Moving to the general form of the one-factor model, we explain Y by X, according to the regression equation, where we drop the subscript "i" designating the asset:

Because the residual and the factor are independent, the variance is:

The standard technique for estimating a factor model is the regression technique, which provides estimates of P0 and P , as well as standard statistics, which tell us whether those coefficients are significantly different from zero. The standard deviation of the residual is called the standard error. The expectation of the error term is zero. The covariance between the error term and x is zero. The ratio of the explained variance by Xto the total variance indicates how much variance of the explained variable is due to the "explaining" variable X. This ratio is called the R square, or R2, of the regression. It is desirable to have high R2.

The formulas for the coefficient depend on all historical observations Y. and x of F and X and on the size of the sample of data, made of a time series of values of Xand Y. The ordinary least square regression technique provides the best fit to the actual relation, defined as the fit that minimizes the squared deviations from the mean of the estimated return of the stock and actual observations over the sample of observations. The significance of the coefficients depends on the sample size. It is possible to have low R2 and still have significant coefficients at some confidence level, such as 5% or 1%.

Using the independence between the factor and the residual, the total risk decomposition of the single asset into general and specific risk is given by the equation:

From the regression analysis, the formula of the coefficient P is:

The covariance between X and F, cov(F, X), results from the independence between the residual and the factor:

By definition of the correlation coefficient, it is:

Using the covariance cov(F, X), the coefficient of the factor model becomes:

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