Test for heteroskedasticity

Question

Suppose that your residual plot indicates the presence of heteroskedasticity. What tests could you perform to formally test this?

Answer 1

In their book, Fox and Weisberg (2011) talk about score tests for nonconstant error variance (in a linear model) suggested by the earlier studies (Breusch and Pagan, 1979; Cook and Weisberg 1983). Basically, it tests the hypothesis of constant error variance against the alternative that the error variance changes with the level of the fitted values (Kabacoff, 2011). Fox and Weisberg explained "the idea is that either the variance is constant or it depends on the mean,

$$\text{Var}(\varepsilon_i)=\sigma^2g[E(y|x)]$$

or on a linear combination of regressors $z_1,\ldots,z_p$,

$$\text{Var}(\varepsilon_i)=\sigma^2g(\gamma_iz_{i1}+\cdots+\gamma_pz_{ip})\text{ "}$$

(Fox and Weisberg, 2011: 316)

You can find an implementation of this test in car R package (by Fox and Weisberg) as ncvTest function. Here is the documentation.

---

Breusch, T. S., & Pagan, A. R. (1979). A simple test for heteroscedasticity and random coefficient variation. Econometrica, 47(5), 1287–1294.

Cook, R. D., & Weisberg, S. (1983). Diagnostics for heteroscedasticity in regression. Biometrika, 70(1), 1–10.

Fox, J., & Weisberg, S. (2011). An R Companion to Applied Regression. Los Angeles: Sage.

Kabacoff, R. I. (2011). R in Action: Data analysis and graphics with R. Shelter Island: Manning.

Answer 2

• I am not sure, If I get your question right. But, I would still try to answer it. Error terms should not have a constant covariance if the residual is heretoskedastic.
  (If there is no heteroskedasticity, then the squared residuals should neither increase nor decrease in magnitude as the other axes increases)
  
• Also the test statistic should be insignificant.

Answer 3

I'll list a few tests out there, but leave out the details for now.

• Omnibus tests: detect presence of heteroscedasticity
  • Goldfeld-Quandt (GQ Test)
  • Szroeter
• Constructive tests: trying to determine the form of heteroscedasticity
  • Breusch-Pagan-Goldfrey-White
  • Glejser
  • Park (least squares or newer GLM version)