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

- The underlying model is described here.
- $$\mathbf{y} = \mathbf{X}\boldsymbol\beta + \boldsymbol\varepsilon$$ $$\hat{\boldsymbol\beta} = (\mathbf{X}^{\rm T}\mathbf{X})^{-1} \mathbf{X}^{\rm T}\mathbf{y} = \big(\, \tfrac{1}{n}{\textstyle\sum} \mathbf{x}_i \mathbf{x}^{\rm T}_i \,\big)^{-1} \big(\, \tfrac{1}{n}{\textstyle\sum} \mathbf{x}_i y_i \,\big).$$ Where:
- $\hat{\boldsymbol\beta}$ is the estimated regression coefficients.

- The sample data may include data points with missing values.
- Each column in the input matrix corresponds to a separate variable.
- Each row in the input matrix corresponds to an observation.
- Observations (i.e., rows) with missing values in X or Y are removed.
- The number of rows of the response variable (Y) must be equal to the number of rows of the explanatory variable (X).
- The MLR_PARAM function is available starting with version 1.60 APACHE.

## Files Examples

## Related Links

## 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.

## Comments

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