Eksamen. ge-506 AdvancedEconometrics. Dato: Varighet 9:00-12:00. Antall sider inkl. forside 5. Tillatte hjelpemidler

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1 Eksamen Emnekode: Emnenavn: ge-506 AdvancedEconometrics Dato: Varighet 9:00-12:00 Antall sider inkl. forside 5 Tillatte hjelpemidler Merknader Dictionary (English-XX,XX-English) pocket calculator (non-programmable) This is an open book exam: textbook and lecture notes can be used. You have to show all of your work. Be explicit! A maximum of 150points can be obtained.

2 Problem 1 [50pts] [a] Let X denote the number countries experiencing the onset of an economic crisis in a 30 pts predefined period of time. Assume the X follows a Poisson-distribution, that is with A > 0. X f (x; { 0 x! if x = 0, 1, 2, 3,... if o.w. Interpret the parameter A in the context of the problem. Given a true random sample X1, X2, X3,... XN derive the maximum likelihood (ML) estimator the method of moment (MM) estimator of À after providing a short concise sketch of the rationale of the respective method. [b] Let X denote the level of household debt and assume that household debt follows a uniform 20 pts distribution u(o, 0) = if x E [0, 0] 0 if o.w. with 0 (0, oo). You have access to a true random sample of household debt measures X1, X2, X3, XIV. Give an interpretation of the parameter 0. Determine the Likelihood function L(0; xl, x2, xs,. xn). Find the ML estimator Ö of the parameter 0. Determine the MM estimator Ö of theta. Compare the two estimators. Problem 2 [50pts] Consider the following AR(p = 1) where ft is i.i.d with E[t] = 0 and V[c] = o-2.

3 (a) Show that 35 pts cov(yt, Yt_j) = and the autocorrelation function for this process is given by rj = for l = 0,1,2, 3,... (b) Provide a sketch of the partial-autocorrelation function for the AR(p =1). (c) The autocorrelation function as well as the partial autocorrelation function for an (real) interest rate process have been estimated using a sample times series from the US (frequency: quarterly; range 1950q1-1970q1). Use the evidence displayed below to respond to the following items: 5 pts 10 pts Would the AR process specified above provide a reasonable model for the US interest rate process during the fifties and sixties? Justify your answer. Estimate the parameter "yr. Outline the procedure you have used to arrive at your result. Assess the stability of the estimated interest rate process Banletasformla for MA(5195%confidencebands 95% Canfidenasbands (ses lisgrt(n)j Figure 1: Estimated acf and pacf for real interest rates (US; 1950q1-1970q1) LAG ac pac

4 Problem 3 [50pts] Suppose one wants to model the relationship between real consumption (C.) and real GDP ( "-Y). Quarterly data for the US are available for the period 1950q1 to 2000q1. The figure given below shows a line plot for the two series of interest. 1950q1 1960q1 1970q1 t 1980q1 1990q1 2000q1 An analyst simply regresses Cs.on i producing the following output: SourceI SS df MS Numberof obs = 204 F( 1, 202) = Model I Prob > F = ResidualI R-squared = Adj R-squared= Total I RootMSE = C I Coef. Std.Err. P>Iti [95 Conf.Interval] Y I _cons I Provide a critical assessment of this modeling approach and motivate the ECM approach. lopts Suppose the long-run equilibrium relationship between the variables consumption ((.Y- ) and GDP (1-.7 ) can be given as 6`t = A or equivalently in log-form as Ct = fio Yt (1) with,80= ln A, Ct = ln C't and Yt = in12. We propose the following short-run (disequilibrium) relationship with ft E (0,1). Ct = -yo+ + -y217t-1+ [ict-1+ et (2) Show that model (2) is equivalent to the model 2Opts ACt = (1 it)fio + Alft (1 + (1 12,),(3117t i + Et. Give a short motivation for this representation. 4

5 The model presented in (b) has been estimated by OLS. The following STATA output lopts has been generated. Obtain the estimates for the long-run and the short-run elasticity of consumption with respect to GDP. Judge the modeling effort on the basis of the default STATA output as well as on the basis lopts of the basic residual analysis carried out (see graphs). Source I SS df MS Number of obs = F( 3, 199) = Model I Prob > F = Residual I R-squared = Adj R-squared = Total I Root MSE = D.lnc I Coef. Std. Err. t P>Iti [95% Conf. Interval] lngdp I D1. I lnc I L1. I lngdp I Ll. I _cons I l 'Ill'1, Bartletrs formula far MA(q)95% cordidencebands 95% Confidencebands [se = Vecl,t[nfi Kernel density estimate Residuals Kernel density estimate Normal density kelnel = epanechnikov.bendeddffi