## 3 Methodology
### Testing and Relaxing the Classical Assumptions
#### 3.3.2 Diagnostics of the Linearity Assumption  
The linearity assumption deals with the selection of regressors and relates to two problems. The first problem is to identify whether all the regressors in our model contribute to the prediction precision instead of only increasing the noise in the regression. If the linearity assumption holds, for an arbitrary regressor in (5) there should be a meaningful linear relation between the dependent variable and the regressor. Let us test for each \(j=1\), \(2,\ldots,p\) the null hypothesis \(H_{0}^{j}\) (that the regressor \(X_{j}\) does not contribute to the prediction precision) against the alternative hypothesis \(H_{1}^{j}\) (that the regressor \(X_{j}\) does contribute to the prediction precision):  
\[H_{0}^{j}\text{: }\beta_{j}=0\text{ against }H_{1}^{j}\text{: }\beta_{j}\
eq 0\text{, for }j=1,2,\ldots,p \tag{28}\]  
The \(t\)-test solves the formulated hypothesis test by using the test statistics \(t_{stats,j}\) (29), where the standard error of the parameter estimates \(b_{j}\) is given in (19):  
\[t_{stat,j}=\frac{b_{j}}{se\big{(}b_{j}\big{)}}\text{, for }j=1,2,\ldots,p \tag{29}\]  
If \(CDF_{t,a}(. )\) is the cumulative distribution function of the Student distribution (aka the \(t\)-distribution) with \(a\) degrees of freedom, then the \(p\)-value of the test (27) is [37] (p. 801):  
\[p-value_{j}=2 CDF_{t,n-p}\big{(}-\big{|}t_{stat,j}\big{|}\big{)}\text{, for }j=1,2,\ldots,p \tag{30}\]  
The test (28) is repeated for each of the regressors in (5), i.e., for \(j=1\), \(2\), \(\ldots,p\). In the ideal case all parameters in the regression model will be significant. However, if some of the coefficients are significant but others are not, we cannot simply keep only the regressors with significant coefficients according to the \(p\) individual \(t\)-tests. The reason is that the model parameters are interconnected and dropping one of them will change both the values and the significance of the other parameters in the new model. The second problem, related to the linear assumption, is to determine the right structure of the model. We would like to have a model where the linearity assumption holds for each of the regressors and we cannot add any available regressor to the model for which the linearity assumption holds. The stepwise regression is an automatic procedure to select the \"correct set\" of regressors. It starts with a set of regressor and adds or drops one regressor at each step based on some selection criterion. There are numerous forms of stepwise regression, but the method as such is subjected to criticism, and is summarized as follows:(a) There is a problem with the significance of individual tests when multiple tests are performed on the same data. For example, of 1000 tests with 5% significance the effect will be false discovered in 50 of them even when the effect is missing. (b) The selection criterion is very often \(R^{2}_{adj}\) which makes the stepwise regression to identify smaller regressors sets than the \"correct set\" of regressors. (c) Due to the partial multicollinearity the regressors are interconnected and the decision in one step may compromise the choices in the previous steps [39] (p. 378). (d) If the regression constant \(b_{1}\) (in front of the unit regressor) is treated similarly to the other parameters, it either can be dropped at an earlier step and the dropping decisions in the subsequent steps can be compromised, or it can be added at a later step and the adding decisions in the previous steps can be compromised. If the regression constant \(b_{1}\) is treated differently to the other parameters, we will always end up with model containing the unit regressor even when the regression constant is insignificant which can produce imprecise values of the slopes. For our model structure we apply a modified stepwise regression algorithm (MSRA) for the MLR, which in our opinion addresses the above issues. That algorithm that follows is based on backward elimination using \(t\)-tests as a selection criterion.