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STAT 1043 Statistical Techniques and Time Series

No Excel spreadsheet should be submitted. Your report should contain all necessary information for a complete understanding of your analysis.

See within.
Assessment: You must provide evidence of understanding. This implies that you are required to discuss the relevant theory behind any of your calculations and implementation.
To achieve full marks, you will need to devise the appropriate amount of discussion behind the theory and formulas used and methodology as applied to the relevant time series data.

Using the package quantmod download S&P500 index price data covering the period (yyyy/mm/dd) 2009/01/01 to 2020/12/31. Use daily data.
Show the 1-liner code responsible for the output
Write the respective mathematical equation i.e., for log-returns and/or for the ARIMA model fit
Explain any hypothesis testing:
(i) What is the underlying hypothesis test null and alternative hypothesis,
(ii) comment on the p-values
Display plots, with caption and explanation
No screenshots of results are allowed [Total: 100 marks]

On the series:
(i) Obtain the time series using the correction code syntax making use of the code below
library(quantmod)
Obj 1:  Load stock prices by symbol getSymbols(symbol) [5 marks]
(ii) Transform your series to log-returns [5 marks]
(iii) Examine the ACF and PACF functions [10 marks]
(iv) Perform the Ljung-Box test and describe the test-hypothesis and report/comment on the result. [10 marks]
(v) Check the data for stationarity using the correct test statistic and comment on the output [10 marks]
(vi) Perform a normality test of your choice on the return series and report the output.
Write down the hypothesis test and comment on the p-value [10 marks]
(vii) Fit an ARIMA model and determine the correct lag order: Show the 1-liner codes for output. [15 marks]
(viii) Report the coefficients for the chosen ARIMA model and show the respective equation given these coefficients [15 marks]
(ix) The residuals from an ARIMA fit require that:
a. The residuals have zero mean = 0
b. Have a finite variance
c. Have zero autocovariance
Using the results from checkresiduals(fitted_model) function comment on the above [20 marks]