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# Empirical Studies in Macroeconomics ECO 5381

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This 30 page Class Notes was uploaded by Kaden Orn on Thursday October 22, 2015. The Class Notes belongs to ECO 5381 at Texas Tech University taught by Summers in Fall. Since its upload, it has received 26 views. For similar materials see /class/226393/eco-5381-texas-tech-university in Economcs at Texas Tech University.

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Date Created: 10/22/15

Some ume senes economemes Some time series econometrics ECO 5381 Dr Peter M Summers January 15 2009 Some time Series econometrics Models and data gt The models we39ll be dealing with will have approximate solutions of the form Xt1 FXt GVt1 gt Xt is one or more time series of interest eg GDP consumption Vt is one or more random disturbance or shock eg technology shocks monetary policy shocks gt F and G are matrices of coefficients usually functions of underlying or deep parameters of the model coefficient of risk aversion disutility of labor etc gt Models impy restrictions on observed time series so to gauge the model39s performance we need to be able to summarize time series behavior Some time Series econometrics Time series basics gt The sequence yt2yt1ytyt1 is a covariancestationary stochastic process if gt Ey u is constant over time gt Vary a is constant over time and gt Covyyk 7k is independent of time it may depend on k gt Suppose at is such that Eet 07 Varet 02 and EEt65 39ys O for all t 7 5 Then at is White noise gt The Lag operator L or backshift operator B shifts a time series forward or backward Lyt ytsl Lzyt yt727 LTlYt Yt17 etc Some time series econometrics ARMA models gt Let at be white noise and define yt at i 0amp4 yt is a first order moving average or MA1 Note that yt 1 i 0Let gt For an MA1 process the mean and variance are Eyt 1 0LE6t 0 03 Hy Eeg 20etet1 0261 02 0 0202 1 0 Some ume senes economemes ARMA models gt The covariances of an MA1 are given by v1 EOthil EKG 06t716t71 05t72 EEt5til 0624 0631 02671272 002 39ys0 sgt1 Some ume serles economemcs ARMA models gt More generally consider the MAq process Vt 6t l 01971 l 02972 l 39 qutiq where 00 1 Some ume senes economemes ARMA models gt The covariances of an MAq are q 39y0 a 02 0 j0 39ys a 05 05101 mm oqoqs s 1q 39ys 07 s gt q Some time Series econometrics ARMA models gt Wold Decomposition Theorem any covariancestationary stochastic process yt can be written as an infiniteorder MA Yr 21094 i Ukt 10 with ELEM1 lt 00 square summability gt We have 00 2 2 2 Uy 21 10 75 Ugh15 s1 1 s2 2 Some time Series econometrics ARMA models gt Now consider the first order autoregressive AR1 model Yr pyH Eh ipi lt1 where the condition on p ensures covariance stationarity gt By the Wold Theorem we can express yt in terms of the innovations std Yr PYt71 6t 1 pLyt 5t 7 1 Yr 71 pLEt Some ume senes economemcs Aside Polynomials in the lag operator gt Recall that if X lt1 then 117X1Xx2 JiOXj gt So if pL lt1 we can use the same trick 1 76 Yr lipL t 1PLP2L2quot E1 lejl jlet em ll M81 g ll 0 Some time Series econometrics Aside V V V Polynomials in the lag operator We have the lag operator polynomial 17 pL Treat L like a number and solve for the value of L such that 17 pL 0 The root of this polynomial is 107 which is greater than 1 in absolute value as long as lpl lt 1 This is what we mean by pL lt 1 in other words the expansion we just did is valid If p 1 the root of 17 pL is 1 unity and in this case yt is said to have a unit root or to be a unit root process In that case the variance of yt doesn39t exist and the process is non stationary Some time Series econometrics Persistence gt The auto covariances of an AR1 model are vs pv07 5 gt 1 gt So the first order autocorrelation is 39y139y0 p and the jth autocorrelation is gt Suppose we have 2 AR1 processes with p1 09 and p2 05 The 2 Wold representations are Y1 2f 09j61tj and ygt 230 05jeh Some time series econometrics Pe rsiste nce gt Suppose all the as are 0 except at time t we get a shock of at 1 Then next period some of that shock39s effect is still present 1 period later 00 Y1t1 ijlet1ij0O9OOquot39 10 00 YQt1ZP 6t17100500 170 Some time Series econometrics Persistence gt After 2 periods 0 y1t2 Z leHH 0 0 081 0 j0 00 mm Zp zw 0 0 025 0 j0 gt The first series is more persistent than the second effects of shocks die out more slowly gt p 1 infinite persistence shocks never die out Some ume senes economemes ARMA models gt The ARp process is written Yr DUtil JD1 72 Pthip 6t pLyt 5t where pL 17 p1L7i ppr Some time Series econometrics ARMA models gt As before we can derive the Wold representation for yt First factor pL as follows 17p1L7p2L277ppr ii1L1i2L17pL gt Then YtpL t 9 gt As long as the roots J71 are all outside the unit circle we have a stationary process Some time Series econometrics Aside complex numbers V Recall that the solutions to the quadratic equation 322 l bz l c O are given by 7b i v b2 7 43c 23 If b2 lt 43c then the solutions are complex That is they have the form 21 X l iy and 22 X 7 iy where i xil 217 22 V gt X and y are the real and imaginary parts of 21 amp 22 gt 21 amp 22 are complex conjugates they have the same real part and their imaginary parts have opposite sign gt Complex roots always occur in conjugate pairs 21 22 Some time Series econometrics Aside complex numbers gt The modulus of a complex number 2 x iy is given by lzl V22 X WW 7 W X2 yZ analogous to the absolute value of a real number gt Complex numbers are represented graphically in 2 dimensions The unit circle is a circle centered at the origin with radius ie modulus equal to 1 Some time series econometrics Back to the ARp model gt Remember that we had 17p1L7p2L277ppr ii1L1i2L17pL gtSo rim 9 and the condition analogous to ipi lt 1 for the AR1 case is that the roots JT1 are all outside the unit circle That is yt is stationary if this condition is satisfied Some time Series econometrics ARMA models gt V V V V We can combine ARp and MAq models to get ARMAp7 q models eg ARMA11 Yr PYtil 6t 0amp71 More generally pLyt 0Let Stationarity is governed by pL need the same condition as for ARp Persistence governed by smallest root of pL closer to 1 a more persistence Some time Series econometrics Vector Autoregressions VARS V A VAR with p lags deonted VARp generalizes the univariate ARp model to describe a vector of In time series V Each series is regressed on p of its own lags and p lags of all other series possibly also a constant trend or other exogenous variables gt Suppose m 2 and consider Xt ctyty where ct and yt are the logs of consumption and gdp Then a VAR1 for Xt is Ct OtCtA BYtil Ea Yt 39th71 6Yt71 Eyt Some time Series econometrics Vector Autoregressions VARS Ct actil BYtil Eat Yr YCt71 571 Eyt This can be written in matrix form as xtltgti iltgtltigt Xt71 6t Some time Series econometrics Vector Autoregressions VARS gt More generally the VARp model is ignoring constants trends etc Xt 1Xt71 2Xt72 pXt7p 6t where each DJ is an m x m matrix In lag operator notation LI7 1L7 2L277 pLP Some time Series econometrics Vector Autoregressions VARS gt If the roots of ClgtL all mp of them are outside the unit circle then the VARp is stationary and we can invert the previous equation to get the multivariate Wold representation Xt Z wJEH IJLet i0 compare DeJong amp Dave39s 427 and 48 Some time Series econometrics VVhy gt Woodford The hypothesis that business fluctuations can be largely attributed to exogenous random variations in monetary policy has few if any remaining adherentsAltig et al 2005 conclude that monetary policy shocks identified by their VAR account for only 14 percent of the variance of fluctuations in aggregate output at business cycle frequencies Smets and Wouters 2007 find that monetary policy shocks account for less than 10 percent of the forecast error variance decomposition for aggregate output at any horizon Modehng Busmess Cyc es Modeling Business Cycles ECO 5381 Dr Peter M Summers January 8 2009 Modehng Busmess Cyc es Readings gt Krueger chs 1amp 2 gt DeJong ch 3 gt Woodford Convergence in Macroeconomics Elements of the New Synthesis American Economic Journal Macroeconomics 1 267 279 Modeling Business Cycles What are business cycles gt NBER a significant decline in economic activity spread across the economy gt RE Lucas Jr JPE 1979 real output undergoing serially correlated movements about trend gt Consensus in the profession that models should be able to handle both long run growth trend and business cycles fluctuations around trend Modeling Business Cycles Separating trend from cycle gt Linear trend gt Easy to do gt No allowance for changes in longrun steady state growth rate gt Hodrick Prescott filter gt Allows for changes in trend gt May induce spurious cycles gt Band pass filter gt Isolates fluctuations in a certain range of duration frequency Modeling Business Cycles Hod rick Prescott H P filter gt Let yt be the natural log of GDP We want to separate the trend component y from the cyclical component ytc So m gt HP filter solves the following problem T T 2 mm W zA23Knlyi 047yL0l YtgtYE t1 t1 gt Convention is to set 1600 for quarterly data 100 for annual data

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