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* Distribution of the ols beta estimate * Use of the CLRM Assumptions * Independence of bols and e [or f(e) = s2=e'e/n-k] * Confidence Interval of β * Oaxaca's Decomposition * Significance of the regression * Marginal distribution of t|X

* Estimating the variance of the ols beta estimate * s2 = e'e / n-k E(s2/X) = σ2 * var(bols/X) = σ2 (X'X)-1 var(bols/X) = [e'e/n-k] (X'X)-1 var(bols/X) = s2 (X'X)-1 * Assumed Homoscedasticity E(εε'/X) = σ2 I

* Gauss Markov Theorem * For constant vector w, in the CLRM, the MVLUE of w'β is w'b, where b = ols estimate * Whether X is stochastic or nonstochastic, the ols beta is the MVLUE of population beta in the CLRM.

* (Un)Conditional mean and variance of the ols beta estimate * population model: y = Xβ ε sample model: y = Xb e * E(bols/X) = β E(bols) = β * var(bols/X) = σ2(X'X)-1 var(bols) = σ2 Ex[(X'X)-1] Assumed Homoscedasticity E(εε'/X) = σ2 I

* Coefficient of Determination (R2), when multiple variables are added to the model * y = X1b1 e1 ; y = X1b1 X2b2 e1.2 e1.2'e1.2 = e1'e1 - b2'X2'M1X2b2 R21.2 = R21 ( 1- R21 ) r*2y2.1 * Amemiya's prediction criterion for comparing different models

* Coefficient of Determination (R2), adjusted R2, and R2 when a variable is added * R2 = SSR/SST = 1 - (SSE/SST) R2 = b'X'M0Xb / y'M0y R2 = 1 - (e'e / y'M0y) * Adj. R2 = 1-[(n-1)/(n-k)](1-R2) * R2Xz = R2X ( 1- R2X ) r*2yz

* Distribution of variation in the dependent variable across the independent variables and the residual term. [ANOVA] * y'M0y = b'X'M0Xb e'e (n-1) (k-1) (n-k) (SST) = (SSR) (SSE)

* Change in the sum of the square of errors when a variable is added *u'u = e'e - c2 z*'z* = e'e - r*2yz y*'y* = e'e - r*2yz e*'e* = e'e (1 - r*2yz ) = e'e- [(z*'y*)2/z*'z*]

* Relation between partial correlation coefficient, beta, and t-score. * r*2yz = t2z / [t2z (n-k-1) ] = c2 z*'z* / [c2 z*'z* u'u ] = r*2yz / [r*2yz (u'u/y*'y*) ] = (z*'y*)2/ [(z*'y*)2 (u'u * z*'z*]

* Correlation Coefficient ryz = cov(y,z) /√var(y) √var(z) * Partial Correlation Coefficient r*yz = cov(ey.x,ez.x) / √var(ey.x) √var(ez.x) * Semi-Partial Correlation Coeff semi r*yz = cov(y,ez.x) / √var(y) √var(ez.x)

* Inverse of a 2x2 partitioned Matrix A= [A11 , A12 \ A21 , A22] * F1 = [A11 - A12A22-1A21]-1 A-1 = [F1 , -F1A12A22-1 \ -A22-1A21F1 , A22-1(I A21F1A12A22-1)] * F2 = [A22 - A21A11-1A12]-1 A-1 = [A11-1(I A12F2A21A11-1) , -A11-1A12F2 \ -F2 A21 A11-1 , F2 ]

* Deviation from mean maker Matrix (M0) = [I - (ii'/i'i)] * Constant term in Regression X = [i X] b' = [b0 b'] y = ib0 Xb e b =(X'MiX)-1 (X'Miy) Mi = I - i(i'i)-1i' = M0 MiX = deviation from the mean of X Miy = deviation from the mean of y

* Partitioned Regression y = X1b1 X2b2 e * b1 = (X1'M2X1)-1 X1'M2y ; b2 = (X2'M1X2)-1 X2'M1y * Frisch-Waugh Theorem * Individual Regression Coefficients c = (z*'z*)-1 (z*'y*) z*=Mz ; y* = My

* Fitted values (Xb) or y^ is the projection of y on col space of X * P = X(X'X)-1X' ; M=I-P Py = Xb^= y^; PX= X; Pe=0 My=e; MX=0; Me=e; PM=MP=0 * e'e = y'y - y^'y^ = e'y = y'e y^'y^ = y^'y = y'y^ * P and M are Idempotent Matrix * y = Py My = Xb e

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