Matrices F and c for F test?

In $\mathcal{F}$ test, the hypotheses involve multiple regression coefficients with the $\mathcal{F}_{0}$ test statistic: \begin{equation} z_{j} = \beta_{0} + \beta_{1} x_{1} + \beta_{2} x_{2} + ... + \beta_{n} x_{j} + \epsilon_{j} \end{equation} where Z is a dependent variable, $X$ the independent variable, and $\beta_{j} \in \mathbb{C}$ for $n \in \mathbb{N}$ \cite{ahlfors, ftests}. The null hypothesis is $H_{0}: \mathbf{L} \beta = \mathbf{c}$ without joint hypothesis.

I think the general form of the matrix $L$ is difficult one for computation. Consider a joint hypothesis $H_{0}: \beta_{1} = \beta_{2} = 0$. This leads to a term $X'X$ that is a diagonal matrix, see these lecture notes. So the corresponding matrix of $\mathbf{L} := X'X$ is a diagonal matrix and $\mathbf{c} := [[0 0 ... 0]^{-1}$ is a null vector.

Joint hypothesis

Here you can consider $\mathbf{L} \hat{\beta} - \mathbf{c} = 0$ for the regression of the observed value. For detailed steps, see the material of Zurich University, here and last part of it. The observed $\mathcal{F}$ test statistic is the following if the null hypothesis has a form $H_{0}: \beta_{1} = \beta_{2} = ... = \beta_{k} = 0$.

\begin{equation} F_{0} = \frac{ \left( SSR_{r} - SSR_{ur} \right) / q }{ SSR_{ur} / \left(n - (k+1) \right)}, \end{equation} where $SSR_{r}$ is the Sum of Squared Residuals of the restricted model, $SSR_{ur}$ is the same for the unrestricted model, and $n$ is the number of observations \cite{ftests} p. 4. This is similar to the one in Zurich University's materials.

\begin{equation} \mathbf{X'X} = \begin{bmatrix} \gamma_{1} & . & . & . & \gamma_{M+1} & 0 & . & . & . & 0 \\[0.3em] 0 & . & \gamma_{1} & . & . & \gamma_{M+1} & 0 & . & . & 0 \\[0.3em] 0 & 0 & 0 & . & . & 0 & \gamma_{1} & . & . & \gamma_{M+1} \end{bmatrix} \end{equation}

and the corresponding vector of $\mathbf{c} = [0 0 ... 0]^{-1}$ is a null vector here under $H_{0}$.

Real values vs Complex values

For me both are ok, but complex values are for useful for future applications. You can assume that the matrix $\mathbf{L}$ and $\mathbf{c}$ are real valued if it makes the analysis easier. Here however something about analysis with complex data.

Matlab

In Matlab, you can do $\mathcal{F}$ test simply by

[h,p,ci,stats] = vartest2(J, model1)
[h2,p2,ci2,stats2] = vartest2(J, model2)


and compare the appropriate models. However, here, Matlab does not give you the values of the matrices. I would like to know in the context of Matlab, for instance, how it handles the matrices $\mathbf{L}$ and $\mathbf{c}$.

How can you present the matrices $\mathbf{L}$ and $\mathbf{c}$ for $\mathcal{F}$ test?

My sources

• Complex analysis, Lars Ahlfors, 1979. (ahlfors)
• Statistical Concepts: A Second Course, Lomax R, 2007. (ftests)
• Some lecture materials from Zurich University Economic Department
• Some lecture materials from Stanford University Political Sciences
• i am vevy curious to know how Ahlfor's stands as a reference for this question. – aginensky Apr 5 '15 at 16:24
• @aginensky He started to use Z for dependent variables and X for independent variables and I use his convention. – Léo Léopold Hertz 준영 Apr 5 '15 at 16:25
• truly not important, but z in Ahlfor's book means a complex variable- z = x+iy typically with x,y real. – aginensky Apr 5 '15 at 16:27
• @aginensky See stats.stackexchange.com/a/66268/3017 – Léo Léopold Hertz 준영 Apr 5 '15 at 16:29