onsider the time series xt = β1 + β2t + wt, where β1 and β2 are known constants and wt is a white noise process with variance σ2 w. (a) Determine whether xt is stationary. (b) Show that the process yt = xt − xt−1 is stationary. (c) Show that the mean of the moving average vt = 1 2q + 1 q j=−q xt−j is β1 + β2t, and give a simplified expression for the autocovariance function.
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Consider the time series xt = β1 + β2t + wt, where β1 and β2 are known constants and wt is a white noise process with variance σ2 w. (a) Determine whether xt is stationary. (b) Show that the process yt = xt − xt−1 is stationary. (c) Show that the
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- Suppose that the response y is generated by y = f(x) + €, where e is a zero-mean Gaussian noise with variance 1. a) Suppose that f(x) = x. Randomly generate 10 x's and generate the corresponding y's; you need to generate two random numbers (i.e., x and e for each of the 10 points). Fit the data with linear regression and plot the scatter points. b) Suppose that f(x) = x². Randomly generate 10 (x, y) pairs. Fit the data with linear regression and plot the scatter points. c) Suppose that f(x) = 1/x. Randomly generate 10 (x, y) pairs. Fit the data with linear regression and plot the scatter points.Given (x + 2y) ,0Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.Compute E(310) for the following model, where & wn(0, 0.16), i.e., a white noise process with mean zero and variance 0.16. Please give the exact answer. 1 Me=Mt-2+ Ct, 30 = 1.There are 3000 stocks in the long-short stock market, which will be traded for 250 days in the next year. Set X and Y to be the daily holding amount of each stock And the return of the next day, and X and Y are respectively sampled from the normal distribution of N (0.100000NO, 2E-2). Use enough data to estimate Calculate the R2 of X and Y linear regression to be 001. Assume that positions among stocks are not correlated, and returns between stocks are not correlated. Define a day’s PNL as the day’s receipt beneficial PNL= 3000∑xy Find the expectation of PN and the sharpe rate of PNL every dayA time series {yt} follows an MA(2) model: Yt = 2 + Ut +0.54t-1 + 0.4ut-2. Assume that ut is a white noise series with a mean of O and a variance of 2. Please calculate Var(yt) (i.e. the variance of) 2.96 1.41 2.82 O 1.98Given the raw data for shipping cost (in Php): 2, 4, 4, 5, 8, 8, 10, 13, 19, 19, 28, 35; smooth the data by applying Equal-Depth Binning. * 6.75, 6.75, 6.75, 6.75, 6.75, 6.75, 19, 19, 31.5, 31.5 3.75, 3.75, 3.75, 3.75, 9.75, 9.75, 9.75 ,9.75, 25.25, 25.25, 25.25, 25.25 6.75, 6.75, 6.75, 3.75, 3.75, 9.75, 9.75, 19, 19, 25.25, 25.25, 31.25 3.75, 3.75, 6.75, 6.75, 9.75, 9.75, 19, 19, 25.25, 25.25, 31.25, 31.252) Let X₁, X2, ..., Xn be a random sample from the pdf f(x;0) = 0xª−¹, 0≤x≤ 1,0 <0 < ∞. Find the MLE of 0 and show that its variance approaches 0 as n approaches ∞o.1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.Let Y < Y, < Y3 be the order statistic of a random sample of size 3 from the uniform distribution having pdff (x;0) = 1/0,0Consider an MA(2) model: STEPS -Write the equation of the M(2) model-Determine the model based on the delay operator.- Find the expected value of the model- Find the variance of the model.- Find the covariances associated with 1,2, and s steps.- Find the associated correlation indices of 1,2, and s steps.- Assume that information is available up to time $h$ and the function that contains the accumulated information f(h, h-1,……), determine the forecasts and the error associated with 1,2, and s steps.(b) Consider the generalised three-parameter beta distribution with pdf 121?x (1 – x)² [1- (1 – 2) x]5* fx (x) = 0Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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