# Stationary Process with autocorrelation in Variance; square root rule

i am currently analyzing a time series of portfolio log returns and have conducted a ADF test with the result, that the series is stationary, but also found significant autocorrelation in the squared return series. Is that not a contradiction, because if the series is stationary, the distribution of the returns should be independent of time and thus have a stable variance over time?

My goal is to find out whether i can use the square root rule to extrapolate the standard deviation from a 1 day to a 10 day. This rule requires zero-autocorrelation in the returns ( which is given) and variance-stationarity. I am not sure, given my test results, whether this is valid. Another question is, whether this square root rule depends on the distribution, does it have to a normal one or is it independent of the distribution?

Thanks.

You are correct in that the series is not stationary. The ADF test isn't designed to test for stationarity outside the center of location. You are not going to be able to use the square root rule to extrapolate because you have significant autocorrelation of the variances.

I do have a suggestion on your problem by noting that returns are not data. Prices are data, but returns are transformations of data. The log return is an even greater transformation of the raw data.

Let's start with a simple AR(1) process of prices $p_{t+1}=Rp_t+\epsilon_{t+1}$, where R is implicitly a return and, for simplicity, $\epsilon_{t+1}\sim\mathcal{N}(0,\sigma_{t+1})$ and $\epsilon_{t}\perp\epsilon_{t+\Delta{t}}$. From Mann and Wald, we know that the OLS estimator is the MLE estimator of $R$ for any distribution of the error term. From White, we know that the sampling distribution of the OLS estimator for $R$ is the Cauchy distribution. Since the Cauchy distribution has no mean, this is the same thing as saying no non-Bayesian solution exists that is also consistent with the thinking behind mean-variance finance. If, on the other hand, returns are defined as $r_t=\frac{p_{t+1}}{p_t}$ and both $p_t$ and $p_{t+1}$ are independent normal random variables then $r_t$ would have a Cauchy distribution. The log version of this would be the hyperbolic secant distribution which does have a mean and a variance.

Let's further assume that prices are locally a function of liquidity and globally a function of discounted cash flows. This makes liquidity a fast function and cash flow a slow function. Although this implies that returns should be a function of liquidity through prices, liquidity itself has two components, the global interest rate and the local adjustment for market-maker specific liquidity needs. The discounting of cash flows shares the global rate. In a separate paper, I argue that the distribution of liquidity is normal or log-normal depending on the model you are using.

The short-run effect is that returns should be centered on the discounted cash flows from dividends, but that the variance of log-returns should be focused on the short-run liquidity. Although liquidity should have a slight effect on the center of location, that should appear in the regression constant because of the form you are using. The variability of prices, which is what the bid-ask spread is, is a short-run process. As liquidity is not idiosyncratic within institutions, but rather a systematic issue, you have to expect serial correlation among the terms.

The number of lags should reflect how long it takes firms to change their overall liquidity levels. This depends on both internal factors, such as margin credit and so forth, and the short-run external popularity of a particular issue.

In looking at the serial correlation of returns you are implying momentum in prices. This implies that prices do not adjust instantaneously, otherwise, there would be no information in historical returns. You should look at both the bid-ask spread and interest rates. I would argue that you have a misspecified model and that there is no fix except to match it to time series of short-term interest rates and the bid-ask spread. You may want to look as Abbott's model of marketability and liquidity in The Valuation Handbook. You should also grab an article on the summation of variables drawn from the hyperbolic secant distribution as that is essentially what the right-hand side of the regression formula is.

for the square-root of time rule you just need uncorrelated returns. Then $$VAR[R_1 + \cdots + R_N] = VAR[R_1] + \cdots + VAR[R_N],$$ then if $VAR[R_i] = VAR[R_j] = \sigma^2$ for $i,j = 1, \ldots, N$ then $$VAR[R_1] + \cdots + VAR[R_N] = N \sigma^2$$ and $$\sqrt{VAR[R_1 + \cdots + R_N]} = \sigma \sqrt{N}.$$

The fact that $R^2_i$ is not uncorrelated shows that the returns are not independent - which is not needed for the above derivation.

• Does Diebold et al. "Converting 1-Day Volatility to h-Day Volatility: Scaling by $\sqrt{n}$ is Worse than You Think" (1998) disagree with what you say, or not really? I have not read it in detail, but perhaps you have? Mar 9, 2017 at 20:15
• I think it depends, i mean if you instead calculate the n-day vola by cummulating 1 day returns over non overlapping intervalls, you will loose quiet a lot data...if you have an accetable variance ratio test and variance stationary based on a long sample, it should be fine.. Mar 10, 2017 at 12:06
• THE OP asks many questions. I am saying that if the returns are uncorrelated and have the same variance then square-root of time rule can be applied. Of course if the variance is not stationary then you don't have this common sigma. The question that is left: how can I test for homogeneous variance. Mar 10, 2017 at 12:10