Research Methods For Economists

Transcription

1 THE BUSINESS SCHOOL Research Methods For Economsts Keshab Bhattara, Research Methods, HUBS

2 Introducton The major objectve of research n economcs s to fnd out the truth about economc questons that s botherng, ndvdual households, communtes, polcy makers n the local and natonal governments or the nternatonal communty as a whole Some questons are quanttatve by nature such as the dstrbuton of ncome, employment level by sectors, prces and costs of commodtes, demand and supply of varous goods and servces n the economy, nternatonal trade, growth rates of output employment, captal stock, nvestment, rate of returns on fnancal assets, prmary secondary or the unversty level educaton Some other questons are qualtatve These range about the welfare and just socety, phlosophcal questons ncludng behavoural and psychologcal analyss of decson makng process by ndvduals and frms or the polcy makers that often nvolve abstract reasonng Economsts have developed many theores regardng how the varous market functon or should functon How the varous peces of economc actvtes make the natonal or nternatonal economy Economc research therefore can be dvded nto two man groups ) theoretcal research ) appled Theoretcal research often nvolves dervaton of demand supply or equlbrum condtons usng some sort of optmsng process Dagrams, equatons or smply the logcal statements can be used for theoretcal deducton Many of the standard mcro or macro economc models or etenson of those n varous felds of economcs eample of theores, consumer optmsaton, producer optmsaton, determnaton of prces n a partcular market for goods and servces or factors of producton or the general equlbrum n the entre economy or ntertemporal models for accumulaton, nvestment and growth or margnal or cumulatve dstrbutons of populatons or samples or law of large numbers are eample of theoretcal research These can be abstract models and requres usng algebra, calculus, matr or real analyss or stochastc probablty theory to represent these theoretcal deas Theores need to be appled n practce to make them useful for mprovement n the welfare of human socety The applcaton nvolves systematc collecton of nformaton on varables dentfed by partcular theory Testng the clams made by those theores usng lnear or non-lnear functons or varous estmaton or computaton technques As amount of nformaton has grown so has the need to processng those nformaton The appled research s bascally about processng these nformaton consstently, coherently, systematcally usng nductve methods Appled research can also vary accordng to the nature of method used n analyss There are manly four categores of appled research: ) statstcal and econometrc analyss ) calbraton and computatons of system of equatons ) strategc analyss 4) epermental analyss Statstcal analyss nvolves desgnng, mplementng and collectng data on economc varables scentfcally n an unbased manner Ths also nvolves determnng the propertes of dstrbuton of those varables, collectng nformaton on central tendences, correlaton and pattern of relatons among them Econometrc analyss further nvolves usng data for testng varous economc theores based on cross sectons and tme seres data Calbraton and computaton of system of equatons nvolves solvng n number of equatons on the bass of certan assumpton about ther behavour, such as market demand and supply functons, or nput-output analyss or a general equlbrum system Lnear, non-lnear or dynamc programmng s often used to determne such a system Game theory s becomng ncreasngly Keshab Bhattara, Research Methods, HUBS

3 popular tool to analyse nter-dependence among economc agents where the acton to be taken by one s determned by the belefs or percepton of that ndvdual about the acton taken by other people n the economy They are appled to analyse the process and outcome of barganng, strategc contngency plannng or just n descrbng the behavours of economc agents Epermental analyss has the concept of usng control groups for testng economc theores, such as mpacts of certan polcy n economc stablty, such as the adopton of euro, effect of certan drugs, or certan measures on productvty, health or educatonal attanment Economcs Subject group on the Research Methods module s to ntroduce students to quanttatve and analytcal tools requred to prepare research proposals n the second year and for the Independent Study or the Dssertaton modules n economcs n the thrd year and to provde basc sklls requred to eecute research programs as a professonal economst takng account of the most relevant economc theory and avalable prmary or secondary dataset n economc ssue after graduaton Sklls learnt n all other modules ncludng the Emprcal and Mcro Economcs n the frst semester mght be very useful n dong tutorals and assgnments n ths module A specal tutoral group wll be formed for students who have not done Emprcal economcs n the frst semester Ths work book ams to llustrate varous technques that can be useful for economcs students n conductng ther research Issues and topcs Basc Statstcal analyss Bascs: Research questons, theory and applcatons and surveys Samplng and dstrbutons: mean and varance, covarance and correlaton, Frequency dstrbutons, Test of statonarty of tme seres Testng the propertes of a random varable: Use of Normal, Standard Normal, t, F and ch-square dstrbutons n decson makng Basc econometrc analyss Propertes of an estmator: unbasedness, consstency and asymptotc effcency Choce of models: Interpretng regresson results: slopes and elastctes Errors n statstcal decsons: Type I and II errors, Transformaton of varables for analyss Predcton and forecastng Heteroscedastcty and Autocorrelaton Mathematcal technques Theoretcal research: Models of demand and supply and strategc decson optmsaton bascs of lnear and non-lnear programmng Questons about the whole Economy: Introducton to the Input-Output and general equlbrum models for an economy Game theory Epermental technques Keshab Bhattara, Research Methods, HUBS

4 Issues and topcs Choce of technques s the most mportant part of the research process There are mllons of topcs that mght be nterestng, see the hard or electronc copes of journals avalable through the lbrary (JSTOR, Econlt, Busness source premer, SSRN) Hundreds of thousands of economsts have wrtten so many thng n so many subject For nstance I can st down and make a lst of topcs that I see needs some further research as followng: Tmely Issues for Research n Economcs UK Economy Energy prces and economy Polcy rules for economc stablty and growth Reform n publc polcy: ta, spendng, trade, regulatons, redstrbuton 4 Can ta cuts fnance budget defct? (endogenous growth model) 5 Productvty growth n manufacturng and servces sectors 6 Human captal, research and development and economc growth 7 Reasons for declne n publc sector nvestment over years 8 Role of publc and prvate sector n fundng of educaton and health sectors 9 Impact of volatlty of echange and nterest rate on eports Unemployment and nflaton: n the long and the short run Two speed economy: growth of ncome of sklled and unsklled workers Provson for penson and socal securty Impact n the economy of rsng ol and energy prces 4 Lberalsaton of the fnancal sector and prvate sector nvestment 5 Regulatons of market for certan products (eg carpets, moble phones, banana, cars, cosmetcs, drugs, cloths, furnture, nursng home, houses,) 6 Arguments for and aganst prvatsaton of sem-publc goods (e ralways, arlnes, telecommuncatons) 7 Determnants of wage and earnng by professons, sklls and regons 8 Wage and ncome of sport clubs and top qualty sport men and women 9 Factors contrbutng to varaton n growth of regonal and local economes Equty and redstrbuton aspects of councl ta, ncome ta or drect and ndrect taes Can growth occur wth redstrbuton New deal and publc and prvate sector partnershp Role of demand sde and supply sde polces n the economy 4 Patterns of consumpton and savng by categores of households 5 Employment and output multplers wth Input-output model of the UK economy 6 Assessment of the relablty of macroeconomc forecasts 7 London Stock Echange and global economy 8 Evaluaton of economc costs and benefts of envronmental levy Economc Growth and Development Issues 9 Why the four-ffth of the World s stll underdeveloped? Why there s a North-South dvde n per-capta ncome? Story of productvty growth: mpact of ndustral to nternet revolutons Eamnaton of poverty allevaton and economc growth How much can human captal contrbute towards economc growth? 4 Problems n transfer and adopton of technology 5 Why cannot all countres grow at the same rate? 6 What s the best technology to acheve hgher rate of growth? 7 Balanced versus unbalanced growth 8 Does economc growth promote economc nequalty? 9 Do hgher envronmental standards reduce the rate of growth? 4 Conflct, coalton and economc growth 4 Economc costs of conflcts and HIV n Afrca 4 How much spendng on research and development promote economc growth? 4 Infrastructure and economc growth Keshab Bhattara, Research Methods, HUBS 4

5 Macroeconomc ssues 44 Why are Keynesan models applcable more n some countres than n others? 45 Why the rates of unemployment are hgher n rgd labour markets? 46 Eamnaton economc problems when savngs are not equal to nvestment 47 Should households save more to make economy grow faster? 48 Resource mbalances and economc crses 49 Frst, second and thrd theores of economc crses 5 What are the best polcy rules for stablty and growth? 5 Trade-off between unemployment and nflaton? 5 Can ndependent central banks do better than government controlled ones? 5 How can echange rate nstablty be harmful for an economy? 54 Credblty of publc polcy and market reactons Mcroeconomc ssue 55 Determnants of consumpton and savng 56 Are consumers soveregn n market for goods and servces? 57 Income and substtuton effects of prce changes 58 Analyss of short and long run cost of a certan frm or ndustry 59 Consequences of factor and product taes n a compettve market 6 Impacts of new technology n costs of producton and supply 6 Market mperfectons, neffcency and regulaton 6 Is there any evdence for ncome and substtuton effects n agrculture, manufacturng or engneerng sectors? 6 What are the welfare consequences of duopoly or olgopoly n the energy markets? 64 Does deregulaton and prvatsaton brng effcency n allocaton of resources? 65 Analyss of ependture pattern of households 66 How elastctes of supply and demand affect the burden to a consumer? 67 Eamnaton of benefts and costs of prvatsaton 68 Applcaton of utlty mamsaton hypothess under uncertanty Trade ssues 69 Eamnaton of Tarff and non-tarff barrers of trade 7 Does global free trade reduce or ncrease ncome and wage nequalty? 7 Does the drect foregn nvestment promote economc growth? 7 Who benefts and who loses from regonal economc cooperaton? 7 Assessment of mpact of ncrease n ol prces n the global ncome 74 Enlargement of EU and Economc prospects of ts new members 75 How can lberal trade reduce pressure of llegal mmgraton to rch countres? 76 Evaluaton of achevements of the WTO and the Doha rounds of trade talk 77 Leontef parado or factor prce equalsaton? Publc Polcy ssues 78 Should budget be balanced all the tme? 79 Should government subsdse educaton or pay more unemployment beneft? 8 How can budget defct create eternal and nternal mbalances? 8 Can lower taes reduce budget defct? 8 Does the Rcardan equvalence apply n modern economes? 8 Eamnaton of optmal ta rate and evdence 84 Optmal amount of publc servces? 85 Optmal allocaton of publc funds between local and central authortes Households and labour market 86 Why current economc polces have created penson crss n the West? 87 Income dynamcs and lfe-cycle profles of ncome 88 Determnants of wage and labour supply 89 Gender nequalty n wage and earnng 9 Lnk between educatonal qualfcaton and earnng 9 New technology, redundancy and structural transformaton of labour market 9 Socal safety net and unemployment: re-eamnaton of Beverdge provsons Envronment and natural resources 9 Economc mpacts of Kyoto agreement Keshab Bhattara, Research Methods, HUBS 5

6 94 Consumpton and producton sde eternaltes and socal welfare 95 Double dvdend hypothess of envronmental taes 96 Do tght envronmental regulatons reduce economc growth? 97 Valuaton and optmal use of non-renewable resources Fnancal market and economy 98 Over or under nvestment, Value of a frm and the optmal stock of captal 99 Why banks tend to accumulate non-performng debt wth weak montorng? Best way of fnancng economc development Analyss of rsks and return n the fnancal market? Volatlty of fnancal markets and economy Can Tobn ta (transacton cost) deter fnancal crss? Educaton economcs 4 Impacts of unversal prmary educaton n skll formaton 5 Who should pay tutons: students or the government? 6 Matchng educaton and job market Country specfc studes 7 Analyss of markets usng mcroeconomc models 8 Model for macroeconomc polcy evaluaton and forecastng 9 Evaluaton of mpacts of ta reforms Forecastng varous polcy scenaros Movement n commodty prces and terms of trade Commodty Markets: agrcultural goods: sugar, potato, cotton, rubber, green vegetables, tomato frut: apples, banana, pears, grapes, oranges, mango, jackfrut, coconut, nuts 4 grans: rce, corn, mllet, wheat, maze, palm ol, peanuts 5 meat market: fsh, beef, pork, lamb 6 drnks: wne, beer, whskey, martn 7 metals and mnerals: gold, slver, alumnum, steel, ron, copper, tn, znc, ol Uncertanty and asymmetrc nformaton 8 rsks and uncertanty and markets for nsurance 9 moral hazards and adverse selecton prncple agent problem and montorng effcent contract and ncentves provsons for contngency Energy sectors Energy prces and trade n the global economy 4 Generaton and dstrbuton of electrcty and polluton 5 Kyoto agreement on clmate change 6 Trade-off between trade and envronment 7 OPEC effect on ol and energy prces 8 Renewable energy and ehauston of non-renewable energy 9 Fuel poverty Role of energy sector n the growth of economy Technologcal factors n promoton of the energy sector Keshab Bhattara, Research Methods, HUBS 6

7 Basc Statstcal analyss Bascs: Research questons, theory and applcatons and surveys Samplng and dstrbutons: mean and varance, covarance and correlaton, Frequency dstrbutons, Test of statonarty of tme seres Testng the propertes of a random varable: Use of Normal, Standard Normal, t, F and ch-square dstrbutons n decson makng Descrptve Statstcs n Economc Research Desgn a plot for a consumer satsfacton survey for a supermarket n your local area wth a questonnare that ncludes about questons Generate a tabulaton plan for the nformaton from the survey Formulate any fve useful hypotheses for the supermarket Dscuss how you could test these hypotheses Do approprate bar, lne, pe, area charts usng Ecel data fles: MBALS, Studentsls, grow_lowncals and ncome_dstrbutonls Prepare a panel data on growth rates, ratos of nvestment, savng, eports and mports and nflaton and populaton growth for any fve low ncome economes usng data on grow_lowncals 4 Read the panel data fle you generated n () above and fnd the mean, varance, Skewness and Kurtoss for the growth rates, ratos of nvestment, savng and trade and populaton growth rates usng the descrptve statstcs package n PcGve 5 Is there any correlaton between growth rates and savng, nvestment, trade and populaton? Bref Instructons to the use of software Ecel Spreadsheets are very user frendly and could be used for algebrac calculatons and statstcal analyses for many knds of economc models Frst prepare an analytcal soluton by hand then use Ecel formula to compute Ecel has constraned optmser routne at tool/goal seek and solver commend Koop () s a brllant tet for analyss of economc data usng ecel Koop G () Analyss of Economc Data, Wley, UK O-GveWn-PcGve-STAMP (wwwometrcsnet) s a very good econometrc software for analysng tme seres and cross secton data Ths software s avalable n all labs n the network of the unversty by sequence of clcks Start/applcatons/economcs/gvewn Followng steps are requred to access ths software a save the data n a standard ecel fle b start gve wn at start/applcatons/economcs/gvewn c open the data fle usng fle/open datafle command d choose PcGve module for econometrc analyss e select the package such as descrptve statstcs, econometrc modellng or panel data models d choose dependent and ndependent varables as asked by the menu Choose optons for output e do the estmaton and analyse the results, generate graphs of actual and predcted seres Consult manuals by Doornk J A and DF Hendry (() PC-Gve Volume I-III, GveWn Tmberlake Consultants Lmted, London or by vstng the web A Batch fle can be wrtten n O for more complcated calculatons usng a tet edtor n GveWn or such as pfeee Such fle contans nstructons for computer to compute several tasks n a gven sequence Keshab Bhattara, Research Methods, HUBS 7

8 Q Tentatve Answers for Tutoral Consumer Satsfacton Survey I Background - questons A Age below or above B Gender Male Female C Employment Employed Unemployed D If Unemployed Student housewfe job seeker 4 dsabled E Employed self employed sem sklled manual 4 Professonal F Accommodaton rent owns a house Councl house G Educaton below GCSE GCSE A-level 4 College 5 Unversty II Income and ependture What s your average monthly ncome? What s your average monthly spendng? Housng cost Food and beverage Transportaton 4 4 Entertanment 5Other III How frequently do you shop n the followng superstores n month and how much do you spend at one tme? Tesco ASDA Morrson Jackson Sansbury Shoppng Tme Amount spent IV Rank these stores accordng to your level of satsfacton Tesco ASDA Morrson Jackson Sansbury Ecellent Good Average Bad Very bad V Rank any one of the above stores on the bass of followng crtera Ecellent Good Average bad Very bad Choces Staff Other servces Adverts Store Credts Sample sze: Potental hypotheses a Does ependture n a superstore depend on ncome of an ndvdual? b Is there any lnk between the average spendng and frequency of shoppng tme? c Is there a postve correlaton among people spendng on housng, food and beverage, transportaton, entertanment and other spendng? d Does the level of spendng depend on the qualty of store credt? e Do unemployed people lke more ASDA than Tesco? Keshab Bhattara, Research Methods, HUBS 8

9 f Fnd the Spearman rank coeffcent between the ratng of superstores made by male/female, employed/unemployed, those who own a house/those who lve n councl houses g Test whether ependture dffers by the level of educaton and age? h How can you show an nteracton effect of age and educaton and gender on average monthly spendng? Testng these hypotheses requres some understandng of descrptve statstcs and use of Normal, T, F and ch-square dstrbutons Tabulaton plan (Statstcal software such as SPSS s very good n tabulatona) Gender based analyss (structure of the sample) Male Below GCSE GDSE A Level College Unversty Female Average ncome (spendng) Below GCSE GDSE A Level College Unversty Male Female Average ncome (spendng) by gender and age Male Below or above Female Average satsfacton rates for a partcular store by gender and age Male Below or above Female Average satsfacton rates for a partcular store by employment category Employed Unemployed Below or above Keshab Bhattara, Research Methods, HUBS 9

10 By accommodaton status Rentng a house Owns a house Stays n a councl house Male Female Satsfacton ratng of stores by age Below or above Tesco ASDA Morrson Jackson Sansbury Satsfacton ratng of stores by the level of educaton Tesco ASDA Morrson Jackson Sansbury Below GCSE GCSE A Level College Unversty Q B Frequency dstrbuton of MBA Students n HUBS Seres 6 Numbers Keshab Bhattara, Research Methods, HUBS

11 Cumulatve Frequency of Age of MBA Students n HUBS 5 Cumulatve Frequency Keshab Bhattara, Research Methods, HUBS

12 Q Composton and number of Home, EU and Overseas Students n Hull, 4 OS, 86, % H/EU OS H/EU, 4, 87% Composton of Home and EU and Overseas Students n the Unversty of Hull n OS, 6, 5% H/EU OS H/EU, 85, 85% Keshab Bhattara, Research Methods, HUBS

13 Consumpton of Households n the UK H 6% H 4% H 6% H 7% H4 7% H5 7% H9 4% H6 8% H8 % H7 9% Steps to make a pe chart: hghlght data of consumpton dstrbuton by hh Clck on chart Wzard n ecel put data labels, ttles and legends as approprate 4 Save the chart n as a new worksheet to look ncer 5 copy and past n a word document lke above Q Lne graphs Savng rato n Four Low Income Countres 5 4 Bangladesh Benn Bhutan Burkna Faso Keshab Bhattara, Research Methods, HUBS

14 Savng ratos n selected OECD economes Australa Germany Unted Kngdom Unted States Q Preparaton of the panel data Year gy nv rnt ppg sv pcg trd gdppc Keshab Bhattara, Research Methods, HUBS 4

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17 Q4 Computng descrptve statstcs usng Ecel (Tools/data analyss/descrptve statstcs) gy Inv rnt Mean Mean 586 Mean 964 Standard Error 6895 Standard Error 87 Standard Error 9495 Medan 8495 Medan 684 Medan 4457 Mode 4 Mode 69 Mode 6874 Standard Devaton 854 Standard Devaton Standard Devaton Sample Varance Sample Varance 585 Sample Varance 456 Kurtoss 799 Kurtoss 4478 Kurtoss 4547 Skewness 599 Skewness 49 Skewness Range 888 Range 68 Range 48 Mnmum -79 Mnmum 58 Mnmum -497 Mamum 79 Mamum 654 Mamum 65 Sum 485 Sum Sum Count 55 Count 55 Count 55 Largest() 79 Largest() 654 Largest() 65 Smallest() -79 Smallest() 58 Smallest() -497 Confdence Level(95%) 67 Confdence Level(95%) 7584 Confdence Level(95%) 5878 Descrptve statstcs calculated from the PcGve Normalty test and descrptve statstcs usng Pcve Normalty tests and descrptve statstcs (usng growth5panel_gwls) Keshab Bhattara, Research Methods, HUBS 7

18 The sample s - 55 Normalty test for gy Observatons 55 Mean 795 StdDevn 96 Skewness 5784 Ecess Kurtoss 69 Mnmum -79 Mamum 79 Asymptotc test: Ch^() 79 [74]* Normalty test: Ch^() 88 [7]* Normalty test for rnt Observatons 55 Mean 964 StdDevn 659 Skewness -465 Ecess Kurtoss 954 Mnmum -497 Mamum 65 Asymptotc test: Ch^() 5596 []** Normalty test: Ch^() 4597 []** Normalty test for nv Observatons 55 Mean 6 StdDevn 479 Skewness 898 Ecess Kurtoss 95 Mnmum 58 Mamum 654 Asymptotc test: Ch^() 69 []** Normalty test: Ch^() 754 []** Normalty test for ppg Observatons 55 Mean 5548 StdDevn 48 Skewness 474 Ecess Kurtoss -96 Mnmum -466 Mamum 99 Asymptotc test: Ch^() 867 [445] Normalty test: Ch^() 74 [94] Normalty test for sv Observatons 55 Mean 947 StdDevn 5744 Skewness 977 Ecess Kurtoss 999 Mnmum 486 Mamum 448 Asymptotc test: Ch^() 98 []** Normalty test: Ch^() []** Normalty test for pcg Observatons 55 Mean 7 StdDevn 998 Skewness 687 Ecess Kurtoss 749 Mnmum -99 Mamum 9464 Asymptotc test: Ch^() 7464 [4]* Normalty test: Ch^() 955 []* Normalty test for trd Observatons 55 Mean 6875 StdDevn 586 Skewness -894 Ecess Kurtoss -5 Mnmum 9 Mamum 665 Asymptotc test: Ch^() 55 [5]** Normalty test: Ch^() 66 []** Ecess Kurtoss 495 Mnmum 87 Mamum 448 Asymptotc test: Ch^() 7477 []** Normalty test: Ch^() 45 []** Normalty test for gdppc Observatons 55 Mean 4469 StdDevn 7775 Skewness 7856 Keshab Bhattara, Research Methods, HUBS 8

19 Q5 Correlaton coeffcents from Ecel gy nv rnt ppg sv pcg trd gdppc Gy Inv 7458 Rnt Ppg Sv Pcg Trd Gdppc From PcGve gy nv rnt ppg sv pcg trd gdppc Gy Inv Rnt Ppg Sv Pcg Trd Gdppc Research Methods, Economcs, HUBS, 6 9

20 Survey on Study and Epenses I would apprecate f you could help me by answerng the followng survey questons Ths study s just for academc purpose and nformaton provded wll strctly be kept confdental Your gender Male Female Age group a5- b-5 c 5- d above How many hours do you study and sleep now-a-days? Study: a less than three b -5 c 5-8 d 8- Sleep: a less than 5 b 5-7 c 7-8 d 8- e more than 4 How many pages do you read everyday? a less than b - c -4 d above 4 5 How many pages do you wrte everyday? a less than 5 b 5- c - d above 6 How many years have you worked so far? a less than Yr b - c -5 d 5-8 e above 8 7 What s your total spendng? a rent b drnks (tea,coffee or mlk) c all other tems 8 Do you smoke? a Yes b No Possble hypotheses: Is number of years of work sgnfcant determnants of how many pages does one read, or wrte or hs or her hours of study? Is the amount of spendng n drnks lnked to numbers of hours of study? Are hours of study sgnfcantly dfferent between males and females? 4 Do pages read everyday vary accordng to age group? 5 Do smokers study longer hours? Statstcal analyss: a proporton of male and female by age group and smokng habts n the sample b Mean, varance, Skewness and Kurtoss and correlaton of age, study hours, pages wrtten and read and years of work epenses on rents, drnks and other thngs 6 Calculate the condtonal probablty of : a a person s smoker gven that he s male b at least page wrtng gven that she s female c the poor student (who spends the least) studyng longer hours 7 Wrte a probt model for decson to smoke as a functon of se, spendng, study hours and pages wrtten and read Research Methods, Economcs, HUBS, 6

21 Tutoral Statstcal Analyss of cross sectons Prepare an ecel fle for the followng data on marks scored by students n two eams and ther monthly earnngs from part tme jobs Use Ecel to calculate varous statstcs as asked below Scores n Eams and Earnngs Observato Eam Eam Earnng Eam Eam Earnng n observaton Represent the data on scores n eams and earnng usng margnal and cumulatve frequency dagrams What are means and varances of scores n eam and eam? What are the coeffcents of varaton of scores n eams and eam? What s the covarance of marks n eams and? 4 What s the correlaton coeffcent of scores between eam and? 5 If eam weghs percent but the scores n eam wegh only percent what would be the weghted aggregate mean score n these two eams? What would be the varance of weghted scores? 6 Eam took place before eam Test whether scores n eam can predct scores n eam? 7 Predct scores n eam for students who scored 6 and 8 n eam 8 Test hypothess whether scores n eam and eam are sgnfcant determnants of earnng Why may earnngs be negatvely related wth ther score n the eams for full tmes students? 9 How can behavours of teachers and students change the dstrbuton of marks? If the true mean was 6 for score and 58 for score fnd whether the current sample reflects the populaton usng t-test Derve the standard normal dstrbuton for score and construct a 99 percent confdence nterval for t Research Methods, Economcs, HUBS, 6

22 4 Densty Eam Densty Eam 5 5 Densty Earnng Densty Eam 4 5 Densty Eam 5 5 Densty Earnng 5 5 Eam Eam 5 Earnng 5 5 Eam Eam Earnng Eam Eam Earnng 5 5 Eam Frequency eam Frequency earnng Frequency More More More Research Methods, Economcs, HUBS, 6

23 Hstogram 5 Frequency 5 Frequency More eam Hstogram 5 Frequency 5 Frequency More eam Hstogram Frequency More earnng Frequency b Research Methods, Economcs, HUBS, 6

24 Mean Sample Estmators of Populaton Parameters Varance Standard Dev Skewness: Kurtoss: var ( ) N ( ) N ( ) s N μ ( ) β μ μ N μ 4 β μ μ 4 N β β Frst Moment: Second Moment: μ μ Skewness (Pearson) (Mean-Mode)/standdev ( ) Normalty mples: Means, standard devatons and correlatons (usng score6ls) The sample s - 48 Means Eam Eam Earnng Standard devatons (usng T-) Eam Eam Earnng Correlaton matr: Eam Eam Earnng Eam Eam Earnng Coeffcent of varaton (CV) s the standard devaton relatve to the mean CV σ Ths s useful for comparng varable n dfferent seres c Coeffcent of varaton: rato of standard devaton/mean Eam Eam Eam 948 Eam d Correlaton matr: Eam Eam Eam Eam e Weghted average: Research Methods, Economcs, HUBS, 6 4

25 ( w w ) w + w where N w, w n eam and eam respectvely The weghted average: 5 Varance of the weghted score: f Eam *Eam (SE) (448) (786) EQ( ) Modellng Eam by OLS-CS (usng Score6newls) The estmaton sample s: to 48 Coeffcent StdError t-value t-prob PartR^ Constant Eam sgma 7565 RSS R^ 5745 F(,46) 66 []** log-lkelhood -645 DW 65 no of observatons 48 no of parameters mean(eam ) 57 var(eam ) 778 Normalty test: Ch^() 8658 [6486] Null of the normalty of errors s not rejected w w and are scores 5 Densty r:eam N(,) hetero test: F(,4) 5956 []** hetero- test: F(,4) 5956 []** RESET test: F(,45) 5797 []** There s statstcal evdence for hetroscedastcty For nstance see below that the error term s sgnfcantly vares wth and square Research Methods, Economcs, HUBS, 6 5

26 EQ( ) Modellng resduals by OLS-CS (usng Score6newls) The estmaton sample s: to 48 Coeffcent StdError t-value t-prob PartR^ Constant esquare Eam sgma 655 RSS 9866 R^ 598 F(,45) 7898 []** log-lkelhood DW 79 no of observatons 48 no of parameters mean(resduals) 66454e-5 var(resduals) 546 Normalty test: Ch^() 447 [88] hetero test: F(,4) 67 [75]* hetero- test: F(4,4) 4964 [579] RESET test: F(,44) 458 [8] resduals *esquare + 974*Eam (SE) (64) (5) (4) g Score n eam : α + β + ε If 6 : Eam *Eam * If 8: Eam *Eam *8 798 h The estmaton sample s: to 48 Earnng *Eam *Eam (SE) (7) (445) (544) EQ( ) Modellng Earnng by OLS-CS (usng Score6newls) The estmaton sample s: to 48 Coeffcent StdError t-value t-prob PartR^ Constant Eam Eam sgma 779 RSS R^ 6755 F(,45) 65 [8] log-lkelhood -647 DW 7 no of observatons 48 no of parameters mean(earnng) 78 var(earnng) Normalty test: Ch^() 4 [995] hetero test: F(4,4) 676 [64] hetero- test: F(5,9) [7] RESET test: F(,44) 967 [6598] Research Methods, Economcs, HUBS, 6 6

27 Any random dstrbuton can be modfed to standard normal dstrbuton: z σ f () Normal Dstrbuton of : Bell Shaped Dstrbuton ( μ) f ( ) ep( ) σ π σ var Propertes: E ( ) μ ( ) E ( μ ) σ Symmetrc: bell-shaped 687% Total area 95% 997% μ σ μ 96σμ σ μ μ + σ μ + 96σ μ + σ Standard Normal z N(,) μ z σ Modfyng the scores to a standard normal dstrbuton Densty stne Densty stnearn Densty - stne 75 - Densty - Eam student wth more efforts can earn more marks and teachers can have more lberal or less lberal approaches to markng Research Methods, Economcs, HUBS, 6 7

28 Hypotheses mean score n eam s 6 and 58 n eam The degrees of freedom ( n ) s Crtcal or theoretcal value for ths s t 45, 65 μ For score n eam : tn 55 s n μ For score n eam : tn 876 s n 4 4 Ths test does not reject the null There we can accept true mean scores n these eams are 6 and 58 4 Densty Stnorm P 75 5 ( 58σ z + 58σ ) ( α ) 99 Densty St norm P ( z ) ( α ) Stnorm Stnorm P ( 55 4 z 658) ( ) Thus the true score les between 554 and 658 wth probablty of error of percent Research Methods (64): Economcs Test Questons Student number: BA or BSc Date: Feb 7, 6 Draw a Venn dagram and show how any four samples of elements can be generated from a populaton that contans 6 elements [8 marks] Illustrate how F statstcs can be used to test whether varances n two dfferent samples are statstcally dfferent or not [8 marks] var( ) m [hnt: F wth m and m degrees of freedom] var m ( ) Research Methods, Economcs, HUBS, 6 8

29 Show unbasedness, effcency and consstency for an estmator usng the followng dagram [8 marks] Margnal probablty densty for a random varable f ( βˆ ) 4 Construct a 95 percent confdence nterval for a Normally dstrbuted varable whose mean 5 and standard devaton s [ Hnt: P ( z σ < μ < + z σ ) ( α ); z α 96 ] [8 marks] α α 5 Draw a densty functon and show how one can do two sded or one sded tests on whether a varable belongs to a normal dstrbuton? [8 marks] 6 Show how one can compute the prce elastcty of supply from the estmates of the followng regresson model [8 marks] Research Methods, Economcs, HUBS, 6 9

30 Postve Lnear Relaton Supply P Y P + e β + β β Y β H H : β > : β 7 Use followng dagrams to show how a researcher can rase probablty of makng type II error whle tryng to mnmse type I error [8 marks] f ( ˆβ ) α E ( ˆ β ) β α ( ˆ β ) β E 8 Observe the followng statstcs on the trade rato (trd) and GDP per capta (gdppc) for 55 countres Are both of them normally dstrbuted base on skewness, Kurtoss statstcs and Ch-square statstc? [8 marks] Normalty test for trd Observatons 55 Mean 6875 StdDevn 586 Skewness -894 Ecess Kurtoss -5 Mnmum 9 Mamum 665 Asymptotc test: Ch^() 55 [5]** Normalty test: Ch^() 66 []** Normalty test for gdppc Observatons 55 Mean 4469 StdDevn 7775 Skewness 7856 Ecess Kurtoss 495 Mnmum 87 Mamum 448 Asymptotc test: Ch^() 7477 []** Normalty test: Ch^() 45 []** Research Methods, Economcs, HUBS, 6

31 9 In a sample of 48 employees average number of tmes an employee takes a publc transport was found to be 55 out of ten possble trps wth standard error of 4 Hypothetcally ths mean s beleved to be 6 Determne usng t-test whether sample mean 55 s statstcally sgnfcant or not [8 marks] Crtcal t-value for the % level of sgnfcance for 48 degrees of freedom s Consder a stuaton where you assumed that true mean of a populaton was 5 nstead of whch was ts true mean If you nfer from sample assumng that 5 was ndeed a true mean what short of error are you makng? [8 marks] What happens when the hypothess s wrong? f ( ) 5 μ μ α E( ) 5 E( ) α Study the followng estmate of demand for a product estmated usng Ecel Based on t, F tests and R-square statstcs determne whether the coeffcents and the overall model s statstcally sgnfcant or not [8 marks] Crtcal value of F for (9,) degrees of freedom s 978 Estmaton of a lnear demand Model n Ecel SUMMARY OUTPUT Regresson Statstcs Keshab Bhattara, Research Methods, HUBS

32 Multple R R Square Adjusted R Square Standard Error Observatons 98 ANOVA df SS MS F Regresson Resdual Total Coeffcents Standard Error t Stat P-value Intercept E-9 Prce E-9 s a random varable, μ s ts mean, and σ ts standard devaton What are the mean and varance of a normalzed random varable? a mean s zero and varance s b mean s and varance s zero c both mean and varance are zero d both mean and varance are [6 marks] Income s normally dstrbuted wth mean μ and varanceσ A surveyor can be 997 percent certan about the mean ncome wthn a one standard errors ( μ ± σ ) b two standard errors ( μ ± σ ) c three standard errors ( μ ± σ ) d four standard errors ( μ ± 4σ ) [6 marks] Keshab Bhattara, Research Methods, HUBS

33 Research Methods Economcs s a random varable, μ s ts mean, and σ ts standard devaton What are the mean and varance of a normalzed random varable? a mean s zero and varance s b mean s and varance s zero c both mean and varance are zero d both mean and varance are Answer: a What s the level of sgnfcanceα n the confdence nterval 96 z 96 α z ~ N,? P ( ) ( ) 95 for a randomzed normal varable, ( ) a % b 5% c 5% d % Answer: b What are the values of skewness and kurtoss for a normally dstrbuted random varable? a and b and c and d, Answer: a 4 There are regons n the UK You want to desgn a sample takng four regons at a tme How many samples can you desgn from these regons? a 46 b c 6 d 4 Hnts: Cn n! ( n r)! r! Answer: b 5 Income s normally dstrbuted wth mean μ and standard error σ A surveyor can be 997 percent certan about the mean ncome wthn a one standard errors ( μ ± σ ) b two standard errors ( μ ± σ ) μ σ μ ± 4σ c three standard errors ( ± ) d four standard errors ( ) Answer: c 6 What happens when you ncrease the level of sgnfcance n a test? Type I Error: Rejectng a Null Hypothess when t s true f ( βˆ ) Truth? f ( ˆβ ) Type II Error: Acceptng a Null Hypothess When t s False Type I error s more serous than type II error But when type I error s mnmsed type II error rases When the Null hypothess s ncorrect Type II Error Type I Error Type I Error Type I Error Type I Error ( ˆ β ) β α E α α E ( ˆ β ) β α ( ˆ β ) β E a Type I error ncreases and Type II error falls b Type I error falls but Type II error rases c Both Type I and Type II error rase d Only Type I error falls but the Type II error remans the same Answer: b 7 When s a strategy domnant for both A and B players n a game? Keshab Bhattara, Research Methods, HUBS

34 Basc Elements of a Game Theory Strategc Interacton among people for economc gan Ratonal Players Strategc Choces A Payoff matr B Strategy Strategy R, p ( ) U, j Strategy Strategy R C R C ( ) ( ) ( ) ( ),,,,,, R C R C,,,,,, payoff to row player when both row and column play strategy Neumann-Morgenstern Utlty functon a When A gans more than B b When B gans more than A c When none of them does better by choosng any other strategy d When A agrees to compensate for loss of B Answer: c 8 General equlbrum n the economy s the system of prces where General Equlbrum: Equalty of Margnal Rate of Substtuton and Margnal Rate of Transformaton + 5 α ( α ) U MRS y U, A MRT P P 5 5 L + λ a where consumer mamze ther utlty but the producers don t b where producers mamze profts but consumers don t c where some markets stll do not clear d where all markets clear and both consumer and producer optmse gven the constrants they face Answer: d 9 μ What best descrbes the t-test, tn k? s n a It tests whether the sample mean compares to populaton mean b It tests whether the sample varance compares the populaton varance c It tests whether both the sample mean varance compare to the populaton mean and varance d It tests whether ether of the sample mean varance compare to the populaton mean or the varance Answer: a What s the χ test used for? χ k ~ N(,) k a It tests whether the sample mean of a square of a normally dstrbuted varable compares to ts populaton counterpart Keshab Bhattara, Research Methods, HUBS 4

35 b It tests whether the sample varance of a square of a normally dstrbuted varable compares to ts populaton counterpart c It tests whether the sample sum of a square of a normally dstrbuted varable compares to ts populaton counterpart d It tests whether ether of the sample mean varance compare to the populaton mean or the varance Answer: c What does the F test, V m tests for? F ~ F V m, m m a It tests whether the means are dfferent between two samples b It tests whether the varances dffer between two samples c It tests whether both the sample mean varance dffer between two samples d It tests whether the mean n one sample compares to varance n another sample Answer: b Problem Estmaton, Hypothess Testng and Confdence Intervals Numbers of years of work among employees n a local labour market s assumed to be normally dstrbuted The mean years of work was found to be 5 wth a standard devaton n a survey of ndvduals Usng ths nformaton construct 9, 95 and 99 percent confdence ntervals for years of work among employees n ths market ( Hnts, Z 645, Z 96, Z 576 for, 5 and percent level of sgnfcance respectvely) Formulate a hypothess regardng possble relaton between consumpton, ncome and the nterest rate Regresson estmate of consumpton on ncome generated slope coeffcent 95 and ts standard error was n a survey of 4 students ) Do you accept or reject your hypothess wth percent level of sgnfcance? ) Construct 9, 95 and 99 percent confdence ntervals for ths slop coeffcent ) What are type I and type II errors n ths eample? Suppose you have the followng data set on number of tckets sold n a football match (Y ), prce of tckets ( ) and ncome of the customers ( )as gven n the followng table and Y are measured n thousand pounds You want to fnd out the eact relaton between tckets sold and prces and ncome of people watchng football games Observatons Y Y Y Y Keshab Bhattara, Research Methods, HUBS 5

36 n 4 n n n n n n Total (a) Wrte a smple regresson model to eplan the number of tckets sold n terms of the prce of the tcket Eplan brefly underlyng assumptons and epected sgns of the parameters n ths model (b) Estmate the slope and ntercept parameters Use cross products and squared terms provded for you n the above table Yˆ, eˆ and the R and R (c) Usng your estmates n (b) fnd the eplaned squared sum (d) Estmate the varance of the error term and the slope coeffcent Eplan ts mportance (e) Test whether the slope term s sgnfcant at 5% confdence level (f) Buld 95 percent confdence nterval for estmate of slope and ntercept terms (g) Dscuss how reducng type I error may cause ncrease n type II errors (h) Calculate the elastcty of demand for football around the mean of Y and () Wrte a multple regresson model to eplan the number of tckets sold n terms of the prce of the tcket and the ncome of ndvduals gong to the football game What addtonal assumpton(s) do you need whle ntroducng an addtonal varable (j) Estmate the parameters of that multple regresson model (k) What s your predcton of the number of tckets sold f 5 and 4? Introduce dummy varables n your multple regresson model to show dfferences n demand for football tcket based on gender dfferences ( for male and for females), four seasons (autumn, wnter, sprng and summer) and nteracton between gender and ncome Answers to Problem μ For a varable wth standard normal dstrbuton, z, where s the σ sample mean σ sample standard devaton and μ the populaton mean the confdence nterval for μ for α level of sgnfcance s gven by P ( z σ < μ < + z σ ) ( α ) The 9 percent confdence nterval: P ( < μ < ) ( ) 9 The 95 percent confdence nterval: P 5 96 < μ < ( ) ( ) 95 The 99 percent confdence nterval: P < μ < ( ) ( ) 99 α α Keshab Bhattara, Research Methods, HUBS 6

37 ˆ β β 95 t calc 879 t 4, 74 var ( ˆ β ) H : β H : β ; t calc > t 4, Therefore reject the null The statstcal evdence does not support the hypothess that the slope coeffcent equals zero In other words the estmated coeffcent s statstcally sgnfcant β β ˆ P t ( α ) c t var( ˆ β ) c P ( ˆ β t SE( ˆ β ) β ˆ β + t SE( ˆ β ) ( α ) c c From t-table crtcal values of t t 74 ; t and t 684 4, 4,5 The 9 percent confdence nterval: P ( β ) ( ) 9 The 95 percent confdence nterval: P ( 95 β 95 + ) ( 5) 95 The 9 percent confdence nterval: P ( β ) ( ) 9 4, Suppose you have the followng data set on number of tckets sold n a football match (Y n thousands), prce of tckets ( ) and ncome of the customers ( n thousand pounds) as gven n the followng table You want to fnd out the eact relaton between tckets sold and prces and ncome of people watchng football games Observatons Y Y Y Y Total Answer all questons from the part A and any fve questons from the Part B (l) Wrte a smple regresson model to eplan the number of tckets sold n terms of the prce of the tcket Eplan brefly underlyng assumptons of your model What are the epected sgns of the parameters n ths model? Answer: A smple regresson Model Y β + + e β, Man assumptons about the error term e are followng: Mean of e s zero for every value of, E [ e ] varance of e s constant var[ e ] σ for every th observaton Keshab Bhattara, Research Methods, HUBS 7

38 cov( e e j ) for all j ; ths also means there s no autocorrelaton or heteroscedastcty; errors are homoscedatc and ndependent of each other there s no correlaton between e and the eplanatory varable ; E [ e ] eplanatory varable,, s eogenous or non-stochastc or t s not random varance of the dependent varable s equal to the varance of the error term var( y ) var[ e ] σ Y Epected sgn of s negatve ( β < ) and β s ntercept term epected to be postve Ths would show the demand for the number of tckets f the prce level was set to zero (m) Estmate the slope and ntercept parameters Use cross products and squared terms provded for you n the above table Answer: The easest way to estmate regresson wth one eplanatory varable s by usng the equaton n devaton form y β + e Ths mples ˆ β y, where usng the devaton formula and the nformaton gven n the table y ( )( Y Y ) Y N Y 97-7(4) (5) where 5 5 and 8 Y Y 4 N 7 N 7 and (, ), N 45 7(5) ; 4 ˆ β 64 7 Use ths value for ˆβ and the means of Y (n thousands), prce of tckets ( ) to estmate ˆ β as followng: β Y ˆ β 4 ( 64) Ths the estmated relaton can be wrtten as: Y ˆ ˆ β + ˆ β, 77 64, Y β 64 < as we epected Ths gves us a downward slopng lnear demand for tckets See estmates form Ecel Keshab Bhattara, Research Methods, HUBS 8

39 Regresson Statstcs Multple R 977 R Square 9467 Adjusted R Square 94 Standard Error 5654 Observatons 7 ANOVA df SS MS F gnfcance F Regresson Resdual Total 6 8 Coeffcentstandard Err t Stat P-value Lower 95%Upper 95% Intercept E Res dual Pl ot 8 Lne Ft Plot 6 Y Y Predcted Y (n) Usng your estmates n (b) fnd the eplaned squared sum Y ˆ, eˆ and the R and R for ths model [] Answer: We can use the estmated for ˆβ to estmate the sum squared errors and sum squared for eplaned varaton as followng: ˆ > y yˆ + e y β, + e y y + e y ˆ β + eˆ > 8 (-64)* 7 + eˆ, ˆ (-64) *7 64 e ˆ 8 64 y and 584 Keshab Bhattara, Research Methods, HUBS 9

40 R R yˆ y eˆ y 64 94% ; 8 ( N K ) ( N ) 584 / 8/ ( 7 ) ( 7 ) 68 9% 4667 (o) Estmate the varance of the error term and the slope coeffcent n ths smple model [] The unbased estmator of the varance of the error term s gven by eˆ 584 var( e ) ˆ σ 68 ; N K 5 The varance of the slope parameter s gven by: ( ˆ ˆ σ 68 var β ) 45 7, (p) Test whether the slope term s sgnfcant at 5% confdence level [] Answer: Calculate t value for the gven estmates to test H : β aganst H : β β ˆ β 64 t calc 967 ; The theoretcal t value for t 5,5 57 var( ˆ β ) 4 Therefore calculate t-value s larger than the table t-value Ths mples that the estmated coeffcent ˆβ s statstcally sgnfcant Based on ths evdence we can reject the null hypothess whch states that H β : f A 95 percent confdence nterval means acceptng the errors of 5 percent Such confdence nterval s gven by the followng formula β β ˆ k P t c tc α var( ˆ β k ) Or by reorganzng ths P ˆ β t SE ˆ β β ˆ β + t SE ˆ β α ( 5) P ( k c ( k ) k k c ( k ) 95 [ ˆ β t SE( ˆ β ), β + t SE( ˆ β )] k c k k c k Keshab Bhattara, Research Methods, HUBS 4

41 4 As shown above ˆ 64 7 SE ˆ ˆ σ 68 var β, 7 β ( ) 45 ( ˆ ) ( ˆ ˆ σ 68 β var β ) 45 67, 7 degrees of freedom ( K ) 7 5 P, N and t 57 ( ˆ β t SE( ˆ β ) β ˆ β + t SE( ˆ β ) k c k k k c k 5,5 ( ) P β ( ) P β P ( ) α ( 5) 95 β k k k Thus we are 95 percent confdent that the true β k les between -446 and -787 (h) Consder followng two dagrams to thnk about type I and Type II error The type I error equals the level of sgnfcance chosen by the researcher When one s happy beng 95 percent confdent then Type I error s 5 percent The Type II error occurs when the Null hypothess n ncorrect In the above test the Null tested was H : β If the rght null was nstead say H : β 5 then havng H β nstead of the true null wll create a Type II error : Type I and Type II Error n Hypothess Testng Accept Reject α True Correct Decson Type I Error Sze of test False Type II Error β Correct Decson P-value: Probablty of Tet statstcs Eceedng that Of the sample Power of test: probablty of rejectng the null whle t s false Power -beta - Prob (type II error) Type II occurs when the Null hypothess s wrong Keshab Bhattara, Research Methods, HUBS 4

42 f ( βˆ ) Type II Error: Acceptng a Null Hypothess When t s False Type I error s more serous than type II error But when type I error s mnmsed type II error rases When the Null hypothess s ncorrect Type II Error Type I Error Type I Error ( ) ˆ β 5 α E α E ( ˆ β ) (h) elastcty of demand The defnton of elastcty of food ependture on ncome s gven by Y Y Y 7 η ˆ β Y Y 4 Wrte a multple regresson model to eplan the number of tckets sold n terms of the prce of the tcket and the ncome of ndvduals gong to the football game What addtonal assumpton(s) do you need when you ntroduce one more varable Answer : Ths multple regresson model s obtaned by addng the ncome term n the above model: β + β, + β, Y + e Y Epected sgn of the slope coeffcent n prces β s negatve as before( β < ) Y and that of ncome coeffcent s postve β >, β s ntercept term epected to be postve Addtonal assumpton requred s that the prce and ncome varables should not be collnear Otherwse t wll create a Multcollnearty problem No estmate of β and β s possble when and are eactly correlated (j) Ths model need to be modfed to estmate the parameters of the multple regresson model gven n () Agan t s convenent to use the regresson model n the devaton form, y β, β,, (g) + β β, (g), y,, + Keshab Bhattara, Research Methods, HUBS 4

43 ˆβ ˆβ,,,,,,,,,, y y,,,, y,,, y, (g) (g4) ˆ β Y ˆ + β β (g5) (, )( Y Y ), Y N Y 97-7(4) (5) , y,, N 45 7(5) ;, y and ( ) ˆ β ˆ (, )( Y Y ), Y N Y 6 7(4)(4) 6 4 (, ), N 6 7(4) 6 4 ; (, )(, ),, N 7 7(5)(4) 7 4,,,,,,,, y,,,, y ( ) 4 4( 4) ( ) ( 7)( 4) ˆ β,,, (,, ) y, y,,, ( ) ( 4) ( ) ( 4) 4(7) ˆ β ˆ ˆ Y β β 4 ( 68)5 ( 996)4 Estmated multple regresson lne s Yˆ ˆ β + ˆ β ˆ, + β, Compare ths result from the results obtaned from the regresson routne contaned n Ecel (tools/data analyss/regresson) as gven below Keshab Bhattara, Research Methods, HUBS 4

44 SUMMARY OUTPUT Regresson Statstcs Multple R 978 R Square Adjusted R Square 9588 Standard Error 696 Observatons 7 ANOVA df SS MS F Sgnfcance F Regresson Resdual Total 6 8 Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Intercept RESIDUAL OUTPUT Observaton Predcted Y Resduals Compare ths wth the results n the PcGve ( ) Modellng Y by OLS-CS (usng Q_football_gwls) The estmaton sample s: to 7 Coeffcent StdError t-value t-prob PartR^ Constant sgma 696 RSS R^ F(,4) 7 []** log-lkelhood DW 8 no of observatons 7 no of parameters mean(y) 4 var(y) 4 Normalty test: Ch^() 8 [946] Hetero test: not enough observatons Hetero- test: not enough observatons RESET test: F(,) 874 [9]* Y * - 996* (SE) (47) () (48) Keshab Bhattara, Research Methods, HUBS 44

45 Matr method could be used to fnd the soluton as followng: ˆ β ˆ β ˆ β ˆ β N,,,,,,,,,, Y Y Y Y,, ( ' ) ' Y (k) What s your predcton of the number of tckets sold f 5 and 4? [5] Answer: Yˆ ˆ ˆ ˆ β + β, + β, () 5 996() 4 4 (l) Introduce dummy varables n your multple regresson model to show dfferences n demand for football tcket based on gender dfferences ( for male and for females), four seasons (autumn, wnter, sprng and summer) and nteracton between gender and ncome Answer: β + β, + β, + γg + θg, + ψ S + ψ S + ψ S Y + e f male G otherwse S S S f summer otherwse f Autumn otherwse f wnt er otherwse Answer any fve questons from ths part Keshab Bhattara, Research Methods, HUBS 45

46 PART B (m) You want to ntroduce taste for the game as a new varable Suppose that the varable gvng the taste for football s eactly correlated wth the ncome levels of ndvduals Show how the OLS procedure breaks down f you regressed Y on and the taste varable ( ) f the relaton between and s gven by [] Answer y y y y,, ˆ β,,,,,,, ( ),,,, Thus the coeffcent becomes ndetermnate,, Keshab Bhattara, Research Methods, HUBS 46

47 Problem 4 Tme Seres Analyss Study monthly data on unemployment, retal prce nde and value of transactons n the stock market as contaned n stocksandprcels fle for the UK economy from 97:4 to 4:8 It s taken from the wwwstatstcsgovuk usng the Navdata Answer followng questons usng PcGve a Represent unemployment, retal prce and value of stock transactons usng lne graphs b Check whether these seres are statonary usng the unt root test What wll be the consequence of runnng a regresson f varables are non-statonary? c Use GveWn calculator to compute the frst dfference of the unemployment rate, retal prce nde and value of stocks d Test for unt root n the frst dfferences Are they statonary? e Determne whether volume of transacton n stocks can be eplaned by retal prces and unemployment rates Do all ths usng frst dfferences f Ft an autoregressve model to forecast volume of stocks n for net twenty months g Wrte 95 percent confdence ntervals for these forecasts There s a concern about the dsparty of ncome among rch and poor households n the UK Study the dstrbutons on ncome and consumpton by decles of households for as gven below H H H H4 H5 H6 H7 H8 H9 H Total Gross ncome Consumpton Source: Economc Trends, a Use Ecel to plot consumpton and ncome n () a pe chart and () a column or a bar chart b Compute means and varances of ncome and consumpton c Calculate the ndvdual and cumulatve shares of ncome decles of households d Represent ths ncome dstrbuton usng a Lorenz curve e What s a Gn (G) coeffcent? Whch values of G represent perfect equalty and whch one represents perfect nequalty? f Calculate the Gn-coeffcents for above dstrbuton of ncome and consumpton (hnt: area of a trangle 5 base hght) g Calculate the beneft and transfers mplct n the above table Brefly eplan how such redstrbuton of ncome occurs n the real world stuaton Study the ncome dstrbuton pattern of the Unted States as gven n the followng table and answer b-d of parts of the above questons Average ncome Populaton Quntle Income share 86 st (lowest) 5 5 nd rd th th (hghest) 5 Source: DeNavas-Walt and Cleveland () and Weal (5) Economc Growth, p 66 How much dfference do you observe n Gn coeffcents between the US and the UK? Why? Keshab Bhattara, Research Methods, HUBS 47

48 Answers a Unemployment, retal prce and the value of stock n the UK 97:4 to 4:8 A tme seres has four components: trend, cycle, season and random element URT RPI Stocks Unt Root Tests of Unemployment Rate U n t ro o t e s ts n th e le v e l o f u n e m p lo y m e n t ra te URT: ADF tests (T7, C onstant; 5% -87 % -45) D-lag t-adf beta Y_ sgm a t-dy_lag t-prob AIC F-prob T h e re s n o u n t ro o t n th e frs t d ffe re n c e DURT: ADF tests (T7, Constant; 5%-87 % -45) D-lag t-adf beta Y_ sgm a t-dy_lag t-prob AIC F-prob ** ** b A seres -59** s non-statonary 77 when 969 ts mean 97 and varance 545 are -4 not constant 68 over tme or when these vary - 5 ** over tme These seres O u tp are u t non-statonary fro m th e P c Gn v ether levels but statonary n the frst dfferences Use the ADF test to determne ths Stocks: ADF tests (T7, Constant; 5%-87 %-45) D-lag t-adf beta Y_ sgma t-dy_lag t-prob AIC F-prob e e e e+4 4 URT: ADF tests (T7, Constant; 5%-87 %-45) D-lag t-adf beta Y_ sgma t-dy_lag t-prob AIC F-prob RPI: ADF tests (T7, Constant; 5%-87 %-45) D-lag t-adf beta Y_ sgma t-dy_lag t-prob AIC F-prob DURT: ADF tests (T7, Constant; 5%-87 %-45) Keshab Bhattara, Research Methods, HUBS 48

49 D-lag t-adf beta Y_ sgma t-dy_lag t-prob AIC F-prob -765** ** ** ** DRPI: ADF tests (T7, Constant; 5%-87 %-45) D-lag t-adf beta Y_ sgma t-dy_lag t-prob AIC F-prob -694** e ** ** ** DStocks: ADF tests (T7, Constant; 5%-87 %-45) D-lag t-adf beta Y_ sgma t-dy_lag t-prob AIC F-prob -79** -578e ** -694e ** e ** -484e+4 88 How to make a Non-Statonary Seres to a Statonary Seres Logs ratos Frst dfference d A spurous regresson occurs Second n dfference tme seres It happens when the coeffcents n a regresson appear to be sgnfcant when there Thrd s no or such hgher relaton order dfference Such coeffcents may just be showng the trend, cycle or season element nherent Contegraton a tme seres Make seres statonary and Error run the Correcton regresson n statonary seres to fnd whether there s any sgnfcant relaton See the followng estmates on whether stock prces are determned by unemployment rate and the retal prce ndces Coeffcents are sgnfcant Temptng to state that there s a sgnfcant relatonshp But from above Stocks, URT and RPI are non-statonary varables Ths s an eample of a spurous regresson Modellng Stocks by OLS (usng stocksandprcels) The estmaton sample s: 97 (5) to 4 (7) Stocks + 7e+5 - e+4*urt - 45e+4*RPI (SE) (e+4) (5e+) (75) Coeffcent StdError t-value t-prob PartR^ Constant 7 7e URT RPI sgma 854 RSS 74968e+ R^ 5499 F(,7) 87 []** log-lkelhood DW 6 no of observatons 75 no of parameters mean(stocks) 684 var(stocks) 5756e+ Check whether ths relaton s true for when seres are made statonary takng the frst dfferences Now t s clear that these relatons are not sgnfcant Coeffcent StdError t-value t-prob PartR^ Constant DURT DRPI Keshab Bhattara, Research Methods, HUBS 49

50 sgma 77 RSS e + R^ 4779 F(,7) 796 [454] log-lkelhood DW 8 no of observatons 75 no of parameters mean(dstocks) 6746 var(dstocks) 766e+9 Ths proves that apparent relaton seen above s ndeed a spurous relaton In PcGve Choose econometrc modellng/sngle equaton dynamc model/ chose lag length of / regress STOCKS on constant and Stocks_/ choose OLS estmaton/ Test graphcs for actual and ftted Forecast perods/ dynamc forecasts and graph To wrte results/ 5 Forecasts Stocks g Fttng an autoregressve model for stock prce wth seasonal elements Stocks + 97*Stocks_ - 9e e+4*Seasonal (SE) (4) (69e+) (8e+) + 95e+4*Seasonal_ + 69e+4*Seasonal_ + 9*Seasonal_ (8e+) (89e+) (8e+) + 85e+4*Seasonal_4 + 46e+4*Seasonal_5 + 58e+4*Seasonal_6 (8e+) (8e+) (8e+) + 468e+4*Seasonal_ e+4*Seasonal_8 + 95e+4*Seasonal_9 (89e+) (8e+) (89e+) + 5e+4*Seasonal_ (89e+) Keshab Bhattara, Research Methods, HUBS 5

51 Modellng Stocks by OLS (usng stocksandprcels) The estmaton sample s: 97 (5) to 4 (7) Coeffcent StdError t-value t-prob PartR^ Stocks_ Constant Seasonal Seasonal_ Seasonal_ Seasonal_ Seasonal_ Seasonal_ Seasonal_ Seasonal_ Seasonal_ Seasonal_ Seasonal_ sgma 58 RSS e+ R^ 96 F(,6) 449 []** log-lkelhood DW 74 no of observatons 75 no of parameters mean(stocks) 684 var(stocks) 5756e+ -step (e post) forecast analyss 4 (8) to 4 (8) Parameter constancy forecast tests: Forecast Ch^() 864 [769] Chow F(,6) 844 [774] Dynamc (e ante) forecasts for Stocks (SE based on error varance only) Horzon Forecast SE Actual Error t-value e e+4 NaN NaN NaN e+4 NaN NaN NaN e+4 NaN NaN NaN e+4 NaN NaN NaN e+4 NaN NaN NaN e+4 NaN NaN NaN e+4 NaN NaN NaN e+4 NaN NaN NaN e+4 NaN NaN NaN e+4 NaN NaN NaN e+4 NaN NaN NaN e+4 NaN NaN NaN e+5 NaN NaN NaN e+5 NaN NaN NaN e+5 NaN NaN NaN 5-8 8e+5 NaN NaN NaN e+5 NaN NaN NaN 6-7 6e+5 NaN NaN NaN 6- e+5 NaN NaN NaN For E-Ante forecast choose test/forecast perods ( above)/opton/ wrte result nstead of graphng Keshab Bhattara, Research Methods, HUBS 5

52 Dstrbuton of Income and Consumpton and Transfers from Hgh to Low Income Households n the UK Gross Income %%% 4% 6% 4% 8% % 9% 4% 4% 9% 5% 49% % Consumpton 4% 6% 6% 7% 7% 7% 4% 8% % 9% Keshab Bhattara, Research Methods, HUBS 5

53 Beneft Transfer and Taes by Households n the UK H H H H4 H5 H6 H7 H8 H9 H Transfer -4-5 B C D E F G H I J Consumpton Share by H Cumshare Dfference Populaton Trangle Rectangle Area equalty H H H H H H H H H H Total Gn G(A/(A+B)) perfect equalty A B Gn Gn perfect nequalty Ecel formulas for calculatons Share of ncome by decles for H: +B/$B$; repeat ths for all ten households Cumulatve share for H: +D+C Dfference of the cumulatve share of H: +D-D Populaton share of H +F Trangle for H+E*(F-F)*5 Area of rectangle for H+(F-F)*D Area under the Lorenz curve +G+H ; Gn coeffcent G +(5-I)/5 Keshab Bhattara, Research Methods, HUBS 5

54 Lorenz Curve of Income Dstrbuton n the UK Concsumpton and Income Seres Seres Populaton Steps for makng a Lorenz curve: prepare data by households fnd cumulatve share of ncome and populaton by decles select Y plot choosng ncome share and populaton shares 4 Select seres and as 4 change to area type graph as above steps to calculate Gn coeffcent: a dvde the are below the Lorenz curve n ten equal parts b dvde each part nto trangles and rectangles c fnd the areas of rectangles and trangles d sum up for the total area under the Lorenz curve area B E Deduct B from 5 to fnd A then use G A/(A+B) A n perfect equalty and A n perfect nequalty; most other cases le between these etremes Lorenz curve of Consumpton Dstrbuton n the UK Share of consumpton A pop Cons B Share of populaton Keshab Bhattara, Research Methods, HUBS 54

55 Appromaton of Area Under the Lorenz Curve Cumulatve ncome share A Cumulatve Populaton share Income group Income Ishare CMIshare Equalty pop cmpop Rectangl e Trangle Total area st (lowest) nd rd th th (hghest) Total 9749 Total area under Lorenz Gn 44 Gn Coeffcent for UK,, s a lot lower than 44 for the US economy Therefore ths data shows UK to have a lot more equal dstrbuton of ncome than the US Is ths a good thng? Keshab Bhattara, Research Methods, HUBS 55

56 Problem 5 Heteroscedastcty and Autocorrelaton Take a smple lnear regresson model of the followng form Y β + β + e (4) Where the varance of the error term dffers for dfferent observatons of (a) Dscuss how the graphcal method be used to detect the heteroscedastcty? (b) Analyse consequences of heteroscedastcty on the BLUE propertes of the OLS estmators (c) Dscuss how the Goldfeld and Quandt and Glesjer tests can be used to determne estence of the heteroscedastcty problem? (d) Illustrate any two remedal measures of removng the heteroscedastcty when the varanceσ σ and σ s known and when t s unknown (e) From a sample of 677 observatons on pay work-hours and taes contaned n PAYHRTLS determne whether heteroscedastcty ests or not on the bass of cross secton estmates from the the PcGve Feel free to use Shazam f you know and prefer t (BSc) Suggest remedal measures to remove heteroscedastcty n a model lke above Consder a smple lnear regresson model Y t β + β t + et Now assume that errors are correlated to each other over tme wth AR() process as: e ρ + v t et t where vt s dentcally and normally dstrbuted error term wth zero mean and constant varance, (, ) v t ~ N σ (a) Illustrate how the graphcal method can be appled to detect autocorrelaton n a smple regresson model lke above? (b) What are consequences of autocorrelaton n a regresson model? Show how the estence of such autocorrelaton among the error terms affects the BLUE propertes of the OLS estmators (c) Defne and derve the Durbn-Watson test statstcs Show how t can test for estence or non estence of autocorrelaton n a gven estmaton? (d) How the autocorrelaton can be removed f the ρ s known? (e) What s a spurous regresson? Why does t arse and how does t affect the usefulness of estmaton from an OLS regresson? What can be done to correct t? Applcaton: Read data on growth rate of per capta GDP, echange rate and nflaton rates from the wwwmforg for year 98 to for Chna, Inda, South Afrca, UK, USA and Brazl as contaned n PERCAP6GLS Test whether nflaton and the echange rate are the sgnfcant varables n eplanng the growth rate of per capta output (n PPP) n these economes Determne whether heteroscedastcty and autocorrelaton est n ths regresson usng PcGve Feel free to use Shazam f you know and prefer t suggest remedy for autocorrelaton n a model lke ths Keshab Bhattara, Research Methods, HUBS 56

57 Tentatve Answers - Tutoral 5 Some eperments wth the Heteroscedastcty Problem n Work Hours and Pay Dataset Take a smple lnear regresson model of the followng form Y β + β + e Where the varance of the error term dffers for dfferent observatons of (f) When s the parameter β unbased? When s t effcent? When s t consstent? Lnear, Unbasedness and Mnmum Varance Propertes of an Estmator (BLUE Property) Lnearty: f ( βˆ ) Y ˆβ w Y f ( βˆ ) Unbasedness ˆ E β β Mnmum Varance var ˆ β ˆ σ ( ˆ β ) β ˆβ E ˆβ Bas Bas An estmator s consstent when the bas n estmaton dsappears as the sample sze goes to nfnty as Bas ˆ followng: ( β ) N (g) Dscuss how the graphcal method be used to detect the heteroscedastcty? Homoscedastcty Hetroscedastcty -Y eˆ eˆ Y Y Keshab Bhattara, Research Methods, HUBS 57

58 Hetroscedastcty - Hetroscedastcty -Y4 eˆ eˆ Y σˆ Hetroscedastcty -5 σˆ Hetroscedastcty -Y5 Y Y Meanng of Homoscedastcty var[ e5 ] σ var e σ Y + e β + β [ ] 4 e var e var e [ ] σ e e 4 e var[ e ] σ [ ] σ Same Varance of error across all e 5 Y Meanng of Heteroscedastcty var [ e5 ] var e σ Y + e β + β [ ] 4 e 4 e var[ e ] σ e var[ e ] σ e var[ ] σ Varance of Error Terms Vares by value of e e 5 Causes: Learnng: reduces errors; drvng practce, drvng errors and accdents typng practce and typng errors, defects n productons; mproved machnes Growth: savng and varance of savng ncreases wth ncome Improved data collecton: better formulas and good software Outlers affect the value of estmates Specfcaton Errors and omtted varables:- n a demand model f you regress demand of a product to only ts own prce, there s a danger varables such as the prces of complements and ncome may appear n the error term More heteroscedastcty ests n cross secton than n tme seres data Keshab Bhattara, Research Methods, HUBS 58

59 (h) Analyse consequences of heteroscedastcty on the BLUE propertes of the OLS estmators Consequences of Hetroscedastcty OLS Estmate s Unbased But t s no longer effcent E E ( ˆ β ) β ( ˆ β ) E[ w y ] E w ( β + β + e ) E [ w β + β w + we ] β var σ ˆ β var σ e not OLS assumpton: varance of e s constant var σ e for every th observaton, var ˆ β ˆ σ but n case of heteroscedastcty varance of error s not constant: σ σ () Dscuss how the Goldfeld and Quandt and Glesjer tests can be used to determne estence of the heteroscedastcty problem? Goldfeld-Quandt test Model Y β + β + e () Steps: Rank observatons n ascendng order of one of the varable Omt c numbers of central observatons leavng two groups wth observatons n c number of Keshab Bhattara, Research Methods, HUBS 59

60 n c Ft OLS to the frst and the last squared errors from both of them 4 Set hypothess n c H : σ σ aganst : σ σ A H ERSS df 5 compute λ t follows F dstrbuton ERSS df Glejser test Y β + β + e There are several tests e β + + v β e β + + v β e β + β + v e β + β + v e β + + v β observatons and fnd sum of the e β + β + v In each case do t-test H : β aganst H A : β If β s sgnfcant then that s the evdence of heteroscedastcty (j) Illustrate any two remedal measures of removng the heteroscedastcty when the varanceσ σ and σ s known and when t s unknown Remedal measures Weghted Least Square and GLS whenσ known, dvde the whole equaton by Apply OLS to transformed varables Y β + β + β + β + β β 4 k σ σ σ σ σ σ Varance of ths transformed model equals Other eamples: Y β + β + e and assume e σ Y β e + β + e σ ; E σ k σ e + σ σ Keshab Bhattara, Research Methods, HUBS 6

61 (k) From a sample of 677 observatons on pay work-hours and taes contaned n PAYHRTLS determne whether heteroscedastcty ests or not on the bass of cross secton estmates from the the PcGve Feel free to use Shazam f you know and prefer t Modellng PAY by OLS-CS (usng PAYHRT_sortls) The estmaton sample s: to 678 Dropped 8 observaton(s) wth mssng values from the sample Coeffcent StdError t-value t-prob PartR^ Constant WHRS TA sgma 649 RSS 6495e+9 R^ 779 F(,6769) 878 []** log-lkelhood -55 DW 96 no of observatons 677 no of parameters mean(pay) 89 var(pay) 7e+6 Normalty test: Ch^() 77e+5 []** hetero test: F(4,6764) 6969 []** hetero- test: F(5,676) 5645 []** RESET test: F(,6768) 64 []** These test reject homoscedastty PAY *WHRS + 565*TA (SE) (6) (79) () Heteroscedastcty consstent standard errors Coeffcents SE HACSE HCSE JHCSE Constant WHRS TA Coeffcents t-se t-hacse t-hcse t-jhcse Constant WHRS TA HACSE heteroscedastcty and autocorrelaton consstent standard error Ths corrected standard errors then can be appled to determne whether a coeffcent s sgnfcant Ths s also called Whte test (l) Suggest remedal measures to remove heteroscedastcty n a model lke above Transform the model to purge the heteroscedastc errors as dscussed n the remedal measure above when σ unknown estmate σ usng the sample nformaton and do the above procedures (Gujarat s a good tet for Heteroscedastcty) Keshab Bhattara, Research Methods, HUBS 6

62 Shazam progam and results for heteroscedastcty ols pay whours ta/ resdres dagnos/chowone 677 OBSERVATIONS DEPENDENT VARIABLE PAY NOTESAMPLE RANGE SET TO:, 677 R-SQUARE 77 R-SQUARE ADJUSTED 77 VARIANCE OF THE ESTIMATE-SIGMA** 847E+6 STANDARD ERROR OF THE ESTIMATE-SIGMA 65 ANALYSIS OF VARIANCE - FROM MEAN SS DF MS F REGRESSION 695E+ 46E ERROR 64E E+6 P-VALUE TOTAL 88966E E+7 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 6769 DF P-VALUE CORR COEFFICIENT AT MEANS WHOURS E- 46E- TA E CONSTANT SEQUENTIAL CHOW AND GOLDFELD-QUANDT TESTS N N SSE SSE CHOW PVALUE G-Q DF DF PVALUE 77 97E E CHOW TEST - F DISTRIBUTION WITH DF AND DF6766 AS the Goldfeld and Quandt test shows varance of the two groups of the sample are not the same Transform the model by dvdng every varable by the square of the resdual genr res res*res ols res whours ta genr pay pay/res genr whourwhours/res genr tata/res *there s no heteroskedastcty n the transformed model ols pay whour ta/ 677 OBSERVATIONS DEPENDENT VARIABLE PAY NOTESAMPLE RANGE SET TO:, 677 R-SQUARE 45 R-SQUARE ADJUSTED 4 VARIANCE OF THE ESTIMATE-SIGMA** 677E+8 STANDARD ERROR OF THE ESTIMATE-SIGMA 676 SUM OF SQUARED ERRORS-SSE 775E+ MEAN OF DEPENDENT VARIABLE 546 LOG OF THE LIKELIHOOD FUNCTION MODEL SELECTION TESTS - SEE JUDGE ET AL (985,P4) AKAIKE (969) FINAL PREDICTION ERROR - FPE 68E+8 (FPE IS ALSO KNOWN AS AMEMIYA PREDICTION CRITERION - PC) AKAIKE (97) INFORMATION CRITERION - LOG AIC 684 SCHWARZ (978) CRITERION - LOG SC 687 MODEL SELECTION TESTS - SEE RAMANATHAN (998,P65) CRAVEN-WAHBA (979) GENERALIZED CROSS VALIDATION - GCV 68E+8 HANNAN AND QUINN (979) CRITERION 69E+8 RICE (984) CRITERION 68E+8 SHIBATA (98) CRITERION 68E+8 SCHWARZ (978) CRITERION - SC 74E+8 AKAIKE (974) INFORMATION CRITERION - AIC 68E+8 ANALYSIS OF VARIANCE - FROM MEAN SS DF MS F REGRESSION 879E+ 644E ERROR 775E E+8 P-VALUE TOTAL 7556E E+8 ANALYSIS OF VARIANCE - FROM ZERO SS DF MS F Keshab Bhattara, Research Methods, HUBS 6

63 REGRESSION 8E+ 7E+ 74 ERROR 775E E+8 P-VALUE TOTAL 7558E E+8 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 6769 DF P-VALUE CORR COEFFICIENT AT MEANS WHOUR TA CONSTANT DURBIN-WATSON VON NEUMANN RATIO 6 RHO -5 RESIDUAL SUM 787E-9 RESIDUAL VARIANCE 677E+8 SUM OF ABSOLUTE ERRORS 54446E+6 Dagnostc suggests that there s no heteroscedastcty _dagnos/acf/het REQUIRED MEMORY IS PAR 6 CURRENT PAR DEPENDENT VARIABLE PAY 677 OBSERVATIONS REGRESSION COEFFICIENTS HETEROSKEDASTICITY TESTS CHI-SQUARE DF P-VALUE TEST STATISTIC E** ON YHAT: 9857 E** ON YHAT**: 9878 E** ON LOG(YHAT**): E** ON LAG(E**) ARCH TEST: 99 LOG(E**) ON (HARVEY) TEST: ABS(E) ON (GLEJSER) TEST: 9985 E** ON TEST: KOENKER(R): B-P-G (SSR) : E** ON ** (WHITE) TEST: KOENKER(R): 4 B-P-G (SSR) : E** ON ** (WHITE) TEST: KOENKER(R): 5 B-P-G (SSR) : Also note that there s no autocorrelaton based on Durbn Watson or LM t-tests RESIDUAL CORRELOGRAM LM-TEST FOR HJ:RHO(J), STATISTIC IS STANDARD NORMAL LAG RHO STD ERR T-STAT LM-STAT DW-TEST BO-PIERCE-LJUNG LM CHI-SQUARE STATISTIC WITH DF IS Resduals do not depend on work hours and ta genr res res*res ols res whours ta het whour pay ta/rstat Keshab Bhattara, Research Methods, HUBS 6

64 Consder a smple lnear regresson model Y t β + β t + et Now assume that errors are correlated to each other over tme wth AR() process as: e + v t ρ et t where vt s dentcally and normally dstrbuted error term wth zero mean and constant varance, (, ) v t ~ N σ (f) Illustrate how the graphcal method can be appled to detect autocorrelaton n a smple regresson model lke above? Negatve autocorrelaton Postve autocorrelaton e e ρe + v ρ < Tme e e ρe + v ρ > Tme e Cyclcal relaton between errors e ρ e + ρ e + v e OLS estmator s lnear and unbased but not effcent n presence Of autocorrelaton Lnearty: f ( βˆ ) Y ˆβ w Y f ( βˆ ) ˆ Unbasedness E β β ( ˆ β ) β Mnmum Varance var ˆ β ˆ σ ˆ β E Bas βˆ Bas (g) What are consequences of autocorrelaton n a regresson model? Show how the estence of such autocorrelaton among the error terms affects the BLUE propertes of the OLS estmators (h) Defne and derve the Durbn-Watson test statstcs Show how t can test for estence or non estence of autocorrelaton n a gven estmaton? Keshab Bhattara, Research Methods, HUBS 64

65 Durbn-Watson (DW) test T ˆ ˆ ˆ ˆ ˆ ˆ + e e ˆ t e t e e t t t e t d ( ρˆ ) eˆ eˆ t t Values of DW depend on number of parameters n the Model, K, and observatons Inconclusve regon No auto-correlaton Inconclusve regon d ; ρ dl du d ; ρ 4 du 4 dl d 4; ρ Postve autocorrelaton Negatve autocorrelaton () How the autocorrelaton can be removed f the ρ s known? When ρ s known transform the model Take a lag of the orgnal model; and multply t by ρ ; and subtract from the orgnal model to fnd a transformed model Gven model: Y t β + t + e β t Where e t et + vt v t ~ N σ Multply by ρ and lag by one perod ρ and ( ) ρ e ρy ρβ ρβ t t t Subtract ths from the orgnal model Y t ρ Y β ρβ + β ρβ + e e t t t t ρ t Transformed model Y * β * + β* * + v t t t Where Y* t Y t ρ Y ; * t t t ρ ; t Keshab Bhattara, Research Methods, HUBS 65

66 e* t e t ρ e ; t β* β ρβ OLS estmates unknown parameters of ths transformed model wll be Best Lnear * and Unbased (BLUE) Retreve β of the orgnal model from estmates of β (j) What s a spurous regresson? Why does t arse and how does t affect the usefulness of estmaton from an OLS regresson? What can be done to correct t? Applcaton of OLS among non-statonary varable may generate a spurous regresson, wth a hgh R and very low Durbn-Watson statstcs (R >d) A gven tme seres{ y t } s statonary when mean and varance are constant or ndependent of tme E μ constant mean () y t var σ y constant varance () t cov y y t t s cov y t y t + s γ tme ndependent covarance () s Tme seres y s non-statonary f the mean and varance of t s not constant It s t non-statonary f the varance s changng over tme even though mean s constant Many economc varables such as GDP, GDP components (C, I, G and ), nflaton, echange rates, labour force evolve over tme It s mportant to check whether these seres have a constant mean and constant varance before they can be used n regresson analyss A meanngful cause and effect relatonshp requres that the concerned seres are statonary Applcaton of OLS procedure n non statonary seres produces a spurous relatonshp A spurous relatonshp mples sgnfcant teststatstcs (t, f, ch-square, R-square) even though there s no relatonshp among the varables Econometrc estmaton usng non-statonary varables may generate meanngless result though t may apparently seem statstcally sgnfcant Applcaton: Read data on growth rate of per capta GDP, echange rate and nflaton rates from the wwwmforg for year 98 to for Chna, Inda, South Afrca, UK, USA and Brazl as contaned n PERCAP6GLS Test whether nflaton and the echange rate are the sgnfcant varables n eplanng the growth rate of per capta output (n PPP) n these economes Determne whether heteroscedastcty and autocorrelaton est n ths regresson usng PcGve Feel free to use Shazam f you know and prefer t Suggest a remedy for autocorrelaton n a model lke ths Keshab Bhattara, Research Methods, HUBS 66

67 A PcGve Batch Fle for Estmaton module("pcgve"); package("pcgve"); usedata("percap6n7"); system { Y DLPCIBrazl; Z Constant, INFBrazl, ERBrazl; } estmate("ols", 98,, 4, ); module("pcgve"); package("pcgve"); usedata("percap6n7"); system { Y DLPCIChna; Z Constant, INFChna, ERChna; } estmate("ols", 98,, 4, ); module("pcgve"); package("pcgve"); usedata("percap6n7"); system { Y DLPCIInda; Z Constant, INFInda, ERInda; } estmate("ols", 98,, 4, ); module("pcgve"); package("pcgve"); usedata("percap6n7"); system { Y DLPCISAfrca; Z Constant, INFSAfrca, ERSAfrca; } estmate("ols", 98,, 4, ); module("pcgve"); package("pcgve"); usedata("percap6n7"); system { Y DLPCIUK; Z Constant, INFUK, ERUK; Keshab Bhattara, Research Methods, HUBS 67

68 } estmate("ols", 98,, 4, ); module("pcgve"); package("pcgve"); usedata("percap6n7"); system { Y DLPCIUS; Z Constant, INFUS, ERUS; } estmate("ols", 98,, 4, ); 5 5 DLPCIBrazl DLPCIChna DLPCIInda DLPCISAfrca DLPCIUK DLPCIUS The above fgures are the log-dfferences, whch gve the growth rates of per capta ncome n these s countres Study and eplan followng results EQ( ) Modellng DLPCIBrazl by OLS (usng Percap6n7) The estmaton sample s: 98 to 4 Coeffcent StdError t-value t-prob PartR^ Constant INFBrazl 46e-5 58e ERBrazl sgma RSS R^ 548 F(,) 67 [557] log-lkelhood 89 DW 7 no of observatons 4 no of parameters mean(dlpcibrazl) 968 var(dlpcibrazl) 498 AR - test: F(,9) 669 [59] ARCH - test: F(,9) 47 [989] Normalty test: Ch^() [44] hetero test: F(4,6) 89 [949] hetero- test: F(5,5) 4676 [978] RESET test: F(,) 68 [455] DLPCIBrazl e-5*INFBrazl - 554*ERBrazl (SE) (65) (5e-5) (95) Keshab Bhattara, Research Methods, HUBS 68

69 EQ( ) Modellng DLPCIChna by OLS (usng Percap6n7) The estmaton sample s: 98 to 4 Coeffcent StdError t-value t-prob PartR^ Constant INFChna ERChna sgma 96 RSS 8989 R^ 984 F(,) 596 [98] log-lkelhood 497 DW 4 no of observatons 4 no of parameters mean(dlpcichna) var(dlpcichna) 956 AR - test: F(,9) 5575 [4]* ARCH - test: F(,9) 895 [668] Normalty test: Ch^() 6565 [48]* hetero test: F(4,6) 56 [5]** hetero- test: F(5,5) [9]** RESET test: F(,) 799 [4] DLPCIChna *INFChna + 95*ERChna (SE) (78) (74) (489) EQ( ) Modellng DLPCIInda by OLS (usng Percap6n7) The estmaton sample s: 98 to 4 Coeffcent StdError t-value t-prob PartR^ Constant INFInda ERInda sgma 667 RSS R^ F(,) 7 [8] log-lkelhood 559 DW 86 no of observatons 4 no of parameters mean(dlpciinda) 59 var(dlpciinda) 4557 AR - test: F(,9) 87 [89] ARCH - test: F(,9) 95 [784] Normalty test: Ch^() 887 [95] hetero test: F(4,6) 48 [6]* hetero- test: F(5,5) 764 [58] RESET test: F(,) 7779 [78] DLPCIInda *INFInda + 69*ERInda (SE) (695) (458) (67) EQ( 4) Modellng DLPCISAfrca by OLS (usng Percap6n7) The estmaton sample s: 98 to 4 Coeffcent StdError t-value t-prob PartR^ Constant INFSAfrca ERSAfrca sgma 6579 RSS 979 R^ F(,) 8 [7] log-lkelhood 58 DW no of observatons 4 no of parameters mean(dlpcisafrca) 9797 var(dlpcisafrca) 8564 AR - test: F(,9) 885 [88] ARCH - test: F(,9) 6675 [957] Normalty test: Ch^() 57 [57] hetero test: F(4,6) 94 [58] hetero- test: F(5,5) 976 [57] RESET test: F(,) 4 [74] DLPCISAfrca *INFSAfrca + 95*ERSAfrca (SE) (47) () (785) Keshab Bhattara, Research Methods, HUBS 69

70 EQ( 5) Modellng DLPCIUK by OLS (usng Percap6n7) The estmaton sample s: 98 to 4 Coeffcent StdError t-value t-prob PartR^ Constant INFUK ERUK sgma 977 RSS R^ F(,) 5497 [585] log-lkelhood 567 DW 9 no of observatons 4 no of parameters mean(dlpciuk) var(dlpciuk) AR - test: F(,9) 88 [88] ARCH - test: F(,9) 84 [776] Normalty test: Ch^() 9 [887] hetero test: F(4,6) 79 [585] hetero- test: F(5,5) 648 [6795] RESET test: F(,) 778 [6] DLPCIUK *INFUK + 9*ERUK (SE) (4) () (454) EQ( 6) Modellng DLPCIUS by OLS (usng Percap6n7) The estmaton sample s: 98 to 4 Coeffcent StdError t-value t-prob PartR^ Constant INFUS ERUS sgma 6444 RSS R^ 4657 F(,) 9 []** log-lkelhood 6675 DW no of observatons 4 no of parameters mean(dlpcius) var(dlpcius) 484 AR - test: F(,9) 65 [59] ARCH - test: F(,9) 476 [865] Normalty test: Ch^() []* hetero test: F(4,6) 784 [55] hetero- test: F(4,6) 784 [55] RESET test: F(,) 8466 [774] DLPCIUS *INFUS + 884*ERUS (SE) (54) (85) (5) Problem 6 Demand and Supply analyss Usng Input Output Table for a Hypothetcal Economy 5 4 Fnd the determnant and nverse of matr A 7 8 An economy, wth two sectors, and, has followng nput-output table F Total 7 5 Labour nput Captal nput Total Keshab Bhattara, Research Methods, HUBS 7

71 Where s the gross producton of sector and s the gross producton of sector, F s the fnal demand that ncludes consumpton, nvestment, government spendng and net eports a Wrte equatons to represent demands by sector for two sectors of the economy b Check how demand and supply, ncome and ependture accounts are balanced for ths economy c Fnd techncal coeffcents, a, j, j j and share of prmary nputs for both sectors Here, represents ntermedate demand for sector good by sector j; j,,, and, represent ntermedate nputs d a, a, Put the technologcal coeffcents n a Leontef matr A a, a, e Propose an nput-output model for ths economy (usng the matr format) f Wth an dentty matr I and ( I A)? a Epress gross output n terms of fnal demand and nverse of ( I A) matr b Fnd out the mpact of a percent ncrease n the fnal demand of sector n outputs of sector and sector General Equlbrum n Markets Household A owns the apple farm and produces quntals of apples and household B owns the orange farm and produces quntals of oranges Both households lke to consume apples and oranges Ther consumpton preferences are gven by Cobb-Douglas Utlty functons, An economy produces two goods; apples and oranges and has two households, h { A, B} h α h ( α h ) h U, h, h where U s the utlty to household h, apples and oranges and h, h, and h are consumptons of α < α represents the weght of each good n the utlty functons, Household A spends 4 percent of ncome n apple and 6 percent n oranges and household B spends 6 percent n apples and 4 percent n oranges Market structure s compettve a Represent the supply of goods usng an approprate Edgeworth bo dagram b Fnd the relatve prce n ths economy that s consstent wth mamzaton of utlty (satsfacton) by both households Choose prce of commodty as a numerare c Determne the ncome for each household d What are demands for apples and oranges by households A and B? e Check whether the market clearng condtons for equlbrum are fulflled Fnd ther levels of utlty at equlbrum f Represent both consumpton and supply and relatve prce of goods n equlbrum usng an approprate Edgeworth bo dagram How would you estmate α n a real world stuaton? h < h Keshab Bhattara, Research Methods, HUBS 7

72 Research Methods 6 Answer 6 Economc theory and economc models: Illustraton of demand and supply curves D a -a P ; S b +b P Smple market: D P S + P P a a b b D P a b Q a a (56) 88 a b What s the deadweght loss of percent advalorem ta? S b +b (-t)p D P S + ( t)p Pt a a b b ( t) + ( ) ) Ps Pt( t) a b Qt a a (664) 7 7 a b ( t) Revenue of the suppler before ta: PQ Revenue of the suppler after ta: Pt Qt Revenue for the government: R t Pt Qt Pt P ΔQ Ps P ΔQ Deadweght loss of taes: Consumers loss + Producer s loss Consumers loss: ( ) 5 ( 66 56) ( 7 88) 6 5 Producers loss: ( ) ( ) ( ) 9 Varables n ths model: P, Q Parameters: a, a, b, b,t How to estmate them? Lecture 8-9 Output, Prce, Welfare and Proft n an mperfectly compettve market Keshab Bhattara, Research Methods, HUBS 7

73 Market demand functon: P q q Cost: C 6q Proft: Π Pq C Duopoly: Cournot Model Π Pq C q q q 6q q q q q 6q Pq C q q q 6q q q q q 6 Frm : ( ) Π q Frm : ( ) Π q Π q q q 6 q q 6 4 q + q 4 q + q Solvng these two equaton q q 8 P q q Π Pq C Π Pq C Consumer surplus from frm product: CS Consumer surplus from frm product: CS Total welfare: Π + Π + CS + CS Stackleberg equlbrum When the Frm s the leader and frm s the follower Π Pq C q q 6q Π q 6 4 q q q Π Π q Pq C q q qq 6 q q 6 q 4 q q q 4 q 6 P q q 6 Π Pq C 6 7 Π Pq C Consumer surplus from frm product: CS 8 8 Consumer surplus from frm product: CS Total welfare: Π + Π + CS + CS Colluson (cartel): mamse ndustry proft Market demand functon: P q q Q Cost: C 6Q Keshab Bhattara, Research Methods, HUBS 7

74 Proft: Π PQ C ( Q) Q 6Q Q Q 6Q Π Q 6 Q 4 Q 6 Q q q 6 P q q Q 8 Consumer surplus from frm product: CS 6 6 Consumer surplus from frm product: CS 6 6 Π Pq C Π Pq C Total welfare: Π + Π + CS + CS Comparson of solutons n an mperfectly compettve market Cournot Stackleberg Colluson P 4 8 q 8 6 q Π Π CS CS TW Keshab Bhattara, Research Methods, HUBS 74

75 Problem 7 Models of demand and supply and strategc decsons Market demand and supply for a normal good are D a bp and S c + dp respectvely, where D and S are quanttes demanded and suppled and a, b, c and d are parameters representng the behavour of buyers and supplers n the market a What are theoretcal assumptons about the sgn of parameters a, b, c and d n ths model? How would you obtan ther numercal values? How demand and supply elastctes can be calculated wth knowledge of b and d respectvely? b The demand for and supply of surfng boards n a market were gven by D P and S 5 + 5P respectvely Determne the prce and quantty n equlbrum c Show demand and supply schedules and equlbrum n a properly labelled and scaled dagram d The government ntroduces a percent sales ta rate on sale of surfng boards Show equlbrum prces and quanttes before and after ths sales ta n another dagram e Calculate the deadweght loss of taes to consumers, to producers and to the entre economy usng partal equlbrum analyss contaned n ths model There are two cnema halls n Hull Objectve of each s to mamse proft The market demand curve for moves s gven byq 5 P, where Q s demand and P s the prce The prce per tcket depends on total sales P ( q + q ), where q and q are quanttes sold by each hall The cost functon for hall s C q for, a) How many tckets does each hall sell to mamse ts proft takng sales of another hall as fed and how much should has one to pay to go to a move? ( hnt: Cournot model) b) Calculate the consumers surplus at that prce c) How bg s the producers surplus? d) What s the sze of welfare to the entre economy at those prces and quanttes? e) Answer questons (a) to (d) when Hall acts as a Stackleberg leader n the market f) Answer questons (a) to (d) when both of these halls collde to mamse jont profts g) Put results on sales, prce, proft and welfare for all three markets n one table h) Make a pay-off matr of profts n all three strategc markets for Cnema and Cnema ) What prce would have prevaled f ths market was perfectly compettve? What would have been the value of welfare then? Two frms are n the telecom market Ther pay-offs by advertsng or not advertsng are as lsted n the followng matr Frm Adv DnotAdv ADV (,5) ( 5,) Frm DnotAdv ( ) ( ) 6,8, Solve ths game usng a domnant strategy What are the pay-offs for frm and frm? 4 (BSc) Consder a zero sum game for a compettve market where one can beneft only at the epense of another S Player S (, ) (,) (,) (, ) S Player S The probablty of player playng S strategy s gven by π Represent epected pay-off of ths game n a dagram Fnd the optmal m of strateges for ths player, e fnd the optmal value of π under the med strategy Keshab Bhattara, Research Methods, HUBS 75

76 Input-output Model Learnng Matr Revenues of two stores are and respectvely The sell two dfferent products and Frm sells 5 unts of and 4 unts of Frm sells 7 unts of and 8 unts of Formulate ths nformaton fst n equaton and then put t n the matr form Then fnd the prces of and R P, + P, R P, + P,, j s the sell by frm of product j R,, P R,, P 5P + 4P 7P + 8P From the gven nformaton P P ; P P P ( ) ( ) P P P These prces satsfy above revenue equatons 5 + 5P + 4P P + 8P ( ) 4( ) ( ) ( ) + 5 Demand and Supply: Input Output Analyss An economy, wth two sectors, and, has followng nput-output table F Total 7 5 Labour nput Captal nput Total Propose an nput-output model for ths economy Gross Supply ntermedate demand plus fnal demand Income (labour ncome plus captal ncome) ependture (F +F) GDP Keshab Bhattara, Research Methods, HUBS 76

77 +, +, F, +, F +, s ntermedate nput from sector to sector j j a, j, a,, a a, share of nput from row sector to sector j,, a,, + a, + a, F a, + a, F + How does the fnal demand affect the level of gross output? ( a, ) a, F a, + ( a, ) F Better to use matr algebra to solve the problem ( a ) a,, ( I A) a, ( a ) F, ( a ) a,, ( I A) F Adj( A) A a F, ( a ) F, ' [ C ], j A ( a, ) a, ( ) a, a, F ( a )( a ) a a F,,, F ( a, ) a, a, ( a, ) ( a, )( a, ) a,a,, a,, a,,,, a, a, Keshab Bhattara, Research Methods, HUBS 77

78 ( )( ) Model s calbrated Fnal demand ncludes demand for consumpton, nvestment, government spendng and net eports If any of these change t has economy wde mpacts, that such mpact can be evaluated usng such a model For nstance f the fnal demand n sector rses by 5 percent what wll be the change n total output 9 5 Δ Δ Thus a 5 percent ncrease n the fnal demand of sector rases output n sector by 7 percent and n sector by 58 percent Prce s constant n ths nput-output model but prces are fleble n the economy Lecture 9 General equlbrum n a pure echange economy wth fleble prces An economy produces apples and oranges Household A owns the apple farm and produces quntal of apples and household B owns the orange farm and produces quntals of oranges Both households lke to consume apple and oranges Ther consumpton preferences by gven by Cobb-Douglas Utlty functons, household A spends 4 percent of ncome n apple and 6 percent n oranges and household B spends 6 percent n apples and 4 percent n oranges Market structure s compettve Fnd the relatve prce n ths economy that s consstent wth mamzaton of utlty (satsfacton) by both households Choose prce of commodty as a numerare Fnd the ncome of each households, ther demands for both apples and oranges Check whether the condtons for equlbrum are fulflled Fnd ther levels of utlty at equlbrum Demand functons: A A A αi A ( α ) I P P B B B βi B ( β ) I P P A B A B Supply of good : ω ω + ω for smplcty ω ω and ω A B A B Supply of good : ω ω + ω for smplcty ω and ω ω A B Further assume that ω ω α 4 β 6 Market clearng condton A B + ω A B + ω I A P ω I B P ω Usng above demand functons Keshab Bhattara, Research Methods, HUBS 78

79 αi P A βi + P A ω A ( α ) I ( β ) P + P I B ω Substtutng above αpω βpω + P P P P βp ω α + ω ω ω P P P ( 4) + ( 6) 6 5 P 4 + P You get the same relatve prce even f you use the second market clearng condton Normalse prce of good ; P I A P ω I B P ω 5 A A A α I 4 A ( α ) I 6 4 P P 5 B B B β I 6 B ( β ) I 4 6 P P 5 A B ω A B ω The market s cleared Change of the numerare does affect the optmal allocaton Try normalsng P GAMS s good partcularly n solvng a general equlbrum model wth many lnear or non-lnear equatons on contnuous or dscrete varables It comes wth a number of solvers that are useful for numercal analyss For economc modellng t can solve very large scale models usng detaled structure of consumpton, producton and trade arrangements on unlateral, blateral or multlateral bass n the global economy where the optmal choces of consumers and producers are constraned by resources and producton technology or arrangements for trade It s a user frendly software Any GAMS programme nvolves declaraton of set, parameters, varables, equatons, ntalsaton of varables and settng ther lower or upper bounds and solvng the model usng Newton or other methods for lnear or non-lnear optmsaton and reportng the results n tables or graphs (eg ISLMgms ) GAMS/MPSGE program s good for large scale standard general equlbrum models GAMS programme s located at N:\specal\ec\gams\gams n the unversty network and can be used by gong through followng steps Frst, create a drectory called models n G:drve G:\> md Models then G:\> cd Models Then wrte or copy a GAMS program fle n that drectory such as G:\models\slmgms Type N:\specal\ec\gams\gams slmgms to run a GAMS program of a model n the network The results of the model computatons can be seen n the lst fle called ISLMLST 8 Keshab Bhattara, Research Methods, HUBS 79

80 The check whether the results are consstent wth the economc theory underlyng the model such as ISLM-ASAD analyss for evaluatng the mpacts of epansonary fscal and monetary polces Use knowledge of growth theory to eplan results of the Solow growth model from Solowgms Consult GAMS and GAMS/MPSGE User Manuals, GAMS Development Corporaton, 7 Potomac Street, Washngton DC or wwwgamscom Addtonal Problem An Introducton to the Matr Algebra Fnd the determnant of the followng matr 5 4 a) A b) A c) B Fnd the nverse of the followng matr A 4 7 Prove that followng matr s a postve defnte matr A Solve followng equatons system usng Cramer s rule () () () Fnd the determnant of the followng matr a) A b) B (a) Keshab Bhattara, Research Methods, HUBS 8

81 Keshab Bhattara, Research Methods, HUBS A (59)+(67)+(84)-(75)-(86)-(94) (b) B (-75)+(4)+(69)-()-(64(-7))-(59) If the determnant of a matr s zero then that matr s called a sngular matr Sngularty reflects lnear dependence among eplanatory varables Fnd the nverse of the followng matr A 7 4 Determnant of ths matr A 99 > the nverse A - ests The cofactor matr ( ) j j j M C,, + where j M, are mnors of each element C ( ) ' A Adj C The desred nverse matr s ( ) A Adj A A Solve followng equatons system usng Cramer s rule + + () + + () 5 + ()

82 Keshab Bhattara, Research Methods, HUBS 8 Answer: Frst trte ths system of three equatons and three unknowns n matr format as followng 5 ; A b A ++(-)--(-) The solutons for,, and s defned as followng: A A A Tetbooks: Research requres thnkng and solvng problems more than readng books Tet books, nevertheless, can epose one to popular tools used by professonal researchers Koop () brllantly shows how to use Ecel for data analyss Dougherty (), Hll-Grffth and Judge () and Studenmund () are good ntroductory tets n econometrcs, any one of these s enough ths module Ecel, GveWn PcGve, STAMP and Shazam are very useful software for handlng data and economc analyss They are avalable through the Start and Applcatons/Economcs menu n the unversty network The GAMS s useful for solvng general equlbrum or lnear and non-lnear programmng models Tets n mcro and macro economcs are helpful n generatng deas about economc ssues Koop G () Analyss of Economc Data, Wley, UK Dougherty C () Introducton of Econometrcs by, Second Edton, Oford Unversty Press Softwares: Doornk J A and DF Hendry (() PC-Gve Volume I-III, GveWn Tmberlake Consultants Lmted, London Shazam (997) User s Reference Manual, Verson 8 GAMS Users Manual, GAMS Development Corporaton, Washngton DC wwwgamscom Problem

83 (q) a Keshab Bhattara, Research Methods, HUBS 8

84 Cross Secton Analyss The marks scored by students n two eams and ther monthly earnngs from a part tme jobs are as gven n the followng table Scores n Eams and Earnngs Observato Eam Eam Earnng Eam Eam Earnng n observaton Keshab Bhattara, Research Methods, HUBS 84

85 Represent scores n eams and earnng data usng frequency table wth ten ntervals What are means and varances n those eams? 4 What s the coeffcent of varatons for scores n eams and? 5 What s the covarance of marks n eams and? 6 What s correlaton coeffcent between scores n eam and? 7 If eam only counts for percent but the scores n eam wegh percent what would be the weghted mean score n these two eams? 8 Eam took place before eam Test whether scores n eam can predct scores n eam? 9 Fnd predcted scores n eam for students who scored 6 and 8 n eam Test hypothess whether scores n eam and eam are sgnfcant determnants of earnng Why may earnngs be negatvely related wth test scores? Keshab Bhattara, Research Methods, HUBS 85

86 Keshab Bhattara, Research Methods, HUBS 86