Applied Statistics and Econometrics Study Essay
Applied Statistics and Econometrics II Bruce McNevin, Ph.D.
Fall 2021 email@example.com
Please submit to NYUBrightspace no later than 4 PM on Monday., 12/20/2021.
Answer all questions (25 points each)
1) Let yi be a binary dependent variable that equals 1 or 0 for i = 1, . . . , n.
Let xi , i = 1, . . . , n, be k-element vectors of explanatory variables. Let β be a k-
element vector of unknown parameters.
- a) Write the functional form of E(yi |xi , β), the conditional mean function,
assuming a logit model.
- b) Derive the marginal effect, or partial derivative ∂E(yi |xi , β)/∂xij , where xij is the
j th element of the xi vector.
- c) Suppose logit model estimation produces the table:
(i) What is the predicted probability that y = 1 when X1 = 2 and X2 = 0.5?
(ii) Compute the change in the predicted probability when X2 increases by
one unit from X2 = 0.5 to X2 = 1.5, holding X1 at X1 = 2.
(iii) Using the derivative result from part (b) and the estimates in the above
table, compute the partial derivative ∂E(y|X1, X2, β)/∂X2 at the X values
given in part b.
2) In a study of the Canadian work force, we seek to predict marital status from x1 =
Age and x2 an indicator for Sex, with x2 = 1 meaning female and x2 = 0 meaning
male. Age is centered by subtracting off the mean for the entire sample, so that a
person of “average” age has x1 = 0. Marital status is coded as (1) Single and never
married, (2) Married, (3) Divorced or separated, and (4) Widowed, meaning the
husband or wife is dead.
- Write the estimation equations for a multinomial logit model. Make “Single
and never married” the reference category that goes in the denominator of
the generalized logit. (Hint: multinomial logit is a set of logit models so you
should have 3 linear equations, pi/pj = …, etc.)
- Define the πj symbols from your model so I know what they mean.
P(Single and never married) = ? P(Married) = ?
P(Divorced or separated) = ? P(Widowed) = ?
- In terms of the parameters from your model, what null hypothesis would you
test to determine whether being married (as opposed to single and never
married) is related to gender?
- Describe the IIA assumption and provide an example that illustrates why it
may be problematic.
3) Suppose that a true data generating process (DGP) is:
(c ) Are yt and xt cointegrated? Explain. If you believe that they are cointegrated
then provide the cointegrating vector.
4) a) Using matrix notation write a first order vector autoregession in structural form.
Can this model be estimated using OLS? Explain.
- b) Re-write the equation in reduced form. Describe the stability condition. Why is
the stability condition necessary?
- c) How would you estimate the model if the stability condition does not hold?
- d) What is Granger causality, and why is it useful?