Econometric take home APPS_Part_41

Econometric take home APPS_Part_41 - (a) x ~ Normal[0,32],...

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(a) x ~ Normal[0,3 2 ], and -4 < x < 4. (b) x ~ chi-squared, 8 degrees of freedom, 0 < x < 16. The inequality given in (3-18) states that Prob[| x - μ | < k σ ] > 1 - 1/ k 2 . Note that the result is not informative if k is less than or equal to 1. (a) The range is 4/3 standard deviations, so the lower limit is 1 - (3/4) 2 or 7/16 = .4375. From the standard normal table, the actual probability is 1 - 2Prob[ z < -4/3] = .8175. (b) The mean of the distribution is 8 and the standard deviation is 4. The range is, therefore, μ ± 2 σ . The lower limit according to the inequality is 1 - (1/2) 2 = .75. The actual probability is the cumulative chi-squared(8) at 16, which is a bit larger than .95. (The actual value is .9576.) 7. Given the following joint probability distribution, X | 0 1 2 --+------------------ 0| .05 .1 .03 Y 1| .21 .11 .19 2| .08 .15 .08 (a) Compute the following probabilities: Prob[ Y < 2], Prob[ Y < 2, X > 0], Prob[ Y = 1, X > 1]. (b) Find the marginal distributions of X and Y . (c) Calculate E [ X ], E [ Y ], Var[ X ], Var[ Y ], Cov[ X , Y ], and E [ X 2 Y 3 ]. (d) Calculate Cov[Y,X 2 ]. (e) What are the conditional distributions of Y given X = 2 and of X given Y > 0? (f) Find E [ Y | X ] and Var[ Y | X ]. Obtain the two parts of the variance decomposition Var[ Y ] = E x [Var[ Y | X ]] + Var x [ E [ Y | X ]]. We first obtain the marginal probabilities. For the joint distribution, these will be X: P(0) = .34, P(1) = .36, P(2) = .30 Y: P(0) = .18, P(1) = .51, P(2) = .31 Then, (a) Prob[ Y < 2] = .18 + .51 = .69. Prob[ Y < 2, X > 0] = .1 + .03 + .11 + .19 = .43. Prob[ Y = 1, X $ 1] = .11 + .19 = .30. (b) They are shown above. (c) E [ X ] = 0(.34) + 1(.36) + 2(.30) = .96 E [ Y ] = 0(.18) + 1(.51) + 2(.31) = 1.13 E [ X 2 ] = 0 2 (.34) + 1 2 (.36) + 2 2 (.30) = 1.56 E [ Y 2 ] = 0 2 (.18) + 1 2 (.51) + 2 2 (.31) = 1.75 Var[ X ] = 1.56 - .96 2 = .6384 Var[ Y ] = 1.75 - 1.13 2 = .4731 E [ XY ] = 1(1)(.11)+1(2)(.15)+2(1)(.19)+2(2)(.08) = 1.11 Cov[ X , Y ] = 1.11 - .96(1.13) = .0252 E [ X 2 Y 3 ] = .11 + 8(.15) + 4(.19) + 32(.08) = 4.63. (d) E[ YX 2 ] = 1(12).11+1(22).19+2(12).15+2(22).08 = 1.81 Cov[ Y , X 2 ] = 1.81 - 1.13(1.56) = .0472. (e) Prob[ Y = 0 * X = 2] = .03/.3 = .1 Prob[ Y = 1 * X = 2] = .19/.3 = .633 Prob[ Y = 1 * X = 2] = .08/.3 = .267 Prob[ X = 0 * Y > 0] = (.21 + .08)/(.51 + .31) = .3537 Prob[ X = 1 * Y > 0] = (.11 + .15)/(.51 + .31) = .3171 Prob[ X = 2 * Y > 0] = (.19 + .08)/(.51 + .31) = .3292. (f) E [ Y * X =0] = 0(.05/.34)+1(.21/.34)+2(.08/.34) = 1.088 E [ Y 2 * X =0] = 1 2 (.21/.34)+2 2 (.08/.34) = 1.559 Var[ Y * X =0] = 1.559 - 1.088 2 = .3751 E [ Y * X =1] = 0(.1/.36)+1(.11/.36)+2(.15/.36) = 1.139 E [ Y 2 * X =1] = 1 2 (.11/.36)+2 2 (.15/.36) = 1.972 Var[ Y * X =1] = 1.972 - 1.139 2 = .6749 E [ Y * X =2] = 0(.03/.30)+1(.19/.30)+2(.08/.30) = 1.167 163
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E [ Y 2 * X =2] = 1 2 (.19/.30)+2 2 (.08/.30) = 1.700 Var[ Y * X =2] = 1.700 - 1.167 2 = .6749 = .3381 E [Var[ Y * X ]] = .34(.3751)+.36(.6749)+.30(.3381) = .4719 Var[ E [ Y * X ]] = .34(1.088 2 )+.36(1.139 2 )+.30(1.167 2 ) - 1.13 2 = 1.2781 - 1.2769 = .0012 E [Var[ Y * X ]] + Var[ E [ Y * X ]] = .4719 + .0012 = .4731 = Var[ Y ]. ~ 8. Minimum mean squared error predictor . For the joint distribution in Exercise 7, compute E [ y - E [ y | x ]] 2 . Now, find the a and b which minimize the function E [ y - a - bx ] 2 . Given the solutions, verify that E [ y - E [ y | x ]] 2 < E [ y - a - bx ] 2 . The result is fundamental in least squares theory. Verify that the a and b which you found satisfy (3-68) and (3-69). ( x =0) ( x =1) ( x =2) E [ y - E[ y | x ]] 2 = ( y =0) .05(0 - 1.088) 2 + .10(0 - 1.139) 2 + .03(0 - 1.167) 2 ( y =1) + .21(1 - 1.088) 2 + .11(1 - 1.139) 2 + .19(1 - 1.167) 2 ( y =2) + .08(2 - 1.088) 2 + .15(2 - 1.139) 2 + .08(2 - 1.167) 2 = .4719 = E [Var[ y | x ]]. The necessary conditions for minimizing the function with respect to a and b are E [ y - a - bx ] 2 / a = 2 E {[ y - a - bx ](-1)} = 0 E [ y - a - bx ] 2 / b = 2 E {[ y - a - bx ](- x )} = 0.
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Econometric take home APPS_Part_41 - (a) x ~ Normal[0,32],...

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