Part 1. De Jong (OLS, eventstudies, abnormal returns, Gauss markov)
1. Explain the OLS estiates iethhd and hhw it can be calculated by linear algebra.
What is the hbjectie uncthn under chnsiderathnn
What kind h assuipthns dh yhu need th acchunt hr in hrder th calculate OLS estiatesn
Using the ordinary least squares approach a functon is determined which minimizes the diference between
the observed value and its value approximated by a functon.
The Beta of the OLS can be calculated by minimizing the Residual’s sums of squares RSS:
1. Take vertcal distances, Ui, between each point in the graph and potental fied line
2. Square the distances/residuals Ui and sums them
3. The objectve functon under consideraton is to minimize the sum of the squared residuals.
Miniiizathn prhcess:
^ = α^ + β^
We know that the fitted ivaluede of dependent variable is Y Xi
^ = α^ + β^ Xi+Ui
We know that the trde ivaluede of dependent variable is Y
^
This means that we can fnd residuals: Ui=Yi− ^
Yi
and minimize the following functon L with respect to α ∧β
N N N
L=∑ U
^i2=∑ (Yi−Yi)
^ 2= ∑ ¿ ¿
i =1 i=1 i=1
To minimize L take derivatve of L with respect to α ∧β , FOC leads to
(Y −Ý )( X− X́)
β= i i i i i i i i i i i i i i i i i i i α^ =Ý − ^β x́
¿¿
Only one assumpton is made (to be tested from the data directly):
b) What are the GM assumptons? What do they imply?
,Assumpton (A1) says that the expected value of the error term is zero, which means that on average the
regression line should be correct.
Assumpton (A2) says The matrix of regressor values X does not provide any informaton about the expected
values of the error terms or their (co)variances. They are independent.
Assumpton (A3) states that all error terms have the same variance, which is referred to as homoskedastcity.
Assumpton (A4) imposes zero correlaton between diferent error terms.
Crhss-secthn assuipthns hr fnite saiple prhpertes 2016/2017
A1 The populaton model is linear in parameters
We cannot use OLS if we have non-linear parameters
A2 We have a random sample from the populaton
We cannot trust OLS if we just select a sample CEO’s with the highest salaries
A3 We have sample variaton in the explanatory variable X
For a multvariate regression model: there are no exact linear relatonships among the explanatory variables.
This refers to no perfect collinearity.
If X varies in the populaton, X must also vary in our sample. X cannot have a variance of zero.
A4 The expected value of error u is zero E ( U|X )=0
This implies that there are no unobserved variables that are infuencing variables in the model.
For any level of X, the average Y is the same. On average the regression line should be correct.
A5 The variance of error u is constant and fnite Var ( U | X ¿=sigm a2
This refers to homoscedastcity
A6 Error u is independent from explanatory variables and is normally distributed U ~ N ( 0, sigm a2)
This refers to normality
Tiie series assuipthn hr large / infnite saiple prhpertes 2016/2017
A1 The populaton model is linear in parameters and Yt and Xt are statonary and weakly dependent
A2 We have sample variaton in our explanatory variables and there is no exact linear relatonship between
them. This refers to no perfect collinearity.
A3 The expected value of error Ut is zero given any value of explanatory variable at the same tme period
This refers to contemporaneous exogeneity: E ( Ut|Xt )=0
A4 The variance of error Ut is constant and fnite given any value of explanatory variables at the same tme
period. This refers to homoscedastcity Var ( Ut| Xt ¿=sigm a 2
A5 Errors in two diferent tme periods are uncorrelated with each other for any value of explanatory variables.
This refers to no serial correlaton or no autocorrelaton. Corr (Ut , Us|X )=0
A6 Populaton Error U is independent of X and is normally distributed U ~ N ( 0, sigm a2)
, c) Under the GM assuipthns, shhw that E(ß) is unbiased hr ß. Dh yhu need all the GM assuipthnsn
No we only need A4: E ( U|X )=0to show that is unbiased, E(β) = β.
E ( U|X )=0,The expectaton of the residual value is zero, so E(β) = β.
d) GM states that β is BLUE. What is ieant by this stateientn
The Gauss-Markov theorem states that under assumptons (A1) – (A4) the OLS estmator β is the best linear
unbiased estiathr (BLUE) for β.
The assumptons guarantee that the OLS estmators of α and β are the most accurate (linear) unbiased
estmators for y.
Best means that the OLS estmators have the smallest variance. There are no other estmators with a smaller
variance.
Linear means that the OLS estmators α and β are linear
Unbiased the estmated values of α and β are equal to the true values of α and β
Estiathrs means that α and β are estmators of the true values of α and β
e) (Stata regressihn is giien, tables h a nhrial regressihn (reg wage ….) and a ln regressihn (lnreg wage …)).
Stateient: “The ihdel with ln(wages) as the dependent iariable is beter because it has a higher R-
squared.” Chiient hn this stateient and explain.
1. Explain the OLS estiates iethhd and hhw it can be calculated by linear algebra.
What is the hbjectie uncthn under chnsiderathnn
What kind h assuipthns dh yhu need th acchunt hr in hrder th calculate OLS estiatesn
Using the ordinary least squares approach a functon is determined which minimizes the diference between
the observed value and its value approximated by a functon.
The Beta of the OLS can be calculated by minimizing the Residual’s sums of squares RSS:
1. Take vertcal distances, Ui, between each point in the graph and potental fied line
2. Square the distances/residuals Ui and sums them
3. The objectve functon under consideraton is to minimize the sum of the squared residuals.
Miniiizathn prhcess:
^ = α^ + β^
We know that the fitted ivaluede of dependent variable is Y Xi
^ = α^ + β^ Xi+Ui
We know that the trde ivaluede of dependent variable is Y
^
This means that we can fnd residuals: Ui=Yi− ^
Yi
and minimize the following functon L with respect to α ∧β
N N N
L=∑ U
^i2=∑ (Yi−Yi)
^ 2= ∑ ¿ ¿
i =1 i=1 i=1
To minimize L take derivatve of L with respect to α ∧β , FOC leads to
(Y −Ý )( X− X́)
β= i i i i i i i i i i i i i i i i i i i α^ =Ý − ^β x́
¿¿
Only one assumpton is made (to be tested from the data directly):
b) What are the GM assumptons? What do they imply?
,Assumpton (A1) says that the expected value of the error term is zero, which means that on average the
regression line should be correct.
Assumpton (A2) says The matrix of regressor values X does not provide any informaton about the expected
values of the error terms or their (co)variances. They are independent.
Assumpton (A3) states that all error terms have the same variance, which is referred to as homoskedastcity.
Assumpton (A4) imposes zero correlaton between diferent error terms.
Crhss-secthn assuipthns hr fnite saiple prhpertes 2016/2017
A1 The populaton model is linear in parameters
We cannot use OLS if we have non-linear parameters
A2 We have a random sample from the populaton
We cannot trust OLS if we just select a sample CEO’s with the highest salaries
A3 We have sample variaton in the explanatory variable X
For a multvariate regression model: there are no exact linear relatonships among the explanatory variables.
This refers to no perfect collinearity.
If X varies in the populaton, X must also vary in our sample. X cannot have a variance of zero.
A4 The expected value of error u is zero E ( U|X )=0
This implies that there are no unobserved variables that are infuencing variables in the model.
For any level of X, the average Y is the same. On average the regression line should be correct.
A5 The variance of error u is constant and fnite Var ( U | X ¿=sigm a2
This refers to homoscedastcity
A6 Error u is independent from explanatory variables and is normally distributed U ~ N ( 0, sigm a2)
This refers to normality
Tiie series assuipthn hr large / infnite saiple prhpertes 2016/2017
A1 The populaton model is linear in parameters and Yt and Xt are statonary and weakly dependent
A2 We have sample variaton in our explanatory variables and there is no exact linear relatonship between
them. This refers to no perfect collinearity.
A3 The expected value of error Ut is zero given any value of explanatory variable at the same tme period
This refers to contemporaneous exogeneity: E ( Ut|Xt )=0
A4 The variance of error Ut is constant and fnite given any value of explanatory variables at the same tme
period. This refers to homoscedastcity Var ( Ut| Xt ¿=sigm a 2
A5 Errors in two diferent tme periods are uncorrelated with each other for any value of explanatory variables.
This refers to no serial correlaton or no autocorrelaton. Corr (Ut , Us|X )=0
A6 Populaton Error U is independent of X and is normally distributed U ~ N ( 0, sigm a2)
, c) Under the GM assuipthns, shhw that E(ß) is unbiased hr ß. Dh yhu need all the GM assuipthnsn
No we only need A4: E ( U|X )=0to show that is unbiased, E(β) = β.
E ( U|X )=0,The expectaton of the residual value is zero, so E(β) = β.
d) GM states that β is BLUE. What is ieant by this stateientn
The Gauss-Markov theorem states that under assumptons (A1) – (A4) the OLS estmator β is the best linear
unbiased estiathr (BLUE) for β.
The assumptons guarantee that the OLS estmators of α and β are the most accurate (linear) unbiased
estmators for y.
Best means that the OLS estmators have the smallest variance. There are no other estmators with a smaller
variance.
Linear means that the OLS estmators α and β are linear
Unbiased the estmated values of α and β are equal to the true values of α and β
Estiathrs means that α and β are estmators of the true values of α and β
e) (Stata regressihn is giien, tables h a nhrial regressihn (reg wage ….) and a ln regressihn (lnreg wage …)).
Stateient: “The ihdel with ln(wages) as the dependent iariable is beter because it has a higher R-
squared.” Chiient hn this stateient and explain.
