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

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Lecture 2:Statistical inference and testing Xin T Tong Sunday 28 th August, 2016 Xin Tong Statistics 1 / 40

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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
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

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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
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

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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
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

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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
History:1801 source: internet Xin Tong Statistics 9 / 40

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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
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 .

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