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Let us say I have a regression equation such as:

$E(y) = \beta_{0} + \beta_{1} \cdot \text{var}_1 + \beta_{2} \cdot \text{var}_2 + \beta_{3} \cdot \text{var}_2^2$

How do I test if the effect of var1 equals the effect of var2? I know how to do it if there is no quadratic term

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  • $\begingroup$ It's not at all clear what your question means in this situation. How are we to interpret "effect of var 2" in this situation? I suppose that one way to interpret the question would be to see if (simultaneously), $\beta_3=0$ and $\beta_2-\beta_1=0$ (assuming var1 and var2 are in the same units) ... but it's not at all clear if that's what you intend at all. Please clarify what "effect of var1 equals effect of var2" really means in this context. $\endgroup$
    – Glen_b
    Oct 25, 2015 at 5:38
  • $\begingroup$ Is $\beta_{1} = \beta_{2} + \beta_{3} . var2$ is a good way to think about it. $\endgroup$ Oct 25, 2015 at 5:41
  • $\begingroup$ Until you clarify what "effect of var1 equals effect of var2" is intended to mean, it's hard to say "yes" or "no". $\endgroup$
    – Glen_b
    Oct 25, 2015 at 5:44
  • $\begingroup$ Let us say $logWage = \beta_0 + \beta_{1} . Education + \beta_{2} . Experience + \beta_{3} . Experience^2$. Is return on education same as return on experience? $\endgroup$ Oct 25, 2015 at 5:50
  • $\begingroup$ How are education and experience measured? What do we mean precisely by "return on" in a situation where the relationship between log-Wage and experience is not linear? You're essentially restating the unclear question in different ways without getting to the central problem in defining what you mean. If you want to test a hypothesis like whether $\beta_1 x = \beta_2 x + \beta_3 x^2$ for all $x$, that's a question with an answer. $\endgroup$
    – Glen_b
    Oct 25, 2015 at 6:04

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I'll work on the assumption that you want to test whether the derivatives with respect to time are the same:

$\text{H}_0: \beta_1 = \beta_2 + 2\beta_3 x\quad$ for all $x$

This will be true for every $x$ if simultaneously $\beta_3=0$ and $\beta_2-\beta_1=0$. If $\beta_3\neq 0$ then it could only be true at one value of $x$ and if $\beta_3=0$ but $\beta_2-\beta_1\neq 0$ then it won't be true at any value of $x$.

Which is to say it's the same as:

$\text{H}_0: \beta_1 = \beta_2\: \text{ & }\: \beta_3 =0$ for all $x$

Since these are linear restrictions the standard partial F approaches could be applied.

You can either (i) fit the full model with all variables free and the reduced model where both restrictions apply and take

$F = \frac{\text{"change in RSS"}/2}{RSS_{\text{full}}/dfe_\text{full}}$

(here "RSS" would be "residual sum of squares" and "dfe" is "degrees of freedom for error")

or (ii) you can also do it by reparameterizing the full model to be in terms of a common linear coefficient and the difference and then an F-test can be constructed from the table of sums of squares of the reparameterized full model.

[I don't know that it really makes sense to do this in the particular context, but I'll take the question on its face and leave any potential issues with that for others to comment on.]

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