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1、Reading 11 Correlation and Regression Copyright FinQ. All rights reserved. 2 0 1 3 2. CORRELATION ANALYSIS Scatter plot and correlation analysis are used to examine how two sets of data are related. 2.1 Scatter Plots A scatter plot graphically shows the relationship between two varaibles. If the poi
2、nts on the scatter plot cluster together in a straight line, the two variables have a strong linear relation. Observations in the scatter plot are represented by a point, and the points are not connected. 2.2 it is a simple number. Example: ?= 47.78 ?= 40 ?= 250 ? =47.78?40?250?= 0.478 The correlati
3、on coefficient can range from -1 to +1. Two variables are perfectly positively correlated if correlation coefficient is +1. If correlation coefficient is -1, that means the asset returns have a perfect inverse (negative) linear relationship to each other. When correlation coefficient equals 0, there
4、 is no linear relationship between the returns on the two assets. The closer the correlation coefficient is to 1, the stronger the relationship between the returns of two assets. Note: Correlation of +/- 1 does not imply that slope of the line is +/- 1. NOTE: Combining two assets having zero correla
5、tion with each other reduces the risk of the portfolio. A negative correlation coefficient results in greater risk reduction. Difference b/w Covariance there could be third variable causing change in both variables.4. Spurious Correlations: Spurious correlation is correlation in the data without any
6、 causal relationship. This may occur when i. Two variables have only chance relationships. ii. Two variables that are uncorrelated but may be correlated if mixed by third variable . iii. Correlation between two variables resulting from a third variable. NOTE: Spurious correlation may suggest investm
7、ent strategies that appear profitable but actually would not be so, if implemented. 2.6 Testing the Significance of the Correlation Coefficient t-test is used to determine if sample correlation coefficient, r, is statistically significant. Two-Tailed Test: Null Hypothesis H0 : the correlation in the
8、 population is 0 ( = 0); Alternative Hypothesis H1: the correlation in the population is different from 0 ( 0); NOTE: The null hypothesis is the hypothesis to be tested. The alternative hypothesis is the hypothesis that is accepted if the null is rejected. The formula for the t-test is (for normally
9、 distributed variables): ? ? ? 21 ? ? ? 2? Correlation and Regression investor about whether the relationship between asset returns is positive, negative or zero, but correlation of relationship between Correlation coefficients are valid only if the means, covariances of X and Y are finite and const
10、ant. When these assumptions dont hold, then the correlations between two different variables Limitations of Correlation Analysis measures linear Correlation also may be an unreliable measure when outliers are present in one or both of Based on Correlation we cannot assume x causes y; there could be
11、third causing change in both variables. Spurious correlation is correlation in the data without any causal Two variables have only chance relationships. Two variables that are uncorrelated but may be Correlation between two variables resulting from gest investment strategies that appear profitable b
12、ut actually would not be so, if Significance of the Correlation correlation the correlation in the population is the correlation in the The null hypothesis is the hypothesis to be tested. The alternative hypothesis is the hypothesis that is distributed where, r is the sample coefficient of correlati
13、on calculated by ? ?,?t = t-statistic (or calculated t) n 2 = degrees of freedom Decision Rule: If test statistic is + t degrees of freedom, (if absolute value of t H0; otherwise Do not Reject HExample: Suppose r = 0.886 and n = 8,and t level i.e. = 5%/2 and 8 2 = 6 degrees of freedom) = 2.4469 t =0
14、.88682 1(0.886)2= 4.68 wereject null hypothsis of no correlation. Magnitute of r needed to reject the null hypothesis (H0: = 0) decreases as sample size n increases. Because as n increases: o The number of degrees of o The absolute value of tc o t-value increases In other words, type II error decrea
15、ses when sample size (n) increases, all else equal. NOTE: Type I error = reject the null hypothesis true. Type II error = do not reject the null it is wrong. Practice: Example 7, 8, 9 otherwise Do not Reject H0. Suppose r = 0.886 and n = 8,and tC (at 5% significance 2 = 6 degrees of freedom) = Since t-value tc, wereject null hypothsis of no correlation. Magnitute of r needed to reject the null hypothesis = 0) decreases as sample size n increases. The number of degrees of freedom increases c decreases. In other words, type II error decreases
