• Xt denotes the observed time series, • n is the sample size; • Zt denotes the white noise...

Question

• Xt denotes the observed time series, • n is the sample size; • Zt denotes the white noise generating ARMA(p,g) model, • Wt denotes the rescaled residuals obtained after the model is fitted. Recall: (i) asymptotically, Wt follow W N(0, 1/n); (ii) asymptotically, their acf ,3 (k) ti N(0,1In); (iii) asymptotically, the x2-statistic (also called Box-Pierce statistic) Qw = nEt_i 4v(j) follows )(2 (h — p — q) distribution. 1. The statistical consultant analyzed a time series of 50 observations and proposed a model of the form: Xt — µ = Zt —0Zt_i. The consultant believes that 0 > 0.9. Is the following statement compatible with the proposed model?
5 E ii2fir(k) = 0.005 k=1
2. You fit 100 observations, using ARMA(1, 2) model. For the 100 residuals, you have determined the first seven autocorrelation coefficients:
k XXXXXXXXXXpw(k XXXXXXXXXX XXXXXXXXXX
Calculate the value of the x2 statistics and determine the number of degrees of freedom.

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David · Accepted Answer

We use  to check if residuals are white noise or not. For this purpose, we apply Box-Pierce 
test which generates a statistic through which we can check if a series of observations recorded over 
time are independent and random. Accordingly we form the null and alternative hypothesis –  
Ho: Residual data is independently distributed 
Ha : The residual data is not independently distributed; there is some serial correlation among them.  
And there Q-statistic is – 
 where n = 50. 
Then                  
The Q statistic follows Chi-square distribution and Chi-square critical value at 5% level of significance and 
49 degrees of freedom is               .  
The calculated              which means that we cannot reject the null hypothesis. Hence,

• Xt denotes the observed time series, • n is the sample size; • Zt denotes the white noise generating ARMA(p,g) model, • Wt denotes the rescaled residuals obtained after the model is fitted. Recall:...

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