# Mathematics Time Series and Spectral Analysis Worksheet

Description

– Double check example questions attached before making an offer, the worksheet questions are difficult.- 80%+ grade needed.- Answer questions on paper or in word.- Notes and exercises will be provided.Topics included:1. Overview. Stationarity, outline of Box-Jenkins approach through identification of model, fitting, diagnostic checking, and forecasting. Mean, autocorrelation function, partial autocorrelation function.

2. Models. Autoregressive (AR) models, moving average (MA) models, ARMA models, their autocorrelation functions, and partial autocorrelation functions. Transformations and differencing to achieve stationarity, ARIMA models.

3. Estimation and diagnostics. Identifying possible models using autocorrelation function, and partial autocorrelation function. Estimation, outline of maximum likelihood, conditional and unconditional least squares approaches. Diagnostic checking, methods and suggestions of possible model modification.

4. Forecasting. Minimum mean square error forecast and forecast error variance, confidence intervals for forecasts, updating forecasts, other forecasting procedures.

5. Seasonality, time series regression.

6.The frequency representation of a stationary time series.

7. The use of a periodiogram to carry out harmonic analysis.

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Module Code: MATH580201
1. Let {Xt } be a stochastic process defined for t 2 Z by
Xt =
p
X
i=0
↵ i “t
i+
q
X
i t i,
i=0
where {“t } and { t } are mutually independent normally distributed white noise processes
with finite variances “2 and 2 respectively. Assume that p, q > 0 and that ↵p , q are
non-zero.
(a) LEEDS5802 Is the process {Xt } strongly stationary, weakly stationary, or not stationary? Justify your answer.
(b) LEEDS5802 Find an expression for the autocovariance function of {Xt } for general
values of p, q, ↵i and i .
Now compute the numerical values of the autocorrelation function if p = 1, q = 2,
↵0 = 1, ↵1 = 1/2, 0 = 1, 1 = 1/2, 2 = 1/4, and “2 = 2 = 2.
(c) LEEDS5802 Show that if p = q = p0 and ↵i = i for i = 0, . . . , q, then {Xt } is a
moving average process, and write down a definition of this process.
(d) LEEDS5802 For the moving average process in (c), assume 2 = 0, explain briefly
what you understand by invertibility. Why it is a desirable property?
(e) LEEDS5802 Let Xt = “t + 1 “t 1 + 0.5″t 2 . Then give two values of 1 , one of
which gives an invertible MA(2) process and one of which gives a non-invertible
MA(2) process. Explain why your values of 1 have the desired properties.
(f) LEEDS5802 Consider the moving average process
X t = “t + “t 1 .
By writing “t in terms of Xt , Xt 1 , Xt 2 , …, verify that Xt is not invertible.
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Module Code: MATH580201
2. (a) LEEDS5802 Define the partial autocorrelation coefficient ↵kk of a stochastic process {Xt }. State the autocorrelation ⇢1 and ⇢2 , and hence find ↵11 and ↵22 , for

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