Sate a condition on the conditional probabilities prx mz

Question

Question 1 -

Let Y denote the profit made by a particular firm's coffee shops in a city in a year. This depends on X, the number of coffee shops opened by a competitor in the city, and U which is an unobserved scalar variable measuring city characteristics associated with high or low profit prospects for coffee shops. There is the following structural equation.

Y = h(X) + U

Here X ∈ {0, 1, 2, . . . , M} and his an unknown function.

Since the competitor is likely to open more stores in cities where U is high it is possible that X and U are correlated. Let Z measure variables which affect the competitor's decision to open coffee shops in a city but are uncorrelated with U (for example distance from its suppliers) and in particular suppose that E[U|Z = z] = 0 for each value z in the list of distinct values: {z₁, . . . , z_R}.

Define γ_m = h(m) for m ∈ {0, 1, 2, . . . , M} and let γ denote the list γ₀, γ₁, . . . , γ_M.

(a) Show how E[Y|Z = z_r] is related to conditional probabilities, Pr[X = m|Z = Z_r].

(b) State a condition on the conditional probabilities Pr[X = m|Z = z_r] under which the value of γ is identified. Explain.

(c) Consider data obtained from observing N cities. Let Z_N be a N x R matrix of ones and zeros with a one in row n and column r if at the nth observation Z takes the value z_r. Let X_N be a N x (M + 1) matrix of ones and zeros with a one in row n and column m + 1 if at the nth observation X takes the value m. Let the N element vector Y_N contain N observations of Y. Under weak conditions the elements of the matrix (N^-1Z'_N Z_N)^-1(N^-1Z'_NX_N) estimate the probabilities appearing in part (a). Explain.

(d) Under weak conditions the expression

((X'_NZ_N)(Z'_NZ_N)^-1(Z'_NX_N))^-1 (X'_NZ_N)(Z'_NZ_N)^-1Z'_NY_N

is a consistent estimator of γ. Explain.

(e) There is data comprising relisations of Y, X and Z obtained in a random sample of N cities. Devise an estimator of γ for the simple case with M = 1 and Z is binary.

Question 2 -

Consider households comprising one man and one woman. Let Y denote a household's annual expenditure on food, let X denote household annual total expenditure and let Z_M and Z_W denote the income of respectively the man and the woman.

(a) There is the model: Y = α + βX + γX² + U and E[U|Z_M = z_M] = E[U|Z_W = z_w] = 0. Under what additional conditions, if any, does this model point identify the values of α, β and γ when only total household income: Z ≡ Z_M + Z_W is observable? Explain. How would your analysis differ if individual incomes Z_M and Z_W are observable?

(b) There is the nonparametric model: Y = h(X, U) with h a strictly monotone function of continuously distributed U at every value of X, and U and Z are independently distributed. The function h is normalized increasing in U and defined in such a way that U is uniformly distributed on the interval [0, 1]. If there are additional conditions such that there is only one function that satisfies

P[Y ≤ h(X, τ)|Z = z] = τ

for all τ ∈ [0,1] and all values z of Z then the resulting model point identifies the function h. Explain.

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