chapter two linear programming in finance.ppt

ADVICE TO ALL EMPLOYEES

1. Build a home earlier. Be it rural home or urban home. Building a house at 50 is not an achievement. Don't get used to government houses. This comfort is so dangerous. Let all your family have good time in your house.

2. Go home. Don't stick at work all the year. You are not the pillar of your department. If you drop dead today, you will be replaced immediately and operations will continue. Make your family a priority.

3. Don't chase promotions. Master your skills and be excellent at what you do. If they want to promote you, that's fine if they don't, stay positive to your personal.
development.

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5. Don't ever compete with your bosses. You will burn your fingers. Don't compete with your colleagues, you will fry your brain.

6. Ensure you have a side business. Your salary will not sustain your needs in the long run.

7. Save some money. Let it be deducted automatically from your payslip.

8. Borrow a loan to invest in a business or to change a situation not to buy luxury. Buy luxury from your profit.

9. Keep your life,marriage and family private. Let them stay away from your work. This is very important.

10. Be loyal to yourself and believe in your work. Hanging around your boss will alienate you from your colleagues and your boss may finally dump you when he leaves.

11. Retire early. The best way to plan for your exit was when you received the employment letter. The other best time is today. By 40 to 50 be out.

12. Join work welfare and be an active member always. It will help you a lot when any eventuality occurs.

13.Take leave days utilize them by developing yr future home or projects..usually what you do during yr leave days is a reflection of how you'll live after retirement..If it means you spend it all holding a remote control watching series on Zee world, expect nothing different after retirement.

14. Start a project whilst still serving or working. Let your project run whilst at work and if it doesn't do well, start another one till it's running viably. When your project is viably running then retire to manage your business. Most people or pensioners fail in life because they retire to start a project instead of retiring to run a project.

15. Pension money is not for starting a project or buy a stand or build a house but it's money for your upkeep or to maintain yourself in good health. Pension money is not for paying school fees or marrying a young wife but to look after yourself.

16. Always remember, when you retire never be a case study for living a miserable life after retirement but be a role model for colleagues to think of retiring too.

17. Don't retire just because you are finished or you are now a burden to the company and just wait for your day t

1. CHAPTER TWO
THE CLASSICAL LINEAR REGRESSION MODEL
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2. Terminology and Notation
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3. .
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4. Dependentvariable:
the variable that is influenced by the independent
variable(s).
Forexample,inaMultipleLinearRegressionModel
(MLRM),outputisinfluencedbyindependentvariables
likefertilizerscost,laborcost,pesticidescostetc.
Independentvariable:
avariable,whosevaluesdoesnotdependuponother
variable,butinfluencesdependentvariable.
Examplesinclude,fertilizerscost,pesticidescostetc.
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5. Regression Analysis
Economictheoriesaremainlyconcernedwiththerelationships
amongvariouseconomicvariables.
Example:demandtheory,supplytheory,consumption
theory,etc.
Theserelationships,whenphrasedinmathematicalterms,can
predicttheeffectofonevariableonanother.
Thefunctionalrelationshipsofthesevariablesdefinethe
dependenceofonevariableupontheothervariable(s)inthe
specificform.
Thespecificfunctionalformsmaybe;
linear,quadratic,logarithmic,exponential,hyperbolic,or
anyotherform.
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6. Cont.…
Theobjectiveoflinearregressionanalysis:istoestimate
and/orpredictthemeanoraveragevalueofthedependent
variableonthebasisoftheknownorfixedvaluesofthe
explanatoryvariables.
Thatistoestimatethepopulationregressionfunction(PRF)on
thebasisofsampleregressionfunction(SRF)asaccuratelyas
possible.
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7. 2.1 The concept of regression Analysis
Regression:isthemostimportanttooltoEconometricians.
Regressionanalysis:
Itisconcernedwiththestudyofthedependenceof
onevariableononeormoreothervariables.
Itiswithaviewtoestimateand/orpredictthe
(population)meanoraveragevalueofthedependent
intermsoftheknownorfixed(inrepeatedsampling)
valuesofthelatter.
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8. Cont…
Simple,ortwo-variable,regressionanalysis:ifwearestudying
thedependenceofavariableononlyasingleexplanatory
variable.
E.g.consumptionexpenditureonrealincome
Multipleregressionanalysis:ifwearestudyingthedependence
ofonevariableonmorethanoneexplanatoryvariable.
Note:intwo-variableregressionthereisonlyoneexplanatoryvariable,
whereasinmultipleregressionsthereismorethanoneexplanatoryvariable.
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9. Cont…
Simple Linear Regression:
Represented by single equation regression model
Y = f(x)
The dependent variable expressed as a function of
only a single explanatory variable
Causal relationship between variables flow in one
direction only.
Example:
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10. Cont…
Multiple Linear Regression:
Dependent variable explained by more than one explanatory
variable.
Example; Y = f(X, Z, K, O)
•Regression equation of Y on X.
Variation in C = systematic variation + random variation.
Consumption = f(Income, Wage rate)
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11. Cont…
Note:afrequentobjectiveinresearchisthespecificationofa
functionalrelationshipbetweentwovariables.
E.g.Y=f(x)
Y–Explainedvariable,ordependentvariable,orpredicted
variable.
X-Explanatoryvariable,orindependentvariable,control
variable,orregressor
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12. Cont…
Inthischapterweshallconsiderasimplelinearregressionmodel.
i.e.arelationshipbetweentwovariablesrelatedinalinear
form.
Weshallfirstdiscusstwoimportantformsofrelation:
i.stochasticand
ii.non-stochastic,
Note:amongwhichweshallbeusingtheformerineconometric
analysis.6/26/2024 Prepared by: Dessie M.

13. 2.2. Stochastic and Non-stochastic Relationships
Econometricianssayrelationshipbetweenvariables(XandY)are
generallyinexact(stochastic).
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14. Cont…
A. Non-Stochastic Model:
ArelationshipbetweenXandY,characterizedasY=f(X)is
saidtobedeterministicornon-stochasticifforeachvalueofthe
independentvariable(X)thereisoneandonlyone
correspondingvalueofdependentvariable(Y).
Example:
Withouttheerror/ordisturbanceterm(u),therelationshipissaid
tobeexact/deterministic,otherwisestochasticorinexact.
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15. Cont.…
B. Stochastic /Inexact Relationship
ArelationshipbetweenXandYissaidtobestochasticiffora
particularvalueofXthereisawholeprobabilisticdistributionof
valuesofY.
Insuchacase,foranygivenvalueofX,thedependentvariable
Yassumessomespecificvalueonlywithsomeprobability.
Stochasticmodel:isamodelinwhichthedependentvariableis
notonlydeterminedbytheexplanatoryvariablebutalsoothers
variableswhicharenotincludedinthemodel.
E.g
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16. Cont…
Existenceofthedisturbanceisjustifiedinthefollowingpoints:
Omissionofothervariables
Measurementerror/datacollectiondifficulties.
Randomnessinhumanbehavior/humansarenotmachines
thatwilldoasinstructed/
Imperfectspecificationofthemodel
Poorproxyvariable
Note:Inregressionanalysisweareconcernedwithastochastic
orstatisticalrelationshipandnotofadeterministicornon
stochasticormathematicalrelationship.
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17. Cont…
Example:assumeasupplyfunction
Thesupplyforacertaincommoditydependsonitsprice(other
determinantstakentobeconstant)andthefunctionbeinglinear,
therelationshipcanbeputas:
ForaparticularvalueofP,thereisonlyonecorrespondingvalue
ofQ.
Thisisadeterministic(non-stochastic)relationshipsinceforeach
pricethereisalwaysonlyonecorrespondingquantitysupplied.
AllthevariationinQisduesolelytochangesinP,andthatthere
arenootherfactorsaffectingthedependentvariable.
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18. Cont…
Ifthisweretrueallthepointsofprice-quantitypairs,ifplottedon
atwo-dimensionalplane,wouldfallonastraightline.
However,ifwegatherobservationsonthequantityactually
suppliedinthemarketatvariouspricesandweplotthemona
diagramweseethattheydonotfallonastraightline.
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19. Cont…
Note:Thederivationoftheobservationfromthelinemaybe
attributedtoseveralfactors.
Omissionofvariablesfromthefunction
Randombehaviorofhumanbeings
Imperfectspecificationofthemathematicalformof
themodel
Errorofaggregation
Errorofmeasurement
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20. Cont…
Totakeintoaccounttheabovesourcesoferrorsweintroducein
econometricfunctionsarandomvariablewhichisusually
denotedbytheletter‘u’or‘ℇ’
Andiscallederrortermorrandomdisturbanceor
stochastictermofthefunction,
Socalledbecauseuissupposedto‘disturb’theexactlinear
relationshipwhichisassumedtoexistbetweenXandY.
Byintroducingthisrandomvariableinthefunctionthemodelis
renderedstochasticoftheform:
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21. Cont…
Thusastochasticmodelisamodelinwhichthedependent
variableisnotonlydeterminedbytheexplanatoryvariable(s)
includedinthemodelbutalsobyotherswhicharenotincluded
inthemodel.
Inordertotakeallthesesourcesoferrorintoaccount,we
introducethestochastic/randomdisturbancetermintoour
econometricmodelsandhencethecompletesimpleeconometric
modelbecomes:
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22. 2.3. Simple Linear Regression model.
Economictheoriesaremainlyconcernedwiththerelationship
amongvarieseconomicvariables.
Thestochasticrelationshipwithoneexplanatoryvariableis
calledsimplelinearregressionmodel.
Asimplelinearregressionmodel:
Itisarelationshipbetweentwovariablesrelatedina
linearform.
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23. Cont…
Thetruerelationshipwhichconnectsthevariables
involvedissplitintotwoparts:
i.apartrepresentedbyalineand
ii.apartrepresentedbytherandomterm‘u’.
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24. Cont…
Thescatterofobservationsrepresentsthetruerelationship
betweenYandX.
Thelinerepresentstheexactpartoftherelationshipandthe
deviationoftheobservationfromthelinerepresentstherandom
componentoftherelationship.
ThesepointsdivergefromtheregressionlinebyU1,U2,…….Un.
ThefirstcomponentisthepartofYexplainedbythechangesinXand
ThesecondisthepartofYnotexplainedbyX,thatistosaythe
changeinYisduetotherandominfluenceofui.
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25. Definition of the simple linear regression model
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Explains variable in terms of variable “

26. .
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27. 2.3.1 Assumptions of the Classical Linear
Stochastic Regression Model.
Theobjectiveofaregressionanalysisisnotonlyestimatethe
unknownparameters,β‘s,(coefficients).
Y=f(X)+U=Β
0+Β
1X+U
i
Theclassicalmadeimportantassumptionintheiranalysisof
regression.
A.SomeassumptionsarerelatedtoYandX
B.SomeassumptionsarerelatedtoXandX
C.SomeassumptionsarerelatedtoU
Themostimportantoftheseassumptionsarediscussedas
folllows.
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28. Assumption 1:
A. The model is linear in parameters
Themodelshouldbelinearintheparametersregardlessof
whethertheexplanatoryandthedependentvariablesarelinearor
not.
Thisisbecauseiftheparametersarenon-linearitisdifficultto
estimatethemsincetheirvalueisnotknownbutyouaregiven
withthedataofthedependentandindependentvariable.
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29. Cont…
Check yourself whether the following models satisfy the above
assumption or not.
Linearityinvariablesimpliesthatanequationislinearmodelif
itisexpressedinastraightline.
Theparametersareraisedtotheirfirstdegree.
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30. Cont…
Note:Linearregressionmeanslinearinparameterbutit
maynotbelinearintheexplanatoryvariable.
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31. Assumption 2:
B. Uiis a Random Real Variable
Thismeansthatthevaluewhichumayassumeinanyone
perioddependsonchance;
itmaybepositive,
negativeor
zero.
Everyvaluehasacertainprobabilityofbeingassumedbyuin
anyparticularinstance.
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32. Assumption 3:
C. Zero Mean Value of the Error term
Thatis;giventhevalueofXthemeanorexpectedvalue
ofthedisturbancetermiszero.
Technically,theconditionalmeanvalueofεiszero.
Mathematically,
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33. Assumption 4:
D.Thevarianceoftherandomvariable(U)isconstantineach
period:(TheassumptionofHomoscedasticity)
Equalvarianceoftheerrorterm.GiventhevalueofX,the
varianceofistheerrorterm(u)thesameforallobservations.
Thevariationofeachℇ
iaroundallvaluesoftheexplanatoryvalue
isthesame.
Thedispersionofthedisturbanceisthesame.
Thisconstantvarianceiscalledhomoscedasticityassumptionand
Theconstantvarianceitselfiscalledhomoscedasticvariance.
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34. Assumption 5:
E.TheRandomVariable(U)hasaNormalDistribution
Thismeansthevaluesofu(foreachx)haveabellshaped
symmetricaldistributionaroundtheirzeromeanandconstant
variance ,
NormalityTest
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35. Assumption 6:
F.Therandomtermsofdifferentobservations(Ui,Uj)
areindependent.
(The assumption of no autocorrelation)
Thismeansthevaluewhichtherandomtermassumedinone
perioddoesnotdependonthevaluewhichitassumedinany
otherperiod.
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36. Assumption 7:
G.Therandomvariable(U)isindependentoftheexplanatoryvariables.
Thereisnocorrelationbetweentherandomvariableandthe
explanatoryvariable.
Iftwovariablesareunrelatedtheircovarianceiszero.
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37. Assumption 8:
H. The explanatory variables are measured without error
Y = f(X) + Ui
Uabsorbstheinfluenceofomittedvariablesandpossiblyerrors
ofmeasurementinthey’s.
i.e.,wewillassumethattheregressorsareerrorfree,whiley
valuesmayormaynotincludeerrorsofmeasurement.
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38. Cont…
Additionally
Theregressionmodeliscorrectlyspecified.
Thereisnoperfectmulticollinearity(thisholdsinthecaseof
multiplelinearregressionmodel).
Thenumberofobservations(n)mustbegreaterthanthenumber
ofparameters(k)tobeestimated(inmultiplelinearregression
Assumptionofdependantvariable:
Wehavetwoassumptionsofdependentvariables:
oThedependentvariableiYisnormallydistributedand
oSuccessivevaluesofthedependentvariableareindependent.
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39. Example 1:
Lety=a+bx:isalinearrelationshipbetweenxandy.
y=a+bx
2
:isanonlinearrelationshipbetweenxandy.
Example:
Firstfindtheinterceptandslopeofafunction
Writethemathematicalrelationshipbetweenxandy
Variable Y Variable X a.Findthefunction.
b.Identifytheslopeandintercept
ofafunction.
c.Interprettheslopeandintercept
ofafunction
2 1
4 2
6 3
8 4
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40. Example 2: you are given a data on saving and income of five
households as follows:
The slope varies. But we need to establish a linear r/hip between x and y.
The relationship is not exact.
So math’s failed to do so.
But econometrics can make it. How?
i Y = saving X = income a.writethefunction
b.Whatdoobserve
c.Istheslopethesame
d.Canwesolvetheabove
equationsusingmath's?
1 200 500
2 100 300
3 600 1000
4 700 800
5 400 450
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41. Cont…
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42. 2.2 Multivariate Case of CLRM
Insimpleregressionwestudytherelationshipbetweena
dependentvariableandasingleexplanatory(independent
variable);assumethatadependentvariableisinfluencedby
onlyoneexplanatoryvariable.
However,manyeconomicvariablesareinfluencedby
severalfactorsorvariables.
Forinstance;
Indecisiontoinvestmentstudieswestudytherelationship
betweenquantityinvested(oreithertoinvestornot)and
interestrate,shareprice,exchangerate,etc.
Thedemandforacommodityisdependentonpriceofthe
samecommodity,priceofothercompetingor
complementarygoods,incomeoftheconsumer,etc.
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43. Cont….
Hencethetwovariablemodelisofteninadequateinpractical
works.
Therefore,weneedtodiscussmultipleregressionmodels.
Themultiplelinearregressionisentirelyconcernedwiththe
relationshipbetweenadependentvariable(Y)andtwoor
moreexplanatoryvariables(X
1,X
2,…,andX
n).
Why do we need multiple regression?
1.Oneofthemotivationformultipleregressionistheomitted
variablebiasinthesimpleregressionanalysis.
Itistheprimarydrawbackofthesimpleregressionbut
multipleregressionallowsustoexplicitlycontrolformany
otherfactorswhichsimultaneouslyaffectthedependent
variable.
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44. Example: wages vs. education
Imaginewewanttomeasurethe(causal)effectofan
additionalyearofeducationonaperson’swage.
Ifwewanttothemodel:wage=β0+β1educ+uand
interpretβ1astheceterisparibuseffectofeduconwage,
wehavetoassumethateducanduareuncorrelated.
Consideradifferentmodelnow:wage=β0+β1educ+
β2exper+u,whereexperisaperson’sworkingexperience
(inyears).
Sincetheequationcontainsexperienceexplicitly,wewill
beabletomeasuretheeffectofeducationonwage,holding
experiencefixed.
Multiple regression analysis is also useful for generalizing
functional relationships between variables
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45. Simple Régression vs. Multiple Régression
Mostofthepropertiesofthesimpleregression
modeldirectlyextendtothemultipleregression.
Wederivedmanyoftheformulasforthesimple
regressionmodel;however,withmultiple
variables,formulascangetdifficultwhen
explanatoryvariablesmorethantwo.
Asfarastheinterpretationofthemodelis
concerned,there’sanewimportantfact:the
coefficientβjcapturestheeffectofjthexplanatory
variable,holdingalltheremainingexplanatory
variablesfixed.
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46. NOTE
Multipleregressionanalysisisanextensionof
simpleregressionanalysistocovercasesinwhichthe
dependentvariableishypothesizedtodependon
morethanoneexplanatoryvariable.
Muchoftheanalysiswillbeastraightforward
extensionofthesimpleregressionmodel.
Inmultiplelinearregression,wehaveonedependent
variableY,andknumberexplanatoryvariables.
Therelationshipbetweenadependent&two/more
independentvariablesislinearinparameters,and
maynotbelinearinvariables.
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47. Cont……
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48. What changes as we move from simple to
multiple regression?
Potentiallymoreexplanatorypowerwithmore
variables;
Theabilitytocontrolforothervariables;(andthe
interactionofvariousexplanatoryvariables:
correlationsandmulticollinearity);
Hardertovisualizedrawingalinethroughthreeor
more(n)-dimensionalspace.
TheR
2
isnolongersimplythesquareofthe
correlationcoefficientbetweenYandX.
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49. Cont……
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50. 2.2.1 Assumptions of the Multiple Linear
Regression
Inordertospecifyourmultiplelinearregression
modelandproceedouranalysiswithregardtothis
model,someassumptionsarecompulsory.
Buttheseassumptionsarethesameasinthesingle
explanatoryvariablemodeldevelopedearlierexcept
theassumptionofnoperfectmulticollinearity.
Theseassumptionsare:
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51. Cont….
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52. Model With Two Explanatory Variables
Inordertounderstandthenatureofmultiple
regressionmodeleasily,westartouranalysiswith
thecaseoftwoexplanatoryvariables,thenextend
thistothecaseofk-explanatoryvariables.
Estimationofparametersoftwo-explanatory
variablesmodel
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53. Cont….
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Sincethepopulationregressionequationisunknowntoany
investigator,ithastobeestimatedfromsampledata.
Letussupposethatthesampledatahasbeenusedtoestimatethe
populationregressionequation.
Weleavethemethodofestimationunspecifiedforthepresent
andmerelyassumethattheequationhasbeenestimatedbysample
regressionequation,whichwewriteas:

54. Given sample observation on Y, X1,, & X2, we
estimate the model using method of least square (OLS
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55. Cont….
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56. Cont….
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57. Cont…..
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58. Cont….
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59. 2.3.2. Methods of Estimation
Specifyingthemodelandstatingitsunderlyingassumptionsare
thefirststageofanyeconometricapplication.
Thenextstepistheestimationofthenumericalvaluesofthe
parametersofeconomicrelationships.
Theparametersofthelinearregressionmodelcanbeestimatedby
variousmethods.
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60. Cont…
Threeofthemostcommonlyusedmethodsare:
Methodofmoments(MM)
Ordinaryleastsquaremethod(OLS)
Maximumlikelihoodmethod(MLM)
But,herewewilldealwiththeOLSmethodsofestimation.
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61. 2.3.2.2. The Ordinary Least Squares
(OLS Method)
Themodel iscalledthetruerelationshipbetween
YandXbecauseYandXrepresenttheirperspectivepopulation
value,andαandβarecalledthetrueparameterssincethey
areestimatedfromthepopulationvalueofYandX.
ButitisdifficulttoobtainthepopulationvalueofYandX
becauseoftechnicaloreconomicreasons.
SoweareforcedtotakethesamplevalueofYandX.
TheparametersestimatedfromthesamplevalueofYandXare
calledtheestimatorsofthetrueparametersαandβandare
symbolizedasiii XY  
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62. Cont.…
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Estimationof byleastsquaremethod(OLS)or
classicalleastsquare(CLS)involvesfindingvaluesforthe
estimates andwhichwillminimizethesumof
squareofthesquaredresiduals(e
2
).

63. Cont.…
Meaning,theresidualsshouldbesmall.
Therefore,whenassessingthefitofaline,theverticaldistancesof
thepointsfromthelinearetheonlydistancesthatmatter.
TheOLSmethodcalculatesthebest-fittinglineforadatasetby
minimizingthesumofthesquaresoftheverticaldeviationsfrom
eachdatapointtotheline(theResidualSumofSquares,RSS)
MinimizeRSS=
wewillusedifferentialcalculus
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64. Cont.…
Whythesumofthesquaredresiduals?
Whynotjustminimizethesumoftheresiduals?
Topreventnegativeresidualsfromcancellingpositive
ones.
Ifweuse,alltheerrortermse
iwouldreceiveequal
importancenomatterhowclosely/widelyscatteredthe
individualobservationsarefromSRF.
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65. Cont.…
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66. Rearranging we will get:
Divide both sides by “n”:
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67. Rewriting:
Rearrangingtheaboveequationweobtain:
Substitutingthevaluesofαweget:
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68. Rewritten in somewhat different way as follows;
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69. 6/26/2024 Prepared by: Dessie M.

70. 2.3. Statistical Properties of Least Square
Estimators
Therearevariouseconometricmethodswithwhich
wemayobtaintheestimatesoftheparametersof
economicrelationships.
Wewouldliketoanestimatetobeascloseasthe
valueofthetruepopulationparametersi.e.tovary
withinonlyasmallrangearoundthetrue
parameter.
Howarewetochooseamongthedifferent
econometricmethods,theonethatgives‘good’
estimates?
Weneedsomecriteriaforjudgingthe‘goodness’of
anestimate.
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Prepared by: Dessie M.

71. Cont.…
‘Closeness’oftheestimatetothepopulation
parameterismeasuredbythemeanandvariance
orstandarddeviationofthesamplingdistribution
oftheestimatesofthedifferenteconometric
methods.
Weassumetheusualprocessofrepeated
samplingi.e.weassumethatwegetaverylarge
numberofsampleseachofsize‘n’;wecompute
theestimatesβ’sfromeachsample,andforeach
econometric method andweformtheir
distribution.
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72. Cont.…
Wenextcomparethemean(expectedvalue)and
thevariancesofthesedistributionsandwechoose
amongthealternativeestimatestheonewhose
distributionisconcentratedascloseaspossible
aroundthepopulationparameter.
6/26/2024 Prepared by: Dessie M.

73. .
6/26/2024 Prepared by: Dessie M.
AccordingtotheGauss-Markovtheorem,theOLSestimatorspossessallthe
BLUEproperties.Thatis:

74. .
6/26/2024 Prepared by: Dessie M.