Consider two time series, {𝑥𝑡}𝑡=1𝑇 and {𝑦𝑡}𝑡=1𝑇, one generated using AR(1) and the other using MA(1), as follows:𝑥𝑡=𝛼+𝛽𝑥𝑡−1+𝜀𝑡, and𝑦𝑡=𝜇+𝜀𝑡+𝜃𝜀𝑡−1.Suppose 𝛽=𝜃=0.5.Based on the provided information, we can claim that:Group of answer choicesNone of the presented answers are correct.The first order autocorrelation of the series following the presented AR model is larger than the first order autocorreation of the series following the presented MA model.The autocorrelation functions of the two models are identical for lags greater than one.The second order autocorrelation of the series following the presented AR model is equal to zeroThe two step-ahead forecast of the series following the presented AR model is equal to the unconditional mean of the series.

AR(1) is equivalent to MA(1) when the autoregressive parameter is equal to 1.Group of answer choicesTrueFalse

Consider a series that follows an MA(1) with zero mean and a moving average coefficient of 0.4. What is the value of the autocorrelation function at lag 1?Group of answer choices0.341It is not possible to determine the value of the autocovariances without knowing the disturbance variance0.4

Suppose that the monthly log return of a security 𝑟" follows the MA(1) model𝑟" = 𝑎" + 0.4𝑎"'& ,where {𝑎" } is a Gaussian white noise series with mean zero and variance 0.03.(a) Compute the mean and variance of the return series. (6 points)(b) Compute the lag-1 and lag-2 autocorrelations of the return series. (10 points)(c) Assume that 𝑎&$$ = 0.5. Compute the 1-step- and 2-step-ahead forecasts of the return at theforecast origin 𝑡 = 100. (8 points)(d) What are the standard deviations of the associated forecast errors? (6 points)

In the context of time series analysis, if the autocorrelation function 𝑅𝑥𝑥(𝜏) exhibits a sharp peak at a specific lag 𝜏=𝑘, what does this suggest about the time series?Select one:a.The time series is highly correlated with its lagged version at lag 𝑘.b. The time series is non-stationary.c.The time series is periodic with period 𝑘.d. The time series is white noise.

If you were to model a financial variable a lot of memory (autocorrelation never zero for any lags), as a time series, how would you choose between an AR(p) and MA(q) process? And which p or q ?