Select all correct statements of the simple linear regression model introduced in this course.

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BlackTom AI · Accepted Answer

To evaluate the statements about the simple linear regression model, I will examine each option on its own merits and relate them to the standard model assumptions.
Option a: Covariance between ε_i and ε_j is zero when i ≠ j. In the classical simple linear regression framework, the error terms ε_i are assumed to be independent with constant variance, which implies Cov(ε_i, ε_j) = 0 for i ≠ j. Therefore, this statement is consistent with the standard assumptions and is considered correct.
Option b: The model can be stated as E(Y_i) = β_0 + β_1 X_i + ε_i. Typically, the model is written as Y_i = β_0 + β_1 X_i + ε_i, and taking expectations gives E(Y_i) = β_0 + β_1 X_i, because E(ε_i) = 0. Including + ε_i inside the expectation is not correct, since E(ε_i) = 0. Hence this statement is incorrect as written.
Option c: The sum of error terms is zero. There is no general requirement in the model that the sum of the ε_i across all observations must be zero. The ε_i are random with mean zero, but their sum does not have to be exactly zero in the sample. This makes the statement incorrect.
Option d: The error terms have constant variance. Homoscedasticity is a key assumption: Var(ε_i) = σ^2 is constant across i. This is a standard and correct property of the classical simple linear regression model.
Option e: Responses are independent and identically distributed. While the errors ε_i are typically assumed independent and identically distributed (i.i.d.) with mean zero and variance σ^2, it is not strictly accurate to claim that the responses Y_i are i.i.d. because Y_i depends on X_i and the variance structure, and X_i may vary across observations. The usual emphasis is on the errors being i.i.d. rather than the Y_i themselves being strictly i.i.d. Therefore, this statement is not correct as a blanket description of the model.

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Select all correct statements of the simple linear regression model introduced in this course.

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