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44 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSconverges asymptotically to a random variable with a well-behaved distributionand we say that ˆρ converges at rate T whereas in the stationarycase we say that convergence takes place at rate √ T . The distributionfor T (ˆρ − ρ) is not normal, however, nor does it have a closedform so that its computation must be done by computer simulation.Similarly, the studentized coefficient or the ‘t-statistic’ for ˆρ reportedby regression packages τ = T ˆρ( P Tt=1 q 2 t )/( P Tt=1 ² 2 t ), also behaves has awell-behaved but non-normal asymptotic distribution. 15Test ProceduresThe discussion above did not include a constant, but in practice one isalmost always required and sometimes it is a good idea also to includea time trend. Bhargava’s [12] framework is useful for thinking aboutincluding constants and trends in the analysis. Let ξ t be the deviationof q t from a linear trendq t = γ 0 + γ 1 t + ξ t . (2.50)If γ 1 6= 0, the question is whether the deviation from the trend is stationaryorifitisadriftlessunitrootprocess.Ifγ 1 =0andγ 0 6=0,the question is whether the deviation of q t from a constant is stationary.Let’s ask the Þrst question–whether the deviation from trend isstationary. Letξ t = ρξ t−1 + ² t , (2.51)iidwhere 0 < ρ ≤ 1and² t ∼ N(0, σ² 2 ). You want to test the null hypothesisH o : ρ = 1 against the alternative H a : ρ < 1. Under the null hypothesis∆q t = γ 1 + ² t ,and q t is a random walk with drift γ 1 . Add the increments to getq t =tXj=1∆q j = γ 1 t +(² 0 + ² 1 + ···+ ² t )=γ 0 + γ 1 t + ξ t , (2.52)(23)⇒where γ 0 = ² 0 and ξ t =(² 1 +² 2 +···+² t ). You can initialize by assuming15 In fact, these distributions look like chi-square distributions so the least squaresestimator is biased downward under the null that ρ = 1.