The quadratic version of the regression showing the effect of advertising on quantity in the textbook is q = 3.95 + 1.2A – 0.04A2.

For this regression, what evidence would you consider in deciding whether the quadratic specification is preferable to the linear specification?

Question

The quadratic version of the regression showing the effect of advertising on quantity in the textbook is q = 3.95 + 1.2A – 0.04A2.

For this regression, what evidence would you consider in deciding whether the quadratic specification is preferable to the linear specification?

BlackTom AI · Accepted Answer

To assess whether the quadratic specification is preferable to the linear one, we should evaluate evidence about the squared term and overall model fit from the quadratic regression.
Option 1: 'The magnitude of the coefficient on the squared term is non-zero.' While a non-zero coefficient suggests that the squared term matters, statistical usefulness requires evidence that it is significantly different from zero rather than just non-zero due to sampling variation. So this alone does not establish statistical significance or superiority of the quadratic form.
Option 2: 'The t-statistic on the squared term in the quadratic regression is large enough in magnitude.' This is the key statistical criterion for whether the squared term meaningfully contributes to explaining the dependent variable beyond chance. A large absolute t-statistic (and thus a small p-value) indicates the squared term is statistically significant, supporting the quadratic specification.
Option 3: 'The coefficient on the squared term is small compared to the coefficient on the linear term.' A small relative size by itself does not determine whether the quadratic term is important. Significance depends on its standard error and whether it is statistically different from zero, not just its absolute magnitude relative to the linear term. Moreover, even a small coefficient can be significant if its standard error is tiny, and the squared term can still improve fit.
Option 4: 'All of these responses are correct.' Since the previous two points are not both correct and the third point is not a sufficient criterion, this option overstates what evidence is necessary. The correct assessment relies on statistical significance of the squared term rather than merely non-zero magnitude or relative size.
In summary, the essential evidence to prefer the quadratic specification is that the squared term's t-statistic is large in magnitude, indicating a substantial and statistically reliable contribution beyond the linear term.

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The quadratic version of the regression showing the effect of advertising on quantity in the textbook is q = 3.95 + 1.2A – 0.04A2. For this regression, what evidence would you consider in deciding whether the quadratic specification is preferable to the linear specification?

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