Calculates the forecast value, error and confidence interval for a regression model.

## Syntax

**SLR_FORE**(

**X**,

**Y**,

**Intercept**,

**Target**,

**Return_type**,

**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.

**Target** is the value of the explanatory variable.

**Return_type** is a switch to select the return output (1 = forecast (default), 2 = error, 3 = upper limit, 4 = lower limit).

Method | Description |
---|---|

1 | Mean Value |

2 | Standard Error |

3 | Upper Limit |

4 | Lower Limit |

**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 SLR_FORE function computes the prediction interval (aka confidence interval) for a given value of the explanatory variable.
- The mean prediction values are computed by:

$$E[Y_f|X_f] = \alpha + \beta \times X_f$$

Where:

- $E[Y_f|X_f]$ is the conditional prediction mean value of $Y$.
- $X_f$ is the value of the explanatory variable.
- $E[.|X_f]$ is the conditional expectation operator.

- The prediction error is driven by the regression mean error and the value of $X_f$ itself.

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

Where:

- $N$ is the number of observations.
- $\bar X$ is the empirical sample average for the explanatory variable ($X$).
- $\mathrm{MSE} = \frac{\mathrm{SSE}}{N-2}= \frac{\sum_{i=1}^N (Y_i - \hat Y_i)^2}{N-2}$

- 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_FORE 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

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