First Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise

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1 First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I-1 Part I - 203A A radom variable X is distributed with the margial desity: >< 2 (1 x) if 0 < x < 1 f X (x) = The coditioal cumulative distributio fuctio of aother radom variable Y give X is 0 if y < x >< F Y jx=x (y) = y x 1 x if x y < 1 1 if y 1 1. Calculate the expectatio of the radom vector (Y; X) : 2. Calculate the covariace of X ad Y: Are X ad Y idepedetly distributed? Explai. 3. Calculate the probability that X 2 [:5; 1] coditioal o Y = :75 4. Let Z = Y X: Calculate the desity of (Z; X) : Questio I-2 A observable radom variable Y is determied by a uobservable radom variable ad a uobservable radom variable "; accordig to the model >< 1 if + " > 0 Y = 1

2 where is a parameter of ukow value. The margial desity of the radom variable " is kow to be N " ; " 2 ; for ukow values of " ad ": 2 The uobservable radom variable is kow to be distributed N 0 ; 0 2 whe Z = 0 ad N 1 ; 1 2 whe Z = 1; for ukow values of 0 ; 1 ; 0 2; ad 1 2 : The radom variable Z is observable. Deote the probability that Z = 0 by the parameter p 0 ad the probability that Z = 1 by the parameter p 1 ; where p 0 + p 1 = 1: Assume that (; Z) ad " are idepedetly distributed. 1. Obtai a expressio for the probability that Y = 1 coditioal o Z = 0 i terms of the ukow parameters. 2. Obtai a expressio for the (margial) probability that Y = 1: 3. Are p 0 ad p 1 ideti ed? Provide a proof for your aswer. 4. Are " ad 2 " ideti ed? Provide a proof for your aswer. Suppose ext that the values of 0 ; 2 0 ad " ; 2 " are kow, with 0 = 0 ad " 6= Determie what parameters are ideti ed. Provide proofs. 6. For the parameters that are ot ideti ed, ca you provide bouds for their values? If your aswer is YES, determie those bouds. If your aswer is NO, explai. Questio I-3 Cosider the followig model: >< 1 if + X + " > 0 Y = where the radom variables Y; X; ad Z are observable, the radom variables ad " are uobservable, ad is a parameter of ukow value. As i Questio 2, the distributio of depeds o the value of Z: The radom variable Z attais the value 0 with probability p 0 ad the value 1 with probability p 1 ; where p 0 +p 1 = 1: Assume, further, that (i) the support of the cotiuous radom variable X is R; (ii) the radom variable " is distributed N (1; 4) ; ad (iii) (; Z) ; X; ad " are mutually idepedet. 1. Suppose rst that the distributio of whe Z = 0 is N(0; 16) ad the distributio of whe Z = 1 is N( 1 ; 2 1 ); where the values of 1 ad 2 1 ; as well as the values of p 0 ad p 1 are ukow. (a) Determie the ideti ed parameters. (b) Give i.i.d. observatios Y i ; X i ; Z i parameters. Provide proofs of your claims. (c) Prove that your proposed estimators i 1.b are cosistet. the asymptotic distributio of your estimators? N ; provide cosistet estimators for the ideti ed i=1 2 What ca you say regardig

3 2. Suppose ow that the distributio F jz=1 of whe Z = 1 ad the distributio F jz=0 of whe Z = 0 do ot ecessarily belog to a parametric family. They are oly kow to be strictly icreasig ad cotiuous fuctios. It is however still kow that V ar(jz = 0) = 16: Aswer the followig questios ad provide proofs. (a) Is ideti ed? (b) Is the distributio of ( + ") ideti ed? (c) Are F jz=1 ad F jz=0 ideti ed? (d) Is the (margial) distributio of ideti ed? 3

4 Part II - 203B Questio II-1 X = (X 1 ; : : : ; X ) 0 is such that X i are iid N ( 1 ; 2 ). = ( 1 ; 2 ) based o X. Let s 2 = 1 X X i X 2 1 i=1 Compute the iformatio for Prove that s 2 is ubiased for 2. Prove that the ite sample variace of s 2 is strictly larger tha the Cramer-Rao boud. (Let = 2 whe you make this compariso.) Derive the asymptotic distributio of p s 2 2 as! 1, ad show that the asymptotic variace of p s 2 2 is idetical to the iverse of the Fisher iformatio. Questio II-2 y i = x i1 1 + x i2 2 + " i such that " i ; x i1, ad x i2 are idepedet of each other with the commo distributio N (0; 1). We assume that every variable is a scalar. Cosider two estimators of 1. The rst estimator e 1 is obtaied by regressig y i o x i1 : e 1 = i=1 x i1y i i=1 x2 i1 The secod estimator b 1 is the rst compoet whe y i is regressed o x i1 ad x i2 : " b1 b 2 # " P = P i=1 x2 i1 i=1 x i1x i2 i=1 x i1x i2 i=1 x2 i2 Compute the asymptotic variaces of p e1 1 ad p b1 which oe is larger? If so, which oe is more e ciet? # 1 " P # Pi=1 x i1y i i=1 x i2y i 1. Is it possible to determie Questio II-3 y i = x i i + " i where " i ; x i, ad i are idepedet of each other. Note that i is a radom variable. We assume that (y i ; x i ) 0 i = 1; 2; : : : ; are observed, ad they are iid. Let = E [ i ], ad propose a cosistet estimator of. Prove why your estimator is cosistet. Derive the asymptotic variace of p b, where b deotes your proposed estimator. 4

5 Part III - 203C Questio III-1 X 1 ; :::; X is a iid sample from a distributio havig desity fuctio of the form ( x 1 if x 2 (0; 1) f(x; ) = : Show that a best critical regio for testig H 0 : = 1 agaist H 1 : = 2 is C = (X 1 ; :::; X ) : c Y o : Questio III-2 i=1 X i y t = y t 1 + u t u t = " t + " t 1 where " t iid 0; 2 ". 1. jj < 1. Is the process fy t g covariace statioary? Derive the autocovariace fuctio of fy t g. 2. Uder the assumptio jj < 1, is the OLS estimate b de ed as b = t=1 y ty t 1 t=1 y2 t 1 (1) a cosistet estimator of? 3. Suppose jj < 1 still hold. Now we estimate by usig the Istrumetal Variable (IV) method, with y t 2 beig the istrumet for y t 1. The IV estimator b IV is de ed as b IV t=2 = P y ty t 2 t=2 y : t 1y t 2 Is the IV estimator b IV a cosistet estimator of? Derive its limitig distributio. 4. Now suppose that = 1, is the OLS estimate b de ed above a cosistet estimator of? 5. Derive the limitig distributio of the OLS estimate b uder = 1. 5

6 Questio III-3 Xt 1 Y t = X t + u t + c where u t iid (0; 1), c ad are some ukow costats ad s=0 u s X t = X t 1 + " t where " t is a iid (0; 1) sequece idepedet of u s for all t ad s, ad X 0 = 0. You ru a regressio of Y t o X t ad get the followig OLS estimate t=1 b = P X ty t : t=1 X2 t 1. c 6= 0. Is b a cosistet estimate of? 2. c = 0. Is b a cosistet estimate of? Derive its limitig distributio. 3. Usig the results i (1) ad (2), costruct a statistic for testig H 0 : c = 0 agaist H 1 : c 6= 0. Why is your test cosistet? 4. Let X t = X t X t 1 ad Y t = Y t Y t 1. What s the limitig distributio of the followig estimate? b t=2 = P X ty t : t=2 X2 t 5. Which estimate, i.e. b or b, do you prefer? Justify your choice (Hit: your aswer may deped o the value of c). 6