{"id":1144,"date":"2019-11-08T17:50:08","date_gmt":"2019-11-09T00:50:08","guid":{"rendered":"http:\/\/www.zhuoyao.net\/?p=1144"},"modified":"2022-11-22T06:53:46","modified_gmt":"2022-11-22T06:53:46","slug":"random-utility-model-and-the-multinomial-logit-model","status":"publish","type":"post","link":"https:\/\/zhuoyao.net\/index.php\/2019\/11\/08\/random-utility-model-and-the-multinomial-logit-model\/","title":{"rendered":"Random utility model and the multinomial logit model"},"content":{"rendered":"\n

Random utility model<\/h2>\n\n\n\n

The utility for alternative l<\/em> is written as: U<\/em>l<\/em>=V<\/em>l<\/em>+\u03f5<\/em>l<\/em> where V<\/em>l<\/em> is a function of some observable covariates and unknown parameters to be estimated, and \u03f5<\/em>l<\/em> is a random deviate which contains all the unobserved determinants of the utility. Alternative l<\/em> is therefore chosen if \u03f5<\/em>j<\/em><(V<\/em>l<\/em>\u2212V<\/em>j<\/em>)+\u03f5<\/em>l<\/em>\u2200j<\/em>\u2260l<\/em><\/p>\n\n\n\n

and the probability of choosing this alternative is then:P(\u03f5<\/em>1<V<\/em>l<\/em>\u2212V<\/em>1+\u03f5<\/em>l<\/em>,\u03f5<\/em>2<V<\/em>l<\/em>\u2212V<\/em>2+\u03f5<\/em>l<\/em>,…,\u03f5<\/em>J<\/em><V<\/em>l<\/em>\u2212V<\/em>J<\/em>+\u03f5<\/em>l<\/em>).<\/p>\n\n\n\n

Denoting F<\/em>\u2212l<\/em> the cumulative density function of all the \u03f5<\/em>s except \u03f5<\/em>l<\/em><\/p>\n\n\n\n

, this probability is:(Pl<\/em>\u2223\u03f5<\/em>l<\/em>)=F<\/em>\u2212l<\/em>(V<\/em>l<\/em>\u2212V<\/em>1+\u03f5<\/em>l<\/em>,…,V<\/em>l<\/em>\u2212V<\/em>J<\/em>+\u03f5<\/em>l<\/em>).<\/p>\n\n\n\n

Note that this probability is conditional on the value of \u03f5<\/em>l<\/em>. The unconditional probability (which depends only on \u03b2<\/em>\n and on the value of the observed explanatory variables) is obtained by \nintegrating out the conditional probability using the marginal density \nof \u03f5<\/em>l<\/em>, denoted f<\/em>l<\/em><\/p>\n\n\n\n

\ud83d\ude1bl<\/em>=\u222bF<\/em>\u2212l<\/em>(V<\/em>l<\/em>\u2212V<\/em>1+\u03f5<\/em>l<\/em>,…,V<\/em>l<\/em>\u2212V<\/em>J<\/em>)+\u03f5<\/em>l<\/em>)f<\/em>l<\/em>(\u03f5<\/em>l<\/em>)d<\/em>\u03f5<\/em>l<\/em>.<\/p>\n\n\n\n

The conditional probability is an integral of dimension J<\/em>\u22121<\/p>\n\n\n\n

and the computation of the unconditional probability adds on more dimension of integration.<\/p>\n\n\n\n

The distribution of the error terms<\/h2>\n\n\n\n

The multinomial logit model (McFadden 1974) is a special case of the model developed in the previous section. It is based on three hypothesis.<\/p>\n\n\n\n

The first hypothesis is the independence of the errors. In this case,\n the univariate distribution of the errors can be used, which leads to \nthe following conditional and unconditional probabilities:(Pl<\/em>\u2223\u03f5<\/em>l<\/em>)=\u220fj<\/em>\u2260l<\/em>F<\/em>j<\/em>(V<\/em>l<\/em>\u2212V<\/em>j<\/em>+\u03f5<\/em>l<\/em>) and Pl<\/em>=\u222b\u220fj<\/em>\u2260l<\/em>F<\/em>j<\/em>(V<\/em>l<\/em>\u2212V<\/em>j<\/em>+\u03f5<\/em>l<\/em>)f<\/em>l<\/em>(\u03f5<\/em>l<\/em>)d<\/em>\u03f5<\/em>l<\/em>,<\/p>\n\n\n\n

which means that the conditional probability is the product of J<\/em>\u22121<\/p>\n\n\n\n

\n univariate cumulative density functions and the evaluation of only a \none-dimensional integral is required to compute the unconditional \nprobability.<\/p>\n\n\n\n

The second hypothesis is that each \u03f5<\/em><\/p>\n\n\n\n

follows a Gumbel distribution, whose density and probability functions are respectively:f<\/em>(z<\/em>)=1\u03b8<\/em>e<\/em>\u2212z<\/em>\u2212\u03bc<\/em>\u03b8<\/em>e<\/em>\u2212e<\/em>\u2212z<\/em>\u2212\u03bc<\/em>\u03b8<\/em> and F<\/em>(z<\/em>)=\u222bz<\/em>\u2212\u221ef<\/em>(t<\/em>)d<\/em>t<\/em>=e<\/em>\u2212e<\/em>\u2212z<\/em>\u2212\u03bc<\/em>\u03b8<\/em>,<\/p>\n\n\n\n

where \u03bc<\/em> is the location parameter and \u03b8<\/em> the scale parameter. The first two moments of the Gumbel distribution are E(z<\/em>)=\u03bc<\/em>+\u03b8<\/em>\u03b3<\/em>, where \u03b3<\/em> is the Euler-Mascheroni constant (\u22480.577) and V(z<\/em>)=\u03c0<\/em>26\u03b8<\/em>2. The mean of \u03f5<\/em>j<\/em> is not identified if V<\/em>j<\/em> contains an intercept. We can then, without loss of generality suppose that \u03bc<\/em>j<\/em>=0,\u2200j<\/em>. Moreover, the overall scale of utility is not identified. Therefore, only J<\/em>\u22121 scale parameters may be identified, and a natural choice of normalization is to impose that one of the \u03b8<\/em>j<\/em><\/p>\n\n\n\n

is equal to 1.<\/p>\n\n\n\n

The last hypothesis is that the errors are identically distributed. \nAs the location parameter is not identified for any error term, this \nhypothesis is essentially an homoscedasticity hypothesis, which means \nthat the scale parameter of the Gumbel distribution is the same for all \nthe alternatives. As one of them has been previously set to 1, we can \ntherefore suppose that, without loss of generality, \u03b8<\/em>j<\/em>=1,\u2200j<\/em>\u22081…J<\/em><\/p>\n\n\n\n

. The conditional and unconditional probabilities then further simplify to:(Pl<\/em>\u2223\u03f5<\/em>l<\/em>)=\u220fj<\/em>\u2260l<\/em>e<\/em>\u2212e<\/em>\u2212(V<\/em>l<\/em>\u2212V<\/em>j<\/em>+\u03f5<\/em>l<\/em>) and Pl<\/em>=\u222b+\u221e\u2212\u221e\u220fj<\/em>\u2260l<\/em>e<\/em>\u2212e<\/em>\u2212(V<\/em>l<\/em>\u2212V<\/em>j<\/em>+t<\/em>)e<\/em>\u2212t<\/em>e<\/em>\u2212e<\/em>\u2212t<\/em>d<\/em>t<\/em>.<\/p>\n\n\n\n

The probabilities have then very simple, closed forms, which \ncorrespond to the logit transformation of the deterministic part of the \nutility.P<\/em>l<\/em>=e<\/em>V<\/em>l<\/em>\u2211J<\/em>j<\/em>=1e<\/em>V<\/em>j<\/em>.<\/p>\n\n\n\n

IIA property<\/h2>\n\n\n\n

If we consider the probabilities of choice for two alternatives l<\/em> and m<\/em>, we have P<\/em>l<\/em>=e<\/em>V<\/em>l<\/em>\u2211j<\/em>e<\/em>V<\/em>j<\/em> and P<\/em>m<\/em>=e<\/em>V<\/em>m<\/em>\u2211j<\/em>e<\/em>V<\/em>j<\/em><\/p>\n\n\n\n

. The ratio of these two probabilities is:P<\/em>l<\/em>P<\/em>m<\/em>=e<\/em>V<\/em>l<\/em>e<\/em>V<\/em>m<\/em>=e<\/em>V<\/em>l<\/em>\u2212V<\/em>m<\/em>.<\/p>\n\n\n\n

This probability ratio for the two alternatives depends only on the \ncharacteristics of these two alternatives and not on those of other \nalternatives. This is called the IIA property (for independence of \nirrelevant alternatives). IIA relies on the hypothesis that the errors \nare identical and independent. It is not a problem by itself and may \neven be considered as a useful feature for a well specified model. \nHowever, this hypothesis may be in practice violated, especially if some\n important variables are omitted.<\/p>\n\n\n\n

Interpretation<\/h2>\n\n\n\n

In a linear model, the coefficients are the marginal effects of the \nexplanatory variables on the explained variable. This is not the case \nfor the multinomial logit model. However, meaningful results can be \nobtained using relevant transformations of the coefficients.<\/p>\n\n\n\n

Marginal effects<\/h3>\n\n\n\n

The marginal effects are the derivatives of the probabilities with \nrespect to the covariates, which can be be choice situation-specific (z<\/em>i<\/em>) or alternative specific (x<\/em>i<\/em>j<\/em><\/p>\n\n\n\n

):\u2202P<\/em>i<\/em>l<\/em>\u2202z<\/em>i<\/em>\u2202P<\/em>i<\/em>l<\/em>\u2202x<\/em>i<\/em>l<\/em>\u2202P<\/em>i<\/em>l<\/em>\u2202x<\/em>i<\/em>k<\/em>===P<\/em>i<\/em>l<\/em>(\u03b2<\/em>l<\/em>\u2212\u2211j<\/em>P<\/em>i<\/em>j<\/em>\u03b2<\/em>j<\/em>)\u03b3<\/em>P<\/em>i<\/em>l<\/em>(1\u2212P<\/em>i<\/em>l<\/em>)\u2212\u03b3<\/em>P<\/em>i<\/em>l<\/em>P<\/em>i<\/em>k<\/em>.<\/p>\n\n\n\n

    \n
  • For a choice situation specific variable, the sign of the \nmarginal effect is not necessarily the sign of the coefficient. \nActually, the sign of the marginal effect is given by (\u03b2<\/em>l<\/em>\u2212\u2211j<\/em>P<\/em>i<\/em>j<\/em>\u03b2<\/em>j<\/em>)<\/li>\n<\/ul>\n\n\n\n

    , which is positive if the coefficient for alternative l<\/em><\/p>\n\n\n\n

      \n
    • \n is greater than a weighted average of the coefficients for all the \nalternatives, the weights being the probabilities of choosing the \nalternatives. In this case, the sign of the marginal effect can be \nestablished with no ambiguity only for the alternatives with the lowest \nand the greatest coefficients.<\/li>\n\n\n\n
    • For an alternative-specific variable, the sign of the coefficient\n can be directly interpreted. The marginal effect is obtained by \nmultiplying the coefficient by the product of two probabilities which is\n at most 0.25. The rule of thumb is therefore to divide the coefficient \nby 4 in order to have an upper bound of the marginal effect.<\/li>\n<\/ul>\n\n\n\n

      Note that the last equation can be rewritten: dP<\/em>i<\/em>l<\/em>\/P<\/em>i<\/em>l<\/em>dx<\/em>i<\/em>k<\/em>=\u2212\u03b3<\/em>P<\/em>i<\/em>k<\/em>. Therefore, when a characteristic of alternative k<\/em> changes, the relative change of the probabilities for every alternatives except k<\/em><\/p>\n\n\n\n

      are the same, which is a consequence of the IIA property.<\/p>\n\n\n\n

      Marginal rates of substitution<\/h3>\n\n\n\n

      Coefficients are marginal utilities, which cannot be interpreted. \nHowever, ratios of coefficients are marginal rates of substitution. For \nexample, if the observable part of utility is: V<\/em>=\u03b2<\/em>o<\/em>+\u03b2<\/em>1x<\/em>1+\u03b2<\/em>x<\/em>2+\u03b2<\/em>x<\/em>3, join variations of x<\/em>1 and x<\/em>2 which ensure the same level of utility are such that: d<\/em>V<\/em>=\u03b2<\/em>1d<\/em>x<\/em>1+\u03b2<\/em>2d<\/em>x<\/em>2=0<\/p>\n\n\n\n

      so that:\u2212d<\/em>x<\/em>2d<\/em>x<\/em>1\u2223d<\/em>V<\/em>=0=\u03b2<\/em>1\u03b2<\/em>2.<\/p>\n\n\n\n

      For example, if x<\/em>2 is transport cost (in $), x<\/em>1 transport time (in hours), \u03b2<\/em>1=1.5 and \u03b2<\/em>2=0.2, \u03b2<\/em>1\u03b2<\/em>2=30<\/p>\n\n\n\n

      \n is the marginal rate of substitution of time in terms of $ and the \nvalue of 30 means that to reduce the travel time of one hour, the \nindividual is willing to pay at most 30$ more. Stated more simply, time \nvalue is 30$ per hour.<\/p>\n\n\n\n

      Consumer\u2019s surplus<\/h3>\n\n\n\n

      Consumer\u2019s surplus has a very simple expression for multinomial logit models, which was first derived by Small and Rosen (1981). The level of utility attained by an individual is U<\/em>j<\/em>=V<\/em>j<\/em>+\u03f5<\/em>j<\/em>, j<\/em> being the chosen alternative. The expected utility, from the searcher\u2019s point of view is then: E(maxj<\/em>U<\/em>j<\/em>)<\/p>\n\n\n\n

      ,\n where the expectation is taken over the values of all the error terms. \nIts expression is simply, up to an additive unknown constant, the log of\n the denominator of the logit probabilities, often called the \u201clog-sum\u201d:E(U<\/em>)=ln\u2211j<\/em>=1J<\/em>e<\/em>V<\/em>j<\/em>+C<\/em>.<\/p>\n\n\n\n

      If the marginal utility of income (\u03b1<\/em>) is known and constant, the expected surplus is simply E(U<\/em>)\u03b1<\/em><\/p>\n\n\n\n

      .<\/p>\n\n\n\n

      Application<\/h2>\n\n\n\n

      Random utility models are fitted using the mlogit<\/code> function. Basically, only two arguments are mandatory, formula<\/code> and data<\/code>, if a mlogit.data<\/code> object (and not an ordinary data.frame<\/code>) is provided.<\/p>\n\n\n\n

      ModeCanada<\/h3>\n\n\n\n

      We first use the ModeCanada<\/code> data set, which was already coerced to a mlogit.data<\/code> object (called MC<\/code>) in the previous section. The same model can then be estimated using as data<\/code> argument this mlogit.data<\/code> object:<\/p>\n\n\n\n

      library(\"mlogit\")\ndata(\"ModeCanada\", package = \"mlogit\")\nMC <- mlogit.data(ModeCanada, subset = noalt == 4, chid.var = \"case\",\n                  alt.var = \"alt\", drop.index = TRUE)\nml.MC1 <- mlogit(choice ~ cost + freq + ovt | income | ivt, MC)<\/code><\/pre>\n\n\n\n
      or a data.frame<\/code>. In this latter case, further arguments that will be passed to mlogit.data<\/code> should be indicated:<\/p>\n\n\n\n
      ml.MC1b <- mlogit(choice ~ cost + freq + ovt | income | ivt, ModeCanada,\nsubset = noalt == 4, alt.var = \"alt\", chid.var = \"case\")<\/code><\/pre>\n\n\n\n
      mlogit<\/code> provides two further useful arguments:<\/p>\n\n\n\n