Financial Econometrics Chapter 2.Statistical Foundations:Overview of OLS Jin Ling School of Finance,Zhongnan University of Economics and Law
Financial Econometrics Chapter 2. Statistical Foundations: Overview of OLS Jin Ling School of Finance, Zhongnan University of Economics and Law 1
Outline Statistical Foundation Review ·Overview of OLS ·Practice of OLS 2
• Statistical Foundation Review • Overview of OLS • Practice of OLS 2 Outline
Statistical Foundation Review ·Probability: P(AB)=P(AnB) 么B为相互独立的随机事件。此时，P48)=P4=PA,故 P(B) P(B) P(A0B)=P(A)P(B) B AnB P(0=∑，P(B,)P(AB) 图2.1条件概率示意图 3
• Probability: 3 Statistical Foundation Review
Statistical Foundation Review Defination of random variable: The conditional value. Probability distribution:Discrete distribution,continuous distribution,joint distribution: ①fxy≥0，xy: Xxx2.装. 间∫f.)dxdy=1: p乃P2.Pk. (曲(x,门落入平面某区域D的概率为P{(x,门eD)=川/x)d山. 连续型随机变量可以取任意实数，其“概率密度函数”(probability density function,简记 d山f)满足， 从二维联合密度函数f(x,),可以计算X的（一维）边缘密度函数(marginal pd的： ①fx)≥0，x: f)-∫"fx (2.6) 间∫f)c=1 这个公式的直观含义与“全概率公式”相似，即给定x,把所有y取值的可能性都“加”起 (G间x落入区间[a,]的概率为P代a《Xs)=∫广f)d, 来（积分的本质就是加总），类似地，可以计算Y的（一维）边缘密度函数： 2.7) 定义“累积分布函数”(cumulative distribution function,简记cd的为 f09=J.f红)d F国)=m<X≤=广f0d 2.5) 定义二维随机向量(X,门的累积分布函数为 其中，1为积分变量。直观来看，下()度量的是，从-至x为止，概念密度函数f)曲线 Fx,川=P-m<X<g-o<y<》=广∫f化，)出山 2.8 下的面积
• Defination of random variable: • The conditional value. • Probability distribution: Discrete distribution, continuous distribution, joint distribution: 4 Statistical Foundation Review
Statistical Foundation Review Defination of random variable: Conditional distribution: flx)= (x) Af(yix) E(Yix) X 5
• Defination of random variable: • Conditional distribution: 5 Statistical Foundation Review
Statistical Foundation Review Characteristics of random variable: ·Expectation:E(X)=∫xfx)dx. Variance:Var(X)=E[X-E(X)]2. Covariance:Cov(X,Y)=E[(X-E(X)(Y-E(Y))]. .Corelation:p=Corr(Yvr Cov(X,Y) .Conditional expectation:E(YX=x)=E(Y)=yf(ylx)dy. .Conditional variance:Var(YIX=x)=Var(Yx)=[y-E(Yx)]2f(ylx)dy. 6
• Characteristics of random variable: ∞− = �� �� :Expectation• +∞ 𝑥𝑓 𝑥 𝑑𝑥. • Variance: 𝑉𝑎𝑟 𝑋 = 𝐸 𝑋 − 𝐸 𝑋 2 . • Covariance: 𝐶𝑜𝑣 𝑋, 𝑌 = 𝐸[ 𝑋 − 𝐸 𝑋 𝑌 − 𝐸 𝑌 ]. • Correlation: 𝜌 = 𝐶𝑜𝑟𝑟 𝑋, 𝑌 = 𝐶𝑜𝑣 𝑋,𝑌 𝑉𝑎𝑟 𝑋 𝑉𝑎𝑟 𝑌 • Conditional expectation: 𝐸 𝑌 𝑋 = 𝑥 = 𝐸 𝑌 𝑥 = ∞− +∞ 𝑦𝑓 𝑦|𝑥 𝑑𝑦. • Conditional variance: 𝑉𝑎𝑟 𝑌 𝑋 = 𝑥 = 𝑉𝑎𝑟 𝑌 𝑥 = ∞− +∞ [𝑦 − 𝐸 𝑌|𝑥 ] 2𝑓 𝑦|𝑥 𝑑𝑦. 6 Statistical Foundation Review
Statistical Foundation Review The concept of independence Mutually independent: f(x,y)=fx(x)f,(y) ·Mean-independent: E(Yx)=E(Y) Linearly independent: Cov(X,Y)=0 Mutually independent Mean-independent Linearly independent
• The concept of independence : • Mutually independent: 𝑓 𝑥, 𝑦 = 𝑓𝑥 𝑥 𝑓𝑦 (𝑦) • Mean-independent: 𝐸 𝑌 𝑥 = 𝐸(𝑌) • Linearly independent: 𝐶𝑜𝑣 𝑋, 𝑌 = 0 • Mutually independent > Mean-independent > Linearly independent. 7 Statistical Foundation Review
Statistical Foundation Review Typical statistical distribution: Normal distribution(Gaussian distribution): f闭=2 xp.-N(4o) Chi-square distribution: Z~N(0,1) ∑r 。t-distribution: ZN(0,1),Y~X2(k) Z ~t(k) .F-distribution:Y().Y2( Y.IkF(k1.k2) 8
• Typical statistical distribution: • Normal distribution (Gaussian distribution): 𝑓 𝑥 = 1 2𝜋𝜎2 exp{ − 𝑥−𝜇 2 2𝜎 2 }, 𝑋~𝑁(𝜇, 𝜎) • Chi-square distribution: 𝑍~𝑁 0,1 𝐼=1 𝐾 𝑍𝑖 2 ~χ 2(𝑘) • t-distribution: 𝑍~𝑁 0,1 , 𝑌~χ 2(𝑘) 𝑍 𝑌/𝑘 ~𝑡(𝑘) • F-distribution: 𝑌1~χ 2(𝑘1 ), 𝑌2~χ 2(𝑘2 ), 𝑌1 /𝑘1 𝑌2 /𝑘2 ~F(𝑘1 ,𝑘2 ) 8 Statistical Foundation Review
Statistical Foundation Review Typical statistical distribution: 0.451 04 0. N(0,1) 0.3 0.25 0.2 0.15 0.1 0.05 (32-一 -4 -3-2-10 图2.3N(0,1)与t(3)的概率密度
• Typical statistical distribution: 9 Statistical Foundation Review
Statistical Foundation Review Typical statistical distribution: 0.25m 0.2i 1x2) 0.154 x(6) 0.1 0.05 1012141618 20 图2.4X2分布的概率密度 10
• Typical statistical distribution: 10 Statistical Foundation Review