(10) (a) Derive and plot the spectrum of y. Discuss how the seasonality in the process is evident in spectrum.

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1 January 01 Final Exam Quesions: Mark W. Wason (Poins/Minues are given in Parenheses) (15) 1. Suppose ha y follows he saionary AR(1) process y = y 1 +, where = 0.5 and ~ iid(0,1). Le x = (y + y 1 )/. (11) (a) Show ha x follows an ARMA(1,1) process and deermine he value of is AR coefficien, MA coefficien, and he variance of is error erm. (4) (b) Is he x ARMA process inverible? Explain. (0). Suppose ha y is a ime series ha is available semiannually (ha is, wice per year), once in he winer and once in he summer; y follows he MA process y = where is iid(0,1). (10) (a) Derive and plo he specrum of y. Discuss how he seasonaliy in he process is eviden in specrum. (10) (b) A researcher proposes o use x = 0.5(1+L), as a seasonally adjused version of y. Compue he gain of he filer 0.5(1+L). Does his filer aenuae and/or eliminae he seasonaliy in y? Explain. (15) 3. Consider he NKPC model ha you sudied in your exercise. Recall ha an implicaion i of he model is ha = x /, where x +i/ denoes he forecas of x +i consruced using i0 i informaion available a ime. Le = excep ha i uses x +i insead of x +i/.) i0 i x i (8) (a) Show ha ( ) is uncorrelaed wih. (7) (b) Show ha var( ) var( ). (which is he same formula as ha used for

2 (5) 4. Suppose ha y = z + u, where u = 0.5u 1 +, and ~ iidn, z (10) (a) Derive he probabiliy limi of ˆOLS, he OLS esimaor of. (10) (b) Suppose ha of using x as an insrumen z ~ iidn 0, 1 3. Le ˆIV x denoe he IV/GMM esimaor Show ha d ( ˆIV T ) N(0, V), and compue he value of V. (5) (c) In a sample size T = 100, you find ˆIV = 0.5. Tes H o : = 0 versus H a : 0 using a es wih a size of 5%. (15) 5. Suppose ha y follows he model y = + v v = 1 T u + u = u 1 + a where a Niid 0 0 1, and u 0 = 0. Le 1 T y T 1 y denoe he sample mean of y. (10) (a) Show ha p y. (5) (b) Wha is he limiing disribuion of T( y )?

3 PART Problem 1. (40 poins) Le y be he variable of ineres and le x be a posiive (scalar) explanaory variable. Assume ha he condiional densiy of y given x is and This implies ha f(yj x) = y x exp ( y / (x)) for y 0 0 for y < 0 E [yj x] = p p x V [yj x] = x Assume ha you have a random sample of size n from he disribuion of (y; x) and ha you are ineresed in esimaing. In answering he following quesions, you do no have o worry abou verifying he regulariy condiions. 1. (8 poins) Derive he asympoic disribuion of he maximum likelihood esimaor of. 1. (8 poins) Find probabiliy limi of he OLS esimaor in a regression of y on x (excluding a consan). Is he esimaor consisen? 3. (8 poins) Derive an expression for he asympoic disribuion of he OLS esimaor in a regression of y on x (excluding a consan). 4. (8 poins) Find he asympoic disribuion of he nonlinear leas squares esimaor de ned by arg min b nx i=1 y i 4 p p xi b 5. (8 poins) How would you esimae using only linear regression? Can you hink of more han one way of doing i? 1

4 Problem. (3 poins) The aached oupu shows he resuls of esimaing a linear regression model, a logi model and a probi model for a binary dependen variable, y, using a scalar explanaory variable, x, and a consan as explanaory variables. 1. (6 poins) Esimae he marginal e ec of x on he probabiliy ha y is 1 for an individual wih x = 0 using each of he hree models (in oher words, repor hree marginal e ecs). How do hey compare?. (1 poins) Repor a 95% con dence inerval for wo of he hree marginal e ecs esimaed above. You can decide which wo o repor. 3. (8 poins) Esimae he marginal e ec of x on he probabiliy ha y is 1 for an individual wih x = using each of he hree models (in oher words, repor hree marginal e ecs). How do hey compare? 4. (6 poins) Brie y discuss he di erence beween he resuls in par 1 and par 3. Problem 7. (14 poins) Consider he panel daa model y i = i + x 0 i + x 0 i 1 + " i ; i = 1; ::; n = 1; ::; T where i is an unobserved individual speci c ( xed ) e ec and E " i j i ; x i ; x i 1 ; x i ; ::: = 0 As usual, denoes ime period and i individual. In he following assume ha n is much bigger han T, so he only ineresing asympoics is for n increasing wih xed T. 1. (5 poins) Wha would be he problem wih esimaing (; ) by regression y i on x i, x i 1 and a dummy for each individual? Is here a problem?. (9 poins) How would you esimae (; )? How many ime periods would you need?

5 Problem 8. (14 poins) This problem is concerned wih bounding average reamen e ecs. Suppose ha D is a random variable ha indicaes wheher an individual has been reaed. If D = 1, he individual has received reamen, and we observe a binary random variable Y 1. If D = 0, he individual has no received reamen, and we observe a binary random variable Y 0. We do no observe Y 0 if D = 1 and we do no observe Y 1 if D = 0. This is he sandard noaion in his lieraure. The only hing new is ha we are assuming ha he oucome is binary. Suppose ha P (D = 1), E [Y 1 j D = 1] = 0:60 and E [Y 0 j D = 0] = 0:45. Use his o bound he average reamen e ec, E [Y 1 Y 0 ], under he assumpion ha P (Y 1 Y 0 ) = 1. 3

6 Oupu for Problem of he second par of he exam. reg y x Source SS df MS Number of obs = F( 1, 198) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE = y Coef. Sd. Err. P> [95% Conf. Inerval] x _cons logi y x Logisic regression Number of obs = 00 LR chi(1) = Prob > chi = Log likelihood = Pseudo R = y Coef. Sd. Err. z P> z [95% Conf. Inerval] x _cons probi y x Probi regression Number of obs = 00 LR chi(1) = Prob > chi = Log likelihood = Pseudo R = y Coef. Sd. Err. z P> z [95% Conf. Inerval] x _cons