Introductory Econometrics – Formula Sheet for the Final Exam

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 = β 0 + β 1 X 1 + . . . + β k X k + U (model)
MLR.2: The observed data {(Y i , X 1i , . . . , X ki ), 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|X 1 , . . . , X k ) = 0
MLR.5: (Homoskedasticity) V ar(U|X 1 , . . . , X k ) = V ar(U) = σ 2
MLR.6: (Normality) U | X 1 , . . . , X k ∼ N(0, σ 2 )
2. The OLS estimator and its algebraic properties
The OLS estimator solves
n∑
min (Y i − ˆβ 0 − ˆβ 1 X 1i − ˆβ 2 X 2i − . . . − ˆβ k X ki ) 2
ˆβ 0 ,..., ˆβ k
i=1
Solution: In the general case no explicit was solution given. If k = 1, i.e.
there is only one explanatory variable, then
ˆβ 1 =
∑ n
i=1 (X 1i − ¯X 1 )(Y i − Ȳ )
∑ n
i=1 (X 1i − ¯X 1 ) 2 and ˆβ0 = Ȳ − ˆβ 1 ¯X1 .
Predicted value for individual i in the sample: Ŷ i = ˆβ 0 + ˆβ 1 X 1i + . . . ˆβ k X ki .
Residual for individual i in the sample: Û i = Y i − Ŷi.
• ∑ n
i=1 Ûi = 0
• ∑ n
i=1 ÛiX ji , j = 1, . . . , k
• estimated regression line goes through ( ¯X 1 , . . . , ¯X k , Ȳ )
3. Properties of the log function:
100∆ log(x) ≈ 100 ∆x
x
= %∆x
1

Predicting Y when the dependent variable is log(Y ):
4. SST, SSE, SSR, R 2
SST = ∑ n
i=1 (Y i − Ȳ )2
SSE = ∑ n
i=1 (Ŷi − Ȳ )2
SSR = ∑ n
i=1 Û 2 i
SST = SSE + SSR
R 2 = SSE/SST = 1 − SSR/SST
Ŷ adjusted = e σ2 /2 e ˆβ 0 + ˆβ 1 X 1 +... ˆβ k X k
5. Variance of the OLS estimator, estimated standard error, etc.
For j = 1, . . . , k:
V ar( ˆβ j ) =
σ 2
SST Xj (1 − R 2 j ),
where SST Xj = ∑ n
i=1 (X ji− ¯X j ) 2 and Rj 2 is the R-squared from the regression
of X j on all the other X variables. The estimated standard error of the ˆβ j ,
j = 1, . . . , k, is given by
√
se(
̂ ˆβ ˆσ
j ) =
2
SST Xj (1 − Rj 2),
where ˆσ 2 =
SSR . n−k−1
6. Simple omitted variables formula:
True model: Y = β 0 + β 1 X 1 + β 2 X 2 + U
Estimated model: Y = β 0 + β 1 X 1 + V
E( ˆβ 1 ) = β 1 + β 2
cov(X 1 , X 2 )
var(X 1 )
2

7. Hypothesis testing
a. t-statistic for testing H 0 : β j = a
ˆβ j − a
ŝe( ˆβ j ) ∼ t(n − k − 1) if H 0 is true
b. The F-statistic for testing hypotheses involving more than one regression
coefficient (“ur” = unrestricted, “r” = restricted, q = # of restrictions
in H 0 , k = # of slope coefficients in the unrestricted regression)
8. Heteroskedasticity
(SSR r − SSR ur )/q
SSR ur /(n − k − 1) ∼ F (q, n − k − 1) if H 0 is true
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(X 1 , . . . , X k ) = V ar(U|X 1 , . . . , X k ). 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|X 1 , . . . , X k ) = (β 0 + β 1 X 1 + . . . β k X k )(1 − (β 0 + β 1 X 1 + . . . β k X k )).
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