Estimation – Log a Squared Variable or Square a Logged Variable: Which To Choose?

estimationlogarithm

I'm estimating an OLS log-log model like so:

\begin{equation}
ln(y) = \alpha + log(x_1) + \varepsilon
\end{equation}

and now want to account for non-linearities in the response of $ln(y)$ to $log(x1)$.

Do I now estimate:

\begin{equation}
ln(y) = \alpha + log(x_1) + log(x_1)^2 + \varepsilon
\end{equation}

or:

\begin{equation}
ln(y) = \alpha + log(x_1) + log(x_1^2) + \varepsilon
\end{equation}

These obviously give me different results – my question is, which criteria do I use to make this decision?

In my case, $x_1$ is GDP and $y$ is carbon emissions.

Best Answer

Note that $\log(x_1^2)=2\cdot\log(x_1)$.

Consequently, $\text{cor}(\log(x_1),\log(x_1^2))=1$.

The log of the square adds no value when the log is already there.

Related Solutions

Solved – Controlling for variables in pooled OLS estimation in EViews

Control variable is also an independent variable. So, it should be listed as independent variable in the model in Eviews. Then, you interpret the coefficient on rgdp of all other countries as the elasticity of exchange rate with respect to the rgdp of all other countries (if you put both in log form), other things remaining the same (i.e. controlling for all other factors).

The model should be as follows:

$$ \begin{align} \log(\mathrm{export})_{it} &= \beta_0+ \beta_1 \log(\mathrm{real gdp})_{it}+ \beta_2 \log(\mathrm{population})_{it} \\ &\quad+ \beta_3\mathrm{political stability}_{it}+ \beta_4\mathrm{realexchange rate}_{it}+ \beta_5\log(\mathrm{chinagdp})_{it}\\ &+\beta_6\log(\mathrm{chinapopn})_{it}+\varepsilon_{it} \end{align} $$ Where $\log(\mathrm{export})_{it}$ is the (log) of the export from china to other countries and $t=1988, \ldots, 2011$.

In Eviews, you have to import the data as panel data and run OLS (under estimate equation) with the following command (assuming that the variables have been log transformed, if necessary) with the variables appearing in the order as in the model (c stands for constant).

logexport c logrealgdp logpopulation politicalstability realexchangerate logchinagdp logchinapopn

Solved – Self-study: Finding the maximum likelihood estimates of the parameters of a density function – UPDATED

[Note: This is my answer to the Dec. 19, 2014, version of the question.]

If you operate the change of variable $y=x^2$ in your density $$f_X(x|\alpha,\beta,\sigma)=\frac{1}{\Gamma \left( \alpha \right)\beta^{\alpha}}\exp\left\{{-\frac{x^2}{2\sigma^{2}}\frac{1}{\beta}}\right\}\frac{x^{2\alpha-1}}{2^{\alpha-1}\sigma^{2\alpha}}\mathbb{I}_{{\mathbb{R}}^{+}}(x) $$ the Jacobian is given by $\dfrac{\text{d}y}{\text{d}x}= 2x = 2y^{1/2}$ and hence \begin{align*} f_Y(y|\alpha,\beta,\sigma)&=\frac{1}{\Gamma \left( \alpha \right)\beta^{\alpha}}\exp\left\{{-\frac{y}{2\sigma^{2}}\frac{1}{\beta}}\right\}\frac{y^{\frac{2\alpha-1}{2}}}{2^{\alpha-1}\sigma^{2\alpha}}\frac{1}{2 y^{1/2}}\mathbb{I}_{{\mathbb{R}}^{+}}(y)\\ &=\frac{1}{\Gamma \left( \alpha \right)\beta^{\alpha}}\exp\left\{{-\frac{y}{2\sigma^{2}}\frac{1}{\beta}}\right\}\frac{y^{{\alpha-1}}}{2^{\alpha}\sigma^{2\alpha}}\mathbb{I}_{{\mathbb{R}}^{+}}(y) \end{align*} This shows that

This is a standard $\mathcal{G}(\alpha,2\sigma^2\beta)$ model, i.e. you observe $$(x_1^2,\ldots,x_n^2)=(y_1,\ldots,y_n)\stackrel{\text{iid}}{\sim}\mathcal{G}(\alpha,\eta);$$
the model is over-parametrised since only $\eta=2\sigma^2\beta$ can be identified;
EM is not necessary to find the MLE of $(\alpha,\eta)$, which is not available in closed form but solution of$$\hat\eta^{-1}=\bar{y}/\hat{\alpha}n\qquad\log(\hat{\alpha})-\psi(\hat{\alpha})=\log(\bar{y})-\frac{1}{n}\sum_{i=1}^n\log(y_i)$$ where $\psi(\cdot)$ is the di-gamma function. This paper by Thomas Minka indicates fast approximations to the resolution of the above equation.

Best Answer

Related Solutions

Solved – Controlling for variables in pooled OLS estimation in EViews

Solved – Self-study: Finding the maximum likelihood estimates of the parameters of a density function – UPDATED

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