Assumptions

Question 1

What are some assumptions/rules for SIMPLE linear regression?

Answer 1

• SR1: Yt = α + βXt + εt
• SR2: E(εt) = 0
• SR3: Var(εt) = σ²
• SR4: Cov (εt, εt+1) = 0
• SR5: Xt is given, and must take at least 2 values
• SR6: εt ~ N (0, σ²)

Question 2

What are some assumptions/rules for MULTIPLE linear regression?

Answer 2

• MR1: Yt = α +ΣβiXit + εt
• MR2: E(εt) = 0
• MR3: Var(εt) = σ²
• MR4: Cov (εi, εj) = 0
• MR5: Xs are given, and are linearly independant
• MR6: εt ~ N (0, σ²)

Question 3

What are some assumptions/rules for Summation?

Answer 3

• S1: ΣXi= X1+X2+…..+Xn, where i is the number of observation [start] and n is the last observations
• S2: If ΣaXi, then this is aΣXi + na
• S3: If Σ(Xi + Yi), then = ΣXi + ΣYi
• S4: If ΣΣ(Xi+Yi), do the first Σ [inside], BUT ΣΣ are interchangeable
• S5: ΣXi = Tx̄

Question 4

What are some assumptions/rules for Expected Values?

Answer 4

• E1: Expectation of Sum = Sum of expectation [E(X+Y) = E(X) + E(Y)]
• E2: Expectation of a constant is a constant [E(C) = C]
• E3: Constant can be taken common [E(cX) = cE(X)]
• E4: Expectations of expected values do not change [E(E(X)) = E(X)]
• E5: E(XY) = E(X)E(Y) + Cov(X,Y)

Question 5

What are some assumptions/rules for Variance?

Answer 5

• V1: Variance of a Constant is 0
• V2: Variance of constant can get taken out [Var(cX) = c²(Var(X))]
• V3: Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y)
• V4: Var(X-Y) = Var(X) + Var(Y) - 2Cov(X,Y)

Question 6

What are some Probability rules?

Answer 6

• JOINT: Probability with multiple variables; f(X,Y) = P(A∩B)
• MARGINAL: Finding outcome for one variable with joint probability; f(X) = Σf(X,Y) [only Y]
• CONDITIONAL: Outcome given a differnet outcome; F(A|B) = P(A∩B) / P(B)

Question 7

How can you calculate Covariance?

Answer 7

• Cov(X,Y) = E(X-x̄)(Y-ȳ)
• Cov(X,Y) = ρX,Y *σXσY

Question 8

What happens if variables are log-linear [mathematical intuition]?

Answer 8

• If relationships are not log-linear (i.e. y=ax), increasing x by one unit increases y by a units
• If relationships are log-linear (i.e. y=alnx), increasing x by 1% will increase y by a units