Cap6-Estimarea Parametrilor in Analiza Econometrica

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CAPITOLUL 6 (67,0$5($3$5$0(75,/25Ì1$1$/,=$(&2120(75,& 6.1. 0RGHOHFXRVLQJXU YDULDELO H[RJHQ 6.2. Modele cu mai multe variabile exogene 6.3. Metode probabilistice 6.3.1. 0HWRGDYHURVLPLOLW LLPD[LPH 6.3.2. $QDOL]DED\HVLDQ 6.3.3. O abordare controversDW &kWHYDSURSULHW LDOHHVWLPDWRULORU

Transcript of Cap6-Estimarea Parametrilor in Analiza Econometrica

Page 1: Cap6-Estimarea Parametrilor in Analiza Econometrica

CAPITOLUL 6

(67,0$5($�3$5$0(75,/25�Ì1�$1$/,=$�(&2120(75,&

6.1. 0RGHOH�FX�R�VLQJXU �YDULDELO �H[RJHQ

6.2. Modele cu mai multe variabile exogene

6.3. Metode probabilistice

6.3.1. 0HWRGD�YHURVLPLOLW LL�PD[LPH

6.3.2. $QDOL]D�ED\HVLDQ

6.3.3. O abordare controversDW

�������&kWHYD�SURSULHW L�DOH�HVWLPDWRULORU

Page 2: Cap6-Estimarea Parametrilor in Analiza Econometrica

ESTIMAREA PARAMETRILOR IN ANALIZA

(&2120(75,&

3DUDPHWULL� j �M ��P��IRORVL L�vQ�FRQVWUXF LD�PRGHOXOXL�HFRQRPHWULF�RIHU �LQIRUPD LL�FX�privire la structura S (��� D� VLVWHPXOXL� HFRQRPLF� ����'H� DFHHD� HVWLPDUHD� ORU� FRQVWLWXLH�unul din obiectivele principale ale analizei econometrice. ÌQGHSOLQLUHD� DFHVWXL� RELHFWLY� SUHVXSXQH� H[LVWHQ D� XQHL� serii de date cu privire la variabile. De asemenea avem nevoie de o func LH� FH�GHVFULH�VXILFLHQW�GH�H[DFW� UHOD LLOH�GLQWUH�YDULDELOD�H[SOLFDW ��GHSHQGHQW ��GH�YDULDELOHOH�H[SOLFDWLYH��LQGHSHQGHQWH�� &HD�PDL�VLPSO �IRUP �D�XQHL�DVWIHO�GH�UHOD LL�VWRFDVWLFH�HVWH�

yt� � 1� 2 xt� t (6-1) ÌQ� FHHD�FH�SULYHúWH�HFXD LLOH� FH� GHVFULX� UHOD LL� vQWUH� YDULDELOH�� GLQ� FDGUXO�PRGHOXOXL�econometric, în particular expresia (6-���WUHEXLH�V �SUHFL] P�

a)� �0 ULPLOH�<�VL�;��FDUH�UHSUH]LQW �SHUHFKL�GH�YDORUL� �[t,yt) pentru fiecare moment t=1,T , se presupun corelate sub forma unei UHOD LL�ELMHFWLYH.�)LHF UHL�YDORUL�GDWH�D�OXL�x îi va corespunde o valoare a lui y DIODW �vQWU-XQ�DQXPLW�LQWHUYDO�FRUHVSXQ] WRU�XQHL�DQXPLWH�SUREDELOLW L�

b) Parametrii 1 si 2� UHSUH]LQW �necunoscutele modelului. Ei se pot determina

servindu-VH� vQ�DFHVW�VHQV�GH�YDORULOH�YDULDELOHORU�FDUH�SULYHVF� vQWUHDJD�SRSXOD LH�VWXGLDW ��QXP UXO�GH�SHUVRDQH��GH�IDPLOLL��GH�VRFLHW L�FRPHUFLDOH�PLFL�úL�PLMORFLL��HWF��

'HRDUHFH� REVHUYDUHD� XQHL� SRSXOD LL� QX� HVWH� vQWRWGHDXQD� QHFHVDU � VDX� SRVLELO �SXWHP� UHFXUJH� OD� XQ� HúDQWLRQ� RE LQXW� SH� ED]D� VRQGDMHORU� VWDWLVWLFH�� 3ULQ� XUPDUH� YRP� IL�QHYRL L�V �GHWHUPLQ P�YDORULOH� 1�úL� 2�FH�FRQVWLWXLH�GH�IDSW�HVWLPD LL�DOH�YDORULORU� 1 úL� 2.

c)��5HOD LD�H[LVWHQW �vQWUH�ILHFDUH�SHUHFKH�Ge variabile (Xt,Yt��FX�W ��7�HVWH�GHVFULV �

�GDU�QX�QHDS UDW�H[SOLFDW ��SULQWU-o IXQF LH�PDWHPDWLF �FRUHVSXQ] WRDUH��$FHDVWD�WUHEXLH�V � GHVFULH� FX� DFXUDWH H� úL� VLPSOLWDWH� UHOD LD� UHVSHFWLY �� ÌQ� DFHVW� VHQV�� PRGHOHOH�econometrice presupun în majoritatea cazurilor IXQF LL� OLQLDUH. La acestea se ajunge fie LQWHUSUHWkQG�FD�DWDUH�UHOD LLOH�GH�GHSHQGHQ �GLQWUH�YDULDELOH� ILH� ³OLQLDUL]kQG´�DFHVWH� UHOD LL�folosind diferite transformate (vezi si CAP.4).

d) � $EDWHUHD� t� �SHUWXUED LLOH�� FRQVWLWXLH� R� YDULDELO � DOHDWRDUH� FH� XUPHD] � R� DQXPLW �OHJH� GH� SUREDELOLWDWH� �GH� UHJXO � GLVWULEX LD� QRUPDO ��� 7HUPHQXO� GH� YDULDELO � DOHDWRDUH� VH�UHIHU �OD�IDSWXO�F �DFHVW�HOHPHQW�SHUWXUE �UHOD LLOH�GHWHUPLQLVWH�GLQWUH�YDULDELOH��UHOD LL�FDUH�GLQ�DFHDVW �FDX] �VH�SUH]LQW �FD�stocastice.�$FHDVWD�vQVHDPQ �F �SHQWUX�ILHFDUH�YDORDUH�D�OXL�X pot rezulta mai multe valori ale lui Y��ILHFDUH�DYkQG�R�DQXPLW �SUREDELOLWDWH�GH�DSDUL LH� Modelele econometrice includ în general UHOD LL�VWRFDVWLFH�� În acest caz descrierea VH�FRPSOLF ��7RWXúL�unele�SRVLELOLW L�GH�testare a ipotezelor * pornind de la variabila t fac QHFHVDU �LQFOXGHUHD�DFHVWHL�YDULDELOH� /HJDW�GH�DFHDVWD��DYHP�vQ�YHGHUH�F �REVHUYD LLOH�SULYHVF�XQ�HúDQWLRQ�úL�QX�vQWUHDJD�SRSXOD LH��3DUDPHWUL�HVWLPD L� j (j=1,m) vor permite pHQWUX�ILHFDUH�YDORDUH�D�OXL�;�RE LQHUHD�unei valori estimate pentru Y. Abaterile valorilor estimate Yt de la valorile Yt (valori care s-DU� IL� RE LQXW� SH� ED]D�XQRU�SDUDPHWUL�GHWHUPLQD L�REVHUYkQG�vQWUHDJD�SRSXOD LH��VH�QXPHVF�reziduuri � t). Aceste reziduuUL�GHQXPLWH�DGHVHD�úL�erori sau abateri��UHSUH]LQW �GH�IDSW�HVWLPD LL�DOH�SHUWXUED LHL�aleatoare � t). *�YH]L�FXUVXULOH�GH�VWDWLVWLF �PDWHPDWLF

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�����0RGHOH�FX�R�VLQJXU �YDULDELO �H[RJHQ

6 � QH� UHIHULP� SHQWUX� vQFHSXW� OD� FHO� PDL� VLPSOX� PRGHO� GH� UHJUHVLH� GHVFULV� GH�expresia (6-1). În etapa de specLILFD LH� WUHEXLH� V � UH]ROY P� QX� QXPDL� SUREOHPD�definirii variabilelor� úL�D� IRUPHL� UHOD LHL�GH�GHSHQGHQ �FL�úL�FHD�D�VSHFLILF ULL� OHJLL�GH�GLVWULEX LH�D�SHUWXUED LHL� 'H�DVHPHQHD�� VH� LPSXQH� LGHQWLILFDUHD�PRGDOLW LORU� GH� respectare a unor ipoteze fundamentale si anume: a) perechile de valori (Xt,Yt��� W� � ��7� �� UHSUH]LQW � P ULPL� RE LQXWH� I U � erori de observare; b) media *�YDULDELOHL�GH�SHUWXUED LH�HVWH�QXO �

(� t) = 0, (6.1-2) FHHD�FH�LQVHDPQ �F �DEDWHULOH�VH�FRPSHQVHD] �I U �a fi corelate cu variabila Xt; c) Abaterea � t)� XUPHD] � R� GLVWULEX LH� LQGHSHQGHQW � GH� Xt� úL� GH� PRPHQWXO�REVHUY ULL��DYkQG�R�GLVWULEX LH�

(� tð�� � ð (6.1-3)

'DF �GLVSHUVLD�YDULDELOHL�� t)�HVWH�FRQVWDQW ��LQGLIHUHQW�GH�³W´��GHFL�SURFHVXO�HVWH�VWD LRQDU��DILUP P�F �HVWH�SUH]HQW�IHQRPHQXO�GH�homoskedasticitate**�� ÌQ�FD]�FRQWUDU��DGLF �GDF �DEDWHULOH�SUHVXSXQ�GLVSHUVLL�GLIHULWH�SHQWUX�DQXPLWH�YDORUL�DOH�OXL�³(´��DWXQFL�OH�FRQVLGHU P�heteroskedastice. d)� LQGHSHQGHQ D�REVHUYD LLORU��$FHDVW � LSRWH] �SUHVXSXQH� LQH[LVWHQ D�DXWRFRUHO ULL�erorilor. Deci:

(� t1��� t2) = 0 cu t1 ��W2 (6.1-4)

e)�DEDWHUHD�XUPHD] �R�GLVWULEX LH�QRUPDO ***; f)� YDULDELOD� H[RJHQ �X� QX� HVWH� VWRFDVWLF �� 'DF �QXP UXO� GH� REVHUYD LL� WLQGH� VSUH�LQILQLW��DGLF �To � DWXQFL�(�[��úL�(�[ð���DGLF �PHGLD�úL�GLVSHUVLD�OXL�;�– tind spre limite finite (nenule). ��ÌQ� YHGHUHD� HVWLP ULL� SDUDPHWULORU� 1� úL� 2� YRP� DYHD� vQ� YHGHUH� UHJUHVLD� OLQLDU �VLPSO ��GH� IRUPD� (6-1).�$úD�FXP�YRP�REVHUYD� vQ�)LJ�������SXQFWHOH�FH� UHIOHFW �SHUHFKile (Xt,Yt)�SUH]LQW �DEDWHUL�GH�OD�GUHDSWD�<� � 1��� 2X; ��6XPD� S WUDWHORU� DFHVWRU� DEDWHUL� YDULD] � vQ� IXQF LH� GH� JUDGXO� GH� vPSU úWLHUH� D�SXQFWHORU�ID �GH�GUHDSW � Y X

Fig. 6.1

*��VH�YD�QRWD�FX�³(´�GH�OD�H[SUHVLD�HQJOH]HDVF �³expectation´��DúWHSWDUH��DQWLFLSDUH��VSHUDQ ��HWF���

**��WHUPHQ�GLQ�OLPED�JUHDF ��DYkQG�VHPQLILFD LD�GH�³HJDO�vPSU úWLDWH´� ***

��YH]L�FXUVXO�GH�VWDWLVWLF �PDWHPDWLF �

yt

yt

εt t21t xˆˆy β+β=

yt

0 xt

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Pentru a estima parametrii 1�úL� 2�YRP�IRORVL�FXQRVFXWD�PHWRG �D�FHORU�PDL�PLFL�S WUDWH� * (MCMMP).Prin urmare, vom determina valorile 1�úL� 2 astfel încât dreapta (r)**: yt� � 1 + 2xt V �VH�DSURSLH�FkW�PDL�PXOW�GH�SHUHFKLOH�GH�YDORUL�HPSLULFH��;t, Yt). De aceea

VH�YD�F XWD�PLQLPL]DUHD�DEDWHULORU�� t��úL�LPSOLFLW�VXPD�S WUDWHORU�ORU��DGLF ��∑=

εT

1t

2t .

6.2. Modele cu mai multe variabile exogene

ÌQ� DFHVW� FD]� SUHVXSXQHP� R� GHSHQGHQ � PXOWLSO � �UHJUHVLH� PXOWLSO �� D� YDULDELOHL�endogene Y. Altfel spus, între variabilele independente X2, X3 ... Xm+1 úL� YDULDELOD�GHSHQGHQW �<�H[LVW �R�UHOD LH�OLQLDU �VWRFDVWLF � de forma: yt� � 1 �� 2 x2t��� 3 x3t��««�� m xmt��� t, cu: t= 1,T (6.2.-1)

Ea se mai poate scrie condensat : m

yt � � j xjt ��� t (6.2.-2) j = 1

FX�W� ���7��úL��;1t = 1; 6LVWHPXO� GH� UHOD LL� �����-��� VH� SRDWH� SUH]HQWD� úL� VXE� IRUP � PDWULFHDO � DYkQG� vQ�YHGHUH�IDSWXO�F �REVHUYD LLOH� I FXWH�DVXSUD� OXL�<�Vi X2, X3….Xm+1 VH�UHIHU � OD�XQ�HúDQWLRQ�IRUPDW� GLQ� ³7´� XQLW L�� 'HFL�� YRP� DYHD� SHQWUX� ILHFDUH� XQLWDWH� XQ� VHW� GH� REVHUYD LL�FRUHVSXQ] WRU� YDULDELOHORU� LQFOXVH� vQ�PRGHO�� &HOH� ³7´� HFXD LL� SRW� IL� H[SULPDWH� FRQGHQVDW�astfel:

1T1mmT1TUBXY××××

+⋅= (6.2.-3)

3DUDPHWUL� j �M� ���P�� �FD�úL�SHUWXUED LD�� ��� IRUPHD] �QHFXQRVFXWHOH�SH�FDUH�YRP�

vQFHUFD�V � OH�HVWLP P��0HQ LRQ P�F �H[LVWHQ D�WHUPHQXOXL� OLEHU� 1 D�I FXW�FD�vQ�PDWULFHD�YDULDELOHORU�H[SOLFDWLYH�;�V LQWURGXFHP�R�FRORDQ �IRUPDW �GLQ�FRQVWDQWD�³�´��ÌQ acest mod QH� YD� IL� SRVLELO� V � RE LQHP� HVWLPDUHD� DFHVWXL� SDUDPHWUX� FRQFRPLWHQW� FX� HVWLPDUHD�SDUDPHWULORU�DVRFLD L�YDULDELOHORU�H[RJHQH�

3HQWUX� D� UHDOL]D� DFHVW� SURFHV� WUHEXLH� V � UHVSHFW P� XUP WRDUHOH� ipoteze fundamentale:

a) )LHFDUH�REVHUYD LH�<t reprezinW �R�IXQF LH�OLQLDU �vQ�UDSRUW�FX�YDULDELOHOH�;jt (j = 2,m) úL�SHUWXUED LD� t . Deci:

Y = X · B + u (6.2.-4)

b)��9DULDELOD�GH�SHUWXUED LH��X��DUH�R�GLVWULEX LH�QRUPDO ��DYkQG�PHGLD�QXO : E(u) = 0 (6.2.-5)

c) 3URGXVXO�GLQWUH�YHFWRUXO�YDULDELOHL�GH�SHUWXUED LH�³X´úL�WUDQVSXVD�VD�³X�´�FRQGXFH�OD�R�PDWULFH�D�F UHL�GLDJRQDO �SULQFLSDO �HVWH�IRUPDW �GLQtr-R�FRQVWDQW �(�Xið�� � ð��L� ���Q�

&RQGL LD�GH�homoskedasticitate apare sub forma:

σσ

σ=

=′

2

2

2

2n2n1n

n12121

000000

)u(E)uu(E)uu(E...................................................)uu(E)uu(E)u(E

)uu(E�

(6.2.-6)

*��YH]L�vQ�DFHVW�VHQV�FXUVXO�GH�VWDWLVWLF �VDX�ELEOLJUDILD�DIHUHQW �DFHVWXLD� **

��QRWD LD�YLQH�GH�OD�UHJUHVLH�

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d)� � 9DULDELOHOH� H[RJHQH� QX� VXQW� VWRFDVWLFH�� $FHDVWD� vQVHDPQ � F � HOH� VXQW�LQGHSHQGHQWH�ID �GH�SHUWXUED LH��ÌQ�FHHD�FH�SULYHúWH�UDQJXO�PDWULFHL�;�VH�FRQVLGHU �F �HO�HVWH�PDL�PLF�GHFkW�³Q´�GHRDUHFH�QXP UXO�REVHUYD LLORU�WUHEXLH�V �GHS úHDVF �QXP UXO�³P´�DO�SDUDPHWULORU�FH�XUPHD] �D�IL�HVWLPD L� E(u/x) = E(u) = 0 E(x’u) = x’ E(u) = 0 (6.2.-7) �������������������(��XX�[�� �(�XX�� � ðÂ,n unde In – matricea unitate de rangul “n”.

e) Variabilele exogene nu sunt corelate între ele.

Acceptând aceste ipoteze, la fel ca în cazul regresiei simple, putem determina o YDULDELO �HQGRJHQ � IXQF LH�GH�PDL�PXOWH�YDULDELOH�H[RJHQH��5HOD LD� �����–���GHILQHúWH�XQ�KLSHUSODQ�� ��vQ�VSD LXO�³P-�´�GLPHQVLRQDO��3R]L LD�OXL�³ ´�YD�IL�vQ�DúD�IHO�GHWHUPLQDW ��vQFkW�S WUDWXO�GLVWDQ HL�GLQWUH�SXQFWHOH�<t,�W� ���7��úL�� ��V �ILH�PLQLP�

3R]L LD�KLSHUSODQXOXL�� ��–GHWHUPLQDW �GH�YDORULOH�SH�FDUH�OH�LDX�SDUDPHWULL� 1�� 2�«��� m

–�VH�VWDELOHúWH�DVWIHO�vQFkW�S WUDWXO�GLVWDQ HL�GLQWUH�SXQFWHOH�<t��W� ��7�úL� �V �ILH�PLQLP� 'DF �GLVWDQ HOH�GH�OD�SXQFWHOH�<t (t ���7��úL�� ��VXQW�FRQVLGHUDWH�SDUDOHOH�FX�D[D�<1

XUPHD] �V �PLQLPL] P�H[SUHVLD�

∑ ∑= =

=εT

1t

T

1t

2t [Yt –�� 1 �� 2 Yt2 �«��� mYtm )]² (6.2.-8)

Anulând derivata în raport cu 1�úL�vPS U LWXO�HOHPHQWHOH�HFXD LHL�OD�Q��RE Lnem: y1 � 1 �� 2 x2��«�� m xm (6.2.-9)

Expresia (6.2-��� H[SULP � FRQGL LD� FD� KLSHUSODQXO� � �� V � WUHDF � SULQ� SXQFWXO� GH�

coordonate (y1 , [���3ULQ�XUPDUH��HD�SRDWH�IL�VFULV �DVWIHO�

(yt – y1�� � 2(x12 – x2���� 3 (x13 - x3���«��� m(x1m – xm ) (6.2.–10) $FHODúL�SURFHGHX�VH�XWLOL]HD] �úL�SHQWUX�GHWHUPLQDUHD�SDUDPHWULORU� j ajungându-se

OD�XQ�VLVWHP�GH�³P´�HFXD LL�QRUPDOH�

���������� 1Q��������� 2 [t2��������«���� m� [tm �� \t

1 [t2 ��� 2 [t2² +….+�� m� \t2xtm� �� \txt2

…………………………………………………………………. (6.2.-11) 1� [tm ��� 2� [t2 xtm ��«���� m� [tmð��� �� \t xtm

1HFXQRVFXWHOH� j� �M� � ��P�� VH� RE LQ� SULQ� UH]ROYDUHD� VLVWHPXOXL� �����-11). În acest sens putem aplica metoda lui Gauss. (VWLPDUHD�SDUDPHWULORU� j (j = 1,m) se poate realiza folosind un procedeu matriceal GH�FDOFXO��9RP�SRUQL�GH�OD�FXQRVFXWD�UHOD LH� ^

y = x·B + u (6.2.-12)

unde: y –�YDULDELOD�HQGRJHQ ��YHFWRU�GH�7�[���HOHPHQWH�� x – variabilele exogene (unde xt1 , t = 1,2….T-Q�SUH]LQW �XQ�YHFWRU��������������������� ����FRORDQ �IRUPDW�GLQ����

B – vectorul parametrilor de regresie (format din m x 1 elemente); u –�YHFWRUXO�SHUWXUED LHL�DOHDWRDUH��GH�GLPHQVLXQH�7�[����

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1H� SURSXQHP� V � HVWLP P� SDUDPHWULL� UHJUHVLHL� PXOWLSOH� �����-���� UHSUH]HQWD L� GH�vectorul B, asWIHO� vQFkW� VXPD� S WUDWHORU� DEDWHULORU� V � ILH�PLQLP �� 'HRDUHFH� X� UHSUH]LQW �YHFWRUXO�FRORDQ �DO�HVWLPDWRULORU�SHUWXUED LHL��DYHP�

^ u = y - x·B (6.2.-13)

ùWLLQG�GLQ�LSRWH]H�F �������������� T

Ht² = uu' (6.2.-14) t = 1

úL�IRORVLQG������-13) vom ajunge la expresia:

BxxByxBBxyyy)Bxy()Bxy(eT

1t

2t ′′+′′−′−′=−⋅∑ ′−=

= (6.2.-15)

6 �REVHUY P�F � yxB ′′ �HVWH�XQ�VFDODU��)RORVLQG�SURSULHW LOH�PDWULFHORU�RE LQHP�

^ ^ (B’x’y) = y’xB (6.2.-16)

Prin urmare (6.2.-15) devine: T ^ ^ ^

� t² = yy'- 2B'x'y + B'x'xB �PLQ�������������������������������������������-17) t = 1

Pentru minimizarea expresiei (6.2.-����VH�XWLOL]HD] �GHULYDWHOH�SDU LDOH��RE LQkQG�VLVWHPXO�

0B'xx2y'x2)'uu('B

=+−=∂

∂ (6.2.-18)

6ROX LD�DFHVWXLD�HVWH�GH�IRUPD� ^ B = (xx’) –1x’y (6.2.-19) În analiza fenomenului studiat, estimarea parametrilor j (j = 2,m) este deosebit de LPSRUWDQW �GHRDUHFH�SHUPLWH�R�LQWHUSUHWDUH�HFRQRPLF �D�DFHVWRUD��$VWIHO��SDUDPHWUXO� j (j > 1) LQGLF � vQ� FH�P VXU � VH� VFKLPE � YDULDELOD� HQGRJHQ � \� FkQG� YDULDELOD� H[RJHQ �xj (j �� ���FUHúWH�Fu o unitate. 3DUDPHWUXO� 1 ar indica, în sens teoretic, nivelul variabilei endogene în cazul în care YDULDELOHOH�VXQW�GH�YDORDUH�QXO ��3UDFWLF��DFHVW�SDUDPHWUX�DUH�R�YDORDUH�GH�FDOFXO� 6SUH� H[HPSOX�� V � QH� UHIHULP� OD� FXQRVFXWD� HFXD LH� D� GHSHQGHQ HL� FRQsumului de GHWHUJHQ L� �&t) de consumul perioadei precedente(Ct-1�� úL� YHQLWXULOH� GLQ� SHULRDGD� FXUHQW �(Vt) –�YH]L�úL�UHOD LD������-3) Altfel spus:

Ct � 1��� 2 Ct-1 �� 3 Vt

Un studiu concret, realizat pentru 1999, ne-D� FRQGXV� OD� XUP WRDUHD� HFXD LH� vQ�regresie:

Ct = 1,328266 + 0,124412 Ct-1 + 0,011395 Vt

6H� REVHUY � F � YDULDELOD� &t-1� UHSUH]LQW � ³WUDGL LD� vQ� FRQVXPXO� GH� GHWHUJHQ L´�� /D� R�FUHúWHUH�FX���D� OXL�&t-1 , Ct FUHúWH�FX������� LDU� OD�R�FUHúWHUH�D�YHQLWXULORU�FX����FRQVXPXO�VSRUHúWH� FX� GRDU� ������� 3ULQ� XUPDUH�� ³WUDGL LD´� LQIOXHQ HD] � PDL� SXWHUQLF� FRQVXPXO� GH�GHWHUJHQ L� GHFkW� YHQLWXULOH�� 'H� DLFL� UH]XOW � R� DQXPLW � LQHODVWLFLWDWH� D� FRQVXPXOXL� DFHVWXL�SURGXV�FKLPLF�vQ�IXQF LH�GH�YHQLW�

Page 7: Cap6-Estimarea Parametrilor in Analiza Econometrica

6.3.Modele probabilistice*

�������0HWRGD�YHURVLPLOLW LL�PD[LPH��090)** Este unul dintre cele mai indicate procedee de estimare a parametrilor de regresie. &RPSDUkQG� DFHDVW � PHWRG � FX� 0&003� REVHUY P� F � HD� DGXFH� vQ� SOXV� XQHOH�FRQVLGHUHQWH� SUREDELOLVWLFH� UHIHULWRDUH� OD� SRSXOD LD� GLQ� FDUH� V-D� H[WUDV� HúDQWLRQXO� VWXGLDW�(respectiv setul de date). &KLDU� úL� vQ� FD]XO� vQ� FDUH� GLVWULEX LD� FRPXQ � D� OXL� \� �YDULDELOD� GHSHQGHQW �� úL�[�YDULDELOD� LQGHSHQGHQW �� HVWH� DSUR[LPDWLY� QRUPDO �� FD]� vQ� FDUH� PHWRGD� YHURVLPLOLW LL�PD[LPH�FRQGXFH� OD� DFHOHDúL� IRUPXOH�GH�FDOFXO�SHQWUX� SDUDPHWULL� GH� UHJUHVLH� j�� VH�RE LQ�WRWXúL� XQHOH� LQIRUPD LL� VXSOLPHQWDUH� DVXSUD� SRSXOD LHL� UHSUH]HQWDWH� vQ� HúDQWLRQXO� SHQWUX�care avem date. Ne vom referi în continuare la un exemplu simplu de aplicare a MVM caracterizat SULQ�XUP WRDUHOH� a)� � )LLQG� GDW � [� �YDULDELOD� H[RJHQ ��� GLVWULEX LD� FRQGL LRQDW � D� OXL� \� �YDULDELOD� HQGRJHQ ��HVWH�QRUPDO � b) Valorile succesive [y – E(y/x)] sunt independente.Prin urmare:

(�\�[�� � ��� [ (6.3.-1) ð�\�[�� � (6.3.-2)

3UHVXSXQkQG�F �GDWHOH�GH�FDUH�GLVSXQHP��FX�SULYLUH�OD�[�úL�y sunt rezultate dintr-un VRQGDM�DOHDWRU��QH�SURSXQHP�V �RE LQHP�HVWLP UL�DOH�SDUDPHWULORU�� �� �úL� ðyx .Acceptând UHSDUWL LD�QRUPDO �úL�DYkQG�vQ�YHGHUH�UHOD LLOH�����-���úL�����-����GHQVLWDWHD�GH�UHSDUWL LH�SRDWH�

IL�VFULV �DVWIHO� 2yx

2tt

2

)]x(y[

yxtt e

2

1)x/y(F

σβ+α−

πσ= (6.3.-3)

)XQF LD�GH�YHURVLPLOLWDWH�UHSUH]LQW �SUREDELOLWDWHD�VLPXOWDQ �D�REVHUYD LLORU�SULYLWH�FD�ILLQG�IXQF LH�GH�XQ�SDUDPHWUX��,PSXQkQG�FRQGL LD�FD�YHURVLPLOLWDWHD�V �ILH�PD[LP ��VH�FHUH��GHFL�� V � DMXQJHP� OD� DFHD� HVWLPDUH� D� SDUDPHWULORU� FDUH� FRQGuce la cea mai mare YHURVLPLOLWDWH�SRVLELO � 'DF �QRW P�SDUDPHWUXO�FX� ��DWXQFL�FRQIRUP�FHORU�DILUPDWH�YRP�DYHD***:

n

/�\��� �� � �f (yt �� � (6.3.-4) t = 1

3HQWUX� D� RE LQH� PD[LPXP� GH� YHURVLPLOLWDWH� UH]ROY P� HFXD LD� GDW � FX� GHULYDWD SDU LDO �vQ�UDSRUW�FX� ��HJDODW �FX�����'HFL�

��/�\� � /��\�� �� � � �� (6.3.-5)

�� ÌQ� DFHVW� IHO� GHWHUPLQ P� H[WUHPXO� IXQF LHL� /� FDUH� YD� DYHD� vQWRWGHDXQD� R� YDORDUH�SR]LWLY ��GHRDUHFH�HD�UH]XOW �GLQWU-XQ�SURGXV�GH�IXQF LL� f(yt �� ��SR]LWLYH��YH]L�����-4). Prin XUPDUH� HD� YD� DYHD� XQ�PD[LP��$YkQG� vQ� YHGHUH� UHOD LD� �����-���� IXQF LD� GH� YHURVLPLOLWDWH�devine: *���YH]L�vQ�DFHVW�VHQV�VL�$O�7DúQDGL��%RJGDQ�7DúQDGL�³&RQWULEX LL�DOH�OXL�7U\JYH�+DDYHlmo la dezvoltarea econometriei,

în suplimentul ziarului Economistul 245/2000. **

���D�IRVW�HODERUDW �GH�5�$�)LVKHU�vQ������ *** s-a notat cu L –IXQF LD�GH�YHURVLPLOLWDWH� LQkQG�VHDPD�GH�FRUHVSRQGHQ D�vQ�HQJOH] �D�GHQXPLULL�VDOH�– Likelihood.

Page 8: Cap6-Estimarea Parametrilor in Analiza Econometrica

2yx

n

1t

2tt

2yx

2tt

2

)]x(y[

2

n2yx

n

1t

2

)]x(y[

yxe)2(e

2

1L

σ

β+α−−−

=

σβ+α−

−∑

σπ=πσ

=

=

∏ (6.3.-6)

Întrucât max L�VH�RE LQH�vQ�DFHODúL�WLPS�FX�PD[�ln L��vQ�SUDFWLF �VH�SURFHGHD] �OD�logDULWPDUHD�IXQF LHL������-���FDUH�YD�F S WD�XUP WRDUHD�IRUP �DGLWLY ��

1 1 n ln L = í� ���OQ�� � ð\[��í�� ����[yt -�� ��� [W�@ð (6.3.-7) 2

� � ���t=1

3DUDPHWUXO� �HVWH�XQ�YHFWRU�GH�FRPSRQHQWH�

� �� �� �� yx)

P ULPL�SH�FDUH�QH�SURSXQHP�V �OH�DOHJHP�vQ�DúD�IHO�vQFkW�V �UHDOL] P�PD[�/��VDX�PD[�OQ�/��FHHD�FH�HVWH�HFKLYDOHQW��3HQWUX�DFHDVWD�DQXO P�GHULYDWHOH�SDU LDOH�GXS �FXP�XUPHD] �

��OQ/��������������Q � � �����>\t -�� ��� [W�@ð� �� (6.3.-8)

�� ������� ðxx t=1

��OQ/���������������Q

� � �����[t[yt -�� ��� [W�@� �� (6.3.-9) �� ������� ðyx

t=1 ��OQ/������������Q�� 1 n

� �í� ��� ������>\t -�� ��� [W�@ð (6.3.-10) �� yx yx ñ\[����W �

'H�DLFL�UH]XOW �HFXD LLOH�

^ ^ Q ��� ��[t = �\t (6.3.-11)

t t

�[t �� ��[t² = ��[tyt (6.3.-12) t t

1 ^ ^ ðyx = ���>\t -�� ��� [W�@ð� (6.3.-13)

n t 3ULPHOH� GRX � HFXD LL� VXQW� LGHQWLFH� FX� FHOH� FDUH� IRUPHD] � VLVWHPXO� GH� HFXD LL�QRUPDOH��RE LQXW�vQ�XUPD�DSOLF ULL�0&003� (FXD LD������-����QH�RIHU � LQIRUPD LL�VXSOLPHQWDUH� UHIHULWRDUH�OD�FRPSRUWDPHQWXO� OXL�yt.(pentru diferite valori particulare ale lui xt) ,Q�VLWXD LL�vQ�FDUH�PHWRGD�YHURVLPLOLW LL�PD[LPH�HVWH�DSOLFDW �XQXL�PRGHO�FX�HFXD LL�VLPXOWDQH��OLQLDU�úL�SHUIHFW�LGHQWLILFDW�YRP�DYHD�vQ�YHGHUH�IRUPD�UHGXV �D�PRGHOXOXL�FDUH�QH�FRQGXFH�OD�R�VWUXFWXU �SHQWUX�FDUH�SURFHGHXO�SRDWH�IL�DSOLFDW�ILHF UHL�HFXD LL�vQ�SDUWH� �������$QDOL]D�ED\HVLDQ 8WLOL]HD] �LQVWUXPHQWH�PDWHPDWLFH�RIHULWH�GH�WHRULD�SUREDELOLW LORU�úL�VH�ED]HD] �SH�GRX �GLUHF LL��A) Postulatul bayesian

%��0HWRGD�ED\HVLDQ �GH�HVWLPDUH

Page 9: Cap6-Estimarea Parametrilor in Analiza Econometrica

A. Postulatul bayesian�VH� UHIHU � OD�SUREDELOLW LOH�DSULRULFH� �GHVHRUL�GH�QDWXU �HPSLULF ��DOH�XQHL�PXO LPL�GH�LSRWH]H�

Considerând un parametru “ ´� DWXQFL� H[LVW � R� GHQVLWDWH� GH� SUREDELOLWDWH� DVRFLDW �DFHVWXLD�SH�FDUH�R�QRW P� � ���(D�HVWH�GHILQLW �SH�PXO LPHD�SDUDPHWULORU�³�´�FRQVLGHUDW �FXQRVFXW �

'HQVLWDWHD� GH� SUREDELOLWDWH� � �� HVWH� FHHD� FH� QXPLP� GHQVLWDWH� DSULRULF . In ipotezD�F �YROXPXO�VHOHF LHL�HVWH� Ä]´�GHQVLWDWHD�GH�SUREDELOLWDWH�D�SDUDPHWUXOXL� ³ ´�R�YRP�numi GHQVLWDWHD�GH�SUREDELOLWDWH�DSRVWHRULF ��QRWDW �FX� � ��3RWULYLW�WHRUHPHL�OXL�%D\HV�ea se poate determina astfel:

������ �]��� �� �

z� �� � (6.3.2.-1) � � p �]�� � �G

unde: p (z) –�GHQVLWDWHD�GH�SUREDELOLWDWH�GHILQLW �SH�PXO LPHD�VHOHF LLORU�GH�YROXP�³Q´�

5HOD LD� ������-��� GHILQHúWH� OHJ WXUD� GLQWUH� SUREDELOLWDWHD� DSULRULF � � � �� úL�probabilitaWHD�DSRVWHRULF � z� ���3RVWXODWXO�ED\HVLDQ��GHFXUJkQG�FD�R�FRQVHFLQ �ORJLF �D�UHJXOLL� GH� vQPXO LUH� D� SUREDELOLW LORU*�� SUH]LQW � R� GHRVHELW � XWLOLWDWH� vQ� GRPHQLXO� OX ULL�GHFL]LLORU��GRPHQLX�vQ�FDUH�SXWHP�DILUPD�F �D�FXQRVFXW�R�DGHY UDW �UHYLJRUDUH�

Având vQ� YHGHUH� GLILFXOW LOH� vQ� GHWHUPLQDUHD� SUREDELOLW LORU� DSULRULFH� %D\HV�VXJHUHD] � FD� DFHVWHD� V � ILH� SUHVXSXVH� GH� P ULPH� HJDO �SULQFLSLXO� HFKLGLVWULEX LHL�LJQRUDQ HL��� ,Q� DERUGDUHD� ED\HVLDQ � D� GHFL]LLORU�� SUREDELOLW LOH� DSULRULFH� vQ� FRQGL LL� GH�incertitudLQH�VXQW�GHWHUPLQDWH�VXELHFWLY�SH�ED]D�H[SHULHQ HL��LQWXL LHL��IDFWRUXOXL�GH�GHFL]LH�

$FHVWH� SUREDELOLW L� SRW� IL� FRQVLGHUDWH� FD� VXVFHSWLELOH� GH� D� IL� PRGLILFDWH� vQ� XUPD�RE LQHULL�GH�QRL�LQIRUPD LL��FD�XUPDUH�D�XQRU�FHUFHW UL�VSHFLDOH�vQ�DFHVW�VHQV���&RPSDrând ³FRVWXO¶� RE LQHULL� DFHVWRU� QRL� LQIRUPD LL�� FX� DYDQWDMHOH� RE LQXWH� vQ� XUPD� ³vPEXQ W LULL´�FRUHVSXQ] WRDUH�D�SUREDELOLW LL�DSULRULFH��VH�DMXQJH�GH�DVHPHQHD�OD�R�SUREOHP �GH�OXDUH�a deciziei.

([SUHVLD� SUREDELOLW LL� DSRVWHRULFH� �DúD� FXP� R� vQWkOQLP� vQ� FHUFHW ULOH� SULYLQG�IXQGDPHQWDUHD�OX ULL�GHFL]LLORU���SRDWH�IL�SUH]HQWDW �DVWIHO�

∑=

=S

1kkk

kkk

)A(P)A/B(P

PA)A/B(P)B/A(P (6.3.2.-2)

unde: A,B –�UHSUH]LQW �VW UL��GLQWUH�FDUH�XQD�SRDWH�IL�FXQRVFXW �DSULRULF�– în cazul nostru B); k = 1,2,……S –�VLWXD LL�SRVLELOH� B) 0HWRGD�ED\HVLDQ �GH�HVWLPDUH

'DF �0&003�SUHVXSXQH�SDUDPHWUL�GH�UHJUHVLH� j (j = 1,m) FD�ILLQG�QLúWH�FRQVWDQWH�

QHFXQRVFXWH�� SURFHGHXO� ED\HVLDQ� GH� HVWLPDUH� FRQVLGHU � FXQRVFXWH� XQHOH� LQIRUPD LL�(referitoare, de exemplu, la legea GH�GLVWULEX LH���SULYLQG�SDUDPHWUL�FD�XUPHD] �D�IL�HVWLPD L�

3URFHGHXO�VH�ED]HD] �SH�XUP WRDUHOH�SRVWXODWH� a)��3DUDPHWUL�QHFXQRVFX L� �DSDU LQ�XQHL�FODVH�GH�GLVWULEX LL�DSULRULF�FXQRVFXWH����GH�UHJXO �GLVWULEX LD�*DXVV�–Laplace). ^ b)� �2ULFH�PHWRG � GH� HVWLPDUH� D� SDUDPHWULORU� � SULQ� � R� FRQVLGHU P�PHWRG � vQWkPSO WRU�DGHY UDW � �SRWULYLW ��� 'H� DVHPHQHD�� FRQVLGHU P� F � DSOLFDUHD� DFHVWHL� PHWRGH� QH� SRDWH�FRQGXFH�OD�DEDWHUL��QHSRWULYLUL��PDL�PLFL�VDX�PDL�PDUL��GHSLQ]kQG�GH�GLVWULEX LD�SDUWLFXODU �

* vezi curVXO�GH�PDWHPDWLF ��UHIHULWRU�OD�WHPHOH�GH�WHRULD�SUREDELOLW LORU

Page 10: Cap6-Estimarea Parametrilor in Analiza Econometrica

D�SDUDPHWULORU� �úL�GH�SURFHGHXO�GH�HVWLPDUH��vQWkPSO WRU�DGHY UDW��SH�FDUH-O�QRW P�FX�'� )XQF LD�DFHVWRU�DEDWHUL��QHSRWULYLUL��DU�SXWHD�IL�GHVFULV �DVWIHO�

^ �'� �� �� �–� �2 (6.3.3-3)

c)�0HWRGD�ED\HVLDQ �GH�HVWLPDUH�GHILQHúWH� vQ�IDSW�XQ�procedeu de selectare a celei mai EXQH�PHWRGH� GH� HVWLPDUH� vQWkPSO WRU� DGHY UDWH� �SRWULYLWH��� GHILQLQG� FODVH�DOWHUQDWLYH� GH�HVWLP UL��úL�HYDOXkQGX-OH�vQ�WHUPHQLL�YDORULL�DúWHSWDWH�D�IXQF LHL�GH�QHSRWULYLUH� �'� ���,QWU-un FD]�LGHDO�DFHDVW �IXQF LH�WUHEXLH�V �vQGHSOLQHDVF �XUP WRDUHD�FRQGL LH�

�'� �� ��

3HQWUX� D� GHWHUPLQD� HVWLPDWRUXO� ED\HVLDQ� DO� SDUDPHWUXOXL� � V � SUHVXSXQHP� XQ�VRQGDM�DOHDWRU�GH�GLPHQVLXQH� ÄQ´�� SHQWUX�FDUH� DYHP�R� IXQF LH� GH� GHQVLWDWH� I�[� ��– spre H[HPSOX�� UHSDUWL LD� QRUPDO �� &HHD� FH� QH� SURSXQHP� HVWH� GH� D� HVWLPD� SH� Ä ´� FDUH� D�GHWHUPLQDW�IXQF LD�GH�GHQVLWDWWH�I�[� ��SHQWUX�VRQGDMXO�UHVSHFWLY� Pentru aceasta avem în vedere GHQVLWDWHD�PDUJLQDO �D�YDULDELOHL� ��S� ��úL� IXQF LD�abaterii ),ˆ( θθλ . 9DORDUHD�PHGLH� D� IXQF LHL� DEDWHULL� SRDWH� IL� FRQVLGHUDW drept „un risc” de a nu se potrivi:

^ (� � � ��� �)�G� �� (6.3.2-4)

Intrucât „ ´� HVWH� R� YDULDELO � DOHDWRDUH�� VXQWHP� LQWHUHVD L� vQ� D� J VL� IXQF LD� G�[1, x2,……….., xn ) unde x1, x2,……….., xn VXQW�YDORULOH�REVHUYDWH�vQ�XUPD�VRQGDMXOXL�úL�DQXPH�acea IXQF LH� FDUH� YD� PLQLPL]D� ÄULVFXO´� DVWHSWDW� SHQWUX� HVWLPDUHD� OXL� �� QRWDW � FX� G�[1, x2,……….., xn ���,Q�DFHVW�FD]��´G´�YD�IL�DúD-QXPLWXO�HVWLPDWRU�ED\QHVLDQ�DO�SDUDPHWUXOXL� � Prin urmare „riscul” mediu va fi de forma: (6.3.2-5)

unde g ( x1, x2,……….., xn � ��–�GHQVLWDWHD�VLPXOWDQ �D�YDORULORU�REVHUYDWH���� x1, x2,……….., xn;

Schimbând ordinea de integrare în (6.3.2-5) în raport cu valorile xt�úL� �RE LQHP� (6.3.2-6) 1H� LQWHUHVHD] � GHWHUPLQDUHD� IXQF LHL� G�[1, x2,……….., xn ) caUH� PLQLPL]HD] �

H[SUHVLD�GLQ�DFRODG �SUH]HQW �vQ�������-���DGLF � (6.3.2.-7)

Notând cu k(x1, x2,……….., xk ���GLVWULEX LD�PDUJLQDO �D�YDULDELOHL�;i (i=1,n), atunci GLVWULEX LD�FRQGL LRQDW �GH� ��GDW �GH�[1, x2,……….., xn va fi:

g(x1, x2,……….., xn � �S� �

+� ��[1, x2,……….., xn ) = �������������������-8) k(x1, x2,……….., xn )

Expresia (6.3.2-7), prezentând similitudini cu (6.3.2-���� UHSUH]LQW � densitatea DSRVWHRULF �

Estimatorul bayesian va fi acea valoare a lui θ FDUH� SHQWUX� RULFH� VHOHF LH������������������x1, x2,……….., xn va minimiza cantitatea.

{ }∫∞∞− ∫∞∞− θθ∫∞∞− θ∫∞∞− θλ=θθθ= )(d)(pdxn.....2dx1dx)/nx,...,2x,1x(g),d(......d)(p),d(F)d(R

[ ]{ } ndx.....- 1dx )nx,....1x(g ),n,....x1d(x ..... R(d) - ∫ ∫ ∫ λ∞∞

∞∞

∞∞−=

[ ] d � )p(g(x1,...xn xn),d(x1,..... ∫ ∞∞− λ

Page 11: Cap6-Estimarea Parametrilor in Analiza Econometrica

^

4� ��[1, x2,……….., xn ) = �R � � ��+� ��[1, x2,……….., xn �G �� (6.3.2-9) 0HWRGD�ED\HVLDQ � GH�HVWLPDUH�SRDWH� IL� DSOLFDW � SHQWUX� D� HVWLPD� SDUDPHWUL� IRUPHL�

UHGXVH�D�XQXL�VLVWHP�GH�HFXD LL�VLPXOWDQH�FX�FRQGL LD�FD�GLVWULEX LD�V �SRDW �IL�DSUR[LPDW �

6.1. �&kWHYD�SURSULHW L�DOH�HVWLPDWRULORU

,Q� VLWXD LD� vQ� FDUH� IHQRPHQXO� HFRQRPLF� GHVFULV� GH�PRGHOXO� HFRQRPLFR-matematic HVWH� vQ� FRQFRUGDQ � FX� LSRWH]HOH� IXQGDPHQWDOH� DYXWH� vQ� YHGHUH� SHQWUX� UHJUHVLD� OLQLDU �PXOWLSO � ED]DW � SH� OHJHD� QRUPDO �� DWXQFL� HVWLPDWRULL� j� �M ��P�� YRU� vQGHSOLQL� XUP WRDUHOH�SURSULHW L��ILLQG�

- QHGHSODVD L� - GH�GLVSHUVLH�PLQLP � - HILFLHQ L� - DVLPWRWLF�HILFLHQ L� - FRQVLVWHQ L� 6 �DQDOL] P�vQ�FRQWLQXDUH�VHPQLILFD LD�DFHVWRU�SURSULHW L�

a) estimator nedeplasat. In cD]XO� vQ� FDUH� SUREDELOLWDWHD� GH� GLVWULEX LH� D� XQXL� HVWLPDWRU� β are o medie m,

DWXQFL��GDF �SHQWUX�RULFH�HúDQWLRQ�PHGLD�H[LVW �úL�HVWH�HJDO �FX�SDUDPHWUXO� ��DILUP P�F �estimatorul β � HVWH� QHGHSODVDW� GDF � DFHDVW � PHGLH� H[LVW �� GDU� GLIHU � GH� �� $WXQFL� P- �

UHSUH]LQW � GHSODVDUHD� GDW � GH� β �� 0HQ LRQ P� F � SULQ� deplasare� vQ HOHJHP� R� DEDWHUH�VLVWHPDWLF �D�XQXL�UH]XOWDW�RE LQXW�SULQ�PHWRGH�VWDWLVWLFH�GH�OD�YDORDUHD�UHDO �

b) HVWLPDWRU�GH�GLVSHUVLH�PLQLP . In cazul în care estimatorul �DUH�R�GLVSHUVLH�YDU� ��úL�SHQWUX�RULFH�HúDQWLRQ�DFHDVW �

UHJUHVLH�H[LVW �úL�HD�HVWH�FHO�PXOW�HJDN �FX�GLVSHUVLD�RULF UXL�DOW�HVWLPDWRU�DO�OXL� ��DWXQFL�

β HVWH�XQ�HVWLPDWRU�GH�GLVSHUVLH�PLQLP �SHQWUX� . c) estimator eficient 5HSUH]LQW �DFHO�HVWLPDWRU� �FDUH�HVWH�QHGHSODVDW�úL�GH�GLVSHUVLH�PLQLP ��'HFL��DFHVWD�

vQWUXQHúWH�DPEHOH�FRQGL LL�GHVFULVH�DQWHULRU��8QHRUL�HO�HVWH�QXPLW�úL�FHO�PDL�EXQ�HVWLPDWRU�nedeplasat.

d) Estimator asimtotic eficient.

Este considerat în acest fel acel estimator ��FDUH�HVWH�FRQVLGHUDW� - consistent; - asimtotic normal distribuit; - GLVSHUVLD�DVLPWRWLF � 0² a statisticii cn( β -�E��HVWH�GH�FHD�PDL�PLF �YDORDUH� 0HQ LRQ P�F � E� UHSUH]LQW � R� FRQVWDQW � LDU� Fn R� IXQF LH� GH� YDULDELO � Q� DVWIHO� vQFkW�

pentru orice n, cn > 0. $OWIHO� VSXV�� GDF � GLVSHUVLD� UHVSHFWLYXOXL� SDUDPHWUX�� HVWLPDW� SULQ� 0&003� HVWH�

DVLPWRWLF� HFKLYDOHQW � FX� GLVSHUVLD� HúDQWLRQXOXL� GH�PD[LP � YHURVLPLOLWDWH�� DWXQFL� β este asimtotic eficient.

e) estimator consistent.

Page 12: Cap6-Estimarea Parametrilor in Analiza Econometrica

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