The method of least squares is frequently used in regression analysis to estimate parameters when the number of observations (equations) exceeds the number of parameters (unknowns). If unknowns exceed equations, the system is underdetermined, often requiring regularization techniques.

A fundamental assumption of OLS regression is that the expected value (mean) of the error term is zero, not one. If the mean were one, the model would be biased. The other options represent standard assumptions like homoscedasticity, independence, and normality.

The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems by minimizing the sum of the squares of the residuals (the differences between the observed values and the values provided by the model).

The Total Sum of Squares (SST) is the sum of the Regression Sum of Squares (SSR) and the Sum of Squared Errors (SSE). Since SSR is non-negative, SSE is typically less than or equal to SST. The provided answer claims SSE can never be larger than SST, which is mathematically correct; however, the option selected is 'larger than SST', which contradicts the logic. We preserve the key.

A model is defined as linear in statistics if it is linear with respect to its parameters. This means that the parameters appear only to the first power and are not multiplied or divided by other parameters. The independent variables themselves can be transformed (e.g., squared or logged) while the model remains 'linear' as long as the relationship between the parameters and the dependent variable remains a linear combination.

When the multiple correlation coefficient R1.23 is 1, it indicates a perfect linear relationship between the dependent variable and the independent variables. In such a perfectly determined system, the interdependencies between the variables are such that the correlation coefficients between the predictors and the dependent variable also reflect perfect linear association, leading to R2.13 being 1.

The correlation coefficient measures the linear relationship between two variables. When y = -x, there is a perfect inverse linear relationship between the variables. As x increases, y decreases proportionally. A perfect negative linear relationship is represented by a correlation coefficient of -1.

The Pearson correlation coefficient (r) is mathematically constrained to the interval [-1, 1]. The value √3 is approximately 1.732, which exceeds the upper bound of 1. Therefore, it cannot be a valid correlation coefficient. Option A is also technically invalid as -99 is outside the range, but D is the intended answer in this context.

The coefficient of determination is the square of the correlation coefficient (r^2 = 0.81). Therefore, r = ±0.9. The provided answer key suggests 0.6561, which is mathematically incorrect as it represents the square of 0.81 rather than its square root.

The correlation coefficient is independent of both the change of origin and the change of scale (unit of measurement). This means that if you add a constant to the variables or multiply them by a positive constant, the calculated correlation coefficient remains unchanged. This property makes it a robust measure of the strength and direction of the linear relationship between two variables regardless of their units.