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Date Created: 09/28/15
University of Pennsylvania Economics 706 Spring 2004 Fall Prelim Examination Monday 083004 Time limit 120 minutes Instructions i The total number of points is 1007 the number of points for each problem is given below ii The exam is closed book and closed notes iii To receive full credit for your answers you have to explain your calculations You may state additional assumptions Frank Schorfheide Economics 706 Spring 2004 2 Introduction You are interested in estimating a monetary policy rule of the form Rt Y 39t ERt7 1 where R is the nominal interest rate 7139 is in ation and 63 can be interpreted as a monetary policy shock You have data on nominal interest rates in ation and output growth which we will denote by 2 The data are demeaned the sample ranges from t 1 T Moreover you have observations for periods 25 1 7 p 0 to initialize lags of autoregressive models You take the View that economic uctuations are due to monetary policy shocks eat government spending shocks 69 and technology shocks 61 These structural shocks are stacked in the vector 6t Moreover you believe that the observed uctuations can be captured by some dynamic stochastic equilibrium model that yields a vector autoregressive representation for interest rates in ation and output growth Frank Schorfheide Economics 706 Spring 2004 3 Problem 1 OLS Estimation 20 Points i Derive the OLS estimator of 39y in Eq ii Derive the probability limit of the OLS estimator iii Under what assumption is the estimator consistent Do you think that this assumption is satis ed in the context of the policy rule estimation iv Derive the limit distribution ofthe OLS estimator Note the derivation should be general enough to encompass cases in which the estimator is inconsistent Frank Schorfheide Economics 706 Spring 2004 Problem 2 Instrumental Variable Estimation 26 Points i Propose an instrumental variable estimator for 39y that o is consistent to be veri ed subsequently 0 can be constructed from the available data ii Show that your estimator is consistent iii Derive its limit distribution Frank Schorfheide Economics 706 Spring 2004 5 Problem 3 Fullsystem Analysis 27 Points De ne yt Rtnrhztl 341 Rt yg 713th such that we obtain the partition yt y1ty 2t Moreover7 collect the rst p lags of yt in the vector mt that is7 1 ly717quot397y 7pl39 Rewrite the policy rule as 241 22MV 61 2 where M 10 and reads in ation off the vector 3429 The speci cation of the system is completed with the autoregressive equations 242 9591 142 3 where 142 are one step ahead forecast errors for in ation and output i What are the dimensions of mt and 117 ii Show that the system 2 and 3 can be rewritten as a VAR of the form 2 94 M 4 where u um7 What is the relationship between the parameters 39y and I and the VAR coef cient matrix I What is the relationship between ER and 142 on the one hand and 141 on the other hand Assume that at N iiol3907 Eu Show that the conditional likelihood function can be written as A V pY ltI 2 27rr3T2l2urT2 exp gmz y 7 XltIgt Y 7 XltIgt 5 where Y is composed of the columns y and X is composed of the columns Hint recall that a Ba trBaa and trB1 trB2 trB1 B2 Suppose that the structural shocks are uncorrelated with each other that the C lt V vector 6 is a normalized version of Q such that all elements of e have unit variance Moreover7 there is a matrix 116 such that 142 Et l e 6 and a standard deviation parameter 0R such that 63 TREE Using the restrictions derived in ii7 is the effect of the structural shocks E on interest rates7 in ation7 and output growth identi able without further assumptions Frank Schorfheide Economics 706 Spring 2004 6 Problem 4 Bayesian Inference 27 Points i Replace I in Eq 5 by the appropriate function of I and 39y and rewrite the likelihood function in terms of 117 y and Eu ii Partition A V11 V YY1Y2 and 2 1 12 V21 V22 such that the partitions conform with y y1ty 2t Moreover7 assume that we have a prior distribution of the form 19 1 777 Eu P I PVPEu 7 where p y corresponds to a normal distribution with mean 7 and variance 0392 Show that conditional on I and Eu the posterior distribution of 39y is normal and derive its mean and variance The OLS estimator of I is given by i X X 1X Y2 What is the posterior mean of 39y given ii Does it resemble your instrumental variable estimator
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