Dear Readers,
I come to you with a technical (and most likely also conceptual) problem concerning gmm estimation.
I am interested in modeling household electricity demand in the presence of a two-step block tariff, primarily with the intention of estimating the price elasticity of demand for electricity - for the country for which I have panel household survey data. The difficulty I am facing is translating the expected electricity demand equation I found in Reiss & White (2005) (which pertains to my case), into Stata code for Generalized Method of Moments (GMM) estimation, through the 'gmm' command.
The expected demand for electricity is the following:
\[
E(x^*|\cdot) = [x(p_1,y,z;\beta)-\sigma\lambda_1]\Phi_1 + \bar{x}\cdot(\Phi_2 - \Phi_1) \\ + [x(p_2,y_2,z;\beta)+\sigma\lambda_2](1-\Phi_2)
\label{eq:estim} \tag{1}
\]
where:
- \(x\) is electricity demand;
- \(p_1,p_2\) are the marginal prices on the two price schedule tiers;
- \(\bar{x}\) is the electricity consumption threshold after which the price switches to the higher tier (700 kWh in my case);
- \(z\) are observable consumer characteristics (control variables I wish to include);
- \(\varepsilon\) are unobserved consumer characteristics (the error term);
- \(y\) is income and \(y_2=y+\bar{x} \cdot (p_2-p_1)\) is the virtual income when consumers lie on the second tier; and
- \(\Phi_j\) is the standard normal distribution evaluated at $\(c_j(\beta)/\sigma\), \(\phi_j\) is the normal density at \(c_j(\beta)/\sigma\), \(\lambda_1=\phi_1/\Phi_1\), and \(\lambda_2=\phi_2/(1-\Phi_2)\).
Would anyone know how to incorporate all the parameters and variables in \eqref{eq:estim} into the gmm command, in order to obtain the unbiased coefficient related to price?
__________________________________________________
Note: the reduced form for the Household's consumption level of electricity \(x^*\) (as a function of the increasing two-tier price schedule), is the following:
\[
x^*=
\begin{cases}
x(p_1,y,z,\varepsilon;\beta) \qquad & \text{if } \varepsilon \le c_1 \\
\bar{x} & \text{if } c_1< \varepsilon < c_2\\
x(p_2,y_2,z,\varepsilon;\beta) \qquad & \text{if } \varepsilon>c_2,
\end{cases}
\label{eq:opt.con.lvl.ecx} \tag{2}
\]
where \(c_j\) (where \(j=1,2\)) is the solution to \(x(p_j,y_j,z,c_j;\beta)=\bar{x}\) with \(y_1=y\). In other words, \(c_j\) is the maximum (for \(j=1\)) or minimum (for \(j=2\)) value of \(\varepsilon\) for which consumption occurs on tier \(j\).
____________________________________
Reiss, P. C., & White, M. W. (2005). Household electricity demand, revisited. The Review of Economic Studies, 72(3), 853-883.
I come to you with a technical (and most likely also conceptual) problem concerning gmm estimation.
I am interested in modeling household electricity demand in the presence of a two-step block tariff, primarily with the intention of estimating the price elasticity of demand for electricity - for the country for which I have panel household survey data. The difficulty I am facing is translating the expected electricity demand equation I found in Reiss & White (2005) (which pertains to my case), into Stata code for Generalized Method of Moments (GMM) estimation, through the 'gmm' command.
The expected demand for electricity is the following:
\[
E(x^*|\cdot) = [x(p_1,y,z;\beta)-\sigma\lambda_1]\Phi_1 + \bar{x}\cdot(\Phi_2 - \Phi_1) \\ + [x(p_2,y_2,z;\beta)+\sigma\lambda_2](1-\Phi_2)
\label{eq:estim} \tag{1}
\]
where:
- \(x\) is electricity demand;
- \(p_1,p_2\) are the marginal prices on the two price schedule tiers;
- \(\bar{x}\) is the electricity consumption threshold after which the price switches to the higher tier (700 kWh in my case);
- \(z\) are observable consumer characteristics (control variables I wish to include);
- \(\varepsilon\) are unobserved consumer characteristics (the error term);
- \(y\) is income and \(y_2=y+\bar{x} \cdot (p_2-p_1)\) is the virtual income when consumers lie on the second tier; and
- \(\Phi_j\) is the standard normal distribution evaluated at $\(c_j(\beta)/\sigma\), \(\phi_j\) is the normal density at \(c_j(\beta)/\sigma\), \(\lambda_1=\phi_1/\Phi_1\), and \(\lambda_2=\phi_2/(1-\Phi_2)\).
Would anyone know how to incorporate all the parameters and variables in \eqref{eq:estim} into the gmm command, in order to obtain the unbiased coefficient related to price?
__________________________________________________
Note: the reduced form for the Household's consumption level of electricity \(x^*\) (as a function of the increasing two-tier price schedule), is the following:
\[
x^*=
\begin{cases}
x(p_1,y,z,\varepsilon;\beta) \qquad & \text{if } \varepsilon \le c_1 \\
\bar{x} & \text{if } c_1< \varepsilon < c_2\\
x(p_2,y_2,z,\varepsilon;\beta) \qquad & \text{if } \varepsilon>c_2,
\end{cases}
\label{eq:opt.con.lvl.ecx} \tag{2}
\]
where \(c_j\) (where \(j=1,2\)) is the solution to \(x(p_j,y_j,z,c_j;\beta)=\bar{x}\) with \(y_1=y\). In other words, \(c_j\) is the maximum (for \(j=1\)) or minimum (for \(j=2\)) value of \(\varepsilon\) for which consumption occurs on tier \(j\).
____________________________________
Reiss, P. C., & White, M. W. (2005). Household electricity demand, revisited. The Review of Economic Studies, 72(3), 853-883.
