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
,Heteroscedasticity For any level of x, the variance of u is not the same or finite. Breusch-Pagan test: Using residuals to estimate the error
Thus u depends on x. 1. Estimate variance as , we can obtain
heteroscedasticity-robust standard
Does not create bias in the beta’s. 2. Obtain residuals, square them, Estimate: errors.
3. Use F-test to test H0 of homoscedasticity
Why not always use robust se’s?
If we reject H0 of homoscedasticity, we need to use
- Small sample size, robust t-
robust standard errors
statistic can have distr. not
very close to t distr.
Reverse causality Causality runs from Y to X
if not ruled out OLS results can show only correlation, but
not causality
Multivariate regression Why it is better than bivariate:
- Almost any variable is likely to be determined by
multiple other variables
- Multiple explanatory variables are likely to improve
goodness of fit
- A4 is more likely to hold in multivariate
Omitted variable bias For some reason a relevant explanatory variable is not included
A4 in the model. Creates biased other explanatory variables. We
get this model:
measures correlation between x1 and x2. To find;
regress x1 on x2 coef is correlation.
Inclusion of irrelevant Doesn’t generate biased OLS estimates, but increased standard You should use adjusted r-squared
variables errors of the other coefficients, thus lowers t-statistic. - Imposes penalty for adding
A5 Due to feature of r squared: extra variables by correcting
- Never decreases, usually increases when another for the DoF.
explanatory variable is added to the model (even if it
is irrelevant)
Multicollinearity Two or more relevant explanatory variables are highly - There is no clear way to solve it as omitting a relevant
A5 correlated. Doesn’t generate biased OLS estimates, but viriable will lead to a bias.
increases standard errors of the correlated variables. - In some cases justifiable to omit weakly significant
variable, if it is highly correlated
, Event studies
What? What is it? How to test it? How to solve it?
Checks for data 1. Negative values for variables that can
only be positive
2. Outliers
3. Missing values
4. Zeros
5. Stock splits
6. Make simple graphs
7. Use sum, d
Event study - We need a reasonable criteria for
estimation period
- Reasonable choice for benchmark
model
- Estimation window preferably
before evaluation period because
event may have lasting impact.
Abnormal returns With CAPM as benchmark model: Significance of abnormal returns: 𝐶𝐴𝐴𝑅
𝑇𝑆 = √𝑁 ~𝑁(0,1)
1. Overage over time for each firm √(𝑁 − 1)−1 ∑(𝐶𝐴𝑅 − 𝐶𝐴𝐴𝑅)2
Is the same as TS2
2. Overage of firms
Test statistic 1 Focus on one dag, disregard time dimension
One period Assumptions:
TS1 = (average abnormal return * √𝑛 )/ Std
Test statistic 2 - Similar to TS1, but without time 1.
Longer event windows index t and with CAR instead of AR
1. Compute CAR for every firm
2. Apply single-period technique on 2.
time series sums (CAR)
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
,Heteroscedasticity For any level of x, the variance of u is not the same or finite. Breusch-Pagan test: Using residuals to estimate the error
Thus u depends on x. 1. Estimate variance as , we can obtain
heteroscedasticity-robust standard
Does not create bias in the beta’s. 2. Obtain residuals, square them, Estimate: errors.
3. Use F-test to test H0 of homoscedasticity
Why not always use robust se’s?
If we reject H0 of homoscedasticity, we need to use
- Small sample size, robust t-
robust standard errors
statistic can have distr. not
very close to t distr.
Reverse causality Causality runs from Y to X
if not ruled out OLS results can show only correlation, but
not causality
Multivariate regression Why it is better than bivariate:
- Almost any variable is likely to be determined by
multiple other variables
- Multiple explanatory variables are likely to improve
goodness of fit
- A4 is more likely to hold in multivariate
Omitted variable bias For some reason a relevant explanatory variable is not included
A4 in the model. Creates biased other explanatory variables. We
get this model:
measures correlation between x1 and x2. To find;
regress x1 on x2 coef is correlation.
Inclusion of irrelevant Doesn’t generate biased OLS estimates, but increased standard You should use adjusted r-squared
variables errors of the other coefficients, thus lowers t-statistic. - Imposes penalty for adding
A5 Due to feature of r squared: extra variables by correcting
- Never decreases, usually increases when another for the DoF.
explanatory variable is added to the model (even if it
is irrelevant)
Multicollinearity Two or more relevant explanatory variables are highly - There is no clear way to solve it as omitting a relevant
A5 correlated. Doesn’t generate biased OLS estimates, but viriable will lead to a bias.
increases standard errors of the correlated variables. - In some cases justifiable to omit weakly significant
variable, if it is highly correlated
, Event studies
What? What is it? How to test it? How to solve it?
Checks for data 1. Negative values for variables that can
only be positive
2. Outliers
3. Missing values
4. Zeros
5. Stock splits
6. Make simple graphs
7. Use sum, d
Event study - We need a reasonable criteria for
estimation period
- Reasonable choice for benchmark
model
- Estimation window preferably
before evaluation period because
event may have lasting impact.
Abnormal returns With CAPM as benchmark model: Significance of abnormal returns: 𝐶𝐴𝐴𝑅
𝑇𝑆 = √𝑁 ~𝑁(0,1)
1. Overage over time for each firm √(𝑁 − 1)−1 ∑(𝐶𝐴𝑅 − 𝐶𝐴𝐴𝑅)2
Is the same as TS2
2. Overage of firms
Test statistic 1 Focus on one dag, disregard time dimension
One period Assumptions:
TS1 = (average abnormal return * √𝑛 )/ Std
Test statistic 2 - Similar to TS1, but without time 1.
Longer event windows index t and with CAR instead of AR
1. Compute CAR for every firm
2. Apply single-period technique on 2.
time series sums (CAR)
