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

- 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 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

## 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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