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hi this is nick in this video i'm going to talk about the conditional logit model this is a model for describing how economic agents choose among a discrete set of unordered outcomes this is one of my favorite econometric frameworks i really like the way that it connects economic theory with econometric estimation it's organized around the concept of utility maximization so the model starts from the idea that there's some discrete set of choices that economic agents can make and those choices are going to be differentiated by a set of characteristics and different economic agents may rank choices differently so this is going to be a lot more general than the ordered logit and probit models we've already talked about examples of where the conditional logic models applied is to understand how consumers make choices in the market what causes different people to choose to purchase different cars or different brands of cereal or choose to live in different neighborhoods the idea of the conditional legit model is that people get utility from making these choices and we can describe the utility they get from individual characteristics comprising the choices that we see them making for instance in the case of cars the idea is that people get utility from the horsepower to weight ratio the number of seats whether it has a luxury interior with a lot of leather and wood and so on in the case of cereal people get utility from its nutritional content the amount of sugar that it has whether there's a toy included in the box and so on in the case of neighborhoods people get utility from the quality of local public schools from access to hiking and biking trails from opportunities for social interaction with their neighbors and so on and as we try to model the process of making utility maximizing choices in these settings the key features of the data that we're going to need as analysts are first we're going to have to observe the choices that economic agents make second we're going to have to see all of the alternative choices that they could have made but didn't and then we're going to have to describe the choices they made and the choices they didn't make according to the prices and characteristics that differentiate those choices and our objective here is to learn something about the decision makers preferences for choice characteristics now while this framework is often described in the context of utility maximization it's important to keep in mind that we don't have to be using this framework to model consumer behavior we can use it to model decisions made by any economic agents and that could include firms it could include governments and so on uh so we can think of this framework more broadly as being like a payoff function and we're seeing agents make choices that maximize their payoffs i'm going to use language organized around the concept of utility maximization but you can alternatively think of this as firms maximizing profits or government leaders making decisions that maximize social welfare so the random utility model is the model that we use to implement the conditional logic framework and it's often credited largely to daniel mcfadden who did a lot of work in developing the model starting in the early 1970s and was awarded the nobel prize for his uh contributions so the random utility model writes down the utility that an individual consumer gets from making a particular choice in terms of notation uh going to be indexing consumers by eye so we're going to have one through capital i consumers and we'll be indexing the choices that they're selecting among by j so one to capital j choices v i j then is going to represent consumer eyes utility from product j and that utility will depend on a set of characteristics the x's so here is characteristic one x one for product j characteristic two for product j characteristic three for product j and so on for instance if we're describing cars this might be the horsepower of car j the number of seats that it has and whether it's a luxury sedan or not and as as before we're going to be using just a big capital xj to represent the full set of those characteristics so notice that this is very much like the hedonic property value model that we've already talked about in ols regression in the sense that we're saying people get utility from a set of characteristics that collectively comprise the product and here those characteristics are the x's and now we're going to directly model the utility that they get and the coefficients in this case the betas represent marginal utilities the marginal utility that a consumer gets from increasing x1 a little bit versus x2 versus xk so beta the betas represent preferences for observed characteristics of the choices people are making when i say observed here what i mean is that characteristics that we're observing as the econometricians and that consumers are observing there may also be characteristics of products that consumers observe that we as the econometricians can't get data on and those unobserved characteristics will be captured by the econometric error term which is typically denoted by epsilon when we're working with the conditional logit model notice that this error term epsilon has both i and j subscripts so it's a consumer product specific preference shock okay so it represents idiosyncratic preferences by a particular consumer for a particular product epsilon can be interpreted a variety of different ways in the random utility model one interpretation is that it's going to capture characteristics of products that consumers observe but we as the econometricians don't so in the automobile example we might see horsepower and number of seats and whether it's a luxury sedan but we may not see higher resolution details like information about the audio system in the car and whether it connects automatically to a smartphone and things like that so those may be things that consumers observe in value but we as the econometricians don't that'll be captured by the error term another interpretation of the error term is that in some situations consumer preferences may truly be random not just from our perspective but from the consumer's perspective we have neurons that are firing in our brains and causing us to make certain types of choices and the neurons may fire differently on different days think about ice cream some days you know i'm in the mood for rocky road other days i'm in the mood for mint chip or strawberry i don't know why but presumably there's there's something that's causing me to prefer one over the other on a particular day and we might think of that as being random both from my own perspective and from the perspective of people analyzing data on my decisions a third interpretation of epsilon is that it represents optimization mistakes in other words people don't always know which choice is best for them when they're making their decisions consumers may not have access to full information when they make their decisions or they may face decisions that are very cognitively complicated decisions about choosing insurance plans or retirement savings plans for instance in other situations consumers may not have full information about the choices they're making i may not uh be fully aware of the negative health effects of ice cream when i choose to have a scoop of it or the naked or the positive health benefits of broccoli perhaps if i better understood how broccoli would improve my health i'd eat more of it and so incomplete information can be captured by the epsilon term as well as any kind of psychological biases that cause people to make choices that they might regret later on a final interpretation of epsilon is that it may capture any parametric misspecification of the shape of the utility function here i've written down this really simple linear additively separable specification where utility is equal to the sum of uh the weighted attributes of the product and there are no interactions between them so perhaps there's some functional form misspecification there perhaps there's an interaction between x1 and x2 that i've omitted and if so it'll be captured by the econometric error epsilon so just keep in mind there are a variety of different things that will be captured by that error now as we develop a framework for estimation we're going to have to make some assumptions about the set of choices that agents face the first assumption is that choices are mutually exclusive what i mean by that is that we have to define the choice set so that the decision makers are only capable of choosing one thing now at first pass that may seem like a restrictive assumption but in reality it's not for instance let's imagine there are three things you can choose a b and c i can define the choice set to be a or b or c or a and b together or b and c together or a and c together or a b and c together at the same time so i've got every possible way of combining these three objects defining my choice set the main thing is that for technical reasons i have to define the choice set in such a way so that consumers can only choose one thing and my point is that that's a technical assumption it's not something that's actually restrictive in practice the second assumption on the choice set is that it's exhaustive we have to define the choice set so that it includes all possible decisions that a consumer could make that again may seem restrictive at first but in fact it's more technical assumption rather than a serious restriction what i mean by that is that we can always define an outside option so for instance i could define the set of choices to be a or b or c or none of the above there may be a lot of different things that are conflated into none of the above you know d through k but the point is that i've kind of closed the model and exhaustively modeled all possible combinations by including this outside option the third restriction on the choice set which is the one that um is the most binding from an economic perspective is that the choice process has to be finite it cannot be the case that the outcome we're interested in modeling can be chosen continuously okay uh so the outcome of interest shouldn't be something like how much of my saving of my earnings am i going to save as opposed to spend that's a number that can be chosen continuously or what is the level of wages that a worker is paid that's another continuously varying variable if we're talking about things that vary continuously then the conditional logit model is simply the wrong framework the set of choices agents can be making fundamentally should be discrete okay so now let's think about moving toward estimation and as with our binary choice model and ordered model and truncated choice model we're going to be thinking about probabilities that agents make certain types of choices so i'm going to let p i j represent the probability that object j maximizes utility for individual i in other words that j is the object that provides the highest utility among all the alternatives in person i's choice set so that's equivalent to the probability that v i j the utility that i gets from j is greater than v i m for all of the objects m in the individual's choice set all m other than j this little symbol here looks kind of like an upside down a this means for all so this statement is true for all m other than j in other words that v i j is greater than the utility for all products in the market other than product j that's what this statement says now we can rewrite that probability just by plugging in this kind of compact notation for the utility that i gets from j x j beta plus epsilon i j i'll just plug that in right here for v i j to rewrite this probability as being x j beta plus epsilon i j is greater than x m beta plus epsilon i m for all m other than j now i'm just going to rearrange terms within this probability statement and put the two error terms to the left of the inequality and the terms that depend on data the x's to the right of the equality so just move epsilon to the left hand side move x j beta to the right hand side so that we rewrite it as the probability that epsilon i i j minus epsilon i m is greater than x m beta minus x j beta for all m other than j so now we have a probability that is based on our data and an assumption about a difference between two errors that is to translate this probability statement into a equation that we can use for maximum likelihood we have to make an assumption about the distribution of the epsilons so that we can say something about the likelihood that this term is going to be less than this difference and in the conditional logit model the standard distributional assumption that's made on the epsilon terms is that all of the epsilons are independent draws from an identical type 1 extreme value distribution now this is a new statistical distribution we haven't talked about it before so let me tell you a little bit about the type 1 extreme value distribution it has a mean of 0.577 its standard deviation is 1.28 it looks a little bit like a normal but it's skewed as you can see in this figure on the right it does have this kind of bell shaped component to it but it's got a longer right tail than the left tail it turns out that this is a natural distribution for modeling the maximum that you would get if you were to take a series of random draws from different distributions then take the maximum of those random draws now i can't read your minds but i can tell you what i was thinking the first time that someone told me that the conditional logit model depends on this uh odd seeming type one extreme value distribution with these properties i was wondering what on earth would have led somebody to choose this particular distributional assumption for the epsilon i mean it's hard to see the intuition for why this statistical distribution would have anything to do with utility maximization well the reason why daniel mcfadden first used this distributional assumption in the conditional logit model is for computational convenience it turns out that if we're willing to make this type one extreme value distribution assumption on the epsilons then the difference between two randomly drawn epsilons follows a logistic distribution and that in turn yields a very simple analytical expression for the probability that a particular consumer chooses a particular object from their choice set so let's come back to these choice probabilities right so we need to we we've made a statistical assumption about the distribution of the epsilons guaranteeing that this difference follows a logistic distribution now we can use that logistic distribution to get a closed form expression for the probability that consumer i chooses product j that is that product j is the utility maximizing product within the consumer's choice set and here's what that expression looks like it's kind of similar to what we originally had in the binary logic framework what we have here is an exponential transformation over the observable part of the utility function x j beta okay this term right here beta 1 x 1 j plus beta 2 x 2 j etc through beta k x k j we're taking an exponential transformation of that and then dividing it by the sum of those same exponential transformations for every single one of the j objects in the choice set so we're saying that the probability that a consumer chooses product j is going to be proportional to uh the relative utility the consumer receives from that product compared to all of the other products in the individual's choice set as the utility from j increases relative to other products so does the probability that i will choose j based on uh this closed form expression and so um what the conditional logic model is going to end up doing when we translate this expression into a maximum likelihood estimator is the maximum likelihood estimator is basically going to choose the betas for this expression for the predicted probability that i chooses j so that these predicted probabilities match market shares that we see in the data if we see that 25 of consumers buy grape nuts breakfast cereal 15 by cheerios and 30 by captain crunch the model is going to choose the betas so that consumers relative preferences for price compared to sugar content compared to marketing strategy yield a set of predicted probabilities that match the market shares seen in the data and once we have the betas which describe marginal utilities you can go back to results from intermediate micro theory that allow us to take the parametric form for utility and use it to calculate demand curves for product characteristics welfare measures measures of consumer surplus that is uh the the welfare to people if we were to change the price of a product increasing it or decreasing it or change the levels of characteristics and we can calculate uh measures of consumer surplus from counter factual policies policies that would remove products from the market that would add new products to the market or that would change the characteristics of products for instance we could calculate how much consumer surplus would increase or decrease if the government were to uh regulate the maximum amount of sugar that were allowed to be contained in breakfast cereals those are the kinds of really interesting uh and policy relevant economic questions that we can ask with the conditional logit model and in the next video i'll go through a detailed example of this in stata