Econometrics Cheat Sheet, Cheat Sheet of Introduction to Econometrics

Download Econometrics Cheat Sheet and more Cheat Sheet Introduction to Econometrics in PDF only on Docsity! Applied Econometrics/Econometrics Mid-term Exam - Reference Sheet Simple linear regression (SLR) model Population model: 𝑦 = 𝛽0 + 𝛽1𝑥 + 𝑢 Population model for a particular observation 𝒊 from a random sample with 𝒏 observations: 𝑦𝑖 = 𝛽0 + 𝛽1𝑥𝑖 + 𝑢𝑖, 𝑖 = 1, 2, . . , 𝑛 The dependent variable expressed in terms of estimates ?̂?′𝒔 and ?̂? for observation 𝒊: 𝑦𝑖 = ?̂?0 + ?̂?1𝑥𝑖 + ?̂?𝑖 Regression residuals: ?̂?𝑖 = 𝑦𝑖 − ?̂?𝑖 The Ordinary Least Squares (OLS) Estimators for SLR model: ?̂?1 = ∑ (𝑦𝑖−?̅?)(𝑥𝑖−?̅?) 𝑛 𝑖=1 ∑ (𝑥𝑖−?̅?) 2𝑛 𝑖=1 , ?̂?0 = ?̅? − ?̂?1?̅? Total Sum of Squares Explained Sum of Squares Residual Sum of Squares SST ≡ ∑ (𝑦𝑖 − ?̅?) 2𝑛 𝑖=1 SSE ≡ ∑ (?̂?𝑖 − ?̅?) 2𝑛 𝑖=1 SSR ≡ ∑ ?̂?𝑖 2𝑛 𝑖=1 SST = SSE + SSR R-squared of the regression: 𝑅2 ≡ SSE SST = 1 − SSR SST Assumptions for the SLR model Assumption SLR.1 (Linear in parameters): In the population, the relationship between 𝑦 and 𝑥 is linear. 𝑦 = 𝛽0 + 𝛽1𝑥 + 𝑢 Assumption SLR.2 (Random sampling): In a random sample with 𝑛 observations {(𝑥𝑖, 𝑦𝑖): 𝑖 = 1, 2, … , 𝑛}, each data point follows the population equation. 𝑦𝑖 = 𝛽0 + 𝛽1𝑥𝑖 + 𝑢𝑖 Assumption SLR.3 (Sample variation in explanatory variable): The values of 𝑥𝑖 are not exactly the same. ∑ (𝑥𝑖 − ?̅?) 2 > 0𝑛𝑖=1 Assumption SLR.4 (Zero conditional mean): E(𝑢𝑖|𝑥𝑖) = 0 Assumption SLR.5 (Homoskedasticity): Var(𝑢|𝑥) = 𝜎2 The unbiased estimator of 𝝈𝟐 for SLR model: ?̂?2 = ∑ 𝑢𝑖 2𝑛 𝑖=1 𝑛−2 = SSR 𝑛−2 Interpretation of 𝜷𝟏 for the functional forms involving logarithms: Model y x Interpretation of 𝜷𝟏 Model y x Interpretation of 𝜷𝟏 Level-level 𝑦 𝑥 ∆𝑦 ∆𝑥 = 𝛽1 Log-level log 𝑦 𝑥 %∆𝑦 ∆𝑥 = (100𝛽1) Level-log 𝑦 log 𝑥 ∆𝑦 %∆𝑥 = ( 𝛽1 100 ) Log-log log 𝑦 log 𝑥 %∆𝑦 %∆𝑥 = 𝛽1 Multiple linear regression (MLR) Model Population model: 𝑦 = 𝛽0 + 𝛽1𝑥1 + 𝛽2𝑥2 + ⋯ + 𝛽𝑘𝑥𝑘 + 𝑢 R-squared of the regression: 𝑅2 ≡ SSE SST = 1 − SSR SST = (∑ (𝑦𝑖−?̅?)(?̂?𝑖−?̅̂?)) 𝑛 𝑖=1 2 (∑ (𝑦𝑖−?̅?) 2)𝑛𝑖=1 (∑ (?̂?𝑖−?̅̂?) 2 )𝑛𝑖=1