M.6.3 Forecasting & Unit Roots

Basic Prerequisites: 02 · Stationarity & ARMA.

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1. Intuition & Practical Objective

Forecasting a stationary process is one thing; forecasting a unit-root process is fundamentally another. This page makes the distinction operational and gives you the test that settles it - the Augmented Dickey–Fuller (ADF) test - plus the disaster that follows if you ignore it, the spurious regression.

The intuition in three lines:

1. The MSE-optimal forecast is the conditional expectation. There is no better predictor (in squared error) than $\mathbb{E}[x_{t+\ell }\mid {\mathcal{F}}_t]$. All of ARMA forecasting is just computing this object cheaply.
2. Stationary processes mean-revert; unit-root processes don’t. An AR(1) with ${\varphi }_1=0.8$ has a half-life of about 3 periods; a random walk’s best forecast of $p_{t+\ell }$ is today’s value forever - and its forecast-error variance grows linearly with $\ell$, not to a finite limit.
3. Regressing one unit-root series on another is a statistical mirage. Two independent random walks, regressed on each other, produce large $t$-stats and inflated $R^2$ - “significance” that means nothing. You must test for unit roots and, when present, difference or find a cointegrating combination (05 · Cointegration).

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2. Mathematical Ground Truth & Derivations

Forecasting (Tsay §2.6). The MSE-optimal $h$-step forecast is $x_{t+h\mid t}=\mathbb{E}[x_{t+h}\mid {\mathcal{F}}_t]$. Via the MA($\infty$) representation, the forecast-error variance is

$$\mathrm{Var}[e_h(\ell )]=\left(1+{\psi }_1^2+\cdots +{\psi }_{\ell -1}^2\right){\sigma }_a^2.$$

• AR(p) forecast → mean. As $\ell \to \infty$ the forecast converges to the unconditional mean ${\varphi }_0/(1-\sum {\varphi }_i)$. For AR(1) the half-life to revert half-way to the mean is $\ell =\frac{\ln (0.5)}{\ln |{\varphi }_1|}$.
• MA(q) forecast → mean after $q$ steps. Once all the past shocks have cleared, the forecast is exactly the mean.
• Random walk forecast. ${\hat{p}}_h(\ell )=p_h$ (no mean reversion), forecast-error variance $=\ell {\sigma }_a^2$ - linear growth. This is why the square-root-of-time rule works for driftless IGARCH and fails under nonzero mean (Tsay §7.2.1, §7.3.1: the 15-day Gauss AR(2)-GARCH VaR was $1,039,191 vs the $1,114,257 the $\sqrt{15}$ rule predicts).

Unit-root taxonomy. Write $x_t$ as an AR(1): stationary if $|{\varphi }_1|<1$; a random walk if ${\varphi }_1=1$ (shock never decays); random walk with drift $p_t=t\mu +p_0+\sum a_i$ (variance still grows linearly, plus a linear trend in the mean); trend-stationary $p_t={\beta }_0+{\beta }_{1}t+r_t$ where $r_t$ is stationary (remove the deterministic trend $\Rightarrow$ stationary, variance finite). The last two look alike in a plot but behave completely differently - that distinction is the heart of unit-root testing.

Differencing. The first difference $\nabla x_t=(1-B)x_t=x_t-x_{t-1}$ removes one unit root; $x_t$ is integrated of order 1, $I(1)$, if $\nabla x_t$ is stationary. An ARIMA(p,1,q) model is one where $(1-B)x_t$ is stationary ARMA(p,q).

Augmented Dickey–Fuller test (Tsay eq. 2.38–2.40). Fit

$$\nabla x_t=c_t+{\beta }_c{x}_{t-1}+\sum _{i=1}^p{\varphi }_i\nabla x_{t-i}+e_t,$$

and test $H_0:{\beta }_c=0$ (unit root, since ${\beta }_c={\varphi }_1-1$) against $H_1:{\beta }_c<0$. The statistic $\mathrm{DF}=({\hat{\varphi }}_1-1)/\mathrm{std}({\hat{\varphi }}_1)$ does not follow a Student-$t$; it follows the Dickey–Fuller distribution (critical values $\approx -3.43,-2.86,-2.57$ at 1%,5%,10% with a constant). The lagged $\nabla x_{t-i}$ terms absorb serial correlation so the test is valid for ARMA errors.

Spurious regression (Granger & Newbold 1974). Two independent $I(1)$ series $Y_t,X_t$ regressed as $Y_t=\alpha +\beta X_t+{\eta }_t$ produce $t$-stats exceeding 10 and $R^2>0.8$ asymptotically despite being unrelated. The residual is itself $I(1)$ - it does not mean-revert.

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3. Computational Implementation - ADF and spurious regression

Run the ADF regression on a stationary AR(1) and on a random walk, and demonstrate the spurious-regression phenomenon averaged over many replications. Stdlib only.

Two independent random walks regressed on each other give mean $R^2=0.248$ and - critically - the residual fails the ADF test in 177/200 replications: you cannot tell them apart from a “real” relationship by $R^2$ or $t$-stats alone. The stationary AR(1) yields a strongly negative ADF stat ($-15.6$, rejects), the random walk a near-zero one ($-0.64$, cannot reject). The ADF on the residual is the discriminator, not $R^2$.

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4. Failure Modes & First-Principles Breakdowns

1. The spurious-regression trap. High $R^2$ + huge $t$ between unrelated unit-root series is a mirage; the residual ADF (or cointegration test) is the only honest verdict. This is the single costliest error in applied finance econometrics.
2. ADF has low power near the boundary. ${\varphi }_1=0.98$ is (barely) stationary but ADF needs huge samples to reject - and a genuinely stationary, highly-persistent series is easily misclassified as a unit root. Complement with economic logic, never rely on the test alone.
3. The square-root-of-time rule silently fails. VaR scaling $\mathrm{VaR}(\ell )=\sqrt{\ell }\mathrm{VaR}(1)$ is only exact for a driftless IGARCH(1,1) (RiskMetrics) model. With a nonzero mean or a different volatility model it misstates risk - Tsay’s own 15-day Gauss AR(2)-GARCH example overstates by ~7% under the rule.
4. Over-differencing. Differencing an already-stationary series (or a trend-stationary one after detrending) injects a non-invertible MA root and destroys estimation. Distinguish random-walk-with-drift (variance grows) from trend-stationary (variance finite) before deciding what to remove.

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5. References

• Tsay, Analysis of Financial Time Series
• Granger, C.W.J. & Newbold, P. (1974), “Spurious Regressions in Econometrics,” Journal of Econometrics
• Hastie, Tibshirani & Friedman, Elements of Statistical Learning
• Campbell, Lo & MacKinlay, The Econometrics of Financial Markets

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6. Connected Graph Bridges

• Back: 02 · Stationarity & ARMA · Index Hub
• Forward: 04 · Volatility Modeling · 05 · Cointegration & Multivariate
• Applied: VaR & Expected Shortfall (square-root-of-time caveat) · Backtesting Hygiene