Which of the following are the assumptions of regression analysis? The relationship between y and x is linear. The errors all have the same variance. The errors are independent of each other. The errors are all normally distributed.[Fill in the blank]

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To analyze the question, I will go through each statement and examine whether it is considered an assumption in regression analysis.

Option a: i, ii and iii. This choice includes linearity, constant variance, and independence, but it excludes normality of the errors. While linearity, homoscedasticity (constant variance), and independence are standard assumptions for estimation, normality of the errors is often required for valid inference (e.g., confidence intervals, hypothesis tests) in small samples. Omitting normality means this option is incomplete if the context requires inference.

Option b: i, ii and iv. This set contains linearity, constant variance, and normality, but it leaves out the independence of errors. Independence is a key assumption to ensure that error terms are not correlated across observations; ignoring it leads to biased standard errors and unreliable inferences. Therefore, this option is missing a crucial assumption.

Option c: ii, iii and iv. Here, the constants variance, independence, and normality are listed, but linearity is not included. Without a linear relationship between y and x, the regression model would not be appropriate or interpretable in the standard linear-regression sense, so this option is also flawed.

Option d: i, ii, iii and iv. This option includes all four statements: (i) the relationship between y and x is linear, (ii) the errors have constant variance (homoscedasticity), (iii) the errors are independent of each other, and (iv) the errors are normally distributed. In many regression frameworks, especially for classical linear regression, these four conditions cover both the estimation properties (linearity, homoscedasticity, independence) and the inference properties (normality of errors for reliable small-sample tests). Although some texts emphasize that normality is not necessary for unbiased coefficient estimation, it is commonly listed as part of the regression assumptions when inference is concerned.

Therefore, comparing all options, the most comprehensive and commonly accepted set that includes all standard assumptions is option d. A careful reader should recognize that the inclusion of normality (iv) strengthens the validity of inferential procedures, even if estimation can proceed under non-normal errors with large samples. The other options each omit at least one essential assumption, which is why they are not preferred as the complete set of regression assumptions.

Which of the following are the assumptions of regression analysis? The relationship between y and x is linear. The errors all have the same variance. The errors are independent of each other. The errors are all normally distributed.[Fill in the blank]

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