我们有线性回归模型y = Xβ + e，其中y，X，β和e分别是响应向量，设计矩阵，参数向量和模型误差项。参数估计器是ˆβ = (XT X)−1XT y，拟合值向量可以写成ˆy = Hy，其中H是帽子矩阵。请证明yT y = ˆyT ˆy + ˆeT ˆe。

1. Break Down the Problem

To prove the relationship $y^T y = \hat{y}^T \hat{y} + \hat{e}^T \hat{e}$, we will:

1. Define each component: $\hat{y}$ (fitted values) and $\hat{e}$ (residuals).
2. Use the hat matrix $H$ to express $\hat{y}$.
3. Establish the relationship between $y$, $\hat{y}$, and $\hat{e}$.

2. Relevant Concepts

• The fitted values in linear regression are given by $\hat{y} = Hy$, where $H = X(X^T X)^{-1} X^T$.
• Residuals are defined as $\hat{e} = y - \hat{y} = y - Hy = (I - H)y$.

3. Analysis and Detail

1. Calculate $\hat{y}^T \hat{y}$: $\hat{y}^T \hat{y} = (Hy)^T(Hy) = y^T H^T H y$ Since $H$ is symmetric ($H = H^T$): $\hat{y}^T \hat{y} = y^T H y$

2. Calculate $\hat{e}^T \hat{e}$: $\hat{e}^T \hat{e} = ((I - H)y)^T((I - H)y) = y^T (I - H)^T (I - H) y$ Again, using the symmetry of $I - H$: $\hat{e}^T \hat{e} = y^T (I - H) (I - H) y = y^T (I - H) y$

3. Expand the total sum of squares: The total sum of squares $y^T y$ can be separated into two parts: $y^T y = \hat{y}^T \hat{y} + \hat{e}^T \hat{e} = y^T H y + y^T (I - H) y$

   Therefore, $y^T y = y^T (H + (I - H)) y = y^T I y = y^T y$

4. Verify and Summarize

The steps show that both fitted values and residuals sum up to the total sum of squares, validating the relationship $y^T y = \hat{y}^T \hat{y} + \hat{e}^T \hat{e}$.

Final Answer

The proof confirms that:
$y^T y = \hat{y}^T \hat{y} + \hat{e}^T \hat{e}$ is valid in the context of linear regression.