lesson 14: model checkinglesson 14: model checking umberto triacca dipartimento di ingegneria e...

Lesson 14: Model Checking

Umberto Triacca

Dipartimento di Ingegneria e Scienze dell’Informazione e MatematicaUniversita dell’Aquila,

[email protected]

Umberto Triacca Lesson 14: Model Checking

Model checking

Given the time series

{xt ; t = 1, ...,T}

suppose that we have estimated the following ARMA model

xt =

p∑j=1

φjxt−j +

q∑j=1

θj ut−j


Model checking

The residuals from fitted model are obtained by applyingrecursively for t = 1, 2, ...,T , the following formula

ut = xt −p∑

j=1

φjxt−j −q∑

j=1

θj ut−j t = 1, 2, ...,T

where xt = 0 and ut = 0 for t < 1.


Model checking

For example, for the MA(1) process with zero mean, we haveut = xt − θut−1. Assuming ut = 0, then we compute theinnovations recursively as follows:

u1 = x1

u2 = x2 − θx1

u3 = x3 − θx2 + θ2x1

and so on. That is,

ut =t−1∑i=0

(−1)i θixt−i


Model checking

The adequacy of the estimated model, can be evaluated byexamining the residuals from fitted model.

Why?


Model checking

We observe that if the time series {xt ; t = 1, ...,T} is arealization of an ARMA(p, q) process

φ(L)xt = θ(L)ut , ut ∼ WN(0, σ2)

then the filter

π(L) =φ(L)

θ(L)

transforms the oservations {xt ; t = 1, ...,T} in a realization ofa Gaussian white noise.


Model checking

Thus if p and q are well specified (the model chosen iscorrect), and if the estimated parameters are close to theactual values, then the residuals should be a realization of awhite noise.

If the diagnostics, such as graphs of the residuals, SACF,SPACF, histogram do not indicate a Gaussian white noise, themodel is found to be inadequate.

In this case it is necessary to go back and try to identify abetter model.


Model checking

In addition to the visual inspection of the graphs, theBox-Pierce statistic

QK = TK∑

k=1

ρ2k

or the the Ljung-Box statistic

QK = T (T + 2)K∑

k=1

ρ2k/(T − k)

can be used for testing the hypothesis that the residuals arerealization of a white noise.


Model checking

In fact, when p and q are well specified and when the numberof observation T is large, these statistics follows a chi-squaredistribution with K − p − q degrees of freedom (if a constantis included, the degrees of fredom are K − p − q − 1).

In practice, K is chosen between 15 and 30.


Model checking

We therefore reject the adequacy of the fitted model at level αif

QK > χ21−α,K−p−q

where χ21−α,K−p−q is the 1− α quantile of the chi-squared

distribution with K − p − q degrees of freedom.


Model checking: some example

Consider the series


Model checking: some examples

Suppose that we have estimated the following model

xt = ut + 0.973ut−1



The residuals



The Box-Pierce statistic

Q20 = 68.7

The p-value is 0.026.



Suppose that we have re-estimated the model obtaining

xt = 0.64xt−1 + ut + 0.81ut−1



The residuals



The Box-Pierce statistic

Q20 = 15.16

The p-value is 0.307.



The histogram of the residuals:



To check whether the residuals are normally distributed, wealso use the chi-square goodness of fit test:

Chi-square(2) = 1.374

with p-value 0.50307


lesson 14: model checkinglesson 14: model checking umberto triacca dipartimento di ingegneria e...

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