class: center, middle, inverse, title-slide # Towards prospectus ### Hans Martinez ### Feb 14, 2022 --- ## Objective - Prospectus plan - Update - Where does it matter? - How to estimate? --- name: prospectus ## Prospectus 1. Clear and original research question 1. Question is interesting (nontrivial problem) and important (of interest to scholars) 1. Feasible 1. Publishable [.small[Document]](#doc) --- class: inverse center middle # Stochastic revealed preferences (SRP) --- layout: true ### SRP --- name: srp - SRP can accommodate not only ME, but also any other latent variables of interest - SRP test can be reformulated using shape restrictions, only convexity of cost function has been exploited in the past: 1. Concavity of the profit function 1. Concavity of the production function - We can do better demand estimates, so far only negative slope (inverse demand) and NSD: 1. Symmetry 1. Iteration methods for M-matrices 1. Taylor approximation [.small[Summer Paper]](#summerp) --- #### Why would applied IO researchers care? - SRP test could be used to: 1. Before estimation, check if a particular model of competition fit certain market of interest 1. After estimation, confirm parametric estimates rationalize data under a competition model hypothesis 1. Obtain bounds for latent variables of interest (marginal costs, markups, Lerner index, competition parameter) --- layout: true ### Stochastic Bernard Rationalizability --- #### Definition A random array `\((\mathbf{q}_t^*,\mathbf{p}_t^*)_{t\in\mathcal T}\)` is *s/B-rationalizable* with market structure `\((\Omega_{jt})_{j \in \mathcal J}\)`, if there exists a tuple `\(\left(\mathbf\Pi, (\mathbf{\lambda}_t)_{t\in\mathcal T}\right)\)`, such that: 1. `\(\mathbf \Pi\)` is a random, continuous and concave profit function; 2. `\(\mathbf{\Pi'}(\mathbf{p}_{jt}^*)-\Omega_{jt}\mathbf{\lambda}_{t}[\nabla \mathbf{q}(\mathbf{p}_t^*)(\mathbf{p}_t^*-\mathbf{mc}_t)+\mathbf{q}(\mathbf{p}_t^*)]\le0\)` a.s. for every `\(j\in\mathcal J\)` and every `\(t\in\mathcal T\)`; 3. `\((\mathbf{\lambda}_t)_{t\in\mathcal T}\)` is a positive random vector supported on or inside a known set `\(\Delta \subseteq\mathbb R_{++}^{|\mathcal T|}\)`, `\(\nabla \mathbf{q}(\mathbf{p}_t^*)\)` is a `\(|L|\times|L|\)` NSD random matrix, and `\(\mathbf{q}(\mathbf{p}_t^*)=\mathbf{q}_t^*\)` for all `\(t\in\mathcal T\)`; --- 4. `\(\Omega_{jt}=[\omega_{jt}^l]\)` is a matrix `\(|L_j|\times|L|\)`, where `\(\omega_{jt}^l = 0\)` if `\(l \not\in L_j\)` and `\(\omega_{jt}^l = 1\)` if `\(l \in L_j\)`; 5. and `\(\mathbf{mc}_{t}\)` is a random vector, where `\(0 \le \mathbf{mc}_{jt}^l \le \mathbf{p}^{*l}_{jt}\)` a.s. if `\(l \in L_j\)` and `\(\mathbf{mc}_{jt}^l=0\)` a.s. if `\(l \not\in L_j\)` for all `\(j\in\mathcal J\)` and `\(t\in\mathcal T\)`. 6. For every `\(j\in \mathcal J\)` and `\(t\in \mathcal T\)`, it must be the case that `$$\begin{equation} \mathbb P\left(\mathbf{p}_{jt}^*\not=0, \mathbf{\Pi'}(\mathbf{p}_{jt}^*)-\Omega_{jt}\mathbf{\lambda}_{t}[\nabla \mathbf{q}(\mathbf{p}_t^*)(\mathbf{p}_t^*-\mathbf{mc}_t)+\mathbf{q}(\mathbf{p}_t^*)]<0\right)=0. \end{equation}$$` --- layout: false **Theorem:** The following statements are equivalent: 1. The random array `\(\{\mathbf{p}_t,\mathbf{q}_t\}_{t\in\mathcal T}\)` is *s/B-rationalizable* with market structure `\((\Omega_{jt})_{t\in\mathcal T}\)` with 2. There exists positive random vectors `\((\mathbf{\pi}_{jt})_{t\in\mathcal T}\)`, non-negative random vector `\((\mathbf{mc}_{jt})_{t\in\mathcal T}\)`, `\((\mathbf{\lambda}_{jt})_{t\in\mathcal T}\)` supported on or inside `\(\Delta\)`, random NSD `\(|L|\times |L|\)` matrix `\((\mathbf{\Lambda}_t)_{t\in\mathcal T}\)`, such that: 1. `\(0 \le \mathbf{mc}_{jt}^l \le \mathbf{p}_{jt}^l\)` a.s. if `\(l \in L_j\)`, `\(\mathbf{mc}_{jt}^l=0\)` a.s. if `\(l \not\in L_j\)`, 1. `\(\mathbf{\Lambda}_t (\mathbf{p}_s^* -\mathbf{p}_t^*)= \mathbf{q}_s^* - \mathbf{q}_t^* + \mathbf\alpha_t\)` 1. `\(\mathbf{\pi}_{jt} - \mathbf{\pi}_{js} \le \mathbf{\lambda}_{t}[\mathbf{\Lambda}_t(\mathbf{p}_{t}^*-\mathbf{mc}_{t}) + \mathbf{q}_t^*]'\Omega_{jt}(\mathbf{p}_{jt}^*-\mathbf{p}_{js}^*)\)` for all `\(t,s\in \mathcal T\)`, and all `\(j\in \mathcal J\)`. --- layout: false ### 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}` --- layout: true ### Moment conditions --- `\begin{equation} g_M(\mathbf{x},\mathbf{e})=log(\mathbf{p}_t)-log(\mathbf{p}_t-\mathbf{w}^p_t) \end{equation}` `\begin{equation} g_{C}(\mathbf{x},\mathbf{e})=\left\{\begin{array}{ll} \mathbb{1}\left(\mathbf{p}_{jt}^l-\mathbf{w}_{jt}^{pl}-\mathbf{mc}_{jt}^l\ge0\right)-1 & \text{ if } l \in L_j \\ \mathbf{mc}_{jt}^l=0 & \text{ if } l \not\in L_j \end{array} \right. \end{equation}` --- `\begin{multline} g_I(\mathbf{x},\mathbf{e})=\\ \mathbb 1 \left[ \mathbf{\pi}_{jt} - \mathbf{\pi}_{js} \le \mathbf{\lambda}_{t}[\mathbf{\Lambda}_t(\mathbf{p}_{t}-\mathbf{w}^p_t-\mathbf{mc}_{t}) + \mathbf{q}_t]'\Omega_{jt}(\mathbf{p}_{jt}-\mathbf{w}^p_{jt}-\mathbf{p}_{js}+\mathbf{w}^p_{js})\right] \\ -1, \; t\ne s \in \mathcal T, j \in \mathcal J \end{multline}` `\begin{equation} g_D(\mathbf{x},\mathbf{e})=\mathbf{\Lambda}_t (\mathbf{p}_s-\mathbf{w}^p_s -\mathbf{p}_t+\mathbf{w}^p_t)-\mathbf{q}_t + \mathbf{q}_s + \mathbf \alpha_t \end{equation}` --- #### Additional instruments `\begin{equation} g_Z(\mathbf{x},\mathbf{e})=\left[\begin{array}{l} \mathbf{z}^{'\pi}_t\mathbf{\pi}_t \\ \mathbf{z}^{'mc}_t\mathbf{mc}_t \\ \end{array}\right] \end{equation}` --- layout: false class: inverse center middle # Immigrant job market and productivity losses in Canada --- layout: true ### IJM + productivity losses --- name: ijm * **Question**: Frictions in the Canadian job market for immigrants (TFW and SIE) and for employers negatively affect firms' productivity. How big? Can we measure them? * **Data**: StatsCan + IRCC linked databases * **Approach**: 1. Search model to calibrate frictions and recover counterfactual matches. 2. 2SGMM procedure for Production function + productivity monotonic in immigrant skills. [.small[Recap]](#ijm_recap) --- layout:true ### Where does it matter? --- For firms, `$$\max_{x\in \mathbb R_+} P\mathbb E[F(x)e^{\omega+\varepsilon}]-wx$$` if employers believe that `\(\mathbb E [\varepsilon| \theta=Native]\ge\mathbb E [\varepsilon| \theta=Immigrant]\)` and, by law, `\(w_N=w_I=w\)`, then, `$$\mbox{FOC become} \left\{ \begin{array}{cl} Pf(x)e^{\omega} = w & \theta=Native \\ Pf(x)e^{\omega-\xi_i} \le w & \theta=Immigrant \end{array} \right.$$` --- Firm bias: - `\(\mathbb E [\varepsilon| \theta=Native]\ge\mathbb E [\varepsilon| \theta=Immigrant]\)` : Firms expect more than average negative shocks when hiring an immigrant versus hiring a native. - `\(\mathbb E [\omega| \theta=Native]\ge\mathbb E [\omega| \theta=Immigrant]\)` : Firms could also take expectations over the candidate's skills. - We usually think of productivity as given and firms taking decisions over inputs. However, firms might expect a certain productivity level by choosing particular inputs (workers' skills, quality of inputs). --- layout: true ### How to estimate ? --- name: ijm-est 1. Assume we observe `\(\mathcal O=\{Y_{jt}, M_{jt}, N_{jt}, I_{jt}, S^N_{jt}, S^I_{jt} \}_{t=1}^T\)` for firms `\(j=1,\dots,J\)`. 1. Assume we have a model such that I can get counter factual `\(\{\tilde N_{jt}, \tilde I_{jt}, \tilde S^N_{jt}, \tilde S^I_{jt}\}_{t=1}^T\)`. I expect the share of immigrant workers and the overall employee's skills to increase. Let `\(X_{jt}=(M_{jt},N_{jt},I_{jt})\)` and `\(S_{jt}=(S^N_{jt},S^I_{jt})\)`, and low case letters are log demeaned variables. [.small[Model]](#ijm_model) --- 1. Fit `\(y_{jt}=\phi(x_{jt},s_{jt})+\varepsilon_{jt}\)`. 1. Assume functional form for the prod fun `\(f(x_{jt})=\alpha' x_{jt} +\omega_{jt}+\varepsilon_{jt}\)` with `\(\mathbb E[\varepsilon_{jt}]=\mathbb E[\omega_{jt}|x_{jt-1}]=0\)`. 1. Then, `\(\hat \omega_{jt}=\hat \phi(x_{jt},s_{jt})-\hat \alpha'x_{jt}\)`. 1. Predict `\(\tilde y_{jt} = \hat\phi(\tilde x_{jt}, \tilde s_{jt})\)`. 1. Keep same functional form, but use counterfactual data `\(f(\tilde x_{jt})=\tilde\alpha' \tilde x_{jt} +\tilde \omega_{jt}+\varepsilon_{jt}\)`. 1. Then, `\(\hat{\tilde{\omega}}_{jt}=\hat \phi(\tilde x_{jt},\tilde s_{jt})-\hat{\tilde{\alpha}}'\tilde x_{jt}\)`. [.small[Estimator]](#estimator) --- layout: false class: inverse center middle # Tax evasion and productivity in Mexico --- layout: true ### Tax evasion and productivity --- * Firms have incentives to under-report profits to avoid taxes. Productivity estimates are likely to be biased. * Mexico: Non-detected tax evasion up to $10 billion USD per year .small[(Zumaya et al., 2021)] * **Question**: Can we recover productivity estimates in the presense of measurement error by using a micro-founded IC as location condition? * **Data**: INEGI-EMIM, Anonymized tax filings (?) .small[(Zumaya et al., 2021)] * **Approach**: Hu & Schennach (2008), Hu (2021), Hu & Yao (*J. Econom., forthcoming*) --- `$$\begin{aligned} Y&=G(x^*)e^{\omega+\varepsilon} \\ X&=X^*+\Delta X(\theta) \\ \Delta X(\theta) &= \left \{ \begin{array}{ll} 0 & (1-\tau)\Pi(x^*) \ge \Pi(x^*)-\tau\Pi(x)-\kappa Pr(a)-c(\Delta x) \\ \mathbf (0,\tilde x] & otherwise \end{array} \right.\\ \end{aligned}$$` --- Can we use a micro-founded IC as location condition? `\(M[f]=x^*\)`, where `\(M[f]\equiv\)` `$$\begin{aligned} \sup\left \{ v: \int^{v}_{x^*} \left \{ \tau [\Pi(x^*)-\Pi(x)]-c(x-x^*)\right\} f_{X|X^*}(x|x^*) dx \le\kappa Pr(a) \right\} \end{aligned}$$` Assume firms only over-report expenses, then `\(\Pi(x)=PY-\rho X\)`. Consequently, `\(\Pi(\tilde x)=0\)`. Once they reach zero profits, firms don't have incentives to over-report more. --- layout: true ### How to estimate? --- `\begin{equation} f(y_{it},x_{it},p_{it}) = \\ f(y_{it},x_{it},p_{it},\omega_{it},\omega_{it-1},x^*_{it})= \\ f(y_{it}| x^*_{it},\omega_{it}) f(p_{it}|x^*_{it},\omega_{it}) f(\omega_{it}|\omega_{it-1}) f(x_{it}|x^*_{it}) f(\omega_{it-1})f(x^*) \end{equation}` --- `\begin{equation} \color{blue}{f(y_{it}| x^*_{it},\omega_{it})} \color{green}{f(p_{it}|x^*_{it},\omega_{it})} \color{violet}{f(\omega_{it}|\omega_{it-1})} \color{red}{f(x_{it}|x^*_{it})} \color{orange}{f(\omega_{it-1})f(x^*_{it})} \end{equation}` `\begin{equation} \color{blue}{ y_{it} = g(x^*_{it})+\omega_{it}+\varepsilon^Y_{it}} \\ \color{green}{ \ln\left(\frac{\rho_t X^*_{it}}{P_{t}Y_{it}}\right)=\ln\left(\frac{G_x(x^*_{it})X^*_{it}}{G(x^*_{it})}\right) - \varepsilon^Y_{it}} \\ \color{violet}{\omega_{it}=\omega_{it-1}+\varepsilon^\omega_{it}} \\ \color{red}{M[f]=x^*} \\ \color{orange}{\text{Flexible input assumption.}} \end{equation}` --- In addition, to calibrate the cost of cheating: `$$\begin{aligned}P(cheated)=&\hat P(cheated|Audited)\hat P(Audited)+\\ &P(cheated|Not Audited)\hat P(Not Audited) \end{aligned}$$` - The cost of cheating could be function of firm characteristics. - Estimate for different values of unobserved `\(P(cheated|Not Audited)\)`. --- - Hu & Yao (Journal of Econometrics, *forthcoming*) use sieve maximum likelihood estimator (MLE) - Mix ELVIS and sieve MLE (?) --- layout: false class: inverse center middle # Aggressive entry, prices and productivity --- name: epp ### Aggressive entry, prices and productivity * Standard economic theory: * `\(p_{it}-mc_{i}(\omega_i)\ge0\)`, * `\(\implies\)` most productive firm, lowest price * `\(\implies\)` market shares inform about firms' productivity and markups. * However, Firms might forgo **today's** profits if the cost of recovering market shares **tomorrow** is greater than immediate losses. * Case of study: [Whirlpool vs LG](#timeline) (US international trade commission) .small[(Flaaen et al., _AER_ 2020 )] --- ### Aggressive entry, prices and productivity * **Key Friction**: Demand dynamics and switching costs. * **Questions**: In the presence of entry and demand switching costs, is it profit-maximizing for firms to set `\(p-mc\ge0\)`? * **Data**: Nielsen .small[Flaaen et al.(_AER_ 2020 ) Gap Intelligence + Traqline Market Research] * **Method**: Structural dynamic model with beliefs and demand switching costs .small[(Aguirregabiria and Jeon, _Rev.Ind.Organ._ 2020)] --- layout: false class: center, middle # Thanks! --- name: ijm_recap ### Recap * Immigrants are key for the economic development of Canada by filling in the gaps in the labor market. By 2036, 30% of pop. .small[(Mortency et al., 2017)] * However, immigrants suffer unemployment and underemployment .small[(Adamuti-Trache, 2016)], and face multiple frictions (financial, signaling, networks). * Furthermore, employers do not actively recruit immigrants, can't assess foreign credentials, don't assess immigrants competencies, have a preference for Canadian experience and domestic candidates .small[(Chhinzer and Oh, 2021)]. Firm side remains understudied. [.small[Back]](#ijm) --- layout: true ### The model --- name: ijm_model Two key frictions: - Workers search friction `\(\implies\)` affects the pool of candidates from which firms randomly draw. - Firms' bias (statistical discrimination) `\(\implies\)` affects the probability of getting a job for candidates. --- Firm bias (statistical discrimination) will affect applicants probabilities to obtain a job. `$$P(\text{getting a job}|N)\ge P(\text{getting a job}|I)$$` > Intuition: If firms get an immigrant in the current period, they would wait another period to get a new draw and hopefully get a native candidate. [.small[Back]](#ijm-est) --- name: estimator layout: false ### How to estimate? `\begin{align} \alpha = [x_{jt-1}'x_{jt1}']^{-1}x_{jt-1}'\hat \phi(x_{jt},s_{jt}) \\ \tilde \alpha = [\tilde x_{jt-1}'\tilde x_{jt1}]^{-1}\tilde x_{jt-1}'\hat \phi(\tilde x_{jt},\tilde s_{jt}) \end{align}` [.small[Back]](#ijm-est) --- name: timeline <img src="tl.png" width="100%" style="display: block; margin: auto;" /> [.small[Back]](#epp) --- name: doc ### Document - Convey that research plan satisfies 1-4. - Present and discuss: 1. Paper's idea 1. Motivation 1. Originality 1. Strategy - Empirical paper: 1. Outline identification and strategy 1. Data to be used 1. Summary statistics or preliminary results [.small[Back]](#prospectus) --- layout: false name: summerp ### Stochastic Cournot Rationalizability and measurement error - **Question:** Do the observed price and quantities in the oil industry arise as an equilibrium of Cournot competition if quantities are mismeasured? - **What I do:** I use tools developed by .small[Schnenach (2014)] and adapted by .small[Aguiar and Kashaev (2020)] to introduce measurement error in .small[Carvajal et al. (2013)]. - **What I find:** The Cournot hypothesis can no longer be rejected at the 5% significance level if we account for measurement error in quantities. - **Limitations:** 1) Silent about Symmetric Market Power 2) Demand 3) Technical issues (iid games, test's power, mismeasured prices) --- - 3 interesting questions to applied IO researchers: 1. Counterfactual demand estimates. Intuition: Products with similar characteristics should attract similar buyers. 1. Recover marginal costs/ markups 1. How much is driven by parametric assumptions? - What's next? 1. Reformulate using profit function 1. Bertrand and product differentiation 1. Define Market and find data (RTE Cereal, airplane trips, yogurts, detergents) --- [.small[Back]](#srp)