(20 points, each for 4 points) This question is about the larger VIX data set {vixlarge.mat} (this is a matlab format file, you can load it by installing package “R.matlab” and use command rmat)that contains the VIX data and the associated dates. introduce and investigate in this proposal. The CBOE VIX is colloquially referred to as the “fear index” or the “fear gauge”. We choose to study the VIX not only on the widespread consensus that the VIX is a barometer of the overall market sentiment as to what concerns investors’ risk appetite, but also on the fact that there are many trading strategies that rely on the VIX index for hedging and speculative purposes.
Plot the VIX data against date. Clearly label the horizontal and vertical axises.
We know that volatility (\(y_{t}\) hearafter) exhibits a high degree of persistence and it’s likely that \(y_{t}\) is better forecast by using more lags, \(y_{t-1}, y_{t-3}, \ldots .\) That makes us think of the model with \(J\) lags: \[ y_{t}=\beta_{0}+\beta_{1} y_{t-1}+\beta_{2} y_{t-2}+\cdots+\beta_{J} y_{t-J}+u_{t} \] where \(u_{t}\) is the error term. But to capture long-range dependence might entail \(J=10\) \(J=20,\) or higher. Let the dependent variable \(y\) be the VIX and the first and the 2nd to 23th columns of the independent variable \(X\) be the intercept term and the 1-22 lag of VIX. Write your own code to imlpement AR(1) to AR(22) model, and pick out the best model by AIC and BIC, are the results same, genarate a table to illustrate your results? if not, why?
Set the window length at 3000 and make forecast on the next period \(y_{t+1}\), start from the beginning and roll until the end. For each roll, we make forecast using AR(1) to AR(22). Compute the mean squared forecast errors and the mean absolute forecast errors for AR(1) to AR(22) and report them in a table.
In (b) and (c), estimating such a large number of coefficients could entail a lot of estimation error and lead to bad forecasting properties. Following Fernandes, Medeiros, and Scharth (2014), a very popular way to model the VIX index is the heterogeneous autoregressive (HAR) model by Corsi (2009) . The HAR model gains great popularity not only because the HAR model well approximates long memory and multiscaling properties of the VIX index, but it is also very easy to implement in practice. The standard HAR model in Corsi (2009) postulates that \(h\) -step-ahead daily volatility \(y_{t+h}\) can be modeled by \[ y_{t+h}=\beta_{0}+\beta_{d} \bar{y}_{t}^{(1)}+\beta_{w} \bar{y}_{t}^{(5)}+\beta_{m} \bar{y}_{t}^{(22)}+\epsilon_{t+h} \] where we define \[ \bar{y}_{t}^{(l)} \equiv l^{-1} \sum_{s=1}^{l} y_{t-s} \] as the averages of the previous \(l\) periods of \(y\) from period \(t\) and \(\left\{\epsilon_{t}\right\}\) is a zero mean innovation process. A typical choice in the literature for the lag index vector \(l\) is \([1,5,22]\) so as to mirror the daily, weekly, and monthly components of volatility process. Using HAR model to do forcasting exercise in (b) and (c) and comparing HAR model with the best AR model in (b) and (c) by AIC, BIC and rolling method, which one is the best?
Try to come up with an algorithm that can beat the best performing method stated in question (d). Clearly describe your motivation, the details of the algorithm, and the results.
(18 points, each for 3 points) (Commodity prices). Consider the daily gold price, London Bullion Market, price per Troy Ounce in U.S. Dollars at 10: 30 AM local time, from January \(2,1992\) to March 31,2015. See file GoLDLBM1030. txt.
Obtain the time plot of the gold price.
Let \(r_{t}\) be the log return of the daily gold price. Obtain the time plot of \(r_{t}\)
Are there serial correlations in the \(r_{t}\) series? You may use \(Q(10)\) to draw the conclusion.
Build an AR model for \(r_{t} .\) Check the adequacy of the model.
Remove any parameter of the AR model with \(t\) -ratio less than 1.645 in absolute value. Write down the file model.
Use the final model to compute 1 -step to 3 -step ahead forecasts of \(r_{t}\) at the forecast origin March \(31,2015\)
Soution: First, we write a function to estimate \(\phi\).
Now,we can do simulation with the function
(32 points, each for 4) If two asset returns \(R_{1, t}\) and \(R_{2, t}\) have correlation \(\rho\) and time varying volatility \(\sigma_{1, t}\) and \(\sigma_{2, t}\)，then their covariance (GARCH covariance) is:
\[ \sigma_{12, t}=\rho \sigma_{1, t} \sigma_{2, t}, \] where we assume the \(\rho\) is a constant. In the following, estimate \(\rho\) between two series.
## [1] "MSFT"## [1] "AMZN"## [1] "GSPC"(30 points, each for 5 points)
移动平均线(MA)是股市中最常用的一种技术分析方法，用来在大行情的波动段找到有效的交易信号。移动平均线不仅简单，而且有效。据金融从业人员称，均线模型能有效地打败大部分的主观策略，是炒股、炒期货的必备基本工具。均线系统市股票市场技术分析的重要组成部分。技术分析其实的核心是统计学，通过对过往历史价格的统计和形成的统计图表来做出对未来走势的预期，并针对预期来制定交易计划。
移动平均线 移动平均线(MA,Moving average）是以道·琼斯的”平均成本概念”为理论基础，采用统计学中”移动平均”的原理，将一段时期内的股票价格平均值连成曲线，用来显示股价的历史波动情况，进而反映股价指数未来发展趋势的技术分析方法。它是道氏理论的形象化表述。
移动平均线的计算方法就是求连续若干天的收盘价的算术平均。天数就是MA的参数。在技术分析领域中，移动平均线是必不可少的指标工具。移动平均线利用统计学上的“移动平均”原理，将每天的市场价格进行移动平均计算，求出一个趋势值，用来作为价格走势的研判工具。
计算公式： MA = (C1+C2+C3+C4+C5+….+Cn)/n ,C为收盘价，n为移动平均周期数。
移动平均线依时间长短可分为三种，即短期移动平均线，中期移动平均线，长期移动平均线。短期移动平均线一般以5日或10日为计算期间，中期移动平均线大多以30日、60日为计算期间；长期移动平均线大多以100天和200天为计算期间。移动均线平滑了数据序列，并有助于识别股市的发展趋势。n值越大，移动均线就越难反映序列中的短期波动，但也更好的把握了整体的趋势。