FE515_MIDEXAM_Shashank-6-7

if (adjusted) { skewness_adj <- ( sum ((x - mean_x) ˆ 3 ) / (n * sd_x ˆ 3 )) return (skewness_adj) } else { skewness_unadj <- ( sum ((x - mean_x) ˆ 3 ) / ((n - 1 ) * sd_x ˆ 3 )) return (skewness_unadj) } } x <- c ( 1 , 2 , 3 , 4 , 5 , 6 , 12 , 71 ) # Calculate the adjusted and unadjusted skewness of daily log returns adjusted_skewness <- sample_skewness (x, adjusted = TRUE ) unadjusted_skewness <- sample_skewness (x, adjusted = FALSE ) cat ( "Adjusted Skewness:" , adjusted_skewness, " \n " ) ## Adjusted Skewness: 1.78443 cat ( "Unadjusted Skewness:" , unadjusted_skewness, " \n " ) ## Unadjusted Skewness: 2.039348 2.2 Without using packages, create a function of 2 variables “x” and “adjusted” that calculates the sample kurtosis of “x” using the formulas on Lecture 6 page 20 and page 23. When “adjusted” = TRUE, it returns the adjusted kurtosis of “x”, and FALSE returns the unadjusted one. sample_kurtosis <- function (x, adjusted = TRUE ) { n <- length (x) mean_x <- mean (x) sd_x <- sd (x) if (adjusted) { kurtosis_adj <- ( sum ((x - mean_x) ˆ 4 ) / n) / (sd_x ˆ 4 ) - 3 return (kurtosis_adj) } else { kurtosis_unadj <- ( sum ((x - mean_x) ˆ 4 ) / n) / (sd_x ˆ 4 ) return (kurtosis_unadj) } } x <- c ( 1 , 2 , 3 , 4 , 5 , 6 ) adjusted_kurt <- sample_kurtosis (x, adjusted = TRUE ) unadjusted_kurt <- sample_kurtosis (x, adjusted = FALSE ) cat ( "Adjusted Kurtosis:" , adjusted_kurt, " \n " ) ## Adjusted Kurtosis: -1.797619 7