# SLR_PARAM - Simple Regression Coefficients Values

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

1. The underlying model is described here.
2. 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 ($Y$).
3. 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.
• $\mathrm{MSE} = \frac{\mathrm{SSE}}{N-2}= \frac{\sum_{i=1}^N (Y_i - \hat Y_i)^2}{N-2}$
4. The sample data may include missing values.
5. Each column in the input matrix corresponds to a separate variable.
6. Each row in the input matrix corresponds to an observation.
7. Observations (i.e. rows) with missing values in X or Y are removed.
8. The number of rows of the response variable (Y) must be equal to the number of rows of the explanatory variable (X).
9. The SLR_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