Equations to Know

Question 1

What is the general equation for the OLS estimator? Give this as an application to demand

Answer 1

• Yt = α + βXt + εt
• Demand => Qt = α - βP

Question 2

What is the equation for the statistical mean? What does this then allow you to do during proof?

Answer 2

• Mean (X̄) = ΣX / n
• This means that ΣX = nX̄

Question 3

What are the equations for statistical variance?

Answer 3

• Variance (σ²) = E(X-X̄)²
• Variance (σ²) = E(X²)-[E(X)]²

Question 4

What is the equation for statistical correlation? What happens if these are estimates?

Answer 4

• Correlation (ρxy) = Cov(x,y) / (σxσy)
• If ρxy^, all variables are with a hat

Question 5

What is the equation for Z-statistic?

Answer 5

• Z = (x - μ) / σ

Question 6

What are the equations for the Sum of Squared errors (SSE)?

Answer 6

• SSE(Σε²t) = Σ(Yt - α - βXt)²
• OR, SSE(Σε²t) = SST (1 - R²)

Question 7

What are the two values for α^ and β^ when deriving the OLS formula?

Answer 7

• α => Ȳ - β^X̄
• β => (1/T[ΣXtYt - X̄Ȳ]) / (1/T[ΣX²t - X̄²)]

Question 8

What is the expected value for β^ under an unbiased estimator like OLS?

Answer 8

• E (β^) = β

Question 9

What is the variance for β^ under an unbiased estimator like OLS? What happens if these are estimates?

Answer 9

• Var(β^) = σ² / ((ΣXt - x̄)²
• If they are all estimates, then Var^(β^) = σ^² / ((ΣXt - x̄)²

Question 10

What is the formula for the variance of εt?

Answer 10

• Var (εt) = Σε²t / T

Question 11

What is the formula for the eqaution of the estimated variance (σ^²)?

Answer 11

• σ^² = Σε²t / T - K

Question 12

What is the formula for Confidence interval level?

Answer 12

• C.I. = [ β^ - Tuσ^ , β^ + Tuσ^ ]

Question 13

What is the formula for the t-statistic calculation?

Answer 13

• T-statistic (t) = (β^ - βH0) / σ^(β)

Question 14

What is the formula for R²?

Answer 14

R² = SSR / SST or 1 - (SSE / SST)

Question 15

What is the formula for the estimated Covariance of (Xt,Yt)?

Answer 15

• Cov^ (Xt,Yt) = Σ(Xt - x̄)(ΣYt - ȳ) / T - 1

Question 16

What is the formula for the estimated variance of Xt?

Answer 16

• Var^(Xt) = Σ(Xt - x̄)² / T - 1

Question 17

What are the equations for the slope and elasticity of a regression?

Answer 17

• Slope: dYt / dXt
• Elasticity: dYt / dXt * Xt / Yt

Question 18

What is the formula for the Jarque-Bera statistic?

Answer 18

• Jarque-Bera Statistic (JB) = T/6 (S² + (K-3)² / 4)

Question 19

Provide the standard deviation, kurtosis and skeweness of the Jarque-Bera statistic

Answer 19

• JB Skewness => 1/T (Σε^t / σ^)³
• JB Kurtosis => 1/T (Σε^t / σ^)4
• JB Standard Deviation => σ^ = SQRT {Σε^t / T - 2}

Question 20

What is the formula for the F-statistic calculation?

Answer 20

• F-statistic (F) = [SSE(R)-SSE(UR) / J] / [SSE(UR) / T - K]

Question 21

What is the equation for adjusted R²?

Answer 21

• RBar² = 1 - ([SSE/T-K] / [SST/T-1])