Solved – How to test the variance in timeseries

covariancevariance

I have a doubt regarding the variance, I try to explain It with an example.

I have two vectors, like:

a <- c(1:10)
1  2  3  4  5  6  7  8  9 10

b <- c(10:1)
10  9  8  7  6  5  4  3  2  1

the variance is obviouly the same:

> var(a)
9.166667
> var(b)
9.166667

Ok, I need to test if the variances are similar, and for this test I use var.test().

The problem is that the variances are equals, OK! but the follow a different direction, the first move from 1 to 10 and the second from 10 to 1. SO the variances are the same and the test pass successfully(obviously), but I need also check the direction, so:

Are the variances similar? Ok…
Are the variances (I know 'variances' here is wrong but try to understand what I mean reading the example above) moving in the same direction?

With the same direction I mean, the variance is equal(similar) BUT are they UP/DOWN togheter?

I need to do those checks because I'm analyzing two financial lists of prices, and I need to know if the variance between their returns is constant and on the same direction.

How Can I do?

Thanks!

Best Answer

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)

Related Solutions

Solved – method to check the homogeneity of variance of a single time series

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 bptest function in the lmtest R package implements the both tests (set studentize=true for 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.

Variance Homogeneity – How to Test Homogeneity of Variance for Two Groups with Different Sample Sizes

I don't know what code you used, but tests do not require equal sample sizes. You can use Levene's test to check for heteroscedasticity. In R, you can use ?leveneTest in the car package:

set.seed(9719)                       # this makes the example exactly reproducible
g1 = rnorm( 50, mean=2, sd=2)        # here I generate data w/ different variances
g2 = rnorm(100, mean=3, sd=3)        #   & different sample sizes
my.data = stack(list(g1=g1, g2=g2))  # getting the data into 'stacked' format

library(car)                         # this package houses the function
leveneTest(values~ind, my.data)      # here I test for heteroscedasticity:
# Levene's Test for Homogeneity of Variance (center = median)
#        Df F value   Pr(>F)   
# group   1  8.4889 0.004128 **
#       148                    
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Levene's test is just a $t$-test ($F$-test) on transformed data. (I discuss tests for heteroscedasticity here: Why Levene test of equality of variances rather than F ratio?) What having unequal sample sizes will do is cause you to have less power to detect a difference. To understand this more fully, it may help to read my answer here: How should one interpret the comparison of means from different sample sizes? Note however, that running a test of your assumptions and then choosing a primary test is not generally recommended (see, e.g., here: A principled method for choosing between t-test or non-parametric e.g. Wilcoxon in small samples). If you are worried that there may be heteroscedasticity, you might do best to simply use a test that won't be susceptible to it, such as the Welch $t$-test, or even the Mann-Whitney $U$-test (which doesn't even require normality). Some information about alternative strategies can be gathered from my answer here: Alternatives to one-way ANOVA for heteroskedastic data.

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Solved – Fligner-Killeen test of homogeneity of variances interpretation