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The regression equation goes like:

$$\text{Wage }= \text{Age}\cdot \beta_1+ \text{Age}^2 \cdot\beta_2 + e, $$ where $E(\text{wage}\times e)=0.$

Wage is in per hour and age is in years.

The question asked me to describe a test to see whether a 40 year old can earn 20$ per hour.

I have tried to use a F-test where $\beta_1 = \beta_2 = 0$ and also tried to use the marginal effect which results in: Marginal effect $ = \beta_1 + 2\text{Age}\cdot \beta_2$.

At some point I thought of using $\theta = 40 \beta_1 + 1600 \beta_2$ and tried to use the null hypothesis as:

$$H_0: \theta = 20$$

$$H_A: \theta \neq 20$$

But didn't feel it was correct to use a restricted regression there.

Is there a better way to do this?

Christian Hennig
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Ayaka
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  • whether a 40 year old can earn 20$ per hour what does that mean? Check if at least one such person exists? Check if the average 40 year old earns 20$ or more/less per hour? – user2974951 Oct 18 '23 at 08:16
  • Check if the person who is 40 years, Age= 40 earns Wage = 20. So more or less E( Wage | Age = 40) = 20 – Ayaka Oct 18 '23 at 08:18

2 Answers2

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The easiest option is probably to "center" the Age variable, however instead of subtracting the mean/median or similar you would subtract 40 from it, that way age 40 is now in your intercept term. Then you get your test of age 40 for free in the intercept.

user2974951
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  • If that's the case, we'd just test for Age = 40 using a regular t-stat? – Ayaka Oct 18 '23 at 08:31
  • @Ayaka maybe, however if you have to keep the model as is, that is age and age^2, then it's not the same (if you still have to control for the "other" variables). – user2974951 Oct 18 '23 at 08:40
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I feel that the aim of the question is not so much about the value of the coefficient but rather the confidence interval associated to them. I.e., I understand the "can a 40-year-old earn 20 dollars per hour" as "is it likely that a 40-year-old earn 20 dollars per hour?".

In that case, I would:

  1. Plug Age=40 in the regression and compute the expectation
  2. Compute the prediction interval. See, for instance, this link. For the multivariate case, please see this link.
  3. Test whether 20 is in the prediction interval via a regular hypothesis testing under Normal assumption, taking into account the properties of your regression (number of observations, number of regressors, etc).
Olivier Hubert
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