Lecture 3 - Lecture 2:Statistical inference and testing Xin...

Info icon This preview shows pages 1–12. Sign up to view the full content.

View Full Document Right Arrow Icon
Lecture 2:Statistical inference and testing Xin T Tong Sunday 28 th August, 2016 Xin Tong Statistics 1 / 40
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Last time Estimator ˆ θ for a parameter θ : plug-in, MLE, moments. Estimation criteria: bias, MSE, consistency. Estimator distribution: N , χ 2 , t distributions. Confidence intervals: [ ˆ θ - c, ˆ θ + c ] Hypothesis testing: reject if p-value is below α . Don’t worry if you couldn’t grasp everything in one class. Important concepts will be reviewed and practiced in assignments. Xin Tong Statistics 2 / 40
Image of page 2
Auto-regressive models AR is one of the most useful model for time series r t = α + βr t - 1 + a t . General AR(k) model: r t = α + k X j =1 β j r t - j + a t . a t : shocks , innovations of the series. Assume a t to be i.i.d. sequence of normal RV. Assume a t to be weakly stationary and uncorrelated . Questions: How to tell the coefficients α, β j ? (Inference) How to tell if the model is good enough? (Testing ) How to tell the model’s order k ? (Next class) Xin Tong Statistics 3 / 40
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Linear regression Simple linear regression model: Y i = α + βX i + i . Multiple linear regression model: Y i = α + K X k =1 β k X k,i + i . Y i is the dependent/response variable. X k,i is the predictor/explanatory model. In most scenarios, you have i = 1 , . . . , n data points ( Y 1 , X 1 , 1 , · · · , X K, 1 ) , ( Y 2 , X 1 , 2 , · · · , X K, 2 ) , · · · ( Y n , X 1 ,n , · · · , X K,n ) . Try to find the coefficients α, β i α is also called the intercept. (If X i = 0 , i = 0, Y i = α ) Xin Tong Statistics 4 / 40
Image of page 4
Wide application Y i = α + K k =1 β k X k,i + i Area happiness dependence: Happy = 2 · Income+1 . 5 · Density - 1 · living price - 0 . 5 · Crime rate Income discrimination: Income = 2 · EQ + 1 . 5 · Education + 1 · Race + 0 . 5 · Gender Asset returns: Return tomorrow = 0 . 8 · Return today+0 . 3 · Return yesterday+ · · · . In general, we concern The coefficients’ value. Positive dependence or negative dependence. Dependence or no dependence. Xin Tong Statistics 5 / 40
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
IBM against value weighted -0.2 0.0 0.2 0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 ibm vw Xin Tong Statistics 6 / 40
Image of page 6
IBM against value weighted -0.2 0.0 0.2 0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 ibm vw Xin Tong Statistics 7 / 40
Image of page 7

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Noise The i in Y i = α + K k =1 β k X k,i + i Called noise, innovation, shocks . Model randomness not within the explanatory variables. Ill-posed if assumed no noise. Usually assumed to be i.i.d. with mean zero. (What if not mean zero?) More general: weakly stationary and uncorrelated E i = 0 , E 2 i = σ 2 , cov( i , j ) = 0 . Xin Tong Statistics 8 / 40
Image of page 8
History:1801 source: internet Xin Tong Statistics 9 / 40
Image of page 9

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
MLE: Gaussian i.i.d. noise Simple linear regression Y i = α + βX i + i . Assume i ∼ N (0 , σ 2 ). Density of Y i given (conditioned on) X i , α, β : p ( Y i | X i , α, β ) = 1 p 2 πσ 2 exp - ( Y i - βX i - α ) 2 2 σ 2 Xin Tong Statistics 10 / 40
Image of page 10
MLE: derivation The joint PDF: p ( Y 1 , . . . , Y n | X 1 , . . . , X n , β, α ) = 1 p 2 πσ 2 n exp - n X i =1 ( Y i - βX i - a ) 2 2 σ 2 ! MLE: maximize p ( Y 1 , . . . , Y n | X 1 , . . . , X n , β, α ) among β, α .
Image of page 11

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 12
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern