Local-To-Unity Asymptotics And Near-Unit-Root Persistence

Why Local-to-Unity Behavior Matters

An autoregressive coefficient of $0.99$ may look safely below one. Yet over a sample of a few hundred observations, a shock can remain influential for a long time, and the data may not clearly distinguish this process from a random walk. The issue is that the coefficient’s distance from one must be considered relative to the sample size.

Local-to-unity asymptotics formalize this situation by letting the coefficient approach one as the sample grows. Unlike a fixed stationary alternative, this sequence remains difficult to distinguish from a unit root in large samples. It provides a useful account of persistent behavior, the finite-sample difficulty of estimating the autoregressive coefficient, and why unit-root tests can have nontrivial power against alternatives close to one.

Definition: A local-to-unity autoregressive sequence has coefficient $\rho_n=1+c/n$, where $n$ is the sample size and $c$ is a fixed real number. For $c<0$, the coefficient is below one for sufficiently large $n$; for $c=0$, it equals one; and for $c>0$, it is above one. The local parameter $c$ measures the coefficient’s distance from one on the $1/n$ scale.

Model and Assumptions

Consider the zero-mean autoregression, with no intercept or trend,

y_t=\rho_n y_{t-1}+u_t,\qquad t=1,\ldots,n,\qquad \rho_n=1+\frac{c}{n}.

Assume $y_0=0$, and that the innovations $u_t$ are independent and identically distributed with mean zero, variance $\sigma^2>0$, and finite $2+\delta$ moment for some $\delta>0$. These assumptions give the functional central limit theorem for the scaled partial sums of the innovations. The zero initial value is a convenient normalization; the same limit holds if the initial value is negligible on the $\sqrt n$ scale, meaning $y_0/(\sigma\sqrt n)\to 0$.

The parameter $c$ is held fixed as $n$ increases. Thus, when $c=-2$, for example, the coefficient is $1-2/n$, not a fixed value such as $0.98$. This distinction is central: a fixed coefficient below one eventually becomes easy to distinguish from a unit root, whereas a coefficient that stays $O(1/n)$ from one does not become separated in the same way.

Worked Example: Reading the Local Parameter

Suppose a sample contains $n=200$ observations and the coefficient is $\rho_n=0.99$. Express this coefficient in local-to-unity form and assess its persistence over 200 periods.

The local parameter is $c=n(\rho_n-1)=200(0.99-1)=-2$, so $\rho_n=1-2/200$. A shock’s coefficient after 200 periods is approximately $\rho_n^{200}$. Using $(1+c/n)^n\to e^c$,

(0.99)^{200}=\left(1-\frac{2}{200}\right)^{200}\approx 0.1340,\qquad e^{-2}\approx0.1353.

Thus, in this illustrative model, the coefficient on a one-time shock is still about $0.1340$ after 200 periods; $0.1353$ is the limiting approximation $e^{-2}$. The approximate half-life is obtained by solving $e^{-2h/200}=1/2$, giving $h=100\log 2\approx 69.31$ periods. These approximations describe persistence under the specified model; they do not establish that a real series follows it.

The Continuous-Time Limit

Scale the observations by $\sigma\sqrt n$ and view them as a path over the unit interval: $X_n(r)=y_{\lfloor nr\rfloor}/(\sigma\sqrt n)$, for $0\le r\le1$. The recursion gives, at observation times,

X_n(t/n)-X_n((t-1)/n) =\frac{c}{n}X_n((t-1)/n)+\frac{u_t}{\sigma\sqrt n}.

The partial-sum process of the last terms converges to standard Brownian motion $W(r)$. The first term accumulates as the Riemann sum $c\int_0^r X_n(s)\,ds$. Consequently, the scaled process converges in distribution to the unique continuous process $J_c$ satisfying the integral equation below, or equivalently its stochastic differential equation.

J_c(r)=W(r)+c\int_0^r J_c(s)\,ds,\qquad dJ_c(r)=cJ_c(r)\,dr+dW(r),\qquad J_c(0)=0.

This is an Ornstein–Uhlenbeck process started at zero. When $c=0$, it is standard Brownian motion, the limit for a unit-root process. When $c<0$, the drift pulls the process toward zero, but on the local scale it does not eliminate the stochastic trend over the sample interval.

Limit result: Under the model and assumptions above, $X_n(r)$ converges in distribution to $J_c(r)$ on $0\le r\le1$. In particular, the scaled sum of lagged squares satisfies $$ \frac{1}{\sigma^2 n^2}\sum_{t=1}^n y_{t-1}^2 \ \Rightarrow\ \int_0^1 J_c(r)^2\,dr. $$

The scaling in this result follows from the path limit: each $y_{t-1}$ is of order $\sqrt n$, there are $n$ squared terms, and hence their sum is of order $n^2$. The integral is random because the limiting path is random.

OLS Estimation Has a Nonstandard Limit

Estimate the autoregressive coefficient by OLS through the origin, $\hat{\rho}=\sum_{t=1}^n y_{t-1}y_t/\sum_{t=1}^n y_{t-1}^2$. Subtracting the true coefficient from this expression yields an exact identity:

\hat{\rho}-\rho_n =\frac{\sum_{t=1}^n y_{t-1}u_t}{\sum_{t=1}^n y_{t-1}^2}.

The denominator is of order $n^2$. Because the lagged observations are themselves accumulated innovations, the numerator is of order $n$, not the order $\sqrt n$ that would arise from summing independent terms with fixed regressors. Applying the process limit jointly with the innovation partial-sum limit gives the following result:

n(\hat{\rho}-\rho_n)\ \Rightarrow\ \frac{\int_0^1 J_c(r)\,dW(r)} {\int_0^1 J_c(r)^2\,dr}.

Since $n(\rho_n-1)=c$, the corresponding limit relative to the unit-root value is

n(\hat{\rho}-1)\ \Rightarrow\ c+\frac{\int_0^1 J_c(r)\,dW(r)} {\int_0^1 J_c(r)^2\,dr} = \frac{\int_0^1 J_c(r)\,dJ_c(r)} {\int_0^1 J_c(r)^2\,dr}.

The last equality uses $dJ_c=cJ_c\,dr+dW$, so $\int J_c\,dJ_c=c\int J_c^2\,dr+\int J_c\,dW$. These integrals are stochastic integrals in the Itô sense where appropriate. The limit is a ratio of random functionals, not a normal distribution. In particular, the estimator’s error is of order $1/n$, and its scaled distribution remains nondegenerate as the sample grows.

Worked Example: Calculating the Local OLS Limit

For a purely algebraic calculation, take $c=-1$, $A=0.40$, and $B=-0.20$. Calculate the limit expression for $n(\hat{\rho}-1)$.

Substituting into the estimator limit gives

c+\frac{B}{A} =-1+\frac{-0.20}{0.40} =-1.50.

Equivalently, the numerator of the integral ratio is $\int J_c\,dJ_c=B+cA=-0.20-0.40=-0.60$, and $(-0.60)/0.40=-1.50$. This is a hypothetical algebraic evaluation, not a claim that these inputs are jointly attainable as functionals of $J_c$.

Why the Unit-Root t-Statistic Has Drift in Its Limit

Consider the usual OLS t-statistic for testing $H_0:\rho=1$, with no deterministic terms. If the residual variance estimate is consistent for $\sigma^2$, the standard error of $\hat{\rho}$ is approximately $\sigma/\sqrt{\sum y_{t-1}^2}$. Combining this with the OLS limit gives

t_{\rho=1}\ \Rightarrow\ T_c= \frac{\int_0^1J_c(r)\,dW(r)+c\int_0^1J_c(r)^2\,dr} {\left(\int_0^1J_c(r)^2\,dr\right)^{1/2}} = \frac{\int_0^1J_c(r)\,dJ_c(r)} {\left(\int_0^1J_c(r)^2\,dr\right)^{1/2}}.

The local-drift term $c\int J_c^2$ must be included: under the model, $\Delta y_t=(c/n)y_{t-1}+u_t$, so the regression numerator contains both its systematic local component and its innovation component. Omitting the drift would give an incorrect limit when $c\ne0$. Under the null $c=0$, the drift term vanishes and $J_0=W$; only then is the numerator $\int W\,dW$.

Worked Example: Evaluating the Limiting t-Statistic

Use the same hypothetical algebraic inputs $c=-1$, $A=0.40$, and $B=-0.20$ from the previous calculation. Evaluate $T_c$.

The full numerator is $B+cA=-0.20+(-1)(0.40)=-0.60$, while the denominator is $\sqrt{A}=\sqrt{0.40}\approx0.6325$. Therefore,

T_c=\frac{-0.60}{0.6325}\approx -0.9487.

The negative sign is consistent with the negative local drift in this illustration. This hypothetical algebraic value is not a realizable evaluation of the limiting statistic or a normal t-statistic with a standard normal reference distribution.

Persistence, Bias, and Test Power

Local-to-unity asymptotics explain why a near-unit-root series can look highly persistent even when its coefficient is below one. For a lag horizon proportional to the sample size, $h=\lfloor nr\rfloor$, the direct effect of a shock is governed by

\rho_n^h=\left(1+\frac{c}{n}\right)^{\lfloor nr\rfloor}\longrightarrow e^{cr}.

If $c<0$, this effect decays, but over a fixed fraction of the sample it decays at a rate represented by $e^{cr}$, rather than disappearing instantly. If $c>0$, it grows over that horizon. Either way, the process has behavior close to the unit-root case when $c$ is near zero.

Near-unit-root estimates also have appreciable finite-sample bias. The limit for $n(\hat{\rho}-\rho_n)$ is not centered at zero in general, and its distribution depends on $c$ and the initial conditions. In the unit-root case, OLS estimates are often biased downward in finite samples. That common pattern is not a universal exact bias formula: the magnitude and even the direction can depend on the model and estimation setup. The practical point is that ordinary approximations designed for a fixed stationary coefficient can be misleading when the coefficient is this close to one.

The same logic explains the power of unit-root tests. Under a fixed stationary alternative, the coefficient stays a positive distance below one as $n$ grows, and a suitable test can increasingly distinguish it from the unit-root null. Under a local alternative $\rho_n=1+c/n$, the test statistic instead has a limiting distribution indexed by $c$. If $q$ is a lower-tail critical value chosen for the unit-root null, the limiting rejection probability at local parameter $c<0$ is

$$ P(T_c<q). $$

This probability can exceed the test’s null rejection probability, but it generally does not have to approach one as the sample grows. The local alternative remains close enough to the null that power is nontrivial rather than automatically perfect. Exact critical values and test specifications depend on the deterministic terms included; those choices must match the fitted regression.

Common pitfall: A common mistake is to treat a coefficient such as $1+c/n$ as a fixed stationary alternative and apply the usual fixed-parameter normal approximation for OLS. But the coefficient changes with $n$, the lagged observations grow on the $\sqrt n$ scale, and $n(\hat{\rho}-1)$ has a random-functional limit. For the t-statistic, it is also incorrect to omit the local drift from the numerator: when $c\ne0$, the limit includes $c\int_0^1J_c(r)^2\,dr$.

A Short Itô-Calculus Check

The t-statistic expression can be checked using Itô’s formula. Since $dJ_c=cJ_c\,dr+dW$ and the quadratic variation of $J_c$ over $[0,1]$ is one, Itô’s formula for $J_c^2$ gives

d\{J_c(r)^2\}=2J_c(r)\,dJ_c(r)+dr.

Integrating from zero to one and using $J_c(0)=0$ yields

\int_0^1J_c(r)\,dJ_c(r) =\frac{J_c(1)^2-1}{2} =\int_0^1J_c(r)\,dW(r)+c\int_0^1J_c(r)^2\,dr.

This identity verifies that the local drift is part of the numerator even though the expression involving $J_c(1)^2$ has a compact form. Under $c=0$, it reduces to the familiar Brownian-motion identity $\int_0^1W\,dW=(W(1)^2-1)/2$.

Check Your Understanding

Use the definitions and limits above to test your understanding.

For $n=400$ and $\rho_n=0.995$, calculate $c$. What is the approximate effect of a shock after 400 periods?
Explain why the denominator $\sum_{t=1}^n y_{t-1}^2$ is of order $n^2$ under the local-to-unity model.
Write the limiting expression for $n(\hat{\rho}-1)$, and identify where the term $c$ enters.
For $c\ne0$, what component would be missing if the unit-root t-statistic limit were written with numerator $\int J_c\,dW$ alone?
Why can a unit-root test have nontrivial rather than perfect asymptotic power against a local alternative?