fakulta matematiky, fyziky a
informatiky univerzity komenskeho
v bratislave
Projekt z Ekonometrie
Bratislava 2008Simona Stefanovicova, Martin Takac
Fakulta Matematiky, Fyziky a Informatiky, Univerzita Komenskeho v Bratislave
Ekonomicka a financna matematika
Modelovanie mortality od znecistenia
Semestralna praca
Martin Takac, Simona Stefanovicova
Bratislava 2008
Obsah
1 Teoreticky uvod 21.1 Testy normality chyb v modeli . . . . . . . . . . . . . . . . . . 31.2 Testy na heteroskedasticitu . . . . . . . . . . . . . . . . . . . . 41.3 Kriteria na odhalenie multikolinearity . . . . . . . . . . . . . . 71.4 Testovanie signifikancie parametrov . . . . . . . . . . . . . . . 81.5 Testovanie linearnych hypotez . . . . . . . . . . . . . . . . . . 101.6 Testovanie signifikancie modelu . . . . . . . . . . . . . . . . . 111.7 Testovanie submodelu . . . . . . . . . . . . . . . . . . . . . . 11
2 Zakladne informacie o datach a modeli 12
3 Vychodiskovy model 143.1
”Pracovne” hypotezy o modeli . . . . . . . . . . . . . . . . . . 14
3.2 Odhady parametrov v prvotnom modeli . . . . . . . . . . . . . 153.3 Whiteov test heteroskedascity . . . . . . . . . . . . . . . . . . 223.4 Goldfeld-Quandtov test heteroskedascity . . . . . . . . . . . . 233.5 Breusch-Paganov test heteroskedasticity . . . . . . . . . . . . 27
4 Testy linearnych hypotez o modeli 294.1 Testovanie hypotezy o koeficientoch . . . . . . . . . . . . . . . 294.2 Testovanie pomocou submodelu . . . . . . . . . . . . . . . . . 29
5 Zaver 32
6 Prılohy 336.1 Popisne statistiky pouzitych premennych . . . . . . . . . . . . 336.2 Korelacna matica pouzitych premennych . . . . . . . . . . . . 39
1 Teoreticky uvod
Ciel’om nasho projektu je vytvorit’ regresny model a testovat’ splnenie Gauss- Markovovych podmienok a roznych hypotez o nom
Vseobecny regresny model
yi = β0 + β1xi1 + β2xi2 + . . . + βk−1xik−1 + ε (*)
Gauss - Markovove podmienky(Basic Assumption)
1. Model je platny.
2. Stredna hodnota chyb modelu je 0, t.j.
E(ε) = 0. (1)
3. Chyby v modeli vykazuju homoskedasticitu a parovu nakorelovanost’,t.j.
V ar(ε) = σ2I. (2)
4. X je nestochasticka (stlpce matice X su stochasticky nekorelovane schybami v modeli), t.j.
E(XTε) = 0. (3)
5. Nahodne chyby maju normalne rozdelenie
ε ∼ N(0, σ2I) (4)
Veta: 1.1. (Gauss - Markovova veta)
1. Za platnosti modelu (*) a platnosti Gauss - Markovovych podmienok (1
- 4) je odhad β zıskany metodou najmensıch stvorcov najlepsi linearnynevychyleny odhad parametra β.
2. Za platnosti modelu (*) a platnosti Gauss - Markovovych podmienok (1
- 5) je odhad β zıskany metodou najmensıch stvorcov efektivny nevy-chyleny odhad parametra β.
2
1.1 Testy normality chyb v modeli
Na testovanie normality nahodnych chyb v modeli sa pouzıva test Jarque -Bera.
H0 : ε ∼ N(0, σ2I) H1 :nahodne chyby nemaju normalne rozdelenie
Testovacia statistika:
JB = n
[β1
6+
β2
24
]∼ χ2
2
kde√
β1 = skewness
β2 = kurtosis
H0 zamietame, ak JB > χ295%(2).
3
1.2 Testy na heteroskedasticitu
Majme model (*) s predpokladom E(ε) = 0.O heteroskedasticite hovorıme, ak platı V ar(ε) = σ2Ω, kde Ω je kladnedefinitna matica.Vo vsetkych z uvedenych testov je H0 formulovana
H0 : v modeli nie je heteroskedasticita H1 : v modeli je heteroskedas-ticitaTesty na odhalenie heteroskedasticity:
1. Whitov test
4
(a) No Cross TermsPomocna regresia
e2 = X∗α + η (5)
kdeX∗ = [1,x1, . . .xk−1,x
21, . . . ,x
2k−1]
Testovacia statistika:
W = n.R2 ∼ χ2p
(b) Cross TermsPomocna regresia
e2 = X∗α + η (6)
kdeX∗ = [1,x1, . . .xk−1,x
21, . . . ,x
2k−1,xi.xj]
Testovacia statistika:
W = n.R2 ∼ χ2p
V oboch prıpadoch H0 zamietame, ak W > χ295%(p).
2. Goldfeld - Quandtov testPouzıva sa, ak predpokladame, ze niektora premenna sposobuje het-eroskedasticitu.
• Data zoradıme podl’a vel’kosti podl’a podozrivej premennej.
• Rozdelıme data do troch skupın.
• Strednu cast’ vynechame, v prvej a tretej urobıme pomocne regre-sie a odhadneme disperzie
σ21 =
RSS1
n1 − k
σ22 =
RSS2
n2 − k,
kde
5
n1 = pocet dat v prvej skupine
n2 = pocet dat v druhej skupine
k = pocet parametrov v povodnom modeli
Testovacia statistika:
GQ =σ2
2
σ21
∼ Fn2−k,n1−k
H0 zamietame, ak GQ > F95%(n2 − k, n1 − k).
3. Breusch - Paganov testPredpokladame, ze disperziu ovplyvnuje spolocne niekol’ko premennych(ozn. z1, . . . , zp).
• Vytvorıme pomocnu premennu
BPi =e2
i
RSSn
• Urobıme pomocnu regresiu
BPi = α0 + α1z1 + α2z2 + . . . + αkzk + η
Testovacia statistika:
BP =1
2ESS ∼ χ2
p
kdeESS = vysvetl’ujuca suma srvorcov v pomocnej regresiiH0 zamietame, ak BP > χ2
95%(p).
V prıpade heteroskedasticity v modeli zostava nevychylenost’ odhadu β
zıskany metodou najmensıch stvorcov zachovana.
V ar(β) = (XTX)−1XTσ2ΩX(XTX)−1
Na odhad XTσ2ΩX sa pouzıva
( XTσ2ΩX) =
n∑
i=1
e2i xix
Ti
6
Whitov odhad kovariancnej matice vyzera
V ar(β) = (XTX)−1( XTσ2ΩX)(XTX)−1
Na testovanie hypotez tvaru
H0 : Rq×kβ = rq×1 H1 : Rq×kβ 6= rq×1
sa v prıpade heteroskedasticity pouzıva Waldov test
W = (Rβ − r)T(R
Var(β)RT)−1(Rβ − r) ∼ χ2q
1.3 Kriteria na odhalenie multikolinearity
1. Cislo podmienenosti matice XTXAk je cislo podmienenosti matice XTX(= λmax
λmin
) > 30, v modeli jemultikolinearita.
2. Korelacny koeficient medzi vysvetl’ujucimi premennymiAk korelacie maju vysoku hodnotu, v modeli je multikolinearita.
3. Regresia na ostatne premenneV prıpade, ze nam v pomocnych regresiach jednotlivych premennychna ostatne vyjde signifikantnost’ regresie, v modeli je multikolinearita.
4. Variancny inflacny faktor V IFi
V IFi =1
1 − R2i
Ak ma V IFi vel’ku hodnotu ( > 5), v modeli je multikolinearita.
Ak v modeli odhalıme multikolinearitu, mame dve moznosti riesenia:situacie
1. Preformulovat’ model
2. Pouzit’ hrebenovu regresiuParameter β = arg min(y − Xβ)T(y −Xβ) − cβTβ
7
1.4 Testovanie signifikancie parametrov
Za predpokladu normality nahodnych chyb testujeme signifikanciu parametrov
H0 : βi = 0 H1 : β 6= 0
pomocou t - statitiky.Testovacia statistika:
T =βi
sd(βi)∼ tn−k
kde
βi = odhad zıskany metodou najmensıch stvorcov
sd(βi) = odhad standardnej odchylky
n = pocet dat
k = pocet parametrovH0 zamietame, ak |T | > t97.5%(n − k).
8
9
1.5 Testovanie linearnych hypotez
Za predpokladu normality nahodnych chyb testujeme linearne hypotezy nasle-dovne:
H0 : Rq×kβ = rq×1 H1 : Rq×kβ 6= rq×1
Testovacia statistika:
F =
(Rbβ−r)T(R(XTX)−1RT)−1(Rbβ−r)q
RSSn−k
∼ Fq,n−k
H0 zamietame, ak F > F95%(q, n − k).
10
1.6 Testovanie signifikancie modelu
Za predpokladu normality nahodnych chyb testujeme signifikanciu modelu:
H0 : β1 = β2 = . . . = βk−1 = 0 H1 : H0 neplatı
pomocou testov na linearnu hypotezu, pricom
R =
0 1 0 . . . 00 0 1 0 . . ....
... . . .. . . 0
0 0 . . . 0 1
, r =
00...0
q = k - 1
Testovacia statistika:
F =
(Rbβ−r)T(R(XTX)−1RT)−1(Rbβ−r)k−1RSSn−k
∼ Fk−1,n−k
H0 zamietame, ak F > F95%(k − 1, n − k).
1.7 Testovanie submodelu
Majme povodny model. Z neho zıskame odhady β. Taktiez mame k dispozıciiRSS (rezidualnu sumu stvorcov). Chceli by sme skumat’ hypotezu
H0 : Rβ = r q-rovnıc
Zostrojıme model s restrikciou a dostaneme novy odhad β∗, RSS∗.Potom mame testovaciu statistiku:F = RSS∗
−RSSRSS
n−kq
∼ Fn−k,q
Hypotezu H0 zamietame, ak F > F95%(n − k, q).
11
2 Zakladne informacie o datach a modeli
Ciel’ nasho projektu je modelovat’ na vek prisposobenu mortalitu (v Amer-ickych mestach) pomocou nasledujucich vysvetl’ujucich premennych
city nazov mestaJanTemp priemerne januarove teploty FarenheitJulyTemp priemerne julove teploty (vo Farenheitoch) FarenheitRelHum relatıvna vlhkost’ percentaRain rocny uhrn zrazok palceMortality na vek prisposobena mortality skalarEducation priemerna vzdelanost’ rokyPopDensity hustota obyvatel’ov pocet l’udı / km2
NonWhite relatıvny pocet nebelochov percentaWC reatıvny pocet robotnıkov bielej rasy percentaPop populacia kusypophouse pocet l’udı na domacnost’ kusyincome priemerny rocny plat USDHCPot znecistenie ovzdusia uhl’ovodıkmi mg v 1lNOxPot znecistenie ovzdusia oxidom dusicnatym mg v 1lSO2Pot znecistene ovzdusia oxidom siricitym mg v 1l
Na vek prisposobena mortalitaVek je asi najdolezitejsı faktor, ktory ovplyvnuje umrtnost’. Aby sme
mohli porovnavat’ umrtnost’ medzi krajinami, musıme zotriet’ rozdiely medzivekovym rozdelenım obyvatel’stva v krajinach. Prave za tymto ucelom sapouzıva na vek prisposobena mortalita (Age Adjusted Mortality) a je defino-vana
AgeAdjustedMortality =
n∑
a=1
iapa
n∑
a=1
pa
× 100000, kde
• ia je miera umrtnosti vekovej skupiny a,
• Pa je vel’kost’ populacie a-tej vekovej skupiny,
12
• n je pocet vekovych skupın (casto ich je 16, kazda s rozsahom 5 rokov).
13
3 Vychodiskovy model
V nasom prvotnom modeli budeme mortalitu modelovat’ pomocou premen-nych:
Education, NonWhite, Income, HCPot, NOxPot, SO2Pot.
Vychodiskovy model:
Mortality ∼ β0 + β1NonWhite + β2Education + β3Income + β4HCPot
β5NOxPot + β6SO2Pot + ε
3.1”Pracovne” hypotezy o modeli
Pri tvorbe modelu sme si stanovili pracovne hypotezy, ktorych platnost’budeme d’alej overovat’.
Logicky nam vychadzaju tieto hypotezy
• so zvysovanım percentualneho podielu nebelosskeho obyvatel’stva v pop-ulacii by sa mala mortalita zvysovat’ v dosledku odlisnej fyziologie nebe-lochov a moznych pretrvavajucich rasistickych a diskriminacnych naladmedzi obyvatel’mi;
• pri vyssıch platoch sa zvysuje zivotna uroven obyvatel’stva, zdravotnastarostlivost’, prıstupnost’ k liekom a to by podl’a nas malo viest’ knizsej mortalite;
• zvysenie vzdelanosti vedie k vacsım platom, a tu mozno uplatnit’ nasuvyssie uvedeu hypotezu;
• znecistenie ovzdusia sposobuje zhorsenie kvality zivota, co by maloviest’ k zvyseniu na vek prisposobenej mortality;
14
3.2 Odhady parametrov v prvotnom modeli
Variable Coefficient Std. Error t-Statistic Prob.
C 1097.101 74.39533 14.74690 0.0000NONWHITE 3.532387 0.574268 6.151115 0.0000
EDUCATION -15.80899 7.434028 -2.126571 0.0382INCOME -0.001101 0.001331 -0.827653 0.4116HCPOT -0.876087 0.454065 -1.929431 0.0591
NOXPOT 1.589532 0.940067 1.690871 0.0968S02POT 0.170180 0.128254 1.326894 0.1903
R-squared 0.673514 Mean dependent var 941.1731Adjusted R-squared 0.635843 S.D. dependent var 62.42133
S.E. of regression 37.66844 Akaike info criterion 10.20652Sum squared resid 73783.39 Schwarz criterion 10.45300
Log likelihood -294.0922 F-statistic 17.87863Durbin-Watson stat 1.915023 Prob(F-statistic) 0.000000
0
2
4
6
8
10
12
14
-80 -60 -40 -20 0 20 40 60 80
Series: RESID
Sample 1 60
Observations 59
Mean -1.21e-13
Median 2.026399
Maximum 84.83812
Minimum -86.73064
Std. Dev. 35.66689
Skewness 0.067954
Kurtosis 3.165032
Jarque-Bera 0.112362
Probability 0.945368
Uz v korelacnej matici vidiet’ vysoku linearnu zavislost’ skupiny pre-mennych HCPOT, NOXPOT, S02POT. Zahrnutım tychto premennychvznika v modeli multikolinearita. Preto sa pokusime identifikovat’ premennesposobujuce multikolinearitu a vylucime ich z modelu.
Zostrojıme tri pomocne regresie:
HCPOT ∼ α0 + α1NOXPOT + α2S02POT + ε (R1)
15
Odhady metodou najmensıch stvorcov
NOX 2.070044 0.034888 59.33428 0.0000S02POT -0.210708 0.025512 -8.259052 0.0000
C 2.396052 1.938264 1.236184 0.2215
R-squared 0.985337 Mean dependent var 37.85000Adjusted R-squared 0.984823 S.D. dependent var 91.97767
S.E. of regression 11.33136 Akaike info criterion 7.741732Sum squared resid 7318.781 Schwarz criterion 7.846449
Log likelihood -229.2519 F-statistic 1915.172Durbin-Watson stat 2.113806 Prob(F-statistic) 0.000000
Zaver: Premenna HCPOT sa da vel’mi dobre vyjadrit’ pomocou zvysnychpremennych.
NOXPOT ∼ α0 + α1HCPOT + α2S02POT + ε (R2)
Odhady metodou najmensıch stvorcov
Variable Coefficient Std. Error t-Statistic Prob.
HCPOT 0.475385 0.008012 59.33428 0.0000S02POT 0.104942 0.011625 9.027126 0.0000
C -1.035672 0.931169 -1.112227 0.2707
R-squared 0.986743 Mean dependent var 22.60000Adjusted R-squared 0.986278 S.D. dependent var 46.35537
S.E. of regression 5.430187 Akaike info criterion 6.270531Sum squared resid 1680.755 Schwarz criterion 6.375248
Log likelihood -185.1159 F-statistic 2121.273Durbin-Watson stat 2.154025 Prob(F-statistic) 0.000000
Zaver: Premenna NOX sa da vel’mi dobre vyjadrit’ pomocou zvysnychpremennych.
S02POT ∼ α0 + α1NOXPOT + α2HCPOT + ε (R3)
16
Odhady metodou najmensıch stvorcov
Variable Coefficient Std. Error t-Statistic Prob.
HCPOT -2.585440 0.313043 -8.259052 0.0000NOXPOT 5.607073 0.621136 9.027126 0.0000
C 24.90574 6.037490 4.125181 0.0001
R-squared 0.621215 Mean dependent var 53.76667Adjusted R-squared 0.607924 S.D. dependent var 63.39047
S.E. of regression 39.69256 Akaike info criterion 10.24891Sum squared resid 89803.46 Schwarz criterion 10.35363
Log likelihood -304.4673 F-statistic 46.74050Durbin-Watson stat 2.031165 Prob(F-statistic) 0.000000
Zaver: Premenna S02POT sa neda vel’mi dobre vyjadrit’ pomocou zvysnychpremennych (R2 = 0.621215).
Teda skusime z modelu vypustit’ premennu NOXPOT.Poznamka: Premenne NOXPOT a HCPOT su silno korelovane kvoli tomu,ze sa do ovzdusia dostavaju sucasne ako produkty spal’ovania a priemyselnejvyroby.
Po vynechanı premennej NOXPOT nas pracovny model bude vyzerat’nasledovne:
Mortality ∼ β0 + β1NonWhite + β2Education + β3Income + β4HCPot
β5SO2Pot + ε
17
Odhady metodou najmensıch stvorcov
Variable Coefficient Std. Error t-Statistic Prob.
C 1105.799 75.50766 14.64487 0.0000NONWHITE 3.695052 0.575996 6.415065 0.0000
EDUCATION -16.88636 7.535452 -2.240922 0.0292INCOME -0.001108 0.001354 -0.818749 0.4166HCPOT -0.115234 0.061851 -1.863093 0.0680S02POT 0.328455 0.089200 3.682219 0.0005
R-squared 0.655563 Mean dependent var 941.1731Adjusted R-squared 0.623069 S.D. dependent var 62.42133
S.E. of regression 38.32338 Akaike info criterion 10.22614Sum squared resid 77840.12 Schwarz criterion 10.43742
Log likelihood -295.6712 F-statistic 20.17489Durbin-Watson stat 1.915109 Prob(F-statistic) 0.000000
P-hodnota pri parametri INCOME naznacuje, ze β3 je nesignifikantnyparameter. Preto ho z modelu vylucime.
Dostavame model:
Mortality ∼ β0 + β1NonWhite + β2Education + β3HCPot + β4SO2Pot + ε
Odhady metodou najmensıch stvorcov
Variable Coefficient Std. Error t-Statistic Prob.
C 1103.646 74.66715 14.78088 0.0000NONWHITE 3.696264 0.571327 6.469610 0.0000
EDUCATION -20.03902 6.596846 -3.037667 0.0036HCPOT -0.121095 0.060619 -1.997636 0.0507S02POT 0.321873 0.087206 3.690940 0.0005
R-squared 0.651776 Mean dependent var 940.3487Adjusted R-squared 0.626450 S.D. dependent var 62.21863
S.E. of regression 38.02724 Akaike info criterion 10.19414Sum squared resid 79533.89 Schwarz criterion 10.36867
Log likelihood -300.8241 F-statistic 25.73602Durbin-Watson stat 1.917584 Prob(F-statistic) 0.000000
18
Normalita rezıduı
0
2
4
6
8
10
12
-80 -40 0 40 80
Series: RESID
Sample 1 60
Observations 60
Mean -1.12e-13
Median 0.353628
Maximum 90.13176
Minimum -88.91262
Std. Dev. 36.71556
Skewness 0.185064
Kurtosis 3.314943
Jarque-Bera 0.590460
Probability 0.744360
Skumajme pridanie d’alsej vysvetl’ujucej premennej do modelu:
• pridanie WC - parameter je nesignifikantny (vid’ prıloha Tabul’ka (6.1)),
• pridanie POP - parameter je nesignifikantny (vid’ prıloha Tabul’ka(6.1)),
• pridanie POPHOUSE - parameter je nesignifikantny (vid’ prılohaTabul’ka (6.1)),
• pridanie POPDENSITY - parameter je nesignifikantny (vid’ prılohaTabul’ka (6.1)),
• pridanie RELHUM - parameter je nesignifikantny (vid’ prıloha Tabul’-ka (6.1)),
• pridanie JULYTEMP - parameter je nesignifikantny (vid’ prılohaTabul’ka (6.1)),
• pridanie JANTEMP - parameter je signifikantny.
Mortality ∼ β0 + β1NonWhite + β2Education + β3HCPot
β4SO2Pot + β5JANTEMP + ε
19
Odhady metodou najmensıch stvorcov
Variable Coefficient Std. Error t-Statistic Prob.
C 1140.671 73.07795 15.60897 0.0000NONWHITE 4.590304 0.657818 6.978077 0.0000
EDUCATION -19.59536 6.319167 -3.100940 0.0031HCPOT -0.046882 0.065487 -0.715891 0.4771S02POT 0.246726 0.088968 2.773188 0.0076
JANTEMP -1.508804 0.616520 -2.447293 0.0177
R-squared 0.686542 Mean dependent var 940.3487Adjusted R-squared 0.657518 S.D. dependent var 62.21863
S.E. of regression 36.41157 Akaike info criterion 10.12229Sum squared resid 71593.34 Schwarz criterion 10.33172
Log likelihood -297.6687 F-statistic 23.65435Durbin-Watson stat 1.842152 Prob(F-statistic) 0.000000
Z tabul’ky si mozeme vsimnut’, ze premenna HCPOT je nesignifikantna,pricom uz v predchadzajucom modeli bola na hranici zamietnutia. Nesig-nifikancia nebola sposobena multikolinearitou (vid’. Tabul’ka (6.1))
Vynechanım premennej HCPOT dostavame nasledovny model:
Mortatilty ∼ β0 + β1NonWhite + β2Education
β3SO2Pot + β4JANTEMP + ε
Odhady metodou najmensıch stvorcov
Variable Coefficient Std. Error t-Statistic Prob.
C 1161.692 66.62196 17.43708 0.0000NONWHITE 4.714142 0.631847 7.460892 0.0000
EDUCATION -21.02681 5.967849 -3.523348 0.0009S02POT 0.216739 0.078142 2.773652 0.0076
JANTEMP -1.713179 0.544012 -3.149157 0.0026
R-squared 0.683567 Mean dependent var 940.3487Adjusted R-squared 0.660554 S.D. dependent var 62.21863
S.E. of regression 36.24984 Akaike info criterion 10.09840Sum squared resid 72272.81 Schwarz criterion 10.27293
Log likelihood -297.9521 F-statistic 29.70309Durbin-Watson stat 1.792685 Prob(F-statistic) 0.000000
20
• pridanie RAIN - parameter je signifikantny.
Vysledny model
Mortatilty ∼ β0 + β1NonWhite + β2Education
β3SO2Pot + β4JANTEMP + β5RAIN + ε
Odhady metodou najmensıch stvorcov
Variable Coefficient Std. Error t-Statistic Prob.
C 1037.169 83.69093 12.39285 0.0000NONWHITE 4.321546 0.631379 6.844613 0.0000
EDUCATION -13.67612 6.563160 -2.083770 0.0419S02POT 0.274285 0.079215 3.462525 0.0011
JANTEMP -1.647633 0.524408 -3.141893 0.0027RAIN 1.125498 0.486050 2.315604 0.0244
R-squared 0.712149 Mean dependent var 940.3487Adjusted R-squared 0.685497 S.D. dependent var 62.21863
S.E. of regression 34.89258 Akaike info criterion 10.03707Sum squared resid 65744.59 Schwarz criterion 10.24650
Log likelihood -295.1120 F-statistic 26.71947Durbin-Watson stat 1.831382 Prob(F-statistic) 0.000000
Normalita rezıduı
0
2
4
6
8
10
12
14
-80 -40 0 40 80
Series: RESID
Sample 1 60
Observations 60
Mean 1.50e-13
Median -1.378287
Maximum 100.8740
Minimum -80.17746
Std. Dev. 33.38136
Skewness 0.408995
Kurtosis 3.723096
Jarque-Bera 2.979936
Probability 0.225380
21
3.3 Whiteov test heteroskedascity
F-statistic 1.808739 Probability 0.083827Obs*R-squared 16.17657 Probability 0.094688
Variable Coefficient Std. Error t-Statistic Prob.
C 85497.74 41262.28 2.072055 0.0435NONWHITE -88.94133 94.27876 -0.943387 0.3501
NONWHITE2 3.389295 2.490578 1.360847 0.1798EDUCATION -14284.23 7552.639 -1.891290 0.0645
EDUCATION2 628.5565 348.4785 1.803717 0.0774S02POT -12.49928 11.18615 -1.117389 0.2693
S02POT2 0.025538 0.045093 0.566340 0.5737JANTEMP -223.2045 138.3630 -1.613180 0.1131
JANTEMP2 2.837456 1.719373 1.650285 0.1053RAIN 56.31228 104.4643 0.539058 0.5923
RAIN2 -0.696136 1.300710 -0.535197 0.5949
R-squared 0.269609 Mean dependent var 1095.743Adjusted R-squared 0.120550 S.D. dependent var 1823.433
S.E. of regression 1709.997 Akaike info criterion 17.89051Sum squared resid 1.43E+08 Schwarz criterion 18.27448
Log likelihood -525.7154 F-statistic 1.808739Durbin-Watson stat 1.925012 Prob(F-statistic) 0.083827
22
3.4 Goldfeld-Quandtov test heteroskedascity
Pozrime si, ako vyzeraju grafy zavislosti jednotlivych regresorov od residuı.
-120
-80
-40
0
40
80
120
8.8 9.2 9.6 10.0 10.8 11.6 12.4
EDUCATION
RE
SID
23
-120
-80
-40
0
40
80
120
10 20 30 40 50 60 70
JANTEMP
RE
SID
-120
-80
-40
0
40
80
120
0 10 20 30 40 50 60 70
RAIN
RE
SID
24
-120
-80
-40
0
40
80
120
0 50 100 150 200 250 300
S02POT
RE
SID
Na poslednom obrazku si mozeme vsimnut’, ze dispezia sa znizuje sovzrastajucim znecistenım.
Preto SO2Pot je kandidatom na vytvaranie heteroskedasticity. Zotried’medata podl’a tejto premennej. Spravıme teraz dva linearne regresne modely.Prvy z prvych 20 dat, druhy z poslednych 20 dat.
Prvych 20 dat:
25
Variable Coefficient Std. Error t-Statistic Prob.
C 1162.689 131.0250 8.873794 0.0000S02POT 2.340530 1.376635 1.700183 0.1112
EDUCATION -30.67021 10.35679 -2.961363 0.0103NONWHITE 7.122130 1.011008 7.044587 0.0000
JANTEMP -1.268644 0.731290 -1.734803 0.1047RAIN 1.094927 0.571712 1.915172 0.0761
R-squared 0.888994 Mean dependent var 911.1390Adjusted R-squared 0.849349 S.D. dependent var 65.83966
S.E. of regression 25.55483 Akaike info criterion 9.562855Sum squared resid 9142.689 Schwarz criterion 9.861574
Log likelihood -89.62855 F-statistic 22.42393Durbin-Watson stat 2.258309 Prob(F-statistic) 0.000003
Poslednych 20 dat:
Variable Coefficient Std. Error t-Statistic Prob.
C 955.4670 108.3441 8.818819 0.0000S02POT 0.348925 0.094836 3.679257 0.0025
EDUCATION -5.859931 7.345457 -0.797763 0.4383NONWHITE 3.474997 0.747566 4.648415 0.0004
JANTEMP -1.684246 0.844507 -1.994355 0.0660RAIN 1.032172 0.729078 1.415722 0.1787
R-squared 0.821838 Mean dependent var 966.5740Adjusted R-squared 0.758209 S.D. dependent var 48.81511
S.E. of regression 24.00349 Akaike info criterion 9.437600Sum squared resid 8066.344 Schwarz criterion 9.736320
Log likelihood -88.37600 F-statistic 12.91605Durbin-Watson stat 1.933599 Prob(F-statistic) 0.000079
F =RSS2
RSS1
n1 − k
n2 − k= 1.1334 < 2.217197 = F95%(18, 18)
Zaver: V modeli nie je heteroskedasticita.
26
3.5 Breusch-Paganov test heteroskedasticity
Z modelu, kde chceme testovat’ heteroskedasticitu, vypocıtame vektor BP.
BP = nresid2
rss
a testujeme nasledovnu regresiu:
BP ∼ α0 + α1S02POT
Odhady metodou najmensıch stvorcov
Variable Coefficient Std. Error t-Statistic Prob.
C 1.314795 0.277948 4.730369 0.0000S02POT -0.005855 0.003360 -1.742412 0.0867
R-squared 0.049741 Mean dependent var 1.000000Adjusted R-squared 0.033357 S.D. dependent var 1.664107
S.E. of regression 1.636116 Akaike info criterion 3.855293Sum squared resid 155.2588 Schwarz criterion 3.925104
Log likelihood -113.6588 F-statistic 3.035999Durbin-Watson stat 1.954562 Prob(F-statistic) 0.086737
1
2ESS =
1
2RSS
(1
1 − R2− 1
)= 4.0635 > 3.841459 = χ95%(1)
Zaver: V modeli je heteroskedasticita.Preto spravıme aj odhad MNS s Whitovym odhadom kovariacnej matice
27
White Heteroskedasticity-Consistent Standard Errors & Covari-ance
Variable Coefficient Std. Error t-Statistic Prob.
C 1037.169 95.34852 10.87766 0.0000S02POT 0.274285 0.058723 4.670799 0.0000
EDUCATION -13.67612 8.025287 -1.704128 0.0941NONWHITE 4.321546 0.707110 6.111562 0.0000
JANTEMP -1.647633 0.525610 -3.134704 0.0028RAIN 1.125498 0.438891 2.564414 0.0131
R-squared 0.712149 Mean dependent var 940.3487Adjusted R-squared 0.685497 S.D. dependent var 62.21863
S.E. of regression 34.89258 Akaike info criterion 10.03707Sum squared resid 65744.59 Schwarz criterion 10.24650
Log likelihood -295.1120 F-statistic 26.71947Durbin-Watson stat 1.918759 Prob(F-statistic) 0.000000
28
4 Testy linearnych hypotez o modeli
4.1 Testovanie hypotezy o koeficientoch
Testujme nasledovnu hypotezu
H0 : β3 = 0.3 H1 : β3 6= 0.3
Vieme, ze
t =β3 − 0.3
std.(β)=
0.274285− 0.3
3.462525= −0.0074267
Ked’ze | − 0.0074267| < 2.004879 = t97.5%(54), takze H0 nezamietame.No jeden test nam nevylucil homoskedasticitu,tak budeme to testovat’ aj
Waldovym testom:
W = (β3 − 0.3)2 ∗ 0.058723−2 = 0.19176 < 3.841459 = χ95%(1)
Teda ani Waldovym testom hypotezu H0 nezamietame.
4.2 Testovanie pomocou submodelu
V tejto casti budeme skumat’ nas model podrobnejsie.
Rozdelme maticu planu nasledovne: X =
(X1
X2
), kde X1 su staty, kde
priemerna januarova teplota bola pod priemernou teplotou (jantemp < 34).Matica planu X2 obsahuje staty, kde priemerna januarova teplota je nadpriemernou teplotou (jantemp > 34).
Potom odhady metodou najmensıch stvorcou s maticou planu X1 su:
29
Variable Coefficient Std. Error t-Statistic Prob.
C 952.3111 120.5031 7.902796 0.0000S02POT 0.301646 0.104946 2.874301 0.0071
EDUCATION -7.156747 8.923859 -0.801979 0.4285NONWHITE 4.787952 1.436440 3.333208 0.0022
JANTEMP -1.669416 1.665427 -1.002395 0.3237RAIN 1.383637 0.710983 1.946092 0.0605
R-squared 0.523532 Mean dependent var 935.5818Adjusted R-squared 0.449083 S.D. dependent var 49.05361
S.E. of regression 36.40943 Akaike info criterion 10.17147Sum squared resid 42420.70 Schwarz criterion 10.43004
Log likelihood -187.2580 F-statistic 7.032159Durbin-Watson stat 1.872787 Prob(F-statistic) 0.000156
Potom odhady metodou najmensıch stvorcou s maticou planu X2 su:
Variable Coefficient Std. Error t-Statistic Prob.
C 1229.749 165.1928 7.444327 0.0000S02POT 0.189494 0.165711 1.143524 0.2707
EDUCATION -29.92393 12.16613 -2.459609 0.0265NONWHITE 3.698913 1.181743 3.130048 0.0069
JANTEMP -1.253748 1.171597 -1.070118 0.3015RAIN 0.732723 0.895016 0.818670 0.4258
R-squared 0.856983 Mean dependent var 947.5400Adjusted R-squared 0.809311 S.D. dependent var 82.61661
S.E. of regression 36.07698 Akaike info criterion 10.24414Sum squared resid 19523.23 Schwarz criterion 10.54258
Log likelihood -101.5635 F-statistic 17.97656Durbin-Watson stat 1.403868 Prob(F-statistic) 0.000007
Testujme nasledovnu hypotezu:
H0 : stacı povodny model H1 : povodny model nestacı
F =RSS∗ − RSS
RSS
n − 2k
k=
65744.59 − (42420.70 + 19523.23)
(42420.70 + 19523.23)
60 − 12
6
= 0.49085 < 3.75712 = F95%(48, 6)
30
Zaver: H0 nezamietame, teda stacı povodny model.Skusime to odtestovat’ aj Waldovym testom.Zostrojme modelkde P je binarna premenna, ktora je 1, pokial’ priemerna januarova
teplota je < 34 a je rovna 0 inak.Potom dostavame nasledovny model:
Mortality ∼ β0 + β1NonWhite + β2Education + β3HCPot
β4SO2Pot + β5JANTEMP + β6P
β7PSO2Pot + β8PJANTEMP + β9PNonWhite
β10PEducation + β11PHCPot + ε
Potom skumame hypotezu:
H0 : β6 = 0; β7 = 0; β8 = 0; β9 = 0; β10 = 0; β11 = 0
Normalized Restriction (= 0) Value Std. Err.
C(7) -266.5603 182.7481C(8) 0.124229 0.128926C(9) 22.10241 15.28975
C(10) 0.944085 1.590042C(11) -0.485389 2.272173C(12) 0.712130 1.023407
Vysledok regresie:
Test Statistic Value df ProbabilityF-statistic 0.537309 (6, 48) 0.7772Chi-square 3.223855 6 0.7803
Zaver: Stacı povodny model.
31
5 Zaver
Na mortalitu maju vplyv pomer nebelosskeho obyvatel’stva k celkovemu poc-tu obyvatel’ov, vzdelanie, mnozstvo oxidu skriciteho v ovzdusı, januaroveteploty a uhrn zrazok.
Nase”pracovne” hypotezy o modeli sa potvrdili.
Ak sa zvysi percento nebelosskej populacie, mortalita stupne. Tento faktmoze byt’ sposobeny na jednej strane roznou fyziologiou, tym i nachylnost’ouk roznym chorobam, na druhej strane spolocenskou atmosferou. Hoci su datazıskane z americkych zdrojov, v USA stale pretrvavaju rasisticke a diskrim-inacne nalady. Nebelosi mozu mat’ vacsı problem pri uplatnenı a zıskanılepsieho zamestnania.
S rastom vzdelanosti rastie pravdepodobnost’ najdenia si lepsie platenehozamestnania. Vyssı plat implikuje lepsie zivotne podmienky, moznost’ kvalit-nejsej zdravotnej starostlivosti a ucinnejsıch liekov. Co v konecnom dosledkuvedie k znızeniu mortality.
So zvysovanım znecistenia ovdzusia sa zvysuje i nachylnost’ k alergiam,pl’ucnym ochoreniam a klesa kvalita zivotneho priestoru. To moze viest’ kzvyseniu umrtnosti.
Cım vyssie su januarove teploty, tym teplejsie zimy l’udia zazıvaju. Primiernych zimach sa ochorenia az tak vel’mi nesıria. Mierne zimy teda mozuimplikovat’ znızenie mortality.
Opacny efekt ma vyssı uhrn zrazok. Ak viac prsı, vlhkost’ pomaha rozm-nozovaniu bakteriı a vırusov. Za nasledok moze byt’ v konecnej faze rastmortality.
32
6 Prılohy
6.1 Popisne statistiky pouzitych premennych
Popisne statistiky premennej education
0
2
4
6
8
10
12
9 10 11 12
Series: EDUCATION
Sample 1 60
Observations 60
Mean 10.97333
Median 11.05000
Maximum 12.30000
Minimum 9.000000
Std. Dev. 0.845299
Skewness -0.219266
Kurtosis 2.211483
Jarque-Bera 2.035174
Probability 0.361466
Popisne statistiky premennej wc
0
2
4
6
8
10
12
14
16
35 40 45 50 55 60
Series: _WC
Sample 1 60
Observations 60
Mean 46.41500
Median 45.60000
Maximum 62.20000
Minimum 33.80000
Std. Dev. 5.031421
Skewness 0.443027
Kurtosis 4.074585
Jarque-Bera 4.849562
Probability 0.088497
Popisne statistiky premennej nonwhite
33
0
2
4
6
8
10
12
0 10 20 30 40
Series: _NONWHITE
Sample 1 60
Observations 60
Mean 11.87000
Median 10.40000
Maximum 38.50000
Minimum 0.800000
Std. Dev. 8.921148
Skewness 1.102651
Kurtosis 3.761424
Jarque-Bera 13.60782
Probability 0.001109
Popisne statistiky premennej hcpot
0
10
20
30
40
50
0 100 200 300 400 500 600
Series: HCPOT
Sample 1 60
Observations 60
Mean 37.85000
Median 14.50000
Maximum 648.0000
Minimum 1.000000
Std. Dev. 91.97767
Skewness 5.452604
Kurtosis 34.76308
Jarque-Bera 2819.542
Probability 0.000000
Popisne statistiky premennej jantemp
0
2
4
6
8
10
12
10 20 30 40 50 60
Series: JANTEMP
Sample 1 60
Observations 60
Mean 33.98333
Median 31.50000
Maximum 67.00000
Minimum 12.00000
Std. Dev. 10.16890
Skewness 0.936525
Kurtosis 3.900900
Jarque-Bera 10.79984
Probability 0.004517
34
Popisne statistiky premennej income
0
1
2
3
4
5
6
7
8
9
28000 32000 36000 40000 44000 48000
Series: INCOME
Sample 1 60
Observations 59
Mean 33246.66
Median 32452.00
Maximum 47966.00
Minimum 25782.00
Std. Dev. 4473.095
Skewness 1.244568
Kurtosis 4.861685
Jarque-Bera 23.75159
Probability 0.000007
Popisne statistiky premennej julytemp
0
2
4
6
8
10
12
64 66 68 70 72 74 76 78 80 82 84 86
Series: JULYTEMP
Sample 1 60
Observations 60
Mean 74.58333
Median 74.00000
Maximum 85.00000
Minimum 63.00000
Std. Dev. 4.763177
Skewness 0.133264
Kurtosis 2.911693
Jarque-Bera 0.197089
Probability 0.906156
Popisne statistiky premennej pop
0
4
8
12
16
20
24
0 2000000 4000000 6000000 8000000
Series: POP
Sample 1 60
Observations 59
Mean 1438037.
Median 914427.0
Maximum 8274961.
Minimum 124833.0
Std. Dev. 1541736.
Skewness 2.815097
Kurtosis 11.66686
Jarque-Bera 262.5835
Probability 0.000000
35
Popisne statistiky premennej mortality
0
2
4
6
8
10
12
14
800 850 900 950 1000 1050 1100
Series: MORTALITY
Sample 1 60
Observations 60
Mean 940.3487
Median 943.6850
Maximum 1113.160
Minimum 790.7300
Std. Dev. 62.21863
Skewness 0.095596
Kurtosis 3.050759
Jarque-Bera 0.097828
Probability 0.952263
Popisne statistiky premennej nox
0
10
20
30
40
50
0 50 100 150 200 250 300
Series: NOX
Sample 1 60
Observations 60
Mean 22.60000
Median 9.000000
Maximum 319.0000
Minimum 1.000000
Std. Dev. 46.35537
Skewness 5.030952
Kurtosis 30.66535
Jarque-Bera 2166.533
Probability 0.000000
Popisne statistiky premennej noxpot
0
10
20
30
40
50
0 50 100 150 200 250 300
Series: NOXPOT
Sample 1 60
Observations 60
Mean 22.60000
Median 9.000000
Maximum 319.0000
Minimum 1.000000
Std. Dev. 46.35537
Skewness 5.030952
Kurtosis 30.66535
Jarque-Bera 2166.533
Probability 0.000000
36
Popisne statistiky premennej s02pot
0
5
10
15
20
25
30
0 50 100 150 200 250 300
Series: S02POT
Sample 1 60
Observations 60
Mean 53.76667
Median 30.00000
Maximum 278.0000
Minimum 1.000000
Std. Dev. 63.39047
Skewness 1.863667
Kurtosis 6.164096
Jarque-Bera 59.76131
Probability 0.000000
Popisne statistiky premennej pophouse
0
2
4
6
8
10
12
14
2.6 2.8 3.0 3.2 3.4
Series: POP_HOUSE
Sample 1 60
Observations 60
Mean 3.246167
Median 3.265000
Maximum 3.530000
Minimum 2.650000
Std. Dev. 0.181398
Skewness -1.650478
Kurtosis 6.417565
Jarque-Bera 56.44017
Probability 0.000000
Popisne statistiky premennej popdensity
0
4
8
12
16
20
2000 4000 6000 8000 10000
Series: POPDENSITY
Sample 1 60
Observations 60
Mean 3876.050
Median 3567.000
Maximum 9699.000
Minimum 1441.000
Std. Dev. 1454.102
Skewness 1.344763
Kurtosis 6.200284
Jarque-Bera 43.68843
Probability 0.000000
37
Popisne statistiky premennej rain
0
2
4
6
8
10
12
14
10 20 30 40 50 60
Series: RAIN
Sample 1 60
Observations 60
Mean 38.38333
Median 38.00000
Maximum 65.00000
Minimum 10.00000
Std. Dev. 11.51578
Skewness -0.148366
Kurtosis 3.808473
Jarque-Bera 1.854199
Probability 0.395700
Popisne statistiky premennej relhum
0
4
8
12
16
20
40 45 50 55 60 65 70 75
Series: RELHUM
Sample 1 60
Observations 60
Mean 57.66667
Median 57.00000
Maximum 73.00000
Minimum 38.00000
Std. Dev. 5.369931
Skewness 0.231529
Kurtosis 6.841923
Jarque-Bera 37.43699
Probability 0.000000
38
6.2 Korelacna matica pouzitych premennych
NONWHITE WC EDUCATION HCPOT INCOME JANTEMP JULYTEMP MORTALITYNONWHITE 1.000000
WC -0.057233 1.000000EDUCATION -0.208875 0.486066 1.000000
HCPOT -0.026188 0.167578 0.291363 1.000000INCOME -0.100769 0.365947 0.507480 0.327506 1.000000
JANTEMP 0.459224 0.207744 0.108194 0.362473 0.198084 1.000000JULYTEMP 0.602237 -0.012766 -0.269484 -0.356892 -0.190628 0.322146 1.000000
MORTALITY 0.646556 -0.289346 -0.508087 -0.184866 -0.283297 -0.015952 0.321828 1.000000NOX 0.019121 0.129406 0.229116 0.983747 0.311683 0.334225 -0.334492 -0.084568
NOXPOT 0.019121 0.129406 0.229116 0.983747 0.311683 0.334225 -0.334492 -0.084568POP 0.115758 0.217839 0.196904 0.529621 0.318484 0.240140 0.021503 0.058614
POPHOUSE 0.352736 -0.346844 -0.389103 -0.489183 -0.295453 -0.325241 0.257080 0.368016POPDENSITY -0.006793 0.253279 -0.236246 0.112698 -0.002990 -0.076006 -0.008833 0.252121
S02POT 0.159657 -0.063471 -0.228976 0.278600 0.067583 -0.093775 -0.071386 0.419118RELHUM -0.119360 0.014788 0.185670 -0.026388 0.127690 0.085522 -0.441397 -0.101074
RAIN 0.302765 -0.114071 -0.472978 -0.494548 -0.362312 0.058566 0.472257 0.433114
NOX NOXPOT POP POPHOUSE POPDENSITY S02POT RELHUM RAINNOX 1.000000
NOXPOT 1.000000 1.000000POP 0.546274 0.546274 1.000000
POPHOUSE -0.449478 -0.449478 -0.314287 1.000000POPDENSITY 0.158495 0.158495 0.334101 -0.166725 1.000000
S02POT 0.406287 0.406287 0.366117 -0.010260 0.421677 1.000000RELHUM -0.052976 -0.052976 -0.143277 -0.143657 -0.149404 -0.116476 1.000000
RAIN -0.459604 -0.459604 -0.234544 0.199056 0.083886 -0.130963 -0.117773 1.000000
39
Variable Coefficient Std. Error t-Statistic Prob.
C 1120.150 75.95486 14.74757 0.0000NONWHITE 3.725593 0.570663 6.528534 0.0000
EDUCATION -16.27575 7.395797 -2.200675 0.0321HCPOT -0.120381 0.060488 -1.990172 0.0516S02POT 0.325910 0.087088 3.742322 0.0004
WC -1.258024 1.127337 -1.115926 0.2694
R-squared 0.659625 Mean dependent var 940.3487Adjusted R-squared 0.628109 S.D. dependent var 62.21863
S.E. of regression 37.94272 Akaike info criterion 10.20467Sum squared resid 77741.11 Schwarz criterion 10.41411
Log likelihood -300.1402 F-statistic 20.92970Durbin-Watson stat 1.926128 Prob(F-statistic) 0.000000
Tabul’ka 1: Nesignifikancia premennej WC
Variable Coefficient Std. Error t-Statistic Prob.
C 1106.218 76.45555 14.46878 0.0000NONWHITE 3.668127 0.584358 6.277189 0.0000
EDUCATION -20.30647 6.799758 -2.986352 0.0043HCPOT -0.135059 0.067486 -2.001282 0.0505S02POT 0.305466 0.092554 3.300395 0.0017
POP 1.79E-06 4.10E-06 0.435954 0.6646
R-squared 0.652453 Mean dependent var 941.1731Adjusted R-squared 0.619666 S.D. dependent var 62.42133
S.E. of regression 38.49602 Akaike info criterion 10.23513Sum squared resid 78543.00 Schwarz criterion 10.44641
Log likelihood -295.9364 F-statistic 19.89948Durbin-Watson stat 1.985412 Prob(F-statistic) 0.000000
Tabul’ka 2: Nesignifikancia premennej POP
40
Variable Coefficient Std. Error t-Statistic Prob.
C 1123.575 150.0078 7.490109 0.0000NONWHITE 3.728997 0.614580 6.067549 0.0000
EDUCATION -20.27240 6.827331 -2.969301 0.0044HCPOT -0.125664 0.068012 -1.847681 0.0701S02POT 0.322156 0.088010 3.660441 0.0006
POPHOUSE -5.421277 35.28766 -0.153631 0.8785
R-squared 0.651928 Mean dependent var 940.3487Adjusted R-squared 0.619699 S.D. dependent var 62.21863
S.E. of regression 38.36934 Akaike info criterion 10.22703Sum squared resid 79499.14 Schwarz criterion 10.43647
Log likelihood -300.8110 F-statistic 20.22803Durbin-Watson stat 1.919996 Prob(F-statistic) 0.000000
Tabul’ka 3: Nesignifikancia premennej POPHOUSE
Variable Coefficient Std. Error t-Statistic Prob.
C 1071.092 79.45243 13.48092 0.0000NONWHITE 3.773965 0.573286 6.583039 0.0000
EDUCATION -18.53541 6.699362 -2.766743 0.0077HCPOT -0.125684 0.060544 -2.075897 0.0427S02POT 0.282103 0.093330 3.022658 0.0038
POPDENSITY 0.004500 0.003848 1.169692 0.2473
R-squared 0.660380 Mean dependent var 940.3487Adjusted R-squared 0.628934 S.D. dependent var 62.21863
S.E. of regression 37.90059 Akaike info criterion 10.20245Sum squared resid 77568.57 Schwarz criterion 10.41188
Log likelihood -300.0735 F-statistic 21.00028Durbin-Watson stat 1.895278 Prob(F-statistic) 0.000000
Tabul’ka 4: Nesignifikancia premennej POPDENSITY
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Variable Coefficient Std. Error t-Statistic Prob.
C 1074.012 87.10071 12.33069 0.0000NONWHITE 3.726404 0.575968 6.469814 0.0000
EDUCATION -20.72530 6.708728 -3.089305 0.0032HCPOT -0.118804 0.061021 -1.946950 0.0567S02POT 0.323633 0.087686 3.690830 0.0005
RELHUM 0.635138 0.947628 0.670239 0.5056
R-squared 0.654648 Mean dependent var 940.3487Adjusted R-squared 0.622671 S.D. dependent var 62.21863
S.E. of regression 38.21908 Akaike info criterion 10.21919Sum squared resid 78877.72 Schwarz criterion 10.42862
Log likelihood -300.5756 F-statistic 20.47248Durbin-Watson stat 1.897587 Prob(F-statistic) 0.000000
Tabul’ka 5: Nesignifikancia premennej RELHUM
Variable Coefficient Std. Error t-Statistic Prob.
C 1304.894 133.1638 9.799166 0.0000NONWHITE 4.259216 0.717754 5.934089 0.0000
EDUCATION -26.69371 6.382094 -4.182595 0.0001S02POT 0.225414 0.085811 2.626880 0.0111
JULYTEMP -1.800711 1.363173 -1.320970 0.1920
R-squared 0.637995 Mean dependent var 940.3487Adjusted R-squared 0.611667 S.D. dependent var 62.21863
S.E. of regression 38.77237 Akaike info criterion 10.23295Sum squared resid 82681.31 Schwarz criterion 10.40748
Log likelihood -301.9884 F-statistic 24.23291Durbin-Watson stat 1.595088 Prob(F-statistic) 0.000000
Tabul’ka 6: Nesignifikancia premennej JULYTEMP
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Variable Coefficient Std. Error t-Statistic Prob.
C -448.3818 137.7880 -3.254143 0.0019NONWHITE -2.641489 1.306790 -2.021357 0.0481
EDUCATION 30.53310 12.34274 2.473769 0.0165S02POT 0.639628 0.161614 3.957756 0.0002
JANTEMP 4.359364 1.125129 3.874545 0.0003
R-squared 0.380636 Mean dependent var 37.85000Adjusted R-squared 0.335591 S.D. dependent var 91.97767
S.E. of regression 74.97216 Akaike info criterion 11.55177Sum squared resid 309145.4 Schwarz criterion 11.72629
Log likelihood -341.5530 F-statistic 8.450195Durbin-Watson stat 2.153862 Prob(F-statistic) 0.000022
Tabul’ka 7: Regresia HCPOT na ostatne vysvetl’ujuce premenne
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