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:

(where lx is ln(x))

I estimate the quadratic specification:

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!

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:

Code:

gen lx_sq=lx*lx

I estimate the quadratic specification:

Code:

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 |

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!

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