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398_39_solutions-instructor-manual_13-introduction-nonstationary-time-series_im_ch13.pdf

16 Pages
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Department
Economics
Course Code
Economics 2122A/B
Professor
Terry Biggs

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Dougherty Introduction to Econometrics 4e Instructors Manual 13 INTRODUCTION TO NONSTATIONARY TIME SERIES131 Demonstrate that the MA1 process X tt2t1is stationary Does the result generalize to higherorder MA processes Answer The expected value of X is 0 and therefore independent of time t EXEEE000 tt2t1t2t1Sinceandare uncorrelated tt122222X1tttand this is independent of time Finally becauseX t1t12t2the population covariance of X and X is given by tt122XX1ttThis is fixed and independent of time The population covariance between X and Xis 0 for tts all s1 because X and X have no elements in common if s1 Thus the third condition for ttsstationarity is also satisfied All MA processes are stationary the general proof being a simple extension of that for the MA1 case 132 A stationary AR1 process XXttt121withhas initial value X where X is defined as 100211X0021122Demonstrate that X is a random draw from the ensemble distribution for X 0AnswerLagging and substituting it was shown equation 1312 that t1tt212XXtttt12221210212With the stochastic definition of X we now have 0t11tt2121Xtttt12221210221112221tt211ttt1222120221122HenceC Dougherty 2011 All rights reserved 2 INTRODUCTION TO NONSTATIONARY TIME SERIES 1XEt12and 121ttvarvarX12221202tttt2122t224222t21222212222tt1222222211122221varXGiven the generating process for X one hasandHence X XE000021122is a random draw from the ensemble distribution Implicitly it has been assumed that the distributions ofand X are both normal This should have been stated explicitly 0 132Spurious regressions133 Consider the model utXYttt321and suppose that X has a strong time trend As a consequence the estimation ofwill be t2subject to multicollinearity and the standard error relatively large Suppose that one detrends the data for Y and X and instead regresses the model ueetXtYt2This is a simple regression and so the problem of multicollinearity does not arise Nevertheless by virtue of the FrischWaughLovell theorem the estimate ofand its 2standard error Explain intuitively why the standard error has not decreased despite the elimination of the problem of multicollinearity Answer The variance of the slope coefficient is given by 212ub22MSD Xn1rXtin the first specification and by 22ub2MSD enXt1The elimination of the factorin the second will be exactly neutralized by the much 21rXtsmaller MSD of e compared with that of X Xt 133 Graphical techniques for detecting nonstationarity C Dougherty 2011 All rights reserved
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