Regression Diagnostics and Specification Tests

Introduction

In many cases of statistical analysis, we are not sure whether our statistical model is correctly specified. For example when using ols, then linearity and homoscedasticity are assumed, some test statistics additionally assume that the errors are normally distributed or that we have a large sample. Since our results depend on these statistical assumptions, the results are only correct of our assumptions hold (at least approximately).

One solution to the problem of uncertainty about the correct specification is to use robust methods, for example robust regression or robust covariance (sandwich) estimators. The second approach is to test whether our sample is consistent with these assumptions.

The following briefly summarizes specification and diagnostics tests for linear regression.

Heteroscedasticity Tests

For these test the null hypothesis is that all observations have the same error variance, i.e. errors are homoscedastic. The tests differ in which kind of heteroscedasticity is considered as alternative hypothesis. They also vary in the power of the test for different types of heteroscedasticity.

het_breuschpagan

Lagrange Multiplier Heteroscedasticity Test by Breusch-Pagan

het_white

Lagrange Multiplier Heteroscedasticity Test by White

het_goldfeldquandt

test whether variance is the same in 2 subsamples

Autocorrelation Tests

This group of test whether the regression residuals are not autocorrelated. They assume that observations are ordered by time.

durbin_watson

  • Durbin-Watson test for no autocorrelation of residuals

  • printed with summary()

acorr_ljungbox

  • Ljung-Box test for no autocorrelation of residuals

  • also returns Box-Pierce statistic

acorr_breusch_godfrey

  • Breusch-Pagan test for no autocorrelation of residuals

missing

  • ?

Non-Linearity Tests

linear_harvey_collier

  • Multiplier test for Null hypothesis that linear specification is correct

acorr_linear_rainbow

  • Multiplier test for Null hypothesis that linear specification is correct.

acorr_linear_lm

  • Lagrange Multiplier test for Null hypothesis that linear specification is correct. This tests against specific functional alternatives.

spec_white

  • White’s two-moment specification test with null hypothesis of homoscedastic and correctly specified.

Tests for Structural Change, Parameter Stability

Test whether all or some regression coefficient are constant over the entire data sample.

Known Change Point

OneWayLS :

  • flexible ols wrapper for testing identical regression coefficients across predefined subsamples (eg. groups)

missing

  • predictive test: Greene, number of observations in subsample is smaller than number of regressors

Unknown Change Point

breaks_cusumolsresid

  • cusum test for parameter stability based on ols residuals

breaks_hansen

  • test for model stability, breaks in parameters for ols, Hansen 1992

recursive_olsresiduals

Calculate recursive ols with residuals and cusum test statistic. This is currently mainly helper function for recursive residual based tests. However, since it uses recursive updating and doesn’t estimate separate problems it should be also quite efficient as expanding OLS function.

missing

  • supLM, expLM, aveLM (Andrews, Andrews/Ploberger)

  • R-structchange also has musum (moving cumulative sum tests)

  • test on recursive parameter estimates, which are there?

Mutlicollinearity Tests

conditionnum (statsmodels.stattools)

  • – needs test vs Stata –

  • cf Grene (3rd ed.) pp 57-8

numpy.linalg.cond

  • (for more general condition numbers, but no behind the scenes help for design preparation)

Variance Inflation Factors

This is currently together with influence and outlier measures (with some links to other tests here: http://www.stata.com/help.cgi?vif)

Normality and Distribution Tests

jarque_bera

  • printed with summary()

  • test for normal distribution of residuals

Normality tests in scipy stats

need to find list again

omni_normtest

  • test for normal distribution of residuals

  • printed with summary()

normal_ad

  • Anderson Darling test for normality with estimated mean and variance

kstest_normal lilliefors

Lilliefors test for normality, this is a Kolmogorov-Smirnov tes with for normality with estimated mean and variance. lilliefors is an alias for kstest_normal

qqplot, scipy.stats.probplot

other goodness-of-fit tests for distributions in scipy.stats and enhancements

  • kolmogorov-smirnov

  • anderson : Anderson-Darling

  • likelihood-ratio, …

  • chisquare tests, powerdiscrepancy : needs wrapping (for binning)

Outlier and Influence Diagnostic Measures

These measures try to identify observations that are outliers, with large residual, or observations that have a large influence on the regression estimates. Robust Regression, RLM, can be used to both estimate in an outlier robust way as well as identify outlier. The advantage of RLM that the estimation results are not strongly influenced even if there are many outliers, while most of the other measures are better in identifying individual outliers and might not be able to identify groups of outliers.

RLM

example from example_rlm.py

import statsmodels.api as sm

### Example for using Huber's T norm with the default
### median absolute deviation scaling

data = sm.datasets.stackloss.load()
data.exog = sm.add_constant(data.exog)
huber_t = sm.RLM(data.endog, data.exog, M=sm.robust.norms.HuberT())
hub_results = huber_t.fit()
print(hub_results.weights)

And the weights give an idea of how much a particular observation is down-weighted according to the scaling asked for.

Influence

Class in stats.outliers_influence, most standard measures for outliers and influence are available as methods or attributes given a fitted OLS model. This is mainly written for OLS, some but not all measures are also valid for other models. Some of these statistics can be calculated from an OLS results instance, others require that an OLS is estimated for each left out variable.

  • resid_press

  • resid_studentized_external

  • resid_studentized_internal

  • ess_press

  • hat_matrix_diag

  • cooks_distance - Cook’s Distance Wikipedia (with some other links)

  • cov_ratio

  • dfbetas

  • dffits

  • dffits_internal

  • det_cov_params_not_obsi

  • params_not_obsi

  • sigma2_not_obsi

Unit Root Tests

unitroot_adf

  • same as adfuller but with different signature