# Interacting with a model

The API allows for:

• build/estimate models
• query models for statistics (e.g., goodness-of-fit statistics), components (parameter estimates)
• export model estimates

## Core Functionality

Econometrics.EconometricModelType
EconometricModel(estimator::Type{<:Union{EconometricModel,ModelEstimator}},
f::FormulaTerm,
data;
contrasts::Dict{Symbol} = Dict{Symbol,Union{<:AbstractContrasts,<:AbstractTerm}}(),
wts::Union{Nothing,Symbol} = nothing,
panel::Union{Nothing,Symbol} = nothing,
time::Union{Nothing,Symbol} = nothing,
vce::VCE = OIM)

Formula has syntax:

@formula(response ~ exogenous + (endogenous ~ instruments) + absorb(highdimscontrols))

Data must implement the Tables.jl API and use CategoricalArrays (CategoricalVector)

Weights are taken as StatsBase.FrequencyWeights

Panel and time indicators are used for longitudinal estimators

Examples

model = fit(EconometricModel, formula, data, kwargs...)
model = fit(BetweenEstimator, formula, data, panel = :panel, kwargs...)
model = fit(RandomEffectsEstimator, formula, data, panel = :panel, time = :time, kwargs...)
source

## Statistical/Regression Model Abstraction

StatsBase.aicFunction
aic(model::StatisticalModel)

Akaike's Information Criterion, defined as $-2 \log L + 2k$, with $L$ the likelihood of the model, and k its number of consumed degrees of freedom (as returned by dof).

StatsBase.aiccFunction
aicc(model::StatisticalModel)

Corrected Akaike's Information Criterion for small sample sizes (Hurvich and Tsai 1989), defined as $-2 \log L + 2k + 2k(k-1)/(n-k-1)$, with $L$ the likelihood of the model, $k$ its number of consumed degrees of freedom (as returned by dof), and $n$ the number of observations (as returned by nobs).

StatsBase.bicFunction
bic(model::StatisticalModel)

Bayesian Information Criterion, defined as $-2 \log L + k \log n$, with $L$ the likelihood of the model, $k$ its number of consumed degrees of freedom (as returned by dof), and $n$ the number of observations (as returned by nobs).

StatsBase.r2Function
r2(model::StatisticalModel)
r²(model::StatisticalModel)

Coefficient of determination (R-squared).

For a linear model, the R² is defined as $ESS/TSS$, with $ESS$ the explained sum of squares and $TSS$ the total sum of squares.

r2(model::StatisticalModel, variant::Symbol)
r²(model::StatisticalModel, variant::Symbol)

Pseudo-coefficient of determination (pseudo R-squared).

For nonlinear models, one of several pseudo R² definitions must be chosen via variant. Supported variants are:

• :MacFadden (a.k.a. likelihood ratio index), defined as $1 - \log (L)/\log (L_0)$;
• :CoxSnell, defined as $1 - (L_0/L)^{2/n}$;
• :Nagelkerke, defined as $(1 - (L_0/L)^{2/n})/(1 - L_0^{2/n})$.
• :devianceratio, defined as $1 - D/D_0$.

In the above formulas, $L$ is the likelihood of the model, $L_0$ is the likelihood of the null model (the model with only an intercept), $D$ is the deviance of the model (from the saturated model), $D_0$ is the deviance of the null model, $n$ is the number of observations (given by nobs).

The Cox-Snell and the deviance ratio variants both match the classical definition of R² for linear models.

StatsBase.adjr2Function
adjr2(model::StatisticalModel)
adjr²(model::StatisticalModel)

For linear models, the adjusted R² is defined as $1 - (1 - (1-R^2)(n-1)/(n-p))$, with $R^2$ the coefficient of determination, $n$ the number of observations, and $p$ the number of coefficients (including the intercept). This definition is generally known as the Wherry Formula I.

adjr2(model::StatisticalModel, variant::Symbol)
adjr²(model::StatisticalModel, variant::Symbol)

For nonlinear models, one of the several pseudo R² definitions must be chosen via variant. The only currently supported variants are :MacFadden, defined as $1 - (\log (L) - k)/\log (L0)$ and :devianceratio, defined as $1 - (D/(n-k))/(D_0/(n-1))$. In these formulas, $L$ is the likelihood of the model, $L0$ that of the null model (the model including only the intercept), $D$ is the deviance of the model, $D_0$ is the deviance of the null model, $n$ is the number of observations (given by nobs) and $k$ is the number of consumed degrees of freedom of the model (as returned by dof).

StatsBase.rssFunction
rss(model::StatisticalModel)

Return the residual sum of squares of the model.

StatsBase.devianceFunction
deviance(model::StatisticalModel)

Return the deviance of the model relative to a reference, which is usually when applicable the saturated model. It is equal, up to a constant, to $-2 \log L$, with $L$ the likelihood of the model.

StatsBase.nulldevianceFunction
nulldeviance(model::StatisticalModel)

Return the deviance of the null model, that is the one including only the intercept.

StatsBase.nullloglikelihoodFunction
loglikelihood(model::StatisticalModel)

Return the log-likelihood of the null model corresponding to model. This is usually the model containing only the intercept.

StatsBase.dofFunction
dof(model::StatisticalModel)

Return the number of degrees of freedom consumed in the model, including when applicable the intercept and the distribution's dispersion parameter.

StatsBase.coeftableMethod
coeftable(obj::EconometricModel;
level::Real = 0.95)
coeftable(obj::EconometricModel{<:LinearModelEstimators};
level::Real = 0.95,
vce::VCE = obj.vce)

Return a table of class CoefTable with coefficients and related statistics. level determines the level for confidence intervals (by default, 95%). vce determines the variance-covariance estimator (by default, OIM).

source
StatsBase.informationmatrixFunction
informationmatrix(model::StatisticalModel; expected::Bool = true)

Return the information matrix of the model. By default the Fisher information matrix is returned, while the observed information matrix can be requested with expected = false.

StatsBase.vcovMethod
vcov(obj::EconometricModel)
vcov(obj::EconometricModel{<:LinearModelEstimators}, vce::VCE = obj.vce)

Return the variance-covariance matrix for the coefficients of the model. The vce argument allows to request variance estimators.

source
StatsBase.stderrorMethod
stderror(obj::EconometricModel)
stderror(obj::EconometricModel{<:LinearModelEstimators}, vce::VCE = obj.vce)

Return the standard errors for the coefficients of the model. The vce argument allows to request variance estimators.

source
StatsBase.confintMethod
confint(obj::EconometricModel; se::AbstractVector{<:Real} = stderror(obj), level::Real = 0.95)

Compute the confidence intervals for coefficients, with confidence level level (by default, 95%). se can be provided as a precomputed value.

source
StatsBase.responseFunction
response(model::RegressionModel)

Return the model response (a.k.a. the dependent variable).

StatsBase.predictFunction
predict(model::RegressionModel, [newX])

Form the predicted response of model. An object with new covariate values newX can be supplied, which should have the same type and structure as that used to fit model; e.g. for a GLM it would generally be a DataFrame with the same variable names as the original predictors.

StatsBase.leverageFunction
leverage(model::RegressionModel)

Return the diagonal of the projection matrix of the model.

StatsBase.nobsFunction
nobs(model::StatisticalModel)

Return the number of independent observations on which the model was fitted. Be careful when using this information, as the definition of an independent observation may vary depending on the model, on the format used to pass the data, on the sampling plan (if specified), etc.

## Estimators

Econometrics.BetweenEstimatorType
BetweenEstimator(effect::Symbol,
groups::Vector{Vector{Int}}) <: LinearModelEstimators

Continuous response estimator collapsing a dimension in a longitudinal setting.

source
Econometrics.RandomEffectsEstimatorType
RandomEffectsEstimator(pid::Tuple{Symbol,Vector{Vector{Int}}},
tid::Tuple{Symbol,Vector{Vector{Int}}},
idiosyncratic::Float64,
individual::Float64,
θ::Vector{Float64}) <: LinearModelEstimators

Swamy-Arora estimator.

source
Econometrics.VCEType
VCE

Variance-covariance estimators.

• Observed Information Matrix (OIM)
• Heteroscedasticity Consistent: HC0, HC1, HC2, HC3, HC4
source