<div dir="ltr"><div>Recomendo tres artigos (existem milhares!):</div><div><br></div><div>Clive W. J. Granger, Robert Engle, “Co-Integration and Error Correction:
Representation, Estimation, and Testing,” Econometrica, 55, no. 2 (1987): 251–76.</div><div><br></div><div>Søren Johansen, “Correlation, Regression, and Cointegration of Nonstationary Economic
Time Series,” Bulletin of the ISI LXII 2007, 2008, 19–26.<br></div><div></div><div><br></div>Michael Murray, “A Drunk and Her Dog: An Illustration of Cointegration and
Error Correction,” American Statistician, 48, no. 1 (1994): 37–39. <br><div><br></div><div><br></div><div>Abs,</div><div>Hedibert</div><div><br></div><div><br></div><div><br></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Fri, Mar 5, 2021 at 10:03 AM Julio Stern <<a href="mailto:jmstern@hotmail.com">jmstern@hotmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">




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<span style="margin:0px;font-size:12pt;background-color:rgb(255,255,255)"><span style="margin:0px;font-size:14pt;font-family:Arial,Helvetica,sans-serif">Caros Redistas </span></span></div>
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<span style="margin:0px;font-size:12pt;background-color:rgb(255,255,255)"><span style="margin:0px;font-size:14pt;font-family:Arial,Helvetica,sans-serif">Conversando com colegas de outras areas sobre a diferenca entre Causalidade e Correlacao, </span></span><span style="color:rgb(0,0,0);font-family:Calibri,Helvetica,sans-serif;font-size:12pt"><span style="font-size:14pt;font-family:Arial,Helvetica,sans-serif">principalmente
 no contexto de vacinas, tratamentos medicos para Covid, etc. </span></span><span style="color:rgb(0,0,0);font-family:Calibri,Helvetica,sans-serif;font-size:12pt"><span style="font-size:14pt;font-family:Arial,Helvetica,sans-serif">acabei conhecendo o</span></span><span lang="PT-BR" style="color:rgb(0,0,0);font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> livro: </span></div>
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<span lang="PT-BR" style="color:rgb(0,0,0);font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span><span lang="PT-BR" style="color:rgb(0,0,0);font-family:Calibri,sans-serif;font-size:11pt"><span style="font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span></div>
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<span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">>>></span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> <span> </span></span></span><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">Tyler
 Vigen (2015).</span><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> <span> </span></span><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">Spurious Correlations.</span><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> <span> </span></span><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">Hachette
 Books.<span> </span></span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span></p>
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<span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">que tem gráficos fantasticos, com series temporais totalmente desconexas que exibem altíssima correlacao.<span> </span></span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span></p>
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<span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">Nao tenho o livro, mas alguns graficos podem ser vistos em<span> </span></span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span></p>
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<span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">>>> <a href="https://www.tylervigen.com/spurious-correlations" target="_blank">https://www.tylervigen.com/spurious-correlations</a></span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span></p>
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<span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">>>> <a href="https://tylervigen.com/discover" target="_blank">https://tylervigen.com/discover</a></span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">  <span> </span></span><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span></p>
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<span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span></p>
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<span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">Comecei a fazer umas continhas rapidas sobre a probabilidade,  p</span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> , </span></span><span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">de
 obter, em duas series temporais, x(t) e y(t) ,<span> </span></span><span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">com dez pontos de abscissa  t={1,2, ... 10}</span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> <span> </span></span></span><span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">e
 ordenada no intervalo  [0,1] , uma alta correlação, i.e.,<span> </span></span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span></p>
<p style="margin:0in 0in 0.0001pt;line-height:16.8667px;font-size:11pt;font-family:Calibri,sans-serif">
<span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"><br>
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<span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">p = Pr ( corr ( x(t) , y(t) )<span> </span></span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span><span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">>=</span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> <span> </span></span></span><span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">(1-alpha)
 ) , para  alpha= 0.1</span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> <span> </span></span></span><span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">ou</span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> <span> </span></span></span><span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">0.05<span> </span></span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span><span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">ou</span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> <span> </span></span></span><span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">0.01</span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">  <span> </span></span><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span></p>
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<span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">Assumindo nao haver estrutura alguma nas series  x(t)  e  y(t) , o resultado eh muito baixo.<span> </span></span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span></p>
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<span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">Mesmo considerando uma enorme biblioteca de graficos aleatorios, nao da...<span> </span></span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span></p>
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<span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">Parece ser necessario assumir "a priori" a existência de alguma "estrutura intrinseca de associacao" entre os pontos de cada uma das series temporais,
 por exemplo, alguma estrutura de auto-correlacao.<span> </span></span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span></p>
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<span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">Antes de sair por ai tentando reinventar a roda, melhor perguntar a quem sabe...<span> </span></span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span></p>
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<span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">Alguem ja deve ter pensado neste problema!<span> </span></span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span><span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">Alguma
 ideia de como modela-lo?<span> </span></span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span></p>
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<span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">Abraco a todos e tudo de bom,<span> </span></span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span></p>
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<span lang="PT-BR" style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">---Julio Stern<span> </span></span><span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif"> </span></span></p>
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<span lang="PT-BR" style="margin:0px"><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">    </span><span style="margin:0px;font-size:14pt;line-height:normal;font-family:Arial,Helvetica,sans-serif">    </span></span></p>
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</blockquote></div><br clear="all"><div><br></div>-- <br><div dir="ltr" class="gmail_signature"><div dir="ltr"><span style="color:rgb(0,0,0);font-family:Calibri,Arial,Helvetica,sans-serif,"Apple Color Emoji","Segoe UI Emoji",NotoColorEmoji,"Segoe UI Symbol","Android Emoji",EmojiSymbols;font-size:small">Hedibert Freitas Lopes, PhD</span><br style="color:rgb(0,0,0);font-family:Calibri,Arial,Helvetica,sans-serif,"Apple Color Emoji","Segoe UI Emoji",NotoColorEmoji,"Segoe UI Symbol","Android Emoji",EmojiSymbols;font-size:small"><span style="color:rgb(0,0,0);font-family:Calibri,Arial,Helvetica,sans-serif,"Apple Color Emoji","Segoe UI Emoji",NotoColorEmoji,"Segoe UI Symbol","Android Emoji",EmojiSymbols;font-size:small">Professor of Statistics and Econometrics</span><br style="color:rgb(0,0,0);font-family:Calibri,Arial,Helvetica,sans-serif,"Apple Color Emoji","Segoe UI Emoji",NotoColorEmoji,"Segoe UI Symbol","Android Emoji",EmojiSymbols;font-size:small"><span style="color:rgb(0,0,0);font-family:Calibri,Arial,Helvetica,sans-serif,"Apple Color Emoji","Segoe UI Emoji",NotoColorEmoji,"Segoe UI Symbol","Android Emoji",EmojiSymbols;font-size:small">INSPER - Institute of Education and Research</span><br style="color:rgb(0,0,0);font-family:Calibri,Arial,Helvetica,sans-serif,"Apple Color Emoji","Segoe UI Emoji",NotoColorEmoji,"Segoe UI Symbol","Android Emoji",EmojiSymbols;font-size:small"><span style="color:rgb(0,0,0);font-family:Calibri,Arial,Helvetica,sans-serif,"Apple Color Emoji","Segoe UI Emoji",NotoColorEmoji,"Segoe UI Symbol","Android Emoji",EmojiSymbols;font-size:small">Rua Quatá, 300 - São Paulo, SP 04546-042 Brazil</span><br style="color:rgb(0,0,0);font-family:Calibri,Arial,Helvetica,sans-serif,"Apple Color Emoji","Segoe UI Emoji",NotoColorEmoji,"Segoe UI Symbol","Android Emoji",EmojiSymbols;font-size:small"><span style="color:rgb(0,0,0);font-family:Calibri,Arial,Helvetica,sans-serif,"Apple Color Emoji","Segoe UI Emoji",NotoColorEmoji,"Segoe UI Symbol","Android Emoji",EmojiSymbols;font-size:small">Phone: +55 11 4504-2343</span><br style="color:rgb(0,0,0);font-family:Calibri,Arial,Helvetica,sans-serif,"Apple Color Emoji","Segoe UI Emoji",NotoColorEmoji,"Segoe UI Symbol","Android Emoji",EmojiSymbols;font-size:small"><span style="color:rgb(0,0,0);font-family:Calibri,Arial,Helvetica,sans-serif,"Apple Color Emoji","Segoe UI Emoji",NotoColorEmoji,"Segoe UI Symbol","Android Emoji",EmojiSymbols;font-size:small"><a href="http://www.hedibert.org" target="_blank">www.hedibert.org</a></span><br></div></div>