Quadratic specification and time-series operators: how to calculate the turning point?

[Mahana Noorma]

Dear all,

I would like to estimate a quadratic equation in order to find the point at which the variable x maximizes the function y (turning point).
I take the natural logarithm of all variables and consider their first-difference because their levels are not stationary. Furthermore, in order to decrease the risk of endogeneity, they are all lagged of one period.

The quadratic specification I would like to estimate is the following:
(lnyt-lnyt-1)= constant + Beta1*(lnxt-1-lnxt-2) + Beta2*((lnxt-1)^2-(lnxt-2)^2)

I use Stata/SE 9.2. I calculate the squared variable in the following way:

gen lx_sq=lx*lx

(where lx is ln(x))

I estimate the quadratic specification:

regress d.ly l.(d.lx) l.(d.lx_sq)

Source SS	df MS	Number of obs = 38
		F( 2, 35) = 5.45
Model	.008934437	2 .004467218	Prob > F = 0.0087
Residual	.028681133	35 .000819461	R-squared = 0.2375
		Adj R-squared = 0.1939
Total	.03761557	37 .001016637	Root MSE = .02863
D.ly	Coef.	Std. Err. t	P>t	[95% Conf. Interval]
lx
LD.	.5400141	.2148788 2.51	0.017	.103787 .9762412
lx_sq
LD.	-.0859535	.0291469 -2.95	0.006	-.1451249 -.0267821
_cons	.0195899	.0057903 3.38	0.002	.007835 .0313449

According to these results, I conclude that there is (quadratic) nonlinearity between lnx and lny. The coefficient of the two independent variables are statistically significant (at the 90% level) and the sign of the squared variable’s coefficient is negative.

Now I would like to calculate the value of the turning point (value of lnxt that maximizes (lnyt-lnyt-1)) but I am confused because of the time-series operators (first-difference and lag). Indeed, on the right-hand side of the equation, there are only lnxt-1 and not lnxt.

Would it be correct to take the derivative of (lnyt-lnyt-1) with respect to lnxt-1?
This would give the following results: Beta1/xt-1+(2*Beta2/xt-1)*lnxt-1=0
Lnxt-1= (-Beta1/2*Beta2), so xt-1=e^(-Beta1/2*Beta2)

I am also confused because I don’t know if the fact that I can only get the value of xt-1 and not xt would be a problem for the interpretation of the results.
Would it be correct to say something like "the value of Y (=(lnyt-lnyt-1)) is maximized when the value of x of the last period was equal to e^(-Beta1/2*Beta2)"? I'm not sure that this makes sense.

Thank you for your help!

[Clyde Schechter]

I would like to estimate a quadratic equation in order to find the point at which the variable x maximizes the function y (turning point).

I don't think it can be done.

The problem is that your model does not give y as a function of x,, quadratic or otherwise. The variables you use in your model are first differences. So if you want to find the value of the first difference in x that maximizes the first difference in y, that would be possible. Using lx = -_b[lx]/(2*_b[lx_sq]) is correct, and then you just exponentiate to get the x that maximizes the first difference in y. But if you need a value of x that maximizes y itself, this approach will not work, and I do not see any other that will. If anything, to the extent that the first difference in y is related to the derivative of y, you might need to find a value of x that makes the first difference of x = 0.

[Mahana Noorma]

I don't think it can be done.

The problem is that your model does not give y as a function of x,, quadratic or otherwise. The variables you use in your model are first differences. So if you want to find the value of the first difference in x that maximizes the first difference in y, that would be possible. Using lx = -_b[lx]/(2*_b[lx_sq]) is correct, and then you just exponentiate to get the x that maximizes the first difference in y. But if you need a value of x that maximizes y itself, this approach will not work, and I do not see any other that will. If anything, to the extent that the first difference in y is related to the derivative of y, you might need to find a value of x that makes the first difference of x = 0.

Dear Clyde Schechter,

Thank you so much for your reply. Your explanations are very clear.
I think I’d rather try to find another methodology in order to find the value of x that maximizes y (in levels).

Kind regards