Unit root

A unit root is a feature of processes that evolve through time that can cause problems in statistical inference involving time series models.

A linear stochastic process has a unit root if 1 is a root of the process's characteristic equation. Such a process is non-stationary. If the other roots of the characteristic equation lie inside the unit circle—that is, have a modulus (absolute value) less than one—then the first difference of the process will be stationary.

Definition

Consider a discrete-time stochastic process $\{y_t,t=1,\ldots,\infty\}$ , and suppose that it can be written as an autoregressive process of order p:

y_t=a_1 y_{t-1}+a_2 y_{t-2} + \cdots + a_p y_{t-p}+\varepsilon_t.

Here, $\{\varepsilon_{t},t=0,\infty\}$ is a serially uncorrelated, zero-mean stochastic process with constant variance $\sigma^2$ . For convenience, assume $y_0 = 0$ . If $m=1$ is a root of the characteristic equation:

m^p - m^{p-1}a_1 - m^{p-2}a_2 - \cdots - a_p = 0

then the stochastic process has a unit root or, alternatively, is integrated of order one, denoted $I(1)$ . If m = 1 is a root of multiplicity r, then the stochastic process is integrated of order r, denoted I(r).

Example

The first order autoregressive model, $y_t=a_{1}y_{t-1}+\varepsilon_t$ , has a unit root when $a_1=1$ . In this example, the characteristic equation is $m - a_1 = 0$ . The root of the equation is $m = 1$ .

If the process has a unit root, then it is a non-stationary time series. That is, the moments of the stochastic process depend on $t$ . To illustrate the effect of a unit root, we can consider the first order case, starting from y₀ = 0:

y_{t}= y_{t-1}+\varepsilon_t.

By repeated substitution, we can write $y_t = y_0 + \sum_{j=1}^t \varepsilon_j$ . Then the variance of $y_t$ is given by:

\operatorname{Var}(y_t) = \sum_{j=1}^t \sigma^2=t \sigma^2 .

The variance depends on t since $\operatorname{Var}(y_{1}) = \sigma^2$ , while $\operatorname{Var}(y_{2}) = 2\sigma^2$ . Note that the variance of the series is diverging to infinity with t.

There are various tests to check stationarity of unit root, some of them are given by:

The ¶ statistics or the Dickey-Fuller test
Testing the significance of more than one coefficients (f-test)
The Phillips Peron Test (PP) unit root test
Dickey Pantula test

Related models

In addition to AR and ARMA models, other important models arise in regression analysis where the model errors may themselves have a time series structure and thus may need to be modelled by an AR or ARMA process that may have a unit root, as discussed above. The finite sample properties of regression models with first order ARMA errors, including unit roots, have been analyzed.^[1]^[2]

Estimation when a unit root may be present

Often, ordinary least squares (OLS) is used to estimate the slope coefficients of the autoregressive model. Use of OLS relies on the stochastic process being stationary. When the stochastic process is non-stationary, the use of OLS can produce invalid estimates. Granger and Newbold called such estimates 'spurious regression' results:^[3] high R² values and high t-ratios yielding results with no economic meaning.

To estimate the slope coefficients, one should first conduct a unit root test, whose null hypothesis is that a unit root is present. If that hypothesis is rejected, one can use OLS. However, if the presence of a unit root is not rejected, then one should apply the difference operator to the series. If another unit root test shows the differenced time series to be stationary, OLS can then be applied to this series to estimate the slope coefficients.

For example, in the AR(1) case, $\Delta y_{t} = y_{t} - y_{t-1} = \varepsilon_{t}$ is stationary.

In the AR(2) case, $y_{t} = a_{1}y_{t-1} + a_{2}y_{t-2} + \varepsilon_{t}$ can be written as $(1 -\lambda_{1}L)(1 - \lambda_{2}L)y_{t} = \varepsilon_{t}$ where L is a lag operator that decreases the time index of a variable by one period: $Ly_{t} = y_{t-1}$ . If $\lambda_{2} = 1$ , the model has a unit root and we can define $z_{t} = \Delta y_{t}$ ; then

z_{t} = \lambda_{1}z_{t-1} + \varepsilon_{t}

is stationary if $|\lambda_1| < 1$ . OLS can be used to estimate the slope coefficient, $\lambda_{1}$ .

If the process has multiple unit roots, the difference operator can be applied multiple times.

Properties and characteristics of unit-root processes

Shocks to a unit root process have permanent effects which do not decay as they would if the process were stationary
As noted above, a unit root process has a variance that depends on t, and diverges to infinity
If it is known that a series has a unit root, the series can be differenced to render it stationary. For example, if a series $Y_t$ is I(1), the series $\Delta Y_t=Y_t-Y_{t-1}$ is I(0) (stationary). It is hence called a difference stationary series.

Unit root hypothesis

The diagram above depicts an example of a potential unit root. The red line represents an observed drop in output. Green shows the path of recovery if the series has a unit root. Blue shows the recovery if there is no unit root and the series is trend-stationary. The blue line returns to meet and follow the dashed trend line while the green line remains permanently below the trend. The unit root hypothesis also holds that a spike in output will lead to levels of output higher than the past trend.

Economists debate whether various economic statistics, especially output, have a unit root or are trend stationary.^[4]^[5]^[6]^[7] A unit root process with drift is given in the first-order case by

y_t = y_{t-1} + c + e_t

where c is a constant term referred to as the "drift" term, and $e_t$ is white noise. Any non-zero value of the noise term, occurring for only one period, will permanently affect the value of $y_t$ as shown in the graph, so deviations from the line $y_t = a + ct$ are non-stationary; there is no reversion to any trend line. In contrast, a trend stationary process is given by

y_t = k \cdot t + u_t

where k is the slope of the trend and $u_t$ is noise (white noise in the simplest case; more generally, noise following its own stationary autoregressive process). Here any transient noise will not alter the long-run tendency for $y_t$ to be on the trend line, as also shown in the graph. This process is said to be trend stationary because deviations from the trend line are stationary.

The issue is particularly popular in the literature on business cycles.^[8]^[9] Research on the subject began with Nelson and Plosser whose paper on GNP and other output aggregates failed to reject the unit root hypothesis for these series.^[10] Since then, a debate—entwined with technical disputes on statistical methods—has ensued. Some economists^[11] argue that GDP has a unit root or structural break, implying that economic downturns result in permanently lower GDP levels in the long run. Other economists argue that GDP is trend-stationary: That is, when GDP dips below trend during a downturn it later returns to the level implied by the trend so that there is no permanent decrease in output. While the literature on the unit root hypothesis may consist of arcane debate on statistical methods, the hypothesis carries significant practical implications for economic forecasts and policies.

Notes

↑ Sargan, J. D.; Bhargava, Alok (1983). "Testing residuals from least squares regressions for being generated by the Gaussian random walk". Econometrica 51 (1): 153–174. doi:10.2307/1912252. JSTOR 1912252.
↑ Sargan, J. D.; Bhargava, Alok (1983). "Maximum Likelihood Estimation of Regression Models with First Order Moving Average Errors when the Root Lies on the Unit Circle". Econometrica 51 (3): 799–820. doi:10.2307/1912159. JSTOR 1912159.
↑ Granger, C. W. J.; Newbold, P. (1974). "Spurious regressions in econometrics". Journal of Econometrics 2 (2): 111–120. doi:10.1016/0304-4076(74)90034-7.
↑ "Trend Stationarity/Difference Stationarity over the (Very) Long Run". Econbrowser. March 13, 2009.
↑ Krugman, Paul (March 3, 2009). "Roots of evil (wonkish)". The New York Times.
↑ "Greg Mankiw Gets Technical". Library of Economics and Liberty. March 3, 2009. Retrieved 2012-06-23.
↑ Verdon, Steve (March 11, 2009). "Economic Cage Match: Mankiw vs. Krugman". Outside the Beltway.
↑ Hegwood, Natalie; Papell, David H. (2007). "Are Real GDP Levels Trend, Difference, or Regime-Wise Trend Stationary? Evidence from Panel Data Tests Incorporating Structural Change". Southern Economic Journal 74 (1): 104–113. JSTOR 20111955.
↑ Lucke, Bernd (2005). "Is Germany‘s GDP trend-stationary? A measurement-with-theory approach" (PDF). Jahrbücher für Nationalökonomie und Statistik 225 (1): 60–76.
↑ Nelson, Charles R.; Plosser, Charles I. (1982). "Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implications". Journal of Monetary Economics 10 (2): 139–162. doi:10.1016/0304-3932(82)90012-5.
↑ Olivier Blanchard with the International Monetary Fund makes the claim that after a banking crisis "on average, output does not go back to its old trend path, but remains permanently below it."

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