Time Series Analysis

Time series analysis is the branch of statistics concerned with data observed sequentially over time. Its methods describe the patterns in a time series, build models of how each observation depends on earlier ones, test hypotheses about those models and use them to predict. It differs from ordinary statistics in that observations cannot be treated as independent of one another.

In finance and economics the subject is central, since prices, returns, interest rates and macroeconomic indicators are all recorded through time. It supplies the tools for measuring volatility, for testing whether assets move together in the long run, and for judging whether an apparent pattern in prices is real. The part of the subject applied to economic data is called time series econometrics.

• Autoregressive and moving average models. Autoregression expresses the current value as a function of past values. Moving average models express it as a function of past random shocks. Combined, and with differencing to remove trends, they form ARIMA.
• Volatility models. ARCH, introduced by Robert Engle in 1982, and its generalization GARCH, introduced by Tim Bollerslev in 1986, model the changing variance of returns.
• Multivariate models. Vector autoregression, proposed for macroeconomics by Christopher Sims in 1980, models several series jointly, each depending on the past of all of them.
• Long-run relationships. Cointegration, formalized by Engle and Clive Granger in 1987, describes series that wander individually but stay tied together.
• State-space models. The observed series is driven by hidden variables that are estimated recursively, most often with the Kalman filter.

The standard procedure was set out by George Box and Gwilym Jenkins in their 1970 book on forecasting, and remains the template.

The first step is to examine the series and transform it so that it shows stationarity, usually by taking differences or returns. Unit root tests such as the augmented Dickey-Fuller test guide this step. The second step is identification: plots of the autocorrelation and partial autocorrelation at each lag suggest which type of model, and how many lags, are appropriate. The third is estimation of the model's coefficients. The fourth is diagnostic checking, in which the residuals are examined to confirm that no pattern is left unexplained. If the check fails, the cycle is repeated.

The first-order autoregressive model is:

• $y_t$ is the value at time $t$.
• $c$ is a constant.
• $\phi$ is the autoregressive coefficient.
• $\varepsilon_t$ is a random shock with mean zero.

Suppose a model fitted to a daily spread gives $c = 0.1$ and $\phi = 0.5$, and the latest value is 0.6. The forecast for the next day is $0.1 + 0.5 \times 0.6 = 0.4$. The forecast for the day after uses the first forecast: $0.1 + 0.5 \times 0.4 = 0.3$. Further forecasts approach the long-run mean, $c / (1 - \phi) = 0.1 / 0.5 = 0.2$. The example shows a general feature: forecasts from a stationary model revert to the mean as the horizon lengthens, and the uncertainty around them widens.

• Risk measurement. Volatility and correlation models feed value-at-risk calculations, margin systems and volatility forecasting.
• Relative-value trading. Tests for cointegration, including the Johansen test, are used to select candidates for pairs trading, and autoregressive models estimate how quickly a spread shows mean reversion.
• Lead and lag relationships. Granger causality tests whether the past of one series improves forecasts of another.
• Economic forecasting. Central banks and investors model inflation, growth and interest rates.
• Testing market efficiency. Checks for autocorrelation in returns assess whether prices depart from a random walk.

Engle and Granger shared the Nobel Memorial Prize in Economic Sciences in 2003 for their methods of analysing time series with changing volatility and with common trends.

Since the 2010s, methods from machine learning have been applied to the same problems. Tree-based models and neural networks, including the long short-term memory network designed for sequences, can capture non-linear relationships that classical models miss. They need more data and are harder to interpret, and on noisy financial series their advantage over simple models is often small.

Regressing one trending series on another produces relationships that look significant but are meaningless, the problem of spurious regression. A model with many lags and parameters can fit the past closely and forecast badly, a form of overfitting. Structural breaks, such as a change in monetary policy or market rules, can invalidate a model fitted to earlier data. Most methods also assume that the statistical character of the series is stable, which financial markets satisfy only approximately.

• The Nobel Prize: Prize in Economic Sciences 2003