Suponha que

!$ x_t = p x_{ t-1} + v_t,\,\,\,\,x_0 = 0, v_t \sim N(0,1),\,\,t=1, \cdots, T\,\,\,\,\,\,\,\,\,\,\,\,(1)\\y_t = \phi y_{t-1} + u_t,\,\,\,\,y_0 = 0,u_t \sim N(0,1),\,\,\,\,t =1, \cdots, T.\,\,\,\,\,\,\,\,\,(2)\\\,\,\,\,E[u_S\,v_t]E [v_S\,v_t]= E[u_S\,u_t]= 0,\,\,\,\forall t\,e\,s, s \neq t !$

Adicionalmente, considere a regressão de y_t em uma constante e x_t:

!$ y_t = a + \gamma x_t + \varepsilon_t,\,\,\,\,t=1, \cdots,T\,\,\,\,\,\,\,\,\,\,\,(3) !$

Item 3 - O estimador de mínimos quadrados ordinários de γ, na equação (3), é consistente se !$ p = 1 !$ e !$ \phi =1 !$;

Suponha que X e Y sejam variáveis aleatórias independentes, com distribuições de Bernoulli com parâmetros p e q, isto é,

!$ X = { \begin{cases} 1\,\,com\,probabilidade\,\,\,\,p\\0\,\,\,com\,probabilidade\,\,\,1-p \end{cases}} !$

e

!$ Y = { \begin{cases} 1\,\,com\,probabilidade\,\,\,\,q\\0\,\,\,com\,probabilidade\,\,\,1-p \end{cases}} !$

Defina Z = aY + bX, para a e b constantes.

E[ ] e V[ ] representam, respectivamente, expectativa e variância. Julgue a afirmativa abaixo:

Item 4 - E[Y|Z = a+b] =1.

Suponha que X e Y sejam variáveis aleatórias independentes, com distribuições de Bernoulli com parâmetros p e q, isto é,

!$ X = { \begin{cases} 1\,\,com\,probabilidade\,\,\,\,p\\0\,\,\,com\,probabilidade\,\,\,1-p \end{cases}} !$

e

!$ Y = { \begin{cases} 1\,\,com\,probabilidade\,\,\,\,q\\0\,\,\,com\,probabilidade\,\,\,1-p \end{cases}} !$

Defina Z = aY + bX, para a e b constantes.

E[ ] e V[ ] representam, respectivamente, expectativa e variância. Julgue a afirmativa abaixo:

Item 1 - V[Z] = a²q + b²p ;

Considere uma economia descrita pelas seguintes equações:

• Produção de bens: !$ Y = K^a (EL_Y)^{ 1- a},\,\,\,\,\,\,\,0 !$

• Produção de “conhecimento”: !$ \dot{E} = \delta L_E E^{\phi},\,\,\,\,\,\,0 \ge \ge 1,\,\,\,0 < \delta < 1 !$

• Acumulação de capital: !$ \dot{K} = sY - dK,\,\,\,\,\,0< s<1,\,\,0< d < 1 !$

• Restrição de trabalho:!$ L = L_Y + L_E,\,\,\,\,\,\,L>0 !$

Em que Y é o produto, K é o estoque de capital, E é o nível de “conhecimento” na economia e L_Y, L_E e L representam, respectivamente, os trabalhadores empregados na produção de bens, na produção de conhecimento e o total de trabalhadores. O número de trabalhadores na produção de conhecimento é uma fração constante, u, da força de trabalho:!$ L_E = uL, 0 < u < 1 !$. A taxa de crescimento da força de trabalho L é constante e igual a n, ou seja, !$ \dot{L} /L = n !$. Com base nessas informações, julgue a seguinte afirmativa:

Item 2 - Se !$ \phi = 1 !$ e !$ n= 0 !$, a economia apresenta uma trajetória de crescimento equilibrado, na qual o produto por trabalhador (Y/L) cresce a uma taxa constante e positiva;

Considere as informações contidas na tabela a seguir, sobre um país hipotético, para os anos de 2006 a 2008. Assuma que sejam produzidos apenas 2 bens finais, chamados X e Y. O preço de cada bem é expresso em unidades monetárias ($). A unidade de medida de cada variável está entre parênteses. Com base nas informações da tabela, julgue as afirmativas a seguir:

Obs.: Para o cálculo do PIB real, não utilize encadeamento.

Item 0 - Houve uma redução de 10% no PIB real, a preços de 2006, entre os anos de 2007 e 2008;

Text 1

Excerpts from:

Fiat's ambitions

The Italian solution

Fiat’s chief executive, Sergio Marchionne,
has gone merger mad

HIS company is among the smallest of the global volume carmakers. But right now Sergio Marchionne is without question the most talked-about car executive in the world. The chief executive of Fiat Group has been alone in seeing an extraordinary opportunity in the meltdown in Detroit. By seeking to take over the running of both Chrysler and Opel, the European arm of General Motors (GM), Mr Marchionne is attempting not only to transform Fiat into a car group almost of the scale of mighty Toyota and Volkswagen (VW), but also to change the face of a perennially troubled industry.

Last December Mr Marchionne said of his stricken industry: “What we are seeing is unprecedented. I have never seen the failure of so many systems at once.” Fiat was in a fight for survival. “We’re just going to slam the brakes on, use as many temporary lay-offs as needed, cut everything back to essentials.” He added an apocalyptic forecast. “By the time we finish with this in the next 24 months, as far as mass producers are concerned, we’re going to end up with one American house [Ford or GM, you presume]; one German of size [VW Group]; one French-Japanese, maybe with an extension in the US [the Renault-Nissan alliance]; one in Japan [Toyota], one in China [several possible candidates] and one potential European player [either Fiat or PSA Peugeot Citroën].”

The details of this vision may be wrong. Despite its present travails and imminent bankruptcy, few believe that GM will vanish and leave Ford as the sole American-owned champion. France’s PSA Peugeot Citroën, though unwieldy, is not about to disappear either. The strength of Hyundai- Kia in emerging markets and North America should ensure that the South Korean producer makes the cut. And in Japan, however great the cull of smaller outfits such as Mitsubishi and Suzuki, Honda and perhaps Mazda will still be around to challenge the dominance of Toyota in its home market. So will Renault’s partner, Nissan.

(…)

Having presided over a near-miraculous turnaround at Fiat since being appointed in 2004, Mr Marchionne saw in Chrysler an opportunity to apply the same lessons. At Fiat he saw a sluggish organisation that lacked leadership and had become accustomed to management by committee. But he also saw, buried within the company, a new generation of leaders.

“The single most important thing was to dismantle the organisational structure,” he recalls. “We tore it apart in 60 days, removing a large number of leaders who had been there a long time and who represented an operating style that lay outside any proper understanding of market dynamics.” In their place he promoted a group of younger executives, many with a background in consumer marketing, who understood and could provide what he wanted: accountability, openness, rapid communication and impatience with hierarchy and internal politics.

Some doubt that Fiat’s lean management has the resources to spread itself across Chrysler, let alone Opel too. Mr Marchionne understands the concern, but rejects it. He believes he already knows who the new leaders at Chrysler will be. He is confident that the same will apply to Opel, should that too fall into his lap.

(…)

(From The Economist print edition, May 7th 2009)

According to the text:

Item 3 - Honda and Mazda are considered smaller outfits in their home market;

Text 1

Excerpts from:

Fiat's ambitions

The Italian solution

Fiat’s chief executive, Sergio Marchionne,
has gone merger mad

HIS company is among the smallest of the global volume carmakers. But right now Sergio Marchionne is without question the most talked-about car executive in the world. The chief executive of Fiat Group has been alone in seeing an extraordinary opportunity in the meltdown in Detroit. By seeking to take over the running of both Chrysler and Opel, the European arm of General Motors (GM), Mr Marchionne is attempting not only to transform Fiat into a car group almost of the scale of mighty Toyota and Volkswagen (VW), but also to change the face of a perennially troubled industry.

Last December Mr Marchionne said of his stricken industry: “What we are seeing is unprecedented. I have never seen the failure of so many systems at once.” Fiat was in a fight for survival. “We’re just going to slam the brakes on, use as many temporary lay-offs as needed, cut everything back to essentials.” He added an apocalyptic forecast. “By the time we finish with this in the next 24 months, as far as mass producers are concerned, we’re going to end up with one American house [Ford or GM, you presume]; one German of size [VW Group]; one French-Japanese, maybe with an extension in the US [the Renault-Nissan alliance]; one in Japan [Toyota], one in China [several possible candidates] and one potential European player [either Fiat or PSA Peugeot Citroën].”

The details of this vision may be wrong. Despite its present travails and imminent bankruptcy, few believe that GM will vanish and leave Ford as the sole American-owned champion. France’s PSA Peugeot Citroën, though unwieldy, is not about to disappear either. The strength of Hyundai- Kia in emerging markets and North America should ensure that the South Korean producer makes the cut. And in Japan, however great the cull of smaller outfits such as Mitsubishi and Suzuki, Honda and perhaps Mazda will still be around to challenge the dominance of Toyota in its home market. So will Renault’s partner, Nissan.

(…)

Having presided over a near-miraculous turnaround at Fiat since being appointed in 2004, Mr Marchionne saw in Chrysler an opportunity to apply the same lessons. At Fiat he saw a sluggish organisation that lacked leadership and had become accustomed to management by committee. But he also saw, buried within the company, a new generation of leaders.

“The single most important thing was to dismantle the organisational structure,” he recalls. “We tore it apart in 60 days, removing a large number of leaders who had been there a long time and who represented an operating style that lay outside any proper understanding of market dynamics.” In their place he promoted a group of younger executives, many with a background in consumer marketing, who understood and could provide what he wanted: accountability, openness, rapid communication and impatience with hierarchy and internal politics.

Some doubt that Fiat’s lean management has the resources to spread itself across Chrysler, let alone Opel too. Mr Marchionne understands the concern, but rejects it. He believes he already knows who the new leaders at Chrysler will be. He is confident that the same will apply to Opel, should that too fall into his lap.

(…)

(From The Economist print edition, May 7th 2009)

According to the text:

Item 4 - Mitsubishi, Susuki and Toyota dominate the Japanese market.

Responda se verdadeiro ou falso:

Item 1 -O estimador de Mínimos Quadrados do coeficiente angular em uma regressão simples, com a constante erroneamente omitida, será consistente se a média da variável explicativa for zero e se o erro for independente da explicativa e possuir média zero;

Julgue a afirmativa:

Item 0 - Seja !$ (a_n)_{n\,\in\,N} !$uma sequência de números reais não nulos, tal que !$ |a_{n+1} | < { \Large { |a_n| \over 2}} !$, para todo !$ n\,\in\,IN !$. Então !$ \underset {n \rightarrow \infty} {lim} a_n = 0 !$;

Considere a curva de Phillips:

!$ \pi_t = \pi_t^e - 2 (u_t - 0,10) !$,

em que !$ \pi_t = \pi_t^e !$ e !$ u_t !$ são, respectivamente, a inflação no ano t, a inflação esperada para t e a taxa de desemprego em t. No ano 1, a economia encontra-se em uma situação em que !$ \pi_1 = \pi_1^e = 0,10 !$. O Banco Central, que controla diretamente a taxa de inflação, anuncia a implementação, a partir do ano 2, de uma política de desinflação visando trazer a inflação para 0,04 (isto é, 4%).

A razão de sacrifício (ou taxa de sacrifício) é o aumento na taxa de desemprego (acumulado ao longo do período de desinflação) dividido pela queda na taxa de inflação.

Com base nessas informações, julgue a seguinte afirmativa:

Item 0 - Se as expectativas são racionais e o Banco Central é capaz de adotar um mecanismo de comprometimento crível, a desinflação é imediata e a razão de sacrifício é zero;