha}_{0} & =\hat{\beta}_{0}+\hat{\beta}_{2}\left(\overline{ability}-\hat{\gamma}_{1}\overline{training}\right)\\ & =\hat{\beta}_{0}+\hat{\beta}_{2}\hat{\gamma}_{0} \end{align*}\]
# load data
df = read.csv(glue::glue("{insert_data_path}/wage1.csv"))
# correlation between variable and its square
cor(df$exper,df$expersq)[1] 0.9609709What do we expect to be the sign of the regression between the two?
If positive correlation,
then positive covariance,
and so sign of OLS estimator (covariance / variance) will be positive.
Check:
summary(lm(exper ~ expersq, data=df))$coefficients[,1](Intercept) expersq
6.99388211 0.02117127 What will happen if we regression log(wages) on experience, without experience squared?
Lets try without computing!
Saw experience squared is positively correlated with experience.
Can guess experience squared is negatively correlated with earnings.
And so we got here a downward-bias!
Check:
summary(lm(lwage ~ exper, data=df)) |> broom::tidy() |> select(term, estimate)# A tibble: 2 × 2
term estimate
<chr> <dbl>
1 (Intercept) 1.55
2 exper 0.00436summary(lm(lwage ~ exper + expersq, data=df)) |> broom::tidy() |> select(term, estimate)# A tibble: 3 × 2
term estimate
<chr> <dbl>
1 (Intercept) 1.30
2 exper 0.0455
3 expersq -0.000944And walla! After controlling for expersq, the estimator for exper has increased.
Consider two equations of regressions of tenure on wages,
once with a first order polynomial: \(wage = \beta_0 + \beta_1 Tenure + \epsilon\)
and once with a second order polynomial: \(wage = \alpha_0 + \alpha_1 Tenure + \alpha_2 Tenure^2 + u\)
Proposition. If the model with a first order polynomial is true, then \(E[\hat{\alpha}_2]=0\).
Proof (True).
If first order model is true,
It must be that the population parameter \(\alpha_2=0\),
And that \(\mathbb{E}[\epsilon \mid Tenure] = 0\).
This implies, that if we estimate the second model, we get
In the model with \(\alpha_2 Tenure^2 + u=e\) in the error term, it holds that \[E[\epsilon\mid Tenure]=0\leftrightarrow E[e\mid Tenure]=0\leftrightarrow E[\alpha_2 Tenure^2 + u\mid Tenure]=0\]
From \(\alpha_2=0\) it follows that \[ E[\alpha_2 Tenure^2 + u \mid Tenure]=E[u\mid Tenure]=0\].
Hence,
The OLS estimators in the second order model are unbiased, and
Specifically this holds for the second estimator, i.e. \(E[\hat{\alpha}_2]=\alpha_2=0\).
Proposition. If the model with a second order polynomial is true, \(\alpha_1>0\) and \(\alpha_2<0\) then \(E[\hat{\beta}_1]\neq\alpha_1\), but we cannot sign the bias of \(\beta_1\) w.r.t \(\alpha_1\).
Proof (Not true).
Tenure squared is positively correlated with tenure, and negatively correlated with wages, and so we can sign the bias as negative. I.e., downward-bias. Did this a-lot above, so we continue without showing equations.
Proposition. The second order model with all variables in logs is estimatable.
Proof (Not true).
If all variables in logs, and it holds that \(log(Tenure^2)=2log(Tenure)\), then the model incorporates full multi-colinearity, and so we can’t estimate the model. In R, the regression would simply omit one of the variables.
Notation: Subjective happiness := \(SH\), some individual := \(i\)
Model: \(SH_i = \alpha + \beta income_i + u_i\)
Finding: \(\hat{\beta}>0\)
Whats missing?
Family background, education, surroundings, within-family correlations, health, …
Whats the sign of the bias? Remember, needs to be w.r.t some OV.
E.g., lets stick with education.
Lets assume educated persons have more income,
But since ignorance is bliss are less happy subjectively.
Whats the sign of the bias?
From all the above, negative (i.e., downward-bias).
Is it easy to control for all these, and other relevant, parameters?
No!
So even if they had controlled for OV, can we give a causal interpretation to the OLS estimator?
No, but probably better than without controls (????????).
Discussion: so where have we got so far?
Discussion: so how can we causally estimate the effect of income on happiness?
A thought experiment:
Lets say we take random people, and divide to two groups
Ask people how happy they are
And then give one of these groups money
And ask how happy they are, afterwards.
Is this good enough?
What could go wrong?
Lab vs. natural setting (external validity?)
Reports vs. actual happiness (measurement error?)
Is there a possible natural experiment, where we can observe actual happiness?