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MAS328 Time Series

School of Mathematical Sciences

Key Objectives

Simple descriptive Techniques

- Understand and be able to point out the important features of the TS plot from the modelling point of view.

Trend and seasonal components

- Understand a TS model with deterministic trend and seasonality and a stochastic component.

- Know the detrending methods (Least Squares regression, differencing, moving average) in the absence of seasonality.

- Be able to calculate a linear filter for removing a polynomial trend of order 1, 2 and 3 from a TS data set.

- Know the methods of eliminating both trend and seasonality (small trend method, classical decomposition, differencing).

- Know the difference operator.

Random Variables and Their Distributions

- Know the definition of a random variable, probability distribution function and cumulative probability function.

- Be able to calculate expectation and variance for simple discrete and continuous random variables. Know the properties of expectation.

- Understand the notion of two-dimensional discrete and continuous random variables and their joint distribution functions.

- Understand the notion of conditional distribution and independence of random variables.

- Know the definitions of covariance and correlation of two random variables and their properties.

- Understand the notion of the distribution of a multivariate random variable.

- Know the definition of a Gaussian Time Series.

Stationary Time Series Models

- Know the definitions of weak stationarity, autocovariance and autocorrelation functions (ACVF and ACF).

- Know the definitions of a white noise process and of an IID process and understand the difference between the two processes.

- Know the definitions of the sample ACVF and sample ACF.

- Know the definition of strict stationarity and the properties of a strictly stationary time series.

- Be able to prove that weak stationarity of a time series does not imply strict stationarity, but strict stationarity does imply weak stationarity.

- Know the definition of a linear process and the form of its autocovariance function.

- Know the definition of the Moving Average model of general order q, MA(q), of the Autoregressive model of a general order p, AR(p),

and of the Autoregressive Moving Average model ARMA(p,q).

- Be able to calculate the autocovariance and autocorrelation functions for MA(1) and MA(2), AR(1) and ARMA(1,1).

- Know the definitions of invertibility and causality of a TS process.

- Be able to obtain an AR representation of an MA(1).

- Be able to calculate the region of admissible parameter values for AR(1) and AR(2) to be causal.

- Show that ARMA(1,1) is invertible and causal if its AR and MA parameters are both in the interval (-1, 1).

- Understand the notion of parameter redundancy in ARMA(p,q) models and be able to check it in low order ARMA models.

- Understand the backshift operator B and be able to write an ARMA model using the B-notation.

- Know the notion of polynomials associated with ARMA models and the notion of homogeneous equations.

- Be able to calculate the autocorrelation function of an AR(1) and AR(2) processes using homogeneous difference equations.

- Know the definition of partial autocorrelation function (PACF) and be able to derive it for an AR(1) model.

- Know the theoretical behaviour of the ACF and PACF of AR(p), MA(q) and ARMA(p,q) processes.

- Understand the notion of forecasting TS.

- Know the definition of the best linear predictor (BLP).

- Derive the BLP for AR(1) and AR(2) models.

- Understand the parameter estimation methods (Method of Moments, Maximum Likelihood, Least Squares) for ARMA processes.

- Be able to use the Method of Moments to calculate estimates of an AR(1) and of an AR(2) model.

Generalizations of ARMA Time Series Models

- Know the definition of ARIMA(p,d,q) model.

- Be able to write ARIMA models using B andoperators.

- Understand the model building steps for ARIMA processes.

The Examination Rubric

You should attempt all questions. Marks awarded are shown next to the questions. Calculators may be used in this examination, but no

programming, graph plotting or algebraic facility may be used. Please state on your answer book the name and type of machine used.