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# lecture 4.pdf - Lecture 4 Linear regression and AR model...

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Lecture 4: Linear regression and AR model inference Xin T Tong Thursday 7 th September, 2017 Xin Tong Statistics 1 / 51

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Last time Statistical properties of financial time series Stationarity Autocorrelation Market eﬃciency Random walk model Market eﬃciency test AR models: theoretical properties Stationarity Mean derivation Yule Walker: ACF and ACVF Forecast Mean reversion, business cycles. Xin Tong Statistics 2 / 51
Auto-regressive models AR is one of the most useful model for time series r t = α + β r t 1 + a t . General AR(K) model: r t = α + K j =1 β j r t j + a t . 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 ? Xin Tong Statistics 3 / 51

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OLS and Linear regression Xin Tong Statistics 4 / 51
Linear regression Simple linear regression model: Y i = α + β X i + ϵ i . Multiple linear regression model: Y i = α + K k =1 β k X k,i + ϵ i . Y i is the dependent/response variable. X k,i is the predictor/explanatory model. In most scenarios, n ( T ) data points are available ( 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 , ϵ i = 0 , Y i = α ) Xin Tong Statistics 5 / 51

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Wide application Y i = α + K k =1 β k X k,i + ϵ i Life quality Quality = 2 · Income +1 . 5 · Density 1 · living price 0 . 5 · Crime rate Income discrimination: Income = 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 6 / 51
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 Years: 1926-2003 Xin Tong Statistics 7 / 51

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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 Years: 1926-2003 Xin Tong Statistics 8 / 51
Noise The ϵ i in Y i = α + K k =1 β k X k,i + ϵ i Called noise, innovation, shocks . Model randomness not within the explanatory variables. Ill-posed if assumed no noise: data more than variables, more equations than unknowns. Usually assumed to be i.i.d. with mean zero. More general: weakly stationary and uncorrelated E ϵ i = 0 , E ϵ 2 i = σ 2 ϵ , cov ( ϵ i , ϵ j ) = 0 . Xin Tong Statistics 9 / 51

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History:1801 source: internet Xin Tong Statistics 10 / 51
MLE: Gaussian i.i.d. noise Simple linear regression Y i = α + β X i + ϵ i . Assume ϵ i N (0 , σ 2 ϵ ) . Density of Y i given (conditioned on) X i , α , β : p ( Y i | X i , α , β ) = 1 2 πσ 2 ϵ exp ( Y i β X i α ) 2 2 σ 2 ϵ Xin Tong Statistics 11 / 51

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MLE: derivation The joint PDF: p ( Y 1 , . . . , Y n | X 1 , . . . , X n , β , α ) = 1 2 πσ 2 ϵ n exp ( n i =1 ( Y i β X i α ) 2 2 σ 2 ϵ ) . MLE: maximize p ( Y 1 , . . . , Y n | X 1 , . . . , X n , β , α ) among β , α . Equivalent: minimize R ( α , β ) = n i =1 ( Y i β X i α ) 2 Xin Tong Statistics 12 / 51
MLE: minimization Want to minimize R ( α , β ) = n i =1 ( Y i β X i α ) 2 : α R = 2 β n i =1 X i + 2 n α 2 n i =1 Y i = 0 β R = 2 n i =1 X i Y i + 2 β n i =1 X 2 i + 2 n i =1 X i α = 0 Linear equation: ( n n i =1 X 2 i ( n i =1 X i ) 2 ) β

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