Predictive model

In the statistics is called forecasting model or prediction model , a model which has a forecast of the dependent variables  provides y and used a functional relationship defined by a regression method was determined. When there are additional x-values ​​with no associated y-value, the fitted model can be used to predict the value of y.

There are other forecast models for time series , see z. B. under linear prediction .

Forecast model

In the multiple linear regression , the forecast model results from

${\displaystyle \mathbf {y} _{0}=\mathbf {X} _{0}{\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }}_{0}}$,

in which

• ${\displaystyle \mathbf {y} _{0}}$represents the vector of future dependent variables and
• ${\displaystyle \mathbf {X} _{0}}$the matrix of explanatory variables at the time . ${\displaystyle T_{0}}$

The forecast is presented as . ${\displaystyle {\hat {\mathbf {y} }}_{0}=\mathbf {X} _{0}\mathbf {b} }$

Forecast error

From the above Representation shows the forecast error with the following properties: ${\displaystyle {\hat {\mathbf {y} }}_{0}-\mathbf {y} _{0}=\mathbf {X} _{0}(\mathbf {b} -{\boldsymbol {\beta }})-{\boldsymbol {\varepsilon }}_{0}}$

• the mean expected value of the forecast error is zero: ${\displaystyle \operatorname {E} ({\hat {\mathbf {y} }}_{0}-\mathbf {y} _{0})=\operatorname {E} (\mathbf {X} _{0}(\mathbf {b} -{\boldsymbol {\beta }})-{\boldsymbol {\varepsilon }}_{0})=\mathbf {0} }$
• the variance-covariance matrix of the forecast error is: . ${\displaystyle \operatorname {E} [({\hat {\mathbf {y} }}_{0}-\mathbf {y} _{0}-\operatorname {E} ({\hat {\mathbf {y} }}_{0}-\mathbf {y} _{0}))(({\hat {\mathbf {y} }}_{0}-\mathbf {y} _{0}-\operatorname {E} ({\hat {\mathbf {y} }}_{0}-\mathbf {y} _{0}))^{\top }]=\sigma ^{2}[\mathbf {X} _{0}(\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} _{0}^{\top }+\mathbf {I} ]}$

One is often interested in estimating the realization of the endogenous (= dependent ) variables for a new value . For example, it could be the planned price of a product and its sales. In this case a simple regression model is assumed. The predicted function value of the exogenous (= independent) variables is then given by ${\displaystyle x_{0}}$ ${\displaystyle y_{0}}$ ${\displaystyle x_{0}}$ ${\displaystyle y_{0}}$ ${\displaystyle {\hat {y}}_{0}}$ ${\displaystyle x_{0}}$

${\displaystyle {\hat {y}}_{0}=b_{0}+b_{1}x_{0}}$

Since one can never exactly predict the value of the endogenous variable, there is always an estimation error. This error is known as the forecast error and results from

${\displaystyle e_{p}:={\hat {y_{0}}}-y_{0}}$

If the true forecast equation is unknown, the forecast error is also unknown. Nevertheless it is possible to make a statement about the precision of the forecast error. Theoretically, the prognosis is considered to be precise, since the mean error is 0:

${\displaystyle \operatorname {E} ({\hat {y}}_{0}-y_{0})=\operatorname {E} (x_{t1}{\hat {\beta }}_{1}+x_{t2}{\hat {\beta }}_{2}+\ldots +x_{tK}{\hat {\beta }}_{K}-(x_{t1}\beta _{1}+x_{t2}\beta _{2}+\ldots +x_{tK}\beta _{K}+\varepsilon _{t}))=0}$.

The averaged sum of the forecast errors gives the mean absolute error .

Forecast interval

In inferential statistics , a forecast interval , also known as a forecast interval or prediction interval , is a range in which the value to be forecast can be assumed ex ante with a certain (high) probability .

Forecast intervals are similar to confidence intervals , but because of their characteristics, they should not be confused with them.

The variance of the forecast error, which reflects the variation of the forecast error and thus the reliability of the forecast, is important for the calculation of a forecast interval. In linear single regression it is given by:

${\displaystyle \sigma _{0}^{2}=\operatorname {Var} ({\hat {y}}_{0}-y_{0})=\sigma ^{2}\left(1+{\frac {1}{n}}+{\frac {(x_{0}-{\bar {x}})^{2}}{\sum \limits _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}\right)}$.

With the help of the variance of the forecast error one then obtains the forecast interval for the forecast value of ${\displaystyle (1-\alpha )}$ ${\displaystyle y_{0}}$

${\displaystyle {\hat {y}}_{0}\pm t_{(1-\alpha /2)}(n-2)\cdot {\sqrt {{\hat {\sigma }}^{2}\left(1+{\frac {1}{n}}+{\frac {(x_{0}-{\bar {x}})^{2}}{\sum \limits _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}\right)}}}$.

Individual evidence

1. ↑ Von Auer: Econometrics. An introduction. 6th edition, p. 135.
2. ↑ L. Fahrmeir, R. Künstler among others: Statistics. The way to data analysis. 8th edition. Springer 2016, p. 448.