Abstract
The price of a stock as a function of time constitutes a financial time series, and as such it contains an element of uncertainty which demands the use of statistical methods for its analysis
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Notes
- 1.
source http://finance.yahoo.com/q/hp?s=AAPL+Historical+Prices
- 2.
Confidence refers to the following: if the probability law of the sample mean is approximately normal, then within these LCL and UCL bounds lie approximately 95 % of sample means taken over nobs observations. We shall discuss normality in the next pages.
- 3.
Campbell et al. (1997 §1.4.2)
- 4.
For a more extensive discussion see Campbell et al. (1997 Chap. 1)
- 5.
We remark that in many classical textbooks of probability and stochastic processes, such as Breiman (1992), strictly stationary is simply termed stationary, but in the modern literature of time series and financial applications, such as Brockwell and Davis (2002); Tsay (2010), it is more common to distinguish different categories of stationarity, in particular strict (as in Def. 2.3) and weak (to be defined later). We adhere to this latter specification of different levels of stationarity, and when using the term by itself –without adverb– is to refer to any, and all, of its possibilities.
- 6.
The following counterexample is from Brockwell and Davis (1991): let \(\{X_t\}\) be a sequence of independent random variables such that \(X_t\) is exponentially distributed with mean 1 when \(t\) is odd and normally distributed with mean 1 and variance 1 when \(t\) is even, then \(\{X_t\}\) is weakly stationary, but \(X_{2k+1}\) and \(X_{2k}\) have different distributions for each \(k>0\), hence \(\{X_t\}\) cannot be strictly stationary.
- 7.
see Box and Jenkins (1976); Breiman (1992) on properties of conditional expectation and best predictor discussed in this section.
- 8.
One that we like very much is the Innovations Algorithm, see Brockwell and Davis (2002).
- 9.
see Hald (1999) for a historical account
- 10.
There other practical reasons to discard the mean: on real financial data the mean is often close to 0; also, a non-centered standard deviation gives a more accurate estimated volatility, even compared to other type of models (see Figlewski (1994)).
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Arratia, A. (2014). Statistics of Financial Time Series. In: Computational Finance. Atlantis Studies in Computational Finance and Financial Engineering, vol 1. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-070-6_2
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DOI: https://doi.org/10.2991/978-94-6239-070-6_2
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