# MLR_PARAM - Multiple Linear Regression Coefficients Values

Calculates the OLS regression coefficients values.

## Syntax

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

X is the independent (explanatory) variables data matrix, such that 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 (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 normally.

Return_type is a switch to select the return output (1 = value (default), 2 = std. error, 3 = t-stat, 4 = P-value, 5 = upper limit (CI), 6 = lower limit (CI))

Method Description
1 Mean Value
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 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.

## 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