What assumption is made related to the error terms in OLS Regression?

Question

BlackTom AI · Accepted Answer

To begin, restating the core idea: in Ordinary Least Squares (OLS) regression, one key assumption about the error terms is that they are centered around zero on average and that their spread (variance) does not depend on the level of the predictor variables.
Option 1: 'The error terms have a mean of zero and a constant variance' matches the standard OLS assumptions of zero-mean errors and homoscedasticity (constant variance across observations). This is foundational for unbiasedness of the estimators and for valid standard errors and inference.
Option 2: 'The error terms have a mean of zero and a non-constant variance' asserts zero mean but heteroscedasticity. This contradicts the homoscedasticity assumption and would lead to inefficiency in OLS estimates and biased standard errors, undermining inference.
Option 3: 'The error terms have a non-zero mean and a constant variance' implies a systematic bias in the errors. A non-zero mean error would bias the estimated intercept (and potentially the slope if the mean error correlates with regressors), violating a core OLS assumption.
Option 4: 'The error terms have a non-zero mean and a non-constant variance' combines both biases and heteroscedasticity, representing a scenario that seriously violates multiple OLS assumptions and would threaten both unbiasedness and the reliability of standard errors.
Putting these together, the intention of the question is to identify the classical, uncontroversial requirement. The only option that aligns with the standard zero-mean, constant-variance assumption is Option 1, while all other choices describe violations of one or both aspects of this assumption.
In terms of conceptual understanding, recognizing zero-mean errors ensures that the regressors capture systematic effects without systematic bias in the residuals, and constant variance ensures that every observation contributes equally to the estimation process, enabling reliable hypothesis testing and confidence intervals.
Additionally, it’s helpful to note that while homoscedasticity is part of the classical assumptions, other important OLS assumptions include no autocorrelation of errors, linearity in parameters, correct model specification, and independence of observations; these are not directly asked about here but provide broader context for why deviations from the stated assumption can affect results.

What assumption is made related to the error terms in OLS Regression?单项选择题

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