Description

Econometrics
Elizabeth Goodwin
1
2
a
If we used the lasso instead of OLS, we would no longer have unbiased coefficient estimates. The lasso is designed to trade biased coefficients for lower variation, and is primarily useful for use cases where you wish to most accurately predict Yˆ. The lasso intentionally biases the coeffients to achieve this.
b
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4
a
b
In this context, a few things would have to be true for RDD to work. Most important is the continuity assumption, or essentially that without the treatment, the outcomes would not have jumped. As we cannot observe the counterfactual reality of those past the cutoff without the treatment or those before the cutoff with the treatment, we must assume that with the jump removed, the trend would be continuous. Continuity removes omitted variable bias, as it assumes that the treatment is what is causing the cutoff. Another assumption is that the values around the cutoff must not be manipulated, or essentially changed in because knowledge of its existence. Also, the people at either side of the cutoff inside the bandwidth are assumed to be exchangeable
d.
5
To answer the question of the impact of the ADA on wages, you can use a differences in differences approach. To test for this, we first have to test the trends of existing data. The theory is, essentially, that disabled and non disabled workers, even if being employed at different rates, would follow the same employment trend. To put this in a regression, you could do something like this. This is a regression with the dependent variable being Employment Rate, a constant term(for both disabled and non disabled), and the other terms being interaction terms between employment rate trends and an indicator for the year. The most important terms, however, are the interaction between year and disability status. The coefficient there represents the difference per year in the employment rate between those disabled and those not. The beta hats with multiple numbers represent the coefficients for each year, as it interacts a year indicator with each beta hat. Disability would be a dummy variable with 1 = disabled and 0 meaning not disabled.
EmpRate = βˆ0 + βˆ1D + βˆ2−22× Y ear + βˆ22−42D × Y ear
You can also do the same thing, but for income as well. You would want to log income though, to show a percent change in income instead of an absolute one. βˆ22−42 represents the difference in percent income per year between disabled and nondisabled workers.
LogWage = βˆ0 + βˆ1D + βˆ2−22× Y ear + βˆ22−42D × Y ear
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a
The bias you would be seeing in this case is omitted variable bias. Essentially, this means your model has endogeneity issues, and there is likely a missing variable from the model. If those relevant variables were added in, the correlation would not exist. It missing means that the model is not accurately showing the effect of x on y, but the effect of x on y and the indirect effect of the omitted variable on y, through x.
b
i.
In this situation, Z passes the relevance test, but not the exogeneity test. As Z is also correlated with the error term, it is facing the same problems as what it is trying to measure, and is an inproper instrument.
ii.
iii.
The OLS estimator is less biased than that of the IV estimator if a weak instrument magnifies the bias to a level above the bias of OLS. Essentially the equation below is the bias of 2SLS in this situation. If it is greater than the bias of the OLS variable, as shown below, 2SLS is more biased than OLS. Weak instruments mean a low covariance in the denominator, which magnifies the bias significantly.
Cov(Z,ϵ)
> Cov(X,ϵ) Cov(Z,X)
iv.
For a fixed non-zero value of Cov(Z,ϵ), weak instruments can make the problem of bias very bad. One issue of weak instruments is simply that it makes the error larger in general, as it will by nature magnify the standard error. In addition to this, the bias being magnified can even make a less biased instrument bias 2SLS more than normal OLS. This means you must be able to justify the exogeneity criteria quite well for weak instruments, as any failure in exogeneity can be very bad.
7
This argument is simply incorrect. The primary reason is it misses the entire point of instruments. If Z is correlated with X, and X is correlated with Y, you would expect Z to be correlated with Y as well. The point of an instrument is to use an instrument to measure the effect of X on Y through Z, to avoid endogeneity issues of using X itself. If you add X and Z together in one regression, so conditioning on X, you are not just conditioning on X, but also the unobserved variable U that is mediated through X. If you could just condition on X to see the causal effect of X on Y, and use that to test Z, you wouldn’t need Z in the first place. The regression of Y on both X and Z would have omitted variable bias, and you would have to be able to condition on X, Z, and U, but U is unobserved, and you are using IV to get around the problem of U.
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a
gen robcap = (robberies / (population/1000) ) gen bacap = (assaults + batteries) / (population/1000) sum population gen popstand = (population – 857.2294)/479.5102 encode community, gen(enc_c)
b
eststo: reg robcap popstand i.enc_c bus_stops i.intersection i.majorstreet pct_black /// pct_hisp pct_hh_w_public_assist pct_residential pct_industrial pct_commercial, vce(robust)
In Table 1, the first regression. Population has a significant negative correlation with robberies per caita, pct residential had a significant negative correlation with robberies per capita. Percent industrial had a significant negative correlation with crime, but less significant than before, with pretty large standard errors. Percent commercial had a huge and very significant positive correlation with crime, with a much greater magnitude than the others.
c
eststo: reg bacap popstand i.enc_c bus_stops i.intersection i.majorstreet pct_black /// pct_hisp pct_hh_w_public_assist pct_residential pct_industrial pct_commercial, vce(robust)
Results, shown in regression 2 of Table 1, were overall pretty similar to before, with a few very notable changes. With batteries and assaults, the negative correlation with population was greater. In addition, the residential correlation was slightly smaller. Most interestingly, the negative correlation with industrial areas is much, much greater and very statistically significant. And the coefficient with percent commercial, which was already big, nearly triples in size.
d
eststo: ivregress 2sls robcap bus_stops i.enc_c i.intersection i.majorstreet /// pct_black pct_hisp pct_hh_w_public_assist (popstand pct_residential pct_industrial /// pct_commercial = pct_low_dens_zoned pct_medium_dens_zoned pct_high_dens_zoned /// pct_highest_den_zoned ct_commercial_zoned pct_industrial_zoned /// pct_residential_zoned), vce(robust)
eststo: ivregress 2sls bacap bus_stops i.enc_c i.intersection i.majorstreet pct_black /// pct_hisp pct_hh_w_public_assist (popstand pct_residential pct_industrial /// pct_commercial = pct_low_dens_zoned pct_medium_dens_zoned pct_high_dens_zoned /// pct_highest_den_zoned pct_commercial_zoned pct_industrial_zoned /// pct_residential_zoned), vce(robust)
After being instrumented, the regressions change significantly. First off, many of the results are just far less signficiant, and many we can’t draw conclusions from at all. These include population, percent industrial, and percent commercial. There was a very significant negative correlation between percent residential and crime, however, and it was larger in the batteries and assaults regression (4). In addition, the hidden variable representing multifamily both end up with a significant positive correlation with both types of crimes. In previous regressions, they have had a insigificant or slightly negative correlation, but not here. With Assults and batteries in particular it is very significant.
e
eststo: ivregress 2sls robcap bus_stops i.enc_c i.intersection i.majorstreet pct_black /// pct_hisp pct_hh_w_public_assist (popstand pct_residential pct_industrial pct_commercial = pct_low_dens_zoned pct_medium_dens_zoned pct_high_dens_zoned pct_highest_den_zoned /// pct_commercial_zoned pct_industrial_zoned pct_residential_zoned pct_low_dens_zoned ///
c.pct_low_dens_zoned#c.pct_residential_zoned c.pct_medium_dens_zoned#c.pct_residential_zoned
c.pct_high_dens_zoned#c.pct_residential_zoned c.pct_highest_den_zoned#c.pct_residential_zoned),
eststo: ivregress 2sls bacap bus_stops i.enc_c i.intersection i.majorstreet pct_black /// pct_hisp pct_hh_w_public_assist (popstand pct_residential pct_industrial pct_commercial = pct_low_dens_zoned pct_medium_dens_zoned pct_high_dens_zoned pct_highest_den_zoned /// pct_commercial_zoned pct_industrial_zoned pct_residential_zoned pct_low_dens_zoned ///
c.pct_low_dens_zoned#c.pct_residential_zoned c.pct_medium_dens_zoned#c.pct_residential_zoned
c.pct_high_dens_zoned#c.pct_residential_zoned c.pct_highest_den_zoned#c.pct_residential_zoned),
///
/// vce(robu
///
/// vce(robu
Table 1: Regression Table (Land Use Only)
(0.0665) (0.168) -0.114
(0.583) 1.850
(1.271) -0.168
(0.511) 1.551
(1.124)
pct_residential -0.790∗∗∗ -0.674∗∗ -2.695∗∗∗ -5.575∗∗∗ -2.672∗∗∗ -5.694∗∗∗
(0.144) (0.261) (0.648) (1.239) (0.618) (1.181)
pct_industrial -0.588∗ -4.543∗∗∗ -2.432 -1.095 -2.573 -2.114
(0.289) (0.611) (2.408) (5.253) (2.153) (4.721)
pct_commercial 2.341∗∗∗ 6.111∗∗∗ 1.153 0.988 1.137 0.796
(0.310) (0.643) (0.774) (1.593) (0.741) (1.528)
_cons 0.184 -1.941∗ 2.550∗∗∗ 5.430∗∗∗ 2.494∗∗∗ 5.349∗∗∗
(0.373) (0.914) (0.640) (1.369) (0.639) (1.364)
N 19330 19330 19330 19330 19330 19330
adj. R2 0.220 0.322 0.207 0.272 0.207 0.277
Standard errors in parentheses
∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001