统计代写|应用时间序列分析代写applied time series analysis代考|Likelihood Function of Returns

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

统计代写|应用时间序列分析代写applied time series anakysis代考|Likelihood Function of Returns

The partition of Eq. (1.15) can be used to obtain the likelihood function of the log returns $\left{r_{1}, \ldots, r_{T}\right}$ of an asset, where for ease in notation the subscript $i$ is omitted from the log return. If the conditional distribution $f\left(r_{t} \mid r_{t-1}, \ldots, r_{1}, \theta\right)$ is normal with mean $\mu_{t}$ and variance $\sigma_{t}^{2}$, then $\boldsymbol{\theta}$ consists of the parameters in $\mu_{t}$ and $\sigma_{t}^{2}$ and the likelihood function of the data is
$$f\left(r_{1}, \ldots, r_{T} ; \boldsymbol{\theta}\right)=f\left(r_{1} ; \boldsymbol{\theta}\right) \prod_{t=2}^{T} \frac{1}{\sqrt{2 \pi} \sigma_{t}} \exp \left[\frac{-\left(r_{t}-\mu_{t}\right)^{2}}{2 \sigma_{t}^{2}}\right]$$
where $f\left(r_{1} ; \theta\right)$ is the marginal density function of the first observation $r_{1}$. The value of $\boldsymbol{\theta}$ that maximizes this likelihood function is the maximum likelihood estimate (MLE) of $\theta$. Since log function is monotone, the MLE can be obtained by maximizing the log likelihood function,
$\ln f\left(r_{1}, \ldots, r_{T} ; \theta\right)=\ln f\left(r_{1} ; \theta\right)-\frac{1}{2} \sum_{t=2}^{T}\left[\ln (2 \pi)+\ln \left(\sigma_{t}^{2}\right)+\frac{\left(r_{t}-\mu_{t}\right)^{2}}{\sigma_{t}^{2}}\right]$
which is casier to handle in practice. Log likelihood function of the data can be obtained in a similar manner if the conditional distribution $f\left(r_{t} \mid r_{t-1}, \ldots, r_{1} ; \theta\right)$ is not normal.

统计代写|应用时间序列分析代写applied time series anakysis代考| Empirical Properties of Returns

The data used in this section are obtained from the Center for Research in Security Prices (CRSP) of the University of Chicago. Dividend payments, if any, are included in the returns. Figure $1.2$ shows the time plots of monthly simple returns and log returns of International Business Machines (IBM) stock from January 1926 to December 1997. A time plot shows the data against the time index. The upper plot is for the simple returns. Figure $1.3$ shows the same plots for the monthly returns of value-weighted market index. As expected, the plots show that the basic patterns of simple and log returns are similar.

Table $1.2$ provides some descriptive statistics of simple and log returns for selected U.S. market indexes and individual stocks. The returns are for daily and monthly sample intervals and are in percentages. The data spans and sample sizes are also given in the table. From the table, we make the following observations. (a) Daily returns of the market indexes and individual stocks tend to have high excess kurtoses. For monthly series, the returns of market indexes have higher excess kurtoses than individual stocks. (b) The mean of a daily return series is close to zero, whereas thatof a monthly return series is slightly larger. (c) Monthly returns have higher standard deviations than daily returns. (d) Among the daily returns, market indexes have smaller standard deviations than individual stocks. This is in agreement with common sense. (e) The skewness is not a serious problem for both daily and monthly

returns. (f) The descriptive statistics show that the difference between simple and log returns is not substantial.

Figure $1.4$ shows the empirical density functions of monthly simple and log returns of IBM stock. Also shown, by a dashed line, in each graph is the normal probability density function evaluated by using the sample mean and standard deviation of IBM returns given in Table 1.2. The plots indicate that the normality assumption is questionable for monthly IBM stock returns. The empirical density function has a higher peak around its mean, but fatter tails than that of the corresponding normal distribution. In other words, the empirical density function is taller, skinnier, but with a wider support than the corresponding normal density.

统计代写|应用时间序列分析代写applied time series anakysis代考|PROCESSES CONSIDERED

Besides the return series, we also consider the volatility process and the behavior of extreme returns of an asset. The volatility process is concerned with the evolution of conditional variance of the return over time. This is a topic of interest because, as shown in Figures $1.2$ and $1.3$, the variabilities of returns vary over time and appear in

clusters. In application, volatility plays an important role in pricing stock options. By extremes of a return series, we mean the large positive or negative returns. Table $1.2$ shows that the minimum and maximum of a return series can be substantial. The negative extreme returns are important in risk management, whereas positive extreme returns are critical to holding a short position. We study properties and applications of extreme returns, such as the frequency of occurrence, the size of an extreme, and the impacts of economic variables on the extremes, in Chapter $7 .$

Other financial time series considered in the book include interest rates, exchange rates, bond yields, and quarterly earning per share of a company. Figure $1.5$ shows the time plots of two U.S. monthly interest rates. They are the 10-year and 1-year Treasury constant maturity rates from April 1954 to January 2001. As expected, the two interest rates moved in unison, but the l-year rates appear to be more volatile. Table $1.3$ provides some descriptive statistics for selected U.S. financial time series. The monthly bond returns obtained from CRSP are from January 1942 to December 1999. The interest rates are obtained from the Federal Reserve Bank of St Louis. The weekly 3 -month Treasury Bill rate started on January 8, 1954, and the 6-month rate started on December 12, 1958. Both series ended on February 16, 2001. For the interest rate series, the sample means are proportional to the time to maturity, but the sample standard deviations are inversely proportional to the time to maturity. For

the bond returns, the sample standard deviations are positively related to the time to maturity, whereas the sample means remain stable for all maturities. Most of the series considered have positive excess kurtoses.

With respect to the empirical characteristics of returns shown in Table $1.2$, Chapters 2 to 4 focus on the first four moments of a return series and Chapter 7 on the behavior of minimum and maximum returns. Chapters 8 and 9 are concerned with moments of and the relationships between multiple asset returns, and Chapter 5 addresses properties of asset returns when the time interval is small. An introduction to mathematical finance is given in Chapter 6 .

统计代写|应用时间序列分析代写applied time series anakysis代考|Likelihood Function of Returns

F(r1,…,r吨;θ)=F(r1;θ)∏吨=2吨12圆周率σ吨经验⁡[−(r吨−μ吨)22σ吨2]

ln⁡F(r1,…,r吨;θ)=ln⁡F(r1;θ)−12∑吨=2吨[ln⁡(2圆周率)+ln⁡(σ吨2)+(r吨−μ吨)2σ吨2]

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