Applied Econometrics – Midterm Summary
,Hypothesis Testing
Confidence interval = β^ ± 1.96*se(se(β)
-> Confidence interval of the difference in expected value between two entities:
(( X ¿ ¿ 11−X 12) ^β 1+(X 21− X 22) β^ 2 )± 1.96∗¿ ¿
T-statistic = β-(target level) / se(β)
-> For statistically different from 0, use target level = 0, for statistically different from 400,
use target level = 400.
OLS Assumptions
1. CM0 or CMI (in case of multiple regression).
2. Variables are independently and identically distributed.
3. Large outliers are unlikely (X, W and Y have non-zero finite moments).
4. There is no perfect multicollinearity.
-> If these hold OLS estimator is a consistent estimator of the causal effect and has a normal
distribution in large samples.
OVB
σu
Omitted variable bias formula: ^β 1 → β 1+ ρ X , where the unobservable part is composed of
σu u
u=θ X 2 +v -> X2 is the omitted variable, v is random noise.
-> If X2 is not related to X, there is no omitted variable bias because:
Cov ( X , u ) Cov ( X ,θ X 2+ v ) cov ( X , θ X 2 ) +cov ( X , v ) θcov ( X , X 2) + 0 θ∗0
ρX = = = = = =0.
u
σ X σu σ X σu σXσu σ X σu σXσu
Potential Outcomes
Model: Y i=Y i ( 0 ) + ( Y i ( 1 )−Y i ( 0 ) ) X i
Counterfactual: Only one of the outcomes can be observed: You either receive treatment or
you do not.
Average Treatment Effect (ATE): Mean of the individual causal effects in the population.
-> Holds as long as the individuals are (1) randomly selected from population and (2)
randomly assigned to treatment.
- Differences in the sample averages for treatment and control groups can be obtained by:
- Binary regression: Y i=β 0 + β 1 X i+ ui.
-> As long as X (treatment) is randomly assigned, CM0 holds: E ( ui|X i) =0.
- Multiple regression: : Y i=β 0 + β 1 X i+ β2 W 1 i +ui.
-> Good control 1: If X is randomly assigned conditional on W (for instance major students
are more likely to be assigned to a homework group than non-major students).
-> Necessary for CMI: E ( ui|X i , W i ) =E ( ui|W i )
-> Good control 2: W helps to explain variation in Y, so that standard error of β1 are
reduced.
-> Good control 3: Heterogenous effects.
-> Bad control: Outcome variables should never be included.=
, Random assignment implies that: E ( Y i ( 1 )|X i=1 ) −E ( Y i ( 0 )|X i=0 ) =E ¿.
Translation to regression coefficients:
Y i=Y i ( 0 ) + ( Y i ( 1 )−Y i ( 0 ) ) X iY i=E ( Y i ( 0 ) ) + ( Y i ( 1 )−Y i ( 0 ) ) X i +Y i ( 0 )−E ( Y i ( 0 ) )- β 0=E ( Y i ( 0 ) )
- β 1=Y i ( 1 )−Y i ( 0 )
- ui=Y i ( 0 )−E ( Y i ( 0 ) )
Validity Threats
Internal:
- Failure to randomize treatment X -> tested by testing hypothesis that coefficients of W are
0 in a regression of X (should be true).
- Partial compliance with treatment protocol -> could cause X to become correlated with u,
even though assignment was random.
-> If there is data on the amount of treatment that is actually received (partial compliance)
and the initial random assignment, the latter can be used as Z variable (instrument for X
(partial compliance)).
- Attrition: Subjects dropping out of the study after random assignment (could cause bias if
the treatment affects the chance of dropping out).
- Hawthorne effect: People can change their behaviour because they are in an experiment.
- Small sample size.
External:
- Non-representative sample.
- Non-representative policy -> policy tested should be sufficiently similar to the policy of
interest.
- General equilibrium effects: If job training is given in small sample, that sample might
benefit, however if the program is made widely available could reduce the benefit of the
programme (trained workers replacing each other).
Unobserved Characteristics
Can be shared across observations -> does not bias effects, though does require standard
errors to be adjusted (using clustered standard errors).
If coefficient does not change much after adding W’s, it is more likely that unobserved
characteristics do not affect random assignment.
Interpretation
Interpreting size of the estimated effects:
- Effect of small classes for kindergarten and grade 1 can be compared by:
-> compare β1K/sdK (test scores for children in kindergarten) and β1G/sdG and also compare
standard errors as a fraction of the standard deviation of test scores (these numbers can be
compared to see for whether small classes affects students in kindergarten and grade 1
similarly).
-> Similar situation: Compare effect of small classes across two different studies by
comparing the coefficient of both studies as a fraction of the respective standard deviation.
,Hypothesis Testing
Confidence interval = β^ ± 1.96*se(se(β)
-> Confidence interval of the difference in expected value between two entities:
(( X ¿ ¿ 11−X 12) ^β 1+(X 21− X 22) β^ 2 )± 1.96∗¿ ¿
T-statistic = β-(target level) / se(β)
-> For statistically different from 0, use target level = 0, for statistically different from 400,
use target level = 400.
OLS Assumptions
1. CM0 or CMI (in case of multiple regression).
2. Variables are independently and identically distributed.
3. Large outliers are unlikely (X, W and Y have non-zero finite moments).
4. There is no perfect multicollinearity.
-> If these hold OLS estimator is a consistent estimator of the causal effect and has a normal
distribution in large samples.
OVB
σu
Omitted variable bias formula: ^β 1 → β 1+ ρ X , where the unobservable part is composed of
σu u
u=θ X 2 +v -> X2 is the omitted variable, v is random noise.
-> If X2 is not related to X, there is no omitted variable bias because:
Cov ( X , u ) Cov ( X ,θ X 2+ v ) cov ( X , θ X 2 ) +cov ( X , v ) θcov ( X , X 2) + 0 θ∗0
ρX = = = = = =0.
u
σ X σu σ X σu σXσu σ X σu σXσu
Potential Outcomes
Model: Y i=Y i ( 0 ) + ( Y i ( 1 )−Y i ( 0 ) ) X i
Counterfactual: Only one of the outcomes can be observed: You either receive treatment or
you do not.
Average Treatment Effect (ATE): Mean of the individual causal effects in the population.
-> Holds as long as the individuals are (1) randomly selected from population and (2)
randomly assigned to treatment.
- Differences in the sample averages for treatment and control groups can be obtained by:
- Binary regression: Y i=β 0 + β 1 X i+ ui.
-> As long as X (treatment) is randomly assigned, CM0 holds: E ( ui|X i) =0.
- Multiple regression: : Y i=β 0 + β 1 X i+ β2 W 1 i +ui.
-> Good control 1: If X is randomly assigned conditional on W (for instance major students
are more likely to be assigned to a homework group than non-major students).
-> Necessary for CMI: E ( ui|X i , W i ) =E ( ui|W i )
-> Good control 2: W helps to explain variation in Y, so that standard error of β1 are
reduced.
-> Good control 3: Heterogenous effects.
-> Bad control: Outcome variables should never be included.=
, Random assignment implies that: E ( Y i ( 1 )|X i=1 ) −E ( Y i ( 0 )|X i=0 ) =E ¿.
Translation to regression coefficients:
Y i=Y i ( 0 ) + ( Y i ( 1 )−Y i ( 0 ) ) X iY i=E ( Y i ( 0 ) ) + ( Y i ( 1 )−Y i ( 0 ) ) X i +Y i ( 0 )−E ( Y i ( 0 ) )- β 0=E ( Y i ( 0 ) )
- β 1=Y i ( 1 )−Y i ( 0 )
- ui=Y i ( 0 )−E ( Y i ( 0 ) )
Validity Threats
Internal:
- Failure to randomize treatment X -> tested by testing hypothesis that coefficients of W are
0 in a regression of X (should be true).
- Partial compliance with treatment protocol -> could cause X to become correlated with u,
even though assignment was random.
-> If there is data on the amount of treatment that is actually received (partial compliance)
and the initial random assignment, the latter can be used as Z variable (instrument for X
(partial compliance)).
- Attrition: Subjects dropping out of the study after random assignment (could cause bias if
the treatment affects the chance of dropping out).
- Hawthorne effect: People can change their behaviour because they are in an experiment.
- Small sample size.
External:
- Non-representative sample.
- Non-representative policy -> policy tested should be sufficiently similar to the policy of
interest.
- General equilibrium effects: If job training is given in small sample, that sample might
benefit, however if the program is made widely available could reduce the benefit of the
programme (trained workers replacing each other).
Unobserved Characteristics
Can be shared across observations -> does not bias effects, though does require standard
errors to be adjusted (using clustered standard errors).
If coefficient does not change much after adding W’s, it is more likely that unobserved
characteristics do not affect random assignment.
Interpretation
Interpreting size of the estimated effects:
- Effect of small classes for kindergarten and grade 1 can be compared by:
-> compare β1K/sdK (test scores for children in kindergarten) and β1G/sdG and also compare
standard errors as a fraction of the standard deviation of test scores (these numbers can be
compared to see for whether small classes affects students in kindergarten and grade 1
similarly).
-> Similar situation: Compare effect of small classes across two different studies by
comparing the coefficient of both studies as a fraction of the respective standard deviation.
