Question 1 Estimating the probability of Labor Force Participation

Let X be the Bernoulli  random variable associated with the labor force participation.  It takes the value 1 when the individual is in the labor force (with probability p) and takes the value zero otherwise (with probability one minus p). You obtain a random sample of 10 individuals from the American population and record their labor force participation status Xi  for i = 1, 2, ..., 10, obtaining this sequence of values: X = (1, 1, 1, 1, 1, 1, 0, 1, 0, 1).

a) Let p be the probability that randomly selected American is in the labor force. Find the maximum likelihood estimate of p.

b) Some alarmists are saying that the labor force participation in the US has reached 53%. Based on the data, do you agree with this statement?

Question 2 Estimating the Unemployment duration

The duration  (in weeks) a worker remains unemployed before finding another job is a random variable denoted by X with an unknown mean. You obtain the following random sample of unemployment duration (in weeks) from the Bureau of Labor Statistics.  X = (X1,X2 ,X3 ,X4 ,X5 ,X6 ,X7 ,X8 ,X9 ,X10 ).

Economic theory suggests that unemployment duration folllows an exponential distribution with rate parameter λ .1  That is, the density of unemployment spells is f(x) = λe−λ北 .

a)    You    observe    the    following    sample    of    unemployment    duration    X      = (6, 8, 12, 4, 8.5, 6.5, 6, 8, 9, 4).  Write  the  likelihood  function  of  the  data  as  a  function of λ .

b) Find the maximum likelihood estimator of λ .

c) The mean of an Exponentially distributed random variable is the inverse of the rate parameter  (as you found in your third assignment).   Can you use this to construct an analogy principle estimator of λ?

Question 3 (Optional) Estimating earnings inequality

In recent years, there has been a growing concern that earnings inequality is on the rise. In particular, it seems that top- end  inequality has been growing.  That is, the share of production that is captured by the very rich seems to be growing.

Economic theory suggests that the tail of the distribution of earnings should have the shape of the Pareto distribution.  That is, if we look, say, only at earnings above 1 million per year, we should have a distribution of earnings that follow this formula: f(x) = θx−θ − 1 .

The parameter θ plays a very important role in the Pareto distribution.  It dictates the amount of inequality in the economy.  In particular, when θ takes the value of 1.16, then the economy presents the so-called “80-20 rule”, which states that 20% of all people receive 80% of all income, and 20% of the most affluent 20% receive 80% of that 80%, and so on, holds precisely when the Pareto index is θ is equal to 1.16.

You randomly select a number N of high-earnings individuals and collect data on their respective earnings. Denote by Xi  the earnings of individual i. For example, if the fourth individual earns 80 million dollars in a year, then X4  = 80. Then X = (X1,X2 , ...,XN ) is a random sample of draws from a Pareto distribution with unknown index θ .

a) You observe the following random sample of earnings of high-earnings individuals X = (2, 2, 1, 1, 10). Find the maximum likelihood estimator of θ .

Question 4 (Law of Large Numbers) Pick a coin.  Define Xn  to be equal to one if the outcome of tossing the coin is “heads” and zero if the outcome is “tails”. The subscript n indexes the order of the experiments: n = 1 means the first time you toss the coin, n = 2 means the second time you tossed the coin, and so on. For example, X3 = 0 is a shorthand for the sentence “when you tossed the coin the third time, the outcome was tails”.

Toss the coin and record the outcome. Call it X1 . Record also X1/1, which is the sample

Experiment    Outcome   Xn     P一n[X = 1]

average up to this point, or the sample proportion of heads.  Call this object 1[X = 1]. Toss the coin again and record the outcome.  Call it X2 .  Record also (X1 + X2)/2.  Call this object 2[X  =  1].   Toss the coin again and record the outcome X3 .   Record also (X1 + X2 + X3)/3. Call this object 3[X = 1].

Continue the experiment in the same fashion until you toss the coin 20 times.  At every

round, record the outcome of tossing the coin and the proportion of heads you have obtained so far – that is, P一n[X = 1]. Put your results in a table with two columns. In the first column, you have Xn, the outcome of the nth time you tossed the coin, and in the second column, you have P一n[X = 1], the current level of your guess for the chance of heads, as obtained by the proportion of heads in the sample. See the table above for an example.

Plot a graph of the estimated proportion as a function of the number of observations. That is, plot a graph where on the y-axis you have P一n[X = 1] and on the x-axis you have the current sample size n. Does the graph present any pattern? Does it seem to have a “long-

run” trend or a limit, as n gets large? Discuss your results. Relate your results to the law of large numbers.

Now, run the same experiment once again, tossing the coin 20 times and recording the outcomes and the sample proportions. Plot your results in a graph. Is your graph identical to the one you draw before?  Explain why or why not.  If not, does it share at least some similar patterns as the one from your first experiment? Discuss your results.

Question 5 You need to make an estimate of monthly earnings wages for these four groups: