B. Modelos Lineales y No Lineales

7/25/2019 B. Modelos Lineales y No Lineales

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Sistema Normal de Ecuaciones (Modelo de regresión lineal)

Año

Xi*Yi Xi^2 yi^2 xi(Xi!X"") yi(Yi!Y"") yi(Yi!Y"")^2 xiyi y^ uY!Y^ (Y#!Y^2)

$%%& ''&&%' &%'+%$+& 2&,'$ $2%&$% --+2, !-'+' 2$2$'& !2+- -&+2, %- '22$%+$% -$& $+%2$

$%%- '%-%+ -+,2'&+,$ 2,&-+2 $''&22 -,2,%' !-$2- $++'% !$'$ &+%%, 2'+, '-,2&,,-2 -' 2$%

$%% '-,,%%% -,2++$'2$ &$& 2&-$% -2,,,$ !&&+ $$22&% !2$'$ -''%$ 2&%2 '%++2%%' !&$ $2&-',

$%%' '%&&'&2' &'&%&$-2 &%-+ &2%+'2 -,+& !2-'' '+,+2 !$' &$-- -&% &%&-2 !--- $%-+-

$%% -$'-$&% '22'-+' -'$,+$ &,2- ++&2 !$% &%+2- !$2%& $'2' 2-% '-2$- !22' $2&,

$%%, '$2$% ''%-%$&$ $%,' --,$%, '+2%$ !$+ 222- !%- ,%22 $-2&% ,%-22- !$&+ $++%

$%%% %22$, $&$,-2--- ','' +,'&2 '&'& !$++ $$- !& -&% ,%- ,$&$'+,&$ !$% 2-+

2+++ ,&%--' %$''', ''--, '2'%2 +-'$ !2,' ,2$ !&$ %%- %+& ,-%%', !$'$ 2+

2++$ ,&-&'&- '-'%-%$% '&,+&- ,-, '%'$'& ! &+,& !&'' $&-+ 2+&& ,-+%,$-&, !+'' +-&%

2++2 ,&-'--2 ,+-22+,$ '2+$ '+%+% '%,++$ !&%, $,- !& $2'$ $-$& ,-%-,2 !$-+ $%'&$

2++& ,-&+&$ ,&-''2&2 ''$, '$%%$ $,++& !&2% $+,$ !2&' , ,&2$-, !+% +&--+

2++- %$++- ,%2+'-& ,2&%$ &,&$2 ,2%$+2 &%+ $2& &%' $' $- ,%2++-2& $, &-22

2++ ,,'%+' ,'''%%%%, ',,++ $2%+ ,'$ -' 2$' $'+ 2' - ,%'$$$' !+%$ +,&'

2++' %'+%+, %,+++-$2, %&+ %'-, %$-$+% $, 2-,%- ,$ 2- $&-&+ %'$,'- !++$ ++++$

2++ %,'$%+- $+$&''- %%%'' $+2$' %22 $%&- &-2$ $$2 $&2' 222,% %--,-% $+ $$&%

2++, $++ $++ $++++++ $++++++ $++++++ $%, &2&2$ $2%+ $''-% 2&$% %',+',& &$% $+$%'&

2++% %2%%''$ ,,2$,-2 ,-+$% ,2+ %+,2+& '2+ &,-+ ,2+ '2, +,& %+--&',,& -,' 2&,+

2+$+ $++$'%'' $+'&--&$-$ $+'2- $$&+%$$ $++&&%' 2-&2 %$ $&+ $+,% &$% $++2&&&' !++' +++-$

2+$$ $+-$-$2- $$+,$&+- $$%,-2 $&2-&2 $+,--+ &&+' $+%2%2 $+- 2%++ '&- $+-%2$,, !+,$ +''

2+$2 $+,2$&'' $2$,,,%' $&$%++& $-,',, $$$+2+ &%, $,%&' 2$$2 --%$ ,-$, $+,'2,,& !+-2 +$2-

2+$& $+%&'-&% $2&%+&,' $&$$$ $2'2& $$%'+ -$ $2,-, 222 -%,& %2' $+%$--% !+$% ++&+

Suma 1829.04 1722.46 156704.93 153654.46 163039.32 0.00 12374.44 0.00 3735.3535 6683.37 1829.04 0.00 125.7043

Media ,$+ ,2+2

.ndice del/i0 (Yi)

.ndice de lasEx1orta (xi)

xi^2(Xi!X"")^2



y^ i Y !̂ym (Y^!ym)̂ 2

22& !2-,, '$,,,

&+22 !222 -%'$+

!2-2-' !$,+% &2-+

!&2,+ !$&&2 $&'

!$&++ !$+' $$&,-

!$+2% !,$- ''2%

!$2%'$ !, &&-2

!$&&& !$ 2&%

!' !&++ ,%%

!$$%+2 !2$ -'2

!++- !$, &$

$'+& 2$$ ---

!,$%$ 2$ '&2

!+%2 ,2 2'2

$+-, $+- $+%$'&+%$2 %$ %-2,

-&%$% && $$2+

!'&, $&$- $2'

!,$$ $,' &$,,+

!-$+ 2$& -'&'2

!2+-% 22- +-2+

0.00 0.00 3609.6443



Sistema Normal de Ecuaciones (Modelo de r

Año Xi*Yi Xi^2 YiX^2 X^& X^- Y^2 xi(Xi!X"")

$%%& ''&&%'$ &%'+%$+& 2&,'$ $2%&$% ,,+,+% -'+-$, $'2&&2,$ --+2, !-'+'

$%%- '%-%+-, -+,2'&+,$ 2,&-+2 $''&22 $$,'2 ',&+'$ 2''&$+' -,2,%' !-$2-

$%% '-,,%%%& -,2++$'2$ &$& 2&-$% $-$&', $$-22-& -22$%, -2,,,$ !&&+

$%%' '%&&'&2 &'&%&$-2 &%-+ &2%+'2 22,$%- $,,'2%- $+,2,$,-$2 -,+& !2-''

$%% -$'-$&, '22'-+' -'$,+$ &,2- 2,2+$ 2-$-2,+ $+&2%&, ++&2 !$%

$%%, '$2$, ''%-%$&$ $%,' --,$%, &-,+&$2$ &+++' 2++,,$--' '+2%$ !$+

$%%% %22$,$ $&$,-2--- ','' +,'&2 -+-%2& &'2-,$' 2,+'2&% '&'& !$++

2+++ ,&%--'-' %$''', ''--, '2'%2 2%%+2 -%%%+$ &%2'$%% +-'$ !2,'

2++$ ,&-&'&&% '-'%-%$% '&,+&- ,-, -,%+$$- --$'$2 &-$%-22%2 '%'$'& !

2++2 ,&-'--$' ,+-22+,$ '2+$ '+%+% +,,-',+ -&22% &+%2-+2' '%,++$ !&%,

2++& ,-&+&$$ ,&-''2&2 ''$, '$%%$ 22,-%2 -,,+, &,-2%-2-$' $,++& !&2%

2++- %$++- ,%2+'-& ,2&%$ &,&$2 '22++- '&-&%-,+ -$+-$-+ ,2%$+2 &%+

2++ ,,'%+'-, ,'''%%%%, ',,++ $2%+ '''&2&, '$$%,& '--&+$& ,'$ -'

2++' %'+%+,& %,+++-$2, %&+ %'-, %$--,'$2 %&--2- %$-,'&$,'% %$-$+% $,

2++ %,'$%+- $+$&''- %%%'' $+2$' $+$&&2'$ $+-$'%, $+,%%&- %22 $%&-

2++, $++ $++ $++++++ $++++++ $++++++++ $++++++++ $++++++++++ $++++++ $%,

2++% %2%%''$2 ,,2$,-2 ,-+$% ,2+ -$''%22 ','%' '+'2-$- %+,2+& '2+

2+$+ $++$'%''& $+'&--&$-$ $+'2- $$&+%$$ $$&2,&++ $2+2'%,, $2,%'+-++- $++&&%' 2-&2

2+$$ $+-$-$2&% $$+,$&+- $$%,-2 $&2-&2 $&%2$% $2-$+'' $&%'$'-' $+,--+ &&+'

2+$2 $+,2$&'% $2$,,,%' $&$%++& $-,',, $'+$$% $,$+,,+- 22+2',-$- $$$+2+ &%,

2+$& $+%&'-&% $2&%+&,' $&$$$ $2'2& $'+'& $,,,+%'- 2&&&'&$-+-+ $$%'+ -$

Suma 1829.04 1722.46 156704.93 153654.46 14471383.55 14603021.18 1456750269.82 163039.32 0.00

Media ,$+ ,2+2

Estimadores Errores estandar 34e 5ariacióngl

6.nsesgadas

a 48.16743 5.378784 4e0ido a la regresión 3616.9708 $ &'$'%+,

0 0.393972 0.138124 4e0ido a lo residual $$,$-,& $% '2$,&&$%

c 0.000904 0.000843 Error total &&&& 2+ $,'''

.ndice del/i0 (Yi)


Suma decuadrados



egresión cuadratico)

yi(Yi!Y"") yi(Yi!Y"")^2 xiyi y^ uY!Y^ (Y!Y^)^2 y^ i Y^!ym (Y^!ym)^2

2$2$'& !2+- -&+2, %- '&+-+'& 2, ,$22+ $,+%, !2&% ''&

$++'% !$'$ &+%%, 2'+, '&,$%, & $-+,$ 2-''% !2$&' -'2+

$$22&% !2$'$ -''%$ 2&%2 '%-$$$- !&%2 $&,&2 !222- !$'% &$2%

'+,+2 !$' &$-- -&% &-$%&- !--$ $%-+%- !&2-,, !$&&' $,&'

&%+2- !$2%& $'2' 2-% '2+-+%& !2+- -$'$- !$- !$+,% $$,''

222- !%- ,%22 $-2&% ,%-'$' !+%- +,%++ !-$ !,+ 22%

$$- !& -&% ,%- ,+,'2%2& !$$- $&+$& !%22- !'2& &,,'

,2$ !&$ %%- %+& ,+$', !$+ $$$- !%$2& !2+, -&2

&+,& !&'' $&-+ 2+&& ,&,+-,- !+$- ++2+, !$2+- !&2 $2&

$,- !& $2'$ $-$& ,--$%' !+, +'2 !&& !2', $

$+,$ !2&' , ,-%+$$ !++' +++&$ !-2 !2&$ &2

$2& &%' $' $- ,,'%&,&, 2&' & 2+%-2 $'+ 2

2$' $'+ 2' - ,%$+-+ !+-$ +$',- !&'' 2+$ -+-

2-,%- ,$ 2- $&-&+ %&--&$ +2' ++++ 222 ,2 ',+2

&-2$ $$2 $&2' 222,% %&%$& $2& $+'& $$%& $+2% $+%,

&2&2$ $2%+ $''-% 2&$% %''+-'& &-+ $$2, &2,+$ %$ %+&%

&,-+ ,2+ '2, +,& ,%%,-+ &- 2,2%+ -,+-% 2,' ,$%

%$ $&+ $+,% &$% $++2, !+$2 ++$&% !$$,2 $&$% $&%%

$+%2%2 $+- 2%++ '&- $+-,% !$&- $,, !$-$+' $,&, &&,,

$,%&' 2$$2 --%$ ,-$, $+%'$,,2 !$-$ $%- !$-+& 222 +2&

$2,-, 222 -%,& %2' $$+'+%$ !$&$ $++ !$--% 2& &

12374.44 0.00 3735.3535 6683.37 1 829.02 0.02 118.1483 0.00 0.00 3616.9708

7once1to Sesgado .nsesgado

74e determin 0.96 837 0.96485 6

74ee correlció +%,-+%2 +%,22+,&,

xi^2(Xi!X"")^2



Sistema Nomal de Ecuaciones (Modelo de regresión 1otencial)

Año 89: Xi 89: Yi log y^

$%%& ''&&%'+% &%'+%$+& $' $,2 2,& 2-2 &&2 !+&- +$$&+ !+$$ ++$2+ $,2+&$ ++- +++$'

$%%- '%-%+-' -+,2'&+,$ $'$ $,- 2% 2% &&% !+2, ++%&$$ !++% +++,'22 $,+',- ++- +++$2

$%% '-,,%%%&& -,2++$'2$ $'% $,2 &+' 2,- &&+ !+2$ ++-2+% !+$2 ++$-++ $,-$$2' !++2 ++++'

$%%' '%&&'&2'% &'&%&$-2 $' $,- &2- &+% &&% !+$& ++$,$$ !++% +++,,+& $,-$% !++& +++$$

$%% -$'-$&,'' '22'-+' $% $, &&' &22 &+ !+$+ +++%2 !++' +++-$2 $,%+&- !++2 ++++-

$%%, '$2$,& ''%-%$&$ $,& $,% &- &&& & !++ +++--+- !++- +++$%%2 $%+-'&, !++$ ++++2

$%%% %22$,$- $&$,-2--- $, $%+ &2 &-& &'2 !++- +++$$& !++& +++$$+& $%$$$ !++2 ++++2

2+++ ,&%--'-'$ %$''', $%+ $%2 &' &'+ &+ ++$ +++++-$ !++$ ++++$$' $%&',% !++$ ++++2

2++$ ,&-&'&&%$ '-'%-%$% $,, $%2 &'2 & &'% !++$ +++++- !++$ ++++$,+ $%&+,& !++$ ++++$

2++2 ,&-'--$, ,+-22+,$ $,% $%2 &'- &, &'% +++ +++++++ !++$ ++++$' $%&-,,% !++$ ++++2

2++& ,-&+&$$- ,&-''2&2 $%+ $%& &'' &'+ &2 +++ +++++$ !++$ +++++- $%&''&2 !++$ ++++$

2++- %$++--' ,%2+'-& $%& $%' &% &- &,- ++- +++$'' ++2 ++++'+$ $%&,& ++$ +++++

2++ ,,'%+'-, ,'''%%%%, $%- $% & &' &% ++ +++2+%, ++$ ++++$2 $%%2 !++$ ++++$

2++' %'+%+,& %,+++-$2, $%% $%, &%- &%' &%2 +$+ +++%'$ ++ +++2+%+ $%%-$$ +++ +++++

2++ %,'$%+-'%% $+$&''- 2+$ $%% -++ -+2 &%, +$$ ++$2%+ ++' +++&+2 $%,'- ++$ ++++$

2++, $++ $++ 2++ 2++ -++ -++ -++ +$$ ++$$'-& ++ +++-2& $%,&%% ++2 ++++&

2++% %2%%''$$ ,,2$,-2 $% $%, &, &% &%2 ++ +++2,, ++- +++$%'& $%%+'% ++2 ++++-

2+$+ $++$'%''& $+'&--&$-$ 2+& 2++ -+ -$$ -++ +$& ++$,$22 ++ +++-&+ $%%%& +++ +++++

2+$$ $+-$-$2&, $$+,$&+- 2+' 2+2 -$' -2 -+ +$ ++2,&+ ++, +++','2 2+$$$2 ++$ +++++

2+$2 $+,2$&', $2$,,,%' 2+% 2+& -2- -& -$- +$% ++&,- +$+ +++%%++ 2+22,-% ++$ ++++$

2+$& $+%&'-&,% $2&%+&,' 2+% 2+- -2 -&, -$' +2+ ++&%%'- +$+ ++$+,& 2+2% ++$ ++++2

Suma 1829.04 1722.46 39.73 40.63 77.07 75.61 78.71 0.00 0.43 0.00 0.0965 4 0.63 0.00 0.0071

Media ,$+ ,2+2 $,% $%&

Estimadores 34e 5ariación gl 7once1to Sesgado .nsesgado

a 11.89267 1.52512 4e0ido a la regresión 0.0895 $ ++,%-22 74e 4 0.92 65 1 0.92 26 42

0 0.45426 0.029351 4e0ido a lo residual ++$ $% ++++2'& 74e 7 +%'2&,% +%'+-2'

.ndice del /i0(Yi)


(89:Xi)(89:Yi)

(89:Xi)^2

(89: Yi)^2

89: Xi!89: X;;

(89: Xi!89:X;)^2

89: Yi!89: Y;;

(89: Yi!89: Y;;)^2

ulogY!logY^

u(logY!logY^)^2

Errores estandarSuma decuadrados

6.nsesgadas



Error #o +%' 2+ ++-,2



log y^! log y;; Y^

!+$ ++2 '+-!+$& ++2 '-$+

!++% ++$ '%&'

!++' +++ -,-

!++- +++ '%

!++& +++ ,+2%

!++2 +++ ,2'&

+++ +++ ,''&

+++ +++ ,2%

+++ +++ ,'+,

+++ +++ ,'-2

++2 +++ ,%%2

++2 +++ %+2,

++- +++ %&

++ +++ %'%&

++ +++ %'&-

++2 +++ %$+$

++' +++ %%+

++, ++$ $+2'%

++% ++$ $+-+

++% ++$ $+'+

0.0000 0.0895 1826.9401

(log y^!log

y;;)^2





Sistema Nomal de Ecuaciones (Modelo de regresión ex1onencial)

Año 89: Xi 89: Yi XiYi Xi^2 Xi89:Yi log y^

$%%& ''&- &%'$ $' $,2 2&,'$ &&2 $2%&$% '$%%2 !+&- +$$&+ !+$$ ++$2+ $,+,' ++$

$%%- '%-%$ -+,& $'$ $,- 2,&-+2 &&% $''&22 $$,'& !+2, ++%&$$ !++% +++,'22 $,2$, ++2

$%% '-,% -,2+ $'% $,2 &$& &&+ 2&-$% ,,$2+$, !+2$ ++-2+% !+$2 ++$-++ $,-&+ !++&

$%%' '%&&' &'- $' $,- &%-+ &&% &2%+'2 $+'+- !+$& ++$,$$ !++% +++,,+& $,'2 !++&

$%% -$'- '22' $% $, -'$,+$ &+ &,2- $$'-22-, !+$+ +++%2 !++' +++-$2 $,,+' !++$

$%%, '$ ''%-, $,& $,% $%,' & --,$%, $2'-+,- !++ +++--+- !++- +++$%%2 $,%& +++

$%%% %22 $&$, $, $%+ ','' &'2 +,'&2 $&'$'&& !++- +++$$& !++& +++$$+& $%+- +++

2+++ ,&%- %$, $%+ $%2 ''--, &+ '2'%2 $22%,'+, ++$ +++++-$ !++$ ++++$$' $%2'% +++

2++$ ,&-&' '-'% $,, $%2 '&,+&- &'% ,-, $-'%2+& !++$ +++++- !++$ ++++$,+ $%$%' +++

2++2 ,&-' ,+-2 $,% $%2 '2+$ &'% '+%+% $-%%%$+$ +++ +++++++ !++$ ++++$' $%2&% +++

2++& ,-& ,& $%+ $%& ''$, &2 '$%%$ $$,+&,% +++ +++++$ !++$ +++++- $%2, +++

2++- %$+ ,%2 $%& $%' ,2&%$ &,- &,&$2 $',&&&2' ++- +++$'' ++2 ++++'+$ $%- ++$

2++ ,,'% ,'' $%- $% ',,++ &% $2%+ $',,&,%& ++ +++2+%, ++$ ++++$2 $%- +++

2++' %'+% %,++ $%% $%, %&+ &%2 %'-, $%&'%2,,, +$+ +++%'$ ++ +++2+%+ $%,+ +++

2++ %,'$% $+$&'' 2+$ $%% %%%'' &%, $+2$' 2+2$2+, +$$ ++$2%+ ++' +++&+2 $%,, ++$

2++, $+++++ $+++++ 2++ 2++ $++++++ -++ $++++++ 2++ +$$ ++$$'-& ++ +++-2& $%,-+ ++2

2++% %&++ ,,2$, $% $%, ,-+$% &%2 ,2+ $-%2&,2 ++ +++2,, ++- +++$%'& $%$, ++&

2+$+ $++$+ $+'&-- 2+& 2++ $+'2- -++ $$&+%$$ 2$2''%2 +$& ++$,$22 ++ +++-&+ 2++$- +++

2+$$ $+-$-$ $$+,$ 2+' 2+2 $$%,-2 -+ $&2-&2 2&2$%+% +$ ++2,&+ ++, +++','2 2+2- !++$

2+$2 $+,2$- $2$,,% 2+% 2+& $&$%++& -$- $-,',, 2-%'$%- +$% ++&,- +$+ +++%%++ 2+--+ !++$

2+$& $+%&'- $2&% 2+% 2+- $&$$$ -$' $2'2& 2$%%%+2 +2+ ++&%%'- +$+ ++$+,& 2+-, !++$Suma 1829.04 1722.46 39.73 40.63 156704.93 78.71 153654.46 3366.50 0.00 0.43 0.00 0.0965 40.63 0.000

Media ,$+ ,2+2 $,% $%&

Estimadores 34e 5ariación gl 6.nsesgadas

a 51.29002 1.256338 4e0ido a la regresión 0.0929 $ ++%2%+$, 7once1to

0 1.006329 0.000288 4e0ido a lo residual +++&'&'&%%' $% ++++$%$&,% 74e 4

.ndice del/i0 (Yi)


(89: Yi)^2

xi(89:Xi!89:X;;)

xi(89: Xi!89: X;)^2

89: Yi!89: Y;;

(89: Yi!89: Y;;)^2

ulogY!logY^

Erroresestandar

Suma decuadrados



Error total ++%' 2+ +++-,2 74e 7



Y^

++++2 !+$& ++2 '-&

++++- !+$$ ++$ ''&-

++++ !++% ++$ '%''

++++ !++ +++ &''

++++$ !++ +++ %

+++++ !++- +++ ,2

+++++ !++& +++ ,+-&

+++++ !++$ +++ ,-$

+++++ !++2 +++ ,&+%

+++++ !++$ +++ ,&%2

+++++ !++$ +++ ,-2%

++++2 ++$ +++ ,,2+

+++++ ++$ +++ ,,'2

+++++ ++- +++ %+'

+++++ ++ +++ %22

++++& ++ +++ %'&%

++++ ++2 +++ ,%-%

+++++ ++ +++ $++&&

++++$ ++% ++$ $+'+$

++++$ +$$ ++$ $$+''

++++$ +$$ ++$ $$$,'

0 .0 03 63 64 0 .0 00 0 0 .0 92 9 1 82 8. 31 85

Sesgado .nsesgado

0.962336 0.960353

u(logY!logY^)^2

log y^! logy;;

(log y^!logy;;)^2



+%,+%,2, +%%%'

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B. Modelos Lineales y No Lineales