Some examples of use-cases using the market models of the package

Setup the environment

Load the required libraries.

library(markets)
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 markets 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

summary(eq_reg@fit$first_stage_model)
#> 
#> Call:
#> lm(formula = first_stage_formula, data = object@data)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -4.8178 -0.9065  0.0739  0.9472  5.0795 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  3.64003    0.01388 262.278   <2e-16 ***
#> Xd1          0.12367    0.01385   8.932   <2e-16 ***
#> Xd2          0.02181    0.01379   1.582    0.114    
#> X1           0.53012    0.01398  37.920   <2e-16 ***
#> X2          -0.14884    0.01392 -10.689   <2e-16 ***
#> Xs1         -0.41736    0.01401 -29.793   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 1.388 on 9994 degrees of freedom
#> Multiple R-squared:  0.2035, Adjusted R-squared:  0.2031 
#> F-statistic: 510.7 on 5 and 9994 DF,  p-value: < 2.2e-16
summary(eq_reg)
#> 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              : 10000
#>   Sample Separation : Not Separated
#>   Quantity Var      : Q
#>   Price Var         : P
#>   Key Var(s)        : id, date
#>   Time Var          : date
#> 
#> Least square estimation:
#>   Method              : 2SLS
#> 
#> First Stage:
#> 
#> Call:
#> lm(formula = first_stage_formula, data = object@data)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -4.8178 -0.9065  0.0739  0.9472  5.0795 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  3.64003    0.01388 262.278   <2e-16 ***
#> Xd1          0.12367    0.01385   8.932   <2e-16 ***
#> Xd2          0.02181    0.01379   1.582    0.114    
#> X1           0.53012    0.01398  37.920   <2e-16 ***
#> X2          -0.14884    0.01392 -10.689   <2e-16 ***
#> Xs1         -0.41736    0.01401 -29.793   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 1.388 on 9994 degrees of freedom
#> Multiple R-squared:  0.2035, Adjusted R-squared:  0.2031 
#> F-statistic: 510.7 on 5 and 9994 DF,  p-value: < 2.2e-16
#> 
#> 
#> Demand Equation:
#> 
#> Call:
#> lm(formula = demand_formula, data = object@data)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -3.4735 -0.5629  0.0282  0.6062  3.1546 
#> 
#> Coefficients:
#>              Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  8.138365   0.077954 104.400   <2e-16 ***
#> P_FITTED    -0.772875   0.021274 -36.330   <2e-16 ***
#> Xd1          0.278803   0.009154  30.455   <2e-16 ***
#> Xd2          0.014726   0.008751   1.683   0.0925 .  
#> X1           0.622276   0.014412  43.177   <2e-16 ***
#> X2          -0.109598   0.009403 -11.656   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 0.8795 on 9994 degrees of freedom
#> Multiple R-squared:  0.1877, Adjusted R-squared:  0.1873 
#> F-statistic: 461.9 on 5 and 9994 DF,  p-value: < 2.2e-16
#> 
#> 
#> Supply Equation:
#> 
#> Call:
#> lm(formula = supply_formula, data = object@data)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -3.4792 -0.5661  0.0298  0.6070  3.1133 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  0.10239    0.25480   0.402    0.688    
#> P_FITTED     1.43487    0.06999  20.502   <2e-16 ***
#> Xs1          0.92130    0.03050  30.205   <2e-16 ***
#> X1          -0.54807    0.03807 -14.398   <2e-16 ***
#> X2           0.21876    0.01376  15.900   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 0.8801 on 9995 degrees of freedom
#> Multiple R-squared:  0.1865, Adjusted R-squared:  0.1862 
#> F-statistic: 572.8 on 4 and 9995 DF,  p-value: < 2.2e-16
summary(eq_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              : 10000
#>   Sample Separation : Not Separated
#>   Quantity Var      : Q
#>   Price Var         : P
#>   Key Var(s)        : id, date
#>   Time Var          : date
#> 
#> Maximum likelihood estimation:
#>   Method              : BFGS
#>   Convergence Status  : success
#>   Starting Values     :
#>        D_P    D_CONST      D_Xd1      D_Xd2       D_X1       D_X2        S_P 
#>   -0.77287    8.13837    0.27880    0.01473    0.62228   -0.10960    1.43487 
#>    S_CONST      S_Xs1       S_X1       S_X2 D_VARIANCE S_VARIANCE        RHO 
#>    0.10239    0.92130   -0.54807    0.21876    0.77312    0.77429    0.99925 
#> 
#> Coefficients:
#>             Estimate Std. Error  z value      Pr(z)
#> D_P        -0.771928    0.03663 -21.0719  1.441e-98
#> D_CONST     8.134649    0.13347  60.9473  0.000e+00
#> D_Xd1       0.278247    0.01598  17.4155  6.296e-68
#> D_Xd2       0.002023    0.01291   0.1568  8.754e-01
#> D_X1        0.621663    0.02456  25.3070 2.676e-141
#> D_X2       -0.109469    0.01599  -6.8452  7.635e-12
#> S_P         1.490002    0.17705   8.4159  3.899e-17
#> S_CONST    -0.098206    0.64480  -0.1523  8.789e-01
#> S_Xs1       0.944989    0.07671  12.3196  7.102e-35
#> S_X1       -0.577267    0.09615  -6.0035  1.931e-09
#> S_X2        0.227037    0.03386   6.7044  2.022e-11
#> D_VARIANCE  2.268433    0.13110  17.3034  4.438e-67
#> S_VARIANCE  4.374334    0.93742   4.6664  3.066e-06
#> RHO        -0.508898    0.04535 -11.2220  3.179e-29
#> 
#> -2 log L: 60383.4
summary(bs_fit)
#> 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
#>   Nobs              : 10000
#>   Sample Separation : Not Separated
#>   Quantity Var      : Q
#>   Price Var         : P
#>   Key Var(s)        : id, date
#>   Time Var          : date
#> 
#> Maximum likelihood estimation:
#>   Method              : Nelder-Mead
#>   Max Iterations      : 1e+05
#>   Convergence Status  : success
#>   Starting Values     :
#>        D_P    D_CONST      D_Xd1      D_Xd2       D_X1       D_X2        S_P 
#>   0.044738   5.161505   0.178662  -0.001769   0.185406   0.015149   0.128093 
#>    S_CONST      S_Xs1       S_X1       S_X2 D_VARIANCE S_VARIANCE        RHO 
#>   4.857007   0.376491   0.143151   0.021754   0.871035   0.775016   0.000000 
#> 
#> Coefficients:
#>            Estimate Std. Error   z value      Pr(z)
#> D_P        -0.04710    0.01220  -3.86141  1.127e-04
#> D_CONST     5.96478    0.05151 115.79794  0.000e+00
#> D_Xd1       0.23523    0.01395  16.86213  8.546e-64
#> D_Xd2      -0.02331    0.01359  -1.71530  8.629e-02
#> D_X1        0.49477    0.02027  24.41089 1.311e-131
#> D_X2       -0.00153    0.01880  -0.08137  9.351e-01
#> S_P         0.36492    0.02044  17.85637  2.579e-71
#> S_CONST     4.81479    0.05047  95.39849  0.000e+00
#> S_Xs1       0.77298    0.02449  31.55999 1.308e-218
#> S_X1       -0.28517    0.02741 -10.40492  2.355e-25
#> S_X2        0.05738    0.02387   2.40392  1.622e-02
#> D_VARIANCE  0.85101    0.02030  41.92772  0.000e+00
#> S_VARIANCE  0.81718    0.02306  35.43429 5.065e-275
#> RHO         0.07948    0.04165   1.90818  5.637e-02
#> 
#> -2 log L: 25178.57
summary(da_fit)
#> 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
#>   Nobs              : 9000
#>   Sample Separation : Demand Obs = 3731, Supply Obs = 5269
#>   Quantity Var      : Q
#>   Price Var         : P
#>   Key Var(s)        : id, date
#>   Time Var          : date
#> 
#> Maximum likelihood estimation:
#>   Method              : BFGS
#>   Convergence Status  : success
#>   Starting Values     :
#>        D_P    D_CONST      D_Xd1      D_Xd2       D_X1       D_X2        S_P 
#>   0.025528   5.258982   0.190255   0.002401   0.224474   0.012186   0.118691 
#>    S_CONST      S_Xs1       S_X1       S_X2     P_DIFF D_VARIANCE S_VARIANCE 
#>   4.900087   0.351146   0.175962   0.018346   0.967398   0.842610   0.769177 
#>        RHO 
#>   0.000000 
#> 
#> Coefficients:
#>             Estimate Std. Error  z value      Pr(z)
#> D_P        -0.282794   0.013288 -21.2816 1.681e-100
#> D_CONST     6.858767   0.057986 118.2837  0.000e+00
#> D_Xd1       0.185900   0.013315  13.9612  2.690e-44
#> D_Xd2       0.008247   0.010004   0.8244  4.097e-01
#> D_X1        0.567684   0.015731  36.0862 3.730e-285
#> D_X2       -0.086836   0.012297  -7.0617  1.645e-12
#> S_P         0.208697   0.013135  15.8886  7.601e-57
#> S_CONST     4.799437   0.047213 101.6543  0.000e+00
#> S_Xs1       0.481558   0.013320  36.1534 3.292e-286
#> S_X1       -0.005406   0.013757  -0.3930  6.944e-01
#> S_X2        0.065150   0.009496   6.8605  6.862e-12
#> P_DIFF      0.614386   0.017231  35.6568 1.848e-278
#> D_VARIANCE  1.277953   0.032091  39.8228  0.000e+00
#> S_VARIANCE  0.760055   0.013867  54.8096  0.000e+00
#> RHO         0.539373   0.022816  23.6406 1.474e-123
#> 
#> -2 log L: 45914.24
summary(sa_fit)
#> 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:   I(D_Q - S_Q) + Xp1
#>   Short Side Rule   : Q = min(D_Q, S_Q)
#>   Shocks            : Correlated
#>   Nobs              : 9000
#>   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      : 1e+05
#>   Convergence Status  : success
#>   Starting Values     :
#>        D_P    D_CONST      D_Xd1      D_Xd2       D_X1       D_X2        S_P 
#>   0.025528   5.258982   0.190255   0.002401   0.224474   0.012186   0.118691 
#>    S_CONST      S_Xs1       S_X1       S_X2     P_DIFF    P_CONST      P_Xp1 
#>   4.900087   0.351146   0.175962   0.018346   0.968928   0.275093  -0.062290 
#> D_VARIANCE S_VARIANCE P_VARIANCE     RHO_DS     RHO_DP     RHO_SP 
#>   0.842610   0.769177   1.600380   0.000000   0.000000   0.000000 
#> 
#> Coefficients:
#>              Estimate Std. Error   z value      Pr(z)
#> D_P        -0.3195795    0.01407 -22.70705 3.816e-114
#> D_CONST     6.9163194    0.06554 105.52937  0.000e+00
#> D_Xd1       0.2874772    0.01236  23.24957 1.437e-119
#> D_Xd2       0.0070640    0.01187   0.59507  5.518e-01
#> D_X1        0.6064175    0.01726  35.14010 1.647e-270
#> D_X2       -0.0980891    0.01253  -7.82584  5.043e-15
#> S_P         0.6163763    0.03164  19.48294  1.532e-84
#> S_CONST     3.9961080    0.07155  55.85101  0.000e+00
#> S_Xs1       0.8989133    0.03142  28.60723 5.462e-180
#> S_X1       -0.4975437    0.03701 -13.44240  3.412e-41
#> S_X2        0.2036846    0.01678  12.13517  6.877e-34
#> P_DIFF      1.2124441    0.04061  29.85630 7.272e-196
#> P_CONST     0.8350155    0.05185  16.10519  2.346e-58
#> P_Xp1      -0.0832652    0.01505  -5.53409  3.129e-08
#> D_VARIANCE  1.0591615    0.03176  33.34445 8.768e-244
#> S_VARIANCE  1.0052351    0.06763  14.86440  5.612e-50
#> P_VARIANCE  1.0188468    0.07944  12.82512  1.186e-37
#> RHO_DS     -0.0252917    0.06887  -0.36726  7.134e-01
#> RHO_DP     -0.0008456    0.04714  -0.01794  9.857e-01
#> RHO_SP     -0.0202279    0.05417  -0.37338  7.089e-01
#> 
#> -2 log L: 45126.03

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
#>                 bs           dr           da           sa
#> B_P   -0.100154381 -0.008859036 -0.155828672 -0.172370413
#> D_Xd1  0.057180496  0.066005139  0.058940070  0.052943279
#> D_Xd2 -0.005665398 -0.000339893  0.002614653  0.001300937
#> B_X1   0.189587066  0.068923076  0.181699873  0.203311137
#> B_X2  -0.014320520 -0.024924848 -0.048187589 -0.055576183
#> S_Xs1 -0.187895816 -0.121583902 -0.152679292 -0.165548468

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
#>                bs            dr           da           sa
#> B_P   -0.12739838 -0.0095704990 -0.195712135 -0.230149573
#> D_Xd1  0.07273474  0.0713059667  0.074025446  0.070690050
#> D_Xd2 -0.00720650 -0.0003671896  0.003283858  0.001737016
#> B_X1   0.24115855  0.0744582412  0.228204924  0.271461735
#> B_X2  -0.01821599 -0.0269265452 -0.060520929 -0.074205513
#> S_Xs1 -0.23900725 -0.1313482223 -0.191756690 -0.221040887

fit <- sa_fit
mdt <- cbind(
  fit@model@data,
  shortage_indicators = c(shortage_indicators(fit)),
  normalized_shortages = c(normalized_shortages(fit)),
  shortage_probabilities = c(shortage_probabilities(fit)),
  relative_shortages = c(relative_shortages(fit))
)

if (requireNamespace("ggplot2", quietly = TRUE)) {
  pdt <- data.frame(
    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 (", 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))])
}

market <- cbind(
  demand = demanded_quantities(fit)[, 1],
  supply = supplied_quantities(fit)[, 1]
)
summary(market)
#>      demand          supply      
#>  Min.   :3.175   Min.   : 2.529  
#>  1st Qu.:5.241   1st Qu.: 5.670  
#>  Median :5.684   Median : 6.366  
#>  Mean   :5.683   Mean   : 6.362  
#>  3rd Qu.:6.129   3rd Qu.: 7.054  
#>  Max.   :8.148   Max.   :10.747

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 <- data.frame(
    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  (", 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
}