Equations to Know Flashcards
What is the general equation for the OLS estimator? Give this as an application to demand
- Yt = α + βXt + εt
- Demand => Qt = α - βP
What is the equation for the statistical mean? What does this then allow you to do during proof?
- Mean (X̄) = ΣX / n
- This means that ΣX = nX̄
What are the equations for statistical variance?
- Variance (σ²) = E(X-X̄)²
- Variance (σ²) = E(X²)-[E(X)]²
What is the equation for statistical correlation? What happens if these are estimates?
- Correlation (ρxy) = Cov(x,y) / (σxσy)
- If ρxy^, all variables are with a hat
What is the equation for Z-statistic?
- Z = (x - μ) / σ
What are the equations for the Sum of Squared errors (SSE)?
- SSE(Σε²t) = Σ(Yt - α - βXt)²
- OR, SSE(Σε²t) = SST (1 - R²)
What are the two values for α^ and β^ when deriving the OLS formula?
- α => Ȳ - β^X̄
- β => (1/T[ΣXtYt - X̄Ȳ]) / (1/T[ΣX²t - X̄²)]
What is the expected value for β^ under an unbiased estimator like OLS?
- E (β^) = β
What is the variance for β^ under an unbiased estimator like OLS? What happens if these are estimates?
- Var(β^) = σ² / ((ΣXt - x̄)²
- If they are all estimates, then Var^(β^) = σ^² / ((ΣXt - x̄)²
What is the formula for the variance of εt?
- Var (εt) = Σε²t / T
What is the formula for the eqaution of the estimated variance (σ^²)?
- σ^² = Σε²t / T - K
What is the formula for Confidence interval level?
- C.I. = [ β^ - Tuσ^ , β^ + Tuσ^ ]
What is the formula for the t-statistic calculation?
- T-statistic (t) = (β^ - βH0) / σ^(β)
What is the formula for R²?
R² = SSR / SST or 1 - (SSE / SST)
What is the formula for the estimated Covariance of (Xt,Yt)?
- Cov^ (Xt,Yt) = Σ(Xt - x̄)(ΣYt - ȳ) / T - 1
What is the formula for the estimated variance of Xt?
- Var^(Xt) = Σ(Xt - x̄)² / T - 1
What are the equations for the slope and elasticity of a regression?
- Slope: dYt / dXt
- Elasticity: dYt / dXt * Xt / Yt
What is the formula for the Jarque-Bera statistic?
- Jarque-Bera Statistic (JB) = T/6 (S² + (K-3)² / 4)
Provide the standard deviation, kurtosis and skeweness of the Jarque-Bera statistic
- 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}
What is the formula for the F-statistic calculation?
- F-statistic (F) = [SSE(R)-SSE(UR) / J] / [SSE(UR) / T - K]
What is the equation for adjusted R²?
- RBar² = 1 - ([SSE/T-K] / [SST/T-1])
