I am trying to estimate residential demand for electricity in a country where electricity is sold (to all households (HH)) at an increasing two-part tariff. By choosing marginal prices as my key independent variable (I am mostly interested in estimating the price elasticity of demand), I have come to understand that OLS regression cannot be pursued because the price variable would be endogenous. Therefore, literature suggests I use either Maximum Likelihood estimation (MLE) or Generalized Method of Moments (GMM) estimation. However, I am struggling with finding by objective function.
In their study, Reiss and White (2005) pursue a moments-based approach and go on to define the reduced form for the HH's consumption level of electricity $x^*$ (as a function of an increasing two-tier price schedule), as such:
$$ x^*= \begin{cases} x(p_1,y,z,\varepsilon;\beta) \qquad & \text{if } \varepsilon \le c_1 \\ \bar{x} & \text{if } c_1< \varepsilon < c_2\\ x(p_2,y_2,z,\varepsilon;\beta) \qquad & \text{if } \varepsilon>c_2, \end{cases} \label{eq:opt.con.lvl.ecx} \tag{1} $$
where:
- $p_1,p_2$ are the marginal prices on the two price schedule tiers (demand is assumed strictly decreasing in $p$);
- $\bar{x}$ is the electricity consumption threshold after which the price switches to the higher tier;
- $z$ are observable consumer characteristics;
- $\varepsilon$ are unobserved consumer characteristics (demand is assumed strictly increasing in $\varepsilon$);
- $y_2=y+\bar{x} \cdot (p_2-p_1)$ is the virtual income when consumers lie on the second tier (hence, $y$ is income); and
- $c_j$ (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 $j$.
Conditional on the observables, the authors integrate \eqref{eq:opt.con.lvl.ecx} piecewise to obtain:
$$ E(x^*|\cdot) = E_{\varepsilon}[x(p_2,y_2,z,\varepsilon;\beta)] + h(p_1,p_2,\bar{x},y,z;\beta) \label{eq:pw.int} \tag{2} $$
where $h(\cdot)\equiv \tau_2 - \tau_1$ is a 'sorting correction function defined by the truncated moments' (I used quotation marks as I do not grasp what this means):
$$ \tau_j=\int_{-\infty}^{c_j(\beta)} [\bar{x}-x(p_j,y_j,z,\varepsilon;\beta)]dF_{\varepsilon}, \label{eq:srt.corr.fcn} \tag{3} $$
with $c_1,c_2$ defined in \eqref{eq:opt.con.lvl.ecx} and $y_1=y$. The authors go on to evaluate 'the moments' in \eqref{eq:srt.corr.fcn} with the error specification in $F_{\varepsilon}$ assumed $N(0,\sigma^2)$ and $\varepsilon$ entering demand \eqref{eq:opt.con.lvl.ecx} additevely, to find:
$$ E(x^*|\cdot) = [x(p_1,y,z;\beta)-\sigma\lambda_1]\Phi_1 + \bar{x}\cdot(\Phi_2 - \Phi_1) \\ + [x(p_2,y_2,z;\beta)+\sigma\lambda_2](1-\Phi_2) \label{eq:estim} \tag{4} $$
where, $\Phi_j$ 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 $\lambda_2=\phi_2/(1-\Phi_2)$.
The authors end up using \eqref{eq:estim} because it fits their data well.
Question(s):
- What math and reasoning did the authors use to move from \eqref{eq:opt.con.lvl.ecx} to \eqref{eq:pw.int}, \eqref{eq:srt.corr.fcn} and \eqref{eq:estim}?
- Is the authors' approach strictly for GMM estimation (as they chose to pursue a moments-based approach) or could MLE be carried out as well?
- Can anyone show me what \eqref{eq:pw.int}, \eqref{eq:srt.corr.fcn} and \eqref{eq:estim} would look like, given that my case has \eqref{eq:my.opt.con.lvl.ecx} (below) as reduced form equation for HH consumption?
- Alternatively to Q.3, could anyone provide me with either textbook/audio/video material which can help me formulate my own objective function and assumptions to find the best fit for my data, for either GMM or MLE?
$$ x^*= \begin{cases} x(p_1,y,z,\varepsilon;\beta) \qquad & \text{if } \varepsilon \leq c_1 \\ x(p_2,y_2,z,\varepsilon;\beta) \qquad & \text{if } \varepsilon>c_2, \end{cases} \label{eq:my.opt.con.lvl.ecx} \tag{5} $$
I am already grateful to those who made it thus far in reading my long question. Any help is deeply appreciated. Thanks!
Gabriele
Reiss, P. C., & White, M. W. (2005). Household electricity demand, revisited. The Review of Economic Studies, 72(3), 853-883.
