Practical MidtermSolQ1S11

# Practical MidtermSolQ1S11 - Time Series Practical Midterm 1...

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Time Series Practical Midterm 1 Solutions 1) The following results are derived from the training set (i.e. December 2000, December 2010). a) Figure : Scatter and Diagnostic Plots of JTUJOL Figure 1 displays the scatter plot (the left hand side) and the ACF plot (right hand side) of the JTUJOL data. Clearly, the data exhibit a trend of unknown form and a seasonal pattern, with approximately a three month period.

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b), c) Two candidate models will be looked at: AR(12) with trigonometric components and AR(8) with dummy variables Model 1: AR(12) with trigonometric components: Mathematically, the model is as follows: The following is the R-Code results: >xxreg=cbind(month,cos(2*pi*month/12),sin(2*pi*month/12),cos(2*pi*4*month/12),sin (2*pi*4*month/12)) > model1=arima(value,order=c(12,0,0),xreg=xxreg,method="ML") > model1Call: arima(x = value, order = c(12, 0, 0), xreg = xxreg, method = "ML") Coefficients: ar1 ar2 ar3 ar4 ar5 ar6 ar7 ar8 0.2570 0.1769 0.4276 0.2967 0.2300 0.1539 -0.0424 -0.2731 s.e. 0.0859 0.0907 0.0849 0.0926 0.0944 0.0919 0.0937 0.0918 ar9 ar10 ar11 ar12 intercept month -0.2894 -0.3631 -0.0853 0.4260 3.2429 -0.0085 -0.1158 s.e. 0.0901 0.0843 0.0921 0.0883 0.2801 0.0039 0.0091 -0.0244 -0.0984 -0.2102 s.e. 0.0091 0.0447 0.0437 sigma^2 estimated as 0.02135: log likelihood = 57.32, aic = -76.63 Diagnostics of Model 1 Homoskedasticity and Zero Mean:
Time Model 1 Residuals 0 20 40 60 80 100 120 -0.4 -0.2 0.0 0.2 0.4 Figure : Residual Plot of Model 1 The residual plot shows that the residuals are roughly centred on zero with a roughly homogenous spread across time. Thus, the residuals can be considered to be homoskedastic with zero mean. Serial Correlation: Standardized Residuals Time 0 20 40 60 80 100 120 -2 0123 0 5 10 15 20 -0.20.20.61.0 Lag ACF Figure : ACF and Ljung-Box Plots of Model 1 The above ACF plots suggest that serial correlation is not significant (the p-value plot is consistently high as well, also reflecting lack of significant of lag coefficients). Normality:

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-2 -1 0 1 2 -0.4 -0.2 0.0 0.2 0.4 Normal Q-Q Plot Theoretical Quantiles Sample Quantiles Figure : QQ Normality Plot for Model 1 The above plot suggests that the residuals are roughly normal. The normality hypothesis is further supported by the Shapiro-Wilk test which gives a p-value of 0.1178 , which is statistically insignificant at the 95% confidence level. Thus, one can conclude Model 1 satisfies the white noise assumptions. Model 2: AR(8) with dummy variables: Mathematically, the model is as follows:
The following is the R-Code results: > year1=matrix(0,nrow=1,ncol=12) > year_subseq=matrix(0,nrow=12,ncol=12) > year1[1,12]=1 > for(i in 1:12) year_subseq[i,i]=1 >year=rbind(year1,year_subseq,year_subseq,year_subseq,year_subseq,year_subseq,year_ subseq,year_subseq,year_subseq,year_subseq,year_subseq) > Jan=year[,1] > Feb=year[,2] > Mar=year[,3] > Apr=year[,4] > May=year[,5] > Jun=year[,6] > Jul=year[,7] > Aug=year[,8]

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## This note was uploaded on 02/23/2012 for the course STAT 443 taught by Professor Yuliagel during the Spring '09 term at Waterloo.

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Practical MidtermSolQ1S11 - Time Series Practical Midterm 1...

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