# Serial correlation vs heteroskedasticity

Please Explain me serial correaltion and heteroskedasticity are the same thing or not? What is the difference btw them?

• It's not clear what's the problem here. Did you try to search the terms in internet? Apr 8, 2018 at 2:28
• I seached. But I cannot fınd ıts difference. There is no problem. I just waiting for an Explaination. @aksakal Apr 8, 2018 at 2:34
• Correlation is a measure that is stronger when two things covary more compactly whether or not they are homoskedastic.. Heteroskedasticity is when two things covary in a different pattern of compactness over the ranges of those two things.
– Carl
Apr 9, 2018 at 4:08

As they say, a picture is worth a thousand words. The attached picture shows what serial correlation and heteroscedasticity would look like in the context of simple linear regression, where we relate an outcome variable Y to a predictor variable X.

If X stands for Time and the values of Y are serially correlated over time, the scatterplot of Y versus X would show that consecutive values of Y tend to clump together above or below the straight line capturing the linear relationship between Y and X.

If X stands for Time and the variability in the values of Y corresponding to the same X value tends to increase over time, that reveals the presence of heteroscedasticity (i.e., the conditional variance of Y given X changes with X).

• To sum up, Dear Student, in my example serial correlation refers to the fact that the value of Y at the current time depends on all or some of the values of Y at previous times. In contrast, heteroscedasticity refers to the vact that the variability of Y at the current time is different from the variability of Y at previous times. Apr 8, 2018 at 15:43
• Thank.you reply. I really understand the difference between them. Apr 9, 2018 at 0:36

Serial correlation or autocorrelation is usually only defined for weakly stationary processes, and it says there is nonzero correlation between variables at different time points.

Heteroskedasticity means not all of the random variables have the same variance.

Note that if you talk about one, you cannot talk about the other:

1. if a series is heteroskedastic, then it cannot be weakly stationarity, and so autocorrelation is not defined,
2. if there is serial correlation, you're assuming weak stationarity, and so heteroskedasticity is impossible.

If you have $n$ random variables $Z_1, \ldots, Z_n$, the covariance matrix potentially has a lot of unique elements. These two terms you're asking about widdle down the number of possible parameters in different ways. The hetero/homoskedastic distinction is mainly concerned with the diagonals, and usually comes with the tacit assumption that all of the off-diagonals are $0$. Serial correlation assumes the diagonals are all the same, but that there are nonzero off-diagonal elements, and that there has to be a lot of these elements matching.

• That’s, $E(u_{t-k}u_t)=a$ is serial correlation. But $E(u_t’u_t)=a$ is heteroskedasticity for any numbers $a$. Am I right? Apr 8, 2018 at 2:59
• Conditional heteroskedasticity of the GARCH type might also be worth mentioning since it is more often discussed than other forms of heteroskedasticity in time series. Apr 8, 2018 at 8:17