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Multiple Regression

#his sa!ple !o$el illustrates how to use Ris% &i!ulator ('nterprise &i!ulator" or)

*. Running a +ultiple Regression nalysis

Model BackgroundFile Name: Multiple Regression.xls

#his ea!ple shows how a !ultiple regression can e run using Ris% &i!ulator. #he raw $ata are arrange$ in the Cross-&ectional /ata wor%sheet,

which contains cross-sectional $ata on all 50 .&. states on the nu!er o aggraate$ assaults (in thousan$s" per year, the nu!er o achelor1s $egrees

awar$e$ per year, police epen$iture per capita population, population sie in !illions, population $ensity (person per s3uare !ile", an$ une!ploy!ent

rate. #he i$ea is to see i there is a relationship etween the nu!er o aggraate$ assaults per year an$ these eplanatory ariales using !ultiple

regression analysis.

Multiple Regression Analysis

#o run this !o$el, si!ply)

Model Results Analysis

4ote that !ore a$ance$ regressions such as lag regressors, stepwise regression, an$ nonlinear regressions can also e run using Ris% &i!ulator.

Results Summary

Reer to the Report wor%sheet or $etails on the regression output. #he report has !ore $etails on the interpretation o speciic statistical results.

It proi$es the ollowing ele!ents) !ultiple regression an$ analysis o ariance output, inclu$ing coeicients o $eter!ination, hypothesis

test results (single ariale t-test an$ !ultiple ariale -test", co!pute$ coeicients or each regressor, itte$ chart, an$ !uch !ore.

Disclaimer

*. In the Cross Sectional Data worksheet , select the area C5:H55 .

2. &elect Simulation l Forecasting l Multiple Regression.

6. Choose Aggraate! Assault as the $epen$ent ariale in the regression an$ clic% on OK.

or $etails on running such regressions as well as the results interpretation, reer to Mo!eling Risk" (iley 2008" y /r. 9ohnathan +un.

D#$#%&'#R S'#C(F(CA%%) D(SC%A(MS A%% &*H#R +ARRAN*(#S" #,'R#SS &R (M'%(#D" (NC%-D(N" /-* N&* %(M(*#D *&" *H# (M'%(#D +ARRAN*(#S&F M#RCHAN*A/(%(*) AND F(*N#SS F&R A 'AR*(C-%AR '-R'&S#. As stan!ar! practice 0or so0tware !eelopment an! en!1user applications" it is important 0orthe %icensee to note that the aluation results attache! herein are accurate to the so0tware Deeloper2s 3est knowle!ge an! are solel4 3ase! on the in0ormation 0urnishe!34 the %icensee or en!1user. +hile the so0tware Deeloper has use! his 3est e00orts in preparing this report" he makes no representations or warranties with respect tothe accurac4 or completeness o0 the contents o0 this mo!el an! speci0icall4 !isclaims an4 implie! warranties o0 merchanta3ilit4 o0 0itness 0or a particular purpose. *he%icensee here34 agrees that the Deeloper is not hel! lia3le 0or an4 loss o0 pro0it or an4 other commercial !amages" inclu!ing" 3ut not limite! to" special" inci!ental"conseuential or other !amages. *his mo!el is onl4 an illustration o0 using the so0tware an! in no wa4 represents the correct an! complete picture o0 an inestor6sinestment" risk" an! return pro0ile. *he user is a!ise! to take great care in using an! interpreting the mo!el an! its results. Mo!el was !eelope! 34 Dr. 7ohnathan Muno0 www.realoptionsaluation.com.

MultipleRegression

Sample Model

© 2005-2007 Copyright./r. 9ohnathan +un.

ll Rights Resere$.

7/23/2019 QTA


© Copyright 2005-2007. Real Options Valuation, Inc. (www.realoptionsaluation.co!"

Multiple Regression Analysis Data Set

52* *:60: *:5 ;.0;* 7<.8 7.2

687 **;: 800 0.55 * :.5;;6 *:08: 672 6.885 62.6 5.7

685 772< *;2 2.65* ;5.* 7.6

8*; *00;:; ;62 2<.78 *<0.: 7.5

6:5 *872: 2<0 6.2<; 6*.: 5

2:8 *;860 6;8 6.2:7 87:.; 8.7

6<7 ;00: 62: 0.888 6;0.: 8.2

78; 6:<27 65; *2.<6: 26<.8 7.6

;27 22622 288 8.;7: ***.< 5

*56 67** 620 *.*0: *72.5 2.:

26* 6*68 *<7 *.007 *2.2 8.*

52; 5050: 288 **.;6* 205.8 7.*

62: 2:::8 *76 5.5;; *5;.8 5.<

2;0 *8<<8 *<0 2.777 ;<.7 ;.82:8 *6065 26< 2.;7: 60.6 ;.;

2:5 *2<76 *<0 6.8:5 <2.: 7.;

58< *860< 2;* ;.22 <8.< 7.*

<8 5227 *:< *.22: 6<.: 7.5

;<: *<265 65: ;.7:* ;:<.2 5.<

;:* ;;;:7 6*5 8.0*8 787.8 <

;8: ;;2*6 606 <.2<5 *86.8 <.2

*77 268*< 22: ;.675 55 5.*

*<: <*08 *6; 2.576 5;.< :.8

;5: 2;<*7 *:< 5.**7 7;.6 8.8

*0: 6:72 *<8 0.7<< 5.5 8.<

2;8 :<;5 *:6 *.57: 20.5 2.7

2<* 2676 ;*7 *.202 *0.< 5.58: 7*2: 266 *.*0< *26.7 7.2

6** 2682; 6;< 7.76 *0;2 8.8

808 52;2 2:; *.5*5 *2.5 8.<

5*2 <282< ;<< *7.<< 6:* 7.2

;28 2:7<5 26* 8.82< *68.* 5.:

;7 ;;:7 *;6 0.86< <.6 ;.*

285 ;:7<< 2;< *0.:;7 28;.< 8.;

670 *;087 *<5 6.*;8 ;5.: 8.7

6*2 *28<6 2:: 2.:;2 2<.8 8

222 82*:; 22< **.::2 285.* 8.<

2:0 <*56 2:7 *.006 <80.6 :.5

75< *;250 22; 6.;:7 **5.: 8.2

**; 68:0 *8* 0.8<8 <.2 6.;;*< *:086 22* ;.:77 **:.6 8.8

;65 85**2 267 *8.<:7 8;.< 8.8

*:8 **6;0 220 *.726 2* ;.<

:7 ;556 *:5 0.586 80.: 8.;

*:: 2:<80 280 8.*:7 *58.6 5.:

606 *<20* 28* ;.:87 76.* 8.6

*02 7566 **: *.7<6 7;.5 *0.5

*27 286;6 28: ;.:<2 <0.* 5.;

25* *8;* 600 0.;5; ;.7 5.*

ggraate$ ssault

=achelor1s/egree

>olice'pen$iture >er

Capita

>opulation in+illions

>opulation /ensity(>ersons?&3 +ile"

ne!ploy!entRate

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© Copyright 2005-2007. Real Options Valuation, Inc. (www.realoptionsa

Regression Analysis Report

Regression Statistics

R1Suare! 8Coe00icient o0 Determination9 .;<=<

A!>uste! R1Suare! .<5?

Multiple R 8Multiple Correlation Coe00icient9 .5=<

Stan!ar! #rror o0 the #stimates 8S#49 @B.=<

n&3serations 5

Regression Results

(ntercept /achelor6s Degree

Coe00icients 5=.B555 1.;5 . <5.<;== 1.? @.55=B

Stan!ar! #rror @?.=B@ .;5 .<5;5 @.@@=< .@@ @.=BB

t1Statistic .5;<= 1@. @.?;@ @.=?== 1.?; @.@@??

p1$alue .5BB 0.3197 0.0738 0.0807 0.9332 0.2693

%ower 5 1@@.<B 1.@ 1. 1;.<@;= 1.<@;< 1@;.<?=

-pper B5 <==.<= .; .B=5; 5;.?B@ .@B@ .;?5

Degrees of Freedom Hypothesis Test

Degrees o0 Free!om 0or Regression 5 Critical t1Statistic 8BB con0i!ence with !0 o0 9 <.

Degrees o0 Free!om 0or Resi!ual Critical t1Statistic 8B5 con0i!ence with !0 o0 9 <.

*otal Degrees o0 Free!om B Critical t1Statistic 8B con0i!ence with !0 o0 9 @.

Analysis of Variance

-&tatistic p-ValueHypothesis Test

Regression =B;??.?B? B5?==.B? .<=BB 0.0029 Critical F1statistic 8BB con0i!ence with !0 o0 an! ;9 ;.Resi!ual B?5=5.@B< <<@.=?B Critical F1statistic 8B5 con0i!ence with !0 o0 an! ;9 <.

*otal @5;.? Critical F1statistic 8B con0i!ence with !0 o0 an! ;9 @.B

*he R1Suare! or Coe00icient o0 Determination in!icates that .;; o0 the ariation in the !epen!ent aria3le can 3e explaine! an! accounte! 0or 34 the in!epen!ent regression anal4sis. Howeer" in a multiple regression" the A!>uste! R1Suare! takes into account the existence o0 a!!itional in!epen!ent aria3les or regressors an!Suare! alue to a more accurate iew o0 the regression6s explanator4 power. Hence" onl4 .<5 o0 the ariation in the !epen!ent aria3le can 3e explaine! 34 the reg

*he Multiple Correlation Coe00icient 8Multiple R9 measures the correlation 3etween the actual !epen!ent aria3le 8)9 an! the estimate! or 0itte! 8)9 3ase! on euation. *his is also the suare root o0 the Coe00icient o0 Determination 8R1Suare!9.

*he Stan!ar! #rror o0 the #stimates 8S# 4 9 !escri3es the !ispersion o0 !ata points a3oe an! 3elow the regression line or plane. *his alue is use! as part o0 the calc

the con0i!ence interal o0 the estimates later.

'olice#xpen!iture

'er Capita'opulation in

Millions

'opulationDensit4

8'ersonsESMile9

-nemplo4ment Rate

*he Coe00icients proi!e the estimate! regression intercept an! slopes. For instance" the coe00icients are estimates o0 the true population 3 alues in the 0ollo

euation ) G β β

@ ,

@ β

< ,

< ... β

n ,

n. *he Stan!ar! #rror measures how accurate the pre!icte! Coe00icients are" an! the t1Statistics are the ratios o0 each pre!ict

its Stan!ar! #rror.

*he t1Statistic is use! in h4pothesis testing" where we set the null h4pothesis 8Ho9 such that the real mean o0 the Coe00icient G " an! the alternate h4pothesis 8Ha9 sumean o0 the Coe00icient is not eual to . A t1test is is per0orme! an! the calculate! t1Statistic is compare! to the critical alues at the releant Degrees o0 Free!om 0ot1test is er4 important as it calculates i0 each o0 the coe00icients is statisticall4 signi0icant in the presence o0 the other regressors. *his means that the t1test stawhether a regressor or in!epen!ent aria3le shoul! remain in the regression or it shoul! 3e !roppe!.

*he Coe00icient is statisticall4 signi0icant i0 its calculate! t1Statistic excee!s the Critical t1Statistic at the releant !egrees o0 0ree!om 8!09. *he three main con0i!enctest 0or signi0icance are B" B5 an! BB. (0 a Coe00icient6s t1Statistic excee!s the Critical leel" it is consi!ere! statisticall4 signi0icant. Alternatiel4" the p1$alue caStatistic6s pro3a3ilit4 o0 occurrence" which means that the smaller the p1$alue" the more signi0icant the Coe00icient. *he usual signi0icant leels 0or the p1$alue are .@" correspon!ing to the BB" B5" an! BB con0i!ence leels.

*he Coe00icients with their p1$alues highlighte! in 3lue in!icate that the4 are statisticall4 signi0icant at the B con0i!ence or .@ alpha leel" while those highlighte!that the4 are not statisticall4 signi0icant at an4 other alpha leels.

&u!s o&3uares

+ean o&3uares

7/23/2019 QTA


© Copyright 2005-2007. Real Options Valuation, Inc. (www.r

Forecasting

>erio$ ctual (@" orecast (" 'rror ('"

* 52* 2<<.5*2; 22*.;:78

2 687 ;:7.*2;6 (*20.*2;6"

6 ;;6 656.27:< :<.72**

; 685 278.62<8 ::.870;

5 8*; 778.*668 (*82.*668"

8 6:5 2<:.<<<6 :8.0007

7 2:8 65;.:7*: (8:.:7*:"

: 6<7 6*2.8*55 :;.6:;5

< 78; 52<.7550 26;.2;50

*0 ;27 6;7.706; 7<.2<88

** *56 288.2528 (**6.2528"

*2 26* 28;.8675 (66.8675"

*6 52; ;08.:00< **7.*<<*

*; 62: 272.2228 55.777;

*5 2;0 26*.7::2 :.2**:

*8 2:8 257.::82 2:.**6:

*7 2:5 6*;.<52* (2<.<52*"

*: 58< 665.6*;0 266.8:80

*< <8 2:2.0658 (*:8.0658"

20 ;<: 670.2082 *27.7<6:

2* ;:* 6;0.:7;2 *;0.*25:

22 ;8: ;27.5**: ;0.;::2

26 *77 27;.52<: (<7.52<:"

2; *<: 2<;.77<5 (<8.77<5"

25 ;5: 2<5.2*:0 *82.7:20

28 *0: 28<.8*<5 (*8*.8*<5"

27 2;8 *<5.5<55 50.;0;5

2: 2<* 68;.500; (76.500;"

2< 8: 2:7.0;28 (2*<.0;28"

60 6** ;6*.758: (*20.758:"

6* 808 626.86<< 2:2.680*

62 5*2 56*.;658 (*<.;658"

66 ;28 625.68;* *00.865<

6; ;7 *<2.6<80 (*;5.6<80"

65 285 67:.*250 (**6.*250"

68 670 2::.808; :*.6<68

67 6*2 6*7.567; (5.567;"

6: 222 655.:075 (*66.:075"

6< 2:0 6*8.82:0 (68.82:0"

;0 75< 60*.*526 ;57.:;77

;* **; *<6.;860 (7<.;860"

;2 ;*< 627.<60; <*.08<8

;6 ;65 ;7;.7662 (6<.7662";; *:8 2;;.676; (5:.676;"

;5 :7 2;7.6:<7 (*80.6:<7"

;8 *:: 628.<*:* (*6:.<*:*"

;7 606 667.8;02 (6;.8;02"

;: *02 60;.560* (202.560*"

;< *27 60*.*87* (*7;.*87*"

50 25* 2:7.6*;< (68.6*;<"

*he Anal4sis o0 $ariance 8AN&$A9 ta3le proi!es an F1test o0 the regression mo!el6s oerall statistical signi0icance. (nstea! o0 looking at in!ii!ual regtest looks at all the estimate! Coe00icients6 statistical properties. *he F1Statistic is calculate! as the ratio o0 the Regression6s Mean o0 Suares to the *he numerator measures how much o0 the regression is explaine!" while the !enominator measures how much is unexplaine!. Hence" the larger the Fthe mo!el. *he correspon!ing p1$alue is calculate! to test the null h4pothesis 8Ho9 where all the Coe00icients are simultaneousl4 eual to Iero" ersus that the4 are all simultaneousl4 !i00erent 0rom Iero" in!icating a signi0icant oerall regression mo!el. (0 the p1$alue is smaller than the .@" .5" or .@regression is signi0icant. *he same approach can 3e applie! to the F1Statistic 34 comparing the calculate! F1Statistic with the critical F alues at arious

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