SLR_PARAM - Simple Regression Coefficients Values

Mohamad

October 20, 2016 19:36

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).

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.

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.

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.