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Question 1:
GBTC is the Greyscale Bitcoin trust and traded in NYSE. Its price has two component (1) value of Bitcoin holdings (determined by Bitcoin price), and (2) premium relative to the Bitcoin holdings. Sometime the premium can be negative. The Excel spread sheet contains daily historical GBTC price, value of its Bitcoin holdings, and GBTC premium. On each day, the GBTC premium is computed as (GBTC price – Value of Bitcoin holdings)/Value of Bitcoin holdings Let’s define
〖BTC〗t= log return of Value of Bitcoin holding on day t, and 〖Premium〗_t=log return GBTC premium on day t. Use Granger Causality to determine whether BTC influences Premium or Premium influences BTC. Please use (1) 1 lag, (2) 2 lags, and (3) 5 lags. For example, for 2 lags, the regression equation should be like this: 〖BTC〗_t=a_1+b_11∙〖BTC〗(t-1)+b_12∙〖BTC〗(t-2)+c_11∙〖Premium〗(t-1)+c_12∙〖Premium〗(t-2)+e_1 〖Premium〗_t=a_2+b_21∙〖BTC〗(t-1)+b_22∙〖BTC〗(t-2)+c_21∙〖Premium〗(t-1)+c_22∙〖Premium〗_(t-2)+ε_2

Question 2:

A. Shanghai Stock Exchange index (SSE) from 1997/7/2 to 2022/04/29
https://finance.yahoo.com/quote/000001.SS/history?p=000001.SS
B. Hang Seng index (HSI) from 1987/01/01 to 2022/04/29
https://finance.yahoo.com/quote/%5EHSI/history?p=%5EHSI
C. Gold price 2000/02/28 to 2022/04/29
https://finance.yahoo.com/quote/GC%3DF/history?p=GC%3DF
D. US oil fund (USO), 2006/04/10 to 2022/04/29
https://finance.yahoo.com/quote/USO/history?p=USO
For above four assets please compute the following
2. Mean, standard deviation of daily returns based on closing prices.
3. t-statistics, p-value, and 5% confidence interval for average daily returns, Ho: average return=0. (10 points)
4. Plot the average daily returns relative to the first trading day of every month from relative day -10 to 10. The X-axis should be the relative days -10 to 10, where 0 indicates the first trading day of a month. (10 points)
5. The average difference between daily returns during TOM (turn of the month) and daily returns during ROM (rest of the month). Please also compute t-stats, p-value, and 5% confidence interval for the difference. (10 points)
6. Repeat above 1-3 using the second half of the sample. (15 points) and fill out the table in the next page

Table for question 2
SSE
HSI
Gold
USO

`` whole   2nd half    whole   2nd half    whole   2nd half    whole   2nd half``

daily mean
Standard deviation
t-stats
pvalue
confidence interval

TOM daily mean
ROM daily mean
diff: TOM – ROM mean
diff t-stats
diff p-value
diff confidence interval