Some examples of use-cases using the market models of the package
In diseq: Estimation Methods for Markets in Equilibrium and Disequilibrium

Package diseq is deprecated. Please use package markets instead.

This short tutorial covers the very basic use cases to get you started with diseq. More usage details can be found in the documentation of the package.

if (requireNamespace("knitr", quietly = TRUE)) {
  knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
}

Setup the environment

Load the required libraries.

library(diseq)
library(magrittr)
library(Formula)

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 <- 1000
tobs <- 10

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 <- 4430

stochastic_adjustment_data <- simulate_data(
  "diseq_stochastic_adjustment", nobs, tobs,
  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
)

Estimate the models

Prepare the basic parameters for model initialization. The simulate_data call uses Q for the simulated traded quantity, P for the simulated prices, id for subject identification, and date for time identification. It automatically creates the demand-specific variables Xd1 and Xd2, the supply-specific variable Xs1, the common (i.e., both demand and supply) variables X1 and X2, and the price dynamics' variable Xp1.

market_spec <- Q | P | id | date ~ P + Xd1 + Xd2 + X1 + X2 | P + Xs1 + X1 + X2

The market specification has to be modified in two cases. For the diseq_directional, the price variable is removed from the supply equation because the separation rule of the model can only be used for markets with exclusively either inelastic demand or supply. For the diseq_stochastic_adjustment, the right-hand side of the price dynamics equation is appended in the market specification.

By default, the models are estimated by allowing the demand, supply, and price equations to have correlated error shocks. The default verbosity behavior is to display errors and warnings that might occur when estimating the models.

By default, all models are estimated using full information maximum likelihood based on the "BFGS" optimization algorithm. The first equilibrium_model call modifies the estimation behavior and estimates the model using two stage least squares. The diseq_basic call modifies the default optimization behavior and estimates the model using the "Nelder-Mead" optimization methods.

Standard errors are by default assumed to be homoscedastic. The second equilibrium_model and diseq_deterministic_adjustment calls modify this behavior by calculating clustered standard errors based on the subject identifier, while the diseq_basic and diseq_directional calls modify it by calculating heteroscedastic standard errors via the sandwich estimator.

eq_reg <- equilibrium_model(
  market_spec, stochastic_adjustment_data,
  estimation_options = list(method = "2SLS")
)
eq_fit <- equilibrium_model(
  market_spec, stochastic_adjustment_data,
  estimation_options = list(standard_errors = c("id"))
)
bs_fit <- diseq_basic(
  market_spec, stochastic_adjustment_data,
  estimation_options = list(
    method = "Nelder-Mead", control = list(maxit = 1e+5),
    standard_errors = "heteroscedastic"
  )
)
dr_fit <- diseq_directional(
  formula(update(Formula(market_spec), . ~ . | . - P)),
  stochastic_adjustment_data,
  estimation_options = list(standard_errors = "heteroscedastic")
)
da_fit <- diseq_deterministic_adjustment(
  market_spec, stochastic_adjustment_data,
  estimation_options = list(standard_errors = c("id"))
)
sa_fit <- diseq_stochastic_adjustment(
  formula(update(Formula(market_spec), . ~ . | . | Xp1)),
  stochastic_adjustment_data,
  estimation_options = list(control = list(maxit = 1e+5))
)

Post estimation analysis

Summaries

All the model estimates support the summary function. The eq_2sls also provides the first-stage estimation, but it is not included in the summary and has to be explicitly asked.

summary(eq_reg@fit[[1]]$first_stage_model)
summary(eq_reg)
summary(eq_fit)
summary(bs_fit)
summary(da_fit)
summary(sa_fit)

Marginal effects

Calculate marginal effects on the shortage probabilities. Diseq offers two marginal effect calls out of the box. The mean marginal effects and the marginal effects ate the mean. Marginal effects on the shortage probabilities are state-dependent. If the variable is only in the demand equation, the output name of the marginal effect is the variable name prefixed by D_. If the variable is only in the supply equation, the name of the marginal effect is the variable name prefixed by S_. If the variable is in both equations, then it is prefixed by B_.

diseq_abbrs <- c("bs", "dr", "da", "sa")
diseq_fits <- c(bs_fit, dr_fit, da_fit, sa_fit)
variables <- c("P", "Xd1", "Xd2", "X1", "X2", "Xs1")

apply_marginal <- function(fnc, ...) {
  function(fit) {
    sapply(variables, function(v) fnc(fit, v, ...), USE.NAMES = FALSE)
  }
}

mspm <- sapply(diseq_fits, apply_marginal(shortage_probability_marginal))
colnames(mspm) <- diseq_abbrs
# Mean Shortage Probabilities' Marginal Effects
mspm

spmm <- sapply(
  diseq_fits,
  apply_marginal(shortage_probability_marginal, aggregate = "at_the_mean")
)
colnames(spmm) <- diseq_abbrs
# Shortage Probabilities' Marginal Effects at the Mean
spmm

Shortages

Copy the disequilibrium model tibble and augment it with post-estimation data. The disequilibrium models can be used to estimate:

Shortage probabilities. These are the probabilities that the disequilibrium models assign to observing a particular extent of excess demand.
Normalized shortages. The point estimates of the shortages are normalized by the variance of the difference of demand and supply shocks.
Relative shortages: The point estimates of the shortages are normalized by the estimated supplied quantity.

fit <- sa_fit
mdt <- tibble::add_column(
  fit@model_tibble,
  shortage_indicators = c(shortage_indicators(fit)),
  normalized_shortages = c(normalized_shortages(fit)),
  shortage_probabilities = c(shortage_probabilities(fit)),
  relative_shortages = c(relative_shortages(fit))
)

How is the sample separated post-estimation? The indices of the observations for which the estimated demand is greater than the estimated supply are easily obtained.

if (requireNamespace("ggplot2", quietly = TRUE)) {
  pdt <- tibble::tibble(
    Shortage = c(mdt$normalized_shortages, mdt$relative_shortages),
    Type = c(rep("Normalized", nrow(mdt)), rep("Relative", nrow(mdt))),
    xpos = c(rep(-1.0, nrow(mdt)), rep(1.0, nrow(mdt))),
    ypos = c(
      rep(0.8 * max(mdt$normalized_shortages), nrow(mdt)),
      rep(0.8 * max(mdt$relative_shortages), nrow(mdt))
    )
  )
  ggplot2::ggplot(pdt) +
    ggplot2::stat_density(ggplot2::aes(Shortage,
      linetype = Type,
      color = Type
    ), geom = "line") +
    ggplot2::ggtitle(paste0("Estimated shortages densities (", model_name(fit), ")")) +
    ggplot2::theme(
      panel.background = ggplot2::element_rect(fill = "transparent"),
      plot.background = ggplot2::element_rect(
        fill = "transparent",
        color = NA
      ),
      legend.background = ggplot2::element_rect(fill = "transparent"),
      legend.box.background = ggplot2::element_rect(
        fill = "transparent",
        color = NA
      ),
      legend.position = c(0.8, 0.8)
    )
} else {
    summary(mdt[, grep("shortage", colnames(mdt))])
}

Fitted values and aggregation

The estimated demanded and supplied quantities can be calculated per observation.

market <- cbind(
  demand = demanded_quantities(fit)[, 1],
  supply = supplied_quantities(fit)[, 1]
)
summary(market)

The package also offers basic aggregation functionality.

aggregates <- aggregate_demand(fit) %>%
  dplyr::left_join(aggregate_supply(fit), by = "date") %>%
  dplyr::mutate(date = as.numeric(date)) %>%
  dplyr::rename(demand = D_Q, supply = S_Q)
if (requireNamespace("ggplot2", quietly = TRUE)) {
  pdt <- tibble::tibble(
    Date = c(aggregates$date, aggregates$date),
    Quantity = c(aggregates$demand, aggregates$supply),
    Side = c(rep("Demand", nrow(aggregates)), rep("Supply", nrow(aggregates)))
  )
  ggplot2::ggplot(pdt, ggplot2::aes(x = Date)) +
    ggplot2::geom_line(ggplot2::aes(y = Quantity, linetype = Side, color = Side)) +
    ggplot2::ggtitle(paste0(
      "Aggregate estimated demand and supply  (", model_name(fit), ")"
    )) +
    ggplot2::theme(
      panel.background = ggplot2::element_rect(fill = "transparent"),
      plot.background = ggplot2::element_rect(
        fill = "transparent", color = NA
      ),
      legend.background = ggplot2::element_rect(fill = "transparent"),
      legend.box.background = ggplot2::element_rect(
        fill = "transparent", color = NA
      ),
      legend.position = c(0.8, 0.5)
    )
} else {

    aggregates
}