14.single_index

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University of California, Los Angeles Department of Statistics Statistics C183/C283 Instructor: Nicolas Christou Single index model Useful formulas The single index model states that R it = α i + β i R mt + ± it where, R it is the return of stock i at time t and R mt is the return of the market at time t . Assumptions and notation: E ( ± i ) = 0 , var ( ± i ) = σ 2 ± i , E ( ± i ± j ) = 0 ,cov ( R m i ) = 0 ,var ( R m ) = σ 2 m ,E ( R m ) = ¯ R m . Therefore, E ( R i ) = α i + β i ¯ R m var ( R i ) = σ 2 i = β 2 i σ 2 m + σ 2 ± i cov ( R i ,R j ) = σ ij = β i β j σ 2 m Here are some useful formulas: a. Estimate of β i (beta of stock i ): ˆ β i = m t =1 ( R it - ¯ R i )( R mt - ¯ R m ) m t =1 ( R mt - ¯ R m ) 2 . b. Estimate of α i (alpha of stock i ): ˆ α i = ¯ R i - ˆ β i ¯ R m . c. Estimate of σ 2 ± i (variance of random error term associated with stock i ): ˆ σ 2 ± i = m t =1 e 2 it m - 2 = m t =1 ( R it - ˆ α i - ˆ β i R mt ) 2 m - 2 . d. Estimate of var ( ˆ β i ). var ( ˆ β i ) = ˆ σ 2 ± i m t =1 ( R mt - ¯ R m ) 2 . e. Correlation between stock i and stock j : ρ ij == σ ij σ i σ j = β i β j σ 2 m σ i σ j . f. Correlation between stock i and market: ρ im = β i σ m σ i β i = ρ im σ i σ m . 1

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Simple R commands: a1 <- read.table("http://www.stat.ucla.edu/~nchristo/statistics_c183_c283/
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Unformatted text preview: stocks5_period1.txt", header=TRUE) #Regression of r11 on rsp1 (index): q <- lm(a1\$r11 ~ a1\$rsp1) #Summary of the regression above: summary(q) #List the names of the results in object q: names(q) #Get the estimates of alpha and beta: q\$coefficients[1] q\$coefficients[2] #List the residuals: q\$residuals #Get the estimate of the variance of the error term (MSE): sum(q\$residuals^2)/(nrow(a1)-2) #Another way: summary(q)\$sigma^2 #variance-covariance matrix of the estimates of the main parameters #of the model: vcov(q) #Get the variance of the estimate of beta: vcov(q)[2,2] #Another way: summary(q)\$coefficients[4]^2 2...
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## This note was uploaded on 06/02/2011 for the course STATS 183 taught by Professor Nicolas during the Spring '11 term at UCLA.

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