Introductory Econometrics - Cheat Sheet for the Final Exam You are allowed to use the following cheat sheet for the final exam. 1. Propert Properties ies of the log functio function: n:
100∆ 100∆ log(x log(x) ≈ 100
∆x = %∆x %∆x x
2. Multiple Multiple linear regression regression assumptions: 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 , . . . , 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 collinearity among the explanatory variables. MLR.4: (Zero conditional mean) E (U |X 1 , . . . , Xk ) = 0 for any possible value of X 1 , . . . , Xk . MLR.5: (Homoskedasticity) V ar( ar(U |X 1 , . . . , Xk ) = V ar( ar(U ) = σ 2 for any possible value of X 1 , . . . , Xk (0, σ2 ) for any possible value of X 1 , . . . , Xk MLR.6: (Normality) U | X 1 , . . . , Xk ∼ N (0, 3. The OLS estimator estimator and its algebraic algebraic properties properties
The OLS estimator solves n
min
ˆ ,..., β ˆ β 0 k
2 ˆ0 − β ˆ1 X 1i − β ˆ2 X 2i − . . . − β ˆk X ki (Y i − β ki )
i=1
Solution: Solution: In the general case no explicit explicit was solution solution given. given. If k = 1, i.e. there is only one explanatory variable, then ˆ1 = β
n i=1
¯ 1 )(Y ¯) (X 1i − X )(Y i − Y ) Y n ¯ 2 i=1 (X 1i − X 1 )
ˆ0 = Y ˆ1 X ¯ − β ¯1. and β
ˆ0 + β ˆ1 X 1i + . . . β ˆk X ki ˆi = β Predicted value for individual i in the sample: Y ki . ˆ ˆ Residual for individual i in the sample: U i = Y i − Y i.
• •
n i=1
ˆi = 0 U
n i=1
ˆi X ji , j = 1, . . . , k U
¯ 1 , . . . , X ¯k , Y ) ¯) • estimated regression line goes through ( X Y 4. SST, SST, SSE, SSR, R2
1
¯ 2 SST = ni=1 (Y i − Y ) ˆi − Y ) ¯ 2 SSE = ni=1 (Y ˆi2 SSR = ni=1 U SST = SSE + SSR R2 = SSE/SST = 1 − SSR/SST
5. Variance of the OLS estimator, estimated standard error, etc.
For j = 1, . . . , k: ˆ j ) = V ar(β
σ2 , SS T Xj (1 − R j2 )
¯ j )2 and R j2 is the R-squared from the regression of X j on all where SST Xj = ni=1 (X ji − X ˆ j , j = 1, . . . , k, is given by the other X variables. The estimated standard error of the β
ˆ j ) = se(β
where σ ˆ2 =
SSR n−k−1
σˆ 2 , SST Xj (1 − R j2 )
.
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 ˆ1 ) = β 1 + β 2 E (β
cov(X 1 , X 2 ) var(X 1 )
7. Hypothesis testing
1. Basic notions – Type 1 error=rejecting a true null hypothesis – Type 2 error=accepting a false null hypothesis – Significance level=probability of a Type 1 error
2. t-statistic for testing H 0 : β j = a ˆ j − a β ∼ t(n − k − 1) if H 0 is true ˆ j ) ˆ β se(
2
3. 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) (SSRr − SS Rur )/q ∼ F (q, n − k − 1) if H 0 is true SSRur /(n − k − 1) or, under certain conditions, 2 (Rur − Rr2 )/q ∼ F (q, n − k − 1) if H 0 is true 2 (1 − Rur )/(n − k − 1)
8. Heteroskedasticity
Testing for heteroskedasticity: Regress squared residuals on all explanatory variables; test for joint significance of the explanatory variables. Correcting for heteroskedasticty: Use heteroskedasticity-robust standard errors; correct Ftest formula. OR: Use GLS 9. Regression with time series data TS. 1: Y t = β 0 + β 1 X 1t + β 2 X 2t + . . . + β k X kt + U t TS. 2: Same as MLR 3.
In the assumptions below let the matrix X contain the values of all explanatory variables in all time periods (past, present and future). TS. TS. TS. TS.
3: 4: 5: 6:
(Zero conditional mean) E [U t | X] = 0, t = 1, 2, . . . , n. (No heteroskedasticity) V ar(U t | X) = V ar(U t ) = σ2 , t = 1, 2, . . . , n. (No autocorrelation) Corr(U t , U s ) = 0 for t = s. 2 (Normality) U t | X ∼ N (0, σ ).
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