Time-Series Analysis

Forecasting sales, inflation, exchange rates or volatility means modelling a variable from its own history. This module covers trend models, autoregressive models and their mean-reverting levels, the tests that tell whether a model is adequate, and the problems that make time-series regressions misleading: unit roots, seasonality, volatility clustering (ARCH) and regressions between nonstationary series, where cointegration decides validity. Level II item sets combine these steps in a single forecasting case.

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What you will learn

1. Fit linear and log-linear trend models, forecast with them, and judge when each is appropriate.

Linear trend: constant absolute change b1 per period; log-linear trend: constant growth rate (e^b1 - 1 per period).
Forecast a log-linear model by exponentiating b0 + b1t.
Use log-linear models for compounding series such as sales, earnings and prices.
A curved residual pattern from a linear trend signals that a log-linear model is needed.
Trend-model residuals are often serially correlated (DW well below 2); then an autoregressive model is more appropriate.

2. Explain covariance stationarity and build autoregressive models, including multi-period forecasts and the mean-reverting level.

Covariance stationarity: constant and finite mean, variance and covariance with its own lags.
AR models are valid only for covariance-stationary series.
Mean-reverting level of AR(1) = b0/(1 - b1), defined only when |b1| < 1.
Chain rule: forecast t + 1, then use that forecast in place of the unknown value to forecast t + 2.
Multi-period forecasts converge to the mean-reverting level and become less certain with the horizon.

3. Test AR models with residual autocorrelations and compare models using out-of-sample root mean squared error.

Use residual autocorrelations, not Durbin-Watson, to test AR models.
Standard error of a residual autocorrelation is about 1/sqrt(T); t = autocorrelation x sqrt(T).
A significant residual autocorrelation means more lags (or a seasonal lag) are needed.
Prefer the model with the lower out-of-sample RMSE for forecasting.
Coefficients can be unstable across periods; balance sample length against regime changes.

4. Explain random walks and unit roots, apply the Dickey-Fuller test, and transform a nonstationary series by first-differencing.

A random walk (with or without drift) has a unit root, b1 = 1, and is not covariance stationary.
With a unit root, the mean-reverting level is undefined and the variance grows over time.
Dickey-Fuller: regress the change on the lagged level; H0: g1 = b1 - 1 = 0 (unit root), using DF critical values.
Failing to reject the unit root means the series must be transformed before AR modelling.
First-differencing a random walk produces a stationary series that can be modelled with an AR model.

5. Model seasonality and ARCH effects, and check variables for unit roots and cointegration before using them in a regression.

Seasonality appears as a significant residual autocorrelation at lag 4 (quarterly) or 12 (monthly); add that lag.
ARCH(1): regress squared residuals on lagged squared residuals; a significant a1 means volatility clustering.
ARCH makes standard errors unreliable; forecast variance = a0 + a1 x latest squared residual.
A regression of one time series on another is valid if neither has a unit root, or if both have unit roots and are cointegrated.
Engle-Granger: apply a Dickey-Fuller test to the residuals of the regression between the two series.

Module at a glance

Linear trends suit constant absolute growth; log-linear trends suit constant percentage growth (exponentiate to forecast).
Trend-model residuals are often serially correlated; AR models then use the series' own past values.
AR models need covariance stationarity; mean-reverting level = b0/(1 - b1); multi-period forecasts use the chain rule.
Test AR models with residual autocorrelations (t = autocorrelation x sqrt(T)); choose models by out-of-sample RMSE.
A unit root (b1 = 1) makes a series nonstationary; test with Dickey-Fuller critical values and first-difference if needed.
Add a lag-4 (quarterly) or lag-12 (monthly) term for seasonality; test ARCH by regressing squared residuals on their lag.
Regressions between two unit-root series are valid only if the series are cointegrated (Engle-Granger test).

This free module has 5 lessons, 17 practice questions with full explanations.

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