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

## Syntax

**SLR_FORE** (**X**, **Y**, Intercept, **Target**, Return, 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 normally.
**Target**- is the value of the explanatory variable.
**Return**- is a switch to select the return output (1 = Forecast (default), 2 = Error, 3 = Upper Limit, 4 = Lower Limit).
Value Return 1 Mean Value ( **default**).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

## 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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