If the multiple correlation coefficient of X 1on X 2and X 3is zero, then:
r12 = 0, r13 = 0
\(R_{1.23}\) is the correlation between \(X_{1}\) and its best linear predictor based on \(X_{2}, X_{3}\). If \(R_{1.23}=0\), no linear combination of \(X_{2}, X_{3}\) correlates with \(X_{1}\).
Using \[R^{2}_{1.23}=\frac{r_{12}^{2}+r_{13}^{2}-2r_{12}r_{13}r_{23}}{1-r_{23}^{2}}=0,\] the numerator must vanish, giving \((r_{12}-r_{13}r_{23})^{2}+r_{13}^{2}(1-r_{23}^{2})=0\). Both squared terms are non-negative, so each is zero, forcing \(r_{13}=0\) and then \(r_{12}=0\).
Hence \(R_{1.23}=0\) requires \(r_{12}=0\) and \(r_{13}=0\).
Similar Questions
If r and R denote correlation and multiple correlation coefficient for the data set for X 1, X 2and X 3. Which option is correct?
The coefficients of the regression β X|y and β Y|x , are known, The coefficient of correlation equals:
The value of simple correlation coefficient lies in the interval:
Which option is correct for the correlation ratio E 2?
Let θ be the angle made by the line of regression of Y on X. If σ Y= 2σ X and the correlation coefficient between X and Y is 0.3, the value θ equals
The multiple correlation coefficient R 1,23 as compared to any simple correlation coefficients between the distinct variable X 1 ,X 2, and X 3is
Important Questions from Correlation Analysis
Given below are two statements: One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): If the securities with less than perfect negative correlation between their price movements are combined, portfolio risk can be reduced significantly.
Reason (R): The term with negative correlation has the effect of reducing the computed value of total portfolio risk, given other terms that are positive.
In the light of the above statements, choose the most appropriate answer from the options given below:
X, Y and Z are three uncorrelated variables having variances \(\sigma_x^2, \sigma_y^2 \:and\:\sigma_z^2\) respectively, then the correlation between X + Y and Y + Z is:
If X ∼ N (0, 1) and Y = X2 then the correlation coefficient r (X, Y) is
Calculate the correlation coefficient between the following values :
x: 3, 5, 1, 7, 5
y: 4, 3, 0, 8, 2
Consider two exponentially distributed random variables X and Y, both having a mean of 0.50. Let Z = X + Y and r be the correlation coefficient between X and Y. If the variance of Z equals 0, then the value of r is _______ (round off to 2 decimal places).
