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Model estimation

Source: R/market_fit.R
estimate.Rd

All models are estimated using full information maximum likelihood. The equilibrium_model can also be estimated using two-stage least squares. The maximum likelihood estimation is based on optim. If no starting values are provided, the function uses linear regression estimates as initializing values. The default optimization method is BFGS. For other alternatives see optim. The implementation of the two-stage least square estimation of the equilibrium_model is based on lm.

Usage

estimate(object, ...)

# S4 method for market_model
estimate(
  object,
  gradient = "calculated",
  hessian = "calculated",
  standard_errors = "homoscedastic",
  ...
)

# S4 method for equilibrium_model
estimate(object, method = "BFGS", optimizer = "optim", ...)

Arguments

object

A model object.

...

Additional parameter used in the model's estimation. These are passed further down to the optimization call. For the equilibrium_model model, the parameters are passed to lm, if the method is set to 2SLS, or to optim for any other method. For the rest of the models, the parameters are passed to optim.

gradient

One of two potential options: "numerical" and "calculated". By default, all the models are estimated using the analytic expressions of their likelihoods' gradients.

hessian

One of three potential options: "skip", "numerical", and "calculated". The default is to use the "calculated" Hessian for the models that expressions are available and the "numerical" Hessian in other cases. Calculated Hessian expressions are available for the basic and directional models.

standard_errors

One of three potential options: "homoscedastic", "heteroscedastic", or a vector with variables names for which standard error clusters are to be created. The default value is "homoscedastic". If the option "heteroscedastic" is passed, the variance-covariance matrix is calculated using heteroscedasticity adjusted (Huber-White) standard errors. If a vector with variable names is supplied, the variance-covariance matrix is calculated by grouping the score matrix based on the passed variables.

method

A string specifying the estimation method. When the passed value is among "Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", and "Brent", the model is estimated using full information maximum likelihood based on optim functionality. When "2SLS" is supplied, the model is estimated using two-stage least squares via lm. In this case, the function returns a list containing the first and second stage estimates. The default value is "BFGS".

optimizer

One of two options: "optim", "gsl". The default value is "optim". If the option "gsl" is set, the equilibrium likelihood is maximized using GSL.

Value

A market fit object holding the estimation result.

Details

The likelihood of the equilibrium model can be optimized either by using optim (the default option) or native GSL routines. The caller can override the default behavior by setting the optimizer argument equal to "gsl", in which case GSL routines are used. This does not necessarily result to faster execution times. This functionality is primarily intended for advanced usage. The optim functionality is a fast, analysis-oriented alternative, which is more suitable for most use case.

When optimizer = "gsl" is used, the only available optimization method is BFGS. Additionally, the caller needs to specify in the control list values for the optimization step (step), the objective's optimization tolerance (objective_tolerance), the gradient's optimization tolerance (gradient_tolerance, and the maximum allowed number of iterations (maxit). If the GSL library is not available in the calling machine, the function returns a trivial result list with convergence status set equal to -1. If the C++17 execution policies are available, the implementation of the optimization is parallelized.

Functions

  • estimate(market_model): Full information maximum likelihood estimation.

  • estimate(equilibrium_model): Equilibrium model estimation.

Examples

# \donttest{
# initialize the model using the houses dataset
model <- new(
  "diseq_deterministic_adjustment", # model type
  subject = ID, time = TREND, quantity = HS, price = RM,
  demand = RM + TREND + W + CSHS + L1RM + L2RM + MONTH,
  supply = RM + TREND + W + L1RM + MA6DSF + MA3DHF + MONTH,
  fair_houses(), # data
  correlated_shocks = FALSE # let shocks be independent
)

# estimate the model object (BFGS is used by default)
fit <- estimate(model)

# estimate the model by specifying the optimization details passed to the optimizer.
fit <- estimate(model, control = list(maxit = 1e+6))

# summarize results
summary(fit)
#> Deterministic Adjustment Model for Markets in Disequilibrium:
#>   Demand RHS        :   D_RM + D_TREND + D_W + D_CSHS + D_L1RM + D_L2RM
#>     + D_MONTH
#>   Supply RHS        :   S_RM + S_TREND + S_W + S_L1RM + S_MA6DSF +
#>     S_MA3DHF + S_MONTH
#>   Short Side Rule   : HS = min(D_HS, S_HS)
#>   Separation Rule   : RM_DIFF analogous to (D_HS - S_HS)
#>   Shocks            : Independent
#>   Nobs              : 129
#>   Sample Separation : Demand Obs = 18, Supply Obs = 111
#>   Quantity Var      : HS
#>   Price Var         : RM
#>   Key Var(s)        : ID, TREND
#>   Time Var          : TREND
#> 
#> Maximum likelihood estimation:
#>   Method              : BFGS
#>   Max Iterations      : 1e+06
#>   Convergence Status  : success
#>   Starting Values     :
#>       D_RM    D_CONST    D_TREND        D_W     D_CSHS     D_L1RM     D_L2RM 
#>  7.528e-02  7.462e+01  2.432e+00  2.395e+00 -1.977e-02  1.512e-01 -3.291e-01 
#>  D_MONTH02  D_MONTH03  D_MONTH04  D_MONTH05  D_MONTH06  D_MONTH07  D_MONTH08 
#>  2.918e+00  3.110e+01  6.173e+01  6.452e+01  5.819e+01  4.851e+01  4.744e+01 
#>  D_MONTH09  D_MONTH10  D_MONTH11  D_MONTH12       S_RM    S_CONST    S_TREND 
#>  4.219e+01  4.566e+01  2.885e+01  8.548e+00  1.465e-01 -5.076e+01 -1.243e-01 
#>        S_W     S_L1RM   S_MA6DSF   S_MA3DHF  S_MONTH02  S_MONTH03  S_MONTH04 
#>  2.669e+00 -8.889e-02  4.754e-02  4.554e-02  3.371e+00  3.254e+01  6.598e+01 
#>  S_MONTH05  S_MONTH06  S_MONTH07  S_MONTH08  S_MONTH09  S_MONTH10  S_MONTH11 
#>  6.660e+01  5.605e+01  4.175e+01  4.719e+01  4.172e+01  4.892e+01  3.026e+01 
#>  S_MONTH12    RM_DIFF D_VARIANCE S_VARIANCE 
#>  1.094e+01 -4.684e-16  1.960e+02  1.035e+02 
#> 
#> Coefficients:
#>                 Estimate  Std. Error       z value     Pr(>|z|) 
#>  D_RM         -6.39287048 0.857591642     -7.454446 9.024631e-14 ***
#>  D_CONST     -20.16533253 0.060974564   -330.717127 0.000000e+00 ***
#>  D_TREND      -3.09530903 0.230919075    -13.404302 5.705752e-41 ***
#>  D_W           2.77909898 1.886047698      1.473504 1.406152e-01  
#>  D_CSHS        0.02878931 0.002627855     10.955440 6.257341e-28 ***
#>  D_L1RM        8.92812266 1.272736617      7.014902 2.301102e-12 ***
#>  D_L2RM       -2.48883659 0.778072698     -3.198720 1.380392e-03 **
#>  D_MONTH02    17.94359360 0.028355163    632.815745 0.000000e+00 ***
#>  D_MONTH03    28.55310947 0.031907690    894.866076 0.000000e+00 ***
#>  D_MONTH04    78.00303909 0.025111880   3106.220640 0.000000e+00 ***
#>  D_MONTH05    84.17025404 0.013734549   6128.359314 0.000000e+00 ***
#>  D_MONTH06    82.50861671 0.008009769  10300.998283 0.000000e+00 ***
#>  D_MONTH07    88.22174395 0.033884480   2603.603317 0.000000e+00 ***
#>  D_MONTH08    52.40940637 0.068537233    764.685185 0.000000e+00 ***
#>  D_MONTH09    49.40057986 0.014656650   3370.523264 0.000000e+00 ***
#>  D_MONTH10    67.05086057 0.025068207   2674.736959 0.000000e+00 ***
#>  D_MONTH11    24.50611800 0.020945815   1169.976811 0.000000e+00 ***
#>  D_MONTH12    30.86396923 0.037121309    831.435376 0.000000e+00 ***
#>  S_RM          0.46442865 0.254953913      1.821618 6.851296e-02 .
#>  S_CONST     -37.46456496 0.017747929  -2110.925994 0.000000e+00 ***
#>  S_TREND      -0.14554797 0.045456126     -3.201944 1.365036e-03 **
#>  S_W           2.08578681 0.667580491      3.124397 1.781696e-03 **
#>  S_L1RM       -0.40511192 0.259778911     -1.559449 1.188902e-01  
#>  S_MA6DSF      0.04693627 0.004759294      9.862022 6.081192e-23 ***
#>  S_MA3DHF      0.03455421 0.007455667      4.634624 3.575865e-06 ***
#>  S_MONTH02     3.81910684 0.038878752     98.231209 0.000000e+00 ***
#>  S_MONTH03    34.38727665 0.044938607    765.205672 0.000000e+00 ***
#>  S_MONTH04    65.75523098 0.048451885   1357.124311 0.000000e+00 ***
#>  S_MONTH05    67.60421681 0.006803688   9936.407793 0.000000e+00 ***
#>  S_MONTH06    57.39161807 0.010350186   5544.984254 0.000000e+00 ***
#>  S_MONTH07    43.90383859 0.094392783    465.118595 0.000000e+00 ***
#>  S_MONTH08    49.34401811 0.033172074   1487.516806 0.000000e+00 ***
#>  S_MONTH09    44.93630427 0.045704348    983.195391 0.000000e+00 ***
#>  S_MONTH10    49.43637822 0.046764866   1057.126482 0.000000e+00 ***
#>  S_MONTH11    32.80682646 0.010520109   3118.487203 0.000000e+00 ***
#>  S_MONTH12    11.27185537 0.090967431    123.910890 0.000000e+00 ***
#>  RM_DIFF       1.53371422 0.744837694      2.059125 3.948223e-02 *
#>  D_VARIANCE 1295.13136810 0.008069301 160501.051989 0.000000e+00 ***
#>  S_VARIANCE  107.76446554 0.007893424  13652.435294 0.000000e+00 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> -2 log L: 1724.665
# }
# \donttest{
# simulate an equilibrium model
model <- simulate_model(
  "equilibrium_model", list(
    # observed entities, observed time points
    nobs = 500, tobs = 3,
    # demand coefficients
    alpha_d = -1.9, beta_d0 = 24.9, beta_d = c(2.3, -1.2), eta_d = c(2.0, -1.5),
    # supply coefficients
    alpha_s = .9, beta_s0 = 8.2, beta_s = c(3.3), eta_s = c(1.5, -2.2)
  ),
  seed = 99
)

# maximize the model's log-likelihood
fit <- estimate(
  model,
  optimizer = "gsl", control = list(
    step = 1e-2, objective_tolerance = 1e-8,
    gradient_tolerance = 1e-2, maxit = 1e+3
  )
)

summary(fit)
#> Equilibrium Model for Markets in Equilibrium:
#>   Demand RHS        :   D_P + D_Xd1 + D_Xd2 + D_X1 + D_X2
#>   Supply RHS        :   S_P + S_Xs1 + S_X1 + S_X2
#>   Market Clearing   : Q = D_Q = S_Q
#>   Shocks            : Correlated
#>   Nobs              : 1500
#>   Sample Separation : Not Separated
#>   Quantity Var      : Q
#>   Price Var         : P
#>   Key Var(s)        : id, date
#>   Time Var          : date
#> 
#> Maximum likelihood estimation:
#>   Method              : BFGS
#>   Max Iterations      : 1000
#>   Convergence Status  : failure (-1)
#>   Starting Values     :
#>        D_P    D_CONST      D_Xd1      D_Xd2       D_X1       D_X2        S_P 
#>   -1.90357   24.92297    2.29870   -1.22392    1.97103   -1.47621    0.83976 
#>    S_CONST      S_Xs1       S_X1       S_X2 D_VARIANCE S_VARIANCE        RHO 
#>    8.57600    3.21044    1.52915   -2.14210    1.10049    0.86999    0.04954 
#> 
#> Coefficients:
#>            Estimate Std. Error z value Pr(>|z|) 
#>  D_P             NaN        NaN     NaN      NaN  
#>  D_CONST         NaN        NaN     NaN      NaN  
#>  D_Xd1           NaN        NaN     NaN      NaN  
#>  D_Xd2           NaN        NaN     NaN      NaN  
#>  D_X1            NaN        NaN     NaN      NaN  
#>  D_X2            NaN        NaN     NaN      NaN  
#>  S_P             NaN        NaN     NaN      NaN  
#>  S_CONST         NaN        NaN     NaN      NaN  
#>  S_Xs1           NaN        NaN     NaN      NaN  
#>  S_X1            NaN        NaN     NaN      NaN  
#>  S_X2            NaN        NaN     NaN      NaN  
#>  D_VARIANCE      NaN        NaN     NaN      NaN  
#>  S_VARIANCE      NaN        NaN     NaN      NaN  
#>  RHO             NaN        NaN     NaN      NaN  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> -2 log L: NaN
# }