Econometrics — Questions 1 & 2
Master Formula Sheet
Question 1 — Core OLS + (Often) OVB + IV/TSLS
1) Model + meaning
yi = β0 + β1 xi + εi . (1)
• yi : outcome (dependent variable)
• xi : regressor (explanatory variable)
• β0 : intercept, β1 : slope/effect
• εi : error/unobservables
A) OLS assumptions + unbiasedness tools
a) OLS assumptions (for unbiasedness; add A5 for BLUE)
• A1 (Linearity in parameters): model is linear in β.
• A2 (Random sampling / iid): (yi , xi ) are iid.
• A3 (No perfect multicollinearity): in simple regression,
Var(x) > 0. (2)
• A4 (Zero conditional mean / exogeneity):
E(εi | xi ) = 0. (3)
This is the key one: rules out correlation between x and unobservables.
• A5 (Homoskedasticity):
Var(εi | xi ) = σ 2 . (4)
BLUE = Best Linear Unbiased Estimator (smallest variance among linear unbiased estimators).
Unbiasedness: A1–A4.
Consistency: A1–A4 + large-sample/LLN conditions.
b) Unbiasedness target
Unbiasedness means:
E(β̂1 ) = β1 . (5)
Unbiasedness example (sample variance with divisor n):
n n
1X 1 X
" # " #
E(S̃ ) = E
2
(yi − ȳ)2 = E y 2 − nȳ 2 .
n i=1 n i=1 i
n
1X
= E(yi2 ) − E(ȳ 2 ).
n i=1
1
, c) LIE (Law of Iterated Expectations)
E[Y ] = E[E(Y | X)] . (6)
It allows you to take expectations in steps: first condition on X, then take the unconditional
expectation. Here, E[·] denotes the population mean (long-run average); E(Y | X) is the mean of
Y holding X fixed.
E(β̂1 | X) = β1 ⇒ E(β̂1 ) = E[E(β̂1 | X)] = β1 ,
so conditional unbiasedness implies unconditional unbiasedness. In exams, if asked to “show
unconditional unbiasedness,” use LIE to move from E(β̂1 | X) to E(β̂1 ).
d) Bias (definition + example)
Bias(θ̂) = E[θ̂] − θ.
Unbiased if E[θ̂] = θ. Illustrative example (mean with divisor n − 1):
n
1 X n µ
E[Ỹ ] = E[Yi ] = µ ⇒ Bias(Ỹ ) = E[Ỹ ] − µ = .
n − 1 i=1 n−1 n−1
Takeaway: using n − 1 instead of n biases this estimator upward.
e) Variance (core rules)
Var(aX) = a2 Var(X).
n n
!
Var = Var(Xi ) if Xi independent.
X X
Xi
i=1 i=1
σ2
Var(Ȳ ) = if Yi i.i.d. with Var(Yi ) = σ 2 .
n
f) Covariance vs. correlation
Cov(X, Y )
Corr(X, Y ) = p .
Var(X)Var(Y )
Correlation is the standardized version of covariance, so it is unitless and lies in [−1, 1].
B) Deriving the OLS estimator (short step-by-step)
Goal: choose (β0 , β1 ) to minimize the sum of squared residuals:
N
S(β0 , β1 ) = (yi − β0 − β1 xi )2 . (7)
X
i=1
F.O.C. (first-order condition): set the partial derivatives of S with respect to β0 , β1 equal to
zero.
2
Master Formula Sheet
Question 1 — Core OLS + (Often) OVB + IV/TSLS
1) Model + meaning
yi = β0 + β1 xi + εi . (1)
• yi : outcome (dependent variable)
• xi : regressor (explanatory variable)
• β0 : intercept, β1 : slope/effect
• εi : error/unobservables
A) OLS assumptions + unbiasedness tools
a) OLS assumptions (for unbiasedness; add A5 for BLUE)
• A1 (Linearity in parameters): model is linear in β.
• A2 (Random sampling / iid): (yi , xi ) are iid.
• A3 (No perfect multicollinearity): in simple regression,
Var(x) > 0. (2)
• A4 (Zero conditional mean / exogeneity):
E(εi | xi ) = 0. (3)
This is the key one: rules out correlation between x and unobservables.
• A5 (Homoskedasticity):
Var(εi | xi ) = σ 2 . (4)
BLUE = Best Linear Unbiased Estimator (smallest variance among linear unbiased estimators).
Unbiasedness: A1–A4.
Consistency: A1–A4 + large-sample/LLN conditions.
b) Unbiasedness target
Unbiasedness means:
E(β̂1 ) = β1 . (5)
Unbiasedness example (sample variance with divisor n):
n n
1X 1 X
" # " #
E(S̃ ) = E
2
(yi − ȳ)2 = E y 2 − nȳ 2 .
n i=1 n i=1 i
n
1X
= E(yi2 ) − E(ȳ 2 ).
n i=1
1
, c) LIE (Law of Iterated Expectations)
E[Y ] = E[E(Y | X)] . (6)
It allows you to take expectations in steps: first condition on X, then take the unconditional
expectation. Here, E[·] denotes the population mean (long-run average); E(Y | X) is the mean of
Y holding X fixed.
E(β̂1 | X) = β1 ⇒ E(β̂1 ) = E[E(β̂1 | X)] = β1 ,
so conditional unbiasedness implies unconditional unbiasedness. In exams, if asked to “show
unconditional unbiasedness,” use LIE to move from E(β̂1 | X) to E(β̂1 ).
d) Bias (definition + example)
Bias(θ̂) = E[θ̂] − θ.
Unbiased if E[θ̂] = θ. Illustrative example (mean with divisor n − 1):
n
1 X n µ
E[Ỹ ] = E[Yi ] = µ ⇒ Bias(Ỹ ) = E[Ỹ ] − µ = .
n − 1 i=1 n−1 n−1
Takeaway: using n − 1 instead of n biases this estimator upward.
e) Variance (core rules)
Var(aX) = a2 Var(X).
n n
!
Var = Var(Xi ) if Xi independent.
X X
Xi
i=1 i=1
σ2
Var(Ȳ ) = if Yi i.i.d. with Var(Yi ) = σ 2 .
n
f) Covariance vs. correlation
Cov(X, Y )
Corr(X, Y ) = p .
Var(X)Var(Y )
Correlation is the standardized version of covariance, so it is unitless and lies in [−1, 1].
B) Deriving the OLS estimator (short step-by-step)
Goal: choose (β0 , β1 ) to minimize the sum of squared residuals:
N
S(β0 , β1 ) = (yi − β0 − β1 xi )2 . (7)
X
i=1
F.O.C. (first-order condition): set the partial derivatives of S with respect to β0 , β1 equal to
zero.
2
