Lecture'Notes'#3'Review'of'Sta4s4cs'ECON'452'–'Fall'2016'The'La4n'American'Economies'Instructor:'Fabrício'd’Almeida'• Economists'rely'on'collected'data'to'run'its'experiments:'– Hard'to'create'a'country'in'a'lab!'• Sta4s4cs'and'Econometrics'provide'valuable'tools'to'mimic'a'lab'environment,'allowing'scien4sts'to'put'their'theories'under'the'test'of'evidence'Empirics'• Review'of'Expecta4on,'Variance'and'Covariance'• The'role'of'Covariance'in'diversifica4on:'– The'case'of'mono-export'countries'vs'diversifica4on'• Test'of'hypothesis:'a'brief'review'• Econometrics:'OLS'– How'to'perform'– How'to'interpret'Road'Plan'• Random'variables:'a'variable'whose'value'is'uncertain'• If'X"is'a'random'variable:''– for'each'possible'outcome'of'X,!set!as'xi",'there'is'an'associated'probability'pi""for'xi""becoming'the'value'of'X"– where'i"is'the'ordering'of'each'possible'outcome'– and'0'≤'pi"≤'1'– and'Σ'pi"='1'Sta4s4cal'Review'Sta4s4cal'Review'• For'a'random'variable'X,'its'expected'value'and'variance'are'defined'and'predicted'as:''– Expecta4on:'measured'by'the'mean'– Variance:''measured'by'the'square'of'the'devia4on'from'the'mean'E[˜X] ⌘ µX=nPi=1pixiV ar[˜X] ⌘ 2X=nPi=1pi(xi µX)21Equations for ECON 452Fabr´ıcio d’Almeida⇤September 9, 20161 Statistical ReviewA ZExpectationE[˜X] ⌘ µX=nXi=1pixiE[a˜X+b˜Y ]=aE[˜X]+bE[˜Y ]Var[a˜X+b˜Y ]=a2Var[˜X]+2abCov[˜X,˜Y ]+b2Var[˜Y ]Variance Var[˜X] ⌘ 2X=nPi=1pi(xi µX)2s.d.⌘ X=snPi=1pi(xi µX)2Covariance Cov[˜X,˜Y ]=nPi=1pi(xi µX)(yi µY)E[0.5Beef +0.5Wool] = 0.5 E[Beef ] + 0.5 E[Wool]E[Beef ] = 0.2 * 0.20 + 0.6 * 0.10 + 0.2 * 0.00 = 0.10 = 10%⇤Department of Finance, University of Illinois at Urbana-Cham p ai g n .1Sta4s4cal'Review'• From'variance,'we'construct'a'related'measure:'standard'devia4on'(s.d.)''• When'referring'to'random'variables'on'finance'such'as'prices,'costs'and''returns,'the'term'“vola4lity”'is'used'in'place'of's.d.''E[˜X] ⌘ µX=nPi=1pixiV ar[˜X] ⌘ 2X=nPi=1pi(xi µX)2s.d.⌘ X=snPi=1pi(xi µX)21Example'• The'change'on'the'value'of'exports'from'a'country'for'the'following'year'is'a'random'variable.'We'can'say'there'is'some'chance'that'it'will'fall,'some'chance'that'will'remain'constant'and'some'chance'that'it'will'increase.'But'we'are'never'certain'of'the'true'value'un4l'the'4me'comes'by.'• How'to'make'a'predic4on'about'it?'We'can’t'rely'on'“some”'to'be'precise.'The'use'of'expecta4on'and'variance'can'help'us.'• Let’s'say'the'country'is'Argen4na'and'that'they'rely'solely'on'the'exports'of'beef.''Example'Equations for ECON 452Fabrício d’Almeida⇤September 7, 2016E[˜X] ⌘ µX=nPi=1pixiVar[˜X] ⌘ 2X=nPi=1pi(xi µX)2s.d.⌘ X=snPi=1pi(xi µX)2dfE[Beef ]=0.2*0.20+0.6*0.10+0.2*0.00=0.10=10%Var[Beef]=0.2 ⇤ (0.20  0.10)2+0.6 ⇤ (0.10  0.10)2+0.2 ⇤ (0.00  0.10)2=0.0024(1)Equation 3 - Intensive analysisLogit(LineofCrediti,t)= ↵ +5Xq=0q i,t,qBetaFXi,t+5Xq=1q i,t,q+ 6P rofitabilityi,t1+ 7Tangibilityi,t+ 8Sizei,t1+ 9Networthi,t1+ 10Qi,t1+ 11ln(Age)i,t+ 12IndSalesV oli,t+ 13P rofitV oli,t, (2)Equation 4 - Extensive analysis⇤Department of Finance, University of Illinois at Urbana-Champaign.1• The'table'below'provides'the'change'in'exports'(in'%)'of'beef'from'Argen4na'condi4onal'on'each'outcome'of'the'world'economy:'boom,'regular'or'recession.'• For'each'state'of'the'economy'“i”,'there'is'a'probability'pi"that'state'“i”'will'happen,'and'the'corresponding'value'of'change'in'exports'of'beef'xi."• What'are'the'expecta4on,'variance'and'standard'devia4on'of'Beef?''pi"! xi!i=1''boom' 20%' +20%'i=2'regular' 60%' +10%'i=3'recession' 20%' 0%'Equations for ECON 452Fabrício d’Almeida⇤September 8, 2016E[˜X] ⌘ µX=nPi=1pixi–E[a˜X+b˜Y ]=aE[˜X]+bE[˜Y ]Var[a˜X+b˜Y ]=a2Var[˜X] + 2 a b Cov[˜X,˜Y ]+b2Var[˜Y ]—E[0.5Beef +0.5Wool]=0.5E[Beef ]+0.5E[Wool]Var[˜X] ⌘ 2X=nPi=1pi(xi µX)2Cov[˜X,˜Y ]=nPi=1pi(xi µX)(yi µY)s.d.⌘ X=snPi=1pi(xi µX)2dfE[Beef ]=0.2*0.20+0.6*0.10+0.2*0.00=0.10=10%Var[Beef ]=0.2 ⇤ (0.20  0.10)2+0. 6 ⇤ (0.10  0.10)2+0. 2 ⇤ (0.00  0.10)2=0.004(1)Beef=p0.0024 = 0.063 = 6.3%(2)⇤Department of Finance, University of Illinois at Urbana-Cham pai gn.1Equations for ECON 452Fabrício d’Almeida⇤September 8, 2 0 1 6E[˜X] ⌘ µX=nPi=1pixi–E[a˜X+b˜Y ]=aE[˜X]+bE[˜Y ]Var[a˜X+b˜Y ]=a2Var[˜X] + 2 a b Cov[˜X,˜Y ]+b2Var[˜Y ]—E[0.5Beef +0.5Wool]=0.5E[ Beef ]+0.5E[Wool]Var[˜X] ⌘ 2X=nPi=1pi(xi µX)2Cov[˜X,˜Y ]=nPi=1pi(xi µX)(yi µY)s.d.⌘ X=snPi=1pi(xi µX)2dfE[Beef ]=0.2*0.20+0.6*0.10+0.2*0.00=0.10=10%Var[Beef ]=0.2 ⇤ (0.20  0.10)2+0.6 ⇤ (0.10  0.10)2+0.2 ⇤ (0.00  0.10)2=0.004(1)Beef=p0.004 = 0. 063 = 6.3%(2)⇤Department of Finance, University of Illinois at Urbana-Champaign.1• We'expect'that'exports'of'Argen4na'will'grow'by'10%'next'year'• If'we'assume'that'expecta4on'and'variance'actually'came'from'a'normal'distribu4on'and'not'from'that'table:'– The's.d'of'6.3'p.p.'indicates'that'the'exports'will 'change'in'a'range'from'-2.4%'to'22.4%'with'95%'of'confidence'– That’s'a'very'broad'range:''• Predic4on'is'difficult'• Argen4na’s'economy'may'suffer'deep'swings'• How'to'minimize'this'vola4lity'to'make'the'economy'more'stable?'Example'• One'op4on'that'mono-product'countries'have'to'minimize'the'effects'of'vola4lity'on'their'economy'is'to'diversify'the'goods'they'produce'• Many'La4n'American'countries'pursued'to'diversify'their'economy'in'the'20th'century'• We'will'study'in'detail'the'policy'adopted'• Before,'we'shall'see'mathema4cally'how'adding'new'products'may'reduce'the'vola4lity'Diversifica4on'• Imagine'that'now'Argen4na'produces'50%'beef'and'50%'wool'• The'table'below'extends'the'possible'outcome'of'the'change'in'value'of'exports'from'beef'(xi)'to'include'wool'exports:'yi"• How'does'the'expecta4on,'variance'and'confidence'of'predic4on'changes'to'Argen4na?'Example'con4nued'pi"! xi! yi!!i=1''boom' 20%' +20%' 0%'i=2'regular' 60%' +10%' 0%'i=3'recession' 20%' 0%' +10%'Example'con4nued'Equations for ECON 452Fabrício d’Almeida⇤September 7, 2016E[˜X] ⌘ µX=nPi=1pixi–E[ a˜X+b˜Y ]=aE[˜X]+bE[˜Y ]—Var[˜X] ⌘ 2X=nPi=1pi(xi µX)2s.d.⌘