CovY

CovY is a statistical notation used to denote the covariance between a response variable Y and one or more predictor variables. Depending on context, CovY may be written as Cov(Y, X) for a single predictor X or as the vector of covariances between Y and each component of a multivariate X. It is not a distinct distribution or model.

Definition and calculation: If (Yi, Xi) is a sample of size n with Yi the observed responses and Xi the corresponding predictors, the sample covariance between Y and X is Cov_hat(Y, X) = (1/(n−1)) sum from i=1 to n of (Yi − mean(Y)) (Xi − mean(X)). For a multivariate X with components X1, ..., Xk, Cov(Y, X) is a 1×k row vector containing the covariances between Y and each Xi.

Properties and interpretation: Covariance is a bilinear measure that captures the degree to which two variables vary together. A zero covariance indicates no linear association between Y and X, though it does not necessarily imply independence unless certain assumptions hold. Covariance is scale dependent, so normalization yields the correlation coefficient ρ(Y, X), which lies between −1 and 1. In simple linear regression, the slope is proportional to Cov(Y, X) divided by Var(X).

Applications: CovY features in regression analysis, time series, and multivariate statistics. In simple regression, Cov(Y, X) relates to the regression coefficient, while in more complex models it contributes to estimators and to understanding the relationships between Y and predictors. Covariances across lags form autocovariance and cross-covariance structures in time-dependent data.

Notational variation: The term CovY may be used differently across texts to denote covariance with Y against a particular covariate or a vector of covariances. Context typically clarifies whether it refers to a single covariate or a set of covariances.