Phillips-Perron Tests And Long-Run Variance Corrections

Why Correct the Dickey–Fuller Statistic?

The Dickey–Fuller test is built around a regression of the change in a series on its lagged level. Its usual reference distribution relies on a particular innovation structure. If the innovations are serially correlated or heteroskedastic, the conventional Dickey–Fuller t statistic need not have the reference distribution used for that structure. The Phillips–Perron (PP) test retains the simple Dickey–Fuller regression but adjusts its statistic using an estimate of the innovations’ long-run variance.

This is a nonparametric correction: instead of adding lagged differences to the regression, as in the augmented Dickey–Fuller approach, the PP method estimates the cumulative effect of serial dependence with a kernel-weighted sum of residual autocovariances. The correction can accommodate serial correlation and certain forms of heteroskedasticity under regularity conditions. It does not make a badly specified unit-root regression, a nonstationary innovation process, or an inappropriate deterministic specification harmless.

Definition: Consider the no-deterministic-term regression $$ \Delta y_t=\gamma y_{t-1}+u_t. $$ The unit-root null is $H_0:\gamma=0$. For an AR(1) model with coefficient $1+\gamma$, stationarity requires $-2<\gamma<0$; the conventional one-sided test uses $\gamma<0$ as its tested direction, not as the full stationarity condition. The PP t statistic adjusts the ordinary t statistic for $\gamma$ to account for serial correlation and heteroskedasticity in $u_t$.

The no-deterministic-term regression makes the correction especially transparent. In applications with an intercept or a time trend, the regression and the limiting Dickey–Fuller distribution change; the statistic must be constructed for that specification and compared with its corresponding critical values. The test is not an ordinary Student t test.

Short-Run and Long-Run Variance

Let $\hat{\sigma}^2$ estimate the short-run variance of the regression innovations, commonly from the OLS residuals. It measures contemporaneous innovation variance. By contrast, the long-run variance is the sum of the innovation autocovariances:

\omega^2=\sum_{j=-\infty}^{\infty}\operatorname{Cov}(u_t,u_{t-j}) =\gamma_u(0)+2\sum_{j=1}^{\infty}\gamma_u(j),

where $\gamma_u(j)=\operatorname{Cov}(u_t,u_{t-j})$. When innovations are uncorrelated, the positive-lag autocovariances are zero and the two variances agree. With serial dependence they can differ. A PP correction estimates $\omega^2$ and uses both it and $\hat{\sigma}^2$; replacing one by the other would remove the adjustment the test is designed to make.

A common HAC (heteroskedasticity-and-autocorrelation-consistent) estimate uses OLS residuals $\hat{u}_t$. Define $\hat{\gamma}_j=T^{-1}\sum_{t=j+1}^{T}\hat{u}_t\hat{u}_{t-j}$ for $j\geq 0$, where $T$ is the number of observations in the regression. For a kernel $k$ and bandwidth $b$, one version is

\hat{\omega}^2 =\hat{\gamma}_0+ 2\sum_{j=1}^{T-1} k(j/b)\hat{\gamma}_j.

For kernels with support on $[-1,1]$, weights are zero when $j/b>1$. The kernel determines how autocovariances are weighted; the bandwidth determines how quickly the weights decline and how many lags contribute. In large-sample consistency arguments, a typical bandwidth sequence grows with $T$ but more slowly than $T$, so that $b\to\infty$ and $b/T\to0$. A bandwidth used in a small worked calculation is not, by itself, a recommendation for practical sample-size selection.

The PP t Correction

Let $t_{DF}$ be the ordinary OLS t statistic on $\hat{\gamma}$ in the Dickey–Fuller regression. Let $\hat{\sigma}$ and $\hat{\omega}$ be the square roots of the short-run and long-run variance estimates. For the no-deterministic-term case, define

Q=T^{-1}\sqrt{\sum_{t=1}^{T}y_{t-1}^{\,2}}.

The PP t statistic can then be written as

Z_t= \frac{\hat{\sigma}}{\hat{\omega}}t_{DF} -\frac{\hat{\omega}^2-\hat{\sigma}^2}{2\hat{\omega}Q}.

The first term rescales the ordinary t statistic. The second corrects its bias contribution. If $\hat{\omega}^2=\hat{\sigma}^2$, the correction term is zero and the rescaling is one, so $Z_t=t_{DF}$. The definition of $Q$ matters: with this convention, the correction contains $1/(\hat{\omega}Q)$. Changing the normalization of $Q$ without changing the formula changes the statistic.

Assumptions and interpretation: The PP asymptotic argument requires a unit-root null with innovations satisfying conditions that yield a functional central limit theorem and a finite, positive long-run variance. The HAC estimator must consistently estimate that variance under the chosen kernel and bandwidth scheme. The method allows serial correlation and forms of heteroskedasticity covered by those conditions; it does not guarantee validity under arbitrary dependence or nonstationary innovations. Under the null, the corrected statistic has the relevant Dickey–Fuller limiting distribution, not a standard normal or Student t distribution. Use critical values for the regression’s deterministic terms.

Why the Correction Has This Form

A short asymptotic calculation shows the purpose of the two pieces. Under the unit-root null, the partial sums of $u_t$ have a Brownian-motion limit when the stated regularity conditions hold. In the simplest no-deterministic-term case, the ordinary statistic has a limiting Dickey–Fuller functional, denoted here by $\tau$, plus a term reflecting the difference between short-run and long-run variances. With $Q$ as defined above, the expansion is

t_{DF} = \frac{\omega}{\sigma}\tau +\frac{\omega^2-\sigma^2}{2\sigma Q} +o_p(1).

Here $\sigma^2$ and $\omega^2$ are the population short-run and long-run variances, and $o_p(1)$ denotes a term that converges to zero in probability. Multiply the expansion by $\sigma/\omega$. The first term becomes $\tau$; the variance-dependent term becomes $\frac{\omega^2-\sigma^2}{2\omega Q}$. Subtracting its sample analogue gives the PP statistic:

\frac{\sigma}{\omega}t_{DF} -\frac{\omega^2-\sigma^2}{2\omega Q} =\tau+o_p(1).

In practice, unknown variances are replaced by their estimates. This argument explains why both the scale adjustment and the additive correction are needed. It also explains why uncertainty in $\hat{\omega}^2$, including uncertainty induced by kernel and bandwidth choices, can affect the calculated statistic.

Worked Example: Applying the PP Correction

Suppose hypothetical calculations from a no-deterministic-term regression give $t_{DF}=-1.750$, $\hat{\sigma}=0.8000$, $\hat{\omega}^2=0.9600$, and $Q=2.000$. Find $Z_t$.

First, $\hat{\omega}=\sqrt{0.9600}=0.9798$. The rescaled first term is

\frac{0.8000}{0.9798}(-1.750)\approx -1.429.

The additive correction to subtract is

\frac{0.9600-0.6400}{2(0.9798)(2.000)} \approx 0.08165.

Therefore $Z_t\approx -1.429-0.08165=-1.511$. For illustration, if the appropriate 5% critical value for the chosen specification were $-1.95$, this statistic would not be more negative than the critical value, so the unit-root null would not be rejected at that level. The actual critical value must match the deterministic specification; the statistic is not interpreted using an ordinary t table.

Kernel Choices in a Concrete Estimate

Two standard kernels illustrate how the autocovariance weights differ. For $0\leq |x|\leq1$, the Bartlett kernel is $k(x)=1-|x|$. The Parzen kernel is

k(x)= \begin{cases} 1-6x^2+6|x|^3, & 0\leq |x|\leq \tfrac12,\\ 2(1-|x|)^3, & \tfrac12<|x|\leq1,\\ 0, & |x|>1. \end{cases}

The absolute value makes both kernels symmetric for negative and positive lags. With bandwidth $b=3$, lags 1 and 2 have arguments $1/3$ and $2/3$, while the weight at lag 3 is zero. For Bartlett, the two weights are $2/3$ and $1/3$. For Parzen, they are approximately $0.5556$ and $0.07407$.

Worked Example: Bartlett and Parzen Long-Run Variances

Suppose hypothetical residual autocovariances are $\hat{\gamma}_0=0.6400$, $\hat{\gamma}_1=0.2000$, and $\hat{\gamma}_2=0.08000$, with bandwidth $b=3$. Using the weights just calculated, find each estimate.

For Bartlett,

\hat{\omega}^2_{\text{Bartlett}} =0.6400+2\left(\frac23(0.2000)+\frac13(0.08000)\right) =0.9600.

For Parzen, the lag-1 weight is $1-6(1/3)^2+6(1/3)^3=0.5556$, and the lag-2 weight is $2(1-2/3)^3=0.07407$. Thus

\hat{\omega}^2_{\text{Parzen}} =0.6400+2\left(0.5556(0.2000)+0.07407(0.08000)\right) \approx 0.8741.

The corresponding long-run standard deviations are approximately $0.9798$ and $0.9349$. The estimates differ because the kernels weight the same positive-lag autocovariances differently. Neither number is automatically the better estimate: the choice should reflect a justified estimation procedure, not which value gives a preferred test result.

Worked Example: Recalculating the Test Under Two Kernels

Use the same hypothetical regression inputs $t_{DF}=-1.750$, $\hat{\sigma}=0.8000$, and $Q=2.000$, along with the Bartlett and Parzen variance estimates from the preceding calculation. Compute the two PP statistics.

For Bartlett, $\hat{\omega}^2=0.9600$ and $\hat{\omega}=0.9798$. Substitution gives

Z_{t,\text{Bartlett}} =\frac{0.8000}{0.9798}(-1.750) -\frac{0.9600-0.6400}{2(0.9798)(2.000)} \approx -1.511.

For Parzen, $\hat{\omega}^2\approx0.8741$ and $\hat{\omega}\approx0.9349$:

Z_{t,\text{Parzen}} =\frac{0.8000}{0.9349}(-1.750) -\frac{0.8741-0.6400}{2(0.9349)(2.000)} \approx -1.560.

The values differ because the estimated long-run variances differ. Each statistic must be compared with the appropriate PP critical value for the same regression specification. A small numerical change should not be described as a change in evidence unless the relevant critical-value comparison supports that conclusion.

Bandwidth, Diagnostics, and Practical Judgment

A kernel alone does not determine the long-run variance estimate. Bandwidth controls how far out in the autocovariance sequence the estimator reaches and how strongly distant lags are downweighted. Too narrow a bandwidth can omit meaningful serial dependence; too wide a bandwidth can add noisy sample autocovariances. Because the PP correction depends on $\hat{\omega}^2$, these choices can influence the test statistic, especially in modest samples.

Worked Example: How Bandwidth Changes Bartlett Weights

Return to the hypothetical autocovariances $0.6400$, $0.2000$, and $0.08000$. With Bartlett bandwidth $b=2$, the lag-1 weight is $1-1/2=0.5$, and lag 2 receives zero weight. The estimate is

\hat{\omega}^2_{b=2} =0.6400+2(0.5)(0.2000)=0.8400.

With $b=3$, the lag-1 and lag-2 weights are $2/3$ and $1/3$, giving $\hat{\omega}^2_{b=3}=0.9600$. Holding the other hypothetical regression inputs fixed, the $b=2$ estimate gives $\hat{\omega}=0.9165$ and

Z_{t,b=2} =\frac{0.8000}{0.9165}(-1.750) -\frac{0.8400-0.6400}{2(0.9165)(2.000)} \approx -1.582.

This arithmetic isolates the effect of the variance estimate; in real data, the regression residuals and their autocovariances determine all inputs. It shows why reporting the kernel and bandwidth is important for reproducibility.

In applied work, assess whether the deterministic terms match the question and series, and whether the innovation assumptions supporting the HAC estimate are plausible. Report the kernel, bandwidth rule or value, and the short-run and long-run variance estimates when these are relevant to interpretation. Sensitivity checks across defensible bandwidth choices can reveal whether a conclusion depends heavily on one arbitrary setting; they do not justify selecting the setting that produces a desired rejection.

Common Mistakes

Using an ordinary t critical value. The PP statistic has a Dickey–Fuller limiting distribution under the null. Use critical values for the specified deterministic terms.
Treating $\hat{\omega}^2$ and $\hat{\sigma}^2$ as interchangeable. The first estimates long-run variance; the second estimates short-run variance. Their difference drives the additive correction.
Changing the definition of $Q$ but keeping the same formula. With $Q=T^{-1}\sqrt{\sum y_{t-1}^2}$, the correction is $-(\hat{\omega}^2-\hat{\sigma}^2)/(2\hat{\omega}Q)$. A different normalization requires a correspondingly changed expression.
Calling the PP method assumption-free. It adjusts for specified forms of serial dependence and heteroskedasticity; it still relies on regularity conditions, a valid deterministic specification, and a consistent long-run variance estimate.
Assuming every negative $\gamma$ implies stationarity. For an AR(1) coefficient $1+\gamma$, stationarity requires $-2<\gamma<0$, even though the conventional one-sided test direction is $\gamma<0$.

Key takeaway: The PP t statistic rescales the Dickey–Fuller statistic by the short-run to long-run standard-deviation ratio and subtracts a variance-dependent correction. Kernel and bandwidth choices enter through the long-run variance estimate, so they are part of the reported testing procedure.

Check Your Understanding

Answer each question using the definitions and formulas above.

What is the distinction between the short-run variance and the long-run variance of the innovations?
For $t_{DF}=-2.00$, $\hat{\sigma}=1.00$, $\hat{\omega}^2=1.00$, and $Q=2.00$, what is the PP t statistic? Why?
With Bartlett kernel and bandwidth $b=4$, what are the weights at lags 1 and 2?
Why must a PP statistic be compared with critical values for the regression’s deterministic specification rather than a standard t table?
Explain one way an overly narrow bandwidth could affect the long-run variance estimate.