1) The following table contains age and weekly earnings (wage) of randomly sampled dependent workers in Germany: Wage Age 962 50 571 23 777 38 692 26 780 31 875 40 723 28 918 52 We are interested in the following relationship between wage and the workers' age:

wagei = β0 + β1 · agei + ϵi i=1,…,n. |

IMPORTANT: Use a dot ('.') to separate decimal numbers and round your results to the THIRD digit after the decimal point. Do NOT use a comma (',') as a decimal separator. Example: Write 1.234 to represent the number 'one point two three four'. 1. Estimate the parameters in the above regression model by OLS. What is the estimated slope coefficient? | | 2. Compute the sample variance of wage. | | 3. What is the predicted wage at age 65? | |

2) We have a dataset of 55 firms containing information on each firm's marketing expenditures X and the total number of products sold Y. The following table reports descriptive statistics for the two variables in the dataset (marketing expenditures are expressed in thousand Euros and the number of products sold are expressed in thousand pieces): | Marketing expenditures X | number of products sold Y | Mean | 6 | 12 | Variance | 15 | 23 |
The correlation between X and Y is 0.71.
Predict the expected number of products for a firm spending 7 (thousand) Euros for marketing purposes. Do so by estimating a linear model with ordinary least-squares.
IMPORTANT: Use a dot ('.') to separate decimal numbers and round your results to the THIRD digit after the decimal point. Do NOT use a comma (',') as a decimal separator. Example: Write 1.234 to represent the number 'one point two three four'.
Answer:

3) Pear Inc. plans the introduction of a new mobile phone at a price of p=100 per phone. The firms' profit function for the production of x units of the new phones is given by G(x)=-87+2x+5 x2 |

for x≥0.
In addition, based on the analysis of the marketing division, you know that the consumers' demand for the new mobile phones is given by the following probability density function:

f(x)=(1.2 x3 -24 x2 +120x)/1000 |

where 0<x<10 is the demanded number of phones. The expected demand implied by this function is equal to four (i.e. E(x)=4) and the variance of demand is equal to four (i.e. Var(x)=4).
IMPORTANT: Use a dot ('.') to separate decimal numbers and round your results to the THIRD digit after the decimal point. Do NOT use a comma (',') as a decimal separator. Example: Write 1.234 to represent the number 'one point two three four'. 1. How much mobile phones do Pear Inc. have to sell in order to break even (in terms of actual profits not (!) expected profits)? | | 2. Compute the expected profits. | |

4) Consider a dataset containing data on birth weights of newborns. The dependent variable we are interested in is "birth weight in g" (bwght) and the explanatory variable is "average number of daily cigarettes mother smoked during pregnancy" (cigs). The following simple regression was estimated using 127 birth observations: Estimate Std. Error (Intercept) 3015.328 7.27066 cigs -15.513 0.30717 IMPORTANT: Use a dot ('.') to separate decimal numbers and round your results to the THIRD digit after the decimal point. Do NOT use a comma (',') as a decimal separator. Example: Write 1.234 to represent the number 'one point two three four'. 1. What is the predicted birth weight for cigs=20? | | 2. Which of the following variables should be also included in the regression? | | 3. What is the implied value cigs in order to predict a birth weight of 2000 g? | | 4. What is the highest predictable birth weight? | |

5) The risk management of financial intermediaries is largely based on the so called "Value at Risk" (VaR). The VaR measures the maximum expected loss on a risky portfolio that is not exceeded, given a certain probability (here we assume 95 percent). For example, suppose the VaR of an asset is equal to -2 percent. This means that the return on the asset will not fall below -2 percent with a 95% probability.
Consider now financial asset X with an expected return of 3 and standard deviation of 2.08.
IMPORTANT: Use a dot ('.') to separate decimal numbers and round your results to the THIRD digit after the decimal point. Do NOT use a comma (',') as a decimal separator. Example: Write 1.234 to represent the number 'one point two three four'. a. Compute the VaR for asset X under the assumption that its returns are normal distributed. b. In reality, returns are often not normally distributed. Compute the VaR for asset X under the assumption that its returns are t-distributed (student-distributed) with 5 degrees-of-freedom.