Introduction To Modern Time Series Analysis

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Introduction to Modern Time Series AnalysisGebhard Kirchgässner, Jürgen Wolters and Uwe HasslerSecond EditionSpringer 2013Teaching MaterialThe following figures and tables are from the above book. They are provided to help instructors andstudents and may be used for teaching purposes as long as a reference to the book is given in class.Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

1 Introduction and Basicsbn 010Figure 1.1: Real Gross Domestic Product of the Federal Republicof Germany in billions of Euro, 1960 – 2011bn ure 1.2: Quarterly Changes of the Real Gross Domestic Product (ΔGDP)of the Federal Republic of Germany, 1960 – 1989Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

gure 1.3: Quarterly Growth Rates of the Real Gross Domestic Product (qgr)of the Federal Republic of Germany, 1960 – 1989bn Euro20151050-5-1019651970197519801985yearFigure 1.4: Annual Changes of the Real Gross Domestic Product (Δ 4GDP) of the Federal Republic of Germany, 1961 – 1989Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

percent1086420-2-41965197019751980year1985Figure 1.5: Annual Growth Rates of Real Gross Domestic Product (agr) of the Federal Republic of Germany, 1960 – 1989bn igure 1.6: ‘Smooth Component‘ and actual values of the Real Gross Domestic Product of the Federal Republic of Germany, 1961 – 2011Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

50403020100-1020406080100-20-30-40-50Figure 1.7:Example of a Random Walk where only the steps 1and –1 are possibleSource: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

CHF/USD3.532.521.510.51974 1978 1982 1986 1990 1994 1998 2002 2006 2010yeara) Exchange Rate CHF/USD 1974 – 20112011percent201510501974 1978 1982 1986 1990 1994 1998 2002 2006 2010-5year-10-15ˆ 1b) Continuous Returns CHF/USD0.750.50.25 0-0.255101520-0.5-0.75-1c) Estimated AutocorrelationsFigure 1.8:Exchange Rate Swiss Franc U.S. Dollar,Monthly data, January 1974 to December 2011Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

2 Univariate Stationary Processesxt7.552.5t0-2.5-5-7.5a) Realisation 10.80.60.40.205-0.2-0.4101520 20 b) Theoretical autocorrelation functionˆ 10.80.60.40.2051015-0.2-0.4c) Estimated autocorrelation functionwith confidence intervalsFigure 2.1: AR(1) process with α 0.9Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

xt42t0-2-4a) Realisation 10.80.60.40.2 0-0.25101520-0.4-0.6-0.8-1b) Theoretical autocorrelation functionˆ 10.80.60.40.2 0-0.25101520-0.4-0.6-0.8-1c) Estimated autocorrelation functionwith confidence intervalsFigure 2.2: AR(1) process with α 0.5Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

xt52.50t-2.5-5a) Realisation 10.80.60.40.2 0-0.25101520-0.4-0.6-0.8-1b) Theoretical autocorrelation functionˆ 10.80.60.40.2 0-0.25101520-0.4-0.6-0.8-1c) Estimated autocorrelation functionwith confidence intervalsFigure 2.3: AR(1) process with α -0.9Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

981a) Popularity of the CDU/CSU, 1971 – 1982 ˆ ( )10.80.60.40.2 0-0.25101520-0.4-0.6-0.8-1b) Autocorrelation ( ) and partial (autocorrelation functions withconfidence intervals) ˆ ( )10.80.60.40.2 0-0.25101520-0.4-0.6-0.8-1c) Estimated autocorrelation function of theresiduals of the estimated AR(1)-processwith confidence intervalsFigure 2.4: Popularity of the CDU/CSU, 1971 – 1982Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

xt105t0-5-10a) Realisation 10.80.60.40.2 05-0.2101520-0.4b) Theoretical autocorrelation functionˆ 10.80.60.40.2 0-0.25101520-0.4c) Estimated autocorrelation functionwith confidence intervalsFigure 2.5: AR(2) process with α1 1.5, α2 -0.56Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

xt52.5t0-2.5-5a) Realisation 10.80.60.40.20-0.2-0.4-0.6-0.8-1 5101520b) Theoretical autocorrelation functionˆ 10.80.60.40.20-0.2-0.4-0.6-0.8-1 5101520c) Estimated autocorrelation functionwith confidence intervalsFigure 2.6: AR(2) process with α1 1.4 and α2 -0.85Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

Percent16141210864201970 ˆ ( )year 19751980198519901995a) Three month money market rate in Frankfurt1970 – 199810.80.60.40.2 0-0.25101520-0.4-0.6-0.8-1b) Estimated autocorrelation ( ) and partialautocorrelation ( ) functions with confidenceintervals ˆ ( )10.80.60.40.2 0-0.25101520-0.4-0.6-0.8-1c) Estimated autocorrelation function of theresiduals of the estimated AR(2)-processwith confidence intervalsFigure 2.7: Three month money market rate in Frankfurt, 1970 – 1998Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

AR(1) process with 0.9 kk10.80.60.40.20-0.2-0.4-0.6-0.8-1k510AR(1) process with -0.9 kk10.80.60.40.20-0.2-0.4-0.6-0.8-1k510AR(2) process with 1 1.5, 2 -0.56 kk10.80.60.40.20-0.2-0.4-0.6-0.8-1k5101520AR(2) process with 1 1.4, 2 -0.85Figure 2.8: Estimated partial autocorrelation functionsTable 2.1: Characteristics of the Autocorrelation and the PartialAutocorrelation Functions of AR and MA ProcessesAutocorrelation FunctionPartial AutocorrelationFunctionMA(q)breaks off with qdoes not break offAR(p)does not break offbreaks off with pSource: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

10.80.60.40.20-0.2-0.4-0.6-0.8-1 5101520152015201520 10.80.60.40.20-0.2-0.4-0.6-0.8-1 510 10.80.60.40.20-0.2-0.4-0.6-0.8-1 510 10.80.60.40.20-0.2-0.4-0.6-0.8-1 510 Figure 2.9: Theoretical autocorrelation functions of ARMA(1,1) processesSource: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

Percent87654321year01994 1996199820002002a) New York three month money market rate,1994 – 200310.80.60.40.2 0-0.25101520-0.4-0.6-0.8-1b) Autocorrelation ( ) and partial ( )autocorrelation functions of the firstdifferences with confidence intervalsˆ 10.80.60.40.2 0-0.25101520-0.4-0.6-0.8-1c) Autocorrelation function of the residualsof the estimated ARMA(1,1)-processwith confidence intervalsFigure 2.10: Three month money market rate in New York, 1994 – 2003Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

Table 2.2: Forecasts of the Council of Economic Expertsand of the Economic Research InstitutesPeriodInstitutesCouncil ofEconomicExpertsR2RMSEMAEMEâ1U1970 – 1995 0.3691.8381.346-0.250*1.005*0.5721970 – 1982 0.4292.2911.654-0.7311.193*0.6251983 – 1995 0.3991.2291.0380.2311.0810.4571970 – 1995 0.502*1.647*1.171*-0.2561.1140.512*1970 – 1982 0.599*2.025*1.477*-0.723*1.3540.552*1983 – 1995 0.472*1.150*0.865*0.212*1.036*0.428*‘*’ denotes the ‘better’ of the two forecasts.Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

3 Granger Causalitypercent86Growth Rate of Real GDP420-2-4Interest Rate Spread (t-4)(GLR - GSR)-61970 1972 1974 1976 1978 1980 1982 1984 1986 1988yearFigure 3.1: Growth rate of real GDP and the four quarters lagged interest rate spread in the Federal Republic of Germany, 1970 – 1989 (in percent)Table 3.1 Test for Granger Causality (I): Direct Granger Procedure1/65 – 4/89, 100 Observationsyxk1k2F(y x)F(y x)F(y–x) 4ln(GDPr) 48910.099**882.521*1.17815.125*** 4ln(GDPr) 4ln(M1r)GLR – GSRGLR – GSR‘(*)’, ‘*’, ‘**’, or ‘***’ denote that the null hypothesis that no causal relation exists canbe rejected at the 10, 5, 1 or 0.1 percent significance level, respectively.Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

Figure 3.2. (The dotted lines are the approximate 95 percent confidence intervals.)ρ̂(k)kFigure 3.2a: Cross-correlations between the residuals of the univariate models of GDP and the quantityof money M1ρ̂(k)kFigure 3.2b: Cross-correlations between the residuals of the univariate models of GDP and theinterest rate spreadSource: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

ρ̂(k)kFigure 3.2c: Cross-correlations between the residuals of the univariate models of the quantity ofmoney M1 and the interest rate differentialTable 3.2: Test for Granger Causality (II): Haugh-Pierce Test1/65 – 4/89, 100 ObservationsYx 4ln(GDPr) 4ln(M1r) 4ln(GDPr) 4ln(M1r)GLR – GSRGLR – GSR ̂(0)0.179(*)0.0760.383***kS(y x)S(y 9.660*36.295***814.424(*) 11.270S(y x)40.362**‘(*)’, ‘*’, ‘**’, or. ‘***’ denote that the null hypothesis that no causal relation existscan be rejected at the 10, 5, 1 or 0.1 percent significance level, respectively.Table 3.3: Optimal Lag Length for the Hsiao ProcedureAkaike Criterion*1kSchwarz Criterion*2k*1k1*k *2k1*Relationk 4ln(M1r) 4ln(GDPr)411111 4ln(GDPr) 4ln(M1r)538404(GLR – GSR) 4ln(GDPr)421121 4ln(GDPr) (GLR – GSR)555505Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

Table 3.4: Models Estimated with the Hsiao Procedure1/65 – 4/89, 100 ObservationsCriterionExplanatory VariableAkaike CriterionSchwarz CriterionDependent Variable 4ln(GDPr,t) 4ln(M1r,t) 4ln(GDPr,t)Constant term0.146(0.67)1.263***(3.42)0.136(0.62) 4ln(GDPr, t-1)0.751***(13.59)-0.195(1.32)0.756***(13.68) 4ln(GDPr,t-2)-0.283(1.65) 4ln(GDPr,t-3)0.369*(2.54) 4.61) 4ln(M1r,t)1.139***(3.94)0.972***(10.12) 4ln(M1r,t-2)-0.173(1.29)-0.135(0.99) 4ln(M1r,t-3)0.185(1.36)0.083(0.61) 4ln(M1r,t-4)-0.478***(3.53)-0.265**(2.72) 4ln(M1r,t-5)0.340*(2.50) 4ln(M1r,t-6)-0.188(1.36) 4ln(M1r,t-7)0.192(1.41) 4ln(M1r,t-8)-0.203*(2.08) ̂ 18The numbers in parentheses are the absolute values of the estimated t statistics. ‘(*)’,‘*’, ‘**’, or ‘***’ denote that the corresponding null hypothesis can be rejected at the10, 5, 1 or 0.1 percent significance level, respectively. m denotes the number of degrees of freedom of the Q statistic.Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

Table 3.5: Models Estimated with the Hsiao Procedure1/65 – 4/89, 100 ObservationsCriterionExplanatory VariableAkaike CriterionSchwarz CriterionDependent Variable 4ln(GDPr,t)(GLR – GSR)t 4ln(GDPr,t)(GLR – GSR)tConstant 3) 4ln(GDPr, t-1)0.730***(12.22)-0.034(0.65)0.733***(12.27) 4ln(GDPr,t-2)-0.132*(2.10) 4ln(GDPr,t-3)0.021(0.32) 4ln(GDPr,t-4)0.154*(2.58) 4ln(GDPr,t-5)-0.083(*)(1.72)(GLR – 8***(12.13)(GLR – (1.42)(GLR – GSR)t-3-0.347**(2.69)-0.316*(2.32)(GLR – GSR)t-40.481***(3.70)0.448**(3.25)(GLR – GSR)t-5-0.274**(2.95)-0.327***(3.53) ̂ 10.6840.7980.7321.3620.7684.82416.6487.1187117The numbers in parentheses are the absolute values of the estimated t statistics. ‘(*)’,‘*’, ‘**’, or ‘***’ denote that the corresponding null hypothesis can be rejected at the10, 5, 1 or 0.1 percent significance level, respectively. m denotes the number of degrees of freedom of the Q statistic.Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

Table 3.6: Test for Granger Causality:Direct Granger Procedure with Three Variables1/65 – 4/89, 100 ObservationsYxzk 4ln(GDPr) 4ln(M1r)GLR – *83.432**1.0098.150*** 4ln(GDPr) 4ln(M1r)GLR – GSRGLR – GSR 4ln(M1r) 4ln(GDPr)F(y x)F(y x)F(y–x)‘(*)’, ‘*’, ‘**’, or ‘***’ denote that the null hypothesis that no causal relation exists canbe rejected at the 10, 5, 1 or 0.1 percent significance level, respectively.Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

4 Vector Autoregressive ProcessesResponse of X1 to X1Response of X1 to X21.21.20.80.80.40.40.00.0-0.4-0.42 4 6 8 10 12 14 16 18 202 4 6 8 10 12 14 16 18 20Response of X2 to X11.21.00.80.60.40.20.0-0.2Response of X2 to X21.21.00.80.60.40.20.0-0.22 4 6 8 10 12 14 16 18 202 4 6 8 10 12 14 16 18 20Figure 4.1: Impulse response functionsSource: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

Accumulated Response of X1 to X1Accumulated Response of X1 to X233221100-1-1-2-2-3-3-4-451015205Accumulated Response of X2 to X16554433221100-1-110151520Accumulated Response of X2 to X26510205101520Figure 4.2: Cumulative impulse response functionsResponse of DLGDPR to DLGDPRResponse of DLGDPR to DLM1RResponse of DLGDPR to 0.4-0.4-0.4-0.8-0.85101520-0.85Response of DLM1R to DLGDPR1015205Response of DLM1R to DLM1R333222111000-1-1-1-2-251015201015205Response of GLSR to -0.8510152020101520Response of GLSR to GLSR1.2-0.815-25Response of GLSR to DLGDPR10Response of DLM1R to GLSR-0.851015205101520Figure 4.3: Impulse response functionsSource: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

Accummulated Responseof DLGDPR to DLGDPRAccummulated Responseof DLGDPR to GLSRAccummulated Responseof DLGDPR to cummulated Responseof GLSR to DLGDPR101586420-2-451015520Accummulated Responseof GLSR to DLM1R86420-2-420152086420-2-452010Accummulated Responseof DLM1R to GLSR86420-2-486420-2-45520Accummulated Responseof DLM1R to DLM1RAccummulated Responseof DLM1R to DLGDPR101520Accummulated Responseof GLSR to GLSR86420-2-42 4 6 8 10 12 14 16 18 205101520Figure 4.4: Cumulative impulse response functionsSource: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

Table 4.1: Variance DecompositionForecast horizonImmediate4 periods8 periods20 74880.252Table 4.2a: Variance Decomposition1/65 – 4/89, 100 ObservationsΔ4ln(GDPr)Forecast horizonimmediate1 year2 years5 yearsinfinityΔ4ln(M1r)GLR – 27.798GLR – 4ln(M1r)8.99441.33649.670GLR – 4ln(M1r)13.89634.91051.194GLR – 4ln(M1r)14.73835.24450.018GLR – Δ4ln(M1r)14.73335.25850.009GLR – GSR15.67713.07971.244Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

Table 4.2b: Variance Decomposition 1/65 – 4/89, 100 ObservationsΔ4ln(GDPr)Δ4ln(M1r)GLR – 000.000GLR – ln(M1r)8.99460.68530.321GLR – 4ln(M1r)13.89650.66935.435GLR – Δ4ln(M1r)14.73850.97034.292GLR – Δ4ln(M1r)14.73350.99934.269GLR – GSR15.67716.13668.188Forecast horizonimmediate1 year2 years5 yearsinfinitySource: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

5 Nonstationary ProcessesFigure 5.1:Linear and quadratic trend, superimposedby a pure random processSource: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

Figure 5.2: Realisations of AR(1) processes α 1.03 (------), α 0.97 (———)Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, UweIntroduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 2013ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

Figure 5.3: Random walk with (-----) and without (––––) driftSource: Kirchgässner,

Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, Uwe Introduction to Modern Time Series Analysis,

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Series 80 Mighty Mouse SuperNine 38999 Type Series 79 Series 792 HS MIL-DTL-83513 Series 88 SuperFly Series 28 HiPer-D Protocol Series 801 Series 802, 803 and 804 Series 805 Series 806 MIL-STD-1560 El Ochito / Quadrax

BRIGGS & STRATTON Fource 0.030RDJ7Y/130-137 . Single cylinder OHV engines — 97700 series, 99700 series, 197700 series, 28E700 series, 287700 series, 28N700 series, 28U700 series, 28W700 . 625e, 675ex, 725ex series, 625-675 series and Quantum 7.5-6.75 gross HP engines 0.020 RJ19LMC/130-065 HONDA