Cross-sectional data
What? What is it? How to test it? How to solve it?
OLS assumptions - A1: Population is linear in parameters A3: when multicollinearity, matrix is not full rank. Cannot
be inverted
- A2: We have a random sample from the population A4 does not hold beta is biased
- A3: We have sample variation in the explanatory Causes A4 fails to hold:
variable (no perfect collinearity) - Omitting relevant variables
- A4: The error u has an expected value of zero, given - Endogeneity
any value of explanatory variable x - Measurement error (Due to misreporting)
A5: two issues variance can increase:
- Inclusion of irrelevant variables (model
If these four assumptions hold than the OLS estimators are
misspecification)
unbiased (finite sample property)
- Multicollinearity
- A5: Homoscedasticity: The variance of error u is
constant and finite for any value of x A6:
Expected Uhati should be zero
Now finite sample property efficiency holds as well
- A6: Normality; the population error u is independent
of the explanatory variables x and is normally
distributed
Now we can use Bhat for hypothesis testing
Endogeneity Endogeneity causes bias and inconsistency if: Instrumental variables (IV) estimation method provides a
- One or more explanatory variables is jointly way to identify causal effect of x on y.
determined with y
- Not only Cov(x1, x2) > 0, but an unobserved or
excluded variable x1 has causal effect on y and x2. you can also use firm fixed effects to
Requires: control for unobserved heterogeneity
X2 becomes endogenous explanatory variable and is
effected by endogeneity bias under the assumption that it is
if not ruled out OLS results can show only correlation, but approximately constant for each firm
Step 1: Check instrument relevance:
not causality
+ other x’s
F-statistic of at least 10. Does landa has sign and
magnitude you expect?
Step 2: Estimate model using z as instrument:
Xhat is predicted x from step 1.
Two stage least squares
Check reduced form relationship: Does z have a direct
effect on y
+other x’s
What? What is it? How to test it? How to solve it?
OLS assumptions - A1: Population is linear in parameters A3: when multicollinearity, matrix is not full rank. Cannot
be inverted
- A2: We have a random sample from the population A4 does not hold beta is biased
- A3: We have sample variation in the explanatory Causes A4 fails to hold:
variable (no perfect collinearity) - Omitting relevant variables
- A4: The error u has an expected value of zero, given - Endogeneity
any value of explanatory variable x - Measurement error (Due to misreporting)
A5: two issues variance can increase:
- Inclusion of irrelevant variables (model
If these four assumptions hold than the OLS estimators are
misspecification)
unbiased (finite sample property)
- Multicollinearity
- A5: Homoscedasticity: The variance of error u is
constant and finite for any value of x A6:
Expected Uhati should be zero
Now finite sample property efficiency holds as well
- A6: Normality; the population error u is independent
of the explanatory variables x and is normally
distributed
Now we can use Bhat for hypothesis testing
Endogeneity Endogeneity causes bias and inconsistency if: Instrumental variables (IV) estimation method provides a
- One or more explanatory variables is jointly way to identify causal effect of x on y.
determined with y
- Not only Cov(x1, x2) > 0, but an unobserved or
excluded variable x1 has causal effect on y and x2. you can also use firm fixed effects to
Requires: control for unobserved heterogeneity
X2 becomes endogenous explanatory variable and is
effected by endogeneity bias under the assumption that it is
if not ruled out OLS results can show only correlation, but approximately constant for each firm
Step 1: Check instrument relevance:
not causality
+ other x’s
F-statistic of at least 10. Does landa has sign and
magnitude you expect?
Step 2: Estimate model using z as instrument:
Xhat is predicted x from step 1.
Two stage least squares
Check reduced form relationship: Does z have a direct
effect on y
+other x’s
