The error term in a regression equation is said to exhibit homoskedasticty if _____.

Question

BlackTom AI · Accepted Answer

The question asks about the error term in a regression equation and when it is said to exhibit homoskedasticity.

First, note that the available answer option provided is: "it has the same variance for all values of the explanatory variable".

To reason through this, recall that heteroskedasticity occurs when the spread (variance) of the error terms differs across levels of the independent variable. In contrast, homoskedasticity means the opposite: the variance of the error term remains constant regardless of the value of the explanatory variable.

So, why is the statement correct? If the error term has constant variance for all X, the model does not exhibit changing spread of residuals across the range of X. This constancy is a key assumption in ordinary least squares (OLS) regression for obtaining efficient estimates and valid standard errors.

If we consider potential misconceptions: 
- Saying the error term has a constant mean would describe a different property (unbiasedness or correct specification in expectation), not homoskedasticity. 
- Suggesting the variance changes with X would point to heteroskedasticity, which can lead to biased standard errors and unreliable hypothesis tests. 
- Limiting homoskedasticity to only small or specific ranges of X would be incorrect; homoskedasticity implies constant variance across the entire range of the explanatory variable.

Since the option explicitly states constant variance across all values of the explanatory variable, it aligns with the formal definition of homoskedasticity and distinguishes it from heteroskedasticity, where variance changes with X.

In summary, the defining feature of homoskedasticity is that the error variance does not depend on the level of the explanatory variable, i.e., it is the same for all X. The alternative views (if present) would typically capture ideas about heteroskedasticity, non-constant variance, or misspecification, which are not consistent with homoskedasticity.

LA-ECON-3503-811-13824-202520 Quiz 1

The error term in a regression equation is said to exhibit homoskedasticty if _____.

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