A C T A U N I 7 E R S I T A T I S L O D Z I E N S I S FOLIA OECOHOMICA 68, 1987
Andrzej Stefan Tomaszewioz* SOME RESULTS CONCERNING EFPICIENCY
OF LINEAR TREND ESTIMATION UNDER HETEROSCEDASTICITY
1. Introduotion
Consider a singla - eąuation eoonomatrio modal of the form y » + E ,
where X ia a fixed nz(k + 1) matriz of rank k + 1 < n, and E£ ■ O.
2 o
One of the alternatiyes for the olassioal aaaumption DC » 6 I about spherioal random oomponant is the assumption of laok of autooorrelation allowing heterosoedastioity:
(1) D 2 £ - 6 2 A ,
where a ls a dlagonal matrix with positlve diagonal elements. If A is a known matriz one oan use the best linear unbiased estimator of the vector ot, whioh ia A i t k e n's [1] GLS-eati- mator
(
2
)
a* - (lT ń ’1i r 1iT ń ’1y
with the variance-covar±ance matriz (3) D 2a* - ef2(lT ft“1X)“1, The OLS-estimator
(4) a
-*Lectarer, Institute of Eoonomatrios and Statistios of the University of Ł6&£,
is generally ineffioient. Moreover tha estimator s 2 (xTx r 1
of the OLSE varianoe-oovarianoe matriz
*
(5) D 2a - B Z(xTl)"1i T flX(iTi)'1 is biased in generał.
We oan take into acoount tha olaes of estimators of the form (6) aCW) » (XTWX )~1XTWy
where W ia a symmetrical and positive definite matriz to whioh the estimators (2) and (4) belongj
(6a) a (a “1) » a * and a(I) » a.
The estimator (6) will be named as WLSE (weighted LS) with weight matriz W. If the matriz W ia non-stoohastlo (at least not oor— related with y) it is easy to derive a formuła
D 2a(W) - ff2(XTWX)"1XTWxj,WX (XTWI)"1
which may be reduced to (3) for W - n. and to (5) if W ■ I. Por praoticał applioations, generally, ao-called two-step estimator
(7) a ( A * 1>,
is suggested, where fi is the estimate of matriz expressed as a function of OLS-residuals
e « y - la
(see f.e. P a r k [13], G l e j s e r [7] and genaralization for three-stop estimator - P r o e h l i o h [5],and J a r m u ł [ 8 ] p. 47-54 as well). The weight matriz is, in this oase, a sto- chaatio matriz and it is diffioułt to ezamine estimator** (7) properties for email c , ’ p l a a .
We oan, however, prove that thia eatimator is asymptotioally eąually efficient as a*. The l&test results for models with heteroscedasticity of random coitponent conoerning maximum like- lihood method optimal method for largo samples, can be found for instanoe in F u l l e r , R a o [6], J o b s o n , P u l i e r [9], C a r r o l l , R u p p e r t [3]*
It seems to U3 that the most generał results referring to OLSE effioiency are Watson ineąualities (see W a t s o n [l7j, B l o o m f i e l d and W a t s o n [ 2 ], K n o t t [10], M i-1 o [i-12]) whioh deflne its lower limit in dependenoe on the matriz eigon values - but independentły on matrix X, similarly aa 3 a t h a * V i n o d [15] ineąualities for biae of variance’a estimator of OLSE. The ineąualities of Watson and Sathe - Vinod, oan be generalized to WLSE with fixed (independent on y weight matrix). Theae results, extremely lmportant from theoretical point of view, don’t have great praotioal meaning, because in a ooncrete problem the matrix X is known and Information about .a is poor. Therefore, methods allowing to evaluate estimation ef- ficiency when the matrix X ia known (or belongs to the defined, relatively n arrow class) are more useful for direct applioations,
2. Problem formulation
Concepta oonneoted with a typical autooorrelation struoture for the linear trend modol
(8) yt -
0*0+ ^ + et*
that is, for the model (1) in whioh k ■ 1 and I
(9) X - X,n
1 1
1 2
were the topio of many works, aee i,e. T o m a s z e w i c z [16 j, P a r k , M i t o h e l l [14], C h i p m a n [4].K r a m e r [i1 ]. In this work we undertake the topio of WLSE-eXficienoy for the model with heteroscedasticity.
As in the oase of autooorrelatlon we consider tbe models in which "non-spherlcity" - in our case heteroscedasticity - is defined by one pararaeter denoted by p «
When we £ix the heteroscedasticity model as
(10) “ 6o ^ t - 1, 2, n
2
(that is when the funotion cf> is given), where ff.j. are the dlago- nal elements of A , thls matrlx is also a funotion of (3, so we will use the symbol A ((3). Let’s aasume that we use the estimator a ( A (-y)"*1) where f is a fixed number (whioh should ba interpreted as an estimate of f3). A measure of YJLSE-effioiency oan be defined (see T o m a s z e w i o z [16]» C h i p m a n [4]) as a ratio o£ varianoes of slope a 1 estimators of model (8). Thls ratio depends on the aample size, so it is a funotion of three para meters i
a n (n((3)~1) (11) tp(n,(3,2r) - —
^(fltj) )
where a.j(A (-y)”1) is the eecond diagonal element of the matrix D 2a(A('y)"’1 ), so (see (6a))
D 2a ^ ( A (-y)”1) **
(1 2 ) - oT (xT ii(|)“1x r 1iT ^ ( ^ r 1 a(|3)a(^)"1i(it a ( ^ r 1x)'1o ’
T r t
for o - L0 1j«
At this Btuge of work we limlted our interest to investlgating soae heteroscedasticity models inoluding a proposition of heteroscedasticity measure (Section 3) &nd numerical analysis of efficiency (12) (seei 3eotlon 4). Some problems conneoted with generalization and a seriea of detailed problems which appeared during the work we hope to solve lat er.
We limit our oonsideration to the most often, up to now, used heterosoedastioity modele (see, for ezampłe J a r m u ł 18] p. 40, 43). These are the following exemplifioations of (10)«
- linear heterosoedastioity
3,...,n. To make possible a coraparlson of different modela we need common heteroscedasticity measure. The most natural measure seems to be the coefficient of variation, which expresses standard deviation of a set 6^, e 2 , in mean value units. This measure is independent of soale coefficient 0 Q and is of the
following formi 3. Heteroscedasticity measure
(
13
)
of - 6o
" 6 o
(1
+
13
n )ł
- parabolic heteroscedasticity (14) - 6^cpp((3, t) - C 20 (1 - 1)2 ), - exponential heterosoedastioity group heteroscedasticity r 1 ■ t sj n/2 t > n/2. (16) - single-obserration heteroscedasticity 2The parameter (3 must be chosen so that 6^ > 0 for t * 1, 2,
where
Por heterosoedastioities (13), (14), (16) and (17) we oan obtaln expliclt formulaa of (18). TheBo are as foliował
Tha rangę of the funotiona (18)-(21) and analogous funotlon vg (f3,n) for ezponentlal heteroaoedaatioity (15), were illuatra- ted on the Plgures 1-5.
We do not lntroduoe here analysls of these funotiona. We cali only one’a attentlon to upper llmlts of yariationa ooef ficient conneoted with restrained valuea of funotiona (18)-(21) rhen (3 -— ► co
1 + — n y* i 4 where k • integer (— -J--) and
Pig. 1. Function vL Pig. 2, Punotion
▼G (co, n) - - 1 1 for even n, M Jor odd n.
/i
v<j (oo 9 n) « / n - T . .
4. Cor.łitional effioienoy of the estimator of slope ooefficient
Derivation of explioite forraulaa for the effioienoy (11) seems to be diffioult in generał oase. However in ąuite siraple way we can oalculate these numerical ąuantitieo for chosen value3
of f3 and ^ for -aorne levela of aample aize. Under the notations
L .
^■oo
*01 .101V
K > i K§
k01 o M l k11 (I* a ( ^ r1
x )‘ 1.xT A(tf
)~1
a((
3
)a (-J-)” 1X f
,1QV W 01 “ 2k01l00l01 + k11lOO (19) D a, U(tf) } - — ---- ---
---u
00x11“
Aov
The efficienciea (11) were calculated on the base of preaented formulas for five distinot raodel3 (13) — (i7) for n = 3, 10, 30, 100 and (3,# chosen ln suoh way that yariation ooeffioient (18) took the values
(20) ▼ - 0, 0.01, 0.05, 0.1, 0.2, 0.3* 0.4.
The resulta are collected in Tables 1-5. Inaide the tablea (row h, coluran j) are efficienciea (11) cnloulated for (3 ■ g(vb , f ■ ■ g W j ) , where g denoteo an inverse funotion in ▼ oorreaponding to one of the functions (I8a)-(18d), and vn (v^ la reapeotively h-th (J-th) ralue from the H a t (20). On the baae of obtained reaulta we oan foimilate some, aa it aeems, ąuite intereating oonoluaiona.
1° The numbera in the tablea above the main diagonal are olightly smaller then oorreaponding onea below thia diagonal. It meana that overeatimation of our heteroacedaaticity measure v causea greater efficiency loaaes than underentimation.
2° Undereatimation or overeatimation of hateroacedasticity meaaure cauaea the greatect loaaes in the caoe of amooth linear hoteroacedaaticity, then exponential and parabollc. Reapective loaaes for group and single-obaervationa heteroacedaaticity are much smaller.
3° With the inoreaae of sample size n, the loaaes in ef ficiency caused by miaestimation of hoteroBcedaaticity meaaure are generally growing up, exeluding the case of alngle-obaerva- tion heteroscedasticity, when we obsertre falling down of ef ficiency.
Oonoluaiona formulated above, espeoially third, seema to ua to be not tri7 ial. They point out some ways in chooaing an eatima tion method for a given heteroacedaaticity model. Thia problem needs more detailed, further inveatigations.
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r-« an H O ie EH L A O N n c - o ^ c o o CO LA t— O o j t - - iA v o O w O ONCTNł- ('Nt-ONO c c oo o n o n c t n o o O O O O O O r -f U ) L A f A i A C T w a O O iavo o n CO CMVO r- CM O OJ cr\cr>cT\<T\cr\0 cx>• • cm«- o r M A ^ W i f i i f i O CM t- 1 ~ O C- «“ O CO CO O (T> O co cd on cr> on c n o O O O O O O t -o -o -o -o -o * - -o ■<ł-r- t- C O CM O O CM 0N«-0 ON O VD CM \£> r - CO r- Q *- nrf\ir\vO CT.O CT» c n on on on c n o on O O O O O r O «- o j v o i a i a c n o 00 r- «i-t— CO ( ^ o C N l A t A 00 VO r - O r » cc o cm no cr> o oo co c n c n o n c n o « • • • • • » O O O O O O r -f i o o c n ^ o •*»■ t - O ON ON t - O '* O LA O t— «- Q r-nrMrwo CnO cn ON ON ON ON ON O CTN • • • • • • • O O O O O r O i n r.<3 s -* p •H O •H oj 0* -c* a> o D O U •P 0> X I O M {►» o a a ■H O •H W T ... .... ... c- "***- o o c—m CM CA O VO Lf\ CM V ^ ( \ l O CM O t— co cn O cn f- j cn cn en ar\ o cn cn i O O O O r - O O CO CTV V i n O I A CTi f ^ C M i r \ f ~ O Cnt-^- CnCMCM CM O r- r - V£> C*- CO Cn O C W O CT\ <T\ CA Cn o c n ch O O O O t- O o c o m ^ t ^ o e - - * * VD <D +~~ CTv O Or- . co «- CM *- O r - r - vO t - CO Cn Q a^yjo cn cn cn cn O cn cn o o o o * - o o i CM "J- T f - o 1- »- «-i O n c M O C n t - O N I n ^ f- C C O Od t~ V 0 i on on <n o cn c - n l o n c ^ o n O c n c n on O O O t- O O O I f - I ^ L T l O C— V C n O J C - O O *- 00 ON CM n c o O CM co o c n c n cn O cr»vo n c n c n c n o c n c n c n • ■ * » • • • O O O r - O O O c n c o t O c n c O * -O U N -O -O -O łi- I A cm n oo O r~ co cn cnaNcno qn\jO c\j c n on on o on c n o n • • • • • • • O O O r - O O O i I n LACD oo co c o c o ON ON c n a ' O TJ- * 3- Tł-<*N o cmc m u) n o CO 'TvOVO ON CO LA »- O ON ON ON ON • • • • • 1 - 0 0 0 0 vo>-c Om? c n t-c n O t — O O i A O J CO 00 O CC' CM CM 00 ON O N O ON CD U N O ON O N O ONONCJNON • • « • • • • O O r - O O O O cm n o r o n * -O t - -O -O CM CM LA CO 00 O CD OJ « - v o ON ON O ON CO LA O ON ON O ON ON ON CJN • • • • « * • O O r - O O O O CO ON ON OJ co t - O N i A n CO 'T " « r CU tD ONONt— 'j-CTv o ncr> o no noo » - o o o o o <NJ O MD n t - O J t— <m O r~ t— n oo m? ON O 0 0 * ^ CM vO CD o n o onon r - n co O N O ON ON ON ON CD o » - o o o o o o j O n a N ^ ^ c o on o t— l a c - n O N O CO f N f lA V O o"> o o ncr> p- n co cn o on o n o n a\ co • • • • • • • O <r- o o O O o
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[1 ] A i t k e n A. C. (1934)* On Least Sąuares and Linear Com- bination of Observations, Prooeedings of the Royal Sooiety Edinburg, 55, p. 42-48.
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between liaximum Likelihood and Generalized Least Sąuares in a Heteroscedastic Linear Model, "Journal of the American Statietical Aosociation", 77, p. 878-881.
[4] C h i p m a n J. 3. (1979 )* Efficiency of Least Sąuares Estiraation of Lineąr Trend Model when Residuala are Autocor- related, "Eoonometrioa", 47, p. 115-128.
[5] F r o e h l i c h B. R. (l973)iSome Estimators for a Random Coeffioient Regression Model, "Journal of the American Sta- tistical Association", 68, p. 329-335.
[6 ] F u l l e r W. A«, R a o J. N. K. (1978 )i Eatimation for a Linear Regression Model with Unknown Diagonal Covariance Matriz, "Annals of Statistios", 6 , p. 1149-1158.
[7 ] G 1 e j a e r H. (1969)* A New Teat for Uomoaoedastioity, "Journal of the American Statiatioal Assooiation",64, p.316- -323.
[8] J a r m u ł M. (1977)* Eatimation of Eoonometrio Modela with Heteroscedasticity of Residuals, Ph.D. theses, Univer- sity of Lublin.
[ 9 j J o ' o s o n J. D., F u l l e r W. A. (1980)* Least Sąuares Eatimation when the Corarianoe Matrix and Parameter Yector aro Punctionally Related, "Journal of tha Amerioan Statiati- cal Association", 75, p. 176-181.
[10] K n o t t M. (1975)i On the Minimum Effioienoy of Leaat Sąuares, "Biometrika", 62, p. 129-132.
[11] K r a m e r V.'. (l982)t Note on Estimating Linear Trend when Residuala are Autocorrelated, "Eooncmetrica", 50, p. 1065- -1067.
[12] M i 1 o V.’. (1977)* Effioienoy of Eatimation of Linear Model Parometora Under Autooorrelation, "Przegląd Statystyczny", 24, ?. 443-454.
1^13] P a r k R. E. (1966)1 Estimation with Heteroaceda3tic Error Terms, "Econometrica", 34, p. 888-900*
[14] P a r k R. Ł , l l ł t c h « l l B. M. (1980 )i Eatimating the Autocorrelated Error. Model with Trended Dnta, "Journal of Econoraetrlcs", 13, p. 185-201.
£.15] S a t h e S. T., V I n o d H, D. (1974): Bounda on the Varlanoe of Regression Coeffioient Due to Hetaro3cedaatic or Autoresresalve Errors, "Econometrica", 42, p. 333-340.
[16] T o m a a z e w i o z A. S. (1975)» Nuraerloal Eyaluatlon of the Efficlenoy of Estlmatlon Methoda for the Modela with Autooorrelation, "Prace Inatytutu Ekonometrii i Statystyki UŁ", Ser. D, 8, Łódź.
[17] W a t a o n G. 3. (1967)» Linear Least Sąuarea Rogreoalon, "Annals of Mathematica] Statistica", 38, p. 1679-1699.
Audrzej S. Tomaszewicz
PEWNE W/NUCI HA TEUAT EFEKTYWNOŚCI ESTYŁLICJI LINIOWEGO TRENDU W PRZYPADKU KETEROSKEDASTYCZNOSCI
W pracy rozważa ai® model trendu postaci:
?t “ °o + “1* + £t t « 1, ..., o
przy założeniu heteroakedaatyoznoścl rozkładu składników losowych,
określonej jakoi * .
" °o t * 1, ..., n,
o
gdzie y oznacza daną funkcję, (3- parametr nieznany, - diago nalne elementy macierzy kowariancji a wektora składników losowyoh A ■ n((3).
Rozważane Jest pięć modeli heteroakedaatyoznośol związanyoh
3 funkojami ip.
Jako miarę heteroskedaatycznoeci przyjmuje flif wapółozynnik
J o O
eetyraa-torów współczynnika kierunkowego o^« Wykorzystywana miara efek tywności będąca stosunkiem warianoji badanych estymatorów .-Jest funKoją trzech parametrów! wielkości próby n, parametru (3 oraz Jego oceny f. Przedstawiono numeryczne wyniki dla wybranych G, <jr oraz kilku poziomów wielkości próby.
