Refer to Ch3 and Ch4. Consider the following multiple regression model: y = β0 + β1x1 + …+ βkxk + u. Which of the following statements is correct?

Note: this question is about assumptions MLR.1 through MLR.6 (which are called the classical linear model (CLM) assumptions).

Question

Refer to Ch3 and Ch4. Consider the following multiple regression model: y = β0 + β1x1 + …+ βkxk + u. Which of the following statements is correct?

Note:  this question is about assumptions MLR.1 through MLR.6 (which are called the classical linear model (CLM) assumptions).

BlackTom AI · Accepted Answer

The question asks us to evaluate statements about the classical linear regression model (MLR) assumptions MLR.1 through MLR.6. We must examine each option and explain why it is correct or incorrect, before identifying which statement matches the CLM assumptions.
Option 1: 'MLR.1, the first assumption of multiple linear regression model, is about how the data used to estimate the parameters βj's are obtained from a random sample.' This description aligns with the notion that the data should be collected via random sampling to avoid bias in estimation, but MLR.1 typically emphasizes that the regression model is correctly specified in terms of the functional form and that the expected value of the error term conditional on the regressors is zero (E[u|X]=0). Depending on the text, some sources describe MLR.1 as requiring random sampling, while others place random sampling as a design requirement outside of the core CLM assumptions. As written, this option captures a plausible aspect but may not precisely define MLR.1 in all curricula, making it potentially misleading if we require the exact formal statement.
Option 2: 'MLR.2, the second assumption of the multiple linear regression model, is about the sample outcomes on any xj, {xij, where i = 1, …, n} not all being the same value, the sample containing at least j+1 observations, and there be no exact linear relationships among the independent variables xj’s.' This mixes several ideas: variation in the regressors (not all xj equal), a minimum sample size relative to the number of regressors, and no exact multicollinearity. While variation in X and absence of perfect multicollinearity are indeed important prerequisites, MLR.2 is typically stated as no perfect collinearity (no exact linear relationships among X variables) and that the independent variables have variation. The inclusion of a requirement about at least j+1 observations is a backhanded way of expressing degrees of freedom but is not a standard textbook articulation of MLR.2. Overall, this option is partially correct in spirit but mixes elements in a way that could mislead about the canonical MLR.2 statement.
Option 3: 'MLR.3, the third assumption of the multiple linear regression model, is about the population model being linear in the parameters βj’s.' This is close to the standard core idea that the model is linear in parameters. However, saying the population model is linear in the parameters is a formal restatement of the linearity in parameters condition, yet MLR.3 more precisely concerns the linearity in β with respect to the regressors and the correct specification of the linear model form. In many texts, MLR.3 is indeed described as the linearity in parameters, so this option is largely accurate but somewhat tautological. Depending on the exact phrasing in the course, some may reserve linearity in x’s while keeping the model not necessarily linear in parameters if interactions/polynomials are included. Overall, it could be acceptable but is not universally the standard labeling for MLR.3 across all curricula.
Option 4: 'MLR.6, the sixth assumption of the multiple linear regression model, is the error u being independent of the explanatory variables and being normally distributed with zero mean and constant variance σ2.' This combines several key aspects of MLR.6: independence between the error term and the regressors, and conditional homoskedasticity (constant variance) B, with an additional explicit normality condition. In the classical linear model, MLR.6 is commonly stated as the error term having E[u|X]=0, being homoskedastic (Var(u|X)=σ2), and sometimes assuming normality for small-sample inference. The exact phrase 'independent of the explanatory variables and being normally distributed with zero mean and constant variance σ2' captures the essential ideas of mean zero, homoskedasticity, and normality conditional on X, which are standard components of MLR.6 in many treatments. Therefore, this option accurately reflects the conventional MLR.6 assumptions as typically presented in CLM frameworks.
In summary, Options 1–3 present partial or variably precise interpretations of the CLM assumptions, while Option 4 clearly aligns with the standard MLR.6 statement that the error term is independent of X and follows a normal distribution with zero mean and constant variance. The correct answer is the statement given in Option 4, which corresponds to the classical MLR.6 assumptions.

Refer to Ch3 and Ch4. Consider the following multiple regression model: y = β0 + β1x1 + …+ βkxk + u. Which of the following statements is correct? Note: this question is about assumptions MLR.1 through MLR.6 (which are called the classical linear model (CLM) assumptions). 单项选择题

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