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Least Squares Estimators for Unit Root Processes with Locally Stationary Disturbance
The random walk is used as a model expressing equitableness and the effectiveness of various finance phenomena. Random walk is included in unit root process which is a class of nonstationary processes. Due to its nonstationarity, the least squares estimator (LSE) of random walk does not satisfy asymptotic normality. However, it is well known that the sequence of partial sum processes of random walk weakly converges to standard Brownian motion. This result is so-called functional central limit theorem (FCLT). We can derive the limiting distribution of LSE of unit root process from the FCLT result. The FCLT result has been extended to unit root process with locally stationary process (LSP) innovation. This model includes different two types of nonstationarity. Since the LSP innovation has time-varying spectral structure, it is suitable for describing the empirical financial time series data. Here we will derive the limiting distributions of LSE of unit root, near unit root and general integrated processes with LSP innovation. Testing problem between unit root and near unit root will be also discussed. Furthermore, we will suggest two kind of extensions for LSE, which include various famous estimators as special cases.
Since the random walk is a martingale sequence, the best predictor of the next term becomes the value of this term. In this sense, the random walk is used as a model expressing equitableness and the effectiveness of various finance phenomena in economics. Furthermore, because the random walk is a unit root process, taking the difference of the random walk, we can recover the independent sequence. However, the information of the original sequence will be lost by taking the difference when it does not include a unit root. Therefore, the testing of the existence of unit root in the original sequence becomes important.
In this section, we review the fundamental asymptotic results for unit root processes. Let be i.i.d. random variables, where , and define the partial sum which is the so-called random walk process. Random walk corresponds to the first-order autoregressive (AR(1)) model with unit coefficient. Therefore, random walk is included in unit root (I(1)) processes which is a class of nonstationary processes. Let be the space of all real-valued continuous functions defined on . For random walk process, we construct the sequence of the processes of the partial sum in as It is well known that the partial sum process converge weakly to a standard Brownian motion on , namely, where denotes the distribution law of the corresponding random elements. This result is the so-called functional central limit theorem (FCLT) (see Billingsley ).
The FCLT result can be extended to the unit root process where the innovation is general linear process. We consider a sequence of a stochastic process in defined by where and is assumed to be generated by Here, is a sequence of i.i.d. random variables, and is a sequence of constants which satisfies ; therefore, becomes stationary process. Using the Beveridge and Nelson  decomposition, it holds (see, e.g., Tanaka )
The asymptotic property of LSE for stationary autoregressive models has been well established (see, e.g., Hannan ). On the other hand, due to its nonstationarity, the LSE of random walk does not satisfy asymptotic normality. However, we can derive the limiting distribution of LSE of unit root process from the FCLT result. For more detailed understanding about unit root process with i.i.d. or stationary innovation, refer to, for example, Billingsley  and Tanaka .
In the above case, the ’s are stationary and hence, have constant variance, while covariances depend on only time differences. This is referred to as the homogeneous case, which is too restrictive to interpret empirical data, for example, empirical financial data. Recently, an important class of nonstationary processes have been proposed by Dahlhaus (see, e.g., Dahlhaus [5, 6]), called locally stationary processes. In this paper, we alternatively adopt locally stationary innovation process, which has smoothly changing variance. Since the LSP innovation has time-varying spectral structure, it is suitable for describing the empirical financial time series data.
This paper is organized as follows. In the appendix, we review the extension of the FCLT results to the cases that the innovations are locally stationary process. Namely, we explain the FCLT for unit root, near unit root, and general integrated processes with LSP innovations. In Section 2, we obtain the asymptotic distribution of the least squares estimator for each case of the appendix. In Section 3, we also consider the testing problem for unit root with LSP innovation. Finally, in Section 4, we discuss the extensions of LSE, which include various famous estimators as special cases.
2. The Property of Least Squares Estimator
In this section, we investigate the asymptotic properties of least squares estimators for unit root, near unit root, and processes with locally stationary process innovations. Testing problem for unit root is also discussed. For the notations which are not defined in this section, refer to the appendix.
2.1. Least Squares Estimator for Unit Root Process
Here, we consider the following statistics: obtained from model (A.3), which can be regarded as the least squares estimator (LSE) of autoregressive coefficient in the first-order autoregressive (AR(1)) model . Define then we have Let us define a continuous function for , where It is easy to check Therefore, the continuous mapping theorem (CMT) leads to and
2.2. Least Squares Estimator for Near Unit Root Process
We next consider the least squares estimator for model (A.11) in the case that is a constant on , namely, with . Then, we have where Let us define a continuous function for , where It is easy to check Therefore, the CMT leads to and
2.3. Least Squares Estimator for Process
Furthermore, we consider the least squares estimator obtained from model , where Let us define a continuous function for , where It is easy to check Therefore, the CMT leads to and The equality above is due to -times differentiability of .
3. Testing for Unit Root
In the analysis of empirical financial data, the existence of the unit root is an important problem. However, as we see in the previous section, the asymptotic results between unit root and near unit root processes are quite different (the drift term appeared in the limiting process of near unit root). Therefore, we consider the following testing problem against the local alternative hypothesis: We should assume that is a unit to identify the models. Let the statistics be constructed in (2.3). Recall that, as , under , where Since are unknown, we construct a test statistic where . A nonparametric time-varying spectral density estimator is given by where , and , . Here, is the local periodogram around time given by where denotes Gauss symbol and, for real number , is the greatest integer that is less than or equal to . Furthermore, we employ the following kernel functions and the orders of bandwidth for smoothing in time and frequency domain, respectively, which are optimal in the sense that they minimize the mean squared error of nonparametric estimator (see Dahlhaus ); however, we simply multiply the orders of bandwidth by the constants equal to one. Then, it can be established that, under , We now have to deal with statistics for which numerical integration must be elaborated. Let be such a statistic, which takes the form . Using Imhof’s  formula gives us distribution function of , where is the characteristic function of , namely, However, so far we do not have the explicit form of the distribution function of the estimator. Therefore, we cannot perform a numerical experiment except for the clear simple cases. It includes the complicated problem in the differential equation and requires one further paper for solution.
4. Extensions of LSE
In this section, we consider the extensions of LSE for near random walk model , .
4.1. Ochi Estimator
Ochi  proposed the class of estimators of the following form, which are the extensions of LSE for autoregressive coefficient: where This class of estimators includes LSE , Daniels’s estimator , and Yule-Walker estimator as the special cases.
Define for , , then we see that is continuous and From the CMT, we obtain , and therefore, where
4.2. Another Extension of LSE
Next, we suggest another class of estimators which are also the extensions of LSE. Define for with continuous derivative , where If we take the taper function as , this estimator corresponds to the local LSE.
Define for , , where , then we see that is continuous and From the CMT, we obtain , and therefore, where The integration by part leads to with . Hence, using Ito’s formula, we have
In this appendix, we review the extensions of functional central limit theorem to the cases that innovations are locally stationary processes, which are used for the main results of this paper.
A. FCLT for Locally Stationary Processes
Hirukawa and Sadakata  extended the FCLT results to the unit root processes which have locally stationary process innovations. Namely, they derived the FCLT for unit root, near unit root, and general integrated processes with LSP innovations. In this section, we briefly review these results which are applied in previous sections.
A.1. Unit Root Process with Locally Stationary Disturbance
First, we introduce locally stationary process innovation. Let be generated by the following time-varying MA model: where is the lag-operator which acts as and , and time-varying MA coefficients satisfy Then, these ’s become locally stationary processes (see Dahlhaus , Hirukawa and Taniguchi ). Using this innovation process, define the partial sum as where , and is independent of .
We consider a sequence of partial sum stochastic processes in defined by Now, we define on Then, we can obtain The integration by parts leads to Note that the time-varying MA process in (A.1) has the spectral representation where is the spectral measure of i.i.d. process which satisfies , and the transfer function is given by Therefore, stochastic differential in (A.7) can be written as
A.2. Near Unit Root Process with Locally Stationary Disturbance
In this section, we consider the following near unit root process with locally stationary disturbance: where is generated from the time-varying MA model in (A.1), , , , and is independent of and . Then, we define a sequence of partial sum processes in as Define on Then, we can obtain The integration by parts and Ito’s formula lead to
A.3. Process with Locally Stationary Disturbance
Let process be generated by with , and being the time-varying MA process in (A.1). Note that the relation (A.16) can be rewritten as Then, we construct the partial sum process as for , , and , where the partial sum process is defined in (A.4). Let us first discuss weak convergence to the onefold integrated process defined by For , the partial sum process in (A.18) becomes Define on Then, we can see that
For the general integer , define the -fold integrated process by From the similar argument in the case of , we can see that the partial sum process satisfies
The authors would like to thank the referees for their many insightful comments, which improved the original version of this paper. The authors would also like to thank Professor Masanobu Taniguchi who is the lead guest editor of this special issue for his efforts and celebrate his sixtieth birthday.
- P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York, NY, USA, 1968.
- S. Beveridge and C. R. Nelson, “A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the ‘business cycle’,” Journal of Monetary Economics, vol. 7, no. 2, pp. 151–174, 1981.
- K. Tanaka, Time Series Analysis: Nonstationary and Noninvertible Distribution Theory, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, New York, NY, USA, 1996.
- E. J. Hannan, Multiple Time Series, John Wiley and Sons, London, UK, 1970.
- R. Dahlhaus, “Maximum likelihood estimation and model selection for locally stationary processes,” Journal of Nonparametric Statistics, vol. 6, no. 2-3, pp. 171–191, 1996.
- R. Dahlhaus, “Asymptotic statistical inference for nonstationary processes with evolutionary spectra,” in Athens Conference on Applied Probability and Time Series Analysis, Vol. II, vol. 115 of Lecture Notes in Statist., pp. 145–159, Springer, New York, NY, USA, 1996.
- J. P. Imhof, “Computing the distribution of quadratic forms in normal variables,” Biometrika, vol. 48, pp. 419–426, 1961.
- Y. Ochi, “Asymptotic expansions for the distribution of an estimator in the first-order autoregressive process,” Journal of Time Series Analysis, vol. 4, no. 1, pp. 57–67, 1983.
- J. Hirukawa and M. Sadakata, “Asymptotic properties of unit root processes with locally stationary disturbance,” Preprint.
- J. Hirukawa and M. Taniguchi, “LAN theorem for non-Gaussian locally stationary processes and its applications,” Journal of Statistical Planning and Inference, vol. 136, no. 3, pp. 640–688, 2006.
Copyright © 2012 Junichi Hirukawa and Mako Sadakata. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.