heteroscedasticitytime seriesvariance
I have a data frame with prices/dates, is there a method that checks if the variance is homogeneous during all the series? I know there are many test like fligner.test, bartlett.test but I need to pass to those methods few groups….I don't have groups I have only one series and I need to know if the variance is constant.
Thank you
If I understand correctly, you have one predictor (explanatory variable $x$) and one criterion (predicted variable $y$) in a simple linear regression. The significance tests rests on the model assumption that for each observation $i$ $$ y_{i} = \beta_{0} + \beta_{1} x_{i} + \epsilon_{i} $$ where $\beta_{0}, \beta_{1}$ are the parameters we want to estimate and test hypotheses about, and the errors $\epsilon_{i} \sim N(0, \sigma^{2})$ are normally-distributed random variables with mean 0 and constant variance $\sigma^{2}$. All $\epsilon_{i}$ are assumed to be independent of each other, and of the $x_{i}$. The $x_{i}$ themselves are assumed to be error free.
You used the term "homogeneity of variances" which is typically used when you have distinct groups (as in ANOVA), i.e., when the $x_{i}$ only take on a few distinct values. In the context of regression, where $x$ is continuous, the assumption that the error variance is $\sigma^{2}$ everywhere is called homoscedasticity. This means that all conditional error distributions have the same variance. This assumption cannot be tested with a test for distinct groups (Fligner-Killeen, Levene).
The following diagram tries to illustrate the idea of identical conditional error distributions (R-code here).
Tests for heteroscedasticity are the Breusch-Pagan-Godfrey-Test (bptest() from package lmtest or ncvTest() from package car) or the White-Test (white.test() from package tseries). You can also consider just using heteroscedasticity-consistent standard errors (modified White estimator, see function hccm() from package car or vcovHC() from package sandwich). These standard errors can then be used in combination with function coeftest() from package lmtest(), as described on page 184-186 in Fox & Weisberg (2011), An R Companion to Applied Regression.
You could also just plot the empirical residuals (or some transform thereof) against the fitted values. Typical transforms are the studentized residuals (spread-level-plot) or the square-root of the absolute residuals (scale-location-plot). These plots should not reveal an obvious trend of residual distribution that depends on the prediction.
N <- 100 # number of observations
X <- seq(from=75, to=140, length.out=N) # predictor
Y <- 0.6*X + 10 + rnorm(N, 0, 10) # DV
fit <- lm(Y ~ X) # regression
E <- residuals(fit) # raw residuals
Estud <- rstudent(fit) # studentized residuals
plot(fitted(fit), Estud, pch=20, ylab="studentized residuals",
xlab="prediction", main="Spread-Level-Plot")
abline(h=0, col="red", lwd=2)
plot(fitted(fit), sqrt(abs(E)), pch=20, ylab="sqrt(|residuals|)",
xlab="prediction", main="Scale-Location-Plot")
I would suggest covariance because it will tell you direction with variance. e.g.,
cov(a,b)= -9.16667
you can also do correlation test in R. e.g., cor.test(a,b)
Best Answer
What you're looking for is a test for homoskedasticity. I haven't need to use this myself yet, so I can't offer much in-depth advice, but the wikipedia page on homoskedasticity lists a couple of tests, including the Breusch–Pagan test, which assumes normality in the data, the Koenker–Basset test which generalises the Breusch–Pagan test for non-normal data.
the
bptestfunction in thelmtestR package implements the both tests (setstudentize=truefor the Koenker–Basset test). Looks like it's only for linear models though, so it might not work for your data.There's a few interesting questions about these test on this site. Check out charlie's answer on Why does the Breusch-Pagan test fail? in particular.