Univariate Unit Root Process (May 14, 2018)

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1 Ch. Univariate Unit Root Process (May 4, 8) Introduction Much conventional asymptotic theory for least-squares estimation (e.g. the standard proofs of consistency and asymptotic normality of OLS estimators) assumes stationarity of the explanatory variables, possibly around a deterministic trend. However, not all economic times series are stationary, as we saw in Chapter 9, and for many important ones, including aggregate consumption and national income, stationarity is not even a sensible approximation. This chapter describes methods of testing for a unit root in an observed series. Both parametric regression tests and non-parametric adjustments to theses test statistics are considered. To begin with, consider OLS estimation of a AR() process, Y t = ρy t + u t, where u t i.i.d.(, σ ), and Y =. The OLS estimator of ρ is given by ( T ) ( ) ˆρ T = Y t Y t T Y = Y t t Y t Y t and we also have ( (ˆρ T ρ) = Y t ) ( ) Y t u t. (-) When the true value of ρ is less than in absolute value, then Y t (so does Y t?) is a covariance-stationary process. Applying LLN for a covariance process (see 9.9 of Ch. 4) we have ( ) Y t /T p E [( Y t ) ] [ T σ /T = ρ ] /T = σ /( ρ ). (-)

2 Ch. INTRODUCTION and Since Y t u t is a martingale difference sequence with variance E(Y t u t ) = E(Y t )E(u t ) = σ σ ρ (since Y t and u t are independent) T [σ σ ρ ] σ σ ρ. Applying CLT for a martingale difference sequence to the second term in the righthand side of () we have ( ) ) L Y t u t N (, σ σ. (-3) T ρ Substituting () and (3) to () we have [( ) ] T (ˆρT ρ) = /T T [( ) ] Y t u t /T Y t [ ] L σ ) N (, σ σ ρ ρ N(, ρ ). (-4) However, Equation(4) is not valid for the case when ρ =. To see this, recall that the variance of Y t when ρ = is tσ, then the LLN as in () would not be valid since if we apply LLN, then it would incur that ( ) [( ) ] /T p E /T Y t Y t = σ T t T. (-5) Similar reason would show that the CLT would not apply for T ( T Y t u t ). (In stead, it is that T ( T Y t u t ) converges.) To obtain the limiting distribution, as we shall prove in the following, for (ˆρ T ρ) in the unit root case, it turn out that we have to multiply (ˆρ T ) by T rather than by T : T (ˆρ T ) = [( Y t ) /T ] [ T ( Y t u t )]. (-6) Thus, the unit root coefficient converge at a faster rate (T ) than a coefficient for stationary regression (which converges at T ). E(Y t u t F t ) = Y t E(u t F t ) = Y t =. Therefore, Y t u t is a martingale difference sequence. R 8 by Prof. Chingnun Lee Ins.of Economics,NSYSU,Taiwan

3 Ch. ASYMPTOTIC THEORIES FOR UNIT ROOT PROCESS Asymptotic Theories for Unit Root Process The fact that unit root process violate the restriction that certain moments of a tim series to be bounded, e.g. E Xt +δ < < for some δ > and for all t had made inapplicable of traditional asymptotic theory to the unit root process. In this section, we develop tools to handle the asymptotics of unit root process.. Random Walks and Wiener Process Consider a random walk, Y t = Y t + ε t, where Y = and ε t is i.i.d. with mean zero and V ar(ε t ) = σ <. By repeated substitution we have Y t = Y t + ε t = Y t + ε t + ε t. = Y + = t ε s. s= t s= ε s Before we can study the behavior of estimators based on random walks, we must understand in more detail the behavior of the random walk process itself. By the assumption of Y =, consider the random walk {Y t }, we can write Y T = ε t. Rescaling, we have T / Y T /σ = T / ε t /σ. (It is important to note here σ should be read as V ar(t / T ε t) = E[T ( ε t ) ] = = σ.) According to the Lindeberg-Lévy CLT, we have T σ T T / Y T /σ L N(, ). R 8 by Prof. Chingnun Lee 3 Ins.of Economics,NSYSU,Taiwan

4 Ch. ASYMPTOTIC THEORIES FOR UNIT ROOT PROCESS More generally, we can construct a variable Y T (r) from the partial sum of ε t [T r] Y T (r) = ε t, where r and [T r] denotes the largest integer that is less than or equal to T r. Applying the same rescaling, we define Now W T (r) T / Y T (r)/σ [T r] = T / ε t /σ. (-7) W T (r) = T / ([T r]) / ([T r]) / [T r] ε t /σ, and for a given r, the term in the brackets { } again obeys the CLT and converges in distribution to N(, ), whereas T / ([T r]) / converges to r /. It follows from standard arguments that W T (r) converges in distribution to N(, r). Suppose that we first choose r t so that for some integer t, T r t = t (i.e., r t = t/t ). Then we will consider what happens as r increase to the value (t + )/T. With r t = t/t we have [T r t ] = t, so t W T (r) = T / ε s /σ, r = t/t. s= For t/t < r < (t + )/T, we still have [T r] = t, so t W T (r) = T / ε s /σ, t/t < r < (t + )/T. s= That is, W T (r) is constant for t/t r < (t + )/T. When r hits (t + )/T, we see that W T (r) jumps to t+ W T (r) = T / ε s /σ, r = (t + )/T. s= Thus, W T (r) is a piecewise constant function that jumps to a new values whenever r = t/t for interger t. In this way we are able to concentrate the original horizontal axis of to T to the close interval [, ], indexing the observations by r. If, for example, T =, the original observation Y 5 will be indexed by r [.5,.5), and so on. R 8 by Prof. Chingnun Lee 4 Ins.of Economics,NSYSU,Taiwan

5 Ch. ASYMPTOTIC THEORIES FOR UNIT ROOT PROCESS We have written W T (r) so that it is clear that W T can be considered to be a function of r. Also, because W T (r) depends on the ε ts, it is random. Therefore, we can think of W T (r) as defining a random function of r, which we write W T ( ). Just as the CLT provides conditions ensuring that the rescaled random walk T / Y T /σ (which we can now write as W T ()) converges, as T become large, to a well-defined limiting random variables (the standard normal), the f unction central limit theorem (FCLT) provides conditions ensuring that the random function W T ( ) converge, as T become large, to a well-defined limit random function, say W ( ). The word Functional in Functional Central Limit theorem appears because this limit is a function of r. Some further properties of random walk, suitably rescaled, are in the following. Proposition. If Y t is a random walk, then Y t4 Y t3 is independent of Y t Y t for all t < t < t 3 < t 4. Consequently, W t (r 4 ) W T (r 3 ) is independent of W t (r ) W T (r ) for all [T r i ] = t i, i =,.., 4. Proof. Note that Y t4 Y t3 = ε t4 + ε t ε t3 +, Y t Y t = ε t + ε t ε t +. Since (ε t, ε t,..., ε t +) is independent of (ε t4, ε t4,..., ε t3 +) it follow that Y t4 Y t3 and Y t Y t are independent. Consequently, W T (r 4 ) W T (r 3 ) = T / (ε t4 + ε t ε t3 +)/σ is independent of W T (r ) W T (r ) = T / (ε t + ε t ε t +)/σ. Proposition. For given a < b, W T (b) W T (a) L N(, b a) as T. R 8 by Prof. Chingnun Lee 5 Ins.of Economics,NSYSU,Taiwan

6 Ch. ASYMPTOTIC THEORIES FOR UNIT ROOT PROCESS Proof. By definition [T b] W T (b) W T (a) = T / t=[t a]+ ε t [T b] = T / ([T b] [T a]) / ([T b] [T a]) / t=[t a]+ The last term ([T b] [T a]) / [T b] t=[t a]+ ε L t N(, ) by the CLT, and T / ([T b] [T a]) / = (([T b] [T a])/t ) / (b a) / L as T. Hence W T (b) W T (a) N(, b a). ε t. In words, the random walk has independent increments and those increments have a limiting normal distribution, with a variance reflecting the size of the interval (b a) over which the increment is taken. It should not be surprising, therefore, that the limit of the sequence of function W T ( ) constructed from the random walk preserves these properties in the limit in an appropriate sense. In fact, these properties form the basis of the definition of the Wiener process. Definition. Let (S, F, P) be a complete probability space. Then W : S [, ] R is a standard Wiener process if each of r [, ], W (, r) is F-measurable, and in addition, (a). The process starts at zero: P[W (, ) = ] =. (b). The increments are independent: if a a... a k, then W (, a i ) W (, a i ) is independent of W (, a j ) W (, a j ), j =,.., k, j i for all i =,..., k. (c). The increments are normally distributed: for a b, the increment W (, b) W (, a) is distributed as N(, b a). In the definition, we have written W (, a) explicitness; whenever convenient, however, we will write W (a) instead of W (, a), analogous to our notation elsewhere. R 8 by Prof. Chingnun Lee 6 Ins.of Economics,NSYSU,Taiwan

7 Ch. ASYMPTOTIC THEORIES FOR UNIT ROOT PROCESS The Wiener process is also called a Brownian motion in honor of Norbert Wiener (94), who provided the mathematical foundation for the theory of random motions observed and described by nineteenth century botanist Robert Brown in 87.. Functional Central Limit Theorems We earlier defined convergence in law for random variables, and now we need to extend the definition to cover random functions. Definition. Let S( ) represent a continuous-time stochastic process with S(r) representing its value at some date r for r [, ]. Suppose, further, that any given realization, S( ) is a continuous function of r with probability. For {S T ( )} T = a sequence of such continuous function, we say that the sequence of probability measure induced by {S T ( )} T = weakly converge to the probability measure induced by S( ), denoted by S T ( ) = S( ) if all of the following hold: (a). For any finite collection of k particular dates, r < r <... < r k, converges in distribu- the sequence of k-dimensional random vector {y T } T = tion to the vector y, where y T S T (r ) S T (r )... S T (r k ), y S(r ) S(r )... S(r k ) ; (b). For each ɛ >, the probability that S T (r ) differs from S T (r ) for any dates r and r within δ of each other goes to zero uniformly in T as δ ; (c). P { S T () > λ} uniformly in T as λ. R 8 by Prof. Chingnun Lee 7 Ins.of Economics,NSYSU,Taiwan

8 Ch. ASYMPTOTIC THEORIES FOR UNIT ROOT PROCESS This definition applies to sequences of continuous functions, though the function in (7) is a discontinues step function. Fortunately, the discountinuities occur at a countable set of points. Formally, S T ( ) can be replaced with a similar continuous function, interpolating between the steps. The Function Central Limit Theorem (FCLT) provides conditions under which W T converges to the standard Wiener process, W. The simplest FCLT is a generalization of the Lindeberg-Lévy CLT, known as Donsker s theorem. Theorem. (Donsker) Let ε t be a sequence of i.i.d. random scalars with mean zero. If σ V ar(ε t ) <, σ, then W T = W. Because pointwise convergence in distribution W T (, r) L W (, r) for each r [, ] is necessary (but not sufficient) for weak convergence W T = W, the L Lindeberg-Lévy CLT (W T (, ) W (, )) follows immediately from Donsker s theorem. Donsker s theorem is strictly stronger than Lindeberg-Lévy however, as both use identical assumptions, but Donsker s theorem delivers a much stronger conclusion. Donsker called his result an invariance principle. Consequently, the FCLT is often referred as an invariance principle. So far, we have assumed that the sequence ε t used to construct W T is i.i.d.. Nevertheless, just as we can obtain central limit theorems when ε t is not necessary i.i.d.. In fact, versions of the FCLT hold for each CLT previous given in Chapter 4. See the following section. X T In Chapter 4 we saw that if {X T } T = is a sequence of random variables with L X and if g : R R L is a continuous function, then g(x T ) g(x). A similar result holds for sequences of random functions. Theorem. (Continuous Mapping Theorem): If S T ( ) = S( ) and g( ) is a continuous functional, then g(s T ( )) = g(s( )). In the above theorem, continuity of a functional g( ) means that for any ς >, there exist a δ > such that if h(r) and k(r) are any continuous bounded functions on [, ], h : [, ] R and k : [, ] R, such that h(r) k(r) < δ for all r [, ], then g(h( )) g(k( )) < ς(δ). R 8 by Prof. Chingnun Lee 8 Ins.of Economics,NSYSU,Taiwan

9 Ch. 3 REGRESSION WITH A UNIT ROOT 3 Regression with a Unit Root 3. Dickey-Fuller Test, Y t is AR() process Consider the following simple AR() process with a unit root, Y t = βy t + u t, (-8) where β =, Y = and u t is i.i.d. with mean zero and variance σ. We consider the three least square regressions: Y t = βy t + ŭ t, (-9) Y t = ˆα + ˆβY t + û t, (-) and Y t = α + βy t + δt + ũ t, (-) where β, (ˆα, ˆβ), and ( α, β, δ) are the conventional least-squares regression coefficients. Dickey and Fuller (979) were concerned with the limiting distribution of the regression in (9), (), and () ( β, (ˆα, ˆβ), and ( α, β, δ)) under the null hypothesis that the data are generated by (8). We first provide the following asymptotic results of the sample moments which are useful to derive the asymptotics of the OLS estimator. Lemma. Let u t be a i.i.d. sequence with mean zero and variance σ and y t = u + u u t for t =,,..., T, (-) with y =. Then (a). T (b). T (c). T 3 u t T Y t Y t L σw (), L σ [W (r)] dr, L σ W (r)dr, R 8 by Prof. Chingnun Lee 9 Ins.of Economics,NSYSU,Taiwan

10 Ch. 3 REGRESSION WITH A UNIT ROOT (d). T T L Y t u t σ [W () ], (e). T 3 (f). T 5 (h). T 3 t L tu t σ[w () W (r)dr], ty t T ty t L σ rw (r)dr, L σ r[w (r)] dr. A joint weak convergence for the sample moments given above to their respective limits is easily established and will be used below. Proof. (a). This is a straightforward results of Donsker s Theorem with r =. (b). First rewrite T T where r t = (t )/T, so that T Y t in term of W T (r t ) T / Y t /σ = T T Y t = σ T W T (r) is constant for (t )/T r < t/t, we have T t / s= u s /σ, W T (r t ). Because T W T (r t ) = = t/t (t )/T W T (r) dr. W T (r) dr The continuous mapping theorem applies to h(w T ) = W T (r) dr. It follows that h(w T ) = h(w ), so that T T Y t = σ W (r) dr, as claimed. (c). The proof of item (c) is analogous to that of (b). First rewrite T 3/ T Y t in term of W T (r t ) T / Y t /σ = T / t s= u s/σ, where r t = (t )/T, so that T 3/ T Y t = σt T W T (r t ). Because W T (r) is constant for (t )/T r < t/t, we have T W T (r t ) = = t/t (t )/T W T (r)dr. W T (r)dr R 8 by Prof. Chingnun Lee Ins.of Economics,NSYSU,Taiwan

11 Ch. 3 REGRESSION WITH A UNIT ROOT The continuous mapping theorem applies to h(w T ) = W T (r)dr. It follows that h(w T ) = h(w ), so that T 3/ T Y t = σ W (r)dr, as claimed. (d). For a random walk, Yt = (Y t + u t ) = Yt + Y t u t + u t, implying that Y t u t = (/){Yt Yt u t } and then T Y t u t = (/){YT Y } (/) T u t. Recall that Y =, and thus it is convenient to write T Y t u t = Y T T u t. From items (a)) we know that T YT (= (T / T u s) L σ W () and T T u p t σ by LLN (Kolmogorov); then, T T Y t u t = σ [W () ]. (e). We first observe that T Y t = [u +(u +u )+(u +u +u 3 )+...+(u +u + u u T )] = [T )u + (T )u + (T 3)u [T (T )u T ] = T (T t)u t = T T u t T tu t, or T tu t = T T u t T Y t. Therefore, T 3 T tu t = T T u t T 3 T Y t. By applying the continuous mapping theorem to the joint convergence of items (a) and (c), we have T 3 tu t σ[w () W (r)dr]. (f). The proofs of item (f) and (g) is analogous to those of (c) and (b). First rewrite T 5/ T ty t in term of W T (r t ) T / Y t /σ = T / t s= u s/σ, where r t = (t )/T, so that T 5/ T ty t = σt T tw T (r t ). Because W T (r) is constant for (t )/T r < t/t, we have T (t/t )W T (r t ) = = t/t (t )/T rw T (r)dr. rw T (r)dr for r = t/t The continuous mapping theorem applies to h(w T ) = rw T (r)dr. It follows that h(w T ) = h(w ), so that T 5/ T ty t = σ rw (r)dr, as claimed. We also write T 3 T ty t in term of W T (r t ) T / Y t /σ = T / t s= u s/σ, where r t = (t )/T, so that T 3 T ty t = σ T T tw T (r t ). Because W T (r) is constant for (t )/T r < t/t, we have T (t/t )W T (r t ) = = t/t (t )/T rw T (r) dr. rw T (r) dr for R 8 by Prof. Chingnun Lee Ins.of Economics,NSYSU,Taiwan

12 Ch. 3 REGRESSION WITH A UNIT ROOT The continuous mapping theorem applies to h(w T ) = rw T (r) dr. It follows that h(w T ) = h(w ), so that T 3 T ty t = σ rw (r) dr. This completes the proofs of this lemma. 3.. No Constant Term or Time Trend in the Regression; True Process Is a Random Walk We first consider the case that no constant term or time trend in the regression model, but true process is a random walk. The asymptotic distributions of OLS unit root coefficient estimator and t-ratio test statistics are in the following. Theorem. Let the data Y t be generated by (8), i.e. Y t = Y t + u t, then as T, for the regression model (9), Y t = βy t + ŭ t, and where σ βt variance: T ( β L T ) /{[W ()] } [W, (r)] dr t = ( β T ) σ βt s T = L /{[W ()] } { [W (r)] dr}, / = [s T T Y t ] / and s T (Y t β T Y t ) /(T ). denote the OLS estimate of the disturbance Proof. Since the deviation of the OLS estimate from the true value is characterized by T ( β T ) = T T T T Y t u t, (-3) Y t R 8 by Prof. Chingnun Lee Ins.of Economics,NSYSU,Taiwan

13 Ch. 3 REGRESSION WITH A UNIT ROOT which is a continuous function of Lemma (b) and (d), it follows that under the null hypothesis that β =, the OLS estimator β is characterized by T ( β T ) /{[W ()] } [W. (-4) (r)] dr This completes the proof of the first part in this Theorem. To prove second part of this theorem, we first show the consistency of s T. Notice that the population disturbance sum of squares can be written (y T y T β) (y T y T β) = (y T y T β + yt β yt β) (y T y T β + yt β yt β) = (y T y T β) (y T y T β) + (yt β yt β) (y T β yt β), where y T = [Y Y... Y T ] and the cross-product have vanished, since (y T y T β) y T ( β β) =, by the OLS orthogonality condition (X e = ). Dividing last equation by T, or (/T )(y T y T β) (y T y T β) = (/T )(y T y T β) (y T y T β) + ( β β) [(y T y T )/T ]( β β) (/T )(y T y T β) (y T y T β) = (/T )( u t ) T / ( β β) [(y T y T )/T ]T / ( β β). (-5) Now (/T )( T u p t ) E(u t ) σ by LLN for i.i.d. sequence, T / ( β β) and (y T y T )/T = σ [W (r)] dr from (4) and Lemma (b), respectively. We thus have T / ( β β) [(y T y T )/T ]T / ( β p β) σ [W (r)] dr =. Substituting these results into (5) we have (/T )(y T y T β) (y T y T β) The OLS disturbance s variance estimator p σ. s T = [/(T )](y T y T β) (y T y T β) = [T/(T )](/T )(y T y T β) (y T y T β) p σ = σ, (-6) R 8 by Prof. Chingnun Lee 3 Ins.of Economics,NSYSU,Taiwan

14 Ch. 3 REGRESSION WITH A UNIT ROOT therefore is a consistent estimator. or Finally, we can express the t statistics alternatively as { } / t T = T ( β T ) T (s t T = Y t T T Y t u t { T } / T Y t (s T )/, T ) / which is a continuous function of Lemma (b) and (d), it follows that under the null hypothesis that β =, the asymptotic distribution of OLS t statistics is characterized by t T L /σ {[W ()] } { σ } / = /{[W ()] } { } /. (-7) [W (r)] dr (σ ) / [W (r)] dr This complete the proof of this Theorem. Statistical tables for the distributions of (4) and (7) for various sample size T are reported in the section labeled Case in Table B.5 and B.6, respectively at Hamilton book. This finite sample result assume Gaussian innovations. 3.. Constant Term but No Time Trend included in the Regression; True Process Is a Random Walk We next consider the case that a constant term is added in the regression model, but true process is a random walk. The asymptotic distributions of OLS unit root coefficient estimator and t-ratio test statistics are in the following. Theorem. Let the data Y t be generated by (8), i.e. Y t = Y t + u t, then as T, for the regression model (), Y t = ˆα + ˆβY t + û t, and T ( ˆβ L T ) /{[W ()] } W () W (r)dr [ ] (-8) [W (r)] dr W (r)dr ˆt = ( ˆβ T ) ˆσ ˆβT L /{[W ()] } W () { [W (r)] dr W (r)dr [ ] } /, (-9) W (r)dr R 8 by Prof. Chingnun Lee 4 Ins.of Economics,NSYSU,Taiwan

15 Ch. 3 REGRESSION WITH A UNIT ROOT Yt denote the OLS esti- [ ] [ where ˆσ ˆβT = s T T Yt Y t mate of the disturbance variance: s T = (Y t ˆα T ˆβ T Y t ) /(T ). ] [ ] and s T Proof. The proof of this theorem is analogous to that of Theorem and is omitted here. Statistical tables for the distributions of (8) and (9) for various sample size T are reported in the section labeled Case in Table B.5 and B.6, respectively. This finite sample result assume Gaussian innovations. These statistics test the null hypothesis that β =. However, a maintained assumption on which the derivation of Theorem was based on is that the true value of α is zero. Thus, it might seem more natural to test for a unit root in this specification by testing the joint hypothesis that α = and β =. Dickey and Fuller (98) derive the limiting distribution of the likelihood ratio test for the hypothesis that (α, β) = (, ) and used Monte Carlo to calculate the distribution of the OLS F test of this hypothesis. Their values are reported under the heading Case in table B Constant Term and Time Trend Included in the Regression; True Process Is a Random Walk With or Without Drift We finally in the section consider the case that a constant term and a linear trend are added in the regression model, but true process is a random walk with a drift. However, the true value of this drift turns out not to matter for the asymptotic distributions of OLS unit root coefficient estimator and t-ratio test statistics in this case. Theorem 3. Let the data Y t be generated by (8), i.e. Y t = Y t + u t, then as T, for the R 8 by Prof. Chingnun Lee 5 Ins.of Economics,NSYSU,Taiwan

16 Ch. 3 REGRESSION WITH A UNIT ROOT regression model (), Y t = α + βy t + δt + ũ t, and where σ βt T ( β ) /{[W () W (r)dr][w () + 6 W (r)dr rw (r)dr] } [W (r)] dr 4[ W (r)dr] + W (r)dr rw (r)dr [ rw (r)dr] t = ( β T ) σ βt T ( β ) Q, ξ t = Y t αt, s T [ ] = s T T ξ t t ξ t ξt tξ t t tξ t t denote the OLS estimate of the disturbance variance:, and s T = (Y t α β T Y t δt) /(T 3), Q [ ] W (r)dr / W (r)dr [W (r)] dr rw (r)dr / rw (r)dr /3. Proof. (a). Let the data generating process be Y t = α + Y t + u t, and the regression model be Y t = α + βy t + δt + u t. (-) Note that the regression model of () can be equivalently rewritten as Y t = ( β)α + β(y t α(t )) + (δ + βα)t + u t α + β ξ t + δ t + u t, (-) R 8 by Prof. Chingnun Lee 6 Ins.of Economics,NSYSU,Taiwan

17 Ch. 3 REGRESSION WITH A UNIT ROOT where α = ( β)α, β = β, δ = δ + βα, and ξ t = Y t α(t ). Moreover, under the null hypothesis that β = and δ =, ξ t = Y + u + u u t, that is, ξ t is the random walk as described in Lemma. Under the maintained hypothesis, α = α, β =, and δ =, which in () means that α =, β = and δ = α. The deviation of the OLS estimate from these true values is given by α β δ α or in shorthand as = T ξ t t ξ t ξt tξ t C = A f. t tξ t t u t ξ t u t,(-) tu t From Lemma, the order of probability of the individual terms in () is as follows, α β δ α = We define a rescaling matrices, Υ T = T T T 3 O p (T ) O p (T 3 ) O p (T ) O p (T 3 ) O p (T ) O p (T 5 ) O p (T ) O p (T 5 ) O p (T 3 ). Multiplying the rescaling matrices on (), we get Υ T C = Υ T A Υ T Υ T f = [ Υ T AΥ T O p (T ) O p (T ) O p (T 3 ). ] Υ T f (-3) Substituting the results of Lemma A. to (3), we establish that b Q h, (-4) R 8 by Prof. Chingnun Lee 7 Ins.of Economics,NSYSU,Taiwan

18 Ch. 3 REGRESSION WITH A UNIT ROOT where b Q h T / α T ( β ), T 3/ ( δ α ) W (r)dr / W (r)dr [W (r)] dr rw (r)dr / rw (r)dr /3 σw () σ {[W ()] } σ[w (). W (r)dr] Thus, the asymptotic distribution of T ( β ) is given by the middle row of (4), that is, T ( β ) /{[W () W (r)dr][w () + 6 W (r)dr rw (r)dr] } [W (r)] dr 4[ W (r)dr] + W (r)dr rw (r)dr [. rw (r)dr] Note that this distribution does not depend on either α or σ; in particular, it doesn t matter whether or not the true value of α is zero.. (b). The asymptotic distribution of the OLS-t statistics can be founded using similar calculation as those in (7). Notice that [ ] T σ βt = T s T = s T [ ] T ξ t t T ξ t t T / T T 3/ ξ t ξt tξ t t tξ t t ξ t ξt tξ t t tξ t t T / T T 3/ R 8 by Prof. Chingnun Lee 8 Ins.of Economics,NSYSU,Taiwan

19 Ch. 3 REGRESSION WITH A UNIT ROOT T 3/ T ξ t T T t [ ] = s T T 3/ ξ t T ξt T 5/ tξ t T t T 5/ tξ t T 3 t σ W (r)dr / σ W (r)dr σ [W (r)] dr σ rw (r)dr / σ rw (r)dr /3 L σ [ ] = σ [ ] σ W (r)dr / W (r)dr [W (r)] dr rw (r)dr σ / rw (r)dr /3 = [ ] W (r)dr / W (r)dr [W (r)] dr rw (r)dr / rw (r)dr /3 Q. From this result it follows that the asymptotic distribution of the OLS t test of the hypothesis that β = is given by t T = T ( β T ) (T σ βt ) / p T ( β T ) Q. This completes the proofs of Theorem 3. Again, this distribution does not dependent on α or σ. The small-sample distribution of the OLS t statistics under the assumption of Gaussian disturbance is presented under case 4 in Table B.6. If this distribution were truly t, then a value below. would be sufficient to reject the null hypothesis. However, Table B.6 reveals that, because of the nonstandard distribution, the t statistic must below 3.4 before the null hypothesis of a unit root could be rejected. The assumption that the true value of δ is equal to zero is again an auxiliary hypothesis upon which the asymptotic properties of the test depend. Thus, as in section 3.., it is natural to consider the OLS F test of the joint null hypothesis that δ = and β =. Though this F test statistic is calculated in the usual way, its asymptotic distribution is nonstandard, and the calculated F statistic should be compared with the value under case 4 in Table B.7. R 8 by Prof. Chingnun Lee 9 Ins.of Economics,NSYSU,Taiwan

20 Ch. 3 REGRESSION WITH A UNIT ROOT Remark. To derive the asymptotic distribution in this chapter, it is useful to use the following result: [ α β ] [ a b c d ] [ α β ] [ α = a αβb αβc β d ]. The unit root tests are not confining themselves to the simple AR() process as the original Dickey and Fuller (979). There are four main directions in the literature devoting to accommodating serially correlated errors and then to alleviating the size distortion and lower power problems. First, using long autoregressions to approximate serially correlated errors was first suggested by Dickey and Fuller (979) and further studied in Said and Dickey (984) and others. That is it is a parametric model that assume Y t in (8) is a AR(p) process or u t is a ARMA(p, q) (Said-Dickey, 984) process. The second one is a non-parametric model that assume u t is a general process which satisfy certain memory and moment constrains. Phillips (987a) and Phillips and Perron (988) constructed consistent estimators of the long- and short-run variances of u t in (8) to remedy the influence of the nuisance parameter. Third, Perron and Ng (996) considered a long-run variance estimator based on estimation of a long autoregression, in which the order of the autoregression is chosen according to the Akaike information criterion (AIC) or the Schwartz (Bayesian) information criterion (BIC). Furthermore, the class of tests suggested by Ng and Perron () was designed to the tests of Perron and Ng (996) by employing a modified form of the AIC to determine suitable lag order for an autoregression. Fourth, in the work of Wang (7) which introduced a consistent estimator for the ratio of long to short-run variances and then used it to modify the DF tests (denoted as R-tests). By using Monte Carlo techniques, Wang (7) showed that the R-tests have improved size properties. The first two modifications of unit root models are discussed in the following. R 8 by Prof. Chingnun Lee Ins.of Economics,NSYSU,Taiwan

21 Ch. 3 REGRESSION WITH A UNIT ROOT 3. Augmented Dickey-Fuller Test, Y t is AR(p) process Instead of (8), suppose that the data were generated from an AR(p) process, ( φ L φ L... φ p L p )Y t = ε t, (-5) where ε t is i.i.d. sequence with zero mean and variance σ and finite fourth moment. It is helpful to write the autoregression (5) in a slightly different form. To do so, define β φ + φ φ p ζ j [φ j+ + φ j φ p ], for j =,,..., p. Notice that for any value of φ, φ,..., φ p, the following polynomials in L are equivalent: ( βl) (ζ L + ζ L ζ p L p )( L) = βl ζ L + ζ L ζ L + ζ L 3... ζ p L p + ζ p L p = (β + ζ )L (ζ ζ )L (ζ 3 ζ )L 3... (ζ p ζ p )L p ( ζ p )L p = [(φ + φ φ p ) (φ φ p )]L [ (φ 3 + φ φ p ) + (φ φ p )]L... [ (φ p ) + (φ p + φ p )]L p (φ p )L p = φ L φ L... φ p L p. Thus, the autoregression (5) can be equivalently be written {( βl) (ζ L + ζ L ζ p L p )( L)}Y t = ε t (-6) or Y t = βy t + ζ Y t + ζ Y t ζ p Y t p+ + ε t. (-7) Example. In the case of p = 3, (5) would be Y t = φ Y t + φ Y t + φ 3 Y t 3 + ε t = (φ + φ + φ 3 )Y t (φ + φ 3 )[Y t Y t ] (φ 3 )[Y t Y t 3 ] = βy t + ζ Y t + ζ Y t. R 8 by Prof. Chingnun Lee Ins.of Economics,NSYSU,Taiwan

22 Ch. 3 REGRESSION WITH A UNIT ROOT Suppose that the process that generated Y t contain only a single unit root, that is suppose one root of ( φ z φ z... φ p z p ) = (-8) is unity, then φ φ... φ p =, (-9) and all other root of (8) are outside the unit circle. Notice that (36) implies that β = in (6). Moreover, when β =, it would imply that φ L φ L... φ p L p = ( L)( ζ L ζ L... ζ p L p ) and all the roots of ( ζ L ζ L... ζ p L p ) = would lie outside the unit circle. Under the null hypothesis that β =, expression (6) could then be written as Y t = ζ Y t + ζ Y t ζ p Y t p+ + ε t. (-3) or Y t = u t (-3) where u t = ψ(l)ε t = ( ζ L ζ L... ζ p L p ) ε t. That is, we may express this AR(p) with an unit root process as the AR() with an unit root process as (3) but with serially correlated u t. Dickey and Fuller (979) propose a test of unit root in this AR(p) model and is known as the Augmented Dickey-Fuller (ADF) test. 3.. Constant Term but No Time Trend included in the Regression; True Process Is a Autoregressive with no Drift Assume that the initial sample is of size T +p, with observation numbered {Y p+, Y p+,..., Y T }, and conditional on the first p observations. We are interested in the properties of OLS estimation of Y t = ζ Y t + ζ Y t ζ p Y t p+ + α + βy t + ε t = x tβ + ε t (-3) R 8 by Prof. Chingnun Lee Ins.of Economics,NSYSU,Taiwan

23 Ch. 3 REGRESSION WITH A UNIT ROOT under the null hypothesis that α = and β =, i.e. the DGP is Y t = ζ Y t + ζ Y t ζ p Y t p+ + ε t, (-33) where β (ζ, ζ,..., ζ p, α, β) and x t ( Y t, Y t,..., Y t p+,, Y t ). The asymptotic distribution of the OLS coefficients estimators, ˆβ T are in the following. Theorem 4. Let the data Y t be generated by (33), i.e. Y t = ζ Y t + ζ Y t ζ p Y t p+ + ε t, then as T, for the regression model (3), Y t = ζ Y t + ζ Y t ζ p Y t p+ + α + βy t + ε t, [ ] [ ] [ Υ T ( ˆβ L V h V T β) = h Q h Q h ], (-34) where Υ T = Q = T... T , V = T... T [ λ ] W (r)dr λ W (r)dr λ [W (r)], dr γ γ... γ p γ γ... γ p γ p γ p 3... γ, [ h N p (, V), h σw () /σλ{[w ()] } ], γ j = E[( Y t )( Y t j )] and λ = σ ψ() = σ/( ζ ζ... ζ p ). (-35) The results reveals that in a regression of I() variables on I() and I() variables, the asymptotic distribution of the coefficient of I() and I() variables are independent. Thus, the asymptotic distribution of T (ˆζ j ζ j ), j =,,.., p and T ( ˆβ β) are independent. This results can be used for showing that the distribution of ˆβ T in the ADF regression is the Dickey-Fuller distribution (taking into account of serially correlated in the u t, see (36)). Also the asymptotic distribution R 8 by Prof. Chingnun Lee 3 Ins.of Economics,NSYSU,Taiwan

24 Ch. 3 REGRESSION WITH A UNIT ROOT of T (ˆζ j ζ j ) is normal. Therefore, the limiting distribution of T ( ˆβ β) is given by the second element of [ ] [ ] T / ˆα T ˆβ T [ L λ ] [ ] W (r)dr λ W (r)dr λ σw () [W (r)] dr /σλ{[w ()] } [ ] [ ] [ σ W (r)dr W () σ/λ W (r)dr [W (r)] dr /{[W ()] } or T ( ˆβ T ) (σ/λ) /{[W ()] } W () W (r)dr [ ]. (-36) [W (r)] dr W (r)dr The parameter (σ/λ) is the factor to correct the serial correlation in u t. When u t is i.i.d., form (35) we have ζ i = and λ = σ, that is (σ/λ) =. This distribution is back to simple Dickey-Fuller distribution. We are now in a position to propose the ADF test statistics which correct for (σ/λ) and have the same distribution as DF. ], Theorem 5. Let the data Y t be generated by (33), i.e. Y t = ζ Y t + ζ Y t ζ p Y t p+ + ε t, then as T, for the regression model (3), Y t = ζ Y t + ζ Y t ζ p Y t p+ + α + βy t + ε t, (a). (b). T ( ˆβ T ) ˆζ ˆζ... ˆζ /{[W ()] } W () W (r)dr [ ], p [W (r)] dr W (r)dr t T = ( ˆβ T ) {s T e p+( x t x t) e p+ } /{[W ()] } W () W (r)dr / { [ ] } /, [W (r)] dr W (r)dr where e p+ = [... ]. Proof. (a). From (36) we have T (λ/σ)( ˆβ T ) /{[W ()] } W () W (r)dr [ ]. (-37) [W (r)] dr W (r)dr R 8 by Prof. Chingnun Lee 4 Ins.of Economics,NSYSU,Taiwan

25 Ch. 3 REGRESSION WITH A UNIT ROOT Recall from (35) that λ/σ = ( ζ ζ... ζ p ). Since ˆζ j is consistent from (34), this magnitude is clearly consistently estimated by λ/σ = ( ˆζ ˆζ... ˆζ p ). (-38) It follows that [T ( ˆβ T ) (λ/σ)] [T ( ˆβ T ) ( λ/σ)] = T ( ˆβ T ) [(λ/σ) ( λ/σ)] = O p () o p () = o p (). Thus [T ( ˆβ T ) ( λ/σ)] and [T ( ˆβ T ) (λ/σ)] have the same asymptotic distribution. This complete the proof of part (a). (b). To prove art (b), we first multiply the numerator and denominator of t T by T results in t T = T ( ˆβ T ) {s T e p+υ T (. (-39) x t x t) Υ T e p+ } / But e p+υ T ( x t x t) Υ T e p+ = e p+ L e p+ = λ [ Υ T ( x t x t)υ T ] e p+ [ V Q { [W (r)] dr ] ep+ [ W (r)dr ] }. (-4) R 8 by Prof. Chingnun Lee 5 Ins.of Economics,NSYSU,Taiwan

26 Ch. 3 REGRESSION WITH A UNIT ROOT Hence from (36) and (4), t T L (σ/λ) /{[W ()] } W () λ [W (r)] dr σ { [W (r)] dr W (r)dr [ W (r)dr ] [ ] } W (r)dr = /{[W ()] } W () W (r)dr { [ ] } /. [W (r)] dr W (r)dr / This is the same distribution as in (9). Thus, the usual t test of β = for OLS estimation of AR(p) can be compared with Theorem and use case of Table B.6 without any correction for the fact u t is serially correlated (or Y are included in the regression). 3.3 Augmented Dickey-Fuller Test, Y t is a ARMA(p, q) process The fact that the distribution of ˆβ T in the ADF regression is the Dickey-Fuller distribution has been extended by Said and Dickey (984) to the more general case in which, under the null hypothesis, the series of first difference are of the general ARMA(p, q) form with unknown p and q. They showed that a regression model, such as (39), is still valid for testing the unit root null under the presence of the serial correlations of error, if the number of lags Y included as regressor increase with the sample size at a controlled rate T /3. Essentially the moving terms are being approximated by including enough autoregressive terms. Consider the general ARIMA(p,, q) model is defined by Y t = βy t + u t, (-4) where ( φ L φ L... φ p L p )u t = ( + θ L + θ L θ q L q )ε t (-4) with Y =, ε t is i.i.d. and β =. R 8 by Prof. Chingnun Lee 6 Ins.of Economics,NSYSU,Taiwan

27 Ch. 3 REGRESSION WITH A UNIT ROOT (4) and (4) is equivalently written as ( φ L φ L... φ p L p ) Y t = ( + θ L + θ L θ q L q )ε t. (-43) Therefore, (3) as a special case of (43) in which θ j =, j =,,..., q. Rewrite (43) as η(l) Y t = ε t (-44) where η(l) = ( η L η L...) = ( + θ L + θ L θ q L q ) ( φ L φ L... φ p L p ). That is, Y t = η Y t + η Y t + η 3 Y t ε t or Y t = Y t + η Y t + η Y t + η 3 Y t ε t. (-45) This motives us to estimate the coefficient in (45) by regression Y t on Y t, Y t, Y t,..., Y t k where k is a suitably chosen integer. To get consistent estimator of the coefficient in (45) it is necessary to let k as a function of T. Consider a truncated version of (45) Y t = α + βy t + η Y t + η Y t η k Y t k + e tk (-46) = x tβ + e tk, where β (α, β, ζ, ζ,..., ζ k ) and x t (, Y t, Y t, Y t,..., Y t k ). Notice that e tk is not a white noise. In this case, the limiting distribution of t statistics of the coefficient on the lagged Y t (i.e., ˆβ T ) from OLS estimation of has the same Dickey-Fuller t-distribution as when u t is i.i.d.. Theorem 6. (Said-Dickey ADF ): Let the data Y t be generated by (4) and (4) and the regression model be (46). We assume that T /3 k and there exist c >, r > such that ck > T /r, then as T, t T = ( ˆβ T ) {s T e k+ ( x t x t) e k+ } / L /{[W ()] } W () W (r)dr { [W (r)] dr [ W (r)dr ] } /, where e k+ = [... ]. R 8 by Prof. Chingnun Lee 7 Ins.of Economics,NSYSU,Taiwan

28 Ch. 3 REGRESSION WITH A UNIT ROOT The intuition behind that k should be a function of T is clear from the fact e tk = ζ k+ Y t k + ζ k+ Y t k ε t. Then as k e tk ε t = ζ k+ Y t k + ζ k+ Y t k +... p from the absolute summability of ζ j, i.e j= ζ j < which imply ζ j and ζ k+ Y t k + ζ k+ Y t k Then, it is expected that a ARIMA(p,, q) process here would have the same asymptotic result with the ARIMA(p,, ) process. Therefore, the asymptotic result should be derived under the condition that k. However, it can not increase quickly then T, i.e. we need the condition that T /3 k Choice of Lag-Length in the ADF Test It has been observed that the size and power properties of the ADF test are sensitive to the number of lagged terms (k) used. Several guidelines have been suggested for the choice of k. Ng and Perron (995) examine these in details. The guidelines are: (a). Rule for fixing k at an arbitrary level independent of T. Overall, choosing a fixed k is not desirable from their detailed simulation. (b). Rule for fixing k as a function of T. A rule commonly used is the one suggested by Schwert (989) which is to choose k = Int{c(T/) /d }. Schwert suggest c = and d = 4. The problem with such a rule is that it need not be optional for all p and q in the ARMA(p, q). (c). Information based rules. The information criteria suggest choosing k to minimize an objective function that trades off parsimony against reduction in sum of squares. The objective function is of the form (see also p.5 of Ch. 6) I k = log ˆσ k + k C T T. The Akaike information criterion (AIC) choose C T =. The Schwarz Bayesian information criterion (BIC) chooses C T = log T. Ng and Perron argue that both AIC and BIC are asymptotically the same with ARMA(p, q) models and that both of them choose k proportional to log T. R 8 by Prof. Chingnun Lee 8 Ins.of Economics,NSYSU,Taiwan

29 Ch. 3 REGRESSION WITH A UNIT ROOT (d). Sequential rules. Hall (994) discusses a general to specif ic rule which is to start with a large value of k (k max ). We test the significance of the last coefficient and reduce k iteratively until a significant statistic is encountered. Ng and Perron (995) compare AIC, BIC, and Hall s general to specific approach through a Monte Carlo study. The major conclusions are: (a). Both AIC and BIC choose very small value of k (e.g. k = 3). This results in high size distortions, especially with M A errors (Remark: The intuition behind this conclusion is that with MA error, this model is AR( ), if you use too small lagged value, your model look not very much like a AR( ), and since this asymptotic results is derived from k, your finite sample distribution is far away from the asymptotic distribution and results in size distortion.) (b). Hall s criterion tends to choose higher values of k. The higher the k max is, the higher is the chosen value of k. This results in the size being at the nominal level, but of course with a loss of power. (Remark: Unit root test statistics sa far we have derived is the distribution under the null. On the other side, we can derive the test statistics under the alternative hypothesis of stationary or fractional difference process. Theses test statistics should go to, say, against the left-tailed hypothesis, when the alternative hypothesis is true to be able to consistently reject the null hypothesis, or what is called power is one. A common results is that the asymptotic distribution of the unit root test statistics under the alternative hypothesis is function of k, and with precisely, (T/k), say. For a fixed T, (T/K) is smaller with a larger k. This cause the unit roots distribution tend to more slower under the alternative hypothesis and has a lower power. See Lee and Schmidt (996) and Lee and Shie (4).) What this study suggests is that Hall s general to specific methods is preferable to the others. DeJong et al. (99) show that increasing k typically results in a modest decrease in power but a substantial decrease in size distortion. If this is the case the information criteria are at a disadvantage because they result in a choice of very small value of k. However, Stock (994) propose opposite evidence arguing in favor of BIC compared with Hall s method. R 8 by Prof. Chingnun Lee 9 Ins.of Economics,NSYSU,Taiwan

30 Ch. 3 REGRESSION WITH A UNIT ROOT 3.4 Phillips-Perron Test, u t is mixing process In contrast to ADF where a AR() unit root process has been extend to AR(p) and ARMA(p, q) with a unit root, Phillips (987) and Phillips and Perron (988) extend the random walk (AR() process with a unit root) to a general setting that allows for weakly dependent and heterogeneously distributed innovations. The model Phillips (987) consider are Y t = βy t + u t, (-47) β = (-48) where Y = (not necessary in Phillips s original paper), β = and u t is a weakly dependent and heterogeneously distributed innovations sequence to be specified below. We consider the three least square regressions Y t = βy t + ŭ t, (-49) Y t = ˆα + ˆβY t + û t, (-5) and Y t = α + βy t + (t δ ) T + ũ t, (-5) where β, (ˆα, ˆβ), and ( α, δ, β) are the conventional least-squares regression coefficients. Phillips (987) and Phillips and Perron (978) were concerned with the limiting distribution of the regression in (49), (5), and (5) ( β, (ˆα, ˆβ), and ( α, δ, β)) under the null hypothesis that the data are generated by (47) and (48). So far, we have assumed that the sequence u t used to construct W T is i.i.d.. Nevertheless, just as we can obtain central limit theorems when u t is not necessary i.i.d., so also can we obtain FCLT when u t is not necessary i.i.d.. Here we present a version of FCLT, due to McLeish (975), under a very weak assumption on u t. Theorem 7. (McLeish) Let u t satisfies that (a). E(u t ) =, (b). sup t E u t γ < for some γ >, (c). λ = lim T E[T ( u t ) ] exists and λ >, and R 8 by Prof. Chingnun Lee 3 Ins.of Economics,NSYSU,Taiwan

31 Ch. 3 REGRESSION WITH A UNIT ROOT (d). u t is strong mixing with mixing coefficients α m that satisfy α /γ m <, then W T = W, where W T (r) T / [T r] u t/λ. These conditions (a) (d) allow for both temporal dependence (by mixing) and heteroscedasticity (as long as λ = lim T E[T ( u t ) ] exist) in the process u t. (Hint: When u t is an i.i.d. process, then λ = lim T E[T ( u t ) ] = σ, and this result is back to ().) We first provide the following asymptotic results of the sample moments which are useful to derive the asymptotics of the OLS estimator. Lemma. Let u t be a random sequence that satisfies the assumptions in Theorem 7, and if sup t E u t γ+η < for some η >, y t = u + u u t for t =,,..., T, (-5) with y =. Then (a) T (b) T (c) T 3 (d) T u t T T u t L λw (), Y t = λ [W (r)] dr, Y t = λ W (r)dr, p σ u = T T E(u t ). (e) T T Y t u t = {λ [W () ] σu}, (f) T 3 (g) T 5 (h) T 3 t T tu t = λ[w () W (r)dr], ty t = λ rw (r)dr, ty t = λ r[w (r)] dr. A joint weak convergence for the sample moments given above to their respective limits is easily established and will be used below. Proof. Proofs of items (a), (b), (c), (f), (g) and (h) are analogous to those proofs at Lemma R 8 by Prof. Chingnun Lee 3 Ins.of Economics,NSYSU,Taiwan

32 Ch. 3 REGRESSION WITH A UNIT ROOT. Item (d) is the results of LLN for a mixing process. (e). For a random walk, Yt = (Y t + u t ) = Yt + Y t u t + u t, implying that Y t u t = (/){Yt Yt u t } and then T Y t u t = (/){YT Y } (/) T Y T u t. Recall that Y =, and thus it is convenient to write T Y t u t = T u t. From items (a) we know that T YT (= (T / T u s) L p σu by LLN (MacLeish); then, T Y t u t = {λ [W () ] λ W () and T u t σu} No Constant Term or Time Trend in the Regression; True Process Is a Random Walk We first consider the case that no constant term or time trend in the regression model, but true process is a random walk. The asymptotic distributions of OLS unit root coefficients estimator and t-ratio test statistics are in the following. Theorem 8. Let the data Y t be generated by (47) and (48); and u t be a random sequence that satisfies the assumptions in Theorem 7, and if sup t E u t γ+η < for some η >, then as T, for the regression model (49), and where σ βt variance: T ( β T ) /([W () ] σ u/λ ) [W (r)] dr t = ( β T ) (λ/σ u){[w ()] σu/λ } σ βt { [W, (r)] dr} / s T = = [s T T Y t ] / and s T (Y t β T Y t ) /(T ). denote the OLS estimate of the disturbance Proof. Since the deviation of the OLS estimate from the true value is characterized by T ( β T ) = T T T T Y t u t, (-53) Y t R 8 by Prof. Chingnun Lee 3 Ins.of Economics,NSYSU,Taiwan

33 Ch. 3 REGRESSION WITH A UNIT ROOT which is a continuous function function of Lemma s (b) and (e), it follows that under the null hypothesis that β =, the OLS estimator β is characterized by T ( β T ) /{λ [W () ] σ u} λ [W (r)] dr = /([W () ] σu/λ ) [W. (-54) (r)] dr To prove second part of this theorem, We first note since β is a consistent estimator of β from (53), s T is a consistent estimator of σ u, from the analogous proofs as in Theorem. Then, we can express the t statistics alternatively as or t T = T ( β T ) t T = { T Y t T T Y t u t { T } / T Y t (s } / (s T )/, T ) / which is a continuous function function of Lemma s (b) and (e), it follows that under the null hypothesis that β =, the asymptotic distribution of OLS t statistics is characterized by t T L (/){[λ W ()] σu} { λ } / = [W (r)] dr (σ u) / (λ/σ u ){[W ()] σ u/λ } { [W (r)] dr} /. (-55) This complete the proof of this Theorem. Theorem 8 extends (4) and (7) to the very general case of weakly dependent and heterogeneously distributed data. When u t is a i.i.d. sequence, λ = T E[( u t ) ] = T E(u t ) = σu, and we see that the results of Theorem 8 reduces to those of Theorem. Several interesting things are noteworthy to the asymptotic distribution results for the least squares estimators, (54). First, note that the scale factor here is T, not T as it previously has been. Thus, β is collapsing to its limits at a much faster rate than before. This is sometimes called superconsistency. Next, note that the limiting distribution is no longer normal; instead, we have a distribution that is somewhat complicated function of a Wiener process. When σu = λ (independence) we have the distribution of J.S. White (958, p.96), apart from an incorrect scaling there, as noted by Phillips (987). For σu = λ, this distribution is also that tabulated by Dickey and Fuller (979) in their famous work on testing for unit root. In the regression setting studied in previous chapters the existence of serial correlation in u t in the presence of a lagged dependent variable regressor leads to the R 8 by Prof. Chingnun Lee 33 Ins.of Economics,NSYSU,Taiwan

34 Ch. 3 REGRESSION WITH A UNIT ROOT inconsistency of β for β as discussed in Chapter. Here, however, the situation is quite different. Even though the regressor is a lagged dependent variables, β is consistent for β = despite the fact that condition (c) and (d) of Theorem 7 permit u t to display considerable correlation. The effect of the serial correlation is that σ u λ. This results is a shift of the location of the asymptotic distribution away from zero (since E(W () σ u/λ ) where E(W () ) = E(χ () ) = ) relative to the σ u = λ case of no serial correlation). Despite this effect of the serial correlation in u t, we no longer have the serious adverse consequence of the inconsistency of β. One way of understanding why this is so is succinctly expressed by Phillips (987, p.83): Intuitively, when the data generating process has a unit root, the strength of the single as measured by the sample variation of the regressor Y t dominates the noise by a factor of O(T ), so that the effect of any regressor-error correlation are annihilated in the regression as T. Note that, however, even when σ u = λ the asymptotic distribution given in (54) is not centered about zero, so an asymptotic bias is still present. The reason for this is that there generally exist a strong (negative) correlation between W () and ( [W (r) ]dr), resulting from the fact that W () and W (r) are highly correlated for each r. Thus, even though E(W () σu/λ ) = with σu = λ, we do not have E [ /([W () ] σ u /λ ) [W (r)] dr ] =. See Abadir (995) for further details Estimation of λ and σ u The limiting distribution given in Theorem 8 depend on unknown parameters σ u and λ. Theses distributions are therefore not directly useable for statistical testing. However, both theses parameters may be consistently estimated and the estimates may be used to construct modified statistics whose limiting distribution are independent of (λ, σ u). As we shall see, these new statistics provide very general tests for the presence of a unit root in (47). As shown in Lemma (d), T T u t σ u. This provides us with the simple estimator s u = T (Y t Y t ) = T u t, R 8 by Prof. Chingnun Lee 34 Ins.of Economics,NSYSU,Taiwan

35 Ch. 3 REGRESSION WITH A UNIT ROOT which is consistent for σu under the null hypothesis β =. Since β by Theorem 8 we may also use s u = T T (Y t βy t ) as a consistent estimator of σu. Consistent estimation of λ = lim T T E( T u t) is more difficult. We start by defining ( λ T = T E u t ) T = T E(u t ) + T and by introducing the approximate τ= t=τ+ E(u t u t τ ) λ T l = T l E(u t ) + T E(u t u t τ ). τ= t=τ+ We shall call l the lag truncation number. For large T and large l < T, λ T l may be expected to very close to λ T if the total contribution in λ T of covariance such as E(u t u t τ ) with long lags τ > l is small. This will be true if u t satisfies the assumption in Theorem 7. Formally, we have the following lemma. Lemma 3. If the sequence u t satisfies the assumption in Theorem 7 and if l as T, then λ T λ T l as T. This lemma suggests that under suitable conditions on the rate at which l as T we may proceed to estimate λ from finite sample of data by sequentially estimating λ T l. We define s T l = T l u t + T u t u t τ. (-56) τ= t=τ+ The following result establishes that s T l is a consistent estimator of λ. Theorem 9. If u t satisfies all the assumptions in Theorem 7 except that part (b) is replaced by the stronger moment condition: sup t E u t γ <, for some γ >, when l as T such that l = o(t /4 ), then s T l λ. R 8 by Prof. Chingnun Lee 35 Ins.of Economics,NSYSU,Taiwan