Time-Series Analysis

A trend regression on quarterly sales can post an R² of 0.96 and still be the wrong model. Time-series work is mostly about earning the right to run the regression at all.

Trend models. A linear trend models the level as $y_t=b_0+b_{1}t+{\epsilon }_t$, with predicted value ${\hat{b}}_0+{\hat{b}}_{1}t$. A log-linear trend models the natural log, $\ln y_t=b_0+b_{1}t+{\epsilon }_t$, with predicted value $e^{{\hat{b}}_0+{\hat{b}}_{1}t}$. Use the linear form when the series grows by a constant amount each period, the log-linear form when it grows at a constant rate (exponential growth). The implied periodic growth rate is $e^{b_1}-1$.

Trend models are limited: the residuals are often serially correlated, meaning the fitted line sits persistently above or below the data. Test with the Durbin-Watson statistic. A value significantly below 2 signals positive serial correlation, and you need an autoregressive model instead.

Covariance stationarity. Three conditions: constant and finite expected value, constant and finite variance, and constant finite covariance with itself at every fixed lag.

Common mistakes

• Using Durbin-Watson on an AR model. It is invalid with lagged dependent variables. Use residual autocorrelation t-tests against $1/\sqrt{T}$.
• Trusting a high R². An R² of 0.96 with residuals negative for seven straight years still means a misspecified trend model.
• Using ordinary t critical values in a unit root test. Dickey-Fuller critical values are larger in absolute value, so an ordinary table over-rejects the unit root.

Bottom line

• Linear trend for constant amount growth, log-linear for constant rate growth; log-linear forecast is e raised to b0 + b1t, growth rate is e^b1 − 1
• Covariance stationary requires constant finite mean, constant finite variance, constant finite covariance at each lag
• AR(p) forecast: one period ahead from the last actual, two periods ahead by the chain rule using the first forecast
• Model fit: residual autocorrelation t-tests with standard error 1 over the square root of T; Durbin-Watson is invalid here

Exam shortcut

Read the residual autocorrelation table before the coefficients. Any t above 2 kills the model, so the answer is "add a lag," and if the offending lag is 4 or 12 the answer is "seasonality." When a vignette gives a Durbin-Watson statistic next to a lagged dependent variable, the intended answer is that the statistic is invalid.

The full lesson (about 1,957 words, 13 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• time series analysis

Browse all free CFA Level II lessons or jump into free CFA Level II practice questions.