Calculates the OLS regression coefficients values.

## Syntax

**SLR_PARAM**(

**X**,

**Y**,

**Intercept**,

**Return_type**,

**Parameter Index**,

**Alpha**)

- X
- is the independent (aka explanatory or predictor) variable data array (one dimensional array of cells (e.g. rows or columns)).
- 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 = mean value (default), 2 = std error, 3 = test score, 4 = p-value, 5 = upper value, 6 = lower value).
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 = explanatory variable).
Method Description 0 Intercept 1 Explanatory Variable - 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.
- The regression slope (i.e. $\beta$) is computed as follows:

$$\beta = \frac{\sum_{i=1}^N (X_i-\bar X)(Y_i-\bar Y)}{\sum_{i=1}^N (X_i-\bar X)^2}$$

And the intercept (i.e. $\alpha$) is expressed as follows:

$$\alpha = \bar Y - \beta \bar X$$

Where:- $N$ is the number of observations.
- $\bar X$ is the empirical sample average for the explanatory variable ($X$).
- $\bar Y$ is the empirical sample average for the dependent variable ($Y$).

- The standard errors in the regression coefficients are expressed as follows:

$$\mathrm{Var}(\beta) = \frac{\mathrm{MSE}}{\sum_{i=1}^N(X_i-\bar X)^2}$$

$$\mathrm{Var}(\alpha)=\mathrm{MSE}\times\left(\frac{1}{N}+\frac{\bar X^2}{\sum_{i=1}^N(X_i-\bar X)^2} \right )$$

Where:

- $\mathrm{Var}(.)$ denotes the statistical variance.
- $\mathrm{MSE} = \frac{\mathrm{SSE}}{N-2}= \frac{\sum_{i=1}^N (Y_i - \hat Y_i)^2}{N-2}$

- 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 SLR_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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