Introduction Logit Generalized Extreme Value (GEV) Qualitative Response Models Michael R. Roberts Department of Finance The Wharton School University of Pennsylvania January 21, 2009 Michael R. Roberts Qualitative Response Models 1/59
The Choice Set & Choice Probabilities Introduction Identification Logit Aggregation Generalized Extreme Value (GEV) Regression Discrete Choice Framework The choice set must exhibit 3 characteristics: Alternatives must be mutually exclusive. 1 Choice set must be exhaustive. 2 # of alternatives must be finite. 3 1 and 2 can usually be satisfied with appropriate classifications. 3 is the defining feature of discrete choice models. Michael R. Roberts Qualitative Response Models 2/59
The Choice Set & Choice Probabilities Introduction Identification Logit Aggregation Generalized Extreme Value (GEV) Regression Random Utility Models Decision maker (agent, firm, person, etc.) i faces J alternatives (alts). Decision maker obtains a certain utility or profit from each alt Utility that agent i obtains from alt j is U ij . Agent chooses alt that provides highest utility U ij > U ik ∀ j � = k . Michael R. Roberts Qualitative Response Models 3/59
The Choice Set & Choice Probabilities Introduction Identification Logit Aggregation Generalized Extreme Value (GEV) Regression Empirical Implimenation Utility for agent i , alternative j , U ij , decomposed into two components: V ( x ij , s j , θ ) = Indirect Utility observed by researcher. Function of 1 alternative attributes x ij , agent attributes s i , and parameters θ . ε ij = unobservable to researcher factors affecting utility. Defined 2 relative to reseracher’s representation of choice situation (i.e., V ). Unobserved components, ε ij , are assumed random according to a distribution f ( ε ij ): Unobserved vector of errors across alternatives is described by joint density f ( ε i ): ε i = ( ε i 1 , ..., ε iJ ) ∼ f ( ε i ) Michael R. Roberts Qualitative Response Models 4/59
The Choice Set & Choice Probabilities Introduction Identification Logit Aggregation Generalized Extreme Value (GEV) Regression Probability of Agent’s Choice With f we can make probabilistic statements about agent’s choice P ij = Pr ( U ij > U ik , ∀ j � = k ) = Pr ( V ij + ε ij > V ik + ε ik , ∀ j � = k ) = Pr ( ε ij − ε ik > V ik − V ij , ∀ j � = k ) This last expression is a CDF � P ij = I ( ε ik − ε ij > V ij − V ik , ∀ j � = k ) f ( ε i ) d ε i ε This is a multidimensional ( J ) integral over the domain of ε i (e.g., R J ). Note : Independence implies f ( ε i ) = f ( ε i 1 ) × · · · × f ( ε iJ ). Different distributions = ⇒ different models. ε i i.i.d. extreme value = ⇒ logit (Closed Form) 1 ε i i.i.d generalized extreme value = ⇒ nested logit (Closed Form) 2 ε i multivariate normal = ⇒ probit 3 Michael R. Roberts Qualitative Response Models 5/59
The Choice Set & Choice Probabilities Introduction Identification Logit Aggregation Generalized Extreme Value (GEV) Regression Model Identification Only Differences in Utility Matter or The level of utility doesn’t matter . The choice probability is: P ij = Pr ( U ij > U ik , ∀ j � = k ) = Pr ( U ij − U ik > 0 , ∀ j � = k ) which depends only on the difference in utility not its absolute level. Similarly, P ij = Pr ( V ij + ε ij > V ik + ε ik , ∀ j � = k ) = Pr ( ε ij − ε ik > V ik − V ij , ∀ j � = k ) which also just depends on differences. In general, the only parameters that can be estimated are those that capture differences across alternatives. Michael R. Roberts Qualitative Response Models 6/59
The Choice Set & Choice Probabilities Introduction Identification Logit Aggregation Generalized Extreme Value (GEV) Regression Alternative-Specific Constants Assume V ij = x ij β + k j , ∀ j k j captures average effect on utility of all factors not in model (like intercept in linear regression). Including k j forces ε ij to have zero-mean “Only differences matter” = ⇒ only differences k j − k k matter, not absolute level of each. Normalize one of the constants to zero. With J alternatives, J − 1 constants can be estimated. Michael R. Roberts Qualitative Response Models 7/59
The Choice Set & Choice Probabilities Introduction Identification Logit Aggregation Generalized Extreme Value (GEV) Regression Sociodemographic Variables Consider choosing between commuting via bus or car U c = α T c + β M c + θ c Y + ε c = α T b + β M b + θ b Y + k b + ε c U b where T =commute time, M =commute cost, Y =income. We can only estimate differences: θ c − θ b (or vice versa). So, either normalize one θ to 0 U c = α T c + β M c + ε c U b = α T b + β M b + θ b Y + k b + ε c where θ b = θ b − θ c , or interact alternative-specific variables U c = α T c + β M c / Y + ε c = α T b + β M b / Y + θ b Y + k b + ε c U b Michael R. Roberts Qualitative Response Models 8/59
The Choice Set & Choice Probabilities Introduction Identification Logit Aggregation Generalized Extreme Value (GEV) Regression Independent Error Terms Recall choice probability is J -dimensional integral: � P ij = I ( ε ik − ε ij > V ij − V ik , ∀ j � = k ) f ( ε i ) d ε i ε Can write in terms of J − 1-dimensional integral P ij = Pr (˜ ε ijk > V ik − V ij , ∀ j � = k ) � = I (˜ ε ijk > V ij − V ik , ∀ j � = k ) g (˜ ε ij ) d ˜ ε ij , ε ˜ ˜ ε ijk = ε ij − ε ik = difference in errors for alt’s j and k 1 ˜ ε ij = (˜ ε ij 1 , ..., ˜ ε ijJ ) = J − 1-dim vector of error differences over all 2 alternatives except j . g (˜ ε ij ) is the J − 1-dimensional density of error differences. 3 Since choice prob’s can be expressed in terms of g (˜ ε ij ), one dimension of f ( ε i ) is not identified and must be normalized. Michael R. Roberts Qualitative Response Models 9/59
The Choice Set & Choice Probabilities Introduction Identification Logit Aggregation Generalized Extreme Value (GEV) Regression Scale of Utility is Irrelevant Multiplying by a positive constant doesn’t affect choice. Following two models are equivalent ∀ λ > 0: U 0 = V ij + ε ij , ∀ j ij U 1 = λ V ij + λε ij , ∀ j ij Address by normalizing the variance of error terms. i.i.d. errors : The two models are equiv 1 U 0 x ij β + ε 0 ij , V ( ε 0 ij ) = σ 2 = ij U 1 x ij ( β/σ ) + ε 1 ij , V ( ε 1 = ij ) = 1 ij Normalizing constant important when comparing coeffs across models (e.g., probit and logit) or datasets where scale varies. Heteroskedastic Errors : Normalize one of the variances = ⇒ 2 normalize variance of error difference. Correlated Errors : Normalize the variance of one of the error 3 differences. Michael R. Roberts Qualitative Response Models 10/59
The Choice Set & Choice Probabilities Introduction Identification Logit Aggregation Generalized Extreme Value (GEV) Regression Average Response Linear model f = ⇒ E [ f ( x )] = f ( E [ x ]) so we can insert aggregate or average values into model, e.g., ¯ y = α + β ¯ x Nonlinear models f : E [ f ( x )] � = f ( E [ x ]) Prob at avg utility can over- or under-estimate depending on where individual choice prob’s are (convex or concave portion of curve). Michael R. Roberts Qualitative Response Models 11/59
The Choice Set & Choice Probabilities Introduction Identification Logit Aggregation Generalized Extreme Value (GEV) Regression Average Marginal Effects Derivative is small a and b , derivative of avg. large. Solution : To get aggregate outcome, average indiv probs. To get at average marginal effect, avg indiv MEs (APE). Michael R. Roberts Qualitative Response Models 12/59
The Choice Set & Choice Probabilities Introduction Identification Logit Aggregation Generalized Extreme Value (GEV) Regression A Regression Perspective Consider binary case: two mutually exclusive outcomes captured by Y ∈ { 0 , 1 } Pr ( Y = 1 | x ) = F ( x , β ) Pr ( Y = 0 | x ) = 1 − F ( x , β ) Assume F ( x , β ) = x ′ β . E ( Y | x ) = 0 ∗ Pr ( Y = 0 | x ) + 1 ∗ Pr ( Y = 1 | x ) = Pr ( Y = 1 | x ) = x ′ β so the regression model is: y = E ( Y | x ) + ( y − E ( Y | x )) = x ′ β + ε Michael R. Roberts Qualitative Response Models 13/59
The Choice Set & Choice Probabilities Introduction Identification Logit Aggregation Generalized Extreme Value (GEV) Regression Two Problems 1 Heteroskedastic errors E ( ε 2 | x ) + ( E ( ε | x ) 2 ) V ( ε | x ) = E (( y − x ′ β ) 2 | x ) = E ( y 2 − 2 yx ′ β + ( x ′ β ) 2 | x ) = E ( y 2 | x ) − E ( y | x )2 x ′ β + ( x ′ β ) 2 = x ′ β − 2( x ′ β ) 2 + ( x ′ β ) 2 = x ′ β (1 − x ′ β ) = (FGLS solves this.) 2 Predicted values not constrained to [0 , 1] implies nonsense probabilities negative variances Michael R. Roberts Qualitative Response Models 14/59
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