noam
Uploaded by
12 SLIDES
300 VUES
120LIKES

Understanding OLS: Theoretical Foundations, Assumptions, and Applications in Econometrics

DESCRIPTION

This comprehensive review delves into Ordinary Least Squares (OLS) by exploring its theoretical underpinnings, key assumptions, and practical applications in econometrics. Key assumptions such as linearity, exogeneity, no multicollinearity, homoscedasticity, and the implications of non-spherical disturbances are discussed. The review further elaborates on the least squares estimator, variance-covariance matrices, and asymptotic normality, providing a clear framework for understanding OLS. Practical examples, including an exploration of the U.S. gasoline market, illustrate real-world applications of these concepts.

1 / 12

Télécharger la présentation

Understanding OLS: Theoretical Foundations, Assumptions, and Applications in Econometrics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

Playing audio...

  1. OLS: Theoretical Review • Assumption 1: Linearity • y = x′b + e(or y = Xb + e) (where x, b: Kx1) • Assumption 2: Exogeneity • E(|X) = 0 • E(x) = limn i xiei/n = limn X′/n = 0 • Var(x) = E(xx′2) = limn i xixi′ei2/n

  2. OLS: Theoretical Review • Assumption 3: No Multicolinearity • rank(X) = K • E(xx') = limnixixi′/n = limnX′X/n • plim X’X/n exists and nonsingular

  3. OLS: Theoretical Review • Non-Spherical Disturbances: General Heteroschedasticity • Var(e) =  • Var(xe) = E(xx′e2) = limnX′X/n • X′e/n dN(0,X′X/n)

  4. OLS: Theoretical Review • Assumption 4: Homoschedasticity andNo Autocorrelation • Var(e) = s2I • Var(xe) = E(xx′e2) = limns2X′X/n • X′e/n dN(0, s2X′X/n)

  5. OLS: Theoretical Review • Least Squares Estimator • b = (X'X)-1X'y • Variance-Covariance Matrix of b • General Heteroscedasticity:Var(b) = (X'X)-1X′X(X'X)-1 • Homoschedasticity: Var(b) = 2(X'X)-1

  6. OLS: Theoretical Review • Least Squares Estimator • b = (X'X/n)-1(X'y/n) • b =  + (X'X/n)-1(X'/n), bp • n(b - ) = (X'X/n)-1(X'/n) • n(b - ) dN(0,A-1BA-1)A = E(xx′) = limnX'X/nB = E(xx′e2) = limnX'X/n • n(b - ) dN(0, s2A-1) under homoschedasticity

  7. OLS: Theoretical Review • Asymptotic Normality • b ~aN(, (X'X)-1X′X(X'X)-1) • b ~aN(, s2(X'X)-1) under homoschedasticity • The unknown  or s2 needs to be consistently estimated.

  8. OLS: Theoretical Review • Estimate of Asymptotic Var(b) • Under Homoschedasticity

  9. OLS: Theoretical Review • Estimate of Asymptotic Var(b) • White Estimator (Heteroschedasticity-Consistent Estmate of Asymptotic Covariance Matrix)

  10. OLS: Theoretical Review • GLS (Generalized Least Squares) • If  is known and nonsingular, then -1/2 y = -1/2 X + -1/2  or y* = X* + * • E(*|X*) = 0, E(**'|X*) = I • bGLS = (X'-1X)-1X'-1y • Var(bGLS) = (X'-1X)-1 • bGLS ~aN(, (X'-1X)-1)

  11. Example • U. S. Gasoline Market, 1953-2004 • EXPG = Total U.S. gasoline expenditure • PG = Price index for gasoline • Y = Per capita disposable income • Pnc = Price index for new cars • Puc = Price index for used cars • Ppt = Price index for public transportation • Pd = Aggregate price index for consumer durables • Pn = Aggregate price index for consumer nondurables • Ps = Aggregate price index for consumer services • Pop = U.S. total population in thousands

  12. Example • y = Xb + e • y = G; X = [1 PG Y]where G = (EXPG/PG)/POP • y = ln(G); X = [1 ln(PG) ln(Y)] • Elasticity Interpretation

More Related