# Compare the explanatory power of parameter.dependent network centrality measures with those of standard measures of network centrality. ```r horse_race( formula = formula(), centralities = c("indegree", "outdegree", "degree", "betweenness", "incloseness", "outcloseness", "closeness", "eigenvector"), directed = FALSE, weighted = FALSE, normalization = FALSE, data = list(), unobservables = list(), G = list(), model = c("model_A", "model_B"), estimation = c("NLLS", "MLE"), endogeneity = FALSE, first_step = NULL, exclusion_restriction = NULL, start.val = NULL, to_weight = NULL, time_fixed_effect = NULL, ind_fixed_effect = NULL, mle_controls = NULL, kappa = NULL, delta = NULL ) ``` ## Arguments - `formula`: an object of class `formula`: a symbolic description of the model to be fitted. The constant (i.e. intercept) and the autogressive parameter needs not to be specified. - `centralities`: - `directed`: logical. `TRUE` if the social network is directed, `FALSE` otherwise. - `weighted`: logical. `TRUE` if the social network is weighted, `FALSE` otherwise. - `normalization`: - `data`: an object of class `data.frame` containing the variables in the model. If data are longitudinal, observations must be ordered by time period and then by individual. - `unobservables`: a numeric vector used to obtain an unbiased estimate of the parameter.dependent centrality when the network is endogenous. See details. - `G`: an object of class `Matrix` representing the social network. Row and column names must be specified and match the order of the observations in `data`. - `model`: string. One of `c("model_A","model_B")`. See details. - `estimation`: string. One of `c("NLLS","MLE")`. They are used to implement respectively a non-linear least square and a maximum likelihood estimator. - `endogeneity`: logical. Default is `FALSE`. If `TRUE`, `net_dep` implements a two-step correction procedure to control for the endogeneity of the network. - `first_step`: - `exclusion_restriction`: an object of class `Matrix` representing the exogenous matrix used to instrument the endogenous social network, if `unobservables` is non-`NULL`. Row and column names must be specified and match the order of the observations in `data`. - `start.val`: an optional list containing the starting values for the estimations. Object names must match the names provided in `formula`. It is also required to specify the value of both the constant and the decay parameter(s). - `to_weight`: an optional vector of weights to be used in the fitting process to indicate that different observations have different variances. Should be `NULL` or a numeric vector. If non-`NULL`, weighted non-linear least squares (if `estimation = "NLLS"`) or weighted maximum likelihood (if `estimation = "MLE"`) is estimated. - `time_fixed_effect`: an optional string. It indicates the name of the time index used in formula. It is used for models with longitudinal data. - `ind_fixed_effect`: an optional string. Default is `NULL`. It indicates the name of the individual index contained in the data. If provided, individual fixed effects are automatically added to the `formula` of the main equation. If `endogeneity = TRUE`, the field `first_step` is overridden, and automatically set equal to `"fe"`. It is used for models with longitudinal data. - `mle_controls`: a list allowing the user to set upper and lower bounds for control variables in MLE estimation and the variance for the ML estimator. See details. - `kappa`: a normalization level with default equals 1 used in MLE estimation. - `delta`: Default is `NULL`. To be used when `estimation = "NLLS"`. It has to be a number between zero (included) and one (excluded). When used, `econet` performs a constrained NLLS estimation. In this case, the estimated peer effect parameter, taken in absolute value, is forced to be higher than zero and lower than the spectral radius of `G`. Specifically, `delta` is a penalizing factor, decreasing the goodness of fit of the NLLS estimation, when the peer effect parameter approaches one of the two bounds. Observe that very high values of `delta` may cause NLLS estimation not to converge. ## Returns A list of two objects: * A list of estimates, each one setting the decay parameter to zero, and adding one of the `centralities` to the specification of `formula`. The last object adds to `formula` all the selected `centralities` and the decay parameter is set different from zero. * An object of class `data.frame` containing the computed centrality measures. * A list of first-step estimations used to correct the effect of centrality measures when the network is endogenous. ## Details A number of different normalization are available to the user: * `bygraph` and `bycomponent` are used to divide **degree** and **closeness** centrality by $n - 1$, and **betweenness** centrality by $(n - 1) * (n - 2)$ if `directed = TRUE`, or by $(n - 1)*(n - 2)/2$ if `directed = FALSE`. In the former case (i.e. `bygraph`), **n** is equal to the number of nodes in the network In the latter case (i.e. `bycomponent`), **n** is equal to the number of nodes of the component in which the node is embedded. * `bymaxgraph` and `bymaxcomponent` are used to divide **degree**, **betweenness** and **closeness** centrality by the maximum value of the centrality of the network (`bymaxgraph`) or component (`bymaxcomponent`) in which the node is embedded. If the network is endogenous, the user is required to run separately `net_dep` and extract from the resulting object the vector of unobservables necessary for obtaining an unbiased estimate of the parameter.dependent centrality. This vector can be passed through the argument `unobservables`. If `endogeneity = TRUE`, a two-step estimation is implemented to control for network endogeneity. The argument `first_step` is used to control for the specification of the first-step model, e.g.: * `first_step = "standard"` is used when agents' connection are predicted by the differences in their characteristics (i.e. those on the right hand side of `formula`), and an `exclusion_restriction`: i.e., their connections in a different network. * `first_step = "fe"` adds individual fixed effects to the `standard` model, as in Graham (2017). * `first_step = "shortest"` adds to the `standard` model, the shortest distance between **i** and **j**, excluding the link between **i** and **j** itself, as in Fafchamps et al (2010). * `first_step = "coauthor"` adds to the `standard` model, the number of shared connections between **i** and **j**, as in Graham (2015). * `first_step = "degree"` adds to the `standard` model, the difference in the degree centrality of **i** and **j**. For additional details, see the vignette (doi:10.18637/jss.v102.i08). ## Examples ```r # Load data data("db_cosponsor") data("G_alumni_111") db_model_B <- db_cosponsor G_model_B <- G_cosponsor_111 G_exclusion_restriction <- G_alumni_111 are_factors <- c("party", "gender", "nchair") db_model_B[are_factors] <- lapply(db_model_B[are_factors], factor) # Specify formula f_model_B <- formula("les ~gender + party + nchair") # Specify starting values starting <- c(alpha = 0.214094, beta_gender1 = -0.212706, beta_party1 = 0.478518, beta_nchair1 = 3.09234, beta_betweenness = 7.06287e-05, phi = 0.344787) # Fit model horse_model_B <- horse_race(formula = f_model_B, centralities = "betweenness", directed = TRUE, weighted = TRUE, data = db_model_B, G = G_model_B, model = "model_B", estimation = "NLLS", start.val = starting) # Store and print results summary(horse_model_B) summary(horse_model_B, centrality = "betweenness") horse_model_B$centrality # WARNING, This toy example is provided only for runtime execution. # Please refer to previous examples for sensible calculations. data("db_alumni_test") data("G_model_A_test") db_model <- db_alumni_test G_model <- G_model_A_test f_model <- formula("les ~ dw") horse_model_test <- horse_race(formula = f_model, centralities = "betweenness", directed = TRUE, weighted = FALSE, normalization = NULL, data = db_model, unobservables = NULL, G = G_model, model = "model_A", estimation = "NLLS", start.val = c(alpha = -0.31055275, beta_dw = 1.50666982, beta_betweenness = 0.09666742, phi = 16.13035695)) summary(horse_model_test) ``` ## References Battaglini M., V. Leone Sciabolazza, E. Patacchini, S. Peng (2020), "Econet: An R package for the Estimation of parameter-dependent centrality measures", Mimeo. ## See Also `net_dep`

horse_race function