Package {netpanel}

netpanel: Maximum Likelihood Estimation Routines for Panel Network Autocorrelation Models

Description

A set of maximum likelihood estimation routines for panel network autocorrelation models. The package contains functionality for fixed and random effects panel network autocorrelation model, in addition to dynamic panel network autocorrelation that contains time lags of the response and networks. Moreover, the package allows for multiple networks to be fitted. For information on the types of models estimated in this package, please see Anselin (1988) doi:10.1007/978-94-015-7799-1, Cook et al. (2023) doi:10.1017/S0003055422000272, Hays et al. (2010) doi:10.1016/j.stamet.2009.11.005, Lee and Yu (2012) doi:10.1111/j.1468-2354.2012.00724.x, Millo (2014) doi:10.1016/j.csda.2013.07.024, Millo and Piras (2012) doi:10.18637/jss.v047.i01, and Wang and Yu (2015) doi:10.1016/j.econlet.2015.01.021.

Author(s)

Maintainer: Kevin Carson kacarson@arizona.edu

See Also

Useful links:

• https://github.com/kevinCarson/netpanel

• Report bugs at https://github.com/kevinCarson/netpanel/issues

Extract the ML parameter estimates from pnam model fits

Description

This function extracts the Maximum Likelihood (ML) parameter estimates for the exogenous covariates from a pnam model fit.

Usage

## S3 method for class 'pnam'
coef(object, ...)

Arguments

Value

The vector of estimated parameter values for the exogenous covariates.

Examples

data("net1", package = "netpanel") #the first panel network
data("simulated.data", package = "netpanel") #the simulated variables
#' a fixed effects panel network autocorrelation model
fe.pnam <- mlpnam(Y~x1+x2+x3, net.formula = ~ net1,
               data =  simulated.data, time = ~panel,
               actor = ~unit, model = "fixed",
               fixed.effect = "actor")
coef(fe.pnam) #the mle parameter values

Fitted values from pnam model fits

Description

This function returns the model fitted values, \hat{y}, for a fitted pnam model. The function returns two types of fitted values to the user: linear regression fitted values and network regression fitted values. The network regression fitted values are formulated as: \hat{y} = (I_{NT} - \sum_i \hat{\rho}_i W^i)^{-1}(X\hat{\beta}), where W^i is the i^{th} included network. In comparison, the linear regression fitted values are formulated as: \hat{y} = X\hat{\beta}.

Usage

## S3 method for class 'pnam'
fitted(object, ...)

Arguments

Value

A data.frame object that stores the fitted values from a pnam model fit, where the column "ols.fitted" stores the linear regression fitted values and "network.fitted" stores the network regression fitted values (see above).

Examples

data("net1", package = "netpanel") #the first panel network
data("simulated.data", package = "netpanel") #the simulated variables

# a fixed effects panel network autocorrelation model
fe.pnam <- mlpnam(Y~x1+x2+x3, net.formula = ~ net1,
               data =  simulated.data, time = ~panel,
               actor = ~unit, model = "fixed",
               fixed.effect = "actor")
fitted(fe.pnam) #the data.frame of fitted values

Extract the estimated model log-likelihood from pnam model fits

Description

This function extracts the estimated model loglikelhood from a pnam model fit.

Usage

## S3 method for class 'pnam'
logLik(object, ..., REML = FALSE)

Arguments

Value

The estimated model log-likelihood.

Examples

data("net1", package = "netpanel") #the first panel network
data("simulated.data", package = "netpanel") #the simulated variables

# a fixed effects panel network autocorrelation model
fe.pnam <- mlpnam(Y~x1+x2+x3, net.formula = ~ net1,
               data =  simulated.data, time = ~panel,
               actor = ~unit, model = "fixed",
               fixed.effect = "actor")
logLik(fe.pnam) #the estimated model log-likelihood.

Fit a Maximum Likelihood Panel Network Autocorrelation Model (PNAM)

Description

A function that contains Maximum Likelihood estimation routines to fit panel network autocorrelation models (MLPNAM). The function allows for (1) fixed and random actor-level effects, (2) the inclusion of time lagged outcome and network terms, (3) multiple networks to construct the network (spatial) filtering matrix, and (4) for the time-varying errors to exhibit network autocorrelation, that is, for the model to become a network mixed effects model (i.e., error + lag model). Please see the Details section for more information.

Usage

mlpnam(
  reg.formula,
  net.formula,
  data,
  time,
  actor,
  model = c("fixed", "random", "dynamic"),
  fixed.effect = c("actor", "time", "two-way"),
  dynamic.lag = c("network", "outcome", "network x outcome"),
  errors = c("idiosyncratic", "autocorrelated"),
  autocorrelated.network = NULL,
  optim.method = "L-BFGS-B",
  optim.control = list(),
  ...
)

Arguments

Details

This function fits a panel network autocorrelation model with time-varying social adjacency matrices via Maximum Likelihood Estimation. The estimated model takes the form:

Y_t = \rho_1 W^{t}_{1}Y_t + \cdots + \rho_k W^{t}_{k}Y_t + X_t\beta + v_t, \ \ t = 1,2,\cdots,T

v_t = \mu_N + e_t,\ \ t = 1,2,\cdots,T

where Y_t is the outcome vector for N actors at time t, \rho_k W^{t}_{k}Y_t is the network autocorrelation term for the k^{th} social network at time t (taken from the net.formula argument), X_t are the set of exogenous time-varying (and invariant) regressors (taken from the reg.formula argument). Additionally, v_t is the time-varying error term that is the sum of the time-invariant unit-specific unobserved factors, \mu_N, thought to explain Y_t and e_t are the time- and unit-varying unobserved factors. Dependent upon the model argument, the function will (1) estimate a fixed-effects network regression, where the within estimator demeans the data by time, actor, or both, (2) estimate a random effects network regression, or (3) estimate a dynamic panel network autocorrelation where the time lags of the outcome and network variables are added to the set of exogenous covariates. Please see the set of references below for more information of network (spatial) cross-sectional and panel models and their respective estimation strategies (e.g., fixed effects).

Moreover, the model can also estimate the network mixed effects model, where the estimated formula is:

Y_t = \rho_1 W^{t}_{1}Y_t + \cdots + \rho_k W^{t}_{k}Y_t + X_t\beta + v_t, \ \ t = 1,2,\cdots,T

v_t = \mu_N + e_t,\ \ t = 1,2,\cdots,T

e_t = \lambda W^e_t e_t + \epsilon_t, \ \ t = 1,2,\cdots,T

where \lambda is the error autocorrelation parameter, W^e_t is the error social adjacency matrix (if applicable, the network included in the autocorrelated.network argument), and \epsilon_t are a set of i.i.d normal errors at time t.

Internally, the function follows the two-step computation strategy discussed in Millo and Piras (2012). In the first step, the set of parameter estimates for \rho, \lambda, and \theta are found via numerical optimization routines. Importantly, within each search, the values for \beta and \sigma^2 are updated via their GLS estimators. In the second step, once the ML parameter estimates for \rho, \lambda, and \theta are found, the final GLS estimates for \beta and \sigma^2 are computed. The asymptotic variance-covariance matrix for \rho, \lambda, and \theta are the inverse of the negative Hessian matrix and the variance-covariance matrix for \beta is found via the standard GLS variance estimator. For sake of brevity, please see the below listed references for more information on these estimators.

Value

An object of class pnam as a list containing the following elements:

• N - The number of unique units in the panel network dataset.

• t - The number of panels (time periods) in the panel network dataset.

• type - The type of model estimated per the model argument.

• error.structure - The assumed structure for the time-varying errors per the errors argument.

• fe.type - If a fixed effects model is estimated, the type of fixed effects demeaning per the fixed.effect argument.

• lag.type - If a dynamic model is estimated, the type of lag structure included in the model per the dynamic.lag argument

• max.eigen - The maximum absolute eigenvalue of the rho*W matrix.

• lambda - If the time-varying errors are assumed to be autocorrelated, the estimated value for lambda.

• rho - The vector of estimated maximum likelihood rho parameters.

• vcov.rho - The asymptotic variance-covariance of the estimated rho parameters.

• coefficients - The vector of estimated exogenous ML parameters.

• vcov - The asymptotic variance-covariance of the exogenous ML parameters.

• sigma2 - The estimated variance of the time-varying residual.

• residuals - The residuals of the fitted model.

• fitted.values - The fitted values from the estimated model.

• logLik - The estimated model log-likelihood.

• convergence - The converge value returned from the optim function. 0 indicates successful convergence.

• netfilter - The filtering matrix of the estimated model (I_{NT} - \{\sum_{j} rho_j*W^j \}), where W^j is the jth network from the net.formula argument.

• k - The total number of estimated parameters.

• df - The model degrees of freedom.

• AIC - The AIC of the estimated PNAM.

• BIC - The BIC of the estimated PNAM.

• response - The outcome vector (Y).

• covariates - The covariate matrix (X).

• optim.information - The returned list from the optim function.

• error.dis - If a random effects model is estimated, a list that contains the residual variances for the time-varying residuals (idiosyncratic), the unit-/actor-specific residual variances (unit), and the estimated ratio of the variances (theta).

• theta - If a random effects model is estimated, the estimated value for theta (the ratio of the unit and idiosyncratic variances.)

• vcov.theta - If a random effects model is estimated, the asymptotic variance of the estimated theta value.

References

Anselin, Luc. 1988. Spatial Econometrics: Methods and Models. Dordrecht: Springer Netherlands.

Cook, Scott J., Jude C. Hays and Robert J. Franzese. 2023. “STADL Up! The Spatiotemporal Autoregressive Distributed Lag Model for TSCS Data Analysis.” American Political Science Review 117(1):59–79.

Doreian, Patrick. 1982. “Maximum Likelihood Methods for Linear Models: Spatial Effect and Spatial Disturbance Terms.” Sociological Methods & Research 10(3):243–269.

Doriean, Patrick, Klaus Teuter and Chi-Hsein Wang. 1984. “Network Autocorrelation Models: Some Monte Carlo Results.” Sociological Methods & Research 13(2):155–200.

Duxbury, Scott. 2023. Longitudinal Network Models. Vol. 192 of Quantitative Applications in the Social Sciences SAGE Publications.

Hays, Jude C., Aya Kachi, and Robert J. Franzese. 2010. "A Spatial Model Incorporating Dynamic, Endogenous Network Interdependence: A Political Science Application." Statistical Methodology 7(3): 406-428.

Lee, Lung-fei and Jihai Yu. 2010. “Estimation of Spatial Autoregressive Panel Data Models with Fixed Effects.” Journal of Econometrics 154(2):165–185.

Lee, Lung-fei and Jihai Yu. 2012a. “QML Estimation of Spatial Dynamic Panel Data Models with Time Varying Spatial Weights Matrices.” Spatial Economic Analysis 7(1):31–74.

Lee, Lung-fei and Jihai Yu. 2012b. “Spatial Panels: Random Components Versus Fixed Effects.” International Economic Review 53(4):1369–1412.

Millo, Giovanni. 2014. "Maximum Likelihood Estimation of Spatially and Serially Correlated Panels with Random Effects." Computational Statistics & Data Analysis 71: 914-933.

Millo, Giovanni and Gianfranco Piras. 2012. "splm: Spatial Panel Data Models in R." Journal of Statistical Software 47: 1-38.

Neuman, Eric J. and Mark S. Mizruchi. 2010. “Structure and Bias in the Network Autocorrelation Model.” Social Networks 32(4):290–300.

Wang, Wei and Jihai Yu. 2015. "Estimation of Spatial Panel Data Models with Time Varying Spatial Weights Matrices.” Economic Letters 128:95-99.

See Also

coef.pnam, vcov.pnam, logLik.pnam, AIC, residuals.pnam, fitted.pnam, netimpacts.

Examples

data("net1", package = "netpanel") #the first panel network
data("net2", package = "netpanel") #the second panel network
data("net3", package = "netpanel") #the third panel network
data("simulated.data", package = "netpanel") #the simulated variables 

# a random effects panel network autocorrelation model
re.pnam <- mlpnam(Y~x1+x2+x3, net.formula = ~ net1,
               data =  simulated.data, time = ~panel,
               actor = ~unit, model = "random")
summary(re.pnam)              

# a fixed effects panel network autocorrelation model 
fe.pnam <- mlpnam(Y~x1+x2+x3, net.formula = ~  net1 + net2,
               data =  simulated.data, time = ~panel,
               actor = ~unit, model = "fixed",
               fixed.effect = "actor")
summary(fe.pnam)   

# a dynamic panel network autocorrelation model with autocorrelated errors
dyn.pnam <- mlpnam(Y~x1+x2+x3, net.formula = ~  net1 + net2,
               data =  simulated.data, time = ~panel,
               actor = ~unit, model = "dynamic",
               dynamic.lag = "outcome")
summary(dyn.pnam)   



# a fixed effects panel network autocorrelation model with autocorrelated errors
fe.pnam.error <- mlpnam(Y~x1+x2+x3, net.formula = ~ net1,
               data =  simulated.data, time = ~panel,
               actor = ~unit, model = "fixed",
               errors = "autocorrelated",
               autocorrelated.network = net3)
summary(fe.pnam.error)   

A list of simulated panel social networks (1)

Description

A list of length 10 where each element is an simulated row-normalized 50 x 50 social adjacency matrix, where a non-zero W_{ijt} entry denotes a directed relationship from i to j at time t.

Usage

data(net1)

Format

net1

A list of length 10 where each element is an simulated row-normalized 50 x 50 social adjacency matrix, where a non-zero W_{ijt} entry denotes a directed relationship from i to j at time t.

A list of simulated panel social networks (2)

Description

A list of length 10 where each element is an simulated row-normalized 50 x 50 social adjacency matrix, where a non-zero W_{ijt} entry denotes a directed relationship from i to j at time t.

Usage

data(net2)

Format

net2

A list of length 10 where each element is an simulated row-normalized 50 x 50 social adjacency matrix, where a non-zero W_{ijt} entry denotes a directed relationship from i to j at time t.

A list of simulated panel social networks (3)

Description

A list of length 10 where each element is an simulated row-normalized 50 x 50 social adjacency matrix, where a non-zero W_{ijt} entry denotes a directed relationship from i to j at time t.

Usage

data(net3)

Format

net3

A list of length 10 where each element is an simulated row-normalized 50 x 50 social adjacency matrix, where a non-zero W_{ijt} entry denotes a directed relationship from i to j at time t.

Compute the direct, indirect, and total impacts from a fitted pnam object

Description

This function returns the estimated average direct, indirect, and total impacts from a fitted panel network autocorrelation model.

Usage

netimpacts(object, return.impact.matrix = FALSE, digits = 5)

Arguments

Details

In classical linear regression, the marginal effect of a regressor, x_1, on the outcome, Y, is \hat{\beta_1}. In panel network (and spatial) effects models, the marginal effect is:

\frac{\partial Y}{\partial x_1} = (I_{NT} - \sum_k \hat{\rho_k}W^{k})^{-1}\hat{\beta_1}

The direct impact is the average of the diagonal value of the above matrix, the indirect impact is the average of the row sum of the off-diagonal values, and the total impact is sum of both. The function will also return the impact matrix for each estimated effect if return.impact.matrix is set to TRUE.

Value

A data.frame object that stores the direct, indirect, and total impact for each estimated effect in the fitted pnam model.

Examples

data("net1", package = "netpanel") #the first panel network
data("simulated.data", package = "netpanel") #the simulated variables

# a fixed effects panel network autocorrelation model
fe.pnam <- mlpnam(Y~x1+x2+x3, net.formula = ~ net1,
               data =  simulated.data, time = ~panel,
               actor = ~unit, model = "fixed",
               fixed.effect = "actor")
netimpacts(fe.pnam) #compute the impact values

Maximum Likelihood Estimation Routines for Panel Network Autocorrelation Models

Description

The netpanel package contains maximum likelihood estimation routines to fit panel network autocorrelation models (PNAMs) with time-varying social adjacency matrices. The package contains functionality for (1) fixed and random effects PNAMs, (2) dynamic PNAMs that include time lags of the outcome and networks, and (3) the estimation of network mixed effects models where, in addition to the lag of the outcome, the time-varying residuals are assumed to exhibit network autocorrelation. Importantly, the package allows for multiple network lags to be fitted simultaneously. Please see the list of below references for more information on panel network (spatial) autocorrelation models.

Details

The netpanel package primarily works through the wrapper function mlpnam. The mlpnam function returns an S3 object of class pnam that stores the relevant results of the fitted PNAM. The full set of functions (including user-helper S3 methods) are:

• mlpnam: fits a panel network autocorrelation model to an empirical panel network dataset via Maximum Likelihood Estimation.

• netimpacts: computes the direct, indirect, and total network impacts for each included regressor from a fitted pnam S3 object.

• coef.pnam: extracts the vector of ML parameter estimates from a fitted pnam S3 object.

• vcov.pnam: extracts the asymptotic variance-covariance matrix from a fitted pnam S3 object.

• logLik.pnam: extracts the estimated log-likelihood from a fitted pnam S3 object.

• residuals.pnam: extracts the model residuals from a fitted pnam S3 object.

• fitted.pnam: extracts the linear and network regression fitted values from a fitted pnam S3 object.

Additionally, the netpanel package includes a set of simulated pseudo data objects for users to start fitting panel network autocorrelation models. Please see the simulated.data netpanel help page for more information on the simulated dataset and respective networks.

Author(s)

Kevin A. Carson kacarson@arizona.edu

References

Anselin, Luc. 1988. Spatial Econometrics: Methods and Models. Dordrecht: Springer Netherlands.

Cook, Scott J., Jude C. Hays and Robert J. Franzese. 2023. “STADL Up! The Spatiotemporal Autoregressive Distributed Lag Model for TSCS Data Analysis.” American Political Science Review 117(1):59–79.

Doreian, Patrick. 1982. “Maximum Likelihood Methods for Linear Models: Spatial Effect and Spatial Disturbance Terms.” Sociological Methods & Research 10(3):243–269.

Doriean, Patrick, Klaus Teuter and Chi-Hsein Wang. 1984. “Network Autocorrelation Models: Some Monte Carlo Results.” Sociological Methods & Research 13(2):155–200.

Duxbury, Scott. 2023. Longitudinal Network Models. Vol. 192 of Quantitative Applications in the Social Sciences SAGE Publications.

Hays, Jude C., Aya Kachi, and Robert J. Franzese. 2010. "A Spatial Model Incorporating Dynamic, Endogenous Network Interdependence: A Political Science Application." Statistical Methodology 7(3): 406-428.

Lee, Lung-fei and Jihai Yu. 2010. “Estimation of Spatial Autoregressive Panel Data Models with Fixed Effects.” Journal of Econometrics 154(2):165–185.

Lee, Lung-fei and Jihai Yu. 2012a. “QML Estimation of Spatial Dynamic Panel Data Models with Time Varying Spatial Weights Matrices.” Spatial Economic Analysis 7(1):31–74.

Lee, Lung-fei and Jihai Yu. 2012b. “Spatial Panels: Random Components Versus Fixed Effects.” International Economic Review 53(4):1369–1412.

Millo, Giovanni. 2014. "Maximum Likelihood Estimation of Spatially and Serially Correlated Panels with Random Effects." Computational Statistics & Data Analysis 71: 914-933.

Millo, Giovanni and Gianfranco Piras. 2012. "splm: Spatial Panel Data Models in R." Journal of Statistical Software 47: 1-38.

Neuman, Eric J. and Mark S. Mizruchi. 2010. “Structure and Bias in the Network Autocorrelation Model.” Social Networks 32(4):290–300.

Wang, Wei and Jihai Yu. 2015. "Estimation of Spatial Panel Data Models with Time Varying Spatial Weights Matrices.” Economic Letters 128:95-99.

See Also

Useful links:

• https://github.com/kevinCarson/netpanel

• Report bugs at https://github.com/kevinCarson/netpanel/issues

Print Method for pnam model objects.

Description

Print Method for pnam model objects.

Usage

## S3 method for class 'pnam'
print(x, digits = 5, ...)

Arguments

Value

No return value. Prints out the main results of a pnam object.

Print Method for summary.pnam Panel Network Autocorrelation Model Summary

Description

Print Method for summary.pnam Panel Network Autocorrelation Model Summary

Usage

## S3 method for class 'summary.pnam'
print(x, digits = 5, ...)

Arguments

Value

No return value. Prints out the summary of a pnam object.

Residuals for pnam model fits

Description

This function returns the model residuals for a pnam model fit.

Usage

## S3 method for class 'pnam'
residuals(object, ...)

Arguments

Value

The vector of pnam model fit residuals.

Examples

data("net1", package = "netpanel") #the first panel network
data("simulated.data", package = "netpanel") #the simulated variables

# a fixed effects panel network autocorrelation model
fe.pnam <- mlpnam(Y~x1+x2+x3, net.formula = ~ net1,
               data =  simulated.data, time = ~panel,
               actor = ~unit, model = "fixed",
               fixed.effect = "actor")
residuals(fe.pnam) #the vector of model residuals

Simulated panel network dataset

Description

A simulated panel network dataset that contains a response variable (Y) and three exogenous regressors (x_1, x_2, and x_3). The exogenous regresors x_1 and x_3 vectors are drawn from standard normal distribution, and x_2 is a vector of random bernoulli variables. The panel variable represents the time of data collection and the unit variable indexes the unique actors across the panels. The outcome is generated from the following deterministic equation:

Y_t = \rho_1 W^1 Y_t + \rho_2 W^w Y_t + \beta_0 + \beta_1 x_t1 + \beta_2 x_t2 + \beta_3 x_t3 + v_t, \ \ t = 1,2,\cdots,10

v_t = \mu_t + e_t, \ \ t = 1,2,\cdots,10

e_t = \lambda W^3 e_t + \epsilon_t, \ \ t = 1,2,\cdots,10

where W^1 is the net1 netpanel data object, W^2 is the net2 netpanel data object, and W^3 is the net3 netpanel data object.

Usage

data(simulated.data)

Format

simulated.data

A simulated panel network dataset with size 500 \times 5 the following variables: Y, x_1, x_2, x_3, panel, and unit.

Summary Method for pnam Objects

Description

Summarizes the results of a fitted pnam model object.

Usage

## S3 method for class 'pnam'
summary(object, digits = 5, ...)

Arguments

Value

A list of summary statistics for the panel network autocorrelation model, such as the estimated log-likelihood and the estimated parameters.

Extract the asymptotic variance-covariance matrix from pnam model fits

Description

This function extracts the asymptotic variance-covariance matrix from pnam model fits for the exogenous covariates. For the endogenous variance matrices (such as the variance-covariance matrix for \rho), please extract those from the model fit. For example, the variance matrix for \rho can be extracted as model$vcov.rho, where model is the pnam object name.

Usage

## S3 method for class 'pnam'
vcov(object, ...)

Arguments

Value

The asymptotic variance-covariance matrix for the exogenous covariates.

Examples

data("net1", package = "netpanel") #the first panel network
data("simulated.data", package = "netpanel") #the simulated variables

# a fixed effects panel network autocorrelation model
fe.pnam <- mlpnam(Y~x1+x2+x3, net.formula = ~ net1,
               data =  simulated.data, time = ~panel,
               actor = ~unit, model = "fixed",
               fixed.effect = "actor")
vcov(fe.pnam) #the asymptotic variance-covariance matrix