TESTING FOR A UNIT ROOT IN LEE–CARTER MORTALITY MODEL

2017 ◽  
Vol 47 (3) ◽  
pp. 715-735 ◽  
Author(s):  
Xuan Leng ◽  
Liang Peng
Keyword(s):  
Unit Root ◽  
Finite Sample ◽  
Asymptotic Results ◽  
Inference Procedure ◽  
Mortality Model ◽  
Mortality Index ◽  
R Packages ◽  
Mortality Projections ◽  
The U.S ◽  
Carter Model

AbstractMotivated by a recent discovery that the two-step inference for the Lee–Carter mortality model may be inconsistent when the mortality index does not follow from a nearly integrated AR(1) process, we propose a test for a unit root in a Lee–Carter model with an AR(p) process for the mortality index. Although testing for a unit root has been studied extensively in econometrics, the method and asymptotic results developed in this paper are unconventional. Unlike a blind application of existing R packages for implementing the two-step inference procedure in Lee and Carter (1992) to the U.S. mortality rate data, the proposed test rejects the null hypothesis that the mortality index follows from a unit root AR(1) process, which calls for serious attention on using the future mortality projections based on the Lee–Carter model in policy making, pricing annuities and hedging longevity risk. A simulation study is conducted to examine the finite sample behavior of the proposed test too.

BIAS-CORRECTED INFERENCE FOR A MODIFIED LEE–CARTER MORTALITY MODEL

2019 ◽  
Vol 49 (2) ◽  
pp. 433-455 ◽  
Author(s):  
Qing Liu ◽  
Chen Ling ◽  
Deyuan Li ◽  
Liang Peng
Keyword(s):  
Unit Root ◽  
Normal Limit ◽  
Mortality Rates ◽  
American Statistical Association ◽  
Mortality Model ◽  
Mortality Index ◽  
The Us ◽  
Modeling And Forecasting ◽  
Near Unit Root ◽  
Carter Model

AbstractAs a benchmark mortality model in forecasting future mortality rates and hedging longevity risk, the widely employed Lee–Carter model (Lee, R.D. and Carter, L.R. (1992) Modeling and forecasting U.S. mortality. Journal of the American Statistical Association, 87, 659–671.) suffers from a restrictive constraint on the unobserved mortality index for ensuring model’s identification and a possible inconsistent inference. Recently, a modified Lee–Carter model (Liu, Q., Ling, C. and Peng, L. (2018) Statistical inference for Lee–Carter mortality model and corresponding forecasts. North American Actuarial Journal, to appear.) removes this constraint and a simple least squares estimation is consistent with a normal limit when the mortality index follows from a unit root or near unit root AR(1) model with a nonzero intercept. This paper proposes a bias-corrected estimator for this modified Lee–Carter model, which is consistent and has a normal limit regardless of the mortality index being a stationary or near unit root or unit root AR(1) process with a nonzero intercept. Applications to the US mortality rates and a simulation study are provided as well.

THE NONSTATIONARY FRACTIONAL UNIT ROOT

1999 ◽  
Vol 15 (4) ◽  
pp. 549-582 ◽  
Author(s):  
Katsuto Tanaka
Keyword(s):  
Unit Root ◽  
Asymptotic Properties ◽  
Test Statistics ◽  
Finite Sample ◽  
Asymptotic Results ◽  
Simulation Experiments ◽  
Normality Assumption ◽  
Asymptotically Efficient ◽  
Uniformly Most Powerful ◽  
Fractional Unit

This paper deals with a scalar I(d) process {yj}, where the integration order d is any real number. Under this setting, we first explore asymptotic properties of various statistics associated with {yj}, assuming that d is known and is greater than or equal to ½. Note that {yj} becomes stationary when d < ½, whose case is not our concern here. It turns out that the case of d = ½ needs a separate treatment from d > ½. We then consider, under the normality assumption, testing and estimation for d, allowing for any value of d. The tests suggested here are asymptotically uniformly most powerful invariant, whereas the maximum likelihood estimator is asymptotically efficient. The asymptotic theory for these results will not assume normality. Unlike in the usual unit root problem based on autoregressive models, standard asymptotic results hold for test statistics and estimators, where d need not be restricted to d ≥ ½. Simulation experiments are conducted to examine the finite sample performance of both the tests and estimators.

ASYMPTOTICALLY UMP PANEL UNIT ROOT TESTS—THE EFFECT OF HETEROGENEITY IN THE ALTERNATIVES

2014 ◽  
Vol 31 (3) ◽  
pp. 539-559 ◽  
Author(s):  
I. Gaia Becheri ◽  
Feike C. Drost ◽  
Ramon van den Akker
Keyword(s):  
Unit Root ◽  
Generalized Least Squares ◽  
Applied Economics ◽  
Unit Root Tests ◽  
Finite Sample ◽  
Asymptotic Power ◽  
Optimal Tests ◽  
Independent Panel ◽  
Local Asymptotic Power ◽  
Uniformly Most Powerful

In a Gaussian, heterogeneous, cross-sectionally independent panel with incidental intercepts, Moon, Perron, and Phillips (2007, Journal of Econometrics 141, 416–459) present an asymptotic power envelope yielding an upper bound to the local asymptotic power of unit root tests. In case of homogeneous alternatives this envelope is known to be sharp, but this paper shows that it is not attainable for heterogeneous alternatives. Using limit experiment theory we derive a sharp power envelope. We also demonstrate that, among others, one of the likelihood ratio based tests in Moon et al. (2007, Journal of Econometrics 141, 416–459), a pooled generalized least squares (GLS) based test using the Breitung and Meyer (1994, Applied Economics 25, 353–361) device, and a new test based on the asymptotic structure of the model are all asymptotically UMP (Uniformly Most Powerful). Thus, perhaps somewhat surprisingly, pooled regression-based tests may yield optimal tests in case of heterogeneous alternatives. Although finite-sample powers are comparable, the new test is easy to implement and has superior size properties.

TESTING FOR SEASONAL UNIT ROOTS IN PERIODIC INTEGRATED AUTOREGRESSIVE PROCESSES

2008 ◽  
Vol 24 (4) ◽  
pp. 1093-1129 ◽  
Author(s):  
Tomas del Barrio Castro ◽  
Denise R. Osborn
Keyword(s):  
Unit Root ◽  
Unit Roots ◽  
Finite Sample ◽  
Test Statistic ◽  
Important Special Case ◽  
Frequency Unit ◽  
Annual Frequency ◽  
Fixed Order ◽  
Fuller Distribution ◽  
Zero Frequency

This paper examines the implications of applying the Hylleberg, Engle, Granger, and Yoo (1990, Journal of Econometrics 44, 215–238) (HEGY) seasonal root tests to a process that is periodically integrated. As an important special case, the random walk process is also considered, where the zero-frequency unit root t-statistic is shown to converge to the Dickey–Fuller distribution and all seasonal unit root statistics diverge. For periodically integrated processes and a sufficiently high order of augmentation, the HEGY t-statistics for unit roots at the zero and semiannual frequencies both converge to the same Dickey–Fuller distribution. Further, the HEGY joint test statistic for a unit root at the annual frequency and all joint test statistics across frequencies converge to the square of this distribution. Results are also derived for a fixed order of augmentation. Finite-sample Monte Carlo results indicate that, in practice, the zero-frequency HEGY statistic (with augmentation) captures the single unit root of the periodic integrated process, but there may be a high probability of incorrectly concluding that the process is seasonally integrated.

Near Observational Equivalence and Theoretical size Problems with Unit Root Tests

1996 ◽  
Vol 12 (4) ◽  
pp. 724-731 ◽  
Author(s):  
Jon Faust
Keyword(s):  
Unit Root ◽  
Sufficient Conditions ◽  
Unit Root Test ◽  
Asymptotic Distributions ◽  
Unit Root Tests ◽  
Finite Sample ◽  
Test Procedures ◽  
Asymptotic Size ◽  
Finite Sample Size ◽  
Topological Sense

Said and Dickey (1984,Biometrika71, 599–608) and Phillips and Perron (1988,Biometrika75, 335–346) have derived unit root tests that have asymptotic distributions free of nuisance parameters under very general maintained models. Under models as general as those assumed by these authors, the size of the unit root test procedures will converge to one, not the size under the asymptotic distribution. Solving this problem requires restricting attention to a model that is small, in a topological sense, relative to the original. Sufficient conditions for solving the asymptotic size problem yield some suggestions for improving finite-sample size performance of standard tests.

QUASI-INDIRECT INFERENCE FOR DIFFUSION PROCESSES

1998 ◽  
Vol 14 (2) ◽  
pp. 161-186 ◽  
Author(s):  
Laurence Broze ◽  
Olivier Scaillet ◽  
Jean-Michel Zakoïan
Keyword(s):  
Diffusion Processes ◽  
Asymptotic Properties ◽  
Estimation Procedure ◽  
Indirect Inference ◽  
Sampled Data ◽  
Finite Sample ◽  
Inference Procedure ◽  
Monte Carlo Experiments ◽  
Simulation Step ◽  
Finite Sample Properties

We discuss an estimation procedure for continuous-time models based on discrete sampled data with a fixed unit of time between two consecutive observations. Because in general the conditional likelihood of the model cannot be derived, an indirect inference procedure following Gouriéroux, Monfort, and Renault (1993, Journal of Applied Econometrics 8, 85–118) is developed. It is based on simulations of a discretized model. We study the asymptotic properties of this “quasi”-indirect estimator and examine some particular cases. Because this method critically depends on simulations, we pay particular attention to the appropriate choice of the simulation step. Finally, finite-sample properties are studied through Monte Carlo experiments.

scholarly journals A POWERFUL TEST OF THE AUTOREGRESSIVE UNIT ROOT HYPOTHESIS BASED ON A TUNING PARAMETER FREE STATISTIC

2009 ◽  
Vol 25 (6) ◽  
pp. 1515-1544 ◽  
Author(s):  
Morten Ørregaard Nielsen
Keyword(s):  
Asymptotic Distribution ◽  
Unit Root ◽  
Linear Time ◽  
Nonparametric Test ◽  
Tuning Parameter ◽  
Ratio Test ◽  
Local Power ◽  
Finite Sample ◽  
Power Of The Test ◽  
The Family

This paper presents a family of simple nonparametric unit root tests indexed by one parameter,d, and containing the Breitung (2002,Journal of Econometrics108, 342–363) test as the special cased= 1. It is shown that (a) each member of the family withd> 0 is consistent, (b) the asymptotic distribution depends ondand thus reflects the parameter chosen to implement the test, and (c) because the asymptotic distribution depends ondand the test remains consistent for alld> 0, it is possible to analyze the power of the test for different values ofd. The usual Phillips–Perron and Dickey–Fuller type tests are indexed by bandwidth, lag length, etc., but have none of these three properties.It is shown that members of the family withd< 1 have higher asymptotic local power than the Breitung (2002) test, and whendis small the asymptotic local power of the proposed nonparametric test is relatively close to the parametric power envelope, particularly in the case with a linear time trend. Furthermore, generalized least squares (GLS) detrending is shown to improve power whendis small, which is not the case for the Breitung (2002) test. Simulations demonstrate that when applying a sieve bootstrap procedure, the proposed variance ratio test has very good size properties, with finite-sample power that is higher than that of the Breitung (2002) test and even rivals the (nearly) optimal parametric GLS detrended augmented Dickey–Fuller test with lag length chosen by an information criterion.

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