Question

(a) State the stochastic assumptions of Multiple Linenr Regression Analysis. (b)State and prove the Gauss-Markov theorem. 2a) Why canonical correlation? (b) In a study of poverty, crime and deterrence, certain summary crime statistics in various states for two periods 2013 and 2016 were reported. A portion of their sample correlation matrix is given below. The variables are: ,(1) (1) X2016 primary homicides 2016 nonprimary homicides 2013 severity of punishment, X=2013 certainty of punishment 1.0 0.62 -0.11 -0.27 0.62 1.0 -0.20 -0.09 R = -0.11 -0.20 1.0 0.27 0.27 -0.09 -0.27 1.0 Carry-out a canonical correlation analysis of the homicide punishment sets of variables and interpret your results.

Answer 1

1)

a)

Multiple linear regression analysis makes several key assumptions:

• There must be a linear relationship between the outcome variable and the independent variables. Scatterplots can show whether there is a linear or curvilinear relationship.
• Multivariate Normality–Multiple regression assumes that the residuals are normally distributed.
• No Multicollinearity—Multiple regression assumes that the independent variables are not highly correlated with each other. This assumption is tested using Variance Inflation Factor (VIF) values.
• Homoscedasticity–This assumption states that the variance of error terms are similar across the values of the independent variables. A plot of standardized residuals versus predicted values can show whether points are equally distributed across all values of the independent variables.

Intellectus Statistics automatically includes the assumption tests and plots when conducting a regression.

b)

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PROOF: Any other linear estimator of B can be written as B A'y [E.15] where A is an n X (k 1) matrix. In order for B to be unbiased conditional on X, A can consist of nonrandom numbers and functions of X. (For example, A cannot be a function of y.) To see what further restrictions on A are needed, write B A'(XBu)(A'X)B A'u [E.16] Then, E(BX) A'XB E(A'uX) =A'XBA'E(u|X) because A is a function of X A'XB because E(ulX) 0. For B to be an unbiased estimator of B, it must be true that EBX) B for all (k1) x 1 vectors B, that is, A'XB B for all (k 1) x 1 vectors B [E.17] 1) matrix, (E.17) holds if, and only if, A'X +1 Because A'X is a (k + 1) x (k Equations (E.15) and (E.17) characterize the class of linear, unbiased estimators for B Next, from (E.16), we have Var(BX) A'IVar(uXJA A'A, by Assumption E.4. Therefore, Var(B)X) Var(BX) IA'A ( X'X ) -'1 A'A - A'X(X'X)-1X'A] because A'X + 1 A'I XX'X)x'A A'MA, where M I X(X'X)X'. Because M is symmetric and idempotent, A'MA is positive semi-definite for any n X (k 1) matrix A. This establishes that the OLS estimator ß is BLUE. Why is this important? Let c be any (k 1) x 1 vector and consider the linear combination c'B Coßo + ciB, + .+ CBk, which is a scalar. The unbiased estimators of c'B are c'B and c'B. But 444 Var(c'B)X) Var(c'BX) cIVar(BX) VarßX)c 0, because IVar(BX) VarBXl is p.s.d. Therefore, when it is used for estimating any linear combination of B, OLS yields the smallest variance. In particular, Var(BX) Var(BX) for any other linear, unbiased estimator of B