TAILIEUCHUNG - Class Notes in Statistics and Econometrics Part 10

CHAPTER 19 Digression about Correlation Coefficients. . A Unified Definition of Correlation Coefficients Correlation coefficients measure linear association. The usual definition of the simple correlation coefficient between two variables ρxy (sometimes we also use the notation corr[x, y]) | CHAPTER 19 Digression about Correlation Coefficients . A Unified Definition of Correlation Coefficients Correlation coefficients measure linear association. The usual definition of the simple correlation coefficient between two variables pxy sometimes we also use the notation corr x y is their standardized covariance cov x y 19-1-1 Pxy . . . v var x var y Because of Cauchy-Schwartz its value lies between 1 and 1. 513 514 19. DIGRESSION ABOUT CORRELATION COEFFICIENTS PROBLEM 254. Given the constant scalars a 0 and c 0 and b and d arbitrary. Show that corr x y corr ax b cy d with the sign being valid if a and c have the same sign and the sign otherwise. ANSWER. Start with cov ax b cy d accov x y and go from there. Besides the simple correlation coefficient pxy between two scalar variables y and x one can also define the squared multiple correlation coefficient py x between one scalar variable y and a whole vector of variables x and the partial correlation coefficient between two scalar variables y1 and y2 with a vector of other variables x partialled out. The multiple correlation coefficient measures the strength of a linear association between y and all components of x together and the partial correlation coefficient measures the strength of that part of the linear association between y1 and y2 which cannot be attributed to their joint association with x. One can also define partial multiple correlation coefficients. If one wants to measure the linear association between two vectors then one number is no longer enough but one needs several numbers the canonical correlations. The multiple or partial correlation coefficients are usually defined as simple correlation coefficients involving the best linear predictor or its residual. But all these . A UNIFIED DEFINITION OF CORRELATION COEFFICIENTS 515 correlation coefficients share the property that they indicate a proportionate reduction in the MSE. See . Rao73 pp. 268-70 . Problem 255 makes this point

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