6 Unit root tests for stationarity. Tests for Stationarity: Dickey-Fuller tests. U/G Econometrics

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1 6 Unit root tests for stationarity U/G Econometrics Because non-stationary processes have such different properties from stationary ones, it is important to be able to distinguish between them. U/G E 16.3 discusses the appearance of the correlogram (the plot of autocorrelations) for both the stationaryar(1)andtherandomwalk. Notes6alsodiscusses the correlogram for a sample from an AR(1). Basically the autocorrelations die away for the stationaryprocessbutstaycloseto1fortherandomwalk. The value of DW calculated for the series y t (rather than for residuals from a regression) (yt y DW = t 1 ) 2. y 2 t also contains information about possible random walking arandomwalkproducesavaluecloseto0. Tests for Stationarity: Dickey-Fuller tests The standard tests are called Dickey-Fuller tests and first became available in They are called unit root tests for a reason explained later. See EViews manualch17sectiononunitroottests. In EViews the tests live in the Quick/Series Statistics/Unit root tests. Their appearance there reflects the view that people using time series will want to investigate the stationarity or otherwise of the series before doing any complicated statistical analysis with them. EViews presents many options Many variations of model and hypotheses are conceivable, so there are many different test statistics.

2 WebeginwiththesimplestsituationofanAR(1) and the question is this a random walk (ρ=1) orisitstationary(ρ<1)? Consider the AR(1) and the hypotheses y t = ρy t 1 +ε t :ε t IN(0,σ 2 ) H 0 : ρ=1:h A :ρ<1. The model can also be written in (first) differences. So, subtractingy t 1 fromeachside of theequation, writing y t =y t y t 1 and introducing α=ρ 1 weget y t = αy t 1 +ε t :ε t IN(0,σ 2 ) H 0 : α=0:h 1 :α<0 where we are just restating the original hypotheses aboutρintermsofα. The natural way of testing whether α = 0 is to regress y t ony t 1 anddoat-test. Thatisform the ratio α est.(s.e of α) where αistheleastsquaresestimateofαandrejectthenullforlargenegativevalues. Ifthenull were α = 0.1 (i.e. ρ = 0.9) and the process were stationary then the t-distribution associated withthenullhypothesiswouldben(0,1)inlarge samples. This fits in which the ordinary regression case where the t-distribution tends to the standard normal when the number of degrees of freedom tends to infinity. Howeverwhenρ 1the t-statistic isnotn(0,1) in large samples. Dickey&Fullerworkedoutthedistributionofthe t-statistic forρ=1(α=0)andtabulatedit.

3 The basic Dickey-Fuller test exists in variants with y t = γ+αy t 1 +ε t y t = γ+αy t 1 +βt+ε t Inthefirstcaseα=0 r. w. withdriftandinthe secondγ=0 r. w. withdrift&timetrend. There is also an Augmented Dickey-Fuller(ADF) test inwhichadditionaltermsin y t i areincludedinthe regression to cover the possibility that the process is ahigherorderar: y t =γ+αy t 1 +β 1 y t β p y t p +ε t Wetestα=0asbefore. Inthisgeneralisedrandom walk y t isexpressedintermsofpast y s. Inpracticewehavenogoodideaabouttheappropriatevalueofp,theorderoftheAR.Tohelpchooseit EViews presents a bundle of model selection criteria, Akaike and others. See U/G E for practical examples of Dickey- Fuller testing. Integration and Differencing Byintroducingthedifference y t =y t y t 1 arandomwalkcanberewrittenas y t = y t 1 +ε t,:ε t IN(0,σ 2 ) y t = ε t, i.e. the(first) difference is white noise. Whitenoiseisastationaryprocessandthereisauseful generalisation of the random walk which requires that the first difference is stationary: y t =ε t, ε t isstationary. Such a y t is said to be integrated of order 1 and denoted I(1). If we have to difference twice to achieve stationarity then the series is called I(2). A stationary series is I(0). Integration is thought of as the opposite of differencing; we get rid of integration by differencing.

4 A useful trick when manipulating time series models is to use operators in particular the difference operator andthelagoperator. Wecandoordinaryalgebrawith these things. The difference operator produces first differences y t =y t y t 1. The difference of the difference is (y t y t 1 ) (y t 1 y t 2 )representedby 2 y t. The lag, or backward shift, operator L (sometimes written B) transforms a time t quantity into a time t 1quantity,thus Ly t =y t 1 It s possible to work algebraically with these operators. Here are some illustrative manipulations L 2 y t = L(Ly t )=Ly t 1 =y t 2. y t y t 1 = (1 L)y t : y t =(1 L)y t. 2 y t = (1 L) 2 y t =(1 2L+L 2 )y t = y t 2y t 1 +y t 2 which agrees with what we had earlier. Bewarethat 2 y t isnotthesameas(1 L 2 )y t = y t y t 2. TheDickey-Fullertestiscalledaunitroottest andis saidtotestwhethertheprocesscontainsaunitroot. The term arises because in the expression (1 ρl)y t =ε t (1 ρl) is a simple example of a lag polynomial. Another, more complicated, lag polynomial is (1 2L+L 2 ). Thequestion, is ρ=1?, is thesame as thequestion does the polynomial 1 ρz hasarootz=1,i.e. doesz=1satisfytheequation 1 ρz=0.

5 This unit root terminology is very natural when dealing with the the Augmented Dickey-Fuller test for higher order ARs. Consider the AR(p) y t =α 0 +α 1 y t α p y t p +ε t and its expression using the lag operator (1 α 1 L... α p L p )y t =α 0 +ε t. In this set-up the unit root question becomes the question,doesthepolynomial(1 α 1 L... α p L p )have a unit root? So can we find a polynomial ϕ(l) for which (1 α 1 L... α p L p )=(1 ϕ(l))(1 L). Ifso,wecanwrite (1 α 1 L... α p L p ) = (1 ϕ(l))(1 L) y t = α 0 +ϕ(l) y t +ε t IntheADFtestweincludeatermγy t 1 ontheright handsideandtestwhetherγ=0.