Information Matrix Test for Model Misspecification

Performs the Information Matrix (IM) test for misspecification on models fitted with the mlmodels package.

Usage

IMtest(object, ...)

# S3 method for class 'mlmodel'
IMtest(object, method = "quad", repetitions = 999, seed = 1234L, ...)

Arguments

object

A fitted model object inheriting from "mlmodel".

...

Further arguments passed to methods (currently not used).

method

Character string. Specifies the version of the test:

"quad" (default): Quadratic form of the Information Matrix test (most common).
"opg": Outer Product of Gradients version (Chesher-Lancaster).
"boot_quad": Analytical chi-square and p-value, plus bootstrap p-value for the quadratic form.
"boot_opg": Analytical chi-square and p-value, plus bootstrap p-value for the OPG version.

repetitions

Integer. Number of bootstrap replications when using a bootstrap method. Default is 999.

seed

Integer. Random seed for reproducibility in bootstrap methods. If NULL, a random seed is generated.

Value

An object of class "IMtest.mlmodel" containing the analytical test statistic, degrees of freedom and p-value, plus the bootstrapped p-value (if a bootstrap method was selected).

Details

The Information Matrix test checks whether the model is correctly specified by testing the equality between the Hessian and the outer product of the gradient (information matrix equality). Rejection of the null hypothesis indicates model misspecification (e.g., incorrect functional form, heteroskedasticity not properly modeled, omitted variables, etc.).

Two main versions are implemented:

Quadratic form ("quad"): Generally preferred for its better finite-sample properties.
OPG version ("opg"): Chesher and Lancaster (1983) version.

Bootstrap versions ("boot_quad" and "boot_opg") provide p-values based on the empirical distribution of the test statistic and are useful when asymptotic approximations may be unreliable.

References

Chesher, A. (1983). The information matrix test: Simplified calculation via a score test interpretation. Economics Letters, 13(1), 45-48.

Lancaster, T. (1984). The covariance matrix of the information matrix test. Econometrica, 52(4), 1051-1053.

White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50(1), 1-25.

Author

Alfonso Sanchez-Penalver

Examples


# Linear model example
data(mroz)
mroz$incthou <- mroz$faminc / 1000

fit <- ml_lm(incthou ~ age + I(age^2) + huswage + educ + unem, 
             data = mroz)

# Default quadratic form test
IMtest(fit)
#> Information Matrix Test
#>  Method: Orthogonalized Quadratic Form 
#>  Model:  Homoskedastic Linear Model 
#> --------------------------------------------
#>  Chisq(28) = 122.732    Pr(>Chisq) = 0.0000
#> --------------------------------------------

# OPG version
IMtest(fit, method = "opg")
#> Information Matrix Test
#>  Method: Chesher/Lancaster OPG 
#>  Model:  Homoskedastic Linear Model 
#> --------------------------------------------
#>  Chisq(28) = 298.563    Pr(>Chisq) = 0.0000
#> --------------------------------------------

# Bootstrap p-value (quadratic form)
IMtest(fit, method = "boot_quad", repetitions = 100, seed = 123)
#> Information Matrix Test
#>  Method: Orthogonalized Quadratic Form + Model-based bootstrap 
#>  Model:  Homoskedastic Linear Model 
#> --------------------------------------------
#>  Repetitions: Total 100 Successful 100 
#>  Chisq(28) = 122.732
#>  P(>Chisq): Analytical   = 0.0000 
#>             Bootstrapped = 0.7700
#> --------------------------------------------

# Heteroskedastic model
fit_het <- ml_lm(incthou ~ age + I(age^2) + huswage + educ + unem,
                 scale = ~ educ, data = mroz)
IMtest(fit_het)
#> Information Matrix Test
#>  Method: Orthogonalized Quadratic Form 
#>  Model:  Heteroskedastic Linear Model 
#> --------------------------------------------
#>  Chisq(36) = 163.110    Pr(>Chisq) = 0.0000
#> --------------------------------------------