COVARIANCE
ABSTRACT
Covariance play a fundamental role in the theory of time series, and they are critical quantities that are needed in both spectral and time domain analysis. Estimation of covariance matrices is needed in the construction of confidence regions for unknown parameters, hypothesis testing, principal component analysis, prediction, discriminant analysis, among others. In this chapter, we consider both low and high-dimensional covariance matrix estimation problems and present a review for asymptotic properties of sample covariances and covariance matrix estimates. In particular, we shall provide an asymptotic theory for estimates of high-dimensional covariance matrices in time series and a consistency result for covariance matrix estimates for estimated parameters.
INTRODUCTION
What is Covariance?
In mathematics and statistics, covariance is a measure of the relationship between two random variables. The metric evaluates how much – to what extent – the variables change together. In other words, it is essentially a measure of the variance between two variables. However, the metric does not assess the dependency between variables.Covariance
Unlike the correlation coefficient, covariance is measured in units. The units are computed by multiplying the units of the two variables. The variance can take any positive or negative values. The values are interpreted as follows:
Positive covariance: Indicates that two variables tend to move in the same direction.
Negative covariance: Reveals that two variables tend to move in inverse directions.
In finnace, the concept is primarily used in portfolio theory. One of its most common applications in portfolio theory is the diversification method, using the covariance between assets in a portfolio. By choosing assets that do not exhibit a high positive covariance with each other, the unsystematic risk can be partially eliminated.
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USES
In current times, the historical definition of covariate has faded somewhat. Many analysts use this term as a synonym for a continuous predictor not only for the specific subset of experimental design cases I describe above.
In current usage, a covariate might be a primary variable of interest in a non-DOE context!
In an analytical sense, the modern usage is valid. Covariates in the stricter context performs the same function as continuous predictors in the broader definition.
Just be aware that some analysts will have an extremely specific context in mind when discussing covariates. Others will be thinking in much broader terms!
FORMULA FOR COVARIANCE
The covariance formula is similar to the formula for correlation and deals with the calculation of data points from the average value in a dataset. For example, the covariance between two random variables X and Y can be calculated using the following formula (for population):
For a sample covariance, the formula is slightly adjusted:
Where:
Xi – the values of the X-variable
Yj– the values of the Y-variable
X̄ – the mean (average) of the X-variable
Ȳ – the mean (average) of the Y-variable
n – the number of data points
PROBLEM AND SOLUTION
John is an investor. His portfolio primarily tracks the performance of the S & P 500 and John wants to add the stock of ABC Corp. Before adding the stock to his portfolio, he wants to assess the directional relationship between the stock and the S&P 500.
John does not want to increase the unsystematic risk of his portfolio. Thus, he is not interested in owning securities in the portfolio that tend to move in the same direction.
John can calculate the covariance between the stock of ABC Corp. and S&P 500 by following the steps below:
1. Obtain the data.
First, John obtains the figures for both ABC Corp. stock and the S&P 500. The prices obtained are summarized in the table below:
2. Calculate the mean (average) prices for each asset.
3. For each security, find the difference between each value and mean price.
4. Multiply the results obtained in the previous step.
5. Using the number calculated in step 4, find the covariance.
In such a case, the positive covariance indicates that the price of the stock and the S&P 500 tend to move in the same direction.
APPLICATIONS OF COVARIANCE
The following are the most common applications of Covariance:
Simulating systems with multiple correlated variables is done using Cholesky decomposition. A covariance matrix helps determine the Cholesky decomposition because it is positive semi-definite. The matrix is decomposed by the product of the lower matrix and its transpose.
To reduce the dimensions of large data sets, principal component analysis is used. To perform principal component analysis, an eigen decomposition is applied to the covariance matrix
CONCLUSION
Ancova is concerned with studying regressions in a set of groups. Models for ANCOVA cater for a wide range of patterns of these regressions and include procedures for selecting among them. As hypothesis testing is the principal method for this, its general limitations, especially in the context of several hypotheses, have to be carefully considered. Extensions of ANCOVA include structures for the groups, such as crossing, nesting, and their combinations, within-group models that are more complex than ordinary regression (factor analysis and generalized linear models), and the groups can be associated with random effects.
REFERENCES
Rice, John (2007). Mathematical Statistics and Data Analysis. Belmont, CA: Brooks/Cole Cengage Learning. p. 138.
Oxford Dictionary of Statistics, Oxford University Press, 2002, p. 104.
Jump to the Park,Kun Il (2018). Fundamentals of Probability and Stochastic Processes with Applications to Communications. Springer.
Yuli Zhang, Huaiyu Wu, Lei Cheng (June 2012). Some new deformation formulas about variance and covariance. Proceedings of 4th International Conference on Modelling, Identification and Control(ICMIC2012). pp. 987–992.