calculus work

Exercise #2

Convention:  if a question is about the linear model, then the notation ‘X’ with or without subscript, stands for a fixed number.  If the question is NOT about the linear model, i.e., the term ‘linear model’ is not in the question, then ‘X’ stands for a random variable or a ‘regular’ random variable (not a fixed number, unless specified in a question).

Other convention: as in Exercise #1, a statement is true only if it is always true.  If it is true only in some cases/scenarios, we classify it as a false statement (of course, if a statement is always false, it is a false statement).

1.Suppose we have a data set (a sample) with 30 observations (30 pairs of  X’s and Y’s), how many equations are there in the corresponding linear model for this data set?  How many unknowns are in these equations? Is it possible to determine the values of the unknowns from the observed data/sample?

  1. Consider a typical equation (without the subscripts) in a linear model:

Y = B1 + B2X + u .

Suppose it is known that Y = 10 and X = 12; and suppose we actually know the values of the B’s to be:  B1 = 2 and B2 = 0.7.  Can we determine the value of the unknown  ‘u’?  Explain.

  1. Let X be a random variable, then:
  2. A) E( E(X) ) = 0.
  3. B) E( X ) = 0.
  4. C) E( E(X) ) = E( X ).
  5. D) None of the above.
  1. Let X be a random variable, then:
  2. A) V( E(X) ) = 0.
  3. B) V( E(X) ) = E(X).
  4. C) V ( E(X) ) = X.
  5. D) None of the above.
  6. Let X1and X2  be any two random variables, then E( Cov( X1, X2) ) =
  7. A) 0.
  8. B)
  9. C) – 1.
  10. D) None of the above.
  1. Let X1 and X2 be any two random variables, then V( Cov( X1, X2 ) ) =
  2. A)
  3. B)
  4. C) -1.
  5. D) None of the above.
  1. In the linear model, E( B2X ) =
  2. A)
  3. B)
  4. C) B2
  5. D) None of the above.
  1. In the linear model, V( B2X ) =
  2. A)
  3. B)
  4. C) B2
  5. D) None of the above.
  1. An estimator is (in most cases or in general):
  2. A) a fixed number.
  3. B) a random variable.
  4. C)
  5. D) None of the above.
  1. In general, an estimator is:
  2. A) an unobservable random variable.
  3. B) an unobservable fixed number.
  4. C) an observable random variable.
  5. D) None of the above.
  1. An estimate is:
  2. A) the value of an estimator in a sample.
  3. B) the value of the unknown parameter/quantity that we are trying to estimate.
  4. C) a random variable.
  5. D) None of the above.
  1. The least squares estimator (for the slope) is:
  2. A) one of the many estimators for the slope parameter in the linear model.
  3. B) the only estimator for the slope parameter in the linear model.
  4. C) always equal to the slope parameter in the linear model.
  5. D) None of the above.
  1. For each of the following statements, state whether it is true or false and explain. Please remember the convention: a statement is true only if it is true in all situations. A statement that is only true in some situations is a false statement.
  1. An estimator in general would give different estimates (of an unknown parameter) in different samples.
  1. The following can be used as an estimator for B2 in the linear model:

(u1 + u2)/ X1.

  1. In the linear model, V(u) = 0.
  1. In the linear model, the covariance between any two u’s is zero.
  1. In the linear model, Cov(u2, u3 ) = E(u3*u2 ).
  1. In the linear model, E(Y) = B2
  1. In the linear model, E(u1 + u2 + … + uN ) = 0.
  1. In the linear model, V(u) = E(u2) = σ2.
  1. In the linear model, the term ‘u’ is called the least squares residual.
  1. In the linear model, the higher the magnitude of u or u-hat, the higher the value of R2.
  1. In the linear model, the more a ‘Y’ can be explained by X, the smaller is the magnitude of the corresponding ‘u’ or u-hat.
  1. In the linear model, E( u12 + u22 ) = 2.
  1. For a linear model, the assumption: V(u1) = V(u2) = … = σ2 is likely to be a feature of cross-sectional data set.
  1. For a linear model, the assumption Cov(any two u’s) = 0 is likely to be satisfied in a time series data set.
  1. Suppose in a linear model, B2 = 0.9, then as X increases by 5, the effect of the increase should be an increase of 5 in the value of Y (holding the value of ‘u’ constant).
  1. Suppose X is a nonnegative random variable (possible values of X are all greater than or equal to zero), and E( X ) = 0.1, then P( X = 1) can be 0.1.
  1. If, in a linear model that describes the relationship between GPA (X) and grade in ECO 4000 (Y), the ‘truth’ is that there is no relationship, does it mean that B2 = 1 in the linear model? Explain.
  1. According to macroeconomic theory in ECO 1002, the general price level, measured by a price index, depends positively on money supply (the total amount of money in the economy). Suppose data (observations) on a price index and money supply are available annually from 1960 to 1988 inclusively,

(a) Specify a linear model to describe the relationship between price and money supply. Please discuss the assumptions you made for each of the term (symbol) in the model, i.e., for each term in your equation, carefully state: (i) whether it is a random variable or a fixed number, (ii) whether it is observable/known or not (in a sample).

(b) If the above macroeconomic theory is indeed true, what would be the sign of B2, i.e., should B2 be a negative or positive number?  If the theory is wrong, what should be the sign of B2?

(c) Derive the expectation of a typical dependent variable, say Yi.

(d) Do all the Y’s have the same expected value? Explain.

(e) Suppose given the values of the Y’s and the X’s in a  certain sample, we have computed  and  510.12. Find the least squares estimate for the slope parameter in your model.

(f) Is the (true) value of B2 the same as the answer in part (e) above?  Explain.  Also, explain whether one would conclude that the value of price index depends positively on money supply (as suggested by economic theory).

  1. Suppose in question 15above, instead of price level and money supply, the two variables are: (total) investment (measured in dollar) in the economy, and interest rate. In macroeconomics theory, we learned that investment depends negatively on interest rate, i.e., the higher the interest rate, the lower the investment level. If we specify a linear model to describe the relationship between investment and interest rate,

(a) Which variable is the dependent variable? Which variable is the independent variable?

(b) Do you expect B2 in the model to be positive?  Explain.

(c ) We can collect a sample of observations and use the least squares estimator to provide an estimate of B2.  Do you expect your estimate to be a negative number?  Explain carefully your answer. [Hint: this is an ‘open’ question. In your answer, you should talk about the different factors: the economic theory, the sample and the linear model with the unknown parameters, and the concept of an estimator and its estimate]

  1. In a linear model and from the definition of covariance, write down an expression for the covariance between ‘B2*X’ (or in short, B2X ) and ‘ u’, i.e., write down an expression for Cov( B2X, u).  What is the value of

E( Cov( B2X, u ) )?  What is the value of V( Cov (B2X, u)  )?

  1. Suppose the random variables X1 and X2 are payoffs of two different investments.  Further , E( X1 ) = E( X2 ) = 1,000;  and  V( X1 ) = V( X2 ) = 50, and the covariance between the two random variables is zero.
  1. Find the expectation of the total payoff, i.e., find E( X1 + X2 ).
  2. Find the variance of the total payoff, i.e., find V( X1 + X2 ).
  3. Find the variance of the average payoff, i.e., find V(( X1 + X2) / 2 ). Is it greater or smaller than the variance of an individual X?
  4. Find the expectation of the average payoff, i.e., find

E( ( X1 + X2) / 2 ).

  1. Suppose in addition to X1 and X2 , we have many more of the X’s, and each X has the same expectation of 1,000 and variance of 50. What is the value of  E ( ( X1 + X2 + … + X500 ) / 500 )?
  1. With the same info as part e), what is the value of

V ( ( X1 + X2 + … + X500 ) / 500 )? [Assume that the covariance between any two X’s is zero]

  1. In part e), we know the expectation of ( X1 + X2 + … + X500 ) / 500 is 1000.  Since your answer in part f) above is a small number, does it mean that the value of  ( X1 + X2 + … + X500 ) / 500 can never be ‘far away’ from 1,000?
  1. What is the implication of the answers in part f) and part g) above? In particular, what can we say about the value of the random variable,

( X1 + X2 + … + X500 ) / 500 ?

  1. Suppose there are a large number (N) of random variables, u1 ,  u2 , … ,uN ,  and E(u1 ) = E( u2 ) = …  =  E( uN ) = 0.  Further,  V(u1) =  V(u2)  = … =  10; and the covariance between any two u’s is zero.  What can we say about the value of the simple average of the u’s when N is large?  Please justify your answer.
  1. (a) Explain, perhaps by a counter example, whether in general, it is true that:  E( X2) =  ( E(X) )2, where X is any given random variable. [Hint: try  X = 1 or -1, the coin tossing example where head is ‘1’ and tail is ‘-1’]

(b) Given your answer in part (a), show that, in general, for any two given random variables, X1 and X2,  E( X1*X2) is not the same as E(X1)*E(X2).  [Hint:  take X1 and X2 to be the same random variable, X, and then apply result of part a)]

  1. Suppose X1 and X2 are payoffs of two different investments and they are random variables. Explain when the ‘risk’ or variance of the total payoff is less or higher than the sum of the ‘risk’ or variances of the individual payoff.  In other words, when will:  V( X1 + X2 ) >  V( X1) + V(X2 )?  Also, when will V( X1 + X2 ) <  V( X1) + V(X2 )?
  1. Suppose for a certain random variable X, E(X) = 0 and V(X) = 10. But suppose we care only about the random variable:  (X – 1).

(a )Find the values of E( X – 1 ) and  V( X – 1).

(b) Suppose we convert the measurement of X into some other unit and the conversion is: X becomes 2*X (for example, an exchange rate type of conversion).  Find the values of  E( 2*(X – 1) ) and V( 2*(X – 1)).

  1. Suppose X1 and X2 are payoffs of two different investments, and the covariance between the two is zero. Suppose they each has a fee of 10, so the net payoffs are: (X1 – 10) and (X2 – 10).  Find the (value of the) covariance of the net payoffs, i.e., findCov(X1 – 10, X2 – 10 ).
  1. Consider the following (simple version of a) linear model:

Y = B*X + u.

We can multiply the equation by ‘X’ and obtain:

X*Y = B*X2 + X*u,   so ‘X*u’ is a ‘new’ random term.

It is assumed that ‘X’ is a fixed number and ‘u’ is a random variable with E(u) = 0, and V(u ) = 1.

  1. What is the value of E( X*u )?
  1. What is V( X*u )?

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