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1、1內容n INTRODUCTIONn THE BASIC Mn SPECIFICATION OF STAR Mn NONLINEAR MLING OF INDUSTRIAL PRODUCTIONn FORECASTING WITH STAR MSCHARACTERIZING NONLINEARITIES IN BUSINESS CYCLES USING SOMOOTH TRANSITION AUTOREGRESSIVE MS.T. TER ASVIRTAH. M. ANDERSONby Chung-Hua Shen2The Basic Myt = p10 + p ' w + (p 20

2、 + p ' w )F (yt - )+ u1 t2 tdtut nid(0,s 2 )p j = (p j1,.,p jp )'w = (y,.y)'tt -1t - pF : transition functionINDRODUCTIONKeynes :The contractions in an economy are more violent but also more short-lived than the expansions, so that GNP follows an asymmetric cyclical process.STARIf busine

3、ss cycles are inherently nonlinear, then theory leading to linear ms of business cycles has to given up as inadequate.3必要時進修正Specification of STAR Ms用LM test 做非線拒絕性檢定拒絕使用線性模型做分析，直接估計報告結果用診斷性檢定與衝擊反應分析模型之適合估計STAR之選擇轉移函F(.)及轉換變設定線性AR(P)LSTAR & ESTAR(1+ exp- g (yt-d - c)-1, g > 0LogisticF =1- exp

4、(- g (yt-d - c)2 ), g > 0Exponential4Linearity TestUsingAICNonlinear Mling of Industrial Productionn DataEurope is an aggregate ofn The Log European countries that are dustrial productiOECD members.ntries andEurope.n The quarterly observations are from the period 1960(1) to 1986(4) and they are s

5、easonally unadjusted values.n Linearity Testsn Discussion of the Estimated M s5United States-1y(t) y(t-4) / y(t-4)n Four-quarter differences of the quarterly logarithmic index of US industrial production, 1962(1) to 1988(4).n Estimated LSTAR m.n Residuals of linearity m(AR6) and LSTAR m.Discussion o

6、f the Estimated Msn Estimated Nonlinearity Mn Compute the roots of the STAR(P) mby solvingpz p - å(p1 j + p2 j F )z p- jfor F=0, 1j=1n Long Run properties of the myt = f (yt-1,.yt- p ;q)q = (p10 ,p' ,p,p' ,g, c)12026United States-3 LSTARyt = -0.021+ 0.3+ 0.021+1.16yt-() + ut(10)s = 0.01

7、76s2 / s2 = 0.64Ls : the residual standard deviation of (10)sL : the residual standard deviation of the AR(6)United States-27Europe-1n Four-quarter differences of the quarterly logarithmic index of US industrial production, 1962(1) to 1988(4).n Estimated LSTAR m.n Residuals of linearity m(AR9) and L

8、STAR m.United States-4n Residual of (10) and an AR(6)8Europe-3 LSTARyt = 0.44 yt -1 -1.17 yt -4 + 0.56 yt-5 - 0.89 yt-8 + 0.26 yt -9+ (0.69 yt -1 + 0.47 yt -4 + 0.59 yt -8 )´ (1+ exp- 5.4 ´ 59.5(yt-3 + 0.0022)-1 + ut(11)s = 0.0165s2 / s2 = 0.93Ls : the residual standard deviation of (10)sL

9、 : the residual standard deviation of the AR(6)Europe-2n Four-quarter difference of the quarterly logarithmic indexof European industrial production.(1962(1)-1988(4)9Japan-1n Four-quarter differences of the quarterly logarithmic index of Japan industrial production, 1962(1) to 1988(4).n Estimated ES

10、TAR m.n Residuals of linearity m(AR5) and LSTAR m.Europe-4n Residuals of (11) and an AR(9).10Japan-3 ESTARyt = 0.0075 + 3.03yt -1 -1.31yt-2 - 0.49Dyt -4+ (-1.68 yt -1 + 0.87 yt -2 - 0.30Dyt-8 )´ (1- exp-1.54 ´196(yt -1 + 0.082)2 )+ ut(12)s = 0.0185s2 / s2 = 0.78Ls : the residual standard d

11、eviation of (12)sL : the residual standard deviation of the AR(6)Japan-211Limiting behavior (t¥)USSP=Unique stable singularU=Upper regime M=Mid-regime O=Outer regimepointL=Lower regimeJapan-4n Residuals of (11) and an AR(9).12Forecasting with STAR ms- 1987(1)Our alternative ihenonlinear mhas su

12、periorforecasting performance.Europe Japan Belgium CanadaFR Germany ItalyUnited StatesUnited statesThe starting values for the series depicted by the joined line are the nine four-quarter differences oft eh logarithmic index of US industrial production preceding 1980(1).The starting values for the s

13、eries depicted by the dashed line are the same , except that the 1979(4) value lhs been perturbed by 0.001.13-2n All of these results are, of course, rather time- specific, because the two domination features during the relatively short observation period are the exogenous shocks associated with the first and second oil crises.n It is possible that linearity is rejected because of outliers, and STAR mling of the series is therefore not appropriate.-1n Most of the ms suggest that the dynamics of the process during recessio