TimeSeriesBook.pdf

Acf is only of limited use to discriminate between an

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ACF is only of limited use to discriminate between an (stationary) ARMA process and a random walk. The above calculation also shows that ρ (1) < 1 so that the expected value of the OLS estimator is downward biased in finite samples: E b φ T < 1. 7.3 Unit-Root Tests The previous Sections 7.1 and 7.2 have shown that, depending on the nature of the non-stationarity (trend versus difference stationarity), the stochastic process has quite different algebraic (forecast, forecast error variance, persis- tence) and statistical (asymptotic distribution of OLS-estimator) properties. It is therefore important to be able to discriminate among these two differ- ent types of processes. This also pertains to standard regression models for which the presence of integrated variables can lead to non-normal asymptotic distributions. The ability to differentiate between trend- and difference-stationary pro-
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154 CHAPTER 7. INTEGRATED PROCESSES 0 5 10 15 20 25 -1 -0.5 0 0.5 1 order theoretical ACF of a random walk estimated ACF of a random walk estimated ACF of an AR(1) process with φ = 0.9 generated with the same innovations as the random walk Figure 7.3: ACF of a random walk with 100 observations cesses is not only important from a statistical point of view, but can be given an economic interpretation. In macroeconomic theory, monetary and demand disturbances are alleged to have only temporary effects whereas supply dis- turbances, in particular technology shocks, are supposed to have permanent effects. To put it in the language of time series analysis: monetary and de- mand shocks have a persistence of zero whereas supply shocks have nonzero (positive) persistence. Nelson and Plosser (1982) were the first to investi- gate the trend properties of economic time series from this angle. In their influential study they reached the conclusion that, with the important ex- ception of the unemployment rate, most economic time series in the US are better characterized as being difference stationary. Although this conclusion came under severe scrutiny (see Cochrane (1988) and Campbell and Perron (1991)), this issue resurfaces in many economic debates. The latest discus- sion relates to the nature and effecct of technology shocks (see Gal´ ı (1999) or Christiano et al. (2003)). The following exposition focuses on the Dickey-Fuller test (DF-test) and the Phillips-Perron test(PP-test). Although other test procedures and vari- ants thereof have been developed in the meantime, these two remain the most widely applied in practice. These types of tests are also called unit-root tests. Both the DF- as well as the PP-test rely on a regression of X t on X t - 1 which may include further deterministic regressors like a constant or a linear
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7.3. UNIT-ROOT TESTS 155 time trend. We call this regression the Dickey-Fuller regression : X t = deterministic variables + φX t - 1 + Z t . (7.1) Alternatively and numerically equivalent, one may run the Dickey-Fuller re- gression in difference form: X t = deterministic variables + βX t - 1 + Z t with β = φ - 1. For both tests, the null hypothesis is that the process is integrated of order one, difference stationary, or has a unit-root.
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