Global F test

The global F test ( English Overall F test ), also global test , overall test , test for overall significance of a model , F test of overall significance , test for the overall context of a model, represents a global test of the regression function. It is checked whether at least one variable provides an explanatory content for the model and the model is therefore significant as a whole. If this hypothesis is rejected, the model is useless. This variant of the F -test is the most common application of the F -test.

Underlying model

The underlying model is that of linear multiple regression , so

${\displaystyle y_{i}=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\dotsc +x_{ik}\beta _{k}+\varepsilon _{i}=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}+\varepsilon _{i}}$.

It is assumed here that the disturbance variables are independent and homoscedastic and that they follow a normal distribution, i.e. H.

${\displaystyle \varepsilon _{i}\sim {\mathcal {N}}(0,\sigma ^{2}),\quad i=1,\ldots ,n}$.

Null and alternative hypothesis

The null hypothesis of the global F test states that all explanatory variables have no influence on the dependent variable. Both the dependent variable and the independent variables can be binary ( categorical ) or metric. The Wald test can then test the global null hypothesis (without including the absolute term ): ${\displaystyle H_{0}}$

${\displaystyle H_{0}\colon \beta _{1}=\beta _{2}=\ldots =\beta _{k}\;=\;0\Rightarrow \rho ^{2}=0}$  against   . ${\displaystyle H_{1}:\beta _{j}\;\neq \;0\;\mathrm {f{\ddot {u}}r\;mindestens\;ein} \;j\in \{1,\ldots ,k\}\Rightarrow \rho ^{2}>0}$

This test can be interpreted as if you were testing the overall quality of the regression, i.e. the population coefficient of determination of the regression. For this reason, the global F test is also known as the goodness of fit test. The term “goodness of fit test” is somewhat misleading because, strictly speaking, it does not check the fit of the regression line to the data, but rather whether at least one of the explanatory variables makes a significant contribution to the explanation. If the null hypothesis applies , the so-called null model results . The null model is a model that consists of only one absolute term . ${\displaystyle \rho ^{2}}$ ${\displaystyle H_{0}}$ ${\displaystyle \beta _{0}}$

Test statistics

The test statistics of this test can be obtained by first looking at the R -square notation of the F -statistics . The general form of the F statistic is given by

${\displaystyle F\equiv {\frac {\left(SQR_{H_{0}}-SQR\right)/q}{SQR/(n-k-1)}}}$,

wherein test restrictions and the number of residual sum of the restricted and represents the residual sum of the full model. In the present case, since the null hypothesis is, restrictions are tested. This means that the test statistic can also be written as and under the null hypothesis ${\displaystyle q}$ ${\displaystyle SQR_{H_{0}}}$ ${\displaystyle SQR}$ ${\displaystyle H_{0}\colon \beta _{1}=\beta _{2}=\ldots =\beta _{k}=0}$ ${\displaystyle q=k}$ ${\displaystyle F={\frac {\left({\mathit {R}}{}^{2}-{\mathit {R}}{}_{H_{0}}^{2}\right)/k}{\left(1-{\mathit {R}}{}^{2}\right)/(n-p)}}}$

${\displaystyle {\begin{aligned}F={\frac {MQE}{MQR}}={\frac {SQE}{SQR}}{\frac {n-p}{k}}={\frac {{\mathit {R}}^{2}}{1-{\mathit {R}}^{2}}}{\frac {n-p}{k}}\;{\stackrel {H_{0}}{\sim }}\;F\left(k,n-p\right)\end{aligned}}}$,

where represents the multiple coefficient of determination . The test statistic of a global F -test is given by the quotient of the "mean square of the explained deviations" and the " mean square of the residuals ". It is distributed under the null hypothesis F -distributed with and degrees of freedom . The calculation of the F test statistic can be summarized in the following table of the analysis of variance : ${\displaystyle {\mathit {R}}^{2}}$ ${\displaystyle k}$ ${\displaystyle (n-p)}$

Source of variation	Squared deviation	Number of degrees of freedom	Mean square of deviation	F test statistics
Regression (explained)	${\displaystyle SQE=\sum \nolimits _{i=1}^{n}({\hat {y}}_{i}-{\overline {\hat {y}}})^{2}}$( explained sum of squares )	${\displaystyle k}$	${\displaystyle MQE={\frac {SQE}{k}}}$	${\displaystyle F={\frac {MQE}{MQR}}}$
Residuals (unexplained)	${\displaystyle SQR=\sum _{i=1}^{n}(y_{i}-{\hat {y}}_{i})^{2}}$( Residual sum of squares )	${\displaystyle (n-p)}$	${\displaystyle MQR={\frac {SQR}{n-p}}}$
total	${\displaystyle SQT=\sum _{i=1}^{n}\left(y_{i}-{\bar {y}}\right)^{2}}$( total sum of squares )

Procedure and interpretation

If the empirical F -value exceeds the critical F -value (the - quantile of the F -distribution with and degrees of freedom) at a significance level determined a priori , the null hypothesis is rejected: ${\displaystyle \alpha }$ ${\displaystyle F_{(1-\alpha )}(k,n-p)}$ ${\displaystyle (1-\alpha )}$ ${\displaystyle k}$ ${\displaystyle (n-p)}$

${\displaystyle F>F_{(1-\alpha )}(k,n-p)\Rightarrow H_{0}\;{\text{verwerfen}}}$.

This is then sufficiently large and at least one explanatory variable probably contributes enough information to explain . It makes sense to reject the null hypothesis in the case of high F values, since a high coefficient of determination leads to a high F value. If the Wald test rejects the null hypothesis for one or more independent variables, then one can assume that the associated regression parameters are not equal to zero, so that the variables should be included in the model. If only one independent variable is involved (   vs.   ) then a t test is used to check whether the parameter is significant. For a single parameter, the result of the Wald statistic agrees with the result of the square of the t statistic. ${\displaystyle {\mathit {R}}^{2}}$ ${\displaystyle y}$ ${\displaystyle H_{0}\colon \beta _{i}=0}$ ${\displaystyle H_{1}\colon \beta _{i}\neq 0}$

Individual evidence

1. ^ ^{A b} Karl Mosler and Friedrich Schmid: Probability calculation and conclusive statistics. Springer-Verlag, 2011, p. 310.
2. ^ Ludwig Fahrmeir , Rita artist, Iris Pigeot , Gerhard Tutz : Statistics. The way to data analysis. 8., revised. and additional edition. Springer Spectrum, Berlin / Heidelberg 2016, ISBN 978-3-662-50371-3 , p. 458.
3. ↑ Jeffrey Marc Wooldridge: Introductory econometrics: A modern approach. 5th edition. Nelson Education, 2015, p. 146.
4. ^ Ludwig Fahrmeir , Rita artist, Iris Pigeot , Gerhard Tutz : Statistics. The way to data analysis. 8., revised. and additional edition. Springer Spectrum, Berlin / Heidelberg 2016, ISBN 978-3-662-50371-3 , p. 458.
5. ^ William H. Greene: Econometric Analysis. 5th edition. Prentice Hall International, 2002, ISBN 0-13-110849-2 , p. 33.