This problem involves a least-squares estimation for a multiple linear regression model. Here's a breakdown of the elements:

1. Observed Data: We have $n$ sets of observed data $(x_{i1}, x_{i2}, \ldots, x_{ip}; y_i)$ where $i = 1, 2, \ldots, n$. Each observation includes $p$ predictor variables and a response variable $y_i$.

2. Design Matrix $X$: The design matrix $X$ is structured to include a column of 1s (intercept term) and then columns for each predictor $x_{ij}$. Its general form is: $X = \begin{bmatrix} 1 & x_{11} & \cdots & x_{1p} \\ 1 & x_{21} & \cdots & x_{2p} \\ \vdots & \vdots & & \vdots \\ 1 & x_{n1} & \cdots & x_{np} \end{bmatrix}$

3. Residual Vector $e$: The residual vector $e = (e_1, e_2, \ldots, e_n)'$ represents the difference between observed values and the values predicted by the model.

4. Objective: The goal is to show that the sum of the residuals $e_1 + e_2 + \ldots + e_n = 0$, assuming that the parameter estimates are derived using the least squares method.

Proof Outline

Using the least squares method, the residuals $e = y - X\hat{\beta}$ are orthogonal to the columns of $X$, which includes a column of ones (to account for the intercept). This orthogonality condition implies that the residuals sum to zero: $1'e = 0$ where $1$ is a column vector of ones. This proves that the total residuals sum to zero under least squares estimation.

Would you like a detailed step-by-step proof of this, or further clarification on the least-squares method?

Follow-up Questions:

1. What are the key conditions for the least squares solution to yield unique estimates?
2. How does adding an intercept term affect the sum of residuals?
3. Why is orthogonality between residuals and predictors essential in regression?
4. How would the proof change if the model lacked an intercept?
5. Can we always assume $\sum e_i = 0$ in generalized linear models?

Tip: In linear regression, the orthogonality of residuals to the design matrix is crucial for minimizing the sum of squared errors effectively.