MATH38032
Statistical Tables to be provided
Two hours
THE UNIVERSITY OF MANCHESTER
Time Series Analysis
04 June 2014
14:00 16:00
Electronic calculators are permitted, provided they cannot store text.
Answer ALL Four questions in Section A (32 marks in all
Example sheet 5
Solutions
Dr G N Boshnakov
Time Series, 2014/15
1. (a) The model is invertible if the roots of the polynomial 1 + 0.7z + 0.6z 2 have moduli
greter than one. The roots are 0.583333 1.15169i and 0.583333 + 1.15169i, both
with modulus 1.29099
Example sheet 7
Estimation
Dr G N Boshnakov
Time Series, 2014/15
1. Let cfw_Xt represent the yearly sunspot numbers. Sunspots are a measure of the solar activity. The R dataset sunspot.year (or file sunspotsyear.txt) contains measurements
of Xt from 1700
Example sheet 6
Solutions
Dr G N Boshnakov
Time Series, 2014/15
1. (a) partial autocorrelations.
(b) (i) Xt = 0.8Xt1 + t No, 2 = 0.
(ii) Xt = 0.8Xt1 + t No, 2 = 0.
(iii) Xt = t + 0.8t1 No, 1 = 1 = 0.8.
(iv) Xt = t 0.8t1 No, 1 = 1 = 0.8.
(v) Xt = Xt1 0.5Xt
Example sheet 7
Solutions
Dr G N Boshnakov
Time Series, 2014/15
1. (a) The Yule-Walker equations for the AR(2) model were given in the answers to ES 2.
Solving them we get 1 = 1.34, 2 = 0.64, 2 = 312.05. We estimate the mean of
the series by the sample me
Example sheet 8
ARIMA exercises
Dr G N Boshnakov
Time Series, 2014/15
1. Consider the seasonal model
(1 0.8B12 )Xt = (1 + 0.5B)t ,
where cfw_t is white noise with variance 1, i.e. cfw_t WN(0, 1).
(a) Write the model in dierence equation form.
(b) This i
Example sheet 9
Multivariate time series and cointegration
Dr G N Boshnakov
Time Series, 2014/15
1. Consider again the model from Question 2(c) in ES 6.
Write it down in state space form using a multivariate AR(1) model. Initially, you could
try with the
Example sheet 8
Solutions
Dr G N Boshnakov
Time Series, 2014/15
1. (a)
Xt = 0.8Xt12 + t + 0.5t1 ,
(Other correct ways to write this down are equally good.)
(b) s = 12, q = 1, ps = 1, the remaining are zeroes.
(c) The root of 1 + 0.5z is 2 whose modulus is
Example sheet 9
Solutions
1. We established before that EXt = 2. Set
Xt 2
1 2 3
Yt = Xt1 2 ,
A = 1 0 0 ,
Xt2 2
0 1 0
Dr G N Boshnakov
Time Series, 2014/15
t
Ut = 0 .
0
The state space form is
Yt = AYt1 + Ut
Xt = 2 + 1 0 0 Yt .
2. Since both time series
Practical Sheet 1
Basic time series computations in R
Dr G N Boshnakov
Applied Time Series, 2010/11
The practical exercises are an integral part of this course. They help you understand the
theory and equip you with model building skills useful not only f
Example sheet 10
Solutions
Dr G N Boshnakov
Time Series, 2014/15
1. In the solution given to the original question we showed that
k k1 + 1 k2 = 0.
2
11
for k 2. Using the values found in the original question we calculate that 2 = 10 and
3
3 = 5 . The roo
Practical Sheet 2
Differencing and backward shift
Dr G N Boshnakov
Applied Time Series, 2010/11
1. Backward shift.
>
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y <- ts(1:12)
# y1[t] for t = 1, . . . , 12, with y[t]=t for convenience
y1 <- lag(y,-1) # y1[t]=By[t]=y[t-1], so y1 is availa
Practical Sheet 3
Simulation, MA, AR, Yule-Walker
Dr G N Boshnakov
Applied Time Series, 2010/11
1. Simulating ARIMA time series. We have seen examples of DIY simulation from simple
models but in practice it is eaisier to use the arima.sim function. For si
Example sheet 10
Dr G N Boshnakov
Time Series, 2014/15
1. Consider again the ARMA(2,1) process cfw_Xt from Qu. 2/ES 5. Show that the autocovariance function of cfw_Xt satisfies a homogeneous difference equation for lags k 2. Solve
the equation to obtain
Standard error of the sample mean
Suppose that we wish to estimate the mean of a population
using a random sample X1 , . . . , Xn .
is used routinely for this purpose but
The sample mean, X
how good is it?
Let be the population mean and 2 the population
Example sheet 5
Autocovariances of ARMA processes
Dr G N Boshnakov
Time Series, 2014/15
1. Consider the causal ARMA(1,2) model
Xt = 0.8Xt1 + t + 0.7t1 + 0.6t2 ,
where cfw_t is white noise with variance 2 .
(a) Show that this model is invertible.
(b) Let
Example sheet 6
Mid-semester revision questions
Dr G N Boshnakov
Time Series, 2014/15
1. Let cfw_Xt be a stationary time series with E Xt = 0.
(a) For any integer j 1 consider the best linear predictor of Xt in terms of Xt1 , . . . , Xtj .
(j)
(j)
Denote
Example sheet 8
Revision ARIMA exercises
Dr G N Boshnakov
Time Series, 2015/16
1. Consider the seasonal model
0.8B12 )Xt = (1 + 0.5B)"t ,
(1
where cfw_"t is white noise with variance 1, i.e. cfw_"t WN(0, 1).
(a) Write the model in dierence equation form
Time series course (2015/2016)
ARIMA modelling example, xsarima2p1a - transcript of
computations
Georgi N. Boshnakov
April 2016
Basic graphical exploration
Exploring xsarima2p1
Exploring \(1B)\)xsarima2p1
Exploring \(1B^cfw_12)(1B)\)xsarima2p1
Fit mo
MATH38032
Statistical Tables to be provided
Two hours
THE UNIVERSITY OF MANCHESTER
Time Series Analysis
19 May 2015
09:45 11:45
Electronic calculators are permitted, provided they cannot store text.
Answer ALL Four questions in Section A (32 marks in all)
Example sheet 9
Multivariate time series and cointegration
Dr G N Boshnakov
Time Series, 2015/16
1. Consider again the model from Question 2(c) in ES 6.
Write it down in state space form using a multivariate AR(1) model. Initially, you could
try with the
Example sheet 7
ARIMA
Dr G N Boshnakov
Time Series, 2015/16
1. A time series, cfw_Xt , follows a seasonal ARIMA (SARIMA) model of order (1, 0, 0)
(0, 1, 1)12 .
(a) Write out the model in operator form (i.e. using the B operator).
(b) Write down the model
MATH38032
Statistical Tables to be provided
Two hours
THE UNIVERSITY OF MANCHESTER
Time Series Analysis
20 May 2013
14:00 16:00
Electronic calculators are permitted, provided they cannot store text.
Answer ALL Four questions in Section A (32 marks in all)
Example sheet 1
Basic techniques
Dr G N Boshnakov
Time Series, 2014/15
1. Autocovariances and autocorrelations. Let cfw_Xt be a time series. Verify the following
properties (even if you do not consider them obvious now, you soon will).
(a) Var(Xt ) = t,0
Example sheet 3
Backward shift and lters
Dr G N Boshnakov
Time Series, 2014/15
1. Backward shift operator.
(a) Write the following equations in operator form.
i. Xt Xt1 Xt2 = t
iii. Xt = Xt1 + t + t1
ii. Xt = Xt1 + t
(b) Write the following equations in d
Example sheet 2
Prediction
Dr G N Boshnakov
Time Series, 2014/15
1. Let X1 and X2 be random variables with means E X1 = 1 , E X2 = 2 , nite variances
2
2
Var(X1 ) = 1 , Var(X2 ) = 2 , and correlation coecient .
(a) Find the constant a0 that minimizes 0 (a
Example sheet 2
Solutions
Dr G N Boshnakov
Time Series, 2014/15
1. (a) We have
0 (a) = E(X2 a)2
= E(X2 2 + 2 a)2
= E(X2 2 )2 2(2 a) E(X2 2 ) + (2 a)2
2
= 2 + (2 a)2 ,
2
with obvious minimum when a0 = 2 with 0 (a0 ) = 2 .
(b) Here
1 (a, b) = E(X2 a bX1 )2
Example sheet 3
Solutions
Dr G N Boshnakov
Time Series, 2014/15
1. (a) There are usually several ways to arrange the terms in the operator form, e.g. the
rst answer could be written also as Xt = (B + B2 )Xt + t . In the forms given
here all terms for a gi
Example sheet 4
Stationarity and autocorrelation
Dr G N Boshnakov
Time Series, 2014/15
1. Let cfw_Xt be a stationary AR(1) process with representation Xt = Xt1 + t , where cfw_t
is white noise satisfying the causality condition (E t Xtk = 0 for k > 0).