Chapter雙變量回歸區(qū)間估計(jì)與假設(shè)檢驗(yàn)

1、Chapter 5 雙變量回歸：區(qū)間估計(jì)與假設(shè)檢驗(yàn)主講：彭紅楓武漢大學(xué)經(jīng)濟(jì)與管理學(xué)院金融系CopyrightHongfeng Peng 2006 Wuhan University9/4/20221Hongfeng Peng Department of Finance, Wuhan University5.1 回顧統(tǒng)計(jì)學(xué)相關(guān)內(nèi)容問(wèn)題：如果 ，使得：則稱9/4/20222Hongfeng Peng Department of Finance, Wuhan University假設(shè)檢驗(yàn)中的兩類錯(cuò)誤第一類錯(cuò)誤：拒絕真實(shí)；第二類錯(cuò)誤：接受錯(cuò)誤。9/4/20223Hongfeng Peng Departm

2、ent of Finance, Wuhan University5.2回歸系數(shù)1和2的置信區(qū)間在ui的正態(tài)性假定下，OLS估計(jì)量0b和1b本身就是正態(tài)分布的，),0(2sNui),(2222ixNsbb)1,0(/2222NxZi-=sbb但是2s很少能知道，在實(shí)踐中用無(wú)偏估計(jì)量2s來(lái)代替，則統(tǒng)計(jì)量t服從自由度為n-2的t分布：)2(/)(2222222-=-=ntxsetisbbbbb其中=222/)(ixsesb表示估計(jì)量2b的標(biāo)準(zhǔn)差（22/ixs）的估計(jì)值。由aaa-=-1)Pr(22ttt得：9/4/20224Hongfeng Peng Department of Finance, W

3、uhan University回歸系數(shù)1和2的置信區(qū)間2的顯著水平為的置信區(qū)間為：同樣，1顯著水平為的置信區(qū)間為：9/4/20225Hongfeng Peng Department of Finance, Wuhan University5.3 2的置信區(qū)間9/4/20226Hongfeng Peng Department of Finance, Wuhan University9/4/20227Hongfeng Peng Department of Finance, Wuhan University5.3 假設(shè)檢驗(yàn)問(wèn)題：某一給定的觀測(cè)或發(fā)現(xiàn)是否與某一聲稱的假設(shè)（statedhypothesi

4、s）相符？此處用“相符”一詞表示觀測(cè)的值與假設(shè)的值“足夠相近”，因而我們不拒絕所聲稱的假設(shè)。虛擬假設(shè)（Null hypothesis）：一種信以為真的、意在維護(hù)的或理論上的假設(shè)，并用H0表示。與之對(duì)立的假設(shè)稱為對(duì)立假設(shè)（alternative hypothesis），記為H1。對(duì)立假設(shè)可以是簡(jiǎn)單的或復(fù)合的。例如，H1：b2=1是一個(gè)簡(jiǎn)單假設(shè)，但是H1：b21則是一個(gè)復(fù)合假設(shè)。方法：有顯著性檢驗(yàn)和置信區(qū)間兩種方法。9/4/20228Hongfeng Peng Department of Finance, Wuhan University5.4 假設(shè)檢驗(yàn)：置信區(qū)間方法9/4/20229Hongfe

5、ng Peng Department of Finance, Wuhan University5.5 假設(shè)檢驗(yàn)：顯著性檢驗(yàn)方法9/4/202210Hongfeng Peng Department of Finance, Wuhan Universityt檢驗(yàn)方法的直接計(jì)算H0：*22bb=；H1：*22bb。計(jì)算 -=-=2222222/)(ixsetsbbbbb比較|t|與2at：|t|2at（t值大）“統(tǒng)計(jì)量的值落入臨界域內(nèi)統(tǒng)計(jì)量在統(tǒng)計(jì)上顯著拒絕H0假設(shè)Pr(t)a（P值?。?。9/4/202211Hongfeng Peng Department of Finance, Wuhan Univ

6、ersity置信區(qū)間方法與顯著性檢驗(yàn)方法的關(guān)系在置信區(qū)間程序中，我們?cè)噲D建立一個(gè)以某種概率包含有真實(shí)但未知的 的一個(gè)范圍或區(qū)間；而在顯著性檢驗(yàn)步驟中，我們假設(shè) 為某值，然后來(lái)看所計(jì)算的 是否位于該假設(shè)值周?chē)哪硞€(gè)合理(置信)范圍之內(nèi)。9/4/202212Hongfeng Peng Department of Finance, Wuhan University顯著性t檢驗(yàn)：決策規(guī)則9/4/202213Hongfeng Peng Department of Finance, Wuhan University2檢驗(yàn)的顯著性（2檢驗(yàn)）9/4/202214Hongfeng Peng Department

7、 of Finance, Wuhan University補(bǔ)充：自由度模型中樣本值可以自由變動(dòng)的個(gè)數(shù)，稱為自由度自由度=樣本個(gè)數(shù)- 樣本數(shù)據(jù)受約束條件（方程）的個(gè)數(shù)例如，樣本數(shù)據(jù)個(gè)數(shù)=n，它們受k+1個(gè)方程的約束（這n個(gè)數(shù)必須滿足這k+1個(gè)方程）那么，自由度df = n-k-19/4/202215Hongfeng Peng Department of Finance, Wuhan University數(shù)據(jù)個(gè)數(shù)與約束方程 Y1+Y2+Y3=7 Y1=7那么Y2、Y3中只有1個(gè)是自由的。又如： Y1+Y2+Y3+Y4=7 Y1=7那么，Y2、Y3、Y4中只有2個(gè)是自由的9/4/202216Hong

8、feng Peng Department of Finance, Wuhan Universityk元模型中隨機(jī)擾動(dòng)項(xiàng)的自由度為什么=n-k-1?9/4/202217Hongfeng Peng Department of Finance, Wuhan University5.6 P valueP value當(dāng)我們對(duì)給定的樣本算出一個(gè)檢驗(yàn)統(tǒng)計(jì)量(如t統(tǒng)計(jì)量)的值時(shí)，為什么不干脆查閱適當(dāng)?shù)慕y(tǒng)計(jì)表，看看得到一個(gè)大到和從樣本得到的檢驗(yàn)統(tǒng)計(jì)量那樣大的數(shù)值的確切概率?這個(gè)概率就叫做P值(P value)9/4/202218Hongfeng Peng Department of Finance, Wuhan

9、 University5.7 回歸分析的應(yīng)用：預(yù)測(cè)問(wèn)題樣本回歸函數(shù)的一個(gè)用途是“預(yù)測(cè)”或“預(yù)報(bào)”對(duì)應(yīng)于給定X的未來(lái)的Y值。均值預(yù)測(cè)已知X的值，去預(yù)測(cè)Y的條件均值個(gè)值預(yù)測(cè)已知X的值，去預(yù)測(cè)Y的一個(gè)個(gè)別值9/4/202219Hongfeng Peng Department of Finance, Wuhan University均值預(yù)測(cè)(mean prediction)9/4/202220Hongfeng Peng Department of Finance, Wuhan University個(gè)值預(yù)測(cè)(individual prediction)9/4/202221Hongfeng Peng De

10、partment of Finance, Wuhan University均值預(yù)測(cè)與各值預(yù)測(cè)之比較9/4/202222Hongfeng Peng Department of Finance, Wuhan University5.8 報(bào)告回歸分析的結(jié)果9/4/202223Hongfeng Peng Department of Finance, Wuhan University5.9 評(píng)價(jià)回歸分析的結(jié)果一些準(zhǔn)則：1、所估系數(shù)的符號(hào)是否與理論或事前預(yù)期相一致?2、系數(shù)在統(tǒng)計(jì)上是否顯著？3、方程的顯著性（回歸模型在多大程度上解釋了因變量的變異）4、殘差的正態(tài)性檢驗(yàn)9/4/202224Hongfeng

11、Peng Department of Finance, Wuhan University正態(tài)性檢驗(yàn)正態(tài)性檢驗(yàn)方法Chi卡方擬合優(yōu)度檢驗(yàn)雅克一貝拉（JB）檢驗(yàn)9/4/202225Hongfeng Peng Department of Finance, Wuhan University補(bǔ)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