# A Robust Unit Root Test That Handles Serial Correlation

Cite as: Gorak, R. (2026). A Robust Unit Root Test That Handles Serial Correlation. Tradicted. https://www.tradicted.com/research/phillips-phillips-perron-1988/
Paper: Peter C. B. Phillips and Pierre Perron — *Testing for a Unit Root in Time Series Regression*
Published in: Biometrika (1988)
Original: https://doi.org/10.1093/biomet/75.2.335
DOI: 10.1093/biomet/75.2.335

Key finding: Phillips and Perron's (1988) Z(α̃) test reached power of .772 against α = 0.85 in 2,000 simulations with T = 100 and i.i.d. errors, versus .557 for the Said-Dickey t(α̂*) test.

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A unit root test determines whether a time series follows a random walk, where shocks accumulate permanently instead of reverting to a trend. Phillips and Perron's (1988) paper, *Testing for a Unit Root in Time Series Regression*, ran 2,000 simulations with T = 100. Their Z(α̃) statistic reached power of .772 against α = 0.85, versus .557 for the competing Said-Dickey t(α̂*) test.

## What the Study Found

With T = 100, i.i.d. errors (θ = 0.0) and lag truncation ℓ = 2, the Z(α̃) test had size .044 against a nominal 5% level. The same Z(α̃) test reached power of .772 against the stationary alternative α = 0.85, exceeding the Said-Dickey t(α̂*) test's .557. Under strongly negative moving-average errors (θ = -0.8), the Z(α̃) test's size rose to .997, far above the 5% nominal level. The Said-Dickey t(α̂*) test held size at .677 under the same θ = -0.8 condition, a smaller distortion than the Z tests showed.

> "As we have seen in Section 6, there is no loss in asymptotic local power in the use of the Z tests for a unit root."
>
> Phillips and Perron (1988), *Testing for a Unit Root in Time Series Regression*, p. 24.

## Methodology

The paper substitutes Monte Carlo simulation for an empirical dataset, running 2,000 replications with T = 100 observations each. Data were generated from a unit-root autoregression with moving-average errors, u_t = e_t + θe_{t-1}, where e_t is i.i.d. N(0,1). The six values of θ tested were 0.0, 0.5, 0.8, -0.2, -0.5 and -0.8. The authors varied the lag length in the augmented autoregression and the lag truncation ℓ in the variance estimator across 2, 4, 6, 8, 12.

## Key Statistics

| Metric | Finding | Context |
|---|---|---|
| Size, Z(α̃) test | .044 | T = 100, θ = 0.0, ℓ = 2, nominal 5% level |
| Power, Z(α̃) test | .772 | T = 100, θ = 0.0, ℓ = 2, alternative α = 0.85 |
| Size, Said-Dickey t(α̂*) test | .068 | T = 100, θ = 0.0, ℓ = 2, nominal 5% level |
| Power, Said-Dickey t(α̂*) test | .557 | T = 100, θ = 0.0, ℓ = 2, alternative α = 0.85 |
| Z(α̂) test statistic | Z(α̂) = T(α̂-1) - (1/2){T⁻²Σ(y_{t-1} - ȳ_{-1})²}⁻¹(σ̂²_Tℓ - ŝ²) | Nonparametric correction to the Dickey-Fuller coefficient statistic |

## Z(α̃) Test vs Said-Dickey t(α̂*) Test

| Measure | Z(α̃) test | Said-Dickey t(α̂*) test |
|---|---|---|
| Size, θ = 0.0, ℓ = 2 | .044 | .068 |
| Power, θ = 0.0, α = 0.85, ℓ = 2 | .772 | .557 |
| Size, θ = -0.8, ℓ = 2 | .997 | .677 |

## Why This Matters

Unit root tests let researchers distinguish stochastic trends from deterministic ones in financial and economic time series. That distinction changes how forecasts, hedges and mean-reversion strategies get built. Because the Z tests need no specific assumption about the error process, they extend the Dickey-Fuller framework to more time series types. The simulations also show the new tests are not a universal replacement. Under negative moving-average errors, the Said-Dickey long-autoregression approach holds size more reliably.

## FAQ

### What is the Phillips-Perron test for a unit root?

The Phillips-Perron test achieved power of .772 against a stationary alternative of α = 0.85 in Phillips and Perron's (1988) simulations with T = 100. It corrects the Dickey-Fuller regression statistics for serial correlation using a nonparametric long-run variance estimator, so it needs no specific error assumption.

### How does the Z(α̃) test compare to the Said-Dickey augmented Dickey-Fuller test?

The Z(α̃) test reached power of .772 against α = 0.85, beating the Said-Dickey t(α̂*) test's .557, when errors were i.i.d. (θ = 0.0) in Phillips and Perron's (1988) simulations. With strongly negative moving-average errors (θ = -0.8), Z(α̃) size rose to .997 versus Said-Dickey's .677, so Said-Dickey performed more reliably there.

### How is the Phillips-Perron test used to test the random walk hypothesis in financial markets?

Phillips and Perron's (1988) Z(α̂) and Z(α̃) statistics test the null hypothesis α = 1, whether a series follows a random walk. The tests stay valid under serially correlated errors. Researchers apply them to stock prices and exchange rates to assess market efficiency.

## Source

Phillips, Peter C. B., and Pierre Perron. (1988). Testing for a Unit Root in Time Series Regression. Biometrika, 75(2), 335-346.

[Read the full paper →](https://doi.org/10.1093/biomet/75.2.335)