Use the explicitly stated examples
The questions refer to the original five-period pairing and a separate symmetric curved construction. They are hypothetical data records, not observations of named investments.
Explain the missing condition
A wrong alternative may confuse separate series moments with paired covariance, absence of a linear relation with independence, or fitted identities with error assumptions.
Try the questions
Rotating one return list leaves its mean and variance unchanged. Does it preserve the fitted slope against the other list?
- A.
Yes, because each series keeps its own moments
- B.
No; pairing can change covariance and slope
- C.
Yes if the lists have the same length
Answer and explanation
Answer: B
Retain actual period keys.
- A
The cross-products depend on the matching pairs.
- B
The original rotation changes beta from 2.1 to zero.
- C
Length does not establish matching periods.
A supplied record has zero linear slope but a curved residual pattern. Which interpretation follows?
- A.
There is no possible relationship
- B.
The straight-line summary leaves a pattern requiring review
- C.
Causation is proven by the curve
Answer and explanation
Answer: B
Inspect the diagnostic evidence without asserting causation.
- A
Zero linear slope does not rule out nonlinear structure.
- B
This is what the original squared-number construction shows.
- C
A pattern alone is not a causal design.
OLS residuals sum to zero when an intercept is fitted. Does this prove the population errors are independent and normal?
- A.
Yes, because the fitting identity tests all assumptions
- B.
No; those assumptions require separate evidence
- C.
Yes whenever there are five observations
Answer and explanation
Answer: B
Keep fit arithmetic separate from inferential conditions.
- A
The zero sum is built into the fitted equations.
- B
Residual identities are not distributional certification.
- C
Sample size alone does not establish the error law.