Minitab Project Report Assignment 3

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3.1.1 Simulation of Gaussian White Noise Minitab Project Report Assignment 3 Time Series Plot of zt Function zt 1 0. 0. zt 0-1 0. 0. -0. -0. - -3 1 0 30 0 50 Index 0 70 0 90 0 1 1 1 1 0 marks The series is a sample from a Gaussian White Noise N(0,1). 1 mark The values oscialate randomly around the zero mean with constant variance. 3 marks The series looks like a stationary process. The correlogram suggests that autocovariances are all equal to zero. 1 mark The sample ACF values are within the 95% insignificance limints.

3.1. Simulating MA(1) Time Series Plot of MA(1)_0.3, MA(1)_-0.3, MA(1)_0.9, MA(1)_-0.9 1 0 0 0 0 0 MA(1)_0.3 MA(1)_-0.3 0 MA(1)_0.9 MA(1)_-0.9 - - 0 - - 1 0 0 0 0 0 Index marks

The series is a combination of a zero mean WN random variables, so it is scattered about zero. 1 mark It is a trend and seasonality free TS. 1 mark There may be a non-constant variance MA(1)_0.9, other plots suggest stationary processes. 1 mark For negative its values alternate about zero, but positive there are small clusters of positive or of negative values. marks The shape of the TS both positive values of is very similar (same both negative values of ), 1 mark but range of values is larger larger absolute value of. 1 mark

Correlograms the simulated four MA(1) processes: Function MA(1)_0.3 Function MA(1)_-0.3 0. 0. 0. 0. 0. 0. -0. -0. 0. 0. -0. -0. 1 1 1 1 0 1 1 1 1 0 Function MA(1)_0.9 Function MA(1)_-0.9 0. 0. 0. 0. 0. 0. -0. -0. 0. 0. -0. -0. 1 1 1 1 0 1 1 1 1 0 marks

For positive we obtained positive (1), while negative we obtained negative (1). 1 mark Other values of the sample ACF (lag > 1) are within the bounds of insignificance. 1 mark The features of all correlograms but the one of MA(1) = -0.3 agree with the features of ACF an MA(1). 1 mark There is a clear cut off in all sample ACFs appart from MA(1) = -0. 3. 1 mark Correlogram of MA(1) = -0.3 suggests a zero-correlated series, this may be a result of a small value of. 1 mark The slightly waivy pattern of the sample ACF positive reflects the shape of the series, that is the clasters of positive or negative values in the data and some local trends. 1 mark The alternating shapes of the sample ACF negative also reflects the shape of the series, now its alternating values. 1 mark

For MA(1) we have ρ( τ) = 1 θ 1 + θ 0 τ = 0 τ = 1 τ > 1 1 mark The values of the sample ACF the four simulated series and the theoretical ACF values are: 0.3-0.3 0.9-0.9 ˆρ 0.335-0.1 0.509-0.05 (1) ρ (1) 0.75-0.75 0.97-0.97 marks The estimates of the ACF at lag 1 agree in sign with the theoretical values. 1 mark We would need to know the variances of the estimates to be able to say if the differences between the estimates and the true values are significant, or if the respective confidence intervals cover the true values. 1 mark For lags > 1 the theoretical ACF values equal to zero, while the sample ACF values of the simulated data are all within the 95% boundaries of insignificance. 1 mark

3.1.3 Simulating MA() Time Series Plot of MA(1)_-0.9_+, MA(1)_0.9_0., MA(1)_-0.9_-,... 1 0 0 0 0 0 MA(1)_-0.9_+0.7 MA(1)_0.9_0.7 5.0.5 -.5 5.0 MA(1)_-0.9_-0.7 MA(1)_0.9_-0.7-5.0.5 -.5-5.0 1 0 0 0 0 0 Index marks

All data look like trend and seasonality free processes with zero mean. marks There may be a non-constant variance the case of both parameters positive. 1 mark There is, however, a clear influence of the signs of the parameters: both parameters positive we have the most meandering data, while when at least one of the parameters is negative we have more oscilating data. marks

Function MA(1)_-0.9_+0.7 Function MA(1)_0.9_0.7 0. 0. 0. 0. 0. 0. -0. -0. 0. 0. -0. -0. 1 1 1 1 0 1 1 1 1 0 Function MA(1)_-0.9_-0.7 Function MA(1)_0.9_-0.7 0. 0. 0. 0. 0. 0. -0. -0. 0. 0. -0. -0. 1 1 1 1 0 1 1 1 1 0 marks

The signs of the first two autocorrelations agree with the sings of the parameters of the respective simulated MA() processes. 1 mark The sample ACF the parameters both positive has waivy pattern; again, it reflects the meandering shape of the series. 1 mark The shapes of the other three sample ACFs agree with the oscilating shapes of the data, although positive 1 we observe slightly waivy pattern too. 1 mark Negative produced (1) < (). 1 mark For lag > the sample ACF values are within nonsignificance limits (with some exceptions both positive). 1 mark

The theoretical ACF MA() is given by ρ( τ) = 1 θ1 + θ1θ 1 + θ1 + θ θ 1 + θ1 + θ 0 τ τ τ τ = 0 = ± 1 = ± > 1 mark (-0.9, 0.7) (0.9, 0.7) (-0.9, -07) (0.9, -0.7) ρˆ(1) 9 0.53-17 0.193 ρ(1) 5 0.5-0.117 0.117 ρˆ() 0.31 0.33-0.39-0.30 ρ() 0.30 0.30-0.30-0.30 marks The comparison of the theoretical ACF with the sample ACF all four series shows the agreement in sign both lags 1 and. 1 mark For lag > values the theoretical ACF is zero, while the sample ACF values are within the 95% boundaries of insignificance, with one exception the case of both parameters positive. marks

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