Time Series

A time series is a set of data points for a single variable, arranged in the order in which they were observed. Daily share prices, hourly exchange rates, quarterly company earnings and monthly unemployment figures are all time series. What distinguishes this kind of data is that the order matters: each observation is related to those before and after it, and rearranging them would destroy information.

Almost all market data take this form, so the time series is the basic object of quantitative finance. It is contrasted with cross-sectional data, which record many subjects at one moment, such as the returns of 500 shares on a single day, and with panel data, which combine the two. The study of such data is time series analysis, and the mathematical model behind an observed series is a stochastic process.

Observations may be regularly spaced, as with daily closing prices, or irregularly spaced, as with individual trades that arrive at random moments. Market data are often condensed into bars, each recording the open, high, low and close for an interval, which is the form shown on a candlestick chart.

A series is commonly described as a combination of parts:

• Trend. A persistent long-run movement upward or downward.
• Seasonality. A pattern that repeats over a fixed period, such as the day of the week or the time of day. Trading volume, for example, is reliably higher near the open and the close.
• Cycles. Fluctuations without a fixed period, such as business cycles.
• Irregular component. The random variation that remains.

The dependence between a series and its own past is measured by autocorrelation, the correlation between values separated by a given number of periods, called the lag:

• $y_t$ is the observation at time $t$, and $\bar{y}$ is the mean of the series.
• $k$ is the lag.
• $n$ is the number of observations.

For the short series 1, 2, 3, 4, 5 the mean is 3 and the deviations are -2, -1, 0, 1 and 2. Their squares sum to 10. The products of each deviation with the one before it are 2, 0, 0 and 2, which sum to 4. The lag-one autocorrelation is $4 / 10 = 0.4$. The positive value reflects the steady rise: a high value tends to follow a high value. A plot of autocorrelation against the lag is the standard first tool for examining a series.

Analysts rarely work with prices directly. A price series usually behaves like a random walk: it has no fixed level and its variability grows with the length of the period examined. It therefore lacks stationarity, the property that most statistical methods require. The series of returns, the percentage changes from one observation to the next, is much closer to stationary.

For prices of 100, 102, 101 and 104, the simple returns are 2.0, -0.98 and 2.97 percent. Logarithmic returns are often preferred because they add up across periods.

Research across many markets has found a set of recurring properties, often called stylized facts.

• Little autocorrelation in returns. Past returns are a poor guide to the direction of the next one, as the efficient market view predicts.
• Volatility clustering. Large moves tend to be followed by large moves of either sign, and calm by calm. The size of returns is predictable even though the sign is not, which is the basis of models such as GARCH.
• Heavy tails. Extreme returns occur far more often than a normal distribution implies, a property known as fat tails.
• Asymmetry. In equity markets, volatility tends to rise more after price falls than after rises.
• Changing behaviour. Relationships that hold in one period may weaken or reverse in another.

Conclusions drawn from a series are only as good as the data. A historical price series must be adjusted for share splits and dividends, or it will show false jumps. Series from different markets need aligned clocks and calendars. Missing values, erroneous prints and changes in how a figure is defined all need handling.

Two errors are particular to the time dimension. Look-ahead bias arises when an analysis uses a value before the moment it was actually known, as with economic figures that are later revised. Survivorship bias arises when the data set contains only the series that still exist, omitting companies that failed.

Time series are used to describe the past, to test whether variables are related and to predict. In markets, typical tasks are estimating volatility and correlation for risk management, smoothing prices with a moving average, testing whether a spread between two assets tends to return to a mean, and time series forecasting of returns or volatility. The finest-grained series, built from tick-by-tick data, support research into trading costs and market microstructure.