Statistical Analysis Of Financial Data 1st Edition James Gentle - Solutions

Unlock the potential of your understanding with our comprehensive solutions for the renowned "Statistical Analysis Of Financial Data 1st Edition" by James Gentle. Access online answers key and detailed solutions manual, including solutions PDF and solved problems. Dive into a rich reserve of questions and answers, complete with step-by-step answers and chapter solutions. Whether you're searching for a test bank or an instructor manual, our textbook resources are designed to enhance your learning experience. Explore these invaluable resources with the option for free download, ensuring you have the tools needed for academic success.

Suppose that $r_{1}, \ldots, r_{n}$ are observations of a return series that follows the AR(1)-GARCH(1,1) model\[ r_{t}=\mu+\phi_{1} r_{t-1}+a_{t}, \quad a_{t}=\sigma_{t} \epsilon_{t}, \quad \sigma_{t}^{2}=\alpha_{0}+\alpha_{1} a_{t-1}^{2}+\beta_{1} \sigma_{t-1}^{2} \]where $\epsilon_{t}$ is
In the equation in Exercise 3.3, assume that $\epsilon_{t}$ follows a standardized Student$t$ distribution with $v$ degrees of freedom. Derive the conditional log-likelihood function of the data.Exercise 3.3:Suppose that $r_{1}, \ldots, r_{n}$ are observations of a return series that
Consider the monthly simple returns of Intel stock from January 1973 to December 2008 in m-intc7308 . txt. Transform the returns into log returns. Build a GARCH model for the transformed series and compute 1-step- to 5-step-ahead volatility forecasts at the forecast origin December 2008.
The file m-mrk4608. txt contains monthly simple returns of Merck stock from June 1946 to December 2008. The file has two columns denoting date and simple return. Transform the simple returns to log returns.(a) Is there any evidence of serial correlations in the log returns? Use autocorrelations and
The file $m-3 m 4608$.txt contains two columns. They are date and the monthly simple return for $3 \mathrm{M}$ stock. Transform the returns to $\log$ returns.(a) Is there any evidence of ARCH effects in the log returns? Use Ljung-Box statistics with 6 and 12 lags of autocorrelations and \(5
The file m-gmsp5008.txt contains the dates and monthly simple returns of General Motors stock and the S\&P 500 index from 1950 to 2008.(a) Build a GARCH model with Gaussian innovations for the log returns of GM stock. Check the model and write down the fitted model.(b) Build a GARCH-M model with
Again, consider the returns in m-gmsp5008.txt.(a) Build a Gaussian GARCH model for the monthly log returns of the S\&P 500 index. Check the model carefully.(b) Is there a summer effect on the volatility of the index return? Use the GARCH model built in part (a) to answer this question.(c) Are
The file d-gmsp9908 . txt contains the daily simple returns of GM stock and the S&P composite index from 1999 to 2008. It has three columns denoting date, GM return, and S&P return.(a) Compute the daily log returns of GM stock. Is there any evidence of ARCH effects in the log returns? You may use
Consider the daily simple returns of the S&P composite index in the file d-gmsp9908.txt.(a) Is there any ARCH effect in the simple return series? Use 10 lags of the squared returns and $5 \%$ significance level to perform the test.(b) Build an adequate GARCH model for the simple return series.(c)
Again, consider the daily simple returns of GM stock in the file d-gmsp9908.txt.(a) Find an adequate GARCH-M model for the series. Write down the fitted model.(b) Find an adequate EGARCH model for the series. Is the "leverage" effect significant at the $5 \%$ level?
Revisit the file d-gmsp9908.txt. However, we shall investigate the value of using market volatility in modeling volatility of individual stocks. Convert the two simple return series into percentage log return series.(a) Build an AR(5)-GARCH(1,1) model with generalized error distribution for the log
Again, consider the percentage daily log returns of GM stock and the S&P 500 index from 1999 to 2008 as before, but we shall investigate whether the volatility of GM stock has any contribution in modeling the S\&P index volatility. Follow the steps below to perform the analysis.(a) Fit a
Consider the daily simple returns of Johnson & Johnson stock from January 1998 to December 2008. The data are in the file d-jnj9808.txt or can be obtained from CRSP. Convert the returns into log returns in percentage.(a) Build a GJR model for the log return series. Write down the fitted model. Is
Consider the monthly simple returns of General Electric (GE) stock from January 1926 to December 2008 with 996 observations. You may download the data from CRSP or use the file m-ge2608.txt on the Web. Convert the returns into log returns in percentages. Build a TGARCH model with GED innovations
Suppose that the monthly log returns of GE stock, measured in percentages, follow a smooth threshold IGARCH $(1,1)$ model. For the sampling period from January 1926 to December 2008, the fitted model is\[ \begin{aligned} r_{t} & =1.14+a_{t}, \quad a_{t}=\sigma_{t} \epsilon_{t} \\
Suppose that the monthly log returns, in percentages, of a stock follow the following Markov switching model:\[ \begin{aligned} r_{t} & =1.25+a_{t}, \quad a_{t}=\sigma_{t} \epsilon_{t}, \\ \sigma_{t}^{2} & = \begin{cases}0.10 a_{t-1}^{2}+0.93 \sigma_{t-1}^{2} & \text { if } s_{t}=1 \\ 4.24+0.10
Consider the monthly simple returns of GE stock from January 1926 to December 2008. Use the last three years of data for forecasting evaluation.(a) Using lagged returns $r_{t-1}, r_{t-2}, r_{t-3}$ as input, build a 3-2-1 feed-forward network to forecast 1 -step-ahead returns. Calculate the mean
Because of the existence of inverted yield curves in the term structure of interest rates, the spread of interest rates should be nonlinear. To verify this, consider the weekly U.S. interest rates of (a) Treasury 1-year constant maturity rate and (b) Treasury 3-year constant maturity rate. As in
The file ibm-d2-dur. txt contains the adjusted durations between trades of IBM stock on November 2, 1990. The file has three columns consisting of day, time of trade measured in seconds from midnight, and adjusted durations.(a) Build an EACD model for the adjusted duration and check the fitted
The file mmm9912-dtp. txt contains the transactions data of the stock of $3 \mathrm{M}$ Company in December 1999. There are three columns: day of the month, time of transaction in seconds from midnight, and transaction price. Transactions that occurred after 4:00 Pм Eastern time are excluded.(a)
Consider again the transactions data of 3M stock in December 1999.(a) Use the data to construct an intraday 5-minute $\log$ return series. Use the simple average of all transaction prices within a 5-minute interval as the stock price for the interval. Is the series serially correlated? You may
The file mmm9912-adur . txt contains an adjusted intraday trading duration of 3M stock in December 1999. There are thirty-nine 10-minute time intervals in a trading day. Let $d_{i}$ be the average of all $\log$ durations for the $i$ th 10 minute interval across all trading days in December
To gain experience in analyzing high-frequency financial data, consider the trade data of Boeing stock from December 1 to December 5, 2008. The data are in five files: taq-td-ba12012008.txt to taq-td-ba12052008.txt. Each file has five columns, namely hour, minute, second, price, and volume. Only
Again, consider the high-frequency data of Boeing stock from December 1 to December 5, 2008. Construct an intraday 5-minute return series. Note that the price of the stock in a 5-minute interval (e.g., 9:30 to 9:35 AM) is the last transaction price within the time interval. For simplicity, ignore
Consider the same problem as in Exercise 5.8, but use 10-minute time intervals.Exercise 5.8:Again, consider the high-frequency data of Boeing stock from December 1 to December 5, 2008. Construct an intraday 5-minute return series. Note that the price of the stock in a 5-minute interval (e.g., 9:30
Again, consider the high-frequency data of Boeing stock. Compute the percentage of consecutive transactions without price change in the sample.
Assume that the $\log$ price $p_{t}=\ln \left(P_{t}\right)$ follows a stochastic differential equation\[ d p_{t}=\gamma d t+\sigma d w_{t} \]where $w_{t}$ is a Wiener process. Derive the stochastic equation for the price $P_{t}$.
Considering the forward price $F$ of a nondividend-paying stock, we have\[ F_{t, T}=P_{t} e^{r(T-t)} \]where $r$ is the risk-free interest rate, which is constant, and $P_{t}$ is the current stock price. Suppose $P_{t}$ follows the geometric Brownian motion $d P_{t}=$ \(\mu P_{t} d
Assume that the price of IBM stock follows the Ito process\[ d P_{t}=\mu P_{t} d t+\sigma P_{t} d w_{t} \]where $\mu$ and $\sigma$ are constant and $w_{t}$ is a standard Brownian motion. Consider the daily log returns of IBM stock in 1997. The average return and the sample standard
Suppose that the current price of a stock is $\$ 120$ per share with volatility $\sigma=50 \%$ per annum. Suppose further that the risk-free interest rate is $7 \%$ per annum and the stock pays no dividend. (a) What is the price of a European call option contingent on the stock with a strike
Derive the limiting marginal effects of the five variables $K, P_{t}, T-t, \sigma$, and $r$ on a European put option contingent on a stock.
A stock price is currently $\$ 60$ per share and follows the geometric Brownian motion $d P_{t}=\mu P_{t} d t+\sigma P_{t} d t$. Assume that the expected return $\mu$ from the stock is $20 \%$ per annum and its volatility is $40 \%$ per annum. What is the probability distribution for the
A stock price is currently $\$ 60$ per share and follows the geometric Brownian motion $d P_{t}=\mu P_{t} d t+\sigma P_{t} d t$. Assume that the expected return $\mu$ from the stock is $20 \%$ per annum and its volatility is $40 \%$ per annum. What is the probability distribution for the
Suppose that the current price of stock A is $\$ 70$ per share and the price follows the jump diffusion model in Eq. (6.26). Assume that the risk-free interest rate is $8 \%$ per annum, the stock pays no dividend, and its volatility $(\sigma)$ is $30 \%$ per annum. In addition, the price on
Consider the European call option of a nondividend-paying stock. Suppose that $P_{t}=\$ 20, K=\$ 18, r=6 \%$ per annum, and $T-t=0.5$ year. If the price of a European call option of the stock is $\$ 2.10$, what opportunities are there for an arbitrageur?
Consider the put option of a nondividend-paying stock. Suppose that $P_{t}=$ $\$ 44, K=\$ 47, r=6 \%$ per annum, and $T-t=0.5$ year. If the European put option of the stock is selling at $\$ 1.00$, what opportunities are there for an arbitrageur?
Consider the daily stock returns of American Express (AXP), Caterpillar (CAT), and Starbucks (SBUX) from January 1999 to December 2008. Thedata are simple returns given in the file d-3stocks9908.txt (date, axp, cat, sbux).(a) Express the simple returns in percentages. Compute the sample mean,
Answer the same questions as in Exercise 1.1 but using monthly stock returns for General Motors (GM), CRSP value-weighted index (VW), CRSP equalweighted index (EW), and S\&P composite index from January 1975 to December 2008. The returns of the indexes include dividend distributions. Data file
Daily foreign exchange rates (spot rates) can be obtained from the Federal Reserve Bank in Chicago. The data are the noon buying rates in New York City certified by the Federal Reserve Bank of New York. Consider the exchange rates between the U.S. dollar and the Canadian dollar, euro, U.K. pound,
Suppose that the simple return of a monthly bond index follows the MA(1) model\[ R_{t}=a_{t}+0.2 a_{t-1}, \quad \sigma_{a}=0.025 \]Assume that $a_{100}=0.01$. Compute the 1 -step- and 2 -step-ahead forecasts of the return at the forecast origin $t=100$. What are the standard deviations of the
Suppose that the daily $\log$ return of a security follows the model\[ r_{t}=0.01+0.2 r_{t-2}+a_{t} \]where $\left\{a_{t}\right\}$ is a Gaussian white noise series with mean zero and variance 0.02 . What are the mean and variance of the return series $r_{t}$ ? Compute the lag- 1 and lag-2
Consider the monthly U.S. unemployment rate from January 1948 to March 2009 in the file m-unrate.txt. The data are seasonally adjusted and obtained from the Federal Reserve Bank of St Louis. Build a time series model for the series and use the model to forecast the unemployment rate for the April,
Consider the monthly simple returns of the Decile 1, Decile 2, Decile 9, and Decile 10 of NYSE/AMEX/NASDAQ based on market capitalization. The data span is from January 1970 to December 2008, and the data are obtained from CRSP.(a) For the return series of Decile 2 and Decile 10, test the null
Consider the daily simple returns of IBM stock from 1970 to 2008 in the file d-ibm3dx7008. txt. Compute the first 100 lags of ACF of the absolute series of daily simple returns of IBM stock. Is there evidence of long-range dependence? Why?
Consider the demand of electricity of a manufacturing sector in the United States. The data are logged, denote the demand of a fixed day of each month, and are in power6.txt. Build a time series model for the series and use the fitted model to produce 1- to 24-step-ahead forecasts.
Consider the daily simple returns of IBM stock, CRSP value-weighted index, CRSP equal-weighted index, and the S\&P composite index from January 1980 to December 2008. The index returns include dividend distributions. The data file is d-ibm3dxwkdays8008.txt, which has 12 columns. The columns are
Consider the data set of the previous question, but focus on the daily simple returns of the S&P composite index. Perform the necessary data analysis and statistical tests using the 5% significance level to answer the following questions:(a) Is there any weekday effect on the daily simple returns
Consider the weekly yields of Moody's Aaa and Baa seasoned bonds from January 5, 1962, to April 10, 2009. The data are obtained from the Federal Reserve Bank of St Louis. Weekly yields are averages of daily yields. Obtain the summary statistics (sample mean, standard deviation, skewness, excess
Consider the monthly Aaa bond yields of the prior problem. Build a time series model for the series.
Again, consider the two bond yield series, that is, Aaa and Baa. What is the relationship between the two series? To answer this question, build a time series model using yields of Aaa bonds as the dependent variable and yields of Baa bonds as independent variable.
Consider the monthly $\log$ returns of CRSP equal-weighted index from January 1962 to December 1999 for 456 observations. You may obtain the data from CRSP directly or from the file m-ew6299.txt on the Web.(a) Build an AR model for the series and check the fitted model.(b) Build an MA model for
This problem is concerned with the dynamic relationship between the spot and futures prices of the S&P 500 index. The data file sp5may. dat has three columns: $\log$ (futures price), $\log$ (spot price), and cost-of-carry $(\times 100)$. The data were obtained from the Chicago Mercantile
The quarterly gross domestic product implicit price deflator is often used as a measure of inflation. The file $q$-gdpdef.txt contains the data for the United States from the first quarter of 1947 to the last quarter of 2008. Data format is year, month, day, and deflator. The data are seasonally
Consider a random variable $X$ in the family of Poisson distributions with parameter $\lambda$. As $\lambda$ varies from 1 to 9 , both the mean and the variance of $X$ increase from 1 to 9 . Now consider the transformation $Y=\log (X+1)$. How do the mean and variance of $Y$ vary as
(a) Write the log-likelihood function for the Poisson distribution, given data $x_{1}, \ldots, x_{n}$, corresponding to the likelihood function in equation (4.26).\(L\left(\lambda ; x_1, \ldots, x_n\right)=\lambda^{\sum_{i=1}^n x_i} \mathrm{e}^{-n \lambda} \prod_{i=1}^n \frac{1}{x_{i}!}
Given a sample $x_{1}, \ldots, x_{n}$ from a $\mathrm{N}\left(\mu, \sigma^{2}\right)$ distribution, determine the method of moments estimate and the MLE of $\sigma^{2}$.
(a) Suppose we have a function $g$ of a random variable $X$ whose PDF is $f_{X}$. We wish to estimate $\mathrm{E}(g(X))$. The first consideration is whether or not $\mathrm{E}(g(X))$ exists. Let us assume that $g$ is such that $\mathrm{E}(g(X))$ exists and is finite. Another issue is
Obtain the daily adjusted closing prices of Intel (INTC) for the period January 1, 2017, through September 30, 2017, and compute the returns. These are the data used in the VaR, although because INTC pays dividends the adjusted closes may be slightly different.Fit a location-scale ("standardized")
To fit the linear model $y=X \beta+\epsilon$, Newton's method leads to the normal equations in one step, and the solution to the normal equations is the solution to the optimization problem. Show that the Gauss-Newton method for the linear least squares problem is the same as Newton's method.
Write out the gradient and the Hessian for the least squares estimation of the parameters in the Gompertz model\[ x_{i}=\alpha \mathrm{e}^{\beta \mathrm{e}^{\gamma t_{i}}}+\epsilon_{i} \]
By minimizing the weighted sum of squared residuals, derive the weighted least squares estimator for the linear model with $\mathrm{V}(\epsilon)=\Sigma$.
Show that the confidence interval (4.70)\[ \left(\bar{X}+t_{n-1,0.025} \sqrt{S^{2} / n}, \quad \bar{X}-t_{n-1,0.025} \sqrt{S^{2} / n}\right) \]is formed from the pivotal quantity $\left(\bar{X}, \sqrt{S^{2} / n}\right)$ as in equation (4.73).\(\left(\bar{X}+t_{n-1,0.025} \sqrt{S^2 / n}, \quad
This exercise requires computation of the individual quantities to test hypotheses and set confidence intervals for $\mu$ and $\sigma^{2}$ in an assumed $\mathrm{N}\left(\mu, \sigma^{2}\right)$ distribution.Given the small set of data $\{10,8,9,13,9,11\}$, do the following.(a) Set a \(90
Consider a hierarchical Bayesian model for a univariate normal distribution $\mathrm{N}\left(\mu, \sigma^{2}\right)$ in which the mean $\mu$ is a realization of a normal random variable $M$ and $\sigma^{2}$ is a scaled realization of a random variable $\Sigma^{2}$ that has an inverted
Obtain the INTC adjusted closing prices for the year 2018 and compute the daily returns. Now determine Bayes estimates of the mean and the standard deviation of the returns, similar to the example illustrated in Figures 4.2 through 4.4. For the prior on the mean use $\mathrm{N}(0,100)$, and for
Obtain the daily closing prices of the S\&P 500 Index for the year 2017 and compute the daily returns.(a) Compute the standard deviation (the volatility), the MAD, and the IQR of the returns.(b) Now produce two side-by-side histograms with normal probability density curves superimposed, where
(a) Let $f_{X}(x)$ be the PDF of a random variable $X$. Suppose that for some $x_{u}0$ and $\alpha>1$Show that the conditional PDF for $x \leq x_{u}$ is $f_{X \leq x_{u}}(x)=$ $\alpha\left|x_{u}\right|^{\alpha}|x|^{-(\alpha+1)}$.(b) Consider a random sample of size $n$ from the
Obtain the daily closes of the S\&P 500 from January 1, 1990, through December 31, 2017, and compute the daily log returns. These are the data used in Figure 4.6.Divide the S\&P 500 daily log returns into four periods, January 1, 1992, through December 31, 1995;January 1, 1997, through
Obtain the daily closing prices of the S\&P 500 Index for the year 2017 and compute the log returns.(a) Compute the Hill estimator of the lower tail index of the distribution of the daily returns of the index for that period, letting the tuning parameter range from 11 to 100.(b) Compute an estimate
Obtain the daily unadjusted closing prices of INTC, GLD, and the Nasdaq Composite for the year 2017 and compute the log returns. These data were shown in the displays in Figure 1.22.(a) Produce a scatterplot of the returns of INTC and versus the returns of the Nasdaq Composite, and another
Obtain the daily adjusted closing prices of Intel (INTC) for the period January 1, 2017, through September 30, 2017, and compute the returns. These are the data used in Exercise 4.5 and in the VaR examples, although because INTC pays dividends the adjusted closes may be slightly different.Assume a
Obtain the daily unadjusted closing prices of INTC and GLD and compute the log returns. Test the hypothesis that the correlation between the returns is 0.
Obtain the daily unadjusted closing prices of each of the 30 stocks in the Dow Jones Industrial Average for the year 2017. There were no stock splits during that period. (There is a text file linked on the website for the book that lists the stock symbols for the stocks in the DJIA during that
Show that for a model fit by least squares, the total sum of squares is equal to the model sum of squares plus the residual sum of squares.That is, show that\[
Consider the models\[ \begin{aligned} y & =\beta_{0,0}+\epsilon_{0} \\ y & =\beta_{0,1}+\beta_{1,1} x_{1}+\epsilon_{1} \\ y & =\beta_{0,2}+\beta_{1,2} x_{1}+\beta_{2,2} x_{2}+\epsilon_{2} \\ y & =\beta_{0,3}+\beta_{1,3} x_{1}+\beta_{2,3} x_{2}+\beta_{3,3} x_{3}+\epsilon_{3} \end{aligned} \]as in
Financial analysts using the common formulas such as the "market model", equation (1.35), or the simple formula for beta, equation (1.36), must make decisions about how to assign values to the terms in the formulas.To use historical data (is there any other way?) to assign values to the terms in
The SPY ETF is designed to track the S\&P 500 Index. Obtain the unadjusted monthly closing prices of SPY and the corresponding monthly closes of the S\&P 500 Index for the year 2017. Note that we should use either the adjusted closes of SPY and the S\&P 500 Total Return as in Figure
The SDS ETF is designed to produce returns twice the negative of the returns of the S\&P 500 Index. Obtain the weekly closes of SDS and of the S\&P 500 Index and compute the weekly returns for each. Note that for short ETFs, we focus on the returns, not on the prices. Plot the returns. Fit a simple
A q-q plot of regression residuals, as in Figure 4.13, is often a part of regression analysis.Why might a q-q plot of the residuals be of interest?What kind of regression residuals should be used in a q-q plot? Why?Figure 4.13: -0.15 Residuals -0.05 0.05 0.08 0.00 Cook's distance 0.04 0 o -0.15
Corporate bond yields and yield changes on US Treasury yields and yield changes.Obtain the weekly yields of Moody's seasoned Aaa corporate bonds and the weekly yields of 3-month Treasury bills for the years 2008 through 2018. (Obtain data from FRED.)(a) Regress the weekly yields of Moody's seasoned
Corporate bond yield changes on US Treasury yield changes.Obtain the weekly yields of Moody's seasoned Aaa corporate bonds, the yields of 3-month Treasury bills, the yields of 2-year Treasury notes, the yields of 10-year Treasury notes, and the yields of 30-year Treasury bonds for the years 2008
Define a dependent variable that takes the values "up" or "down", representing the direction of the price move of the S\&P 500 Index for the following day from the previous day. Now define regressors that represent the difference in the current (closing) price and the 50-day moving average and the
Obtain the closing prices of the S\&P 500 Index for 2017, and compute the simple returns.Test the returns for a normal distribution using the Anderson-Darling test, the Jarque-Bera test, and the Shapiro-Wilk test.
Generate samples of size 100 from four different t distributions, one each with 100 df, 20 df, 5 df, and 3 df. Also generate samples of size 100 from two different gamma distributions both with rate equal to 1 , one with shape equal to 5 and one with shape equal to 100.Now perform the
(a) It was stated that the LOOCV estimator $\mathrm{CV}_{(n)}$ may have large variance "because the individual terms have positive covariances".Why do they have positive covariances and why does that cause the variance of $\mathrm{CV}_{(n)}$ to be large?(b) It was stated that in $k$-fold
Let $\left\{x_{t}\right\}$ and $\left\{y_{t}\right\}$ be univariate random walk processes with $\operatorname{Cor}\left(x_{s}, y_{t}\right)=0$ for all $s$ and $t$. Now for a constant $a$, let $z_{t}=a x_{t}+y_{t}$. Show that $\left\{z_{t}\right\}$ is a random walk process.
Show that $h(x)=x_{1}^{2}+x_{2}$ is not linear. (Here, $x=\left(x_{1}, x_{2}\right)$ is a real 2 -vector.)
(a) Show that $\left(1-\mathrm{B}^{2}\right)\left(x_{t}\right)=(1-\mathrm{B})(1+\mathrm{B})\left(x_{t}\right)$.(b) Show that $\triangle^{k}\left(x_{t}\right)=\sum_{h=0}^{k}(-1)^{h}\binom{k}{h} x_{t-h}$ (equation (5.26)).(c) Show that for any integer
Consider a specific instance of the difference equation (5.33): $\left(1-a_1 \mathrm{~B}-\cdots-a_p \mathrm{~B}^p\right)\left(f_t\right)-a_0=0 . \tag{5.33}$\[ \left(1-3 \mathrm{~B}+2 \mathrm{~B}^{2}\right)\left(x_{t}\right)=0, \quad \text { for } t=2,3, \ldots \](a) What is the
Generate 100 observations in a random walk process starting at $x_{0}=10$ with a drift term of $\delta=2$.Use a white noise process with variance 16, as in Figure 5.2, but distributed as a multiple of a $t$ random variable with $4 \mathrm{df}$. (a) Plot the observations and fit a least
(a) Suppose $\left\{w_{t}\right\}$ is a white noise with constant variance $\sigma_{w}^{2}$. What is the variance of the process $\left\{\triangle\left(w_{t}\right)\right\}$ ?Why is this so?(b) Generate 100 realizations of a white noise sequence with a variance of 1 .Compute the variance of
Suppose that $\left\{x_{t}\right\}$ is an $\mathrm{I}(d)$ process for $d>k$, for some integer $k>1$.(a) Show that $\left\{x_{t}\right\}$ is an $\mathrm{I}(d+k)$ process(b) Show that $\triangle^{k}\left(\left\{x_{t}\right\}\right)$ is an $\mathrm{I}(d-k)$ process.
Show that the process in equation (5.110), $x_{t}=-\sum_{j=0}^{\infty} \phi_{1}^{-j} w_{t+j}$ where \(\left|\phi_{1}\right|
(a) Show that the ACF of a moving average of a white noise process $x_{t}=\frac{1}{k} \sum_{j=1}^{k} w_{t-j}$ is 0 outside the window of the moving average; that is, for $x_{s}$ and $x_{t}$ in a $k$-moving average of a white noise, $ho(s, t)=0$ if $|s-t| \geq k$.(b) Show that the ACF of
Consider a moving average of white noise, $x_{t}=\frac{1}{k} \sum_{j=1}^{k} w_{t-j}$, where the $w_{i}$ have independent $\mathrm{t}$ distributions with $4 \mathrm{df}$.Generate random samples of length 1,000 of this process with $k=1$ (which is just white noise) and with $k=2, k=4$,
Let\[ P_{t}=P_{0} \exp \left(r_{t}+\cdots+r_{1}\right) \]where the $r_{i}$ are distributed iid as $\mathrm{N}\left(\mu, \sigma^{2}\right)$. Show that $P_{t}$ is distributed as lognormal with parameters $t \mu+\log \left(P_{0}\right)$ and $t \sigma$. (If $\mu=0$, this is a lognormal
Suppose $x_{t}$ is a time series with a linear and a quadratic trend, with white noise superimposed:\[ x_{t}=\beta_{0}+\beta_{1} t+\beta_{2} t^{2}+w_{t}, \quad \text { for } t=1,2, \ldots \]where $\beta_{0}, \beta_{1}$, and $\beta_{2}$ are constants, and $w_{t}$ is a white noise with
Let $x_{t}$ be a time series with linear and quadratic trend.\[ x_{t}=\beta_{0}+\beta_{1} t+\beta_{2} t^{2}+w_{t}, \quad \text { for } t=1,2, \ldots \]where $\beta_{0}, \beta_{1}$, and $\beta_{2}$ are constants, and $w_{t}$ is a white noise with variance $\sigma^{2}$.Let
For the jointly stationary bivariate process $\left\{\left(x_{t}, y_{t}\right)\right\}$, show that\[ \gamma_{x y}(h)=\gamma_{y x}(-h) \]and\[ ho_{x y}(h)=ho_{y x}(-h) \]
Obtain the closing prices of MSFT and the S\&P 500 Index for the period January 1, 2017, through December 31, 2017, and compute the daily log returns for this period.(a) Compute and plot the ACFs of the returns of MSFT and of the S\&P 500 Index for this period.(b) Test the hypothesis that the ACF

Showing 100 - 200 of 277

Statistical Analysis Of Financial Data 1st Edition James Gentle - Solutions

Step by Step Answers