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Lecture# Multivariate Normal Distribution: Correlation and Covariance

Description

This lecture covers the concept of correlation matrix related to the covariance matrix, the calculation of the correlation matrix, and the properties of the correlation matrix. It also discusses the standard empirical estimates for mean and covariance, the accuracy of standard estimates, and the eigenvalues distribution. Additionally, it explores the normality testing, the Jarque-Bera statistic, and the QQ plot. The lecture concludes with discussions on multivariate normal distribution, normal variance mixtures, and factor models.

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FIN-609: Asset Pricing

This course provides an overview of the theory of asset pricing and portfolio choice theory following historical developments in the field and putting
emphasis on theoretical models that help our unde

Related concepts (116)

Normal distribution

In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter is its standard deviation. The variance of the distribution is . A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.

Multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem.

Normality test

In statistics, normality tests are used to determine if a data set is well-modeled by a normal distribution and to compute how likely it is for a random variable underlying the data set to be normally distributed. More precisely, the tests are a form of model selection, and can be interpreted several ways, depending on one's interpretations of probability: In descriptive statistics terms, one measures a goodness of fit of a normal model to the data – if the fit is poor then the data are not well modeled in that respect by a normal distribution, without making a judgment on any underlying variable.

Correlation

In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve.

Truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated normal distribution has wide applications in statistics and econometrics. Suppose has a normal distribution with mean and variance and lies within the interval . Then conditional on has a truncated normal distribution. Its probability density function, , for , is given by and by otherwise.

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