On the Bickel-Rosenblatt test of goodness-of-fit for the residuals of autoregressive processes

We investigate in this paper a Bickel-Rosenblatt test of goodness-of-fit for the density of the noise in an autoregressive model. Since the seminal work of Bickel and Rosenblatt, it is well-known that the integrated squared error of the Parzen-Rosenblatt density estimator, once correctly renormalized, is asymptotically Gaussian for independent and identically distributed (i.i.d.) sequences. We show that the result still holds when the statistic is built from the residuals of general stable and explosive autoregressive processes. In the univariate unstable case, we prove that the result holds when the unit root is located at $-1$ whereas we give further results when the unit root is located at $1$. In particular, we establish that except for some particular asymmetric kernels leading to a non-Gaussian limiting distribution and a slower convergence, the statistic has the same order of magnitude. We also study some common unstable cases, like the integrated seasonal process. Finally we build a goodness-of-fit Bickel-Rosenblatt test for the true density of the noise together with its empirical properties on the basis of a simulation study.


Introduction and Motivations
For i.i.d. sequences of random variables, there is a wide range of goodness-of-fit statistical procedures in connection with the underlying true distribution. Among many others, one can think about the Kolmogorov-Smirnov test, the Cramér-von Mises criterion, the Pearson's chi-squared test, or more specific ones like the whole class of normality tests. Most of them have become of frequent practical use and directly implemented on the set of residuals of regression models. For such applications the independence hypothesis is irrelevant, especially for time series where lagged dependent variables are included. Thus, the crucial issue that naturally arises consists in having an overview of their sensitivity facing some weakened assumptions. This paper focus on such a generalization for the Bickel-Rosenblatt statistic, introduced by the eponymous statisticians [3] in 1973, who first established its asymptotic normality and gave their names to the associated testing procedure. The statistic is closely related to the L 2 distance on the real line between the Parzen-Rosenblatt kernel density estimator and a parametric distribution (or a smoothed version of it).
Namely it takes the form of R f n (x) − f (x) 2 a(x) dx with notation that we will detail in the sequel. Some improvements are of interest for us. First, Takahata and Yoshihara [20] in 1987 and later Neumann and Paparoditis [14] in 2000 extended the result to weakly dependent sequences (with mixing or absolute regularity conditions). As they noticed, these assumptions are satisfied by several processes of the time series literature. Then, Lee and Na [12] showed in 2002 that it also holds for the residuals of an autoregressive process of order 1 as soon as it contains no unit root (we will explain precisely this fact in the paper). Such a study leads to a goodness-of-fit test for the distribution of the innovations of the process. Bachmann and Dette [1] went further in 2005 by putting the results obtained by Lee and Na into practice. Their study also enables to get an asymptotic normality of the correctly renormalised statistic under some fixed alternatives.
The main purpose of this paper is the generalization of the results of Lee and Na to autoregressive processes of order p (p 1) besides refining the set of hypotheses, to discuss on the effect of unit roots on the statistic and to derive a goodness-of-fit test in the same way. On finite samples, it has been observed that the Gaussian behavior is difficult to reach and that, instead, an asymmetry occurs for dependent frameworks (see, e.g., Valeinis and Locmelis [21]). In the simulation study, we will use the configurations suggested in this previous paper, and the ones of Fan [9] and Ghosh and Huang [10] to try to minimize this effect. In the end of this section, we introduce the context of our study and present both notation and vocabulary used in the sequel. Moreover, we recall the well-known asymptotic behavior of the Bickel-Rosenblatt statistic for i.i.d. random variables. In Section 2, we give an overview of the existing results about the least-squares estimation of the autoregressive parameter, depending on the roots of its characteristic polynomial, since it is of crucial interest for our reasonings. Section 3 is dedicated to our results that are proved in the Appendix. To sum up, we establish the asymptotic behavior of the Bickel-Rosenblatt statistic applied to the residuals of stable and explosive autoregressive processes of order p (p 1), together with some results related to common unstable autoregressions (like the random walks and the seasonally integrated processes). We also suggest some considerations to deal with general unstable processes or mixed processes (for example, the unstable ARIMA(p − 1,1,0) process would deserve a particuliar attention due to its widespread use in the econometric field). In Section 4, we build a goodness-of-fit test in keeping with our context and discuss on some empirical bases.
To start with, let us consider an autoregressive process of order p (AR(p)) defined by for any t 1 or equivalently, in a compact form, by where θ = (θ 1 , . . . , θ p ) T is a vector parameter, Φ 0 is an arbitrary initial random vector, Φ t = (X t , . . . , X t−p+1 ) T and (ε t ) is a strong white noise having a finite positive variance σ 2 and a marginal density f (positive on the real line). The corresponding characteristic polynomial is defined, for all z ∈ C, by and the companion matrix associated with Θ (see, e.g., [8,Sec. 4.1.2]) is given by It follows that the process may also be written as It is well-known that the stability of this p-dimensional process is closely related to the eigenvalues of the companion matrix that we will denote and arrange like In particular, according to [8,Def. 2.3.17], the process is said to be stable when |λ 1 | < 1, purely explosive when |λ p | > 1 and purely unstable when |λ 1 | = |λ p | = 1. Among the purely unstable processes of interest, let us mention the seasonal model admitting the complex s-th roots of unity as solutions of its autoregressive polynomial Θ(z) = 1 − z s for a season s ∈ N\{0, 1}. In the paper, this model will be shortened as seasonal unstable of order p = s, it satisfies θ 1 = . . . = θ s−1 = 0 and θ s = 1. In a general way, it is easy to see that det(C θ ) = (−1) p+1 θ p so that C θ is invertible as soon as θ p = 0 (which will be one of our hypotheses when p > 0). In addition, a simple calculation shows that which implies (since Θ(0) = 0) that each zero of Θ is the inverse of an eigenvalue of C θ . Consequently, the stability of the process may be expressed in the paper through the eigenvalues of C θ as well as through the zeroes of Θ. We will also consider that is the minimal polynomial of C θ (which, in the terminology of [8], means that the process is regular ). Now, assume that X −p+1 , . . . , X 0 , X 1 , . . . , X n are observable (for n ≫ p) and let be the least-squares estimator of θ (for p > 0). The associated residual process is for all 1 t n, or simply ε t = X t when p = 0. Hereafter, K is a kernel and (h n ) is a bandwidth. That is, K is a non-negative function satisfying and (h n ) is a positive sequence decreasing to 0. The so-called Parzen-Rosenblatt estimator [15,17] of the density f is given, for all x ∈ R, by The local behavior of this empirical density has been well studied in the literature. However, for a goodness-of-fit test, we focus on the global fitness of f n to f on the whole real line. From this viewpoint, we consider the Bickel-Rosenblatt statistic that we define as a is a positive piecewise continuous integrable function and * denotes the convolution operator, i.e. (g * h)(x) = R g(x − u) h(u) du. A statistic of probably greater interest and easier to implement is Bickel and Rosenblatt show in [3] that under appropriate conditions, if T n is the statistic given in (1.8) built on the strong white noise (ε t ) instead of the residuals, then, as n tends to infinity, where the centering term is and the asymptotic variance is given by The aforementioned conditions are summarized in [10, Sec. 2], they come from the original work of Bickel and Rosenblatt later improved by Rosenblatt [18]. In addition to some technical assumptions that will be recalled in (A 0 ), we just notice that a choice of a continuous and positive kernel defined on R and a bandwidth h n = h 0 n −κ must coincide with 0 < κ < 1/4, |K ′ (x)| |x| 3 ln ln |x| dx < +∞ and If K is chosen to be bounded on a compact support, then 0 < κ < 1 in the bandwidth is a sufficient condition. We can find in later references (like [20], [14] or [1]) some alternative proofs of the asymptotic normality (1.10) with a(x) = 1 and appropriate assumptions. However in this paper we will keep a as an integrable function, so as to make the calculations easier and to follow the original framework of Bickel and Rosenblatt.
Remark 1.1. It is convenient to have short expressions for terms that converge in probability to zero. In the whole study, the notation o P (1) (resp. O P (1)) stands for a sequence of random variables that converges to zero in probability (resp. is bounded in probability) as n → ∞.

Preliminary results
We start by giving some (already known) preliminary results related to the behavior of (X t ), depending on the eigenvalues of C θ and, in each case, the asymptotic behavior of the least-squares estimator. In the sequel, we assume that Φ 0 shares the same assumptions of moments as (ε t ). We only focus on results that will be useful for our reasonings.

Proposition 2.3 (Purely unstable case).
Assume that (ε t ) is a strong white noise having a finite variance σ 2 . If (X t ) satisfies (1.1)-(1.4) with p = 1 and is unstable (that is |λ| = 1 or, equivalently, Θ(z) = 0 for all |z| = 1), then for k ∈ N, In addition, The same rates are reached in the seasonal case for p = s.
These results are detailed in Lemma A.4 and then proved.
Proposition 2.6 (Purely unstable case). In the purely unstable case (|λ 1 | = |λ p | = 1), the least-squares estimator θ n of θ is strongly consistent. In the univariate case (p = 1), we have in addition is a standard Wiener process and sgn(θ) stands for the sign of θ.
In the seasonal case (p = s), since Θ(z) = 0 is an equation admitting the complex s-th roots of unity as solutions, where S(W s ) is a functional of a standard Wiener process (W s (t), t ∈ [0, 1]) of dimension s that can be explicitly built following the given reference.

The Bickel-Rosenblatt statistic
In this section, we derive the limiting distribution of the test statistics T n and T n given by (1.8) and (1.9), based on the residuals in the stable, some unstable and purely explosive cases. For the whole study, we make the following assumptions. (A 0 ) The strong white noise (ε t ) has a bounded density f which is positive, twice differentiable, and the second derivative f ′′ is itself bounded. The weighting function a is positive, piecewise continuous and integrable. The kernel K is bounded, continuous on its support. The kernel K and the bandwidth (h n ) are chosen to satisfy the hypotheses of Bickel and Rosenblatt summarized at the end of Section 1. Some additional hypotheses are given below, not simultaneously needed. (A 1 ) The kernel K is such that K ′′′ exists and is bounded on the real line, (A 4 ) The noise (ε t ) and the initial vector Φ 0 have a finite moment of order ν.
Remark 3.1. To be rigorous in the technical tools used in the proofs, we have in addition to consider that any kernel with compact support must be approximated arbitrarily well by kernels satisfying the above hypotheses, around the discontinuities. Lee and Na [12] also made this observation in their simulation study.
Then, as n tends to infinity, where µ and τ 2 are given in (1.11) and (1.12), respectively. In addition, the result is still valid for T n if (A 3 ) with β = 9/2 holds.
Remark 3.5. Theorem 3.4 holds for h n = h 0 n −κ as soon as 2/9 < κ < 1 if K has a compact support, the usual bandwidth h n = h 0 n −1/4 is appropriate. For a kernel positive on R, we must restrict to 2/9 < κ < 1/4 and a standard choice may be h n = h 0 n −1/4+ǫ for a small ǫ > 0.
We now investigate the asymptotic behavior of the statistics for some unstable cases. We establish in particular that the analysis of [12] is only partially true.
where µ and τ 2 are given in (1.11) and (1.12), respectively. If λ = 1 and R K ′ (s) ds = 0 then The results are still valid for T n if (A 3 ) with β = 9/2 holds. Finally, if λ = 1 and R K ′ (s) ds = 0 then, as n tends to infinity, is a standard Wiener process. In the seasonal case with p = s, we also reach depending on whether R K ′ (s) ds = 0 or R K ′ (s) ds = 0, respectively, where S(W s ) is defined in Proposition 2.6, H(W s ) will be clarified in the proof and (W s (t), t ∈ [0, 1]) is a standard Wiener process of dimension s.
Remark 3.7. This last result needs some observations.
A standard choice may be h n = h 0 n −1/4+ǫ for a small ǫ > 0. (2) One can observe that the value of R K ′ (s) ds is crucial to deal with the unstable case. In fact, all usual kernels are even, leading to R K ′ (s) ds = 0. It seems that the unstable case always holds in applications (except for some very particular asymmetric kernels having different bound values). (3) At the end of the associated proof, we show that, for λ = 1 and a choice of kernel such that R K ′ (s) ds = 0, we also need R s K(s) ds = 0 to translate the result from T n to T n under (A 3 ) with β = 6. In this case, the result holds for h n = h 0 n −κ as soon as 1/6 < κ < 1/4. (4) In [13, Thm. 3.2], Lee and Wei had already noticed that only the unit roots located at 1 affect the residual process. Our result is coherent from that point of view, and the link between both studies lies in the fact that where F is the cumulative distribution function of the noise and where, for all 0 u 1, To conclude this part, we draw the reader's attention to the fact that the purely unstable case is not fully treated. The general results may be a challenging study due to the phenomenon of compensation arising through unit roots different from 1. Lemma A.8 at the end of the Appendix is not used as part of this paper but may be a trail for future studies. It illustrates the compensation via the fact that ( t k=1 X k ) is of the same order as (X t ) in a purely unstable process having no unit root located at 1. Mixed models (|λ 1 | 1 and |λ p | 1) should lead to similar reasonings, they also have to be handled to definitively lift the veil on the Bickel-Rosenblatt statistic for the residuals of autoregressive processes. As a priority, it seems that unstable ARIMA(p − 1,1,0) processes would deserve close investigations due to their widespread use in the econometric field. The difficulty arising here is that the estimator converges at the slow rate of stability while the process grows at the fast rate of instability: a compensation will be needed.

A goodness-of-fit Bickel-Rosenblatt test
Our objective is now to derive a goodness-of-fit testing procedure from the results established in the previous section. First, one can notice that the choice of for any δ > 0 leads to the simplifications Bickel and Rosenblatt [3] suggest a similar choice for the weight function a, with [0 ; 1] for compact support. Nevertheless, it seems more reasonable to work on a symmetric interval in order to test for the density of a random noise. In addition, µ and τ 2 become independent of f , which will be useful to build a statistical procedure based on f . For a couple of densities f and f 0 such that f 0 does not cancel on [−δ ; δ], let us define Hence, ∆ δ (f, f 0 ) = 0 means that f and f 0 coincide almost everywhere on [−δ ; δ], and everywhere under our usual continuity hypotheses on the densities. On the contrary, ∆ δ (f, f 0 ) > 0 means that there exists an interval I ⊆ [−δ ; δ] with nonempty interior on which f and f 0 differ. Accordingly, let The natural test statistic is therefore given by where T 0 n is the statistic T n reminded above built using f 0 instead of f . Proof. The proof is immediate using our previous results. The consistency under H 1 is reached using the fact that For any level 0 < α < 1, we reject H 0 as soon as Z 0 n > u 1−α where u 1−α stands for the (1 − α)-quantile of the N (0, 1) distribution. For our simulations, we focus on a normality test (probably the most useful in regression, for goodness-of-fit). We have trusted the observations of [10], [9] or [21]. In particular, only the N (0, 1) and the U([−1 ; 1]) kernels are used (in fact a smoothed version of the latter, to satisfy the hypotheses), with obviously R K ′ (s) ds = 0. The bandwidth is h n = h 0 n −1/4+ǫ for a small ǫ > 0, where h 0 is calibrated to reach an empirical level close to α = 5% for the neutral model (p = 0), and the true distribution of the noise is N (0, 1). We only give an overview of the results in Table 1 for some typical models: •  Table 1. Empirical level of the test under H 0 , for the configurations described above. We used δ = 2, n ∈ {50, 100, 500} and 1000 replications. ( * ) Simulations that needed more than one numerical trial, due to the explosive nature of the process and the large value of n.
For model M 4 , it is important to note that Proposition 4.1 may not hold. Nevertheless, we know by virtue of Proposition 3.6 that the statistic has the same order of magnitude, thus it seemed interesting to look at its empirical behavior in comparison with the other models (and we observe that it reacts as well). We now turn to the empirical power of the test, for the configuration n = 100 and the N (0, 1) kernel. In Figures 1-2 below, we represent the percentage of rejection of H 0 against the N (m, 1) and N (0, σ 2 ) alternatives, for different values of the parameters, to investigate the sensitivity towards location and scale. We also make experiments in Figure 3 with different distributions as alternatives (Student, uniform, Laplace and Cauchy). First of all, the main observation is that all models give very similar results (all curves are almost superimposed) even if n is not so large. That corroborates the results of the paper: residuals from stable, explosive or some (univariate) unstable models satisfy the Bickel-Rosenblatt original convergence. Our procedure is roughly equivalent to the Kolmogorov-Smirnov one to test for location or scale in the Gaussian family (in fact it seems to be slightly less powerful for location and slightly more powerful for scale). However, it appears that our procedure better managed to discriminate some alternatives with close but different distributions. The objective of the paper is mainly theoretical and of course, a much more extensive study is needed to give any permanent conclusion about the comparison (role of n, h 0 , κ, K, δ, the true distribution of (ε t ), etc.).

Remark 4.2.
In the unstable case with p = 1 and a positive unit root (namely, the random walk), and a kernel satisfying R K ′ (s) ds = 0, even if it is of lesser statistical interest, it is also possible to exploit Proposition 3.6 to derive a statistical procedure. Indeed, let with an adjustment of σ 2 0 if f 0 is not centered. Then, we can choose and compare it with the quantiles associated with the distribution of Acknowledgments. The authors warmly thank Bernard Bercu for all his advices and suggestions during the preparation of this work.

Appendix. Proofs of the main results
In this section, we prove our results.
Using the mean value theorem, we can write for all 1 t n, for some 0 < ζ < 1. Note that and obviously that On the one hand, we consider the first term (say, I n ) of the right-hand side of (A.3) that can be bounded like At this step, we need two technical lemmas.
Lemma A.1. We have Proof. One clearly has Now let us consider J 1,n and J 2,n . First, But, under our assumptions, we recall that E[X 2 n ] = O(1) from the asymptotic stationarity of the process and for some random variable Z and by Assumptions (A 0 ) and (A 1 ). Thus J 1,n = O P (h n ) via Proposition 2.4. Then, by a direct calculation, Using Proposition 2.1, it follows that J 2,n = O P (h 2 n ).
Lemma A.2. We have Proof. We directly get From the mean value theorem, under our hypotheses, h n for some 0 < ζ, ξ < 1. We deduce that Consequently, since K ′′′ is bounded, for some arbitrary constants. Then, K 1,n = O P (n h −2 n ) as soon as we suppose that (ε t ) has a finite moment of order ν = 4 (by virtue of Proposition 2.1). Now we proceed as for J 1,n to get which shows K 2,n = O P (n h n ), since E X 4 n = O(1) under the hypotheses of stability and fourth-order moments. Finally, for a constant C. Whence we deduce that I 2,n = O P (h 2 n + n −1 h −2 n ). We are now ready to conclude the proof of Theorem 3.
which ends the first part of the proof. The second part makes use of a result of Bickel and Rosenblatt [3]. Indeed, denote byT n the statistic given in (1.9) built on the strong white noise (ε t ) instead of the residuals. They show that as soon as n h 9/2 n → 0. A similar calculation leads to T n = I n +T n +R n whereR n = 2 n h n We deduce that as soon asT n satisfies the original Bickel-Rosenblatt convergence and (A.12) and (A.13) hold, that is n h 9/2 n → 0 and n h 4 n → +∞. It only remains to note that each term being o P (1).
A.2. Proof of Theorem 3.4. Let I n be the first term of the right-hand side of (A.3), like in the last proof, and note that where for an arbitrary rate that we set to v 2 n = ln(n h n ), Under our hypotheses, this choice of (v n ) ensures Let us look at I n,1 . By Cauchy-Schwarz, Here we recall that |λ p | > 1. Consequently, ρ(C −1 θ ) = 1/|λ p | < 1. It follows that there exists a matrix norm · * = sup(| · u| * ; u ∈ C p , |u| * = 1) satisfying C −1 θ * < 1 (see e.g. [8,Prop. 2.3.15]), and a constant k * such that, with C * = 2 B k * a ∞ K ∞ , we obtain But Proposition 2.2 ensures that sup t |C −t θ Φ t | * is a.s. bounded for a sufficiently large n, and we also have We deduce that, for some constant k * , where [8,Cor. 1.3.21] shows that ε ♯ n = sup t |ε t | = o( √ n) a.s. under our conditions of moments on (ε t ). Proposition 2.2 and the fact that C −1 θ * < 1 lead to (A.17) a.s.
for some 0 < R < 1. Let us now turn to I n,2 for which the same strategy gives for some 0 < R < 1, as a consequence of the properties of (v n ) in (A.16). The cross term R n is treated in the same way as in the proof of Theorem 3.2. Indeed, since our choice of (v n ) also ensures that I n = o(h n ) a.s., the same reasoning leads to the conclusion. It follows from (A.15) and (A.19) that which ends the first part of the proof. The second part merely consists in noting that (A.14) still holds.
A.3. Proof of Proposition 3.6. In this proof, the notation =⇒ refers to the weak convergence of sequences of random elements in D([0, 1]), the space of right continuous functions on [0, 1] having left-hand limits, equipped with the Skorokhod topology (see Billingsley [4]). One can find in Thm. 2.7 of the same reference the statement and the proof of the continous mapping theorem. In Lemmas A.4 and A.8 below, (Y n ) n 1 is a sequence of random variables satisfying, for some δ > 0, Y t =⇒ L P (·) and 1 n δ where L P and Λ are random paths in D( . Let X 0 = 0 and, for 1 t n, consider where P n ⊂ {1, . . . , n} is the set of even integers (resp. odd integers) between 1 and n, for an even (resp. odd) n. Then we conclude to the first result. Finally, We now turn to the proof of Proposition 3.6. Following the idea of [12], we make once again the decomposition where µ is the centering term (1.11) of the statistic, I n is given in (A.4), T n is the original Bickel-Rosenblatt statistic and R n is the cross term in (A.3) such that as we have seen earlier. Going on with the decomposition of I n in (A.4) and using the same notation, we establish the two following lemmas. In all the sequel, the case p = s refers to the seasonal process θ 1 = . . . = θ s−1 = 0 and θ s = 1.
Lemma A.5. When p = 1 or p = s, Lemma A.2 still holds in the unstable case.
Proof. First, with p = 1 and θ = ±1, we note that under our assumptions on the noise, and the same results obviously hold for p = s where X n still has a random walk behavior. Thus, K 2,n in (A.10) is O P (n 3 h n ). and, accordingly with Proposition 2.6, K 1,n in (A.9) is O P (n 3 h −2 n ) and K 3,n in (A.11) is O P (n 4 h 2 n ). Finally, (A.8) concludes the proof. Lemma A.6. Let p = 1. Then, Lemma A.1 holds for λ = θ = −1. On the contrary, for λ = θ = 1 and R K ′ (s) ds = 0, I 1,n = O P (n h 2 n ) whereas for R K ′ (s) ds = 0, I 1,n = O P (n h 4 n ). The last two results also hold for p = s.
Proof. On the one hand, let θ = −1. The reasoning of (A.6) shows that J 1,n is still O P (h n ), using Proposition 2.6 and E[X 2 n ] = O(n). From Lemma A.4, we know that Then, J 2,n in (A.7) must be O P (n −1 h 2 n ) and I 1,n = O P (h n ). On the other hand, we have to investigate the case where θ = 1. Since E[X 2 n ] does not depend on θ, J 1,n has exactly the same behavior. For a better readability, let Under our assumptions, it is easy to see that which shows that J 2,n is O P (n h 2 n ) or O P (n h 4 n ), depending on R K ′ (s) ds. The latter reasoning still applies for p = s, via the second part of Proposition 2.3. The proof is achieved using (A.5) and the hypotheses of Proposition 3.6.
The combination of Lemmas A.5 and A.6 is sufficient to establish that holds for θ = −1, despite the instability, and replacing T n by T n is possible without disturbing the asymptotic normality, as a consequence of (A.14) for n h which unfortunately prevents us from concluding to the Bickel-Rosenblatt convergence in this case. However, we still have the order of magnitude Finally for R K ′ (s) ds = 0, the same lines also imply, together with (A.21), Lemmas A.5 and A.6 and the asymptotic normality (1.10), that h n ( T n − µ) = O P (1).
It remains to study the asymptotic behavior of the correctly renormalized statistic using the explicit expression of J 2,n in (A.7). From the proof of Lemma A.4, Proposition 2.6 and the continuous mapping theorem, we get for the univariate unstable case with θ = 1, n ( θ n − θ) Remark A.7. To go beyond, note that for θ = 1 and R K ′ (s) ds = 0, we get h n | T n − T n | 2 h n T n U n + h n U n where U n = n h n R (K hn * f )(x) − f (x) 2 a(x) dx.
A straightforward calculation shows that U n = O(n h 3 n ) if R s K(s) ds = 0 whereas U n = O(n h 5 n ) if R s K(s) ds = 0. In the second case, T n may be replaced by T n as soon as n h 6 n → 0. But it the first case, one needs n h 4 n → 0 which contradicts Assumption (A 3 ) with α = 4. This last lemma is not used as part of this paper, but it may be a trail for future studies. It is related to the conclusion of Section 3 and illustrates the phenomenon of compensation in purely unstable processes.
Moreover, if the process is generated by then n t=1 X t = O P (n δ+1 ).
In the last case, let Z t = X t − X t−1 and note that Z t = −Z t−1 + Y t . Then, Lemma A.4 gives n t=1 Z t = O P (n δ ).
We conclude by applying again Lemma A.4 to the relation X t = X t−1 + Z t .