## 3 Methodology
### Classical Normal Linear Regression Model  
Let \(X_{j}\) be \(p\) different functions (called regressors) of the five independent variables:  
\[X_{j}=g_{j}(x_{1},\,x_{2},\,x_{3},\,x_{4},\,x_{5})\text{, for }j=1,\,2,\, \ldots,\,p \tag{4}\]  
Almost always the regressor \(X_{1}\) is the unity function \(g_{1}(x_{1},\,x_{2},\,x_{3},\,x_{4},\,x_{5})=1\). Then, we can form a general stochastic regression connection between the efficiency of the OWC and the values of the regressors:  
\[Y=\beta_{1}+\beta_{2}X_{2}+\beta_{2}X_{2}+\cdots+\beta_{p}X_{p}+\varepsilon= \sum_{j=1}^{p}\beta_{j}X_{j}+\varepsilon \tag{5}\]  
In (5) the dependent variable \(Y\) (the hydrodynamic efficiency) is a random variable (r.v. ), unlike the regressors \(X_{j}\) (\(j=1,\,2,\,\ldots,\,p\)) that are non-random observable variables. The error variable, \(\varepsilon\), is also considered to be an r.v. and is theoretically responsible for the randomness of \(Y\) (although in practice there are other components to this randomness, as discussed after (17) below). The partial regression coefficients \(\beta_{1}\), \(\beta_{2}\),..., \(\beta_{p}\) are non-random and unobservable quantities often called _slopes_. They can be organized in a \(p\)-dimensional column vector known as the parameter vector:  
\[\overset{\rightarrow}{\beta}=\big{(}\beta_{1},\beta_{2},\ldots,\beta_{p} \big{)}^{T} \tag{6}\]  
When the regressor \(X_{1}\) is the unity function, then \(\beta_{1}\) is also known as the regression constant. Equation (5) shows how the probability distribution of the r.v. in the left-hand side can be estimated using the right-hand side. The primary task in a MLR is to find a point estimate \(\overset{\rightarrow}{b}=\big{(}b_{1},b_{2},\ldots,b_{p}\big{)}^{T}\) for the unobservable parameter vector \(\overset{\rightarrow}{\beta}=\big{(}\beta_{1},\beta_{2},\ldots,\beta_{p} \big{)}^{T}\). That can be solved with the ordinary least square (OLS) method if the following assumptions hold [37] (p. 791):  
1. Nullity assumption: the expected value of the error variable is zero, i.e., \(E[\varepsilon]=0\). 2. Homoskedasticity assumption: the variance of the error variable is constant for any independent variables' vector \(\overset{\rightarrow}{x}=(x_{1},x_{2},x_{3},x_{4},x_{5})^{T}\), i.e., \(V\Big{[}\varepsilon\Big{|}\overset{\rightarrow}{x}\Big{]}=\sigma_{\varepsilon}^ {2}\). 3. Normality assumption: the probability distribution of the error variable is normal, i.e., \(\varepsilon\sim N\big{(}0,\sigma_{\varepsilon}^{2}\big{)}\). 4. Correlation assumption: the errors of the dependent variables are not linearlycorrelated for any two different independent variables' vectors, i.e., \(corr\Big{[}Y_{A}-E\Big{(}Y_{A}\Big{|}\overset{\rightarrow}{x}_{A}\Big{)},Y_{ B}-E\Big{(}Y_{B}\Big{|}\overset{\rightarrow}{x}_{B}\Big{)}\Big{]}=0\). 5. Multicollinearity assumption: not a single regressor can be expressed as a linear combination of the rest. If we denote \(p\) arbitrary sets, each containing \(p\) real numbers with \[S_{j}=\left\{a_{1}^{j},\ldots,a_{j-1}^{j},a_{j+1}^{j},\ldots,a_{p}^{j}\right\}\,\ \text{for}\ j=1,2,\ldots,p,\ \text{then the multicollinearity assumption}\] boils down to: \[\exists\ S_{j}\ \text{such that}\ X_{j}=\sum\limits_{ \begin{array}{c}k=1\\ k\
eq j\end{array} }^{p}a_{k}^{j}X_{k}\,\text{for}\ j=0,1,2,\ldots,p.\]  
6. Linearity assumption: the conditional expected value of the dependent variable given some independent variables' vector \(\overrightarrow{x}\) is a linear combination of the regressors with coefficients equal to the components of the parameter vector, i.e., \(E\!\left[Y\middle|\overrightarrow{x}\right]=\sum\limits_{j=1}^{p}\beta_{j}X_{j}\). The sixth assumption, formulated in [38], is often omitted in the set of classical OLS assumptions, but it is an important one. If we use OLS to construct a model like (4) that complies with the six classical assumptions, we will obtain a classical normal linear regression model (CNLRM) [39] (pp. 107-117).