Describe the statistical properties of prices, returns and volatility over time and scale volatility across horizons.
A series is stationary if its mean, variance and autocorrelations do not change over time. Most statistical tools assume it. Prices are not stationary: they wander and trend. Returns are approximately stationary, which is another reason analysis is done on returns.
If price changes are unpredictable from past prices, the price follows a random walk: tomorrow's price is today's plus a random step. This is the statistical statement of weak-form market efficiency (RS 302, Module 7). The best forecast of tomorrow's price is roughly today's price.
The correlation of a series with its own past values. Daily and monthly stock returns show autocorrelation close to zero, which is why past returns are weak predictors. Small positive autocorrelation at horizons of several months is the momentum effect; small negative autocorrelation over several years is long-run reversal.
Although returns are nearly uncorrelated, their size is not. Large moves follow large moves and calm follows calm. The absolute or squared values of returns are strongly autocorrelated. Volatility is therefore partly predictable even though direction is not. ARCH and GARCH models, introduced by Robert Engle and Tim Bollerslev in the 1980s, capture this.
The exponentially weighted moving average (EWMA) gives recent observations more weight:
with lambda often set to 0.94 for daily data.
If returns are independent, variance grows in proportion to time and volatility with its square root:
Daily volatility of 1.2% is 1.2% x square root of 252 = 19.0% a year. Weekly volatility of 2% is 2% x square root of 52 = 14.4% a year. The rule understates long-horizon risk when returns trend and overstates it when they mean-revert, and it does not allow for fat tails.
Some series are pulled back toward a long-run level: interest rate spreads, valuation ratios, volatility itself. Mean reversion can be slow and unreliable; a ratio can stay far from its average for longer than an investor can stay solvent.
Economic series often have a trend and a seasonal pattern that must be removed before analysis; statistical agencies publish seasonally adjusted data. A structural break is a permanent change in the process, such as a new monetary regime, which makes older data misleading.
Regressing one trending series on another produces high R-squared and large t-statistics even when the two are unrelated. The fix is to work with changes or returns.
A moving average smooths a series to show its trend. It describes the past with a lag; any claim that it predicts must be tested like any other trading rule (Module 10).
Yesterday's daily volatility estimate was 1.0% (variance 0.0001). Today the market moved 3%. With lambda = 0.94:
New volatility = square root of 0.000148 = 1.22% a day. One large move lifts the estimate by more than a fifth; it then decays gradually if markets calm down. This is how many risk systems respond to shocks overnight.
If a series follows x(t) = phi x x(t minus 1) + noise around its mean, a deviation shrinks by the factor phi each period. Its half-life is ln(0.5) / ln(phi). With phi = 0.9 for monthly data, the half-life is 6.6 months; with phi = 0.98 it is about 34 months. A valuation spread with a three-year half-life can stay wide far longer than a trader's patience or funding lasts.
If returns are a random walk, the variance of k-period returns is k times the variance of one-period returns. The variance ratio compares the two: above 1 suggests trending (positive autocorrelation), below 1 suggests mean reversion. Stock indexes have shown ratios slightly below 1 at multi-year horizons in some periods, but the evidence is weak and unstable.
The augmented Dickey-Fuller (ADF) test asks whether a series has a unit root (is non-stationary, like a price) or reverts to a stable mean. Prices almost always fail to reject a unit root; returns almost always reject it. Testing first prevents spurious regressions.
Daily volatility of 1.2% implies a 10-day volatility of 1.2% x square root of 10 = 3.8%, the scaling banks use for 10-day value at risk. After a volatility shock, scaling the latest EWMA estimate gives a very different answer from scaling a long-run average, which is the point of using a responsive estimate.
Claimed calendar effects, such as the January effect or 'sell in May', are weak, unstable and often vanish after costs once widely known. Seasonality is real in economic data (retail sales, employment, heating oil demand) and must be removed before analysis; in asset returns, treat it with Module 6's scepticism.
For your index, compute the autocorrelation of daily returns at lags 1 to 5 and the autocorrelation of squared daily returns at the same lags. Then compute rolling 21-day volatility, annualize it and chart it. Describe what the chart shows about clustering.
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