金融计量学——ARMA模型

金融计量学作业1对中国人均GDP同比建立ARMA模型：数据来源：国家统计局，中国人均GDP同比数据从1990年-2018年。折线图如图所示，是1990-2018年的中国人均GDP同比的折线图，1990年到1993年中国人均GDP同比一路增长，1993年到2000年中国人均GDP同比一路下跌，而后又增长，一直到2007年，2007年后中国人均GDP同比震荡下跌。CorrelogramofR2468101214909294969800020406081012141618R从中国人均GDP同比增长的SACF、SPACF以及Q统计量中的SPACF在一阶后出现截尾，并且ACF快速收敛到0，说明时间序列平稳，可以得到模型有AR（1）过程。于是构建AR（1）模型。并分别进行显著性检验、平稳性检验和残差自相关检验。显著性检验中，p值小于0.05，说明显著，解释变量对被解释变量有较好的解释意义。初步回归模型：（Rt-9）=0.6（Rt-1-9）+μtμ是随机扰动项，R是中国人均GDP同比。Date:10/09/19Time:19:00Sample:19902018Includedobservations:29AutocorrelationPartialCorrelationACPACQ-StatProb10.5820.58210.8700.00120.180-0.24011.9440.00330.0280.06111.9700.0074-0.095-0.16012.2940.0155-0.203-0.10213.8370.0176-0.279-0.15016.8810.0107-0.289-0.08520.2870.0058-0.236-0.06422.6690.0049-0.176-0.07424.0680.00410-0.114-0.05124.6860.00611-0.040-0.03224.7650.010120.0690.04725.0200.015DependentVariable:RMethod:ARMAConditionalLeastSquares(Gauss-Newton/Marquardtsteps)Date:10/09/19Time:19:05Sample(adjusted):19912018Includedobservations:28afteradjustmentsConvergenceachievedafter2iterationsCoefficientcovariancecomputedusingouterproductofgradientsVariableCoefficientStd.Errort-StatisticProb.C9.0354750.78688311.482610.0000AR(1)0.6018640.1288714.6702720.0001R-squared0.456197Meandependentvar8.835714AdjustedR-squared0.435282S.D.dependentvar2.185341S.E.ofregression1.642234Akaikeinfocriterion3.898742Sumsquaredresid70.12028Schwarzcriterion3.993899Loglikelihood-52.58238Hannan-Quinncriter.3.927832F-statistic21.81144Durbin-Watsonstat0.921676Prob(F-statistic)0.000080InvertedARRoots.60特征方程的根在单位圆内，说明平稳。-1.5-1.0-0.50.00.51.01.5-1.5-1.0-0.50.00.51.01.5ARrootsInverseRootsofAR/MAPolynomial(s)Breusch-GodfreySerialCorrelationLMTest:F-statistic8.302936Prob.F(2,24)0.0018Obs*R-squared11.45067Prob.Chi-Square(2)0.0033TestEquation:DependentVariable:RESIDMethod:LeastSquaresDate:10/09/19Time:19:09Sample:19912018Includedobservations:28CoefficientcovariancecomputedusingouterproductofgradientsPresamplemissingvaluelaggedresidualssettozero.VariableCoefficientStd.Errort-StatisticProb.C-0.4865830.647594-0.7513700.4597AR(1)-0.4231810.160628-2.6345460.0145RESID(-1)0.7975730.2057513.8764050.0007RESID(-2)0.2169130.2185630.9924550.3309R-squared0.408952Meandependentvar-1.43E-16AdjustedR-squared0.335072S.D.dependentvar1.611536S.E.ofregression1.314097Akaikeinfocriterion3.515740Sumsquaredresid41.44442Schwarzcriterion3.706055Loglikelihood-45.22036Hannan-Quinncriter.3.573921F-statistic5.535291Durbin-Watsonstat2.113876Prob(F-statistic)0.004932对AR（1）模型的残差自相关作LM检验，由F统计量的p值小于0.05,拒绝原假设，说明存在残差自相关。CorrelogramofResiduals从残差的ACF、PACF及Q统计量图中的PACF可知，ARMA模型中MA的阶数是1。于是，可以构建ARMA（1，1）模型。Date:10/09/19Time:19:14Sample:19902018Includedobservations:28Q-statisticprobabilitiesadjustedfor1ARMAtermAutocorrelationPartialCorrelationACPACQ-StatProb10.4740.4746.994320.142-0.1077.64500.00630.1230.1308.15530.01740.037-0.0878.20440.0425-0.172-0.2049.28370.0546-0.243-0.09711.5430.0427-0.235-0.10713.7610.0328-0.240-0.08516.1800.0249-0.189-0.01517.7650.02310-0.0660.03717.9700.03611-0.072-0.11318.2250.05112-0.0080.04118.2280.076从图中可知，解释变量对中国人均GDP同比有较好的解释意义。由于特征方程的根都落在单位圆内，说明模型平稳，并且可逆。DependentVariable:RMethod:ARMAConditionalLeastSquares(Gauss-Newton/Marquardtsteps)Date:10/09/19Time:19:20Sample(adjusted):19912018Includedobservations:28afteradjustmentsFailuretoimprovelikelihood(non-zerogradients)after15iterationsCoefficientcovariancecomputedusingouterproductofgradientsMABackcast:1990VariableCoefficientStd.Errort-StatisticProb.C8.9186370.80170511.124590.0000AR(1)0.4792440.1844132.5987550.0155MA(1)0.5320380.2040962.6068010.0152R-squared0.595868Meandependentvar8.835714AdjustedR-squared0.563538S.D.dependentvar2.185341S.E.ofregression1.443752Akaikeinfocriterion3.673324Sumsquaredresid52.11046Schwarzcriterion3.816060Loglikelihood-48.42654Hannan-Quinncriter.3.716960F-statistic18.43052Durbin-Watsonstat1.682752Prob(F-statistic)0.000012InvertedARRoots.48InvertedMARoots-.53-1.5-1.0-0.50.00.51.01.5-1.5-1.0-0.50.00.51.01.5ARrootsMArootsInverseRootsofAR/MAPolynomial(s)最后，对ARMA（1，1）模型作LM检验，发现F统计量不显著，接受原假设，说明没有残差自相关存在。中国人均GDP同比的ARMA(1，1)模型如下：（Rt-8.92）=0.48（Rt-1-8.92）+μt+0.53μt-1μ是随机扰动项，R是中国人均GDP同比。Breusch-GodfreySerialCorrelationLMTest:F-statistic1.958758Prob.F(2,23)0.1639Obs*R-squared4.075057Prob.Chi-Square(2)0.1304TestEquation:DependentVariable:RESIDMethod:LeastSquaresDate:10/09/19Time:19:24Sample:19912018Includedobservations:28CoefficientcovariancecomputedusingouterproductofgradientsPresamplemissingvaluelaggedresidualssettozero.VariableCoefficientStd.Errort-StatisticProb.C-0.3745830.795469-0.4708960.6422AR(1)-0.5014310.312828-1.6028960.1226MA(1)0.5121900.6341790.8076420.4276RESID(-1)0.0253280.5950810.0425630.9664RESID(-2)0.7166000.4871181.4711030.1548R-squared0.145538Meandependentvar0.000160AdjustedR-squared-0.003064S.D.dependentvar1.389251S.E.ofregression1.391378Akaikeinfocriterion3.658898Sumsquaredresid44.52642Schwarzcriterion3.896792Loglikelihood-46.22458Hannan-Quinncriter.3.731625F-statistic0.979379Durbin-Watsonstat2.308995Prob(F-statistic)0.438105指标名称人均GDP同比频度年年单