that between x and z), and it is downward otherwise. Detailed analysis As an example, consider a linear model of the form where xi is a 1 × p row vector of values of p independent variables observed at time i or for the i th study participant; β is a p × 1 column vector of unobservable parameters (the response coefficients of the dependent variable to each of the p independent variables in xi) to be estimated; zi is a scalar and is the value of another independent variable that is observed at time i or for the i th study participant; δ is a scalar and is an unobservable parameter (the response coefficient of the dependent variable to zi) to be estimated; ui is the unobservable error term occurring at time i or for the i th study participant; it is an unobserved realization of a random variable having expected value 0 (conditionally on xi and zi); yi is the observation of the dependent variable at time i or for the i th study participant. We collect the observations of all variables subscripted i = 1, ..., n, and stack them one below another, to obtain the matrix X and the vectors Y, Z, and U: and If the independent variable z is omitted from the regression, then the estimated values of the response parameters of the other independent variables will be given by the usual least squares calculation, (where the "prime" notation means the transpose of a matrix and the -1 superscript is matrix inversion). Substituting for Y based on the assumed linear model, On taking expectations, the contribution of the final term is zero; this follows from the assumption that U is uncorrelated with the regressors X. On simplifying the remaining terms: The second term after the equal sign is the omitted-variable bias in this case, which is non-zero if the omitted variable z is correlated with any of the included variables in the matrix X (that is, if X′Z does not equal a vector of zeroes). Note that the bias is equal to the weighted portion of zi which is "explained" by xi. Effect in ordinary least squares The Gauss–Markov theorem states that regression models which fulfill the classical linear regression model assumptions provide the most efficient, linear and unbiased estimators. In ordinary least squares, the relevant assumption of the classical linear regression model is