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Asymptotic Convergence Properties of Autoregressive Moving
Average (Arma) Model Estimators
Dayo, Kayode Vincent, Olanrewaju Samuel Olayemi, Nasiru Mukaila Olakorede
Department of Statistics, University of Abuja.
DOI: https://doi.org/10.51583/IJLTEMAS.2026.15020000096
Received: 18 February 2026; Accepted: 23 February 2026; Published: 19 March 2026
ABSTRACTS
This study investigates the empirical convergence thresholds of the Gaussian Estimation Procedure (GEP),
Generalised Least Squares (GLS), and Exact Maximum Likelihood (EML) for ARMA processes. While large-
sample asymptotic equivalence is theoretically established, the specific data requirements for numerical
reconciliation in higher-order models remain under-researched. Using a generalised Fibonacci-based sampling
recurrence
to determine the non-linear sample interval from to , estimator
stability across six distinct data-generating processes was evaluated. The findings demonstrated a ‘complexity-
dependent convergence’: while lower-order processes achieved numerical reconciliation at , higher-
order ARMA (2,2) specifications require to achieve harmonization. These results identify a critical
transition zone where estimator choice become neural, providing a structural blueprint for selection based on
modal dimensionality and available sample size.
Keywords: ARMA Models, Likelihood, Asymptotic Theory, Stationarity, Score, Approximation
INTRODUCTION
Several existing procedures exhibit distinctive asymptotic properties while estimating the parameters of time
series models. This study examines the comparative asymptotic convergence properties of alternative estimation
procedures for estimating the parameters of an Autoregressive Moving Average (ARMA) process.
, [1.1]
Where
, and
is a white noise process, typically assumed to be independent and identically
distributed (i.i.d.) with mean zero and variance
. If the backshift operator , is considered
denotes the
random variable
at lag k, such that,
, then [1.1] can compactly be written as:
[1.2]
The asymptotic properties of the parameter estimators of [1,1] are intimately tied to the concepts of stationarity
and invertibility in time series analysis. A process is strictly stationary if the joint distribution of
is
identical to that of
for all . In the context of linear ARMA models, however, weak stationarity
(or covariance stationarity) is usually sufficient, requiring that the first two moments are time-invariant. This
condition is met iff all roots of the characteristic equation
, lie outside the unit circle in the complex
plane. Where:
, [1.3]
and
. [1.4]
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Conversely, invertibility requires that the innovations,
admit a convergent linear combination of current and
past observations of
, which requires that all roots of the moving average polynomial
, lie outside
the unit circle. An ARMA process is stable if the AR polynomial satisfies
for all
; equivalently,
in state-space (companion) form, stability holds if all eigenvalues of the companion matrix lie strictly inside the
unit circle, i.e., formally, if is the companion matrix,
.
Generally, patterns in a random set of data can be modelled to draw inferences about the process or population
being observed. The science that deals with the collection and interpretation of these patterns is popularized in
time series. As conceptualised in the recent literature on stochastic processes (Montgomery et al., 2024), a time
series is not merely a data sequence but a realisation of a generating mechanism where temporary dependency
dictates the information value of each successive observation. In a discrete-time series, these observations are
recorded at a specified time interval, where , the continuous-time series, on the other hand,
observations are recorded continuously over some time interval, e.g.,
.
The analysis of random variables,
as outlined in (Kitagawa, 2020), is captured in the selection of a
suitable probability model for the data. A complete probabilistic time series model for the sequential
would
specify all the joint distributions of the random vector
, or equivalently, all the
probabilities
.
Exact Maximum Likelihood (EML) has long been the gold standard for ARMA estimation (Wei, 2006). Recent
studies by Shumway, Robert H., and Stoffer (2025) have highlighted its computational limitations in high-
dimensional settings. Consequently, there has been a resurgence in investigating Generalised Estimation
Procedure (GEP), as seen in the work of (Athanasopoulos et al., 2024). An ARMA (p, q) process is a sequence
of random variables
, capturing the present and future dynamics.
Several existing procedures exhibit distinctive asymptotic properties; recent comparative studies have focused
primarily on large-scale datasets, often overlooking the nuanced transition from small to moderate sample sizes
Chen & Yao, (2024). The focus of this research is to bridge this gap by examining the comparative asymptotic
convergence properties of GEP, GLS, and EML estimators specifically within the Fibonacci sequential
framework. By identifying the precise thresholds where these estimators achieve numerical reconciliation,
particularly in higher-order ARMA (2, 2) specifications, this study establishes a structural blueprint for estimator
selection in environments characterised by varying data availability.
The use of finitely parameterized models to characterize the behaviuors of time series data assumed to be
generated by a stochastic process has received a wide coverage in both statistical and engineering literature
(Salau, 2000). An important class of such stochastic models is the autoregressive moving average (ARMA)
models. One reason for the great empirical importance of ARMA models is that every regular stationary process
can be approximated with arbitrary accuracy by an ARMA process, and that only a finite number of (real-valued)
parameters is needed to describe (up to second moment) an ARMA process.
Arma Representation
In modelling applications, such as forecasting and control, it is commonplace to represent such a process using
a stationary and invertible autoregressive moving average (ARMA) model of the form:
[1.5]
Where
is an unobservable stochastic disturbance, assumed to be white noise or more generally, a sequence of
stationary martingale differences, so that, almost surely
,
, [1.6]
And symbol is the expectation operator. Here,
is the of events determined by the
,
and
is the prediction variance. The condition in [1.6] provides a natural relaxation of the model
assumption that the innovation sequence,
, are serially independent. For a complete exposition of these
moment assumptions, see, for example, (Box et. al., 1994), (Tsay, 2019). The essential feature of [1.6] in our
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context is that the classical theory still goes through, even though the innovation sequence,
, is no longer
serially independent and identically distributed. Thus, [1.6] is a natural condition with which the linear model
[2.4] has little meaning.
These later assumptions ensure that
is realized from a stationary and invertible ARMA (p,q) process. Thus,
[1.5] can be represented as an infinite autoregression.
,
[1.7]
Where the generating polynomial
,
,
and that
since
decreases to zero at a geometric rate. The significance of [2.5] in the analysis of linear time series has
been thoroughly discussed in the literature, see for example, (Koreisha & Pukkila, 1990). The assumption
means that in practice, the data will need to be mean-corrected; however, since that adjustment will
not affect the asymptotic results we shall obtain later, it is ignored for simplicity. The integers, p, q (or other
integers introduced to replace them), will be called the order of the system. The coefficients,
, specifying model [2.4] will be, called system parameters.
Interpreting as a unit backshift operator, that is
and
, [2.4] can be succinctly rewritten
in the form of [1.2], where
and
,
. The generating polynomial
(or z-transforms), that is,
and
defined in the convention of Box, Jenkins
& Reinsel (1994), are assumed to be relatively prime and also satisfy:
,
,
[1.8]
LITERATURE REVIEW
The estimation of Autoregressive Moving Average (ARMA) models is grounded in classical asymptotic theory.
Foundational results by Whittle, (1953) and Walker, (1964) established that, under regularity conditions,
likelihood-based and Least Squares type estimators are consistent and asymptotically normal, and converge to
the same probability limit. This asymptotic equivalence underpins the conventional preference for Exact
Maximum Likelihood (EML), which fully exploits the innovation covariance structure (See William W. S. Wei
2006).
However, asymptotic equivalence does not imply finite-sample equivalence. In moderate samples, particularly
in higher-order ARMA specifications, the curvature of the likelihood surface may exhibit multinormality or near-
flat regions, especially when parameters approach the boundary of invertibility or stationarity regions. As
discussed in Time Series and Its Applications, numerical evaluation of the likelihood function becomes
increasingly sensitive to nonlinear root configurations.
This distinction between asymptotic theory and finite-sample realization is critical. While asymptotic efficiency
rankings are well established, the rate at which estimators reconcile numerically as sample size increases remains
less precisely characterized. In practice, forecasters operate in finite samples, and the transition from estimator
divergence to effective equivalence may be nonlinear and model-dependent.
Monte Carlo investigation of ARMA estimation traditionally evaluates performance at a small set of discrete
sample sizes. Early simulation studies, such as Ansley & Newbold, (1980) and Burnside, (1994), provide
important comparative insights, yet their experimental designs typically contrast widely spaced values of . Such
binary or coarse benchmarking implicitly assumes smooth convergence toward asymptotic equivalence.
For higher-order models such as ARMA (2, 2), this assumption may be restrictive. The parameter-to-sample bias
analytical treatments of finite-sample properties (e.g., M. O. Salau 2001; Zinde-walsh, (1994) indicate that
estimator dispersion may decay at a rate slower than asymptotic approximations suggest, particularly when roots
lie close to the unit circle.
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METHODOLOGY
This section presents some procedures for estimating the parameters of the postulated ARMA model. For our
exposition, the integers (order) p and q in [2.1] are assumed known, and after observing
, the
system parameters,
, and
are to be estimated. Thus, this section is devoted to the straightforward part of
the modelling procedure, namely, the estimations, for fixed values p and q of the parameters
;
are white noise variance
. It will be assumed throughout that the data
have been adjusted for the means. A variety of estimation procedures have been suggested in the literature (Wei,
2006). For ease of presentation, we have organized our discussion in this section into three subsections: the
Gaussian Estimation procedure, the Generalised Least Squares method, and the Exact Maximum Likelihood
technique. First, we introduce some notations, employing the conventions adopted in Salau (1998), we set for
any integer
are the symbol has been used (Kronecker) matrix product. To fix ideas,
we introduce as a general parameter vector, so that:
. [3.1]
Gaussian Estimation Procedure
In ARMA theory, the Gaussian Estimation Procedure refers to conditional Gaussian Likelihood, where initial
observations are conditional upon. In R, the procedure is implemented as conditional Sum of Squares (CSS) or
CSS-ML.
Under the standard assumption that the innovations
of an ARMA process, are independently distributed as
Gaussian random variables with mean zero and variance
, the probability density function of each innovation
is given by
[3.2]
This assumption provides a natural justification for least squares and likelihood-based estimation procedures,
since minimizing the sum of squared residuals is equivalent to maximizing the Gaussian likelihood. Hardin &
Hilbe (2021) has very reliable discussion on GEP .
Generalised Least Squares Method
In the standard linear model, estimating the parameters of an ARMA process may be approached via least squares
or likelihood-based procedures under the assumption of Gaussian innovations. Let the linear model be given as:
,
, [3.3]
For which the ordinary least squares estimation procedure is obtainable
. [3.4]
More generally, when the distribution vector satisfies
, with a symmetric positive definite covariance
matrix , the Generalised Least Squares estimator provides an efficient gain by accounting for the covariance
structure included by serial dependence.
In-depth discussion on this subject can be seen in the approaches of (Hamilton, 1994), (Koutsoyiannis, 2009),
(Greene, 2024), among others.
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Exact Maximum Likelihood Technique
The Exact Maximum Likelihood estimation of ARMA models is derived by constructing the joint density of the
conditional innovations on initial observations. Under the assumption of Gaussian and independent innovations,
the conditional likelihood for observations . is given as:
[3.5]
Maximization of this likelihood yields the exact maximum likelihood estimators of the ARMA parameters, see
(Olajide et al., 2012), (Box et at., 2015), and (Shumway & Stoffer, 2025).
Stability Consideration in Arma
To introduce the notion of invertibility (stability), consider the MA (1) process,
[3.6]
The only non-zero autocorrelation associated with this process is
. [3.7]
Suppose that is replaced with
, then [3.4] becomes
. [3.8]
Defining [3.3] in terms of the back shift operator, ,
[3.9]
By inverting [3.6], we have:
[3.10]
Expressing
as a linear filter of
and taking a formal power series expansion of the term,
, gives
:
[3.11]
The easiest way to obtain the autocorrelation function in a specific class is to express
as a general linear
process.
[3.12]
[3.13]
The coefficient
being determined by a formal power series expansion of
.
Thus if
is thought of as being determined by the present and the past
, it is noticeable that the remote past
has a vanishingly small influence if and only if .
In a similar discussion to (Salau, 2000), a simple adaptation of the technique in spectral analysis can be used to
derive the second-order properties of an ARMA process
defined equivalently by [2.4] or [1.2]. It can
immediately be deduced from [3.8] that the spectrum,
, must satisfy:
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[3.14]
Thus,
[3.15]
[3.16]
Assuming that
is stationary, the form of
suggests that the values of
are critical in determining
stationarity, and this is indeed the case. The precise condition is the same as for the autoregressive process
, that all roots of
must be greater than 1 in absolute value. More specifically, from [4.10],
the stationarity condition follows from the requirement that
be bounded away from zero for all
, which equivalently satisfies that all roots of
lying outside the unit circle.
Theorem I:
Let
be the true spectral density of
and,
under the stationarity and
invertibility conditions,
. Then there exists an integer N and a constant k dependent
on
such that
, and the truncated autoregressive estimator satisfies:
, where
is the modulus of the zeros of
nearest
.
Proof:
By Baxter’s inequality (Baxter, 1962),
[3.17]
Obtaining the bound for the term on the RHS of [4.12], and by the invertibility property of the stationary ARMA
process,
satisfies the inequality
. The roots of the power series expansion of
shows that
decreases geometrically at a rate determined by
. That is
. For large however,
.
We may now write that:
[3.18]
[3.19]
[3.20]
Hence, it is concluded that [4.12] above is bounded by
.
Model Selection
Correct specification of the ARMA (p, q) order is a prerequisite for meaningful estimator comparison. Following
the iterative model identification procedure described in (Box et al., 2015), and recent developments in
automated order selection (Lin et al., 2024), the determination of and precedes parameter estimation.
Underfitting induces asymptotic misspecification bias, as estimators converge to pseudo-true values that fail to
represent the full dynamic structure. Overfitting, by contrast, increases estimators variance and may introduce
near-cancelation between autoregression and moving average roots, compromising numerical stability through
weak identification.
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Information criteria, such as the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC)
provide a likelihood-based selection rule.
[3.21]
[3.22]
Where . The asymptotic properties of these criteria diverge significantly. The BIC is a consistent
selector when the true model is contained within the candidate set (the probability of selection approaches 1 as
), whereas AIC is asymptotically efficient for prediction but not consistent, whether the maintained
specification reflects the true data-generating process.
Sampling Framework
To trace estimator reconciliation across increasing sample sizes, this study employs a Fibonacci-type sequential
sampling selection defined recursively by:
[3.23]
This nonlinear expression produces dense convergence in smaller samples, where bias and curvature effects are
strongest, while gradually widening intervals as convergence stabilizes. Such adaptive spacing enables more
precise identification of the transition region between finite-sample divergence and effective asymptotic
equivalence.
Recent discussions in stochastic process and time series estimation literature (e.g., Franq & Zakoian, 2024;
Zhang and Ling, 2024; Shumway & Stoffer, 2025) emphasized the importance of examining convergence
behaviour beyond fixed sample benchmarks. The present design operationalizes this principle within a structural
framework.
Law of Large Numbers
The most fundamental requirement for an estimator, as discussed in (Hamilton, 1994), is the property that the
estimator converges to the true parameter value as more data are collected (consistency). Specifically, for a
stationary and ergodic ARMA process
with mean .
[3.24]
The consistency of an estimator is typically established using the concept of uniform convergence. See more on
uniform convergence in (Sanchez, 2010).
The normalized log-likelihood
converges uniformly in probability to a non-stochastic function
.
For ARMA models, the compactness of the parameter space is usually satisfied by restricting the roots of
and
to lie in a closed set outside the unit circle, ensuring stationarity and invertibility.
Equivalence of Estimators
Theorem II
Let
be a causal and invertible ARMA (p. q) process with true parameter vector
, [3.25]
where the autoregressive and moving-average polynomials
and
have no common zeros, and
. Supposed that is a preliminary estimator of
such that
,
and that is the estimator constructed from
as outlined in the estimation procedures of Section Three earlier.
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If
,
, and
denote the estimators obtainable via the Gaussian Estimation Procedure, Generalized
Least Squares, and Exact Maximum Likelihood, respectively, then
i.
ii.
[3.26]
iii.
Here
denotes the Euclidean norm. Since all norms on a finite-dimensional space are equivalent, the choice
of norm does not affect the asymptotic order.
Proof:
Each of the estimators
,
, and
may be expressed as a solution to estimating equations of the form
[3.27]
Where
denotes the corresponding score or normal equations.
By construction, these estimating equations coincide up to terms of order
when evaluated in a
-
neighbourhood of the true parameter . Expanding each estimator around the preliminary estimator using a
first-order Taylor expansion yields
[3.28]
Under the assumed regularity conditions (i.e., causality, invertibility, finite fourth moment and non-singularity
of information matrix), the leading terms of the expressions are identical for GEP, GLS, and EML. Consequently,
their pairwise differences satisfy:
. [3.29]
, which establishes asymptotic equivalence.
Numerical Application
The procedure for various estimation procedures for the investigation is applied to the simulated data points. For
each of these processes, the sample size was determined via the Fibonacci sequence,
,
, the range of
. Replication for each is given by the relation
for the integer
part of the relation.
; As a measure of convergence, the average squared differences between GEP and GLS, GLS and EML, and
GEP and EML were captured by:
[4.1]
And is compared with the criterion measure
, where, for convenience, we have used and to denote
the pair of estimates used in the evaluation of
while
, as before, denotes the parameter
estimates at the replication.
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Data-Generating Process
The data-generating process used in the simulation study is presented in Error! Reference source not found..
Table 1: Data Structure
Process
Structure
Parameter Value
P1
ARMA (1, 0)
P2
ARMA (0, 1)
P3
ARMA (1,1)
P4
ARMA (2,1)
P5
ARMA (1,2)
P6
ARMA (2,2)
Table 2: Estimated Parameter Values and Sample Sizes
Proce
ss
Parameter
True
Value
Estimator
T=75(Mean)
T=125(Mean)
T=200(Mean)
T=325(Mea
n)
T=525(Mea
n)
T=850(Mean
)
1
0.7
GEP
0.66117944
0.676854126
0.67973859
0.6884713
0.6955236
0.6972699
2
GLS
0.6854949
0.69131673
0.6886923
0.69327329
0.6994132
0.6990736
3
EML
0.66253758
0.67763406
0.6801831
0.68805141
0.6961638
0.69707419
4
0.4
GEP
0.41646304
0.405868294
0.39195877
0.4005283
0.4022999
0.3966196
5
GLS
0.4299250
0.41097835
0.3955579
0.40263239
0.4035263
0.3973998
6
EML
0.42044837
0.40604285
0.3924096
0.40072115
0.4023788
0.39668798
7
0.7
GEP
0.65555884
0.672510009
0.68838887
0.6976498
0.6904089
0.6947891
8
GLS
0.6778366
0.68728898
0.6977506
0.70272793
0.6937755
0.6970958
9
EML
0.65085997
0.67112715
0.6877177
0.69660846
0.6900087
0.69476456
10
0.4
GEP
0.42870028
0.412090866
0.3982597
0.3994185
0.4088747
0.4026696
11
GLS
0.4258462
0.40883597
0.3958234
0.39798633
0.4086827
0.4018921
12
EML
0.43581313
0.41481648
0.3994608
0.4001733
0.4099883
0.40272220
13
0.5
GEP
0.74729542
0.698346975
0.60270501
0.5830821
0.5800875
0.5607545
14
GLS
0.6196919
0.6257646
0.5579713
0.52206532
0.5218179
0.5303176
15
EML
0.60268130
0.59386990
0.5185011
0.52503356
0.5116059
0.53424541
16
0.2
GEP
-0.0608097
0.005661657
0.09550125
0.1154162
0.1298494
0.1496292
17
GLS
0.0682709
0.07761254
0.1428440
0.16994660
0.1800150
0.1773297
18
EML
0.05731817
0.08893925
0.1643769
0.16104726
0.1843502
0.17161058
19
0.4
GEP
0.15279453
0.182474081
0.29494190
0.3126248
0.3114914
0.3414853
20
GLS
0.3021933
0.27480916
0.3454526
0.37795349
0.3718250
0.3719911
21
EML
0.29957737
0.29616436
0.3788311
0.37155445
0.3793551
0.36674232
22
0.7
GEP
0.64209803
0.663612821
0.66949443
0.6887492
0.6822076
0.6949135
23
GLS
0.6925140
0.68811499
0.6832942
0.69550947
0.6863627
0.6978062
24
EML
0.63586773
0.65627115
0.6677945
0.68712898
0.6811649
0.69461578
25
0.3
GEP
0.33374164
0.324440565
0.32626738
0.3138208
0.3201734
0.3007742
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26
GLS
0.1292696
0.12299916
0.1279157
0.11972310
0.1207768
0.1067530
27
EML
0.33863283
0.33260504
0.3282570
0.31479910
0.3212135
0.30141462
28
0.1
GEP
0.13942311
0.110499635
0.11607845
0.1097090
0.1116613
0.1006137
29
GLS
0.4437183
0.48081055
0.4582843
0.43088931
0.4347899
0.4006575
30
EML
0.14393842
0.11573372
0.1176162
0.11068653
0.1124858
0.10109678
31
0.5
GEP
0.49919022
0.553030094
0.60742611
0.5883734
0.6044689
0.5754640
32
GLS
0.4564667
0.58491049
0.4726875
0.49454986
0.4823419
0.5400178
33
EML
0.38451568
0.45953029
0.3591297
0.40020291
0.4468448
0.46238272
34
0.2
GEP
0.08642173
0.114701816
0.09719208
0.1188177
0.1127602
0.1339194
35
GLS
0.1719496
0.11220136
0.2124507
0.19773765
0.2161714
0.1636760
36
EML
0.18640371
0.19629044
0.2921534
0.26858634
0.2408711
0.22552130
38
0.3
GEP
0.27921655
0.235098686
0.18571668
0.2101895
0.1876904
0.2198918
39
GLS
0.3271343
0.20466686
0.3244308
0.30773466
0.3121061
0.2561541
40
EML
0.38603628
0.32275311
0.4341254
0.39867916
0.3453440
0.33223466
41
0.1
GEP
0.18842684
0.139641002
0.11858267
0.1000197
0.1022251
0.1034287
42
GLS
0.1640100
0.14500102
0.1174242
0.09987044
0.1005205
0.1029733
43
EML
0.17013016
0.14333271
0.1084024
0.09977264
0.1000577
0.09989274
Table 3: Bias and MSE for Each Estimation Procedure
T
Process
Bias_GEP
Bias_GLS
Bias_EML
MSE_GEP
MSE_GLS
MSE_EML
75
P1
-0.0488
-0.0232
-0.0461
0.00239
0.000540
0.00212
125
P1
-0.0118
0.00170
-0.0120
0.000139
0.00000291
0.000144
200
P1
-0.0124
-0.00368
-0.0122
0.000154
0.0000135
0.000149
325
P1
-0.0155
-0.00942
-0.0146
0.000242
0.0000888
0.000215
525
P1
-0.00667
-0.00340
-0.00664
0.0000444
0.0000116
0.0000441
850
P1
0.000246
0.00217
0.000169
0.0000000607
0.00000472
0.0000000286
75
P2
0.00352
0.0114
0.00282
0.0000124
0.000131
0.00000795
125
P2
-0.00837
-0.00299
-0.00805
0.0000700
0.00000897
0.0000647
200
P2
0.00143
0.00505
0.00204
0.00000204
0.0000255
0.00000415
325
P2
0.00272
0.00485
0.00295
0.00000738
0.0000235
0.00000870
525
P2
-0.00531
-0.00411
-0.00529
0.0000282
0.0000169
0.0000280
850
P2
0.00230
0.00312
0.00240
0.00000528
0.00000972
0.00000576
75
P3
-0.0539
-0.0296
-0.0568
0.00290
0.000877
0.00323
125
P3
-0.0400
-0.0257
-0.0417
0.00160
0.000659
0.00174
200
P3
-0.0220
-0.0135
-0.0235
0.000485
0.000182
0.000553
325
P3
-0.00526
0.000549
-0.00560
0.0000277
0.000000302
0.0000313
525
P3
0.00356
0.00681
0.00304
0.0000127
0.0000464
0.00000921
850
P3
-0.00788
-0.00604
-0.00837
0.0000621
0.0000365
0.0000700
75
P4
0.153
0.108
0.115
0.0235
0.0118
0.0131
125
P4
0.166
0.0346
0.0148
0.0274
0.00119
0.000219
200
P4
0.153
0.0971
0.0853
0.0235
0.00943
0.00728
325
P4
0.144
0.0967
0.0941
0.0209
0.00935
0.00886
525
P4
0.0736
0.00888
0.0113
0.00542
0.0000788
0.000128
850
P4
0.0830
0.0447
0.0516
0.00689
0.00200
0.00266
75
P5
-0.0695
-0.0206
-0.0855
0.00484
0.000424
0.00731
125
P5
-0.0420
-0.0211
-0.0468
0.00176
0.000444
0.00219
200
P5
-0.0312
-0.0194
-0.0340
0.000974
0.000377
0.00115
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325
P5
-0.00534
0.00147
-0.00699
0.0000285
0.00000215
0.0000489
525
P5
-0.0208
-0.0170
-0.0221
0.000431
0.000287
0.000488
850
P5
-0.00682
-0.00438
-0.00756
0.0000465
0.0000192
0.0000571
75
P6
0.176
-0.0392
-0.0446
0.0310
0.00154
0.00199
125
P6
0.158
0.0296
-0.0632
0.0249
0.000876
0.00400
200
P6
0.195
0.0850
-0.0142
0.0379
0.00723
0.000201
325
P6
0.0439
-0.0245
-0.0563
0.00192
0.000601
0.00317
525
P6
0.0951
0.0348
-0.0320
0.00904
0.00121
0.00102
850
P6
0.0305
-0.0568
-0.100
0.000931
0.00323
0.0101
Table 4: Convergence Thresholds (D vs C)
Process
T_val
Param
D
C
Converged
P1
75
ar1
1.493106e-04
0.0094220126
YES
P1
125
ar1
5.289990e-05
0.0033124115
YES
P1
200
ar1
3.522878e-05
0.0025225950
YES
P1
325
ar1
1.179219e-05
0.0016164380
YES
P1
525
ar1
3.422185e-06
0.0010587146
YES
P1
850
ar1
1.042063e-06
0.0006540241
YES
P2
75
ma1
1.052965e-04
0.0137012959
YES
P2
125
ma1
6.452275e-05
0.0068361929
YES
P2
200
ma1
8.109736e -06
0.0052452223
YES
P2
325
ma1
4.732902e-06
0.0020500065
YES
P2
525
ma1
1.333455e-06
0.0019661002
YES
P2
850
ma1
1.006547e-06
0.0008515338
YES
P3
75
ar1
2.755184e-04
0.0160599796
YES
P3
75
ma1
2.889659e-04
0.0146185451
YES
P3
125
ar1
9.171017e-05
0.0071936179
YES
P3
125
ma1
2.503606e-04
0.0106358788
YES
P3
200
ar1
2.879243e-05
0.0038605966
YES
P3
200
ma1
1.101488e-04
0.0053895968
YES
P3
325
ar1
1.034999e-05
0.0024219560
YES
P3
325
ma1
1.490384e-05
0.0037135382
YES
P3
525
ar1
6.874336e-06
0.0011912826
YES
P3
525
ma1
1.321776e-05
0.0023214317
YES
P3
850
ar1
1.646092e-06
0.0007264899
YES
P3
850
ma1
4.537375e-06
0.0010187526
YES
P4
75
ar1
3.420026e-01
0.3072018256
NO
P4
75
ar2
1.996112e-01
0.1957135467
NO
P4
75
ma1
4.152028e-01
0.3028746536
NO
P4
125
ar1
2.669891e-01
0.2164452655
NO
P4
125
ar2
1.726361e-01
0.1446557172
NO
P4
125
ma1
2.851509e-01
0.2042712564
NO
P4
200
ar1
2.890677e-01
0.2371224787
NO
P4
200
ar2
1.883616e-01
0.1647818128
NO
P4
200
ma1
2.954760e-01
0.2473319448
NO
P4
325
ar1
4.369652e-02
0.1607622777
YES
P4
325
ar2
2.694771e-02
0.1087164706
YES
P4
325
ma1
4.610328e-02
0.1496021705
YES
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P4
525
ar1
4.239244e-02
0.1283222189
YES
P4
525
ar2
2.787307e-02
0.0843635641
YES
P4
525
ma1
4.102681e-02
0.1213877662
YES
P4
850
ar1
2.741151e-02
0.0738167978
YES
P4
850
ar2
1.784693e-02
0.0482923042
YES
P4
850
ma1
2.822225e-02
0.0729458484
YES
P5
75
ar1
6.425307e-03
0.0298653266
YES
P5
75
ma1
9.198733e-03
0.0379876074
YES
P5
75
ma2
7.442244e-03
0.0287535307
YES
P5
125
ar1
8.004057e-04
0.0199091435
YES
P5
125
ma1
1.063077e-03
0.0245159348
YES
P5
125
ma2
8.476869e-04
0.0241792397
YES
P5
200
ar1
1.566959e-04
0.0104618020
YES
P5
200
ma1
1.727998e-04
0.0127901935
YES
P5
200
ma2
1.315822e-04
0.0100145274
YES
P5
325
ar1
4.386340e-05
0.0048297281
YES
P5
325
ma1
4.971498e-05
0.0054908511
YES
P5
325
ma2
4.027359e-05
0.0048265603
YES
P5
525
ar1
1.031019e-05
0.0028525433
YES
P5
525
ma1
2.028162e-05
0.0042154993
YES
P5
525
ma2
1.549325e-05
0.0038667733
YES
P5
850
ar1
4.793569e-06
0.0017050822
YES
P5
850
ma1
7.854964e-06
0.0021204804
YES
P5
850
ma2
7.187170e-06
0.0023276628
YES
P6
75
ar1
4.470805e-01
0.3292313312
NO
P6
75
ar2
2.654512e-01
0.2334360308
NO
P6
75
ma1
5.138451e-01
0.3924679603
NO
P6
75
ma2
2.264582e-02
0.0442964111
YES
P6
125
ar1
2.239885e-01
0.2488812293
YES
P6
125
ar2
1.436606e-01
0.1743962483
YES
P6
125
ma1
2.460867e-01
0.2861316257
YES
P6
125
ma2
1.041010e-02
0.0222345687
YES
P6
200
ar1
3.270347e-01
0.2290212239
NO
P6
200
ar2
2.175085e-01
0.1550506697
NO
P6
200
ma1
3.389386e-01
0.2484301869
NO
P6
200
ma2
2.724779e-03
0.0128367576
YES
P6
325
ar1
1.811885e-01
0.1784909371
NO
P6
325
ar2
1.160543e-01
0.1189935996
YES
P6
325
ma1
1.849996e-01
0.1811415612
NO
P6
325
ma2
9.926453e-04
0.0058761309
YES
P6
525
ar1
1.839834e-01
0.1838276324
NO
P6
525
ar2
1.203909e-01
0.1214142733
YES
P6
525
ma1
1.885004e-01
0.1914593077
YES
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Table 5: Convergence Thresholds (D vs C)
Figure 1: Estimator Convergence: D < C intersection
Figure 2: ARMA (2,2) Convergence Intersection Analysis
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Design of Experiment
For each estimator, the sample size, parameter estimates, standard error, bias, and mean square error were
obtained by averaging the estimates over the number of replications, where
with
denoting the estimate of
in the r
th
replication. Following the standard practise, the accuracy of the
parameter estimate is judged by the mean square error. The outcomes of this investigation are summarized
inError! Reference source not found. Table 2 and Error! Reference source not found., respectively.
As a measure of convergence, the average square differences between GEP and GLS, GEP and EML, GLS and
EML were computed by:
[4.2]
And is compared with the criterion measure
where, for instance, we have used S and to denote the
pair of estimates used in the evaluation of
while
,
as before, denotes the parameter estimates
at the r
th
replication.
CONCLUSION
The ARMA (2, 2) process
constitutes the upper bound of structural complexity in the simulation study,
comprising four parameters
, and provided a stringent test of asymptotic approximation. While
the autoregressive components exhibited stable and accurate estimation even at small sample sizes, the moving
average parameters displayed pronounced finite sample instability, characterized by elevated variance and
delayed convergence. The convergence diagnostics showed that the Difference MSE (D) exceeded the Reference
MSE (C) through T =200, with consistent interaction achieved for all parameters, only at . This
identified a clear “indifference threshold,” beyond which the Gaussian Estimation Procedure (GEP) became
numerically indistinguishable from the Exact Likelihood Estimator (EML). More broadly, the results revealed a
systematic complexity lag: as model dimensionality increased, the intersection shifted markedly to the
right, indicating that estimator choice depends jointly on sample size and the parameter-to-sample ratio rather
than on T alone.
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