Lecture 16 Unit Root Tests

1 RS EC2 - Lecture 1611 Lecture 16 Unit Root Tests A shock is usually used to describe an unexpected change in a variable or in the value of the error terms at a particular time period. When we have a stationary system, effect of a shock will die out gradually. But, when we have a non-stationary system, effect of a shock is permanent. We have two types of non-stationarity. In an AR(1) model we have:- Unit root: | 1| = 1: homogeneous non-stationarity-Explosive root: | 1| > 1: explosive non-stationarity In the last case, a shock to the system become more influential as time goes on. It can never be seen in real life. We will not consider Unit RootRS EC2 - Lecture 162 Consider the AR(p) process:As we discussed before, if one of the rj s equals 1, (1)=0, or We say ythas a unit root.

2 In this case, ytis non-stationary. Example: AR(1): Unit root: 1= (ytnon-stationarity): 1= 1 (or, 1 1 = 0)H1(ytstationarity): 1< 1(or, 1 1 < 0) A t-test seems natural to test H0. But, the ergodic theorem and MDS CLT do not apply: the t-statistic does not have the usual Unit RootppttLLLLyL ..1)( where )( p 11tttyy Now, let s reparameterize the AR(1) process. Subtract yt-1from yt: Unit root test:H0: 0 = 1 1 = 0 H1: 0 < 0. Natural test for H0: t-test. We call this test the Dickey-Fuller(DF) test. But, what is its distribution? Back to the general, AR(p) process:We rewrite the process using the Dickey-Fuller reparameterization:: Both AR(p) formulations are equivalent.

3 Autoregressive Unit Root ..)1(1221110tptpttttyyyyy tttttttyyyyy 10111 )1(ttyL )(RS EC2 - Lecture 163 AR(p) lag (L): DF reparameterization: Both parameterizations should be equal. Then, (1)=- 0. unit root hypothesis can be stated as H0: 0= : The model is stationary if 0< 0 natural H1: 0 < 0. Under H0: 0=0, the model is AR(p-1) stationary in yt. Then, if ythas a (single) unit root, then yt is a stationary AR process. We have a linear regression framework. A t-test for H0is the Augmented Dickey-Fuller(ADF) test. )(..)()()1(11322210pppLLLLLLL Autoregressive Unit Root TestingppLLLL ..1)(2211 The Dickey-Fuller(DF) test is a special case of the ADF: No lags are included in the regression.

4 It is easier to derive. We gain intuition from its derivation. From our previous example, we have: If 0= 0, system has a unit root:: We can test H0with a t-test: There is another associated test with H0, the -test:. (T-1)( 1).Autoregressive Unit Root Testing: DF tttttyyy 1011)0|(|0:0:00100 HH 1 1 SEtRS EC2 - Lecture 164 Kolmogorov Continuity Theorem If for all T> 0, there exist a, b, > 0 such that: E(|X(t1, ) X(t2, )|a) |t1 t2|(1 + b) Then X(t, ) can be considered as a continuous stochastic process. Brownian motion is a continuous stochastic process. Brownian motion (Wienerprocess): X(t, ) is almost surely continuous, has independent normal distributed (N(0,t-s)) increments and X(t=0, ) =0 ( a continuous random walk ).

5 Review: Stochastic Calculus8 Let the variable z(t)be almost surely continuous, with z(t=0)=0. Define ( ,v)as a normal distribution with mean and variance v. The change in a small interval of time tis z Definition: The variable z(t)follows a Wiener process if z(0) = 0 z= t,where N(0,1) It has continuous paths. The values of zfor any 2 different (non-overlapping) periods of time are : W(t), W(t, ), B(t).Example: Review: Stochastic Calculus Wiener process]1,0[);..(1)(][321 rTrWTrTRS EC2 - Lecture 1659 What is the distribution of the change in z over the next 2 time units?The change over the next 2 units equals the sum of:- The change over the next 1 unit (distributed as N(0,1)) plus- The change over the following time unit --also distributed as N(0,1).

6 - The changes are The sum of 2 normal distributions is also normally , the change over 2 time units is distributed as N(0,2). Properties of Wiener processes: Mean of zis 0 Variance of zis t Standard deviation of zis t Let N=T/ t, then Review: Stochastic Process: Wiener process niitzTz1)0()( 10 Example: If Tis large, WT(.) is a good approximation to W(r); r [0,1], defined:W(r) = limT WT(r) E[W(r) ] =0 Var[W(r) ] =t Check Billingsley (1986) for the details behind the proof that WT(r) converges as a function to a continuous function W(r). In a nutshell, we need- tsatisfying some assumptions (stationarity, E[| t|q < for q>2, etc.)-a FCLT (Functional CLT).]

7 - a Continuous Mapping Theorem. (Similar to Slutzky s theorem). Review: Stochastic Calculus Wiener process]1,0[;1)..(1)(][][321 rSTTrWTrTrTRS EC2 - Lecture 16611 Functional CLT(Donsker s FCLT)If tsatisfies some assumptions, then WT(r) W(r),where W(r) is a standard Brownian motion for r [0, 1].Note: That is, sample statistics , like WT(r),do not converge to constants, but to functions of Brownian motions. A CLT is a limit for one term of a sequence of partial sums {Sk}, Donsker s FCLT is a limit for the entire sequence {Sk} instead of one : Stochastic Calculus Wiener process D12 Example: yt= yt-1+ t(Case 1). Get distribution of (X X/T2)-1for : Stochastic Calculus Wiener process 1022201102/1010222011/)1(][2/101/)1(2][2 2011112/101121212010212120121211021212.

8 ,)()(2)(1212]2)[(][][)(TdrrWyTdrrXTydrrX yTdrSTTydrSTyTTTSTyTTSySySTySTyTyTdTTTtT tTtTrTtTtTtTrTttTttTtttTttTttiitTttRS EC2 - Lecture 16713 The integral a Brownian motion, given by Ito s theorem (integral): f(t, ) dB = f(tk, ) Bkwhere tk* [tk,tk + 1) as tk + 1 tk we increase the partitions of [0,T], the sum pto the integral. But, this is a probability statement: We can find a sample path where the sum can be arbitrarily far from the integral for arbitrarily large partitions (small intervals of integration). You may recall that for a Rienman integral, the choice of tk*(at the start or at the end of the partition) is not important. But, for Ito s integral, it is important (at the start of the partition).]

9 Ito s Theorem result: B(t, ) dB(t, ) = B2(t, )/2 : Stochastic Calculus Ito s Theorem We continue with yt= yt-1+ t(Case 1). Using OLS, we estimate : This implies: From the way we defined WT(.), we can see that yt/sqrt(T) converges to a Brownian motion. Under H0, ytis a sum of white noise Unit Root Testing: Intuition TtTtttTtTttttTtTttttttyyyyyyyyyy12111121 11112111111)( TttTtttTtTtttTyTTTyyyyTTt1211112111)/(1) /)(/()1 (1RS EC2 - Lecture 168 Intuition for distribution under H0: - Think of ytas a sum of white noise errors. - Think of tas dW(t). Then, using Billingley (1986), we guess that T(-1) converges to We think of tas dW(t). Then, k=0 to t k, which corresponds to 0to(t/T)dW(s)=W( s/T) (for W(0)=0).

10 Using Ito s integral, we havedttWtdWtWTd 10210)()()()1 ( dttWWTd 1022)(1)1(21)1 ( Autoregressive Unit Root Testing: Intuition Note: W(1) is a N(0,1). Then, W(1)2is just a 2(1) RV. Contrary to the stable model the denominator of the expressionfor the OLS estimator , (1/T) txt2--does not converge to a constant , but to a RV strongly correlated with the numerator. Then, the asymptotic distribution is not normal. It turns out that the limiting distribution of the OLS estimator is highly skewed, with a long tail to the 1022)(1)1(21)1 (Autoregressive Unit Root Testing: IntuitionRS EC2 - Lecture 169 DF distribution relative to a Normal. It is skewed, with a long tail to the Unit Root Testing: Intuition Back to the AR(1) model.