This chapter’s objectives are to: Formalize simple models of variables with a time-dependent mean, compare models with deterministic versus stochastic trends, show that the so-called unit root problem arises in standard regression and in timesseries models,. | Chapter 4 Applied Econometric Time Series 4th ed. Walter Enders 1 2 The Random Walk Model yt = yt–1 + et (or Dyt = et). Given the first t realizations of the {et} process, the conditional mean of yt+1 is Etyt+1 = Et(yt + et+1) = yt Similarly, the conditional mean of yt+s (for any s > 0) can be obtained from Hence var(yt) = var(et + et–1 + . + e1) = ts2 var(yt–s) = var(et–s + et–s–1 + . + e1) = (t – s)s 3 Random Walk Plus Drift yt = yt–1 + a0 + et Given the initial condition y0, the general solution for yt is Etyt+s = yt + a0s. E[(yt – y0)(yt–s – y0)] = E[(et + et–1+.+ e1)(et–s+ et–s–1 +.+e1)] = E[(et–s)2+(et–s–1)2+.+(e1)2] = (t – s)s2 The autocorrelation coefficient = [(t – s)/t] Hence, in using sample data, the autocorrelation function for a random walk process will show a slight tendency to decay. 5 6 7 Table : Selected Autocorrelations From Nelson and Plosser 8 Worksheet Consider the two random walk plus drift processes yt = + yt 1 + yt zt = + zt 1 + zt Here {yt} and {zt} series are unit-root processes with uncorrelated error terms so that the regression is spurious. Although it is the deterministic drift terms that cause the sustained increase in yt and the overall decline in zt, it appears that the two series are inversely related to each other. The residuals from the regression yt = are nonstationary. Scatter Plot of yt Against zt Regression Residuals Worksheet 3. UNIT ROOTS AND REGRESSION RESIDUALS yt = a0 + a1zt + et Assumptions of the classical model: both the {yt} and {zt} sequences be stationary the errors have a zero mean and a finite variance. In the presence of nonstationary variables, there might be what Granger and Newbold (1974) call a spurious regression A spurious regression has a high R2 and t-statistics that appear to be significant, but the results are without any economic meaning. The regression output “looks good” because the least-squares estimates are not consistent and the customary tests | Chapter 4 Applied Econometric Time Series 4th ed. Walter Enders 1 2 The Random Walk Model yt = yt–1 + et (or Dyt = et). Given the first t realizations of the {et} process, the conditional mean of yt+1 is Etyt+1 = Et(yt + et+1) = yt Similarly, the conditional mean of yt+s (for any s > 0) can be obtained from Hence var(yt) = var(et + et–1 + . + e1) = ts2 var(yt–s) = var(et–s + et–s–1 + . + e1) = (t – s)s 3 Random Walk Plus Drift yt = yt–1 + a0 + et Given the initial condition y0, the general solution for yt is Etyt+s = yt + a0s. E[(yt – y0)(yt–s – y0)] = E[(et + et–1+.+ e1)(et–s+ et–s–1 +.+e1)] = E[(et–s)2+(et–s–1)2+.+(e1)2] = (t – s)s2 The autocorrelation coefficient = [(t – s)/t] Hence, in using sample data, the autocorrelation function for a random walk process will show a slight tendency to decay. 5 6 7 Table : Selected Autocorrelations From Nelson and Plosser 8 Worksheet Consider the two random walk plus drift processes yt = + yt 1 + yt zt = + zt 1 + .
