Lesson M08.L04: Stationarity, Trends, and Unit Root Tests

Field	Detail
Module	08 — Time Series Basics
Level	Graduate (ECMT5001 S1 2027)
Duration	30–35 min
Learning Objective	Define covariance stationarity; distinguish deterministic from stochastic trends; understand the ADF unit root test; know when to detrend versus first-difference.
Provenance	Wooldridge, Introductory Econometrics, Ch 10.4–10.5

Explanation

1. Stationarity

A time series $\{y_t\}$ is covariance stationary (weakly stationary) if its first two moments are time-invariant:

\[\begin{aligned} E(y_t) &= \mu \quad &&\text{(constant mean)} \\ \text{Var}(y_t) &= \sigma^2 < \infty \quad &&\text{(constant, finite variance)} \\ \text{Cov}(y_t, y_{t-h}) &= \gamma(h) \quad &&\text{(depends only on lag } h\text{, not on } t\text{)} \end{aligned}\]

Covariance stationarity is also called weak stationarity (as opposed to strict stationarity, which requires the entire joint distribution to be time-invariant).

Why does stationarity matter? - Standard asymptotic theory (LLN, CLT) for time series requires stationarity or near-stationarity. - Non-stationary series have moments that change over time, making the past a poor guide to the future and standard inference invalid. - Spurious regressions (M08.L03) arise from regressing non-stationary series on each other.

Examples: - Stationary: stock returns, GDP growth rate (%), CPI inflation (if in a stable policy regime). - Non-stationary: GDP level, stock prices, nominal wages — all trend upward over time.

2. Deterministic Trend (Trend-Stationary Process)

A deterministic trend model:

\[y_t = \alpha + \beta t + u_t, \quad u_t \sim \text{stationary}\]

Here the mean changes linearly over time ($E(y_t) = \alpha + \beta t$), but the variance is constant ($\text{Var}(y_t) = \sigma_u^2$). The series "wanders" above and below the trend line but always reverts to it.

This series is NOT stationary (non-constant mean), but it is trend-stationary: once you remove the linear time trend, the residuals $u_t$ are stationary.

Solution: include $t$ as a regressor (detrending):

Regress $y_t$ on a constant and $t$: $y_t = \hat{\alpha} + \hat{\beta} t + \hat{u}_t$.
Use the detrended residuals $\hat{u}_t$ (or the detrended $y$) in subsequent regressions.

Alternatively, simply include $t$ as a control variable in the regression of interest. This soaks up the common trend and avoids spurious correlation between $y_t$ and other trending series.

3. Stochastic Trend / Unit Root

A random walk (the canonical stochastic trend model):

\[y_t = y_{t-1} + e_t, \quad e_t \sim \text{i.i.d.}(0, \sigma_e^2)\]

Starting from $y_0$:

\[y_t = y_0 + \sum_{s=1}^t e_s\]

Key properties: $$E(y_t) = y_0 \quad \text{(constant mean)}$$ $$\text{Var}(y_t) = t \sigma_e^2 \quad \text{(grows without bound)}$$

The variance increases linearly with $t$ — this series has a stochastic trend: it wanders further and further from its starting point over time, with no tendency to revert. The shocks are permanent: a shock $e_t$ shifts the level of $y$ permanently.

Why "unit root"? Express the random walk as:

\[y_t = \rho y_{t-1} + e_t\]

with $\rho = 1$. In the AR(1) representation, the autoregressive coefficient equals 1 (the "unit root"). For $|\rho| < 1$, the process is stationary (mean-reverting). For $\rho = 1$, the process is non-stationary.

Random walk with drift:

\[y_t = \mu + y_{t-1} + e_t\]

Now $E(y_t) = y_0 + \mu t$ — trending upward (or downward) on average, with a variance still growing as $t\sigma_e^2$. Most economic levels (GDP, price indices) follow a random walk with positive drift.

4. Contrast: Deterministic vs Stochastic Trend

Feature	Deterministic Trend	Stochastic Trend (Unit Root)
Mean	$\alpha + \beta t$ (non-constant)	$y_0$ (random walk) or $y_0 + \mu t$ (drift)
Variance	Constant $\sigma_u^2$	Grows: $t \sigma_e^2$
Shock persistence	Temporary (shocks decay)	Permanent (shocks accumulate)
Remedy	Detrend (subtract $\hat{\alpha} + \hat{\beta} t$)	First-difference ($\Delta y_t$)
After remedy	Stationary	Stationary (if one unit root)

Critical point: If a series has a unit root, detrending does not make it stationary — you must first-difference. Applying the wrong remedy leaves you with a non-stationary series and invalid inference.

5. Integrated Processes

A process $y_t$ that becomes stationary after $d$ first-differences is said to be integrated of order $d$, written $y_t \sim I(d)$:

$I(0)$: already stationary.
$I(1)$: one unit root; $\Delta y_t \sim I(0)$.
$I(2)$: two unit roots; $\Delta^2 y_t \sim I(0)$ (rare in economics).

Most macroeconomic levels (GDP, prices, nominal wages) are $I(1)$. Their growth rates (log differences) are $I(0)$.

6. Augmented Dickey-Fuller (ADF) Test

The ADF test is the standard formal test for a unit root.

Null hypothesis: $H_0$: $y_t$ has a unit root (is non-stationary, $I(1)$). Alternative: $H_1$: $y_t$ is stationary ($I(0)$).

Regression form:

\[\Delta y_t = \alpha + \theta y_{t-1} + \sum_{j=1}^p \gamma_j \Delta y_{t-j} + \varepsilon_t\]

The augmentation terms $\sum_{j=1}^p \gamma_j \Delta y_{t-j}$ absorb any serial correlation in the errors (the Dickey-Fuller test without augmentation assumes i.i.d. errors; the ADF is more general).

Test statistic: $t_\theta = \hat{\theta} / SE(\hat{\theta})$

Decision rule: - Reject $H_0$ (conclude stationary) if $t_\theta < \text{ADF critical value}$ (e.g. $-2.89$ at 5% for no trend; $-3.45$ at 5% with trend). - The ADF critical values are not standard normal — they are more negative (require a larger magnitude to reject) because the distribution shifts under the unit-root null. - Fail to reject $H_0$ → conclude $I(1)$; take first differences.

Lag length selection for $p$: Use AIC/BIC or start with a long lag and drop insignificant ones.

Specification choices: - No constant, no trend (rarely appropriate in economics). - With constant (most common for stationary-around-mean series). - With constant and linear trend (for potentially trend-stationary series).

7. Solution: First Differencing

If a series is $I(1)$, first-differencing produces a stationary $I(0)$ series:

\[\Delta y_t = y_t - y_{t-1}\]

Examples: - $\ln(\text{GDP}_t)$ is $I(1)$ → $\Delta \ln(\text{GDP}_t) \approx g_t$ (GDP growth rate) is $I(0)$. - $\ln(\text{CPI}_t)$ is $I(1)$ → $\Delta \ln(\text{CPI}_t) \approx \pi_t$ (inflation rate) is $I(0)$. - Nominal interest rate $r_t$ is often $I(1)$ → $\Delta r_t$ is $I(0)$.

Trade-off of differencing: You lose information about long-run levels. If you suspect a long-run cointegrating relationship between two $I(1)$ series, first-differencing discards that relationship — you should instead use a vector error correction model (VECM) (advanced topic).

Worked Example

Is Australian GDP stationary? Levels vs. growth rates.

Setup. Consider quarterly Australian real GDP ($Y_t$) and the quarterly GDP growth rate ($g_t = 100 \times \Delta \ln Y_t$), 1960Q1–2023Q4 ($T = 256$).

Step 1. Visual inspection.

$Y_t$ (level): trends steadily upward from ~$100bn to ~$600bn (2015 AUD). Clearly non-constant mean.
$g_t$ (growth rate): fluctuates around approximately 0.8% per quarter with no obvious trend. Plausible stationarity.

Step 2. ADF test — GDP level ($Y_t$).

Specification: with constant and linear trend (because $Y_t$ trends upward).

\[\Delta Y_t = \alpha + \beta t + \theta Y_{t-1} + \sum_{j=1}^4 \gamma_j \Delta Y_{t-j} + \varepsilon_t\]

Suppose $t_\theta = -1.85$. ADF 5% critical value (with trend) = $-3.45$.

Since $-1.85 > -3.45$: fail to reject $H_0$. Conclude $Y_t$ has a unit root — it is $I(1)$.

Step 3. ADF test — GDP growth rate ($g_t$).

Specification: with constant only (growth rate fluctuates around a positive mean).

\[\Delta g_t = \alpha + \theta g_{t-1} + \sum_{j=1}^4 \gamma_j \Delta g_{t-j} + \varepsilon_t\]

Suppose $t_\theta = -5.42$. ADF 5% critical value (with constant) = $-2.89$.

Since $-5.42 < -2.89$: reject $H_0$. Conclude $g_t$ is stationary — it is $I(0)$.

Step 4. Conclusion and implication.

Series	ADF stat	5% CV	Decision	Order
$Y_t$ (level)	−1.85	−3.45	Fail to reject $H_0$	$I(1)$
$g_t$ (growth)	−5.42	−2.89	Reject $H_0$	$I(0)$

Implication: Regressions using GDP levels are at risk of spurious regression unless the other variable is also $I(1)$ and the two series are cointegrated. Regressions using GDP growth rates are valid under standard OLS asymptotics (given stationarity).

Common Misconception

"Detrending (removing a linear time trend) removes all non-stationarity."

This is false. Detrending (subtracting $\hat{\alpha} + \hat{\beta} t$) only works if the series is trend-stationary — i.e. non-stationary only because of a deterministic trend, with stationary errors.

If the series has a stochastic trend (unit root), detrending does not produce a stationary series. The detrended series will still have a variance that grows over time (because the unit-root shocks accumulate even around a linear trend). You must first-difference, not detrend.

How to tell which applies: Use the ADF test. Include a time trend in the ADF regression. If you reject the unit-root null, the series is trend-stationary (detrend). If you fail to reject, it has a unit root (first-difference).

Practical cost of the wrong remedy: - First-differencing a trend-stationary series: over-differencing → introduces unnecessary moving-average structure in errors. - Detrending a unit-root series: fails to achieve stationarity → spurious regression, invalid inference.

Practice Prompts

Stationarity check. Consider three time series:
$A_t$: Australian CPI level (1990–2023, monthly).
$B_t$: Australian CPI inflation rate (monthly change in log CPI).
$C_t$: 10-year government bond yield (%).

(a) For each series, state whether you would expect it to be stationary or non-stationary, and why. (b) For any non-stationary series, suggest the appropriate transformation to induce stationarity. (c) What specification (no trend, constant only, constant + trend) would you use in the ADF test for each series? Justify.

ADF test interpretation. A researcher tests whether the monthly Australian unemployment rate ($U_t$) is stationary using an ADF test with constant and no trend. She finds $t_\theta = -2.31$ and the 5% ADF critical value is $-2.89$. (a) State the null and alternative hypotheses of the ADF test. (b) What is the conclusion at the 5% level? (c) What should she do before running a regression of $U_t$ on other economic variables? Give two specific steps.
Random walk properties. A random walk is defined by $y_t = y_{t-1} + e_t$, $e_t \sim \text{i.i.d.}(0, 4)$, $y_0 = 0$. (a) Compute $\text{Var}(y_{25})$. (b) Explain why this series is not covariance stationary. (c) Show that $\Delta y_t = y_t - y_{t-1}$ is stationary. What are its mean and variance? (d) Explain in words what it means for shocks to be "permanent" in a random walk, vs "transitory" in a stationary AR(1).

Further Resources

Wooldridge, Introductory Econometrics (7e), §10.4–10.5 — Trending Time Series and Stationarity.
Dickey, D.A. & Fuller, W.A. (1979). "Distribution of the estimators for autoregressive time series with a unit root." Journal of the American Statistical Association, 74(366), 427–431. — Original Dickey-Fuller paper.
Elliott, G., Rothenberg, T.J. & Stock, J.H. (1996). "Efficient tests for an autoregressive unit root." Econometrica, 64(4), 813–836. — More powerful DF-GLS test.
ABS (2024). Australian National Accounts: National Income, Expenditure and Product (Cat. 5206.0). — Source for Australian GDP series.
See also: M08.L03 (Spurious Regression — motivation for why stationarity matters) and future cointegration module.

Feature	Deterministic Trend	Stochastic Trend (Unit Root)
Mean	\(\alpha + \beta t\) (non-constant)	\(y_0\) (random walk) or \(y_0 + \mu t\) (drift)
Variance	Constant \(\sigma_u^2\)	Grows: \(t \sigma_e^2\)
Shock persistence	Temporary (shocks decay)	Permanent (shocks accumulate)
Remedy	Detrend (subtract \(\hat{\alpha} + \hat{\beta} t\))	First-difference (\(\Delta y_t\))
After remedy	Stationary	Stationary (if one unit root)

Series	ADF stat	5% CV	Decision	Order
\(Y_t\) (level)	−1.85	−3.45	Fail to reject \(H_0\)	\(I(1)\)
\(g_t\) (growth)	−5.42	−2.89	Reject \(H_0\)	\(I(0)\)