# Why is volatility said to be persistent?

Persistence in volatility of stock returns is one of the common 'stylized facts' when it comes to analyzing time series. However, I am wondering for theoretical arguments why (estimated) volatility should have a long memory. One of the ideas I came across is to assume that information flow is slow and therefore news coming into the markets are not absorbed immediately but with some latency, leading to 'long-term-adjustments'. This explanation does not satisfy me completely, as the speed of trading and information processing should be much faster nowadays. In my eyes, the picture has also changes with the financial crisis. Using the publicly available data from Oxford-Man Institute I computed the auto-correlations of daily RV as proxies for volatility for 21 assets, divided into the period before the Lehman Crash and afterwards. Clearly, the persistence in the Realized Volatility decreased a lot, giving me a hard time to accept the persistence of volatility as something given... So, what are different channels driving persistence in volatility, that can also explain such changes over time?

Edit after 2.5 years (thanks for your comment @Jared:

A list of used assets is provided here. For the sake of brevity I used the estimates based on Realized Variance (10-min Sub-sampled). The figures can be replicated by running the following R-code (I updated it so it now also contains data until 2018 but the figures did not change at all). The code directly downloads the data from the realized library (thanks to Heber, Gerd, Asger Lunde, Neil Shephard and Kevin Sheppard (2009) to provide this rich database!).

url <- "https://realized.oxford-man.ox.ac.uk/images/oxfordmanrealizedvolatilityindices-0.2-final.zip"
temp <- tempfile()
unzip(temp, "OxfordManRealizedVolatilityIndices.csv")

library(tidyverse)
data <- data%>%select(matches('.rv10ss|DateID')) %>% na.omit()

fit_before <- apply(data%>%filter(DateID<20060917)%>%select(-DateID),2 ,function(x) fit <- acf(x, lag=30))
fit_after <- apply(data%>%filter(DateID>=20060917)%>%select(-DateID),2 ,function(x) fit <- acf(x, lag=30))

fit_before%>%
map(function(x)x$acf) %>% bind_rows() %>% mutate(lag = 0:30) %>% gather(Asset, Autocorrelation, -lag) %>% ggplot(aes(x=lag, y = Autocorrelation, group=Asset)) + geom_line() + theme_bw() fit_after%>% map(function(x)x$acf) %>%
bind_rows() %>%
mutate(lag = 0:30) %>%
gather(Asset, Autocorrelation, -lag) %>%
ggplot(aes(x=lag, y = Autocorrelation, group=Asset)) + geom_line() + theme_bw()

• Can you mention what asset classes are used and what years specifically the two graphs include? Also it seems to me your hypothesis matches the phenomenon observed: you predict that an increase in the speed of trading and information processing would decrease "long term adjustments" as you call it. This appears to be the case. Although the financial crisis could be a catalyst, your hypothesis may be correct and you are comparing the years from 2000-2008, to 2008-present which ought to show a decrease in realized volatility autocorrelation. – Jared Aug 28 '18 at 17:04

Two theoretical explanations regarding the long memory are given by:

1. The mixture of distributions hypothesis of Tauchen and Pitts (1983). Essentially this hypothesis states that trading volume and return are driven by the same information flow process, therefore trading volume and return volatility should share the same long range dependence. ( see Bollerslev and Jubinski, 1999; Fleming and Kirby, 2011).
2. A more general (statistic oriented) approach (not only valid for volatility): which has been firstly developed by Granger (1980) (see also Zaffaroni (2004)) , explains long memory as the result of the aggregation of micro-economic linear dynamic models. As a simple example, if we assume that a serie S corresponds to the aggregation of several AR(1) processes, this serie (S) may exhibit long memory even those if the AR(1) are short memory processes. Now there is still some room to explain why volatility can be seen as an aggregated process…

Regarding the sudden change in persistence I think it is still an ongoing research issue but the two aformentioned theories can give you some ideas.

Ref :

• Granger, C., 1980. Long memory relationships and the aggregation of dynamic models. Journal of econometrics 14, 227 .

• Zaffaroni, P., 2004. Contemporaneous aggregation of linear dynamic models in large economies. Journal of Econometrics 120, 75 .

• Tauchen, G. E., Pitts, M., Mar. 1983. The Price Variability-Volume Relationship on Speculative Markets. Econometrica 51 (2),

• Bollerslev, T., Jubinski, D., 1999. Equity Trading Volume and Volatility: Latent Information Arrivals and Common Long-Run Dependencies. Journal of business and economic statistics

• Fleming, J., Kirby, C., Jul. 2011. Long memory in volatility and trading volume. Journal of Banking & Finance 35

EDIT :

A very recent paper entitled “Long Memory Through Marginalization of Large Systems and Hidden Cross-Section Dependence” -Revise/Resubmit to the Journal of Econometrics. (Guillaume Chevillon, Alain Hecq, and Sébastien Laurent) propose another econometric reason based on the marginalization of a large dimensional multivariate system.

Moreover they list five reasons to explain long memory:

-Aggregation across series

-Linear modeling of a nonlinear underlying process

-Structural change

-Learning (bounded rationality) by economic agents in forward looking models of expectations

-Network effects

For details see here the paper.

• Thank you @Malick for providing the additional evidence, this paper answers a number of interesting questions! – muffin1974 May 23 '16 at 6:57

Check out the book of Teyssière & Kirman (2007) entitled "Long Memory in Economics". For instance, the model of Gaunersdorfer & Hommes features heterogeneous agents: fundamentalists believe that prices move to their fundamental rational expectations value, while chartists simply look at deviations of actual traded prices. The latter thus feature a simple trend following trading rule. If the ratio of chartists is high, then the reinforcement of price movements is stronger and volatility more persistent.

One possible explanation for your observation might therefore be a decline in "uneducated traders" after the crisis.

There are many models that do not assume constant volatility. For example, GARCH, OU, COX.

• Thank you @JOHN. I am aware the toolbox for volatility estimation is huge, nothing was said in my question regarding constant volatility assumptions. But the existence of those models is also related to the persistence in volatility: Given, we would not observe strong persistence in the data, there would be no mean in estimating a GARCH model, especially not RiskMetrics or - more general - iGARCH. My question refers to the reasons why the persistence occur, not to the statistical methods able to capture this behavior. – muffin1974 Feb 29 '16 at 14:22