Unit Roots in White Noise?!

Transcription

1 Unit Roots in White Noise?! A.Onatski and H. Uhlig September 26, 2008 Abstract We show that the empirical distribution of the roots of the vector auto-regression of order n fitted to T observations of a general stationary or non-stationary process, converges to the uniform distribution over the unit circle on the complex plane, when both T and n tend to infinity so that (ln T ) /n 0 and n 3 /T 0. In particular, even if the process is a white noise, the roots of the estimated vector auto-regression will converge by absolute value to unity. Researchers are often inclined to interpret the presence of an estimated root with a near-unit absolute value as evidence for nonstationarity in the data. Should they? Granger and Jeon (2006) have found that the roots of auto-regressions fittedtousmacroeconomicserieswhenplottedonthe complex plane lie in an indistinct milky-way band or halo, with modulus around 0.8. They speculate that such a strange pattern is due to the overfitting and suggest a heuristic partial explanation of the phenomenon.

2 In this paper, we shed light on these issues. We study the roots of the characteristic polynomials of VAR fitted either to stationary or to non-stationary data. We show that the empirical distribution of the roots converges to the uniform distribution over the unit circle when both the sample size T and the order n of the fitted VAR tend to infinity so that (ln T ) /n 0 and n 3 /T 0. This convergence is independent from the covariance structure of the process approximated by VAR. In particular, even if the process is a white noise, the roots of the estimated vector auto-regression will converge by absolute value to unity. We consider r-dimensional processes y t = (yt,y 0 2t) 0 0 such that its r - dimensional component y t and r 2 -dimensional component y 2t satisfy: y t = C y 2t + u t, y 2t = u 2t, () where u t =(u 0 t,u 0 2t) 0 has a VAR( ) representation: u t + H u t + H 2 u t = η t. (2) Here..., η,η 0,η,... ª is a sequence of i.i.d. random r vectors with mean Eη t =0, positive definite covariance matrix and finite fourth moments. We assume that the r r coefficient matrices H j are such that P j= j kh jk <, where kh j k is defined as p tr H j Hj 0, and H(z) I r +H z +H 2 z satisfies det H (z) 6= 0for z. Note that the above DGP spans a wide range of 2

3 processes from stationary invertible ARMA, when the dimensionality of y 2t is zero, to general cointegrated processes. Let Â,...,Ân be the OLS estimates of the coefficient matrices of a vector auto-regression of n-th order fitted to T observations of y t. Consider the estimated characteristic polynomial ˆPn,T (z) =det I r z n P n j= Âjzn j. Letusdenotethenumberoftherootsof ˆP n,t (z) that belong to a subset Ω of the complex plane as N n,t (Ω). For any 0 <δ< and 0 θ<ϕ 2π, let C δ = {z C : δ< z < +δ} be an annulus in the complex plane thatcontainstheunitcircleandletd θ,ϕ = {z C : θ Arg(z) ϕ} be a sector in the complex plane. Our result is as follows. Theorem. Let {y t } satisfy (), and assume that n is chosen as a function of T so that n 3 /T 0, (ln T ) /n 0, and T (kh n k + kh n+ k +...) 0 as T. Then, for any 0 <δ< and any 0 θ<ϕ 2π, as T : i) N nr n,t (D θ,ϕ ) p ϕ θ, 2π ii) N nr n,t (C δ ) p. Figure illustrates the result. It shows the roots of ˆP n,t (z) for T = 00,n =2(00 MC replications) and for T =000,n =48(33 MC replications). The upper panel of the Figure corresponds to y t which is a univariate white noise, the lower panel of the Figure corresponds to y t which is a univariate random walk. As T and n become larger, the roots stick to the unit circle in a uniform way for both the white noise and the random walk. Note that ˆP n,t (z) can be interpreted as a polynomial with random co- 3

4 y t =η t, T=00,n=2 y t =η t, T=000,n= y t =y t +η t, T=00,n=2 y t =y t +η t, T=000,n= Figure : Characteristic roots of VAR(n) fitted to T observations of different DGPs. Left panel: 00 MC replications, right panel: 33 MC replications. efficients. Shparo and Schur (962) prove an equivalent of Theorem for polynomials with i.i.d. coefficients under very general assumptions. For a beautiful geometric discussion of the properties of the roots of random polynomials which provides a piece of intuition for the Shparo and Schur s result see Edelman and Kostlan (995). The contribution of this paper is to extend Shparo and Schur (962) to ˆP n,t (z) whose coefficients are functions of OLS estimates of the auto-regressive parameters, and therefore not i.i.d. Let us now prove Theorem. First, we will describe useful asymptotic properties of Â,...,Ân. As shown, for example, in Saikkonen and Lütkepohl (996), y t has the following VAR representation: y t = A y t +...+A n y t n +e t, 4

5 where e t = R η t P j=n H ju t j, A = R (Q H ) R, A j = R ( H j + H j Q) R for j =2, 3,..., n, (3) A n = RH n QR, and R I r C 0 I r2,q I r2. We have the following: Lemma. Under the conditions of Theorem, we have: i)  A p = n Op T, where  [Â,...,  n ] and A [A,...,A n ], T ii) Pr σ r Ân A n >δ T for any sequence δ T such that δ T 0 as T. Here σ r (M) denotes the r-th singular value of a matrix M, that is the square root of the r-th largest eigenvalue of MM 0. A proof of Lemma is available from us upon request. It uses the same techniques as proofs in Saikkonen and Lütkepohl (996). For stationary DGP, the lemma follows from the proof of Theorem and from Theorem 4 of Lewis and Reinsel (985). Now we are ready to prove statement i) of Theorem. Our main technical apparatus is the following lemma: Lemma 2. (Erdös and Turan, 950) Let a k,k=0,,..., rn, be arbitrary complex numbers not all of which are equal to zero, and let N (θ, ϕ) denote the number of zeros of F rn (z) = P rn k=0 a kz k that lie in the sector 0 θ 5

6 h arg z ϕ. Then, for a 0 a rn 6=0: N (θ, ϕ) (ϕ θ)rn rn i /2 2π < 6 rn ln k=0 a k a 0 a rn. /2 Taking F rn (z) P rn k=0 a kz k =det z n I r P n j= Âjzn j, we have: a 0 a rn = T T det Ân. Note that det  n /r σr Ân A n T ka n k. The second term in the latter difference converges to zero by the assumption that T (kh n k + kh n+ k +...) 0. The first term satisfies Lemma ii) with, say, δ T = n /2 + T ka n k.therefore, Pr a r/2 0 a rn > (nt ). (4) By definition of the determinant, F rn (z) = P τ ( ) τ P τ() (z)...p rτ(r) (z), where the summation is over all permutations of, 2,..., r and P ij (z) z n Â,ijz n... Ân,ij. P Such a representation implies that nr a k Ã! k=0 P rq P + n Âj,iτ(i) P à rq +! n  A np + ka j k, where the τ i= j= τ i= j= latter inequality uses the fact that for any vector v =(v,..., v n ), P n j= v j P n kvk. But formulas (3) and the assumption that j= j kh jk < imply that P j= ka jk <, and by Lemma i) n  A = op (). Therefore, µ nr P there exists a constant M such that Pr a k M. Combining k=0 r/4 rn the latter convergence with (4), we obtain: Pr k=0 a k <M(nT ). a 0 a rn /2 This fact and Lemma 2 imply that q ln T +ln n Pr < 6 + which proves statement i) of N(θ,ϕ) rn (ϕ θ) 2π ln M rn Theorem because ln T/n 0 by assumption. 4n Turningtotheproofofstatementii),define Z =[z I r,z 2 I r,..., z n I r ] 0. 6

7 Then ˆP n,t (z) =z rn det I A)Z r AZ (Â, and therefore ˆP n,t (z) /r. Using (3), we get: I r AZ =(I r z n σ r (I r AZ) σ ( A)Z Qz ) I r + P j n j= H jz. Note that the second term in the latter product converges to H(z ) uniformly outside the unit circle. Since det H (z) 6= 0for z and since σ r (I r Qz ) z, there exists a positive constant c such that for any z > +δ and large enough T, σ r (I r AZ) >c.further, A)Z q σ (  A σ (Z) =  A r z 2n  A p r = z 2 2δ o p () uniformly over z > +δ. Summing up, min z >+δ ˆP n,t (z) /r min z >+δ z n (c o p ()) > 0 with probability arbitrarily close to one for large enough T. Hence, for any δ>0: Pr (N n,t (B +δ )=rn), (5) where B +δ is the ball of radius +δ in the complex plane. It remains to be shown that nr N n,t (B δ ) p 0, or, in other words, that for any ε>0, Pr N nr n,t (B δ ) <ε as T. Let us fix anε>0 and let τ>0be such that ln ( + τ) / ln( δ) =ε/2. (6) Let z,..., z rn be the roots of ˆP n,t (z) so that ˆP n,t (z) = Q rn i= (z z i). Note that det Ân equals ( ) rn Q rn i= z Q i, and therefore, det Ân = rn i= z i. Replacing δ by τ in (5), we see that all of z i are no larger than +τ with 7

8 probability arbitrarily close to one for large enough T. Furthermore, by definition, there are N n,t (B δ ) of z i which are less than or equal δ. Thus, T Pr det rn  n <T r/2 ( δ) N n,t (B δ ) ( + τ). Using this convergence and (4), we have: Pr n rn r/2 <T r/2 ( δ) N n,t (B δ ) ( + τ). Taking logarithms of the both sides of the latter inequality, rearranging and recalling (6), we get: Pr N nr n,t (B δ ) < ε + ln T +ln n which implies 2 2n ln( δ) that Pr N nr n,t (B δ ) <ε. Q.E.D. In conclusion, we would like to point out that the striking ubiquity of unit roots established by Theorem does not have negative implications for the econometric procedures not directly based on the estimated roots. For example, univariate stationary processes that satisfy the conditions of Theorem would satisfy Berk s (974) conditions for the consistency and asymptotic normality of the auto-regressive spectral estimates. For another example, the critical coefficient in the long augmented Dickey-Fuller regression would not behave peculiarly because it is related to the characteristic roots of the regression only through their sum. What Theorem does imply, is that inference regarding the presence of unit roots and nonstationarity by looking at the largest roots in an autoregression can be highly misleading: with enough lags, one is bound to detect many roots near unity, even if the data is white noise. 8

9 References [] Berk, K. N. (974) Consistent autoregressive spectral estimates, Annals of Statistics 2, No3, [2] Edelman, A. and Kostlan, E. (995) How many zeros of a random polynomial are real?, Bulletin of the American Mathematical Society 32, pp [3] Erdös, P. and Turan, P. (950) On the distribution of roots of polynomials, Annals of Mathematics 5, [4] Granger, C.W.J. and Y. Jeon (2006) Dynamics of Model Overfitting Measured in Terms of Autoregressive Roots, Journal of Time Series Analysis 27, [5] Lewis, R. and G.C. Reinsel (985) Prediction of Multivariate Time Series by Autoregressive Model Fitting, Journal of Multivariate Analysis 6, [6] Saikkonen, P., and H. Lütkepohl (996) Infinite-Order Cointegrated Vector Autoregressive Processes: Estimation and Inference, Econometric Theory 2, [7] Shparo, D.I. and M.G. Schur (962) On the Distribution of Roots of Random Polynomials, Vestnik Moskovskogo Universiteta, Series : Mathematics and Mechanics, no.3, pp