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# 数学代写|随机过程Stochastic Porcesses代考|STAT507 Data

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## 数学代写|随机过程代写Stochastic Porcesses代考|Data

The task of setting up an appropriate dataset for my purposes involves decisions on the commodities to include, whether to use short-term future contracts or spot price data, what time period to analyse, and what data frequency to use.

Following Marshall, Nguyen, and Visaltanachoti (2012), I focus on the components of the S\&P Goldman Sachs Commodity Index (S\&P GSCI). As this index comprises 24 important commodities spanning energy, industrial metal, precious metal, agriculture, and livestock, it suits my desire to obtain conclusions of preferably broad, cross-sectoral applicability and to account for the heterogeneity of different commodities (Kat \& Oomen, 2007).

With respect to the type of price quote to use, I would generally prefer to use spot instead of future prices for my purposes, however, Fama and French (1987) suggest that “good spot-price data are not available for most commodities” (p. 57) and Schwartz (1997) raises similar concerns about their liquidity. As a result, these authors propose the use of short-term futures data instead. Since this view is not shared by Brooks and Prokopczuk (2013) and would generally complicate the analysis and interpretation of results throughout the paper, I conjecture that in the wake of “tremendous growth in commodity […] markets” (Tsekrekos et al., 2012 , p. 543), these earlier findings are not necessarily reflective of spot price data quality in the more recent past. ${ }^9$ Since the use of justifiable data can be pivotal for the results of any empirical study (Gujarati, 2003), I feel it is necessary to assess the relative data quality in spot and futures markets based on a short, back-of-theenvelope liquidity calculation.

## 数学代写|随机过程代写Stochastic Porcesses代考|Empirical analysis

As a first step to a better understanding of commodity price dynamics, it is useful to exploit the rich set of information contained in historical price series via a range of econometric and statistical tests. Although, one may be sceptical that a backward looking analysis can yield valuable insights for the decision of how to model seemingly random price movements in the future, more than half a century of empirical studies have revealed that the statistical properties of such price variations are indeed common across numerous asset classes in different markets and time periods (Cont, 2001; Mantegna \& Stanley, 2000). For my purposes, it is logical to split the empirical analysis into two main blocks. First, it is analysed whether commodity prices are mean-reverting. The answer to this question is a corner stone in this thesis, as it resembles a central assumption to distinguish between stochastic processes. Second, we will assess whether commodity returns are normally distributed.

Following from the previous discussion, the question of mean-reverting prices is the principal distinctive assumption between commonly used GBM and meanreverting processes. Broadly speaking, a stochastic process is mean-reverting (interchangeably we may refer to such a process as stationary or one without unit root) if its mean and variance are constant over time and the covariance between two observations depends only on the lag between them but not on the time when it is computed (Gujarati, 2003). ${ }^{11}$ On the contrary, a non-stationary price process is characterised by a growing and unlimited variance over time that allows prices to rise without bound. To see more explicitly how the property of mean reversion enters a stochastic process let us consider
$$S_t=\theta S_{t-1}+\varepsilon_t,$$
where $S_t$ denotes, for instance, the price of a commodity at time $t, \theta$ the autoregressive coefficient, and $\varepsilon_t$ a white noise process ${ }^{12}$. In this framework, the absolute value of $\theta$ is governing the mechanics of mean reversion. In particular, we distinguish between two cases. First, the non-stationary Random Walk (RW) process, where $|\theta|=1$ and, second, the stationary $\operatorname{AR}(1)$ process where $|\theta|<1$. ${ }^{13}$ As a deeper understanding of the link between $\theta$, process variance, and stationarity is very beneficial for subsequent statistical testing, let us study these cases in greater detail. To begin with, it is useful to apply repeated backward substitution to rewrite our expression for $S_t$ as
$$S_t=\theta\left[\theta\left[\theta S_{t-3}+\varepsilon_{t-2}\right]+\varepsilon_{t-1}\right]+\varepsilon_t=\theta^3 S_{t-3}+\theta^2 \varepsilon_{t-2}+\theta \varepsilon_{t-1}+\varepsilon_t$$

## 数学代写|随机过程代写Stochastic Porcesses代考|Empirical analysis

$$S_t=\theta S_{t-1}+\varepsilon_t,$$

$$S_t=\theta\left[\theta\left[\theta S_{t-3}+\varepsilon_{t-2}\right]+\varepsilon_{t-1}\right]+\varepsilon_t=\theta^3 S_{t-3}+\theta^2 \varepsilon_{t-2}+\theta \varepsilon_{t-1}+\varepsilon_t$$

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