class: center, middle, inverse, title-slide # Lecture 5 ## GEV ### Tyler Ransom ### ECON 6343, University of Oklahoma --- # Attribution Many of these slides are based on slides written by Peter Arcidiacono. I use them with his permission. These slides also heavily follow Chapter 4 of Train (2009) --- # Plan for the day 1. Nested Logit - two-step estimation 2. Generalized Extreme Value Distributions - multinomial logit and nested logit as special cases 3. Bresnahan, Stern, and Trajtenberg (1997) - allows for correlations across multiple nests --- # Red bus / blue bus choice set - As we discussed last time, adding another bus results in odd substitution patterns - This is because of the IIA property of multinomial logit models <center> <div id="htmlwidget-4c9aa05829b34740fa7d" style="width:504px;height:504px;" class="DiagrammeR html-widget"></div> <script>HTMLWidgets.pymChild = new pym.Child();HTMLWidgets.addPostRenderHandler(function(){ setTimeout(function(){HTMLWidgets.pymChild.sendHeight();},100); });</script> <script type="application/json" data-for="htmlwidget-4c9aa05829b34740fa7d">{"x":{"diagram":"\ngraph TD;\nA(Commute Method) --> B(Car);\nA-->C(Red Bus);\nA-->D(Blue Bus);\n"},"evals":[],"jsHooks":[]}</script> </center> --- # Nesting the choice set - One way to get around IIA is to .hi[nest] the choice set - Nesting explicitly introduces correlation across alternatives within the same nest <center> <div id="htmlwidget-f9fb15f7b36beb634e97" style="width:504px;height:504px;" class="DiagrammeR html-widget"></div> <script>HTMLWidgets.pymChild = new pym.Child();HTMLWidgets.addPostRenderHandler(function(){ setTimeout(function(){HTMLWidgets.pymChild.sendHeight();},100); });</script> <script type="application/json" data-for="htmlwidget-f9fb15f7b36beb634e97">{"x":{"diagram":"\ngraph TD;\nA(Commute Method) --> B(Car);\nA-->C(Bus);\nC-->D(Red Bus);\nC-->E(Blue Bus);\n"},"evals":[],"jsHooks":[]}</script> </center> --- # Other cases where nesting is useful - Red bus / blue bus came from the transportation economics literature - But nesting is used in all fields of economics - Elections - Suppose we have two candidates, A and B - If we introduce C, whose platform resembles B, what will new vote shares be? - (Primary elections are a kind of nesting) - Product markets - Nesting "branded" and "generic" products (e.g. branded vs. micro-brewed beer) - But some consumers won't purchase either type of the product - Ignoring non-buyers, will give misleading price elasticities of demand --- # Nested Logit - Coming back to the red bus/blue bus problem, we would like some way for the errors for the red bus to be correlated with the errors for the blue bus - The .hi[nested logit] allows a nest-specific error: .small[ `\begin{align*} U_{i,RedBus}&=u_{i,RedBus\phantom{e}}+\nu_{i,Bus}+\lambda\epsilon_{i,RedBus}\\ U_{i,BlueBus}&=u_{i,BlueBus}+\nu_{i,Bus}+\lambda\epsilon_{i,BlueBus}\\ U_{i,Car}&=u_{i,Car\phantom{eBus}}+\nu_{i,Car}+\lambda\epsilon_{i,Car} \end{align*}` ] where: 1. `\(\nu_{ik}+\lambda\epsilon_{ij}\)` is distributed Type I Extreme Value 2. The `\(\nu\)`'s and the `\(\epsilon\)`'s are independent 3. `\(\epsilon_{ij}\)` is distributed Type I extreme value 4. Distribution of `\(\nu_k\)`'s is derived in Theorem 2.1 of Cardell (1997) --- # Nested Logit 2 - Composite error term for car is independent from either the red bus error or the blue bus error - If we added a yellow bus, all errors in the bus nest would be independent conditional on choosing to take a bus (i.e. .hi[IIA within nest]) - But the bus nest errors are correlated from the viewpoint of the top level (i.e. before conditioning on nest choice) - Note: adding two extreme value errors does .hi[not] give back an extreme value error - But the difference between two T1EV errors is distributed logistic --- # Nested Logit 3 - More important than the exact error distribution is the choice probabilities: `\begin{align*} P_{iC}&=\frac{\exp(u_{iC})}{\left[\exp\left(\frac{u_{iRB}}{\lambda}\right)+\exp\left(\frac{u_{iBB}}{\lambda}\right)\right]^{\lambda}+\exp(u_{iC})}\\ P_{iRB}&=\frac{\exp\left(\frac{u_{iRB}}{\lambda}\right)\left[\exp\left(\frac{u_{iRB}}{\lambda}\right)+\exp\left(\frac{u_{iBB}}{\lambda}\right)\right]^{\lambda-1}}{\left[\exp\left(\frac{u_{iRB}}{\lambda}\right)+\exp\left(\frac{u_{iBB}}{\lambda}\right)\right]^{\lambda}+\exp(u_{iC})} \end{align*}` - Not particularly intuitive, but can break it down into parts `\(P(B)P(RB|B)\)`: `\begin{align} P_{iRB}&=\left(\frac{\left[\exp\left(\frac{u_{iRB}}{\lambda}\right)+\exp\left(\frac{u_{iBB}}{\lambda}\right)\right]^{\lambda}}{\left[\exp\left(\frac{u_{iRB}}{\lambda}\right)+\exp\left(\frac{u_{iBB}}{\lambda}\right)\right]^{\lambda}+\exp(u_{iC})}\right)\times\label{eq:pbus}\\ &\phantom{\times\times}\left(\frac{\exp\left(\frac{u_{iRB}}{\lambda}\right)}{\exp\left(\frac{u_{iRB}}{\lambda}\right)+\exp\left(\frac{u_{iBB}}{\lambda}\right)}\right)\nonumber \end{align}` --- # Nested Logit Estimation The log likelihood can then be written as: .small[ `\begin{align*} \ell&=\sum_{i=1}^N\sum_{j\in J}(d_{ij}=1)\ln(P_{ij})\\ &= \sum_{i=1}^N\left[(d_{iC}=1)\ln(P_{iC})+\sum_{j\in J_B}(d_{ij}=1)\ln(P_{iB}P_{ij|B})\right]\\ &=\sum_{i=1}^N\left[(d_{iC}=1)\ln(P_{iC})+\sum_{j\in J_B}(d_{ij}=1)(\ln(P_{iB})+\ln(P_{ij|B}))\right]\\ &=\sum_{i=1}^N\Bigg[(d_{iC}=1)\ln(P_{iC})+(d_{iBB}=1+d_{iRB}=1)\ln(P_{iB})\\ &\qquad+\left.\sum_{j\in J_B}(d_{ij}=1)\ln(P_{ij|B})\right] \end{align*}` ] --- # Nested Logit Estimation 2 - Could estimate a nested logit by straight maximum likelihood. An alternative follows from decomposing the nests into the product of two probabilities: `\(P(RB|B)P(B)\)` - In order to do this, however, first decompose `\(u_{RB}\)` into two parts: `\begin{align*} u_{iRB}&=u_{iB}+u_{iRB|B} \end{align*}` - We also need to choose normalizations: - `\(u_{iC} = 0\)` - `\(u_{iBB|B} = 0\)` - So we will estimate `\((\beta_{B},\beta_{RB}, \gamma,\lambda)\)` where `\(\gamma\)` corresponds to the `\(Z\)`'s (alt-specific) --- # Nested Logit Estimation 3 - Note that our normalizations imply the following observable components of utility `\begin{align*} u_{iC}&=0\\ u_{iBB}&=\beta_{B}X_{i}+\gamma (Z_{BB}-Z_{C})\\ u_{iRB}&=(\beta_{B}+\beta_{RB})X_{i}+\gamma (Z_{RB}-Z_{C}) \end{align*}` - Now estimate `\(\beta_{RB}\)` and `\(\gamma\)` in a 1st stage using only observations that chose bus, `\(N_B\)`: `\begin{align*} \ell_1&=\sum_{i=1}^{N_B}(d_{iRB}=1)(u_{iRB|B}/\lambda)+\ln\left(1+\exp(u_{iRB|B}/\lambda)\right) \end{align*}` - The `\(1\)` in the `\(\ln()\)` operator corresponds to `\(\exp(u_{iBB|B}/\lambda)\)` since `\(u_{iBB|B} = 0\)` --- # Nested Logit Estimation 4 - Now consider the term in the numerator of `\(P(B)\)` in \eqref{eq:pbus}. We can rewrite this as: `\begin{align*} \left[\exp\left(\frac{u_{iRB}}{\lambda}\right)+\exp\left(\frac{u_{iBB}}{\lambda}\right)\right]^{\lambda}&= \exp(u_{iBB})\left[\exp\left(\frac{u_{iRB|B}}{\lambda}\right)+1\right]^{\lambda}\\ &=\exp(u_{iBB}+\lambda I_{iB}) \end{align*}` where `\(I_{iB}\)` is called the .hi[inclusive value] and is given by: `\begin{align*} I_{iB}&=\ln\left(\exp\left(\frac{u_{iRB|B}}{\lambda}\right)+1\right) \end{align*}` Note: looks like `\(E\left(\text{utility}\right)\)` associated with a particular nest (minus Euler's constant) --- # Nested Logit Estimation 5 - Taking the estimates of `\(u_{iRB|B}\)` as given and calculating the inclusive value, we now estimate a second logit to get `\(\beta_B\)`: `\begin{align*} \ell_2&=\sum_i(d_{iB}=1)(u_{iBB}+\lambda I_{iB}-u_{iC})+\ln(1+\exp(u_{iBB}+\lambda I_{iB}-u_{iC})) \end{align*}` - Could do all this because log of the probabilities was additively separable. Consider the log likelihood contribution of someone who chose red bus: `\begin{align*} \ln(P_{iB}(\beta_{B},\beta_{RB},\gamma,\lambda))&+\ln(P_{iRB|B}(\beta_{RB},\gamma)) \end{align*}` - We get estimates of `\(\beta_{RB}\)` and `\(\gamma\)` only from the second part of log likelihood - Then we take these as given when estimating `\(\beta_{B}\)` and `\(\lambda\)` --- # The Nested Logit as a Dynamic Discrete Choice Model - Instead of having individuals know their full error, consider the case where the error is revealed in stages - First individuals choose whether or not to ride the bus and there is an extreme value error associated with both the bus and the car option - Individuals take into account that if they choose the bus option they will get to make a choice about which bus in the next period (option value) - With the errors in the second choice also distributed Type I extreme value, independent from each other, and independent from the errors in the first period, the expectation on the value of the second period decision is `\(\lambda I_{iB}\)` plus Euler's constant. --- # Proposition 1 (McFadden, 1978) Let `\(Y_{j}=e^{u_{j}}\)`. Suppose we have a function `\(G(Y_{1},...,Y_{{J}})\)` that maps from `\(R^{{J}}\)` into `\(R^1\)` If `\(G\)` satisfies: 1. `\(G\geq 0\)` 2. `\(G\)` is homogeneous of some degree `\(k\)` 3. `\(G\rightarrow \infty\)` as `\(Y_{j}\rightarrow \infty\)` for any `\(j\)` 4. Cross partial derivatives weakly alternate in sign, beginning with `\(G_{i}\geq 0\)` --- # Proposition 1 (Continued) then: `\begin{align*} F(u_1,...,u_\mathcal{J})&=\exp\left[-G(Y_1,....,Y_{J})\right] \end{align*}` is the cumulative distribution of a multivariate extreme value function and: `\begin{align*} P_{i}&=\frac{Y_{i}G_{i}}{G} \end{align*}` where `\(G_i\)` denotes the derivative of `\(G\)` with respect to `\(Y_i\)` --- # Logit from GEV - Another way of thinking about the last statement is that: `\begin{align*} P_i&=\frac{\partial \ln(G)}{\partial u_i} \end{align*}` - For the multinomial logit case, the `\(G\)` function is: `\begin{align*} G&=\sum_{j=1}^{{J}}\exp(u_j) \end{align*}` with the derivative of the log of this giving multinomial logit probabilities - But `\(\ln(G)\)` (plus Euler's constant) is .hi[also] expected utility - In fact, for all GEV models `\(\ln(G)\)` is expected utility! --- # Nested Logit from GEV - Suppose a nested logit model with two nests `\((F,NF)\)` and a no-purchase option `\(N\)` - The `\(G\)` function is then: `\begin{align*} G&=\left(\sum_{j\in F}\exp(u_j/\lambda_F)\right)^{\lambda_F}+\left(\sum_{j\in NF}\exp(u_j/\lambda_{NF})\right)^{\lambda_{NF}}+\exp(u_N) \end{align*}` - Differentiating `\(\ln(G)\)` (the expected utility function) with respect to `\(u_j\)` where `\(k\in F\)` yields the probability `\(k\)` is chosen: `\begin{align*} P_k&=\frac{\exp(u_k/\lambda_F)\left(\sum_{j\in F}\exp(u_j/\lambda_F)\right)^{\lambda_F-1}}{\left(\sum_{j\in F}\exp(u_j/\lambda_F)\right)^{\lambda_F}+\left(\sum_{j\in NF}\exp(u_j/\lambda_{NF})\right)^{\lambda_{NF}}+\exp(u_N)} \end{align*}` --- # Overlapping nests (Bresnahan et al., 1997) - We can also come up with more general nesting structures - Bresnahan, Stern, and Trajtenberg (1997) model 4 overlapping nests for computers: 1. Branded but not Frontier `\(\{B,NF\}\)` 2. Generic but Frontier `\(\{NB,F\}\)` 3. Branded and Frontier `\(\{B,F\}\)` 4. Generic but not Frontier `\(\{NB,NF\}\)` - Use the model to understand market power in PC sector in late 1980s - Overlapping nests explain coexistence of imitative entry and innovative investment --- # References .smaller[ Bresnahan, T. F, S. Stern, and M. Trajtenberg (1997). "Market Segmentation and the Sources of Rents from Innovation: Personal Computers in the Late 1980s". In: _The RAND Journal of Economics_ 28.0, pp. S17-S44. DOI: [10.2307/3087454](https://doi.org/10.2307%2F3087454). Cardell, N. S. (1997). "Variance Components Structures for the Extreme-Value and Logistic Distributions with Application to Models of Heterogeneity". In: _Econometric Theory_ 13.2, pp. 185-213. URL: [https://www.jstor.org/stable/3532724](https://www.jstor.org/stable/3532724). McFadden, D. (1978). "Modelling the Choice of Residential Location". In: _Spatial Interaction Theory and Planning Models_. Ed. by A. Karlqvist, L. Lundqvist, F. Snickers and J. W. Weibull. Amsterdam: North Holland, pp. 75-96. Train, K. (2009). _Discrete Choice Methods with Simulation_. 2nd ed. Cambridge; New York: Cambridge University Press. ISBN: 9780521766555. ]