STA457H5S, Winter 2023 Applied Time Series Analysis Assignment 2
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Department of Mathematical and Computational Sciences
Applied Time Series Analysis
STA457H5S, Winter 2023
Assignment 2
Due Tuesday March 21, 2023 at 11:59 pm
Problem 1 (10 pts). Suppose that {Xt } is a stationary process with mean u. Show that the best linear predictor Xˆt+l of Xt+l follows the form aXt + b, where a = pl and b = u(1 _ pl ).
Problem 2 (20 pts). Consider the model
Xt = Xt − 1 + cXt −2 + wt
where {wt }~wn(0, 7⑪(2)) and c e R is a constant.
a) Show that the process is stationary provided _1 < c < 0.
b) Find the ACF and PACF of {Xt } when c = _3/16.
Solve the following problems assuming c = _3/16.
c) Find the one- and two-step ahead forecasts of this time series.
d) Derive a recursive relation to make three or more steps ahead forecasts.
e) Find the variance of three-steps-ahead forecast errors.
f) If zt = 6, zt − 1 = 4, wt = 1 and 7⑪(2) = 1.2, obtain 
t (2) and the standard error of its corresponding forecast error.
Problem 3 (20 pts). Consider the ARIMA(0,1,1) process
(1 _ B)Xt = (1 _ 9B)wt
where {wt }~wn(0, 7⑪(2)).
a) Show that
t (1) = zt _ 9wt
and
t (l) = 
t (l _ 1) for l ≥ 2.
b) Express 
t (1) in terms of zt and 
t − 1 (1) using T-weights.
c) By considering the w-weights of the process, show that the variance of l-steps-ahead prediction error is [1 + (l _ 1)(1 _ 9)2]7⑪(2) .
d) Discuss the difference between Confidence Intervals (CI) and Prediction Intervals (PI). Thereafter, derive a 100(1 _ a)% Prediction Interval for Xt+l .
Problem 4 (50 pts). The provided dataset contains is a univariate time series of monthly infectious disease uncertainty index from January 1985 - December 2022. This index quantifies the degree of uncertainty in government policies regarding infectious diseases, which is constructed by text mining and extracting the daily count of newspaper articles that contain at least one term from the following set of terms across more than 3000 U.S. newspapers:
1. {epidemic, pandemic, virus, flu, disease, coronavirus, mers, sars, ebola, H5N1, H1N1}
2. {volatility, volatile, uncertain, uncertainty, risk, risky}
3. {economic, economy, financial}
4. {equity, equities, ”Standard and Poors”}
By adhering to the Box-Jenkins framework, perform each of the following steps to build an appropriate model for the given time series:
1. Model formulation (or model specification)
2. Model estimation (or model fitting)
3. Model checking (or model diagnostics)
Employ a proper set of graphical and computational tools to analyze the data at each stage. In case you believe the given time series is a nonstationary one, you may use differencing or any other transformation technique for converting the time series into a stationary one and then fitting an ARMA model to it. Note that if you decide differencing is required, you need to perform the ADF unit root test and KPSS stationarity tests to avoid over-differencing.
Please demonstrate all the steps of your work and present the results by following the steps depicted schematically on the posted Box-Jenkins diagram.
2023-03-21