MLR_PARAM - Multiple Linear Regression Coefficients Values

Calculates the OLS regression coefficient values.

Syntax

MLR_PARAM (X, Mask, Y, Intercept, Return, Parameter Index, Alpha)

X

is the independent (explanatory) variables data matrix, so each column represents one variable.

Mask

is the boolean array to choose the explanatory variables in the model. If missing, all variables in X are included.

Y

is the response or the dependent variable data array (a one-dimensional array of cells (e.g., rows or columns)).

Intercept

is the constant or the intercept value to fix (e.g., zero). If missing, an intercept will not be fixed and is computed typically.

Return

is a switch to select the return output (1 = Mean Value (default), 2 = Std. Error, 3 = T-Stat, 4 = P-Value, 5 = Upper Limit (CI), 6 = Lower Limit (CI)).

Value	Return
1	Mean Value (default).
2	Standard Error.
3	T-Stat.
4	P-Value.
5	Upper Limit.
6	Lower Limit.

Parameter Index

is a switch to designate the target parameter (0 = intercept (default), 1 = first variable, 2 = 2nd variable, etc.).

Alpha

is the statistical significance of the test (i.e., alpha). If missing or omitted, an alpha value of 5% is assumed.

Remarks

1. The underlying model is described here.
2. $$\mathbf{y} = \mathbf{X}\boldsymbol\beta + \boldsymbol\varepsilon$$ $$\hat{\boldsymbol\beta} = (\mathbf{X}^{\rm T}\mathbf{X})^{-1} \mathbf{X}^{\rm T}\mathbf{y} = \big(\, \tfrac{1}{n}{\textstyle\sum} \mathbf{x}_i \mathbf{x}^{\rm T}_i \,\big)^{-1} \big(\, \tfrac{1}{n}{\textstyle\sum} \mathbf{x}_i y_i \,\big).$$ Where:
   • $\hat{\boldsymbol\beta}$ is the estimated regression coefficients.
3. The sample data may include data points with missing values.
4. Each column in the input matrix corresponds to a separate variable.
5. Each row in the input matrix corresponds to an observation.
6. Observations (i.e., rows) with missing values in X or Y are removed.
7. The number of rows of the response variable (Y) must be equal to the number of rows of the explanatory variable (X).
8. The MLR_PARAM function is available starting with version 1.60 APACHE.

Files Examples

Related Links

• Wikipedia - Linear regression.
• Wikipedia - Regression analysis.
• Wikipedia - Ordinary least squares.

References

• Hamilton, J.D.; Time Series Analysis, Princeton University Press (1994), ISBN 0-691-04289-6.
• Kenney, J. F. and Keeping, E. S. (1962) "Linear Regression and Correlation." Ch. 15 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 252-285.