Improved Point Distribution Model with Curvature for Boundary Finding

《统计学》_各章关键术语（中英⽂对照）第⼆部分各章关键术语（中英⽂对照）第1章统计学（statistics）随机性（randomness）描述统计学（descriptive statistics）推断统计学（inferential statistics）总体（population）母体（parent）(parent population）样本、⼦样（sample）调查对象总体（respondents population）有限总体（finite population）调查的理论总体（survey’s heoretical population）超总体（super population）变量（variable）数据（data）原始数据（original data）派⽣数据（derived data）定类尺度（nominal scale）定类尺度变量（nominal scale level variable）定类尺度数据（nominal scale level data）定序尺度（ordinal scale）定序尺度变量（ordinal scale level variable）定序尺度数据（ordinal scale level data）定距尺度（interval scale）定距尺度变量（interval scale level variable）定距尺度数据（interval scale level data）定⽐尺度（ratio scale）定⽐尺度变量（ratio scale level variable）定⽐尺度数据（ratio scale level data）分类变量（categorical variable）定性变量、属性变量（qualitative variable）数值变量（numerical variable）定量变量、数量变量（quantitative variable）绝对数变量（absolute number level variable）绝对数数据（absolute number level data）⽐率变量（ratio level variable）⽐率数据（ratio level data）实验数据（experimental data）调查数据（survey data）观察数据（observed data）第2章随机性（randomness）随机现象（random phenomenon）随机试验（random experiment）事件（event）基本事件（elementary event）复合事件（union of event）必然事件（certain event）不可能事件（impossible event）基本事件空间（elementary event space）互不相容事件（mutually exclusive events）统计独⽴（statistical independent）统计相依（statistical dependence）概率（probability）古典⽅法概率（classical method probability）相对频数⽅法概率（relative frequency method probability）主观⽅法概率（subjective method probability）⼏何概率（geometric probability）条件概率（conditional probability）全概率公式（formula of total probability）贝叶斯公式（Bayes’ formula）先验概率（prior probability）后验概率（posterior probability）随机变量（random variable）离散型随机变量（discrete type random variable）连续型随机变量（continuous type random variable）概率分布（probability distribution）特征数（characteristic number）位置特征数（location characteristic number）数学期望（mathematical expectation）散布特征数（scatter characteristic number）⽅差（variance）标准差（standard deviation）变异系数（variable coefficient）贝努⾥分布（Bernoulli distribution）⼆点分布（two-point distribution） 0-1分布（zero-one distribution）贝努⾥试验（Bernoulli trials）⼆项分布（binomial distribution）超⼏何分布（hyper-geometric distribution）正态分布（normal distribution）正态概率密度函数（normal probability density function）正态概率密度曲线（normal probability density curve）正态随机变量（normal random variable）卡⽅分布（chi-square distribution）F_分布（F-distribution）t_分布（t-distribution） “学⽣”⽒t_分布（Student’s t-distribution）列联表（contingency table）联合概率分布（joint probability distribution）边缘概率分布（marginal probability distribution）条件分布（conditional distribution）协⽅差（covariance）相关系数（correlation coefficient）第3章统计调查（statistical survey）数据收集（collection of data）统计单位（statistical unit）统计个体（statistical individual）社会经济总体（socioeconomic population）调查对象总体（respondents population）有限总体（finite population）标志（character）标志值（character value）属性标志（attributive character ）品质标志（qualitative character ）数量标志（numerical indication）不变标志（invariant indication）变异（variation）调查条⽬（item of survey）指标（indicator）统计指标（statistical indicator）总量指标（total amount indicator）绝对数（absolute number）统计单位总量（total amount of statistical unit ）标志值总量（total amount of indication value）（total amount of character value）时期性总量指标（time period total amount indicator）流量指标（flow indicator）时点性总量指标（time point total amount indicator）存量指标（stock indicator）平均指标（average indicator）平均数（average number）相对指标（relative indicator）相对数（relative number）动态相对指标（dynamic relative indicator）发展速度（speed of development）增长速度（speed of growth）增长量（growth amount）百分点（percentage point）计划完成相对指标（relative indicator of fulfilling plan）⽐较相对指标（comparison relative indicator）结构相对指标（structural relative indicator）强度相对指标（intensity relative indicator）基期（base period）报告期（given period）分组（classification）（grouping）统计分组（statistical classification）（statistical grouping）组（class）（group）分组设计（class divisible design）（group divisible design）互斥性（mutually exclusive）包容性（hold）分组标志（classification character）（grouping character）按品质标志分组（classification by qualitative character）（grouping by qualitative character）按数量标志分组（classification by numerical indication）（grouping by numerical indication）离散型分组标志（discrete classification character）（discrete grouping character）连续型分组标志（continuous classification character）（continuous grouping character）单项式分组设计（single-valued class divisible design）（single-valued group divisible design）组距式分组设计（class interval divisible design）（group interval divisible design）组界（class boundary）（group boundary）频数（frequency）（frequency number）频率（frequency）组距（class interval）（group interval）组限（class limit）（group limit）下限（lower limit）上限（upper limit）组中值（class mid-value）（group mid-value）开⼝组（open class）（open-end class）（open-end group）开⼝式分组（open-end grouping）等距式分组设计（equal class interval divisible design）（equal group interval divisible design）不等距分组设计（unequal class interval divisible design）（unequal group interval divisible design）调查⽅案（survey plan）抽样调查（sample survey）有限总体概率抽样（probability sampling in finite populations）抽样单位（sampling unit）个体抽样（elements sampling）等距抽样（systematic sampling）整群抽样（cluster sampling）放回抽样（sampling with replacement）不放回抽样（sampling without replacement）分层抽样（stratified sampling）概率样本（probability sample）样本统计量（sample statistic）估计量（estimator）估计值（estimate）⽆偏估计量（unbiased estimator）有偏估计量（biased estimator）偏差（bias）精度（degree of precision）估计量的⽅差（variance of estimates）标准误（standard error）准确度（degree of accuracy）均⽅误差（mean square error）估计（estimation）点估计（point estimation）区间估计（interval estimate）置信区间（confidence interval）置信下限（confidence lower limit）置信上限（confidence upper limit）置信概率（confidence probability）总体均值（population mean）总体总值（population total）总体⽐例（population proportion）总体⽐率（population ratio）简单随机抽样（simple random sampling）简单随机样本（simple random sample）研究域（domains of study）⼦总体（subpopulations）抽样框（frame）估计量的估计⽅差（estimated variance of estimates）第4章频数（frequency）（frequency number）频率（frequency）分布列（distribution series）经验分布（empirical distribution）理论分布（theoretical distribution）品质型数据分布列（qualitative data distribution series）数量型数据分布列（quantitative data distribution series）单项式数列（single-valued distribution series）组距式数列（class interval distribution series）频率密度（frequency density）分布棒图（bar graph of distribution）分布直⽅图（histogram of distribution）分布折线图（polygon of distribution）累积分布数列（cumulative distribution series）累积分布图（polygon of cumulative distribution）位置特征（location characteristic）位置特征数（location characteristic number）平均值、均值（mean）平均数（average number）权数（weight number）加权算术平均数（weighted arithmetic average）加权算术平均值（weighted arithmetic mean）简单算术平均数（simple arithmetic average）简单算术平均值（simple arithmetic mean）加权调和平均数（weighted harmonic average）加权调和平均值（weighted harmonic mean）简单调和平均数（simple harmonic average）简单调和平均值（simple harmonic mean）加权⼏何平均数（weighted geometric average）加权⼏何平均值（weighted geometric mean）简单⼏何平均数（simple geometric average）简单⼏何平均值（simple geometric mean）绝对数数据（absolute number data）⽐率类型数据（ratio level data）中位数（median）众数（mode）耐抗性（resistance）散布特征（scatter characteristic）散布特征数（scatter characteristic number）极差、全距（range）四分位差（quartile deviation）四分间距（inter-quartile range）上四分位数（upper quartile）下四分位数（lower quartile）在外截断点（outside cutoffs）平均差（mean deviation）⽅差（variance）标准差（standard deviation）变异系数（variable coefficient）第5章随机样本（random sample）简单随机样本（simple random sample）参数估计（parameter estimation）矩（moment）矩估计（moment estimation）修正样本⽅差（modified sample variance）极⼤似然估计（maximum likelihood estimate）参数空间（space of paramete）似然函数（likelihood function）似然⽅程（likelihood equation）点估计（point estimation）区间估计（interval estimation）假设检验（test of hypothesis）原假设（null hypothesis）备择假设（alternative hypothesis）检验统计量（statistic for test）观察到的显著⽔平（observed significance level）显著性检验（test of significance）显著⽔平标准（critical of significance level）临界值（critical value）拒绝域（rejection region）接受域（acceptance region）临界值检验规则（test regulation by critical value）双尾检验（two-tailed tests）显著⽔平（significance level）单尾检验（one-tailed tests）第⼀类错误（first-kind error）第⼀类错误概率（probability of first-kind error）第⼆类错误（second-kind error）第⼆类错误概率（probability of second-kind error）P_值（P_value）P_值检验规则（test regulation by P_value）经典统计学（classical statistics）贝叶斯统计学（Bayesian statistics）第6章⽅差分析（analysis of variance，ANOVA）⽅差分析恒等式（analysis of variance identity equation）单因⼦⽅差分析（one-factor analysis of variance）双因⼦⽅差分析（two-factor analysis of variance）总变差平⽅和（total variation sum of squares）总平⽅和SST（total sum of squares）组间变差平⽅和（among class(group) variation sum of squares），回归平⽅和SSR（regression sum of squares）组内变差平⽅和（within variation sum of squares）误差平⽅和SSE（error sum ofsquares）⽪尔逊χ2统计量（Pearson’s chi-statistic）分布拟合（fitting of distrbution）分布拟合检验（test of fitting of distrbution）⽪尔逊χ2检验（Pearson’s chi-square test）列联表（contingency table）独⽴性检验（test of independence）数量变量（quantitative variable）属性变量（qualitative variable）对数线性模型（loglinear model）回归分析（regression analysis）随机项（random term）随机扰动项（random disturbance term）回归系数（regression coefficient）总体⼀元线性回归模型（population linear regression model with a single regressor）总体多元线性回归模型（population multiple regression model with a single regressor）完全多重共线性（perfect multicollinearity）遗漏变量（omitted variable）遗漏变量偏差（omitted variable bias）⾯板数据（panel data）⾯板数据回归（panel data regressions）⼯具变量（instrumental variable）⼯具变量回归（instrumental variable regressions）两阶段最⼩平⽅估计量（two stage least squares estimator）随机化实验（randomized experiment）准实验（quasi-experiment）⾃然实验（natural experiment）普通最⼩平⽅准则（ordinary least squares criterion）最⼩平⽅准则（least squares criterion）普通最⼩平⽅（ordinary least squares，OLS）最⼩平⽅（least squares）最⼩平⽅法（least squares method）第7章简单总体（simple population）复合总体（combined population）个体指数：价⽐（price relative），量⽐（quantity relative）总指数（general index）（combined index）统计指数（statistical indices）类指数、组指数（class index）动态指数（dynamic index）⽐较指数（comparison index）计划完成指数（index of fulfilling plan）数量指标指数（quantitative indicator index）物量指数（quantitative index）（quantity index）（quantum index）质量指标指数（qualitative indicator index）价格指数、物价指数（price index）综合指数（aggregative index）（composite index）拉斯贝尔指数（Laspeyres’ index）派许指数（Paasche’s index）阿斯·杨指数（Arthur Young’s index）马歇尔—埃奇沃斯指数（Marshall-Edgeworth’s index）理想指数（ideal index）加权综合指数（weighted aggregate index）平均指数（average index）加权算术平均指数（weighted arithmetic average index）加权调和平均指数（weighted harmonic average index）因⼦互换（factor-reversal）购买⼒平价（purchasing power parity，PPP）环⽐指数（chain index）定基指数（fixed base index）连环替代因素分析法（factor analysis by chain substitution method）不变结构指数、固定构成指数（index of invariable construction）结构指数、结构影响指数（structural index）第8章截⾯数据（cross-section data）时序数据（time series data）动态数据（dynamic data）时间数列（time series）发展⽔平（level of development）基期⽔平（level of base period）报告期⽔平（level of given period）平均发展⽔平（average level of development）序时平均数（chronological average）增长量（growth quantity）平均增长量（average growth amount）发展速度（speed of development）增长速度（speed of growth）增长率（growth rate）环⽐发展速度（chained speed of development）定基发展速度（fixed base speed of development）环⽐增长速度（chained growth speed）定基增长速度（fixed base growth speed）平均发展速度（average speed of development）平均增长速度（average speed of growth）平均增长率（average growth rate）算术图（arithmetic chart）半对数图（semilog graph）时间数列散点图（scatter diagram of time series）时间数列折线图（broken line graph of time series）⽔平型时间数列（horizontal patterns in time series data）趋势型时间数列（trend patterns in time series data）季节型时间数列（season patterns in time series data）趋势—季节型时间数列（trend-season patterns in time series data）⼀次指数平滑平均数（simple exponential smoothing mean）⼀次指数平滑法（simple exponential smoothing method）最⼩平⽅法（leas square method）最⼩平⽅准则（least squares criterion）原资料平均法（average of original data method）季节模型（seasonal model）（seasonal pattern）长期趋势（secular trends）季节变动（变差）（seasonal variation）季节波动（seasonal fluctuations）不规则变动（变差）（erratic variation）不规则波动（random fluctuations）时间数列加法模型（additive model of time series）时间数列乘法模型（multiplicative model of time series）。

Graduate School of Business,University of ChicagoBusiness41202,Spring Quarter2008,Mr.Ruey S.TsaySolutions to MidtermProblem A:(30pts)Answer briefly the following questions.Each question has two points.1.Describe two methods for choosing a time series model.Answer:Any two of(a)Information criteria such as AIC or BIC,(b)Out-of-sample forecasts,and(c)ACF and PACF of the series.2.Describe two applications of volatility infinance.Answer:Any two of(a)derivative(option)pricing,(b)risk management,(c)portfolio selection or asset allocation.3.Give two applications of seasonal time series models infinance.Answer:(a)Earnings forecasts and(b)weather-related derivative pricing or risk man-agement.4.Describe two weaknesses of the ARCH models in modelling stock volatility.Answer:Any two of(a)symmetric response to past positive and negative shocks,(b)restrictive,(c)Not adaptive,and(d)provides no explanation about the source ofvolatility clustering.5.Give two empirical characteristics of daily stock returns.Answer:any two of(a)heavy tails,(b)non-Gaussian distribution,(c)volatility clus-tering.6.The daily simple returns of Stock A for the last week were0.02,0.01,-0.005,-0.01,and0.025,respectively.What is the weekly log return of the stock last week?What is theweekly simple return of the stock last week?Answer:Weekly log return is0.03938;weekly simple return is0.04017.7.Suppose the closing price of Stock B for the past three trading days were$100,$120,and$100,respectively.What is the arithmetic mean of the simple return of the stock for the past three days?What is the geometric average of the simple return of the stockfor the past three days? Answer:Arithmetic mean=12120−100100+100−120120=0.017.and the geometric mean is120×100−1=0.8.Consider the AR(1)model r t=0.02+0.8r t−1+a t,where the shock a t is normally distrib-uted with mean zero and variance1.What are the variance and lag-1autocorrelation function of r t?Answer:Var(r t)=11−0.82=2.78and the lag-1ACF is0.8.19.For problems6and7,suppose the daily return r t,in percentages,of Stock A followsthe model r t=1.0+a t+0.3a t−1,where a t=σt t withσ2t =1.0+0.4a2t−1and t beingstandard normal.What is the unconditional variance of a t?What is the variance of r t?Answer:Var(a t)=11−0.4=1.67.Var(r t)=(1+0.32)σ2a=1.82.10.Suppose that a n=3.0,what is the1-step ahead forecast for r n+1at the forecast originn?What is the1-step ahead volatility forecast of r t at the forecast origin n?Answer:r n(1)=1+0.3a n=1.9,andσ2n (1)=1+0.4a2n=4.6.11.Consider the simple AR(1)model r t=100+0.8r t−1+a t,where a t is normally distributedwith mean zero and variance10.Is the r t series mean-reverting?If yes,what is the half-life of the series?Answer:Yes,r t series is mean-reverting.The half-life is ln(0.5)/ln(0.8)=3.11.12.Describe two test statistics for testing the ARCH effect of an asset return series.Writedown the associated null hypotheses.Answer:(a)The Ljung-Box statistic Q(m)of the squared shocks,i.e.a2t .The nullhypothesis is H o:ρ1=ρ2=···=ρm=0,whereρi is the lag-i ACF of a2t .(b)TheEngle F-test for the regression a2t =β0+β1a2t−1+···+βm a2t−m+e t.The null hypothesisis H o:β1=β2=···=βm=0.13.Consider the following two IGARCH(1,1)models for percentage log returns:Model A:σ2t =1.0+0.1a2t−1+0.9σ2t−1Model B:σ2t =0.1a2t−1+0.9σ2t−1.Suppose thatσ2100=20and a100=−2.0.What are the3-step ahead volatility forecastsfor Models A and B?Answer:For model A:3-step ahead volatility forecast isσ2100(3)=2+(1.0+0.1×(−2.0)2+0.9×20)=21.4.For model B,the3-step ahead volatility forecast isσ2100(3)=0.1(−2.0)2+0.9×20=18.4.14.Consider the following two models for the log price of an asset:Model A:p t=p t−1+a tModel B:p t=0.00001+p t−1+a twhere the shock a t is normally distributed with mean zero and varianceσ2>0.Suppose further that p100=5.Let p n( )be the -step ahead forecast at the forecast origin n.What are the point forecasts p100( )for both models as →∞?Answer:For model A,p100( )=5for all .For model B,p100( )converges to infinity as →∞.215.Suppose that we have T =1000daily log returns for the Decile 1portfolio.Supposefurther that the sample autocorrelation at lag-12is ˆρ12=0.15.Test the hypothesis H o :ρ12=0against the alternative hypothesis H a :ρ12=pute the test statistic and draw your conclusion.Answer :t =0.151/√1000=√1000×0.15=4.74,which is highly significant.Thus,the lag-12ACF is not zero.Problem B .(20pts)It is well-known in economics that growth rate of the domestic gross product (GDP)is negatively correlated with the change in unemployment rate.Consider the U.S.quarterly real GDP and unemployment rate from the first quarter of 1948to the first quarter of 2008.Let dgdp t be the growth rate of the GDP,i.e.dgdp t =ln(GDP t )−ln(GDP t −1),and dun t be the change in unempolyment rate,i.e.dun t =U t −U t −1with U t being the civilian unemployment rate.The data were seasonally adjusted and obtained from the Federal Reserve Bank at St.Louis.The sample size after the differencing is e the attached R output to answer the following questions.1.(5points)Write down the fitted linear regression model with dgdp t and dun t representing the dependent and independent variable,respectively,including residual standard error.What is the R 2of the linear regression?Is the fitted model adequate?Why?Answer :The fitted linear regression isdgdp t =0.017−0.017dun t +e t ,ˆσe =0.0088.The R 2is 0.37.The model is not adequate because the Q (m )statistics of the residuals show that the residuals have serial correlations,i.e.Q (12)=219.4with p-value close to zero.2.(5points)To take care of the serial correlations in the residuals,a linear regression model with time-series errors is built for the two variables.Write down the fitted model,including the residual variance.Answer :The fitted linear regression model with time-series errors is(1−0.21B −0.12B 2)(1−0.86B 4)(dgdp t −0.017+0.018dun t )=(1−0.72B 4)a t ,ˆσ2a =6.01×10−5.3.(2points)Is the model in Question 2adequate?Why?Answer :Yes,the model is adequate.The Q (m )statistics of the residuals fail to indicate the existence of any serial correlations.We have Q (12)=17.53with p-value 0.13.4.(4points)Based on the fitted model in Question 2,is the growth rate of GDP negatively correlated with the change in unempolyment rate?Why?Answer :Yes,the growth rate of GDP is negatively related to the change in unemploy-ment rate.The estimated coefficient is −0.018which is highly significant,because it standard error 0.0014is small,resulting in a large t -ratio.35.(4points)To check the predictive power of the model,it was re-estimated using thefirst236data points.This re-fitted model is used to produce1-step to4-step ahead forecasts at the forecast origin t=236.The actual value of the GDP growth rates are also given.Construct the1-step ahead95%interval forecast of the model.Is the actual growth rate in the forecasting interval?Answer:The95%interval forecast is0.012±1.96×0.0077,i.e.[−0.0031,0.027].The actual value is0.0159,which is in the interval.Problem C.(16pts)Consider the quarterly earnings per share of the Microsoft stock from thefirst quarter of1992to thefirst quarter of2008.The data were obtained from First Call. To take the log transformation,we add0.5to all data points.The R output is attached. Let x t=ln(y t+0.5)be the transformed earnings,where y t is the actual earnings per share.1.(5points)Write down thefitted model for x t,including the variance of the residuals.Answer:Thefitted model is(1−B)r t=(1−0.70B+0.39B2)(1+0.39B4)a t,ˆσ2a=0.0016, where r t=ln(x t+0.5)with x t being the earnings per share.2.(2points)Is there any significant serial correlation in the residuals of thefitted model?Why?Answer:No,the Q(m)statistics of the residuals give Q(12)=9.65with p-value0.65.3.(4points)Let T=65be the forecast origin,where T is the sample size.Based on thefitted model,and,for simplicity,use the relationship y t=exp(x t)−0.5,what are the 1-step and2-step ahead forecasts of earnings per share for the Microsoft stock?Answer:The1-step and2-step earnings forecasts are0.56and0.54,respectively.4.(2points)Test the null hypothesis H o:θ4=0vs H a:θ4=0.What is the test statistic?Draw your conclusion.Answer:The test statistic is t=0.39120.1442=2.71with two-sided p-value0.0067.Thus,the seasonal MA coefficientθ4is significantly different from zero.5.(3points)Consider the regular(i.e.,non-seasonal)part of the MA model.Is it invertible?Why?Answer:Yes,it is invertible,because the polynomial1−0.6953x+0.3889x2has roots0.89±1.33i so that the absolute value of the roots(Mod in R)is1.6,which is greaterthan1.[If you compute the roots of x2−0.6953x+0.3889,the the absolute value of the roots is less than1.]Problem D.(34pts)Consider the daily log returns of the Starbucks stock,in percentages, from January1993to December2007.The relevant R output is attached.Answer the following questions.41.(2points)Is the mean log return significant different from zero?Why?Answer:No,the basic statistics show the95%confidence interval of the mean is [−0.0103,0.1624],which contains zero.2.(2points)Is there any serial correlation in the log return series?Why?Answer:Yes,the Q(m)statistics show Q(15)=38.39with p-value0.0008.3.(2points)An MA model is used to handle the mean equation,which appears to beadequate.Is there any ARCH effect in the return series?Why?Answer:Yes,because the Q(m)statistics of the squared residuals show Q(15)=112.61 with p-value close to zero.4.(6points)A GARCH(1,1)model with Student-t distribution is used for the volatilityequation.Write down thefitted model,including the degrees of freedom of the Student-t innovations and mean equation.Answer:Thefitted model isr t=0.037+a t−0.043a t−1−0.048a t−2,a t=σt t, ∼t5.27.σ2 t =0.012+0.026a2t−1+0.973σ2t−1.5.(4points)Since the constant term of the GARCH(1,1)model is not significantly differentfrom zero at the1%level,an IGARCH(1,1)model is used.Write down thefitted IGARCH(1,1)model,including the mean equation.Answer:Thefitted IGARCH(1,1)model isr t=0.077+a t−0.029a t−1−0.044a t−2,a t=σt t, t∼N(0,1).σ2 t =0.022a2t−1+0.978σ2t−1.6.(3points)Is the IGARCH(1,1)model adequate?Why?What is the3-step aheadvolatility forecast with the last data point as the forecast origin?Answer:Yes,the Q(m)statistics for the standardized residuals give Q(10)=1.77, Q(15)=10.79,and Q(20)=18.66.The p-values of these statistics are all greater than0.05.In addition,the Q(m)statistics of the squared standardized residuals also havelarge p-values.The3-step ahead volatility forecast is √3.779=1.94.7.(5points)A GJR(or TGARCH)model with Student-t distribution is alsofitted to thelog return series.Write down thefitted model,including the mean equation and all parameters.Answer:Thwfitted GJR model isr t=0.032+a t−0.043a t−1−0.048a t−2,a t=σt t, t∼t5.31.σ2 t =0.015+(0.021+0.017N t−1)a2t−1+0.970σ2t−1,where N t−1=0if a t−1≥0and=1,otherwise.58.(2points)Is the fitted GJR (or TGARCH)model adequate?Why?Answer :Yes,the Q (m )statistics of the standardized residuals and those of the squred standardized residuals all have large p-values.9.(2points)Among the GARCH(1,1),IGARCH(1,1)and GJR(1,1)models,which one is preferred?Why?Answer :The GJR(1,1)model because it has the smallest AIC value.10.(2points)Is the leverage effect of the GJR model significant?Why?Answer :Yes,the t -ratio of the leverage parameter is 2.01,which is significant at the 5%level.11.(4points)To better understand the leverage effect,use the fitted GJR to calculate theratio σ2t (a t −1=−5.10)σ2t(a t −1=5.10),assuming σ2t −1=7.5.Answer :σ2t (a t −1=−5.10)σ2t (a t −1=5.10)=0.0154+0.0379×(−5.10)2+0.97×7.50.0154+0.0208×(5.10)2+0.97×7.5=1.057.6。

训练集（trainset）验证集（validationset）测试集（testset）。

训练集(train set) 验证集(validation set) 集(test set)。

⼀般需要将样本分成独⽴的三部分训练集(train set)，验证集(validation set)和测试集(test set)。

其中训练集⽤来估计模型，验证集⽤来确定⽹络结构或者控制模型复杂程度的参数，⽽测试集则检验最终选择最优的模型的性能如何。

⼀个典型的划分是训练集占总样本的50％，⽽其它各占25％，三部分都是从样本中随机抽取。

样本少的时候，上⾯的划分就不合适了。

常⽤的是留少部分做测试集。

然后对其余N个样本采⽤K折交叉验证法。

就是将样本打乱，然后均匀分成K份，轮流选择其中K－1份训练，剩余的⼀份做验证，计算预测误差平⽅和，最后把K次的预测误差平⽅和再做平均作为选择最优模型结构的依据。

特别的K取N，就是留⼀法（leave one out）。

这三个名词在机器学习领域的⽂章中极其常见，但很多⼈对他们的概念并不是特别清楚，尤其是后两个经常被⼈混⽤。

Ripley, B.D（1996）在他的经典专著Pattern Recognition and Neural Networks中给出了这三个词的定义。

Training set: A set of examples used for learning, which is to fit the parameters [i.e., weights] of the classifier.Validation set: A set of examples used to tune the parameters [i.e., architecture, not weights] of a classifier, for example to choose the number of hidden units in a neural network.Test set: A set of examples used only to assess the performance [generalization] of a fully specified classifier.显然，training set是⽤来训练模型或确定模型参数的，如ANN中权值等； validation set是⽤来做模型选择（model selection），即做模型的最终优化及确定的，如ANN的结构；⽽ test set则纯粹是为了测试已经训练好的模型的推⼴能⼒。

英文原文：10The Markowitz Investment Portfolio Selection ModelThe first nine chapters of this book presented the basic probability theory with which any student of insurance and investments should be familiar. In this final chapter, we discuss an important application of the basic theory: the Nobel Prize winning investment portfolio selection model due to Harry Markowitz. This material is not discussed in other probability texts of this level; however, it is a nice application of the basic theory and it is very accessible.The Markowitz portfolio selection model has a profound effect on the investment industry. Indeed, the popularity of index funds (mutual funds that track the performance of an index such as the S&P 500 and do not attempt to “beat the market”) can be traced to a surprising consequence of the Markowitz model: that every investor, regardless of risk tolerance, should hold the same portfolio of risky securities. This result has called into question the conventional wisdom that it is possible to beat the market with the “right” investment manager and in so doing has revolutionized the investment industry.Our presentation of the Markowitz model is organized in the following way. We begin by considering portfolios of two securities. An important example of a portfolio of this type is one consisting of a stock mutual fund and a bond mutual fund. Seen from this perspective, the portfolio selection problem with two securities is equivalent to the problem of asset allocation between stocks and bonds. We then consider portfolios of two risky securities and a risk-free asset, the prototype being a portfolio of a stock mutual fund, a bond mutual fund, and a money-market fund. Finally, we consider portfolio selection when an unlimited number of securities is available for inclusion in the portfolio.We conclude this chapter by briefly discussing an important consequence of the Markowitz model, namely, the Nobel Prize winning capital asset pricing model due to William Sharpe. The CAPM, as it is referred to, gives a formula for the fair return on a risky security when the overall market is in equilibrium. Like the Markowitz model, the CAPM has had a profound influence on portfolio management practice.10.1 Portfolios of Two SecuritiesIn this section, we consider portfolios consisting of only two securities, 1S and 2S . These two securities could be a stock mutual fund and a bond mutual fund, in which case the portfolio selection problem amounts to asset allocation, or they could be something else. Our objective is to determine the “best mix” of 1S and 2S in the portfolio.Portfolio Opportunity SetLet's begin by describing the set of possible portfolios that can be constructed from 1S and 2S . Suppose that the current value of our portfolio is d dollars and let 1d and 2d be the dollar amounts invested in 1S and 2S , respectively. Let 1R and 2R be the simple returns on 1S and 2S over a future time period that begins now and ends at a fixed future point in time and let R be the corresponding simple return for the portfolio. Then, if no changes are made to the portfolio mix during the time period under consideration,()()()2211111R d R d R d +++=+.Hence, the return on the portfolio over the given time period is()211R x xR R -+=, where d d x 1= is the fraction of the portfolio currently invested in 1S . Consequently, by varying x , we can change the return characteristics of the portfolio.Now if 1S and 2S are risky securities, as we will assume throughout this section, then 1R , 2R , and R are all random variables. Suppose that 1R and 2R are both normally distributed and their joint distribution has a bivariate normal distribution. This may appear to be a strong assumption. However, data on stock price returns suggest that, as a first approximation, it is not unreasonable. Then, from the properties of the normal distribution, it follows that R is normally distributed and that the distributions of 1R , 2R , and R are completely characterized by theirrespective means and standard deviations. Hence, since R is a linear combination of 1R and 2R , the set of possible investment portfolios consisting of 1S and 2S can be described by a curve in the μσ- plane.To see this more clearly, note that from the identity ()211R x xR R -+= and the properties of means and variances, we have()211R R R x x μμμ-+=, ()()222222211112R R R R R x x x x σσρσσσ-+-+=, where ρ is the correlation between 1R and 2R , Eliminating x from these two equations by substituting ()()212R R R R x μμμμ--=, which we obtain from theequation for R μ, into the equation for 2R σ, we obtain()()()()()()()222222222211212*********R R R R R R R R R R R R R R R R R RR σμμμμσρσμμμμμμσμμμμσ--+---+--=, which describes a curve in the R R μσ plane as claimed.Notice that R σ and R μ change with x , while 1R σ, 1R μ, 2R σ, 2R μ, ρ remain fixed. To emphasize the fact that R σ and R μ are variables, let’s drop thesubscript R from now on. Then, the preceding equation for 2R σ can be written as()20202σμμσ+-=A , where A , 0μ, 20σ are parameters depending only on 1S and 2S with 0>A and 020≥σ. Indeed,(){}22222112121R R R R R R A σσρσσμμ+--= ()()(){}21212112122R R R R R R σσρσσμμ-+--=0>(the inequality holding since 11≤≤-ρ), and()()()212121121222220R R R R R R σσρσσρσσσ-+--=(again since 11≤≤-ρ). Further,()()()212112212121122220R R R R R R R R R R R R σσρσσσμσρσμμσμμ-+-++-=.Consequently, the possible portfolios lie on the curve()20202σμμσ+-=A ,0≥σ， which we recognize as being the right half of a hyperbola with vertex at ()00,μσ. (Figure 10.1).Notice that the hyperbola ()20202σμμσ+-=A describes a trade-off between risk (as measured by σ) and reward (as measured by μ). Indeed, along the upper branch of the hyperbola, it is clear that to obtain a greater reward, we must invest in a portfolio with greater risk; in other words, “no pain, no gain.” The portfolios on the lower branch ofthe hyperbola, while theoretically possible, will never be selected risk level σ, the portfolio on the upper branch with standard deviation σ will always have higher expected return (i.e., higher reward) than the portfolio on the lower branch with standard deviation σ and, hence, will always be preferred to the portfolio on the lower branch. Consequently, the only portfolios that need be considered further areFIGURE10.1 Set of Possible Portfolios consisting of 1S and 2Sthe ones on the upper branch. These portfolios are referred to as efficient portfolios. In general, an efficient portfolio is one that provides the highest reward for a given level of risk.Determining the Optimal PortfolioNow let’s consider which portfolio in the efficient set is best. To do this, we need to consider the investor’s tolerance for risk. Since different investors in general have different risk tolerances, we should expect each investor to have a different optimal portfolio. We will soon see that this is indeed the case.Let’s consider one particular investor and let’s suppose that this investor is able to assign a number ()R F U to each possible investment return distribution R F with the following properties:1.()()b a R R F U F U > if and only if the investor prefers the investment with return a R to the investment with return b R .2. ()()b a R R F U F U = if and only if the investor is indifferent to choosing between theinvestment with return a R and the investment with return b R .The functional U , which maps distribution functions to the real numbers, is called a utility functional. Note that different investors in general have different utility functionals.There are many different forms of utility functionals. For simplicity, we assume that every investor has a utility functional of the form()2σμk F U R -=,where 0>k is a number that measures the investor’s level of risk aversion and is unique to each investor. (Here, μ and σ represent the mean and standard deviation of the return distribution R F .) There are good theoretical reasons for assuming a utility functional of this form. However, in the interest of brevity, we omit the details. Note that in assuming a utility functional of this form, we are implicitly assuming that among portfolios with the same expected return, less risk is preferable.The portfolio optimization problem for an investor with risk tolerance level kcan then be stated as follows:Maximize:2σμk U -= Subject to: ()20202σμμσ+-=A . This is a simple constrained optimization problem that can be solved by substituting the condition into the objective function and then using standard optimization techniques from single variable calculus. Alternatively, this optimization problem can be solved using the Lagrange multiplier method from multivariable calculus.Graphically, the maximum value of U is the number u such that the parabolau k =-2σμ is tangent to the hyperbola ()20202σμμσ+-=A . (See Figure 10.2.The optimal portfolio in this figure is denoted by O .) Clearly, the optimal portfolio depends on the value of k , which specifies the investor’s level of risk aversion.Carrying out the details of the optimization, we find that when 1S and 2S are both risky securities (i.e.01≠R σ and 02≠R σ), the risk-reward coordinates of theoptimal portfolio O are 20241*σσ+=Ak, kA 21*0+=μμ. Since ()211R R x x μμμ-+=, it follows that the portion of the portfolio that should beFIGURE10.2 Portfolio with Greatest Utilityinvested in 1S is212*R R R x μμμμ--=. Comment We have assumed that short selling without margin posting is possible (i.e., we have assumed that x can assume any real value, including values outside the interval[0,1]). In the more realistic case, where short selling is restricted, the optimal portfolio may differ from the one just determined.EXAMPLE 1: The return on a bond fund has expected value 5% and standard deviation 12%, while the return on a stock fund has expected value 10% and standard deviation 20%. The correlation betwee n the returns is 0.60. Suppose that an investor’sutility functional is of the form 21001σμ-=U . Determine the investor’s optimal allocation between stocks and bonds assuming short selling without margin posting is possible.It is customary in problems of this type to assume that the utility functional is calibrated using percentages. Hence, if 1R ,2R represent the returns on the bond and stock funds, respectively, then()()56.3121001521=-=R F U , ()()61210011022=-=R F U . Note that such a calibration can always be achieved by proper selection of k . From the formulas that have been developed, the expected return on the optimal portfolio iskA 21*0+=μμ, where 1001=k ,()()()212112212121122220R R R R R R R R R R R R σσρσσσμσρσμμσμμ-+-++-=()()()()()()()()()()()()201240.0220121210201260.010*******+-++-= 875.21=and()()(){}21212112122R R R R R R A σσρσσμμ-+--= ()()()()(){}201240.022*********+--= 24.10=. Hence, the portion of the portfolio that should be invested in bonds is212*R R R x μμμμ--= 105107578125.26--= 3515625.3-=.Thus, for a portfolio of $1000, it is optimal to sell short $33351.56 worth of bonds andinvest $4351.56 in stocks.■ Special Cases of the Portfolio Opportunity SetWe conclude this section by high lighting the form of the portfolio opportunity set in some special cases. Throughout, we assume that 1S and 2S are securities such that21R R μμ< and 21R R σσ<. (The situation where 21R R μμ< and 21R R σσ> is not interesting since then 2S is always preferable to 1S .) We also assume that no short positions are allowed. Assets Are Perfectly Positively Correlated Suppose that 1=ρ (i.e. 1R and 2R are perfectly positively correlated). Then the set of possible portfolios is a straight line, as illustrated in Figure 10.3a.Assets Are Perfectly Negatively Correlated Suppose that 1-=ρ (i.e, 1R and 2R are perfectly negatively correlated). Then the set of possible portfolios is as illustrated in Figure 10.3b. Note that, in this case, it is possible to construct a perfectly hedged portfolio (i.e., portfolio with 0=σ).Assets Are Uncorrelated Suppose that 0=ρ. Then the portfolio opportunity set has the form illustrated in Figure 10.3c. From this picture, it is clear that starting from a portfolio consisting only of the low-risk security 1S , it is possible to decrease risk and increase expected return simultaneously by adding a portion of the high-risk security 2S to the portfolio. Hence, even investors with a low level of risk tolerance should have a portion of their portfolios invested in the high-risk security 2S . (See also the discussion on the standard deviation of a sum in §8.3.3.)One of the Assets Is Risk Free Suppose that 1S is a risk-free asset (i.e., 01=R σ) a.b.c.d. One of the Assets Is Risk FreeFIGURE 10.3 Special Cases of the Portfolio Opportunity Setμ 0μ μand put f R r =1μ, the risk-free rate of return. Further, let S denote 2S and write S σ, S μ in place of 2R σ, 2R μ. Then the efficient set is given by σσμμ⋅-+=Sf S f r r , 0>σ. This is a line in risk-reward space with slope ()S f S r σμ- and μ-intercept f r (see Figure 10.3d).10.2 Portfolios of Two Risky Securities and a Risk-Free Asset Suppose now that we are to construct a portfolio from two risky securities and a risk-free asset. This corresponds to the problem of allocating assets among stocks, bonds, and short-term money-market securities. Let 1R , 2R denote the returns on the risky securities and suppose that 21R R μμ< and 21R R σσ<. Further, let f r denote the risk-free rate.The Efficient SetFrom our discussion in §10.1, we know that the portfolios consisting only of the two risky securities 1S , 2S must lie on a hyperbola of the type illustrated in Figure 10.4. We claim that when a risk-free asset is also available, the efficient set consists of the portfolios on the tangent line through (0,f r ) (Figure 10.5). Note that f r in this figure is the μ-intercept of the tangent line through T .FIGURE 10.4Portfolio Opportunity Set for Two SecuritiesFIGURE 10.5Efficient Set as a Tangent LineFIGURE 10.6Portfolios Containing the Tangency Portfolio Dominate All OthersTo see why this is so, consider a portfolio P consisting only of 1S and 2S and let T be the tangency portfolio (i.e., the portfolio which is on both the hyperbola and the tangent line). From our discussion in §10.1, we know that every portfolio consisting of the risky portfolio P and the risk-free asset lies on the straight line through P and (0,f r ), and every portfolio consisting of the tangency portfolio T and the risk-free asset lies on the tangent line through T and (0,f r ) (Figure 10.6). Hence, from Figure 10.6, it is clear that every portfolio consisting of P and the risk-free asset is dominated by a portfolio consisting of T and the risk-free asset. Indeed, for any given risk level σ', there is a portfolio on the line through T and (0,f r ) with greater μ than the corresponding portfolio on the line through P and (0,f r ). Hence, given a choice between holding P as the risky part of our portfolio and holding T as the risky part, we should always choose T .Consequently, the efficient portfolios lie on the line through (0,f r ) and T as claimed. Note, in particular, that the efficient portfolios all have the same risky part T ; the only difference among them is the portion allocated to the risk-free asset. This surprising result, which provides a theoretical justification for the use of index mutual funds by every investor, is known as the mutual fund separation theorem. In view of this separation theorem, the portfolio selection problem is reduced to determining the fraction of an investor’s portfolio that should be invested in the risk -free asset. This is a straightforward problem when the utility functional has the form 2σμk U -=(Figure 10.7). The investor’s optimal portfolio in this figure is denoted by O . Details are left to the reader.Determining the Tangency PortfolioThe tangency portfolio T has the property of being the portfolio on the hyperbola for which the ratioσμf r - is maximal. (Convince yourself that this is so.) Hence, one method of determining the coordinates of T is to solve the following optimization problem:Maximize:σμθf r -= Subject to: ()20202σμμσ+-=A . We will determine the coordinates of T in a slightly different way, which is more easily adapted when the number of risky securities is greater than two.Recall that the efficient portfolios are the ones with the least risk (i.e., smallest σ) for a given level of expected return μ. Hence, the efficient set, which we already know is the line through (0,f r ) and T , can be determined by solving the following collection of optimization problems (one for each μ):Minimize: ()R VarFIGURE 10.7 Optimal Portfolio for a Given Utility FunctionalSubject to: []μ=R E .Let 1y , 2y , 3y be the amounts allocated to 1S , 2S , and the risk-free asset, respectively. Then the return on such a portfolio isf r y R y R y R 32211++=,and so()2222211221212y y y y R Var σσσ++=and[]32211y r y y R E f ++=μμ,where []j j R E =μ, ()j j R Var =2σ, and ()j i j i R R Cov ,=σ. Note that ()R Var does not contain any terms in 3y ! Consequently, the optimization problem can be written asMinimize: 2222211221212y y y y σσσψ++= Subject to: μμμ=++32211y r y y f ,1321=++y y y .Note that the conditions in the optimization are equivalent to the conditions()()f f f r y r y r -=-+-μμμ2211,1321=++y y y .(Substitute 1321=++y y y into the first condition.) Since the only place that 3y now occurs is in the condition 1321=++y y y , this means that we can solve the general optimization problem by first solving the simpler problemMinimize: ψSubject to: ()()f f f r y r y r -=-+-μμμ2211and then determining 3y by 2131y y y --=. Indeed, ψ will still be minimized because the required 1y , 2y will be the same in both optimization problems.The simpler optimization problem can be solved using the Lagrange multipliermethod. In general, we will haveg ∇=∇τψ and 0=g ,where ()()()f f f r y r y r g ---+-=μμμ2211 and τ is the Lagrange multiplier. The letter λ is generally reserved in investment theory for the reward-to-variability ratio ()2σμf r - and, hence, will not be used to represent a Lagrange multiplier here. Performing the required differentiation, we obtain ⎪⎪⎭⎫ ⎝⎛--⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛⎪⎪⎭⎫ ⎝⎛ff r r y y 2121221212212μμτσσσσ, or equivalently, ⎪⎪⎭⎫ ⎝⎛--=⎪⎪⎭⎫ ⎝⎛⎪⎪⎭⎫ ⎝⎛ff r r z z 212122121221μμσσσσ, where112y z τ=, 222y z τ=.Note that the Lagrange multiplier τ will depend in general on μ.Now the tangency portfolio T lies on the efficient set and has the property that 03=y (i.e., no portion of the tangency portfolio is invested in the risk-free asset). Hence, the values of 1y and 2y for the tangency portfolio are given by2111z z z y +=, 2122z z z y +=, where (1z ,2z ) is the unique solution of the preceding matrix equation. Indeed, since T lies on the efficient set, we must have (1y ,2y )=2τ(1z ,2z ), and since 11321=-=+y y y , we must have ()2112z z +=τ. The risk-reward coordinates (T R σ,T R μ) for the tangency portfolio are then determined using the equations2211μμμy y TR +=, 22221221212122σσσσy y y y TR ++=, where 1y , 2y are the fractions just calculated.中文译文：第十章：Markowitz 投资组合选择模型这本书前面九个章节提出了保险和投资任一名学生应该熟悉的基本的概率理论。