class: center, middle, inverse, title-slide # Cournot rationalizability and measurement error ### Hans Martinez ### Feb 03, 2022 --- ### Overview - **What’s the question?**: Do the observed prices and quantities arise as an equilibrium of the Cournot model when quantities are mismeasured? - **Why does it matter?**: Measurement Error (ME) is ubiquitous. Ignoring ME leads to an over-rejection of the Revealed Preference (RP) deterministic test. .small[(Aguiar & Kashaev, 2020)]. - **How do I do it?**: I design a stochastic test of the Cournot model by introducing ME to Carvajal et al. (.small[*ECTA*, 2013]). I use *Entropic Latent Variable Integration via Simulation* (ELVIS) .small[(Schennach, 2018)] to estimate the test following Aguiar & Kashaev .small[ ( *RESTUD*, 2020)]. - **What do I find?**: The Cournot hypothesis can no longer be rejected at the 5% significance level once ME in quantities is introduced. --- layout: true ### Introduction --- - Long-standing question in economics .small[(Afriat, 1967)]. - In contrast to the parametric efforts, the RP approach relies only on minimal assumptions. - Until recently the RP tests could not accommodate measurement error. - On the consumer side, ignoring ME leas to an over-rejection of the RP of the Utility Maximization Theory (UMT) .small[(Aguiar & Kashaev, 2020)]. - This result has not been extended to the producer side. --- - Carvajal et al. (2013) develops a RP test for the Cournot model. - Using data from *Monthly Energy Review* on oil-producing countries, the authors reject the Cournot Hypothesis. - However, in this setting, ME in quantities could arise from: - Consolidation process - Coordination mistakes --- layout: false ### Roadmap 1. Define deterministic RP test 1. Define Stochastic Cournot Rationalizability (SCR) and its corresponding stochastic RP test 1. Introduce ME and centering condition 1. Cast the empirical content of the SCR into a set of moment conditions 1. Redefine SCR in terms of the maximum-entropy moment (MEM) to use ELVIS 1. Define test statistic with the sample analogues of the MEM --- ### The Cournot RP deterministic test - Consider an industry consisting with `\(I\)` firms, `\(\mathcal I= \{1,2,..., I\}\)`. - The data set `\(\mathcal O\)` consists of `\(T\)` observations indexed by `\(t\in \mathcal T =\{1,2,...,T\}\)`. - The econometrician observes the output of each firm, `\(q_{it}\in\mathbb R_+\)` and the price `\(p_t\in \mathbb R_{++}\)`. - One homogeneous good. - Firms have convex costs functions that do not vary across observations and - Demand is a downward-sloping function. --- ### THEOREM 1 .small[(Carvajal et al., 2013)] > The following statements are equivalent: > A. The data set `\(\mathcal O = \{p_t, (q_{it})_{i \in \mathcal I} \}_{t\in \mathcal T}\)` is *Cournot rationalizable* with convex cost functions. > B. There exists a set of non-negative numbers `\((\delta_{it})_{(i,t)\in\mathcal{I\times T}}\)` that satisfy: > i. `\(\frac{p_t-\delta_{it}}{q_{it}}=\frac{p_t-\delta_{jt}}{q_{jt}}\ge0\)` *(common ratio property)*, and > ii. `\((\delta_{it}-\delta_{is})(q_{it}-q_{is})\ge0\)` *(co-monotone property)* > for every `\(t,s\in \mathcal T\)`, and every `\(i, j\in \mathcal I\)`. --- ### Stochastic Cournot Rationalizability Let `\(\mathbf{ p}_t^* \in \mathbf P_t \subseteq \mathbb R_{++}^L\)` and `\(\mathbf{ q}_t^* \in \mathbf Q_t \subseteq \mathbb R_{+}^L \setminus \{0\}\)` denote random vectors of true prices and true quantities at time `\(t\)`, respectively. ### Measurement Error *Measurement error*, `\(\mathbf{w}=(\mathbf{w}_t)_{t\in\mathcal T}\in W\)`, is the difference between the observed random variables and their true values. `\begin{equation} \mathbf{w}=\left( \begin{array}{c} \mathbf{w}^q_t \\ \mathbf{w}^p_t \end{array} \right) = \left( \begin{array}{c} \mathbf{q}_t-\mathbf{q}_t^* \\ \mathbf{p}_t-\mathbf{p}_t^* \end{array} \right), \quad \forall t \in \mathcal T \notag \end{equation}` --- > **DEF**: A random array `\(\{(\mathbf{q}_{it}^*)_{i\in\mathcal I},\mathbf{p}_t^*\}_{t\in\mathcal T}\)` is **stochastic Cournot rationalizable (SCR)** if there exists a tuple `\(\left(\mathbf c, (\mathbf{\lambda}_t)_{t\in\mathcal T}\right)\)`, such that: > 1. `\(\mathbf c\)` is a random, continuous, and convex cost function; > 2. `\(\mathbf{c'}(\mathbf{q}_{it}^*)-\lambda_{t}[\mathbf{q}_{it}^*\nabla \mathbf{p}(\mathbf{q}_{t}^*)+\mathbf{p}(\mathbf{q}_t^*)]\le0\)` a.s. for every `\(i\in\mathcal I\)` and every `\(t\in\mathcal T\)`; > 3. `\((\lambda_t)_{t\in\mathcal T}\)` is a positive random vector supported on or inside a known set `\(\Delta \subseteq\mathbb R_{++}^{|\mathcal T|}\)`, `\(\nabla \mathbf{p}(\mathbf{q}_t^*)\)` is a random negative number, and `\(\mathbf{p}(\mathbf{q}_t^*)=\mathbf{p}_t^*\)` for all `\(t\in\mathcal T\)`; > 4. For every `\(i\in \mathcal I\)` and `\(t\in \mathcal T\)`, it must be the case that > `$$\mathbb P\left(\mathbf{q}_{it}^*\not=0, \mathbf{c'}(\mathbf{q}_{it}^*)-\lambda_{t}[\mathbf{q}_{it}^*\nabla \mathbf{p}(\mathbf{q}_{t}^*)+\mathbf{p}(\mathbf{q}_t^*)]<0\right)=0.$$` --- ### Lemma > The following statements are equivalent: > 1. The random array `\(\{(\mathbf{q}_{it}^*)_{i\in\mathcal I},\mathbf{p}_t^*\}_{t\in\mathcal T}\)` is *stochastic Cournot rationalizable (SCR)*. > 2. There exists nonnegative random vectors `\((\delta_{it})_{(i,t)\in\mathcal{I\times T}}\)` that satisfy: > i. `\(\frac{\mathbf{p}^*_t-\delta_{it}}{\mathbf{q}^*_{it}}=\frac{\mathbf{p}^*_t-\delta_{jt}}{\mathbf{q}^*_{jt}}\ge0\)` *(common ratio property)*, and > ii. `\((\delta_{it}-\delta_{is})(\mathbf{q}^*_{it}-\mathbf{q}^*_{is})\ge0\)` *(co-monotone property)* > a.s., `\(\forall t,s\in \mathcal T\)`, and every `\(i, j\in \mathcal I\)`. --- layout: true ### Characterization by moment conditions --- - We can summarize the empirical content of the SCR by a set of moment conditions. - Let `\(\mathbf e=(\delta,\mathbf{w})'\in E|X\)` denote the vector of random latent variables and - `\(\mathbf{x}=(\mathbf{q},\mathbf{p})'\)` the vector of observed random variables. - Moreover, let `\(\mathcal{P}_X\)`, `\(\mathcal{P}_{E,X}\)`, and `\(\mathcal{P}_{E|X}\)` denote the set of all probability measures defined over the support of `\(\mathbf{x}, (\mathbf{e},\mathbf{x}), \text{ and } \mathbf{e}|\mathbf{x}\)`, respectively. --- Define the following moment conditions: `$$\begin{aligned} g_M(\mathbf{x},\mathbf{e})&=\mathbf{p}'_{t}\mathbf{w}^q_{t}, \\ g_{R}(\mathbf{x},\mathbf{e})&= \mathbf 1 \left[ \frac{\mathbf{p}_t-\delta_{it}}{\mathbf{q}_{it}-\mathbf{w}^q_{it}}=\frac{\mathbf{p}_t-\delta_{jt}}{\mathbf{q}_{jt}-\mathbf{w}^q_{jt}}\ge0\right]-1, \\ g_{C}(\mathbf{x},\mathbf{e})&= \mathbf 1 \left[(\delta_{it}-\delta_{is})(\mathbf{q}_{it}-\mathbf{w}^q_{it}-\mathbf{q}_{is}+\mathbf{w}^q_{is})\ge0\right]-1,\\ \forall i\not=j \in \mathcal I, t\not=s \in \mathcal T \notag \end{aligned}$$` --- Define `\(\mathbb E_{\mu\times\pi}[g(\mathbf x, \mathbf e)]=\int_X \int_{E|X} g(x.e)d_\mu d_\pi\)`, where `\(\mu\in \mathcal{P}_{E|X}\)` and `\(\pi\in \mathcal{P}_{X}\)`, and `\(g(\mathbf{x},\mathbf{e})=(g_M(\mathbf{x},\mathbf{e})',g_R(\mathbf{x},\mathbf{e})',g_{C}(\mathbf{x})')'\)`. #### Theorem > A random vector `\(\mathbf{x}=(\mathbf{p}_t,\mathbf{q}_t)_{t\in\mathcal T}\)` is approximately **stochastic Cournot rationalizable (SCR)** iff > `$$\inf_{\mu\in\mathcal P_{E|X}} \|\mathbb E_{\mu\times\pi_0}[g(\mathbf{x},\mathbf{e})] \| =0$$` > where `\(\pi_0\in \mathcal P_X\)` is the observed distribution of `\(\mathbf x\)`. --- layout: true ### Using ELVIS --- - We can solve the problem of having a latent random set of variables `\(\mathbf e\)` with unknown `\(\mu\in\mathcal P_{E|X}\)` using the Entropic Latent Variable Integration via Simulation (ELVIS) .small[(Schennach, 2014)]. - Intuitively, there might be many possible conditional distributions `\(\mu\in\mathcal P_{E|X}\)` of the latent variables that satisfy the moment conditions. - ELVIS ranks them by entropy and selects the *least favourable*, thus, converting an *existence* problem into an optimization problem. --- 1. Define a maximum-entropy moment, `\(h(x;\gamma)\)`, as the marginal moment of the function `\(g\)` at which the latent variable has been integrated out. 1. This maximum-entropy moment requires an arbitrary user-defined distribution `\(\eta\in \mathcal P_{E|X}\)` that needs to comply with certain characteristics, but it does not affect the set of values that `\(\mathbb E_{\pi_0}[h(\mathbf x;\cdot)]\)` can take. 1. Finally, we can state the necessary and sufficient conditions to be approximately SCR in terms of the maximum-entropy moment, `\(\tilde h(x;\gamma)\)`, with distribution `\(d\tilde \eta(\cdot|x)=\mathbb 1(g_R(x,\cdot)=0)\mathbb 1 (g_C(x,\cdot)=0)d\eta(\cdot|x)\)`. --- - In practice, we need to use Markov Chain Monte Carlo (MCMC) methods to compute `\(\tilde h(x;\gamma)\)` by sampling from `\(\eta\)` and reject if the draw if it does not satisfy `\(\mathbb 1(g_R(x,\cdot)=0)\mathbb 1 (g_C(x,\cdot)=0)\)`. - In the application, I used a hit-and-run algorithm adapted from Aguiar & Kashaev (2020) to sample directly from `\(\tilde \eta\)`. --- layout: true ### Test statistic --- `\begin{equation} TS_n=T \inf_{\gamma\in\mathbb R^q} \hat{\tilde{h}}_M(\gamma)'\hat{\tilde{\Omega}}^{-1}\hat{\tilde{ h}}_M(\gamma) \end{equation}` where `\(\hat{\tilde{ h}}_M(\gamma)\)` and `\(\hat{\tilde{\Omega}}\)` are the sample analogues of the maximum-entropy moment and its variance, defined as: `$$\begin{aligned} \hat{\tilde{h}}_M(\gamma)&=\frac{1}{T}\sum_{i=1}^{n}\tilde h_M(\mathbf x;\gamma) \\ \hat{\tilde{\Omega}} (\gamma)&=\frac{1}{T}\sum_{i=1}^{n}\tilde h_M(\mathbf x;\gamma)\tilde h_M(\mathbf x;\gamma)'-\tilde h_M(\gamma)\tilde h_M(\gamma)' \end{aligned}$$` --- Under the null hypothesis that the data is approximately consistent with SCR, then: `\begin{equation} \lim_{n\rightarrow\infty} \mathbb P \left( TS_n > \chi_{q,1-\alpha}^{2} \right)\le\alpha .\notag \end{equation}` --- layout: false ### Application - Data from .small[Carvajal et al. (2013)], two sources: the *Monthly Energy Review* (MER), published by the U.S. Energy Information Administration, and the St. Louis Federal Reserve. - MER provides a series of monthly crude oil production in thousands of barrels per day by 12 OPEC members and 7 nonmembers - The data covers January 1973 to April 2009. --- ### Results <table class="table" style="margin-left: auto; margin-right: auto;border-bottom: 0;"> <caption>\(TS_n\) statistic and p-values for the six major oil producers</caption> <thead> <tr> <th style="text-align:left;"> Period </th> <th style="text-align:right;"> \(TS_n\) </th> <th style="text-align:right;"> \(p\) value </th> <th style="text-align:left;"> </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;"> 1973-2008 </td> <td style="text-align:right;"> 0.90 </td> <td style="text-align:right;"> 0.9890 </td> <td style="text-align:left;"> </td> </tr> <tr> <td style="text-align:left;"> 1973-1980 </td> <td style="text-align:right;"> 0.80 </td> <td style="text-align:right;"> 0.9920 </td> <td style="text-align:left;"> </td> </tr> <tr> <td style="text-align:left;"> 1981-1990 </td> <td style="text-align:right;"> 1.68 </td> <td style="text-align:right;"> 0.9469 </td> <td style="text-align:left;"> </td> </tr> <tr> <td style="text-align:left;"> 1991-2000 </td> <td style="text-align:right;"> 2.66 </td> <td style="text-align:right;"> 0.8502 </td> <td style="text-align:left;"> </td> </tr> <tr> <td style="text-align:left;"> 2001-2008 </td> <td style="text-align:right;"> 4.27 </td> <td style="text-align:right;"> 0.6400 </td> <td style="text-align:left;"> </td> </tr> </tbody> <tfoot> <tr><td style="padding: 0; " colspan="100%"><span style="font-style: italic;">Note: </span></td></tr> <tr><td style="padding: 0; " colspan="100%">  The six producers are Saudi Arabia, U.S.A, Iran, China, Mexico and Venezuela. Countries are assumed to play semiannual independent games.</td></tr> </tfoot> </table> --- ### Discussion and future work 1. MC simulations to check if test has any bite 1. Better demand estimates 1. Bertrand and product differentiation 1. Reformulate using profit function 1. Reformulate using production function --- ### Conclusions - I develop a stochastic RP test for the Cournot model that accounts for ME. - The test builds on .small[Carvajal et al. (2013)] and uses recently developed techniques first by .small[Schennach (2014)] and then applied in the consumer case to test the UMT by .small[Aguiar & Kashaev (2020)]. - Using data from .small[Carvajal et al. (2013)] on the major oil-producing countries, we can no longer reject the Cournot hypothesis at the 5 % significance level. - Consequently, as in .small[Aguiar & Kashaev (2020)]; ignoring measurement error leads to an over-rejection of the RP tests. --- layout: false class: center, middle # END