Financial Econometrics Homework 1

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Note: For Problem #1, write your answers into a Jupyter Notebook …le and submit the …

1. (2) Using Python, perform the following for the KOSPI index returns during the period 2021:01:02-2021:12:30 (on the basis of daily, closing prices; data available at https://ecos.bok.or.kr/‡ex/EasySearch.jsp).

1. 1. Plot the sample autocorrelation function of the simple returns of the KOSPI index (log-di¤erences of the index). Do they indicate serial corre-lation?

1. 2. Test the null of no serial correlation using the Ljung-Box test at the 5% level. Set the lag length at 10.

2. (3) ¹Suppose that the daily log return of a security follows the model

r_t = 0:01 + 0:2r_{t 1} + a_t

where fa_tg is a Gaussian white noise series with mean zero and variance 0.02.

1. 1. What are the mean and variance of the return series r_t?

1. 2. Compute the lag-1 and lag-2 autocorrelations of r_t .

1. 3. Assume that r₁₀₀ = 0:01, and r₉₉ = 0:02. Compute the 1- and 2-step ahead forecasts of the return series at the forecast origin T = 100. What are the associated standard deviations of the forecast errors?

3. (1) Suppose that r_t is represented by the AR(2) process

r_t = 1:1r_{t 1} 0:4r_{t 2} + a_t; a_t W N(0; ²):

Is this process stationary?

4. (1) Consider the process

X¹

r_t = 1 + 0:2^j a_{t j} ; a_t W N(0; ²):

j=0

Is this process stationary?

2

5. (1) Find the ACF function of the MA(2) process

r_t = a_t + ₁a_{t 1} + ₂a_{t 2}; a_t W N(0; ²):