Calculates the OLS regression coefficient values.

## Syntax

**SLR_PARAM**(

**X**,

**Y**, Intercept, Return, Parameter Index, Alpha)

**X**- is the independent (aka explanatory or predictor) variable data array (a one-dimensional array of cells (e.g., rows or columns)).
**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 = Test Score, 4 = P-Value, 5 = Upper Value, 6 = Lower Value).
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 = Explanatory Variable).
Value Parameter Index 0 Intercept ( **default**).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 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 equal 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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