Model initialization and estimation details

This tutorial examines model initialization and estimation in some detail. Models can be initialized and estimated with a single function call (see basic_usage), which is the recommended approach for most usage cases. However, it is convenient to separate model estimation and initialization on some occasions. This is particularly relevant when estimating the same model using different methods and/or options without re-initializing.

The operations of this vignette cover the many but not all use initialization cases. More usage details can be found in the documentation of the package.

Setup the environment

library(diseq)

Prepare the data. Normally this step is long and depends on the nature of the data and the considered market. For this example, we will use simulated data. Although we could simulate data independently from the package, we will use the top-level simulation functionality of diseq to simplify the process. See the documentation of simulate_data for more information on the simulation functionality. Here, we simulate data using a data generating process for a market in disequilibrium with stochastic price dynamics.

nobs <- 2000
tobs <- 5

alpha_d <- -0.3
beta_d0 <- 6.8
beta_d <- c(0.3, -0.02)
eta_d <- c(0.6, -0.1)

alpha_s <- 0.6
beta_s0 <- 4.1
beta_s <- c(0.9)
eta_s <- c(-0.5, 0.2)

gamma <- 1.2
beta_p0 <- 0.9
beta_p <- c(-0.1)

sigma_d <- 1
sigma_s <- 1
sigma_p <- 1
rho_ds <- 0.0
rho_dp <- 0.0
rho_sp <- 0.0

seed <- 443

alpha_d, beta_d0, beta_d, eta_d,
alpha_s, beta_s0, beta_s, eta_s,
gamma, beta_p0, beta_p,
sigma_d = sigma_d, sigma_s = sigma_s, sigma_p = sigma_p,
rho_ds = rho_ds, rho_dp = rho_dp, rho_sp = rho_sp,
seed = seed
)

Initialize the model

The constructor sets the basic parameters for model initialization and constructs a model object. The needed arguments for a construction call are configured as follows:

• The fields that uniquely identify simulated data records are id (for subjects) and date (for time). These variable names are automatically set for the data that simulate_data generates.

• The quantity variable is automatically named Q by the simulate_data function. The quantity variable is observable. For the equilibrium models, it is equal to both the demanded and supplied quantities. The observed quantity represents either a demanded or a supplied quantity for the disequilibrium models. Each disequilibrium model resolves that state of the observation in a different way.

• The price variable is set as P by the simulate_data call.

• The right-hand sides of the demand and supply equations. Simply include the factor variables here as in a usual lm formula. Indicator variables and interaction terms will be created automatically by the constructor. For the diseq_directional model, the price cannot go in both equations. For the rest of the models, the price can go in both equations if treated as exogenous. The diseq_stochastic_adjustment also requires the specification of price dynamics. The simulate_data call generates the demand-specific variables Xd1 and Xd2, the supply-specific variable Xs1, the common (i.e.Â both demand and supply) variables X1 and X2, and the price dynamicsâ€™ variable Xp1.

• Set the verbosity level. This controls the level of messaging. The verbosity level here is set so that the constructed objects display basic information in addition to errors and warnings.

verbose <- 2
• Should the estimation allow for correlated demand and supply shocks?
correlated_shocks <- TRUE

Using the above parameterization, construct the model objects. Here, we construct an equilibrium model and four disequilibrium models, using in all cases the same data simulated by the process based on the stochastic price adjustment model. Of course, this is only done to simplify the exposition of the functionality. The constructors of the models that use price dynamics information in the estimation, i.e., diseq_directional, diseq_deterministic_adjustment, and diseq_stochastic_adjustment, will automatically generate lagged prices and drop one observation per subject.

eq <- new(
"equilibrium_model",
quantity = Q, price = P,
demand = P + Xd1 + Xd2 + X1 + X2,
supply = P + Xs1 + X1 + X2,
subject = id, time = date,
correlated_shocks = correlated_shocks, verbose = verbose
)
#> Info: This is 'Equilibrium with correlated shocks' model
bs <- new(
"diseq_basic",
quantity = Q, price = P,
demand = P + Xd1 + Xd2 + X1 + X2,
supply = P + Xs1 + X1 + X2,
subject = id, time = date,
correlated_shocks = correlated_shocks, verbose = verbose
)
#> Info: This is 'Basic with correlated shocks' model
dr <- new(
"diseq_directional",
quantity = Q, price = P,
demand = P + Xd1 + Xd2 + X1 + X2,
supply = Xs1 + X1 + X2,
subject = id, time = date,
correlated_shocks = correlated_shocks, verbose = verbose
)
#> Info: This is 'Directional with correlated shocks' model
#> Info: Dropping 2000 rows to generate 'LAGGED_P'.
#> Info: Sample separated with 2773 rows in excess supply and 5227 in excess demand state.
da <- new(
quantity = Q, price = P,
demand = P + Xd1 + Xd2 + X1 + X2,
supply = P + Xs1 + X1 + X2,
subject = id, time = date,
correlated_shocks = correlated_shocks, verbose = verbose
)
#> Info: This is 'Deterministic Adjustment with correlated shocks' model
#> Info: Dropping 2000 rows to generate 'LAGGED_P'.
#> Info: Sample separated with 2773 rows in excess supply and 5227 in excess demand state.
sa <- new(
quantity = Q, price = P,
demand = P + Xd1 + Xd2 + X1 + X2,
supply = P + Xs1 + X1 + X2,
price_dynamics = Xp1,
subject = id, time = date,
correlated_shocks = correlated_shocks, verbose = verbose
)
#> Info: This is 'Stochastic Adjustment with correlated shocks' model
#> Info: Dropping 2000 rows to generate 'LAGGED_P'.

Estimation

First, we need to set the estimation parameters and choose an estimation method. The only model that can be estimated by least squares is the equilibrium_model. To estimate the model with this methodology call diseq::estimate with method = 2SLS set. The equilibrium_model can also be estimated using full information maximum likelihood, as it is the case for all the disequilibrium models. One may choose an optimization method and the corresponding optimization controls. The available methods are:

• "Nelder-Mead": Does not require the gradient of the likelihood to be known.

• "BFGS": Uses the analytically calculated gradients. By default, the diseq package uses this method.

• "L-BFGS-B": Constrained optimization.

optimization_method <- "BFGS"
optimization_options <- list(REPORT = 10, maxit = 10000, reltol = 1e-6)

Then, estimate the models. The eq model is estimated with two different methods, namely two stage least squares and full information maximum likelihood. Moreover, the bs is estimated using two different optimization options; these are the gradient-based "BFGS" method and the simplex-based "Nelder-Mead" methods. Lastly, the models estimated with maximal likelihood use different estimation options regarding the calculation of standard errors. See the documentation for more options.

estimate(eq, method = "2SLS")
#> 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
estimate(eq,
control = optimization_options, method = optimization_method,
standard_errors = c("id")
)
#> 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
estimate(bs,
control = optimization_options, method = optimization_method,
standard_errors = "heteroscedastic"
)
#> Basic Model for Markets in Disequilibrium
#>   Demand RHS        : D_P + D_Xd1 + D_Xd2 + D_X1 + D_X2
#>   Supply RHS        : S_P + S_Xs1 + S_X1 + S_X2
#>   Short Side Rule   : Q = min(D_Q, S_Q)
#>   Shocks            : Correlated
estimate(bs,
control = optimization_options, method = "Nelder-Mead",
standard_errors = "heteroscedastic"
)
#> Basic Model for Markets in Disequilibrium
#>   Demand RHS        : D_P + D_Xd1 + D_Xd2 + D_X1 + D_X2
#>   Supply RHS        : S_P + S_Xs1 + S_X1 + S_X2
#>   Short Side Rule   : Q = min(D_Q, S_Q)
#>   Shocks            : Correlated
estimate(dr,
control = optimization_options, method = optimization_method,
standard_errors = "heteroscedastic"
)
#> Directional Model for Markets in Disequilibrium
#>   Demand RHS        : D_P + D_Xd1 + D_Xd2 + D_X1 + D_X2
#>   Supply RHS        : S_Xs1 + S_X1 + S_X2
#>   Short Side Rule   : Q = min(D_Q, S_Q)
#>   Separation Rule   : P_DIFF >= 0 then D_Q >= S_Q
#>   Shocks            : Correlated
estimate(da,
control = optimization_options, method = optimization_method,
standard_errors = c("id")
)
#> Deterministic Adjustment Model for Markets in Disequilibrium
#>   Demand RHS        : D_P + D_Xd1 + D_Xd2 + D_X1 + D_X2
#>   Supply RHS        : S_P + S_Xs1 + S_X1 + S_X2
#>   Short Side Rule   : Q = min(D_Q, S_Q)
#>   Separation Rule   : P_DIFF analogous to (D_Q - S_Q)
#>   Shocks            : Correlated
estimate(sa,
control = optimization_options, method = optimization_method
)
#> Stochastic Adjustment Model for Markets in Disequilibrium
#>   Demand RHS        : D_P + D_Xd1 + D_Xd2 + D_X1 + D_X2
#>   Supply RHS        : S_P + S_Xs1 + S_X1 + S_X2
#>   Price Dynamics RHS: (D_Q - S_Q) + Xp1
#>   Short Side Rule   : Q = min(D_Q, S_Q)
#>   Shocks            : Correlated