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If I am taking the log of a variable $X$ and also a square term of the variable $X$ on the right-hand side. Say, the equation is

$$\log Y_{it} = α + β_1\log X + β_2 \log X^2 + u_i$$

where $Y =$ Waste Generation, $X =$ Economic Development, measured by GDP. In this case, while regressing the equation using Stata, the system does not provide appropriate results owing to collinearity in the variable $X$. The system mentions that there is collinearity in variable $X$. What should I do?

Just to inform, that in my case, $T=7$, $N=30$

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    $\begingroup$ Maybe the second regressor should be "$\left(\log X\right)^2$"?? $\endgroup$ – whuber Oct 5 '20 at 21:10
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Your problem stems from the simple fact that $\log X_i^2 = 2\log X_i$. Thus, your model reduces to: $\log Y_i = \alpha + \beta_1 \log X_i + 2 \beta_2 \log X_i + u_i = \alpha + (\beta_1 + 2 \beta_2) \log X_i + u_i $.

Due to the perfect collinearity, $\beta_1$ and $\beta_2$ cannot be distinguished.


As @whuber mentioned in a comment:

Maybe the second regressor should be "$(\log X)^2$"??

Which would, indeed solve your problem with collinearity. Perhaps you should check you modeling and see if the second one was the one intended.

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  • $\begingroup$ Yes (log X)2 was intended. $\endgroup$ – shubham sharma Oct 6 '20 at 7:02
  • $\begingroup$ I tried estimating the model with (logX)2 as suggested by experts here and this time the problem of collinearity is not there but my results are still insignificant. Please suggest something. Should I try estimating it using the Generalised method of moments (GMM) since my T<N ie, the time period is less than the number of observations. $\endgroup$ – shubham sharma Oct 8 '20 at 11:43
  • $\begingroup$ @shubhamsharma Non-significant (i.e. null) results are still results. There is no problem with that. If you knew the alternative hypothesis to be true, then why even test it? $\endgroup$ – Firebug Oct 8 '20 at 12:39
  • $\begingroup$ Yes, good suggestion @Firebug $\endgroup$ – shubham sharma Oct 12 '20 at 20:06

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