EQC7006 : Time Series Analysis 2018/2019

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ACADEMIC SESSION 2018/2019: SEMESTER II

EQC7006: Time Series Analysis

1.  Answer the following questions.

a)  Given below is an annual data for 12 years. Estimate trend-cycle component of the series using a single 3-period moving average (3 MA) and a 3-term weighted moving average (3 WMA) with weights of 0.25, 0.5, and 0.25.

Year

Yt

3 MA

3 WMA

1

42

2

69

3

100

4

115

5

132

6

141

7

154

8

171

9

180

10

204

11

228

12

247

(4 marks)

Write formulae used to estimate irregular component ( ) for an annual time series data (additive and multiplicative trend).

(1 mark) (Total: 5 marks)

2.  The following tables give estimated coefficients for three regression models. Y denotes the monthly temperature (in Celsius) series. T represents time trend (T = 1, 2, 3,  …,  192).  Di’s are seasonal dummies where  D1=1 for January and 0 otherwise, and D2, D3, D4, D5, D6, D7, D8, D9, D10, D11 and D12 are similarly defined for February, March until December, respectively.

Model 1

Dependent Variable: Y

Method: Least Squares

Sample: 2000M01 2015M12

Included observations: 192

Variable

Coefficient

Std. Error

t-Statistic

Prob.

D1                        25.23673 0.073214         344.6961         0.0000

D2                        25.40154 0.073214         346.9472         0.0000

D3                        25.82834 0.073214         352.7767         0.0000

D4                        26.19186 0.073214         357.7418         0.0000

D5                        26.38059 0.073214         360.3195         0.0000

D6                        26.17270 0.073214         357.4801         0.0000

D7                        25.95264 0.073214         354.4745         0.0000

D8                        25.96539 0.073214         354.6485         0.0000

D9                        25.82726 0.073214         352.7619         0.0000

D10                       25.69261 0.073214         350.9227         0.0000

D11                       25.50648 0.073214         348.3805         0.0000

D12                       25.48380 0.073214         348.0708         0.0000

R-squared 0.584144    Mean dependent var                25.80333

Adjusted R-squared 0.558731 S.D. dependent var                  0.440864

S.E. of regression 0.292858 Akaike info criterion 0.442202

Sum squared resid 15.43781    Schwarz criterion                     0.645795

Model 2

Dependent Variable: Y

Method: Least Squares

Sample: 2000M01 2015M12

Included observations: 192

Variable

Coefficient

Std. Error

t-Statistic

Prob.

T 0.000446         0.000382         1.168131         0.2443

D1                        25.14800 0.105445         238.4929         0.0000

D2                        25.31237 0.105721         239.4266         0.0000

D3                        25.73873 0.105997         242.8257         0.0000

D4                        26.10179 0.106273         245.6101         0.0000

D5                        26.29008 0.106551         246.7382         0.0000

D6                        26.08174 0.106828         244.1462         0.0000

D7                        25.86124 0.107107         241.4526         0.0000

D8                        25.87354 0.107386         240.9395         0.0000

D9                        25.73497 0.107666         239.0263         0.0000

D10                       25.59987 0.107946         237.1538         0.0000

D11                       25.41330 0.108227         234.8142         0.0000

D12                       25.39017 0.108509         233.9916         0.0000

R-squared 0.587291    Mean dependent var                25.80333

Adjusted R-squared 0.559623 S.D. dependent var                  0.440864

S.E. of regression 0.292562 Akaike info criterion 0.445024

Sum squared resid 15.32102    Schwarz criterion                     0.665584

Model 3

Dependent Variable: Y

Method: Least Squares

Sample: 2000M01 2015M12

Included observations: 192

Variable

Coefficient

Std. Error

t-Statistic

Prob.

C

25.80333

0.031817

811.0009

0.0000

R-squared 0.000000    Mean dependent var                25.80333

Adjusted R-squared 0.000000 S.D. dependent var                  0.440864

S.E. of regression 0.440864 Akaike info criterion 1.205036

Sum squared resid 37.12302    Schwarz criterion                     1.222002

a)  What estimates are given by the twelve coefficients of Model 1?

(1 mark)

b)  In between Model 1 and Model 2, which one should be used? Why?

(2 marks)

c)  Based on Model 1, perform a seasonal variation test at 5% significance level. State the null and alternative hypotheses of the test.

(4 marks)

d)  Given  below  are  out-sample  (January  2016  –  December  2016)  forecast accuracy measures produced by Model 1. Interpret test set MAPE, ACF1 and Theil’s U values.

(3 marks)

(Total: 10 marks)

3.  The following time  series  plot  represents  monthly  data  of  rainfall  amount  (in millimeters) in Malaysia for the period of 16 years (January 2000 – December 2015).

RAINFALL

600

500

400

300

200

100

0

00    01    02    03    04    05    06    07    08    09    10    11    12    13    14    15


















2022-05-19