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Comparisons between MDCEV model and EMDC model

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TWayne
Posts: 4
Joined: 04 Dec 2023, 06:49

Comparisons between MDCEV model and EMDC model

Post by TWayne »

Hi authors,

I have been thinking about the MDCEV model and the EMDC model. If I understand correctly, the EMDC model is a special case of the MDCEV model (or should I say it is MDCEV model with complementarity/substitution). That is to say, if I restrict the complementarity/substitution parameters delta's in an EMDC model, the estimation results/prediction results would be the same as the MDCEV model. Thus, I tried to make some changes to the sample code provided on the website and compare. However, I found that they actually generated very different estimates. I provided my code here.

I am wondering:

1. Why might these two models generate different estimates even when the delta's are fixed in the emdc model? I also tried to do a prediction using the same dataset to see whether the predicted outcome is the same as the observation. Seems that these two models give different prediction outcomes too.
2. I did realize that the MDCEV model needs to specify alpha, while the EMDC model does not need alpha information. Why is that? Could this cause the inconsistency?
3. Related to the above point, I realized that in the MDCEV model, if I do not fix alpha parameter, it does not produce s.e.

MDCEV model code:

Code: Select all

#################################MDCEV model with outside good


# ################################################################# #
#### LOAD LIBRARY AND DEFINE CORE SETTINGS                       ####
# ################################################################# #

### Clear memory
rm(list = ls())

### Load Apollo library
library(apollo)

### Initialise code
apollo_initialise()

### Set core controls
apollo_control = list(
  modelName  ="MDCEV_with_outside_good",
  modelDescr ="MDCEV model on time use data, alpha-gamma profile with outside good and socio-demographics",
  indivID    ="indivID", 
  outputDirectory = "output"
)

# ################################################################# #
#### LOAD DATA AND APPLY ANY TRANSFORMATIONS                     ####
# ################################################################# #

### Loading data from package
### if data is to be loaded from a file (e.g. called data.csv), 
### the code would be: database = read.csv("data.csv",header=TRUE)
database = apollo_timeUseData
### for data dictionary, use ?apollo_timeUseData

### Create consumption variables for combined activities
database$t_outside = rowSums(database[,c("t_a01", "t_a06", "t_a10", "t_a11", "t_a12")]) # outside good: time spent at home and travelling
database$t_leisure = rowSums(database[,c("t_a07", "t_a08", "t_a09")])

# ################################################################# #
#### DEFINE MODEL PARAMETERS                                     ####
# ################################################################# #

### Vector of parameters, including any that are kept fixed in estimation
apollo_beta = c(alpha_base         = -50,
                gamma_work         = 10.425,
                gamma_school       = 9.955,
                gamma_shopping     = 2.065,
                gamma_private      = 4.419,
                gamma_leisure      = 6.927,
                delta_work         = -3.372,
                delta_school       = -5.250,
                delta_shopping     = -3.663,
                delta_private      = -3.973,
                delta_leisure      = -3.3,
                delta_work_FT      = 0.708,
                delta_work_wknd    = -1.6,
                delta_school_young = 0.948,
                delta_leisure_wknd = 0.155,
                sig                = 1.944)

### Vector with names (in quotes) of parameters to be kept fixed at their starting value in apollo_beta, use apollo_beta_fixed = c() if none
apollo_fixed = c("sig" , "alpha_base")

# ################################################################# #
#### GROUP AND VALIDATE INPUTS                                   ####
# ################################################################# #

apollo_inputs = apollo_validateInputs()

# ################################################################# #
#### DEFINE MODEL AND LIKELIHOOD FUNCTION                        ####
# ################################################################# #

apollo_probabilities=function(apollo_beta, apollo_inputs, functionality="estimate"){
  
  ### Attach inputs and detach after function exit
  apollo_attach(apollo_beta, apollo_inputs)
  on.exit(apollo_detach(apollo_beta, apollo_inputs))
  
  ### Create list of probabilities P
  P = list()
  
  ### Define individual alternatives
  alternatives  = c("outside",
                    "work",
                    "school",
                    "shopping",
                    "private", 
                    "leisure")
  
  ### Define availabilities
  avail = list(outside  = 1, 
               work     = 1, 
               school   = 1, 
               shopping = 1, 
               private  = 1,
               leisure  = 1)
  
  ### Define continuous consumption for individual alternatives
  continuousChoice = list(outside  = t_outside/60,
                          work     = t_a02/60,
                          school   = t_a03/60,
                          shopping = t_a04/60,
                          private  = t_a05/60,
                          leisure  = t_leisure/60)
  
  ### Define utilities for individual alternatives
  V = list()
  V[["outside"]]  = 0
  V[["work"]]     = delta_work     + delta_work_FT * occ_full_time + delta_work_wknd * weekend
  V[["school"]]   = delta_school   + delta_school_young * (age<=30)
  V[["shopping"]] = delta_shopping
  V[["private"]]  = delta_private
  V[["leisure"]]  = delta_leisure  + delta_leisure_wknd*weekend
  
  ### Define alpha parameters
  alpha = list(outside  = 1 /(1 + exp(-alpha_base)), 
               work     = 1 /(1 + exp(-alpha_base)), 
               school   = 1 /(1 + exp(-alpha_base)), 
               shopping = 1 /(1 + exp(-alpha_base)), 
               private  = 1 /(1 + exp(-alpha_base)),
               leisure  = 1 /(1 + exp(-alpha_base)))
  
  ### Define gamma parameters
  gamma = list(work     = gamma_work,    
               school   = gamma_school,
               shopping = gamma_shopping,
               private  = gamma_private,
               leisure  = gamma_leisure)
  
  ### Define costs for individual alternatives
  cost = list(outside  = 1, 
              work     = 1, 
              school   = 1, 
              shopping = 1, 
              private  = 1,
              leisure  = 1)
  
  ### Define settings for MDCEV model
  mdcev_settings <- list(alternatives      = alternatives,
                         avail             = avail,
                         continuousChoice  = continuousChoice,
                         utilities         = V,
                         alpha             = alpha,
                         gamma             = gamma, 
                         sigma             = sig, 
                         cost              = cost,
                         budget            = 24)
  
  ### Compute probabilities using MDCEV model
  P[["model"]] = apollo_mdcev(mdcev_settings, functionality)
  
  ### Take product across observation for same individual
  P = apollo_panelProd(P, apollo_inputs, functionality)
  
  ### Prepare and return outputs of function
  P = apollo_prepareProb(P, apollo_inputs, functionality)
  return(P)
}

# ################################################################# #
#### MODEL ESTIMATION                                            ####
# ################################################################# #

model = apollo_estimate(apollo_beta, apollo_fixed, apollo_probabilities, apollo_inputs)

# ################################################################# #
#### MODEL OUTPUTS                                               ####
# ################################################################# #

# ----------------------------------------------------------------- #
#---- FORMATTED OUTPUT (TO SCREEN)                               ----
# ----------------------------------------------------------------- #

apollo_modelOutput(model)

# ----------------------------------------------------------------- #
#---- FORMATTED OUTPUT (TO FILE, using model name)               ----
# ----------------------------------------------------------------- #

apollo_saveOutput(model)





# ################################################################# #
#### PREDICTION                                                  ####
# ################################################################# #
#############################Use the model estimates to predict and see whether it is close to what is observed



model <- apollo_loadModel(apollo_control$modelName)
apollo_inputs <- apollo_validateInputs(database=database)
apollo_inputs$apollo_control$nCores <- 4
pred <- apollo_prediction(model, apollo_probabilities, apollo_inputs)


XObs <- cbind(work     =     database$t_a02/60,
              school   =     database$t_a03/60,
              shopping =     database$t_a04/60,
              private  =     database$t_a05/60,
              leisure  = database$t_leisure/60)
XPre <- pred[,4:8]

round(sqrt(colMeans((XObs - XPre)^2)),2) # RMSE per product: 3.61 0.81 1.97 1.85 3.73
round(sqrt(mean((colSums(XObs) - colSums(XPre))^2)),2) # RMSE for sample: 975.87
Here is the EMDC model code：

Code: Select all

# ################################################################# #
#### LOAD LIBRARY AND DEFINE CORE SETTINGS                       ####
# ################################################################# #

### Clear memory and initialise
rm(list = ls())
library(apollo)
apollo_initialise()

### Set core controls
apollo_control = list(
  modelName  ="eMDC_with_budget",
  modelDescr ="Extended MDC with complementarity and substitution, with observed budget and socio-demographics",
  indivID    ="indivID",
  outputDirectory="output"
)

# ################################################################# #
#### LOAD DATA AND APPLY ANY TRANSFORMATIONS                     ####
# ################################################################# #

### Load data from within the Apollo package
database = apollo_timeUseData

### Create consumption variables for combined activities
# outside good: time spent at home and travelling
database$t_outside = rowSums(database[,c("t_a01", "t_a06", "t_a10", "t_a11", "t_a12")]) 
database$t_leisure = rowSums(database[,c("t_a07", "t_a08", "t_a09")])

# ### Randomly split dataset into estimation (70%) and validation (30%)
# set.seed(1)
# database$validation <- runif(nrow(database))>0.7
# dbVal    <- database[ database$validation,] # validation sample
# database <- database[!database$validation,] # estimation sample

# ################################################################# #
#### DEFINE MODEL PARAMETERS                                     ####
# ################################################################# #

### Parameters starting values c(name1=value1, name2=value2, ...)
apollo_beta = c(  sigma = 0.71,   ###Here I remove aFemale
                  # Satiation
                  gWork    = 10.425, 
                  gSchool  = 9.955, 
                  gShopping= 2.065, 
                  gPrivate = 4.419, 
                  gLeisure = 6.927, 
                  # Base utility
                  bWork     =-3.372, bSchool   =-5.250, 
                  bShopping =-3.663, bPrivate  =-3.973, 
                  bLeisure  =-3.300, bWork_FT  = 0.708, 
                  bWork_wknd=-1.600, bSchool_young= 0.948, 
                  bLeisure_wknd= 0.155, 
                  # Compl/subst
                  # dWorkScho=-0.008, dWorkShop= 0.000, 
                  # dWorkPriv= 0.000, dWorkLeis= 0.000, 
                  # dSchoShop= 0.000, dSchoPriv= 0.000, 
                  # dSchoLeis= 0.000, dShopPriv= 0.010, 
                  # dShopLeis= 0.012, dPrivLeis= 0.012
                  
                  dWorkScho=-0.000, dWorkShop= 0.000, 
                  dWorkPriv= 0.000, dWorkLeis= 0.000, 
                  dSchoShop= 0.000, dSchoPriv= 0.000, 
                  dSchoLeis= 0.000, dShopPriv= 0.000, 
                  dShopLeis= 0.000, dPrivLeis= 0.000)



### Names of fixed parameters
apollo_fixed = c('dWorkShop', 'dWorkPriv', 'dSchoShop', 
                 'dSchoPriv', 'dSchoLeis', 'dWorkLeis'   ,
                 "dWorkScho" , "dShopPriv" , "dShopLeis" , "dPrivLeis")  #############Here I also fix sigma to the true value 0.71

# ################################################################# #
#### GROUP AND VALIDATE INPUTS                                   ####
# ################################################################# #

apollo_inputs = apollo_validateInputs()

# ################################################################# #
#### DEFINE MODEL AND LIKELIHOOD FUNCTION                        ####
# ################################################################# #

apollo_probabilities=function(apollo_beta, apollo_inputs, 
                              functionality="estimate"){
    
  ### Initialise
  apollo_attach(apollo_beta, apollo_inputs)
  on.exit(apollo_detach(apollo_beta, apollo_inputs))
  P = list()
  
  ### Prepare Inputs
  alts  = c("work", "school", "shopping", "private", "leisure")
  nAlt = length(alts)
  ones = setNames(as.list(rep(1, nAlt)), alts)
  continuousChoice = list(work     =     t_a02/60,
                          school   =     t_a03/60,
                          shopping =     t_a04/60,
                          private  =     t_a05/60,
                          leisure  = t_leisure/60)
  utilities = list(
    work     = bWork     + bWork_FT*occ_full_time + bWork_wknd*weekend,
    school   = bSchool   + bSchool_young*(age<=30), 
    shopping = bShopping, 
    private  = bPrivate, 
    leisure  = bLeisure  + bLeisure_wknd*weekend
  )
  gamma = list(work     = gWork,    
               school   = gSchool,
               shopping = gShopping,
               private  = gPrivate,
               leisure  = gLeisure)
  delta <- c(0,                 0,         0,         0, 0,
             dWorkScho,         0,         0,         0, 0,
             dWorkShop, dSchoShop,         0,         0, 0,
             dWorkPriv, dSchoPriv, dShopPriv,         0, 0,
             dWorkLeis, dSchoLeis, dShopLeis, dPrivLeis, 0)
  delta <- matrix(delta, nrow=nAlt, ncol=nAlt, byrow=TRUE)
  emdc_settings <- list(continuousChoice = continuousChoice, 
                        avail            = ones,
                        utilityOutside   = 0, 
                        utilities        = utilities, 
                        budget           = 24,
                        sigma            = sigma, 
                        gamma            = gamma, 
                        delta            = delta, 
                        cost             = ones)
  P[["model"]] = apollo_emdc(emdc_settings, functionality)
  
  ### Comment out as necessary
  P = apollo_panelProd(P, apollo_inputs, functionality)
  P = apollo_prepareProb(P, apollo_inputs, functionality)
  return(P)
}

# ################################################################# #
#### MODEL ESTIMATION & OUTPUT                                   ####
# ################################################################# #

model = apollo_estimate(apollo_beta, apollo_fixed, 
                        apollo_probabilities, apollo_inputs)

apollo_modelOutput(model)

apollo_saveOutput(model)


# ################################################################# #
#### PREDICTION                                                  ####
# ################################################################# #
#############################Use the model estimates to predict and see whether it is close to what is observed

model <- apollo_loadModel(apollo_control$modelName)
apollo_inputs <- apollo_validateInputs(database=database)
apollo_inputs$apollo_control$nCores <- 4
pred <- apollo_prediction(model, apollo_probabilities, apollo_inputs)


XObs <- cbind(work     =     dbVal$t_a02/60,
              school   =     dbVal$t_a03/60,
              shopping =     dbVal$t_a04/60,
              private  =     dbVal$t_a05/60,
              leisure  = dbVal$t_leisure/60)
XPre <- pred[,4:8]

round(sqrt(colMeans((XObs - XPre)^2)),2) # 
round(sqrt(mean((colSums(XObs) - colSums(XPre))^2)),2) #
Thank you in advance!

Best,
TW
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dpalma
Posts: 227
Joined: 24 Apr 2020, 17:54

Re: Comparisons between MDCEV model and EMDC model

Post by dpalma »

Hi TW,

Apologies for the long delay in answering.

The eMDC is not a special case of the MDCEV. While the eMDC is inspired on the MDCEV, it is not a subset or special version of MDCEV (nor the other way around). There are important differences between both models:
  • The MDCEV model uses extreme value disturbances (error terms), while the eMDC uses normal disturbances.
  • The MDCEV model includes a random error term in the utility of the outside good, while the eMDC does not.
  • The MDCEV model allows for different parametrisations (alpha, gamma, and alpha-gamma profiles), while the eMDC only allows for the gamma profile (it could be extended, but the aren't many benefits to doing it).

Therefore, even if you fix all complementarity and substitution parameters of the eMDC to zero, you will still not get the same parameters than the MDCEV, and both models will predict differently.

Concerning your last point about alpha on MDCEV models, it usually happens that alpha tends towards zero (or alpha_base tends to -infinity if you use the transformation alpha = exp(alpha_base) ). So it is common for people to set alpha to something very close to zero (or alpha_base to a large negative value), to avoid estimation issues.

Best wishes
David
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