# Testing if the volatility of single stocks and/or indices have risen in the past

I'm currently writing my bachelor thesis and the main goal of my paper is to test if the volatility of single stocks and indices have risen in the past. My data consists of all stocks of the SMI and the DAX. In total, I have 50 stocks with monthly volatility data tested between 2005-2015. So, I have 50 $\times$ 12 $\times$ 10 = 6000 data points. Now I've heard of the time series analysis, ARCH, GARCH(1.1) and GARCH(1.2). I have read a bit about those models, but until now, I have only had 2 statistics courses and 1 econometrics course. And with the knowledge I have at the moment, I cannot understand which model would suit the best and/or is the simplest to model in R/Stata.

I've read that the models ARCH/GARCH are good to model volatility, but it is not the tool to test my hypothesis. Also I've read that those models are mainly to predict future volatility, but my goal is to analyse past volatility, do I still choose ARCH/GARCH?

So my questions are:

1. Which model should I use for this kind of hypothesis (hypothesis: Has the volatility of the financial market represented through the SMI and DAX indices risen significantly in the past?)
2. If I'd use a time series model like ARCH/GARCH, how do I test my hypothesis?
• What do you mean by "test if the volatility of stocks and indices have risen in the past?" Do you mean: (1) test if volatility is constant over time? (i.e. does volatility ever change?) or (2) test if some notion of "average volatility" has changed over time? (eg. the unconditional average volatility implied by a GARCH model). Or (3) something completely different? Aksakai's answer goes to (1) Richard Hardy's answer is more about question (2). – Matthew Gunn Sep 26 '16 at 19:03
• I want to test (1), that volatility is constant over time. Basically my idea came from reading an american paper which based of the discussion, that the financial market has become more volatile in the past years. They've tested it with the ARCH model and some regressors and came to the conclusion, that there is no significant change, and that the market was rather constant in the "long-term". Now, I want to do the same thing, in a simpler way for the european market, if that makes sense? – RazorLazor Sep 26 '16 at 19:07
• You say you're interested in whether volatility of the market was constant over the long-term. That's actually question (2). Volatility may more up or down over the short run, but what's it's long-term average over a period? – Matthew Gunn Sep 26 '16 at 19:15
• That is what i want to show. Regarding the volatility clusters, investors seem to be biased by the more volatile periods, which they wheigh stronger subjectively. And I want to show that in the short run, there are low-volatility and high-volatility periods, but in the long run, the volatility should be constant. Is that still under 2? – RazorLazor Sep 26 '16 at 19:42
• @RazorLazor, when you talk about the long-run, it is about 2. When about the short-run, it is about 1. 1 is more about changes in conditional volatility, while 2 in unconditional volatility. – Richard Hardy Sep 27 '16 at 5:05

You don't need a model to show that volatilities are changing. Simply show the time series of squared returns, you'll be able to spot the clusters of high and low volatilities easily. If you want to fit a model, then GARCH(1,1) will do. GARCH is usually not an undergrad topic, by the way, so you can do simpler analysis and get away with it.

UPDATE: The simple analysis would be plots and trivial regressions. For instance, plot the squared return (y-axis) against their lagged values (x-axis). This was the GARCH idea to start with. If you see the significant slope, then you can claim autocorelation of returns, i.e. if the stocks were moving a lot recently then they're likely to be moving a lot in near future. That's volatility clustering in its simplest form

UPDATE 2:

I decided to write your thesis. I got SPX Index prices from Bloomberg:

You can't see much from prices, so we get the daily returns:

It turns out that the returns are not very infomative either, so let's look at the squared returns:

Bingo! There's clearly volatility clustering in effect. So, let's scatter the squared return against itself one day ago:

We can now fit the simple regression: $$r_t^2=\beta_0+\beta_1r_{t-1}^2+e_t$$ Here are the results. Note that the slope is significant, so you can claim that you tested the volatility changes.

SUMMARY OUTPUT

Regression Statistics
Multiple R  0.305793227
R Square    0.093509498
Adjusted R Square   0.091085726
Standard Error  0.000175742
Observations    376

ANOVA
df  SS  MS  F   Significance F
Regression  1   1.19156E-06 1.19156E-06 38.58016389 1.4001E-09
Residual    374 1.15511E-05 3.08853E-08
Total   375 1.27427E-05

Coefficients    Standard Error  t Stat  P-value Lower 95%   Upper 95%   Lower 95.0% Upper 95.0%
Intercept   6.72079E-05 1.02634E-05 6.548334991 1.92453E-10 4.70268E-05 8.73891E-05 4.70268E-05 8.73891E-05
X Variable 1    0.305193577 0.049135271 6.211293254 1.4001E-09  0.208577558 0.401809596 0.208577558 0.401809596


You could also show the simple correlation of lagged squared returns:

Correlation:

    Column 1    Column 2
Column 1    1
Column 2    0.305793227 1


Now, all you need is to get a few gallons of water to fill in the space between the Title and the bibliography of the thesis. Don't forget: this is bachelor level work :)

• The question is about testing for a change. – Richard Hardy Sep 26 '16 at 18:57
• What simpler analysis would you recommend then? I thought of that too, but you cannot do a simple regression, that would not show any results at all. Are there any plain simple time series models out there?(I kinda feel bad, since I read, that ARCH/GARCH are already simple models..) – RazorLazor Sep 26 '16 at 19:00
• Okay thanks for the graph, and the extended answer. I have to look up more theory about GARCH in order to understand what you are trying to teach me. It's simply because I don't know that much about Time Series Analysis yet. – RazorLazor Sep 26 '16 at 19:44
• I'd try to stay away from GARCH. Just stick to volatility changes topic. – Aksakal Sep 26 '16 at 19:45
• Holy moly, I just saw the Update 2 now. What kind of god are you?! You did that in like 15-20 mins? Wish I'd the same knowledge as you. I will try to understand it deeper with some research. I hope I can use these steps for my thesis. Thanks, have a nice day! – RazorLazor Sep 26 '16 at 19:52

ARCH/GARCH models are appropriate if there is autoregressive conditional heteroskedasticity in the data. So if there is, and if you also know the shape of change in volatility you want to test for, you may include corresponding terms as extra regressors in the conditional variance equation, something like $$\sigma^2_t=\omega+\alpha_1 e^2_{t-1}+\beta_1\sigma^2_{t-1}+\gamma x_t.$$ For example, you could include a linear time trend $x=(1,2,\dots)$ if you want to test for a constant increase in volatility over time; or a level-shift variable with zeros followed by unities after the suspected change point, $x=(0,\dots,0,1,\dots,1)$, if you want to test for an abrupt jump. Then you may test the significance of those regressors ($\text{H}_0\colon \, \gamma=0$) and this way test for the rise in volatility.

• Could you maybe explain it in a more amateur-ish language? English isn't my native language and speaking about statistics in a foreign language for me is even harder. I think that there is autoregressive conditional heteroskedasticity in my data. But I want to keep it univariat, without any extra regressors. Is there another possibility in that way? – RazorLazor Sep 26 '16 at 18:42
• Is it clearer now that I included the example equation and example variables? I don't know if you can get away with using a plain ARCH/GARCH model without the extra regressors. If you only look for an abrupt change in the level of volatility, you could estimate GARCH models on two subsamples (one before the jump and one after) and compare the implied unconditional variances. But I don't know if you could easily make a formal test out of that. – Richard Hardy Sep 26 '16 at 19:17
• I understand everything, but the part with level-shift variable with zeros. What do you mean with that and how does it help my model to explain more? – RazorLazor Sep 26 '16 at 19:46
• If the coefficient on the level-shift variable is significant, you will know the volatility has increased in the recent period (where unities start). Also note the difference between the problems addressed in my answer vs. Aksakal's answer. He considers volatility clustering rather than whether (unconditional) volatility is rising or has risen at some point in the past. Meanwhile, I focus on long-term changes in volatility (but also allow for clustering in the model), and that is how I understand your question [whether] the volatility of single stocks and/or indices have risen in the past. – Richard Hardy Sep 27 '16 at 5:09
• You've understood my question right. And I also got most of your points. I nearly got all the points considering how to build such a model for my thesis, but it is only theoretical. I don't have a clue how to implement that knowledge into Stata/R. If you could explain it like Aksakal without any actual Data used(that would be too much asked for), I could probably use your ideas better. + would you recommend Stata or R for a beginner in econometrics/time series analysis? – RazorLazor Sep 27 '16 at 8:16