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2021

Financial Econometrics

Question 1 [25 marks]

Consider the case where the log-prices follow a Gaussian diﬀusion process with time- varying local variance process ct , over the time interval [0, 1]:

dXt = ′ct dWt

Show that

|∆i(n)X |〇 →

iī＋

＋

cs ds

Question 2 [25 marks]

Suppose we have excess returns of N stocks at time t denoted as Rit(e), where i = 1, ..., N

(a) Consider a cross-sectional regression of the N excess stock returns on a constant (normalized to unity). Show that the regression coeﬃcient is the return on a portfolio. What portfolio? (10 marks)

(b) Suppose that, for each stock, we have a measure of the stock’s historical beta with the market. We run a cross-sectional regression of excess returns onto a constant and historical betas. Show that both the intercept and the slope coeﬃcient are the returns on some portfolios. Characterize each of these portfolios by stating the sum of the portfolio weights and the historical beta of the portfolio with the market. (15 marks)

Question 3 [25 marks]

Consider a problem of a representative household who maximizes her utility subject to the inter-temporal budget constraint for consumption ct and ct←＋ in each period. In period t, she receives an endowment yt and chooses to buy ξ shares of assets at price of 1 which pay 1 + rt←＋ in the following period t + 1. This problem is then characterized as

follows:

max u(ct ) + βEt [u(ct←＋)]

s.t., ct = yt 一 ξ

ct←＋ = yt←＋ + (1 + rt←＋)ξ

Now, we assume that her utility is u(ct ) = log(ct ).

(a) Write down the Euler equation of this problem and interpret it. What is the corresponding moment condition for this Euler equation? (10 marks)

(b) Let zt = ln(ct /ct ≥＋) and mt = β exp(一zt ). Also, let µz be the unconditional mean of zt . A ﬁrst-order Taylor series expansion of mt around the point zt = µz takes the form m˜t = ξ[1 一 (zt 一 µz )]. What is the value of ξ? Justify your answer. (8 marks)

0 = Et (Rt(e)←＋m˜t←＋) (1)

where Rt(e)←＋ is a vector of excess returns on assets and m˜t←＋ is derived in the last question. Could you obtain a unique value of ξ based on equation (1) or not? If yes, justify your answer. If not, state the reason. (7 marks)

Question 4 [25 marks]

A Gaussian linear state-space model is written in the form

xt←＋ = Fxt + wt←＋

zt = A + H|xt + vt

where xt is an r × 1 vector of latent state variables, zt is an n × 1 observed vector. Besides, vt and wt are two white noises with the following joint distribution:

┌ ┐ ╱┌ ┐ ┌ ┐\

Here, F, A, H, Q, R are known matrices. Also, Σt|t ≥＋ is the predicting error variance covariance matrix of xt given the history of zt ≥＋, while Ωt|t ≥＋ is the predicting error variance covariance matrix of zt given the history of zt ≥＋. They could be explicitly expressed as

Σt|t ≥＋ = E[(xt 一 xt|t ≥＋)(xt 一 xt|t ≥＋)| |zt ≥＋]

Ωt|t ≥＋ = E[(zt 一 zt|t ≥＋)(zt 一 zt|t ≥＋)| |zt ≥＋]

zt = µ + ∈t + θ＋∈t ≥＋ + θ〇 ∈t ≥〇

with ∈t ～ i.i.d. N (0, σ〇 ). Show that this can be written in state-space form

xt←＋ = Fxt + wt←＋

zt = A + H|xt + vt

by specifying xt = (∈t , ∈t ≥＋, ∈t ≥〇 )| . Find the values of F, A, H, Q, R for this representation (Note: Q and R are deﬁned in the beginning of this question). (9 marks)