Create Presentation
Download Presentation

Download Presentation
## Models with limited dependent variables

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Models with limited dependent variables**Doctoral Program 2006-2007 Katia Campo**3. Nested Logit Model**(K.Train, Ch.4, Franses and Paap Ch.5)**3. Nested Logit Model**• Choice between J>2 alternatives which can be grouped into subsets based on differences in substitution pattern • Example: Choice of transportation mode Car Transit Carpool Bus Train Car alone**3. Nested Logit Model**• Assumption cumulative distribution of error terms • Probability**3. Nested Logit Model**• Probability can be decomposed into 2 logit models (1) (2)**3. Nested Logit Model**• Link between Pni | Bk * Pn Bk (upper and lower model level) through inclusive value Ink • To comply with RUM k must be in the [0,1] interval • IIA within, not across nests**3. Nested Logit Model**Estimation • Joint estimation • Sequential estimation • Estimate lower model • Compute inclusive value • Estimate upper model with inclusive value as explanatory variable Disadvantages sequential estimation • Add fourth step, using the parameter estimates as starting values for joint estimation**3. Nested Logit Model**• Example: Location choices by French firms in Eastern and Western Europe • Dependent var.: probability of choosing location j Pj=P(πj> πk k≠j ) • Location choices are likely to have a nested structure (non-IIA) • First, select region (East or Western Europe) • Next, select country within region Disdier and Mayer (2004)**3. Nested Logit Model**• Example 1: Location choices by French firms in Eastern and Western Europe Location choice E.Eur W.Eur ……… ……… CN C1 CJ+1 CJ Disdier and Mayer (2004)**3. Nested Logit Model**• Location choices: Data • 1843 location decisions in Europe from 1980 to 1999 (official statistics) • 19 host countries (13 W.Eur, 6 E.Eur) Disdier and Mayer (2004)**3. Nested Logit Model**• Location choices: Data • NF French firms already located in the country • GDP GDP • GPP/CAP GDP per capita • DIST Distance France – host country • W Average wage per capita (manufacturing) • UNEMPL unemployment rate • EXCHR Exchange rate volatility • FREE Free country • PNFREE Partly free and not free country • PR1 Country with political rights rated 1 • PR2 Country with political rights rated 2 • PR345 Country with political rights rated 3,4,5 • PR67 Country with political rights rated 6,7 • LI Annual liberalization index • CLI Cumulative liberalization index • ASSOC =1 if an association agreement is signed Disdier and Mayer (2004)**3. Nested Logit Model**Disdier and Mayer (2004)**3. Nested Logit Model**Disdier and Mayer (2004)**3. Nested Logit Model**• Example 2: Shopping centre choice**3. Nested Logit Model**• Example 2: Shopping centre choice**3. Nested Logit Model**• Example 2: Shopping centre choice**3. Nested Logit Model**• (Purchase) incidence and choice models can be linked in the same way Include incl.value Purchase decision No purchase Purchase ... (See above: Bucklin & Gupta) Altern. J Altern.1**3. Nested Logit Model**• Other examples • Private label versus nationals brands • Same versus other brand • Fixed rate (low risk) versus variable rate (high risk) investments • Choice of transportation mode • ....**4. Probit Model**(K.Train, Ch.5 ; Franses and Paap, Ch.4-5)**4. Probit Model**• Based on the general RUM-model • Ass.: error terms are distributed normal with a mean vector of zero and covariance matrix • density function:**4. Probit Model**Logit or Probit? • Trade-off between tractability and flexibility • Closed-form expression of the integral for Logit, not for Probit models • Probit allows for random taste variation, can capture any substitution pattern, allows for correlated error terms and unequal error variances Dependent on the specifics of the choice situation**4. Probit Model**Estimation • Approximation of the multidimensional integral • Non-simulation procedures (see Kamakura 1989) Can usually only be applied to restricted cases and/or provide inaccurate estimations • Simulation procedures (see Geweke et al.1994, Train)**4. Probit Model**Simulation-based estimation(binary probit, CFS) • Step 1 • For each observation n=1, ..., N draw r ~ N(0,1), (r = 1, ......., R: repetitions) • Initialize y_count= 0, =mt (starting values) • Compute y*rn = xn mt + L r ; L= choleski factor (LL’= ) • Evaluate: y*rn >0 y_count= y_count+1 • Repeat R times Weeks (1997)**4. Probit Model**Simulation-based estimation(binary probit, CFS) • Step 2: calculate probabilities Pn| mt= y_count/R • Step 3: Form the simulated LL function SLL= n yn ln(Pn|mt)+(1-yn) ln(1-Pn|mt) • Step 4: Check convergence criteria (SLL(mt)- SLL(mt-1)) • Step 5: Update mt: mt+1 = mt + v • Step 6: Iterate (until convergence) Weeks (1997)**4. Probit Model**Simulation-based estimation: MNProbit • Based on same principles • More efficient simulation procedures (see Train) • Identification: normalization of level and scale • Re-express model in utility differences • Normalization of varcov matrix (see Train)**4. Probit Model**• Random taste variation • Model with random coefficients • E.g.: n~N(b,W) • Unj = b’xnj+ *’n xnj + nj • = b’xnj+ nj • nj : correlated error terms (dep.on xnj, see Train)**4. Probit Model**• Substitution patterns • Full covariance matrix (no parameter restrictions) unrestricted substitution patterns • Structured covariance matrix (restrictions on some covariance parameters) • the structure imposed on determines the substitution pattern and may allow to reduce the number of parameters to be estimated**4. Probit Model**• Example (Kamakura and Srivastava 1984): random utility components ni, nj are more (less) highly correlated when i and j are more (less) similar on important attributes (dij = weighted eucledian distance between i & j)**4. Probit Model**• Examples • Choice models at brand-size level: correlation between ≠ sizes of same brand (Chintagunta 1992) MNL model gives biased estimates of price elasticity**4. Probit Model**• Examples • Firm innovation (Harris et al. 2003) • Binary probit model for innovative status (innovation occurred or not) • Based on panel data correlation of innovative status over time: unobserved heterogeneity related to management ability and/or strategy**Model (2)-(4) account for unobserved heterogeneity (ρ) ->**superior results**4. Probit Model**• Examples • Dynamics of individual health (Contoyannis, Jones and Nigel 2004) • Binary probit model for health status (healthy or not) • Survey data for several years correlation over time (state dependence) + individual-specific (time-invariant) random coefficient**4. Probit Model**• Examples • Choice of transportation mode (Linardakis and Dellaportas 2003) Non-IIA substitution patterns**5. Ordered Logit Model**• Choice between J>2 ordered ‘alternatives’ • Ordinal dependent variable y = 1, 2, ... J, with rank(1) < rank(2) < ... < rank(J) • Example: • Purchase of 1, 2, ... J units • Evaluation on a J-point scale ranging from, e.g., ‘dislike very much’ to ‘like very much’**5. Ordered Logit Model**• Suppose yi* is a continous latent variable which • is a linear function of the explanatory variables yn* = Xn + n • and can be ‘mapped’ on an ordered multinomial variable as follows: yn= 1 if 0 < yn* 1 yn= j if j-1 < yn* j yn= J if J-1 < yn* J 0 < 1 < …. < j < … < J**5. Ordered Logit Model**Ordered logit (see above) • 0 , J and 0: set equal to zero**5. Ordered Logit Model**Interpretation of parameters (marginal effects)**5. Ordered Logit Model**Estimation: ML**5. Ordered Logit Model**• Disadvantages (Borooah 2002) • Assumption of equal slope k • Biased estimates if assumption of strictly ordered outcomes does not hold • treat outcomes as nonordered unless there are good reasons for imposing a ranking**5. Ordered Logit Model**Example: Effectiveness of better public transit as a way to reduce automobile congestion and air polution in urban areas • Research objective: develop and estimate models to measure how public transit affects automobile ownership and miles driven. • Data: Nationwide Personal Transportation Survey (42.033 hh): socio-demo’s, automobile ownership and use, public transportation avail. Kim and Kim (2004)**5. Ordered Logit Model**• Dependent variable ownership model = number of cars (k = 0, 1, 2, 3) ordinal variable • C*i = latent variable: automobile ownership propensity of hh i • Relation to observed automobile ownership: Ci=k if k-1 < ’xi + < k • P(Ci=k)=F(k- ’xi) - F(k-1 - ’xi) Kim and Kim (2004)**5. Ordered Logit Model**• Examples • Occupational outcome as a function of socio-demographic characteristics (Borooah) • Unskilled/semiskilled • Skilled manual/non-manual • Professional/managerial/technical • School performance (Sawkins 2002) • Grade 1 to 5 • Function of school, teacher and student characteristics • Level of insurance coverage**D.Heterogeneity**• Observed heterogeneity • Unobserved heterogeneity • Over decision makers • Random coefficients Models • E.g. Mixed Logit Model (see Train) • Over segments • Latent class estimation**D.Heterogeneity: Latent class est.**• Ass.: Consumers can be placed into a small number of – homogeneous - segments which differ in choice behavior ( response parameters) • Relative size of the segment s (s=1, 2, ..., M) is given by fs = exp(s) / s’exp(s’) Kamakura and Russell (1989)