Introductory Econometrics – Formula Sheet for the Final Exam
You are allowed to use the following formula sheet for the final exam.
1. Multiple linear regression assumptions:
MLR.1: Y = β0 + β1 X1 + . . . + βk Xk + U (model)
MLR.2: The observed data {(Yi , X1i , . . . , Xki ), i = 1, . . . , n} is a random
sample from the population
MLR.3: In the sample, none of the explanatory variables has constant values
and there is no perfect linear relationships among the explanatory variables.
MLR.4: (Zero conditional mean) E(U |X1 , . . . , Xk ) = 0
MLR.5: (Homoskedasticity) V ar(U |X1 , . . . , Xk ) = V ar(U ) = σ 2
MLR.6: (Normality) U | X1 , . . . , Xk ∼ N (0, σ 2 )
2. The OLS estimator and its algebraic properties
The OLS estimator solves
n
X
min
(Yi − β̂0 − β̂1 X1i − β̂2 X2i − . . . − β̂k Xki )2
β̂0 ,...,β̂k
i=1
Solution: In the general case no explicit was solution given. If k = 1, i.e.
there is only one explanatory variable, then
Pn
(X − X̄1 )(Yi − Ȳ )
Pn 1i
and β̂0 = Ȳ − β̂1 X̄1 .
β̂1 = i=1
2
i=1 (X1i − X̄1 )
Predicted value for individual i in the sample: Ŷi = β̂0 + β̂1 X1i + . . . β̂k Xki .
Residual for individual i in the sample: Ûi = Yi − Ŷi .
Pn
•
i=1 Ûi = 0
Pn
•
i=1 Ûi Xji , j = 1, . . . , k
• estimated regression line goes through (X̄1 , . . . , X̄k , Ȳ )
3. Properties of the log function:
100∆ log(x) ≈ 100
1
∆x
= %∆x
x
Predicting Y when the dependent variable is log(Y ):
Ŷadjusted = eσ
2 /2
eβ̂0 +β̂1 X1 +...β̂k Xk
4. SST, SSE, SSR, R2
P
SST = ni=1 (Yi − Ȳ )2
P
SSE = ni=1 (Ŷi − Ȳ )2
P
SSR = ni=1 Ûi2
SST = SSE + SSR
R2 = SSE/SST = 1 − SSR/SST
5. Variance of the OLS estimator, estimated standard error, etc.
For j = 1, . . . , k:
V ar(β̂j ) =
σ2
,
SSTXj (1 − Rj2 )
P
where SSTXj = ni=1 (Xji − X̄j )2 and Rj2 is the R-squared from the regression
of Xj on all the other X variables. The estimated standard error of the β̂j ,
j = 1, . . . , k, is given by
s
σ̂ 2
\
se(
β̂j ) =
,
SSTXj (1 − Rj2 )
where σ̂ 2 =
SSR
.
n−k−1
6. Simple omitted variables formula:
True model: Y = β0 + β1 X1 + β2 X2 + U
Estimated model: Y = β0 + β1 X1 + V
E(β̂1 ) = β1 + β2
2
cov(X1 , X2 )
var(X1 )
7. Hypothesis testing
a. t-statistic for testing H0 : βj = a
β̂j − a
se(
ˆ β̂j )
∼ t(n − k − 1) if H0 is true
b. The F-statistic for testing hypotheses involving more than one regression coefficient (“ur” = unrestricted, “r” = restricted, q = # of restrictions in H0 , k = # of slope coefficients in the unrestricted regression)
(SSRr − SSRur )/q
∼ F (q, n − k − 1) if H0 is true
SSRur /(n − k − 1)
8. Heteroskedasticity
Breusch-Pagan test for heteroskedasticity: Regress squared residuals on all
explanatory variables and constant; test for joint significance of the explanatory variables.
(F)GLS estimator: Let h(X1 , . . . , Xk ) = V ar(U |X1 , . . . , Xk ). Divide the
original model by the square root of h and do OLS. If h is unknown, one
needs to model and estimate it as well.
9. Linear probabiliy model
V ar(U |X1 , . . . , Xk ) = (β0 + β1 X1 + . . . βk Xk )(1 − (β0 + β1 X1 + . . . βk Xk )).
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