# Check unity in a regression

I have the following question:

The equation from part c:

# My Solution

I'm thinking of solving this problem using the RESET test so something along the lines of:

$$yl_t = -0.027 + 0.537 kl_t + \hat{y}_t^2o$$

where $$o$$ is the coefficient.

That being said I know this is wrong, I have absolutely no idea on how to approach this problem.

• It is needed point (c).
– Nicg
Commented May 28, 2019 at 13:38
• @Nicg Thank you very much for your answer. very sorry for this, I completely forgot to add it. I just updated my question. Commented May 28, 2019 at 13:43
• Ok now makes sense: do you have familiarity with T and F Tests?
– Nicg
Commented May 28, 2019 at 13:53
• @Nicg Yes, I'm familiar with both tests Commented May 28, 2019 at 13:56
• Great! So, at me it seems you need a T - test for $\hat{\beta_{l}}$ $+$ $\hat{\beta_{k}}$ $=$ 1. Although the fact that it instead gives you the values for the F test leaves me a bit puzzled.
– Nicg
Commented May 28, 2019 at 14:02

Ok, now I see. So, in point (c), you have:

$$log(y) = \beta_{0} + \beta_{k}log(k) + \beta_{l}log(l) + \varepsilon$$

and estimate it: we will call this the $$unrestricted$$ model, since we aren't making any restriction on the parameters. We find that $$SSE_{unrest}$$ $$=$$ $$0.085$$

Let's now move to the $$restricted$$ model. In Economic terms, you are assuming constant returns to scale: have $$\beta_{k}$$ $$+$$ $$\beta_{l}$$ $$=$$ $$1$$. Working out the algebra we obtain the model you have in point (e):

$$log(\frac{y}{l}) = \beta_{0} + \beta_{kl}log(\frac{k}{l})$$.

We know that, for this model, $$SSE_{rest}$$ $$=$$ $$0.115$$. Considering that we have, de facto, put one only restriction, the F Test will be:

$$F = \frac{\frac{SSE_{rest} - SSE_{unrest}}{1}}{\frac{SSE_{unrest}}{n - k}} = \frac{0.115 - 0.085}{\frac{0.085}{n - 3}}$$

where $$n$$ is the sample size. You then have to check whether the computed F is larger or not of $$F_{1, n - 3}$$. If yes, Reject the hypothesis $$\beta_{k}$$ $$+$$ $$\beta_{l}$$ $$=$$ $$1$$.

• Thank you very much for your answer, why is $𝛽_k + 𝛽_l = 1$? Commented May 28, 2019 at 14:52
• Because that's the hypothesis that is required to be tested by point (e), your actual question.
– Nicg
Commented May 28, 2019 at 15:13