Co-Variance

Examples of Co-Variance in the following topics:

• Portfolio Risk

  • In order to calculate the variance of a portfolio of three assets, we need to know that figure for apples, bananas, and cherries, and we also need to know the co-variance of each.
  • Co-variances can be thought of as correlations.
  • If every time bananas have a bad day, so do apples, their co-variance will be large.
  • If bananas do great half of the time when cherries do bad and bananas do terrible the other half, their co-variance is zero.
  • The formula to compute the co-variance between returns on X and Y:

• Regression Analysis for Forecast Improvement

  • The errors are uncorrelated, that is, the variance– co-variance matrix of the errors is diagonal, and each non-zero element is the variance of the error.
  • The variance of the error is constant across observations (homoscedasticity).

• Introduction to quantitative analysis

  • Using information about the co-variation among the multiple measures, we can infer an underlying dimension or factor; once we've done that, we can locate our observations along this dimension.
  • That is, we could "scale" or index the similarity of the actors in terms of their participation in events - but weight the events according to common variance among them.
  • Similarly, we could "scale" the events in terms of the patterns of co-participation of actors -- but weight the actors according to their frequency of co-occurrence.
  • UCINET includes two closely-related factor analytic techniques (Tools>2-Mode Scaling>SVD and Tools>2-Mode Scaling Factor Analysis) that examine the variance in common among both actors and events simultaneously.
  • Once the underlying dimensions of the joint variance have been identified, we can then "map" both actors and events into the same "space."

• Variance Sum Law II

  • Compute the variance of the sum of two variables if the variance of each and their correlation is known
  • Compute the variance of the difference between two variables if the variance of each and their correlation is known
  • which is read: "The variance of X plus or minus Y is equal to the variance of X plus the variance of Y."
  • The variance of the difference is:
  • If the variances and the correlation are computed in a sample, then the following notation is used to express the variance sum law:

• Two-mode correspondence analysis

  • Factoring methods operate on the variance/covariance or correlation matrices among actors and events.
  • We do see, however, that this method also can be used to locate the initiatives along multiple underlying dimensions that capture variance in both actors and events.
  • The result is showing that there is a cluster of issues that "co-occur" with a cluster of donors - actors defining events, and events defining actors.
  • Event coordinates for co-participation of donors in California initiative campaigns
  • Actor coordinates for co-participation of donors in California initiative campaigns

• Hypotheses about the means of multiple groups

  • The approach to estimating difference between the means of two groups discussed in the previous section can be extended to multiple groups with one-way analysis of variance (ANOVA).
  • If we examine the network of connections among donors (defined by co-participating in the same campaigns), we anticipate that the worker's groups will display higher eigenvector centrality than donors in the other groups.
  • The differences in group means account for 78% of the total variance in eigenvector centrality scores among the donors.

• Variance

  • The variance of a data set measures the average square of these deviations.
  • Calculating the variance begins with finding the mean.
  • Once the mean is known, the variance can be calculated.
  • The variance for the above set of numbers is:
  • The population variance can be very helpful in analyzing data of various wildlife populations.

• Variance Sum Law I

  • The question is, "What is the variance of this sum?"
  • where the first term is the variance of the sum, the second term is the variance of the males and the third term is the variance of the females.
  • Therefore, if the variances on the memory span test for the males and females respectively were 0.9 and 0.8, respectively, then the variance of the sum would be 1.7.
  • More generally, the variance sum law can be written as follows:
  • which is read: "The variance of X plus or minus Y is equal to the variance of X plus the variance of Y."

• Test of Two Variances

  • Another of the uses of the F distribution is testing two variances.
  • Let $\sigma ^2_1$ and $\sigma ^2_2$ be the population variances and $s^2_1$ and $s^2_2$ be the sample variances.
  • It depends on Ha and on which sample variance is larger.
  • The first instructor's grades have a variance of 52.3.
  • The second instructor's grades have a variance of 89.9.

• Variance Estimates

  • The $F$-test can be used to test the hypothesis that the variances of two populations are equal.
  • Notionally, any $F$-test can be regarded as a comparison of two variances, but the specific case being discussed here is that of two populations, where the test statistic used is the ratio of two sample variances.
  • The expected values for the two populations can be different, and the hypothesis to be tested is that the variances are equal.
  • It has an $F$-distribution with $n-1$ and $m-1$ degrees of freedom if the null hypothesis of equality of variances is true.
  • Discuss the $F$-test for equality of variances, its method, and its properties.