Write the model with $p$ parameters (intercept included).

Error mean square. The residual sum of squares satisfies $\mathbb{E}(\mathrm{SSE})=(n-p)\sigma^{2}$ , so

$\mathbb{E}\bigl(\mathrm{MSE}\bigr)=\mathbb{E}\!\left[\frac{\mathrm{SSE}}{n-p}\right]=\sigma^{2}$

always — whatever the true $\uline{B}$ . Unbiased.

Regression mean square. Here

$\mathbb{E}\bigl(\mathrm{MSReg}\bigr)=\sigma^{2} +\frac{\uline{B}_1'X_c'X_c\uline{B}_1}{p-1},$

where $\uline{B}_1$ collects the slope coefficients and $X_c$ is the centred design matrix. The second term is a non-negative quadratic form that vanishes only when all slopes are zero. So MReg ss is unbiased for $\sigma^{2}$ only under $H_0$ , and biased upward otherwise — which is precisely why large $F=\mathrm{MSReg}/\mathrm{MSE}$ is evidence against $H_0$ .

Total mean square. $\mathrm{SST}=\mathrm{SSReg}+\mathrm{SSE}$ inherits the same non-centrality, so $\mathbb{E}(\mathrm{SST}/(n-1))>\sigma^{2}$ unless all slopes vanish. Biased.

Hence only MError ss is unbiased in general.