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Classical Linear Regression Model Notation and Assumptions Model Estimation –Method of Moments –Least Squares –Partitioned Regression Model Interpretation

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Notations y: Dependent Variable (Regressand) –y i, i = 1,2,…n X: Explanatory Variables (Regressors) –x i ', i = 1,2,…,n –x ij, i = 1,2,…,n; j = 1,2,…,K

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Assumptions Assumption 1: Linearity y i = 1 x i1 + 2 x i2 +…+ K x iK + i (i = 1,2,…,n) Linearity in Vector Notation y i = x i ' + i (i = 1,2,…,n)

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Assumptions Linearity in Matrix Notation y = X +

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Assumptions Assumption 2: Exogeneity E( i |X) = E( i |x j1,x j2,…,x jK ) = 0 (i,j = 1,2,…,n) Implications of Exogeneity E( i ) = E(E( i |X)) = 0 E(x jk i ) = E(E(x jk i |X)) = E(x jk E( i |X)) = 0 Cov( i,x jk ) = E(x jk i ) - E(x jk )E( i ) = 0

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Assumptions Assumption 3: No Multicolinearity rank(X) = K, with probability 1. Assumption 4: Spherical Error Variance E( i 2 |X) = 2 > 0, i = 1,2,…,n E( i j |X) = 0, i,j = 1,2,…,n; i j In matrix notation: Var( |X) = E( '|X) = 2 I n

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Assumptions Implication of Spherical Error Variance (Homoscedasticity and No Autocorrelation) –Var( ) = E(Var( |X))+Var(E( |X)) = E(Var( |X)) = 2 I n –Var(X' |X) = E(X' 'X|X) = 2 E(X'X) Note: E(X' |X) = 0 followed from Exogeneity Assumption

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Discussions Exogeneity in Time Series Models Random Samples The sample (y,X) is a random sample if {y i,x i } is i.i.d (independently and identically distributed). Fixed Regressors X is fixed or deterministic.

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Discussions Nonlinearity in Variables g(y i ) = 1 f 1 (x i1 )+ 2 f 2 (x i2 )+…+ K f K (x iK )+ i (i = 1,2,…,n) –Linearity in Parameters and Model Errors

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Method-of-Moments Estimator From the implication of strict exogeneity assumption, E(x jk i ) = 0 for i = 1,2,…,n and k = 1,2,…,K. That is, Moment Conditions: E(X' ) = 0 X'(y-X ) = 0 or X'y=X'X The method-of-moments estimator of : b = (X'X) -1 X'y

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Least Squares Estimator OLS: b = argmin SSR( ) SSR( ) = ' = (y-X )'(y-X ) = i=1,2…,n (y i -x i ' ) 2 b = (X'X) -1 X'y

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The Algebra of Least Squares Normal Equations: X'Xb = X'y b = (X'X) -1 X’y

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The Algebra of Least Squares b = (X'X) -1 X'y = + (X'X) -1 X' b = (X'X/n) -1 (X'y/n) = S xx -1 s xy S xx is the sample average of x i x i ', and s xy is the sample average of x i y i. OLS Residuals: e = y – Xb X'e = 0 s 2 = SSR/(n-K) = e'e/(n-K)

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The Algebra of Least Squares Projection Matrix P and “Residual Maker” M: –P = X(X'X) -1 X', M = I n - P –PX = X, MX = 0 –Py = Xb, My = y-Py = e, M = e SSR = e'e = 'M h = Vector of diagonal elements in P can be used to check for the influential observation: h i > K/n, with 0 h i 1.

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The Algebra of Least Squares Fitted Value: Uncentered R 2 : R uc 2 = 1 – e'e/y'y = Centered R 2 : R 2 =

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Analysis of Variance TSS = ESS = RSS = e'e (or SSR) TSS = ESS + RSS Degrees of Freedom –TSS: n-1 –ESS: K-1 –RSS: n-K

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Analysis of Variance R 2 Adjusted R 2

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Discussions Examples –Constant or Mean Regression y i = + i (i=1,2,…,n) –Simple Regression y i = + x i + i (i=1,2,…,n)

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Partitioned Regression y = X + X1 1 + X2 2 + X = X1 X2], 1' 2' ' Normal Equations X'Xb = X'y, or

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Partitioned Regression b1 = (X1'X1) -1 X1'y - (X1'X1) -1 (X1'X2)b2 = (X1'X1) -1 X1'(y-X2b2) b2 = (X2'M1X2) -1 X2'M1y = (X2 * 'X2 * ) -1 X2 * 'y * M1 = [I-X1(X1'X1) -1 X1'] X2 * = M1X2 (matrix of residuals obtained from each column of X2 regressed on X1) y * = M1y (residuals of y regressed on X1) b2 is LS estimator of y * on X2 *

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Partitioned Regression Frisch-Waugh Theorem If y = X1 1 + X2 2 + , then b2 is the LS estimator for the regression of residuals from a regression of y on X1 alone are regressed on the set of residuals obtained when each column of X2 is regressed on X1. If X1 and X2 are independent, X1'X2 = 0. b1= (X1'X1) -1 X1'y b2 = (X2'X2) -1 X2'y

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Partitioned Regression Applications of Frisch-Waugh Theorem –Partial Regression Coefficient –Trend Regression y t = + t + t

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Model Interpretation Marginal Effects (Ceteris Paribus Interpretation): y = Xb + e, where Elasticity Interpretation

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Example U. S. Gasoline Market, –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

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

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Example (Continued) y = X1 + X2 + y = ln(G); X1 = [1 YEAR]; X2 = [ln(PG) ln(Y)] where G = (EXPG/PG)/POP Frisch-Waugh Theorem

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