Lesson 14: Model Checking

Transcription

1 Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@univaq.it

2 Model checking Given the time series {x t ; t = 1,..., T } suppose that we have estimated the following ARMA model ˆx t = p ˆφ j x t j + j=1 q ˆθ j û t j j=1

3 Model checking The residuals from fitted model are obtained by applying recursively for t = 1, 2,..., T, the following formula û t = x t p ˆφ j x t j q ˆθ j û t j t = 1, 2,..., T j=1 j=1 where x t = 0 and û t = 0 for t < 1.

4 Model checking For example, for the MA(1) process with zero mean, we have û t = x t ˆθu t 1. Assuming û t = 0, then we compute the innovations recursively as follows: û 1 = x 1 û 2 = x 2 ˆθx 1 and so on. That is, û 3 = x 3 ˆθx 2 + ˆθ 2 x 1 t 1 û t = ( 1) i ˆθi x t i i=0

5 Model checking The adequacy of the estimated model, can be evaluated by examining the residuals from fitted model. Why?

6 Model checking We observe that if the time series {x t ; t = 1,..., T } is a realization of an ARMA(p, q) process then the filter φ(l)x t = θ(l)u t, u t WN(0, σ 2 ) π(l) = φ(l) θ(l) transforms the oservations {x t ; t = 1,..., T } in a realization of a Gaussian white noise.

7 Model checking Thus if p and q are well specified (the model chosen is correct), and if the estimated parameters are close to the actual values, then the residuals should be a realization of a white noise. If the diagnostics, such as graphs of the residuals, SACF, SPACF, histogram do not indicate a Gaussian white noise, the model is found to be inadequate. In this case it is necessary to go back and try to identify a better model.

8 Model checking In addition to the visual inspection of the graphs, the Box-Pierce statistic Q K = T K k=1 ˆρ 2 k or the the Ljung-Box statistic Q K = T (T + 2) K ˆρ 2 k/(t k) k=1 can be used for testing the hypothesis that the residuals are realization of a white noise.

9 Model checking In fact, when p and q are well specified and when the number of observation T is large, these statistics follows a chi-square distribution with K p q degrees of freedom (if a constant is included, the degrees of fredom are K p q 1). In practice, K is chosen between 15 and 30.

10 Model checking We therefore reject the adequacy of the fitted model at level α if Q K > χ 2 1 α,k p q where χ 2 1 α,k p q is the 1 α quantile of the chi-squared distribution with K p q degrees of freedom.

11 Model checking: some example Consider the series

12 Model checking: some examples Suppose that we have estimated the following model x t = u t u t 1

13 Model checking: some examples The residuals

14 Model checking: some examples

15 Model checking: some examples The Box-Pierce statistic The p-value is Q 20 = 68.7

16 Model checking: some examples Suppose that we have re-estimated the model obtaining x t = 0.64x t 1 + u t u t 1

17 Model checking: some examples The residuals

18 Model checking: some examples

19 Model checking: some examples The Box-Pierce statistic The p-value is Q 20 = 15.16

20 Model checking: some examples The histogram of the residuals:

21 Model checking: some examples To check whether the residuals are normally distributed, we also use the chi-square goodness of fit test: Chi-square(2) = with p-value