Exa Applied Statistics and Econometrics II Bruce McNevin, Ph.D.

Fall 2021 bm76@nyu.edu

Final Examination

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.

a. 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, i/j = , etc.)

b. 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) = ?

c. 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?

d. 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?