Forecast a persistent series honestly

Learning goals

• Calculate the two-period-ahead AR forecast.
• Explain why 5 applies to this case.
• Identify the case-specific assumption or failure mode before taking action.

Draft · approximately 80 minutes, excluding further revision.

Shared case and decision context

A stationary AR(1) approximation is x_t=2+0.6x_(t−1)+e_t, with current x=10 and zero expected innovations. A separate price-level series shows a suspected unit root. Forecasts must be dated before future observations arrive.

Economic or professional mechanism

An AR forecast recursively substitutes the last observation and then the first forecast; it does not reuse today’s value at every horizon. With |b|<1 a simple AR(1) has long-run mean a/(1−b), provided the error process satisfies stationarity conditions. Trend-stationary and difference-stationary series require different treatments.

Apply, challenge and communicate

Here the mean is 5, the one-step forecast 8 and the two-step forecast 6.8. Examine residual autocorrelation and changes in variance. Use chronologically ordered validation and compare RMSE to a simple benchmark. Differencing may address a unit root but does not automatically solve seasonality, structural breaks or cointegration issues.

Compute linear and exponential trends

A linear trend y=10+2t implies 14 at t=2; a log-linear trend ln(y)=2+0.1t implies exp(2.2), about 9.025 at t=2. The log-linear slope 0.1 is a continuously compounded growth rate of 10% per period, corresponding to discrete growth exp(0.1)−1≈10.517%; it is not a constant 0.1-unit increase in y. Transforming back from log forecasts requires care: the exponential of the expected log is not generally the arithmetic expected level.

Choose the trend scale from economics

Use a linear trend when constant absolute increments are plausible; use a log-linear trend when proportional growth is plausible and values are positive. Residual diagnostics and holdout forecast error should challenge both choices. Neither deterministic trend removes business-cycle shocks or structural breaks. Selecting a log transformation solely because it raises in-sample fit can conceal economic misspecification.

Test coefficient stability across regimes

A policy change or market structure shift can make coefficients estimated on an earlier period unreliable. Compare rolling estimates, economically justified subsamples and genuine later forecasts; formal break tests supplement judgment. Shorter windows adapt faster but raise estimation noise. Do not select the break date after seeing returns without accounting for the search and revalidation.

Recognize a random walk

A random walk x_t=x_(t−1)+e_t has a unit autoregressive coefficient. Its conditional mean forecast is the latest observation, or that observation plus accumulated drift if drift is specified. With independent shocks its unconditional variance grows with time, so the level is not covariance stationary. White-noise increments can be stationary even while the accumulated level is not.

Difference a unit-root level when justified

Regressing unrelated persistent levels can produce spurious significance. If x_t has a unit root, first differences delta x_t may be stationary and suitable for a model of changes. Differencing changes the economic target and can discard long-run information; investigate cointegration before automatically differencing all variables. Unit-root testing has limited power in small samples and around structural breaks.

Use the unit-root test distribution

An augmented Dickey–Fuller style test rewrites the AR relation in differences and tests whether the lagged-level coefficient is zero, including suitable deterministic terms and lagged differences. The null is a unit root. Its critical values are nonstandard; ordinary two-sided regression t critical values are not a safe substitute. Failing to reject is not proof that a unit root exists with certainty.

Add seasonal dependence explicitly

A quarterly model can include a lag-four term when the same quarter last year carries information beyond recent quarters. If forecast y_t=2+.5y_(t−1)+.2y_(t−4), with lag values10 and20, the prediction is11. Inspect seasonal residual autocorrelation and validate against a seasonal-naive benchmark. Seasonal indicators and seasonal autoregression represent different mechanisms and should be chosen from data and economics.

Forecast conditional variance with ARCH

ARCH models let current conditional error variance depend on past squared innovations. If h_t=.0001+.2e_(t−1)² and the last innovation is.05, the variance forecast is.0006, giving volatility about2.449%. This predicts variance, not the sign of the next return. Ensure nonnegative variance conditions and examine whether longer volatility persistence requires a richer specification such as GARCH.

Test whether persistent variables share a stable relation

Two nonstationary series can be cointegrated if a suitable linear combination is stationary. This can support a long-run equilibrium relation with short-run error correction, rather than a naive levels regression or automatic deletion of all levels. Test the integration order and residual relation using appropriate methods. Cointegration is not causality or a guarantee that a trading spread will revert before funding expires.

Select the model against the actual forecast target

A model for the next return, a long-run spread and next-period variance solves different problems. Diagnose trend, persistence, seasonality, structural breaks and volatility before selecting a specification. Compare a simple baseline and a richer model on the same chronologically held-out target and loss measure. A more complex model earns its place only through defensible structure and reliable validation, not a larger equation.

Formula and assumptions

xhat(t+2)=a+b*(a+b*x_t); long-run mean=a/(1−b)

Case analysis with explicit assumptions

1. A stationary AR(1) approximation is x_t=2+0.6x_(t−1)+e_t, with current x=10 and zero expected innovations. A separate price-level series shows a suspected unit root. Forecasts must be dated before future observations arrive.
2. Method: xhat(t+2)=a+b*(a+b*x_t); long-run mean=a/(1−b)
3. Case result: 6.80. Here the mean is 5, the one-step forecast 8 and the two-step forecast 6.8. Examine residual autocorrelation and changes in variance. Use chronologically ordered validation and compare RMSE to a simple benchmark. Differencing may address a unit root but does not automatically solve seasonality, structural breaks or cointegration issues.

Errors to catch

• Do not replace the stated assumptions with an unstated shortcut. Reconcile units, timing and the claim being valued or duty being assessed.
• Only the best in-sample fit is not a defensible conclusion for these case facts.

Original knowledge check

Answer every question before checking. Review explanations after submitting; retry starts a new attempt.

Sources and coverage status

These original lessons are drafts. Public outlines are used to check scope; the official learning-outcome text and proprietary question banks are not reproduced here. Follow the source for exam-year requirements and updates.

• Reference · www.cfainstitute.org
• Reference · www.itl.nist.gov
• Reference · www.cfainstitute.org