Chapter双变量回归区间估计与假设检验

1、Chapter 5 双变量回归：区间估计与假设检验主讲：彭红枫武汉大学经济与管理学院金融系CopyrightHongfeng Peng 2006 Wuhan University9/4/20221Hongfeng Peng Department of Finance, Wuhan University5.1 回顾统计学相关内容问题：如果 ，使得：则称9/4/20222Hongfeng Peng Department of Finance, Wuhan University假设检验中的两类错误第一类错误：拒绝真实；第二类错误：接受错误。9/4/20223Hongfeng Peng Departm

2、ent of Finance, Wuhan University5.2回归系数1和2的置信区间在ui的正态性假定下，OLS估计量0b和1b本身就是正态分布的，),0(2sNui),(2222ixNsbb)1,0(/2222NxZi-=sbb但是2s很少能知道，在实践中用无偏估计量2s来代替，则统计量t服从自由度为n-2的t分布：)2(/)(2222222-=-=ntxsetisbbbbb其中=222/)(ixsesb表示估计量2b的标准差（22/ixs）的估计值。由aaa-=-1)Pr(22ttt得：9/4/20224Hongfeng Peng Department of Finance, W

3、uhan University回归系数1和2的置信区间2的显著水平为的置信区间为：同样，1显著水平为的置信区间为：9/4/20225Hongfeng Peng Department of Finance, Wuhan University5.3 2的置信区间9/4/20226Hongfeng Peng Department of Finance, Wuhan University9/4/20227Hongfeng Peng Department of Finance, Wuhan University5.3 假设检验问题：某一给定的观测或发现是否与某一声称的假设（statedhypothesi

4、s）相符？此处用“相符”一词表示观测的值与假设的值“足够相近”，因而我们不拒绝所声称的假设。虚拟假设（Null hypothesis）：一种信以为真的、意在维护的或理论上的假设，并用H0表示。与之对立的假设称为对立假设（alternative hypothesis），记为H1。对立假设可以是简单的或复合的。例如，H1：b2=1是一个简单假设，但是H1：b21则是一个复合假设。方法：有显著性检验和置信区间两种方法。9/4/20228Hongfeng Peng Department of Finance, Wuhan University5.4 假设检验：置信区间方法9/4/20229Hongfe

5、ng Peng Department of Finance, Wuhan University5.5 假设检验：显著性检验方法9/4/202210Hongfeng Peng Department of Finance, Wuhan Universityt检验方法的直接计算H0：*22bb=；H1：*22bb。计算 -=-=2222222/)(ixsetsbbbbb比较|t|与2at：|t|2at（t值大）“统计量的值落入临界域内统计量在统计上显著拒绝H0假设Pr(t)a（P值小）。9/4/202211Hongfeng Peng Department of Finance, Wuhan Univ

6、ersity置信区间方法与显著性检验方法的关系在置信区间程序中，我们试图建立一个以某种概率包含有真实但未知的 的一个范围或区间；而在显著性检验步骤中，我们假设 为某值，然后来看所计算的 是否位于该假设值周围的某个合理(置信)范围之内。9/4/202212Hongfeng Peng Department of Finance, Wuhan University显著性t检验：决策规则9/4/202213Hongfeng Peng Department of Finance, Wuhan University2检验的显著性（2检验）9/4/202214Hongfeng Peng Department

7、 of Finance, Wuhan University补充：自由度模型中样本值可以自由变动的个数，称为自由度自由度=样本个数- 样本数据受约束条件（方程）的个数例如，样本数据个数=n，它们受k+1个方程的约束（这n个数必须满足这k+1个方程）那么，自由度df = n-k-19/4/202215Hongfeng Peng Department of Finance, Wuhan University数据个数与约束方程 Y1+Y2+Y3=7 Y1=7那么Y2、Y3中只有1个是自由的。又如： Y1+Y2+Y3+Y4=7 Y1=7那么，Y2、Y3、Y4中只有2个是自由的9/4/202216Hong

8、feng Peng Department of Finance, Wuhan Universityk元模型中随机扰动项的自由度为什么=n-k-1?9/4/202217Hongfeng Peng Department of Finance, Wuhan University5.6 P valueP value当我们对给定的样本算出一个检验统计量(如t统计量)的值时，为什么不干脆查阅适当的统计表，看看得到一个大到和从样本得到的检验统计量那样大的数值的确切概率?这个概率就叫做P值(P value)9/4/202218Hongfeng Peng Department of Finance, Wuhan

9、 University5.7 回归分析的应用：预测问题样本回归函数的一个用途是“预测”或“预报”对应于给定X的未来的Y值。均值预测已知X的值，去预测Y的条件均值个值预测已知X的值，去预测Y的一个个别值9/4/202219Hongfeng Peng Department of Finance, Wuhan University均值预测(mean prediction)9/4/202220Hongfeng Peng Department of Finance, Wuhan University个值预测(individual prediction)9/4/202221Hongfeng Peng De

10、partment of Finance, Wuhan University均值预测与各值预测之比较9/4/202222Hongfeng Peng Department of Finance, Wuhan University5.8 报告回归分析的结果9/4/202223Hongfeng Peng Department of Finance, Wuhan University5.9 评价回归分析的结果一些准则：1、所估系数的符号是否与理论或事前预期相一致?2、系数在统计上是否显著？3、方程的显著性（回归模型在多大程度上解释了因变量的变异）4、残差的正态性检验9/4/202224Hongfeng

11、Peng Department of Finance, Wuhan University正态性检验正态性检验方法Chi卡方拟合优度检验雅克一贝拉（JB）检验9/4/202225Hongfeng Peng Department of Finance, Wuhan University补充： Moments of a Random VariableThe l-th moment of a continuous random variable X is defined as9/4/202226Hongfeng Peng Department of Finance, Wuhan UniversityT

12、he first moment is called the mean or expectation of X. It measures the central location of the distribution.We denote the mean of X by x.The l-th central moment of X is defined as9/4/202227Hongfeng Peng Department of Finance, Wuhan UniversityThe second central moment, denoted by , measures the vari

13、ability of X and is called the variance of X. The positive square root, x ,of variance is the standard deviation of X.9/4/202228Hongfeng Peng Department of Finance, Wuhan UniversityThe third central moment measures the symmetry of X with respect to its mean,whereas the 4th central moment measures th

14、e tail behavior of X. In statistics, skewness and kurtosis, which are normalized 3rd and 4th central moments of X, are often used to summarize the extent of asymmetry and tail thickness. 9/4/202229Hongfeng Peng Department of Finance, Wuhan UniversitySpecifically, the skewness and kurtosis of X are de