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A centered continuous Gaussian process that is also a martingale has independent increments. Every future increment has zero covariance with every finite vector of past values by the martingale property; uncorrelated jointly normal variables are independent.

Let $Q={\sum }_i{x}_i^2>0$. Under the usual nondegenerate regression assumption $n>2$, maximizing the normal likelihood is equivalent to minimizing the residual sum of squares. The normal equations and ${\sum }_i{x}_i=0$ give the maximum-likelihood estimatorsWrite $A=\overline{\epsilon }$ and $B={\sum }_i{x}_i{\epsilon }_i/Q$. Then $\hat{\alpha }=\alpha +A$, $\hat{\beta }=\beta +B$, and the residual is $r_i={\epsilon }_i-A-x_{i}B$. Independence of the errors givesFor each $i$,Thus the three requested variables are pairwise uncorrelated.

They are linear functions of the jointly normal error vector. More strongly, the whole vector $(A,B,r_1,\dots ,r_n)$ has a multivariate normal distribution, and its block covariances show that $A$, $B$ and the residual vector are independent, by uncorrelated jointly normal variables are independent. Independence is preserved when applying a measurable function to the residual block. HenceThe residual projection is $I-11^{\mathsf{T}}/n-x{x}^{\mathsf{T}}/Q$, of rank $n-2$, so additionally ${\sum }_i{r}_i^2/{\sigma }^2\sim {\chi }_{n-2}^2$. This is the orthogonal projection of a Gaussian vector interpretation of the same result. The printed design conditions alone allow $n=2$; then the fit is exact, the positive-variance likelihood has no finite maximizer, and the residual test in part (b) is undefined. The standard testing question therefore needs $n>2$.

LetThey are jointly normal because they are linear combinations of a Gaussian vector. When ${\sigma }_1={\sigma }_2=\sigma$,Hence uncorrelated jointly normal variables are independent. Their means and variances areandThereforeindependently.

Fix $0\le r\le t$. Since $M$ is a martingale,Every finite vector consisting of $M_{t+s}-M_t$ and past values $M_{r_1},\dots ,M_{r_n}$ has a multivariate normal distribution. Therefore uncorrelated jointly normal variables are independent, so the increment is independent of every finite vector of past values. A Monotone class theorem then extends this to independence from ${\mathcal{F}}_t=\sigma (M_r:0\le r\le t)$. This is the independent increments of a Gaussian martingale.