ON MULTIPLE PORTMANTEAU TESTS

Transcription

1 ON MULIPLE PORMNEU ESS BY NOY KYM Faculty of Economics, Kyushu University, -19-1, Hakozaki, Higashi-ku, Fukuoka 1-51, Japan bstract he portmanteau statistic based on the first m residual autocorrelations is used for diagnostic checks on the adequacy of fitting a model with varying m In this paper, we propose an approximation of the joint probability of multiple portmanteau tests with different degrees of freedom DF) his distribution is easy to compute when all DF are even integers; its empirical behavior is clarified in terms of asymptotic theory Keywords utoregressive-moving average model; Portmanteau test; Residual autocorrelations; Goodness-of-fit; Multiple tests 1 INRODUCION his paper provides a practical application of portmanteau tests In time series analysis, the portmanteau test statistic is known as a goodness-of-fit test statistic and is defined by the sum of squares of the first m residual autocorrelations, r i, i = 1,,, m First presented by Box & Pierce 197), this statistic is supposed to behave as the sum of squares of the first m sample autocorrelations of the white noise process: r i, i = 1,,, m Li ) reviews applications of portmanteau statistics in various time series models From a practical standpoint, portmanteau tests are conducted by various values of m because we are unable to determine the optimal number of m for the following reasons: i) he number of m needs to be moderately large in order to conduct chi-squared tests ii) For diagnosis of the residual process, we need to check many r i s, meaning m has to be large iii) Conversely, if the number of m is unnecessarily large, the size of this test is unstable and its power will decrease For example, in the S-PLUS statistical software, the S-PLUS function arimadiag computes portmanteau statistics for an autoregressive integrated moving average RIM) model and plots the p-values of the portmanteau tests for any possible value of m o overcome i), Ljung 19) and Katayama ) propose modifications of the test herefore, in practical analysis we can conduct multiple portmanteau tests for small to medium values of m regarding ii) and iii) eg, m < ) However, there has been no attempt to establish the joint probability of the multiple tests For this reason we provide an approximated joint distribution of the portmanteau test statistics in Section In addition, we review the recursive formula in Hosoya 19) to compute the joint probability including the significance levels and p-values his formula is useful when the degrees of freedom DF) are even integers he empirical performance in finite samples suggest that the approximation of the joint significance level of multiple portmanteau tests performs well when we conduct tests for small to medium even integer values of DF MIN RESULS 1 Univariate chi-squared approximation of the portmanteau statistic First, we review the basic framework found in previous work Suppose that a univariate time series {x t } is generated by an autoregressive-moving average, RMp, q), model: αl; α )x t = βl; β )ε t, t =, ±1, ±,, 1) Revision Date: June 1, 1

2 where p + q >, L i x t = x t i, {ε t } is iid, σ ), αl; α ) = 1 α 1L α pl p, βl; β ) = 1 + β 1L + + β q L q, α = α 1,, α p), and β = β 1,, β q ) It is assumed that this model is stationary, invertible, and not redundant, so that the polynomials αz; α ) = and βz; β ) = have no roots in common, and that all the roots are outside the unit circle Given a process, {x t } n t=1, defined in 1), the nonlinear least squares estimator of θ = α, β ), θ = α, β ), is obtained by minimizing the sum of squared residuals Box & Jenkins 197, Chapter ) point out that it is important to check the assumption of the independence of {ε t } by using the residuals, ê t = e t θ), for t = 1,, n, which should behave in a manner that is consistent with the model It follows that it is reasonable to check the adequacy of model fit by examining the residual autocorrelations as follows: r j = r j {ê t }) = n t=j+1 / n ê t ê t j ê t, j = 1,,, n 1 ) he asymptotic joint distribution of r m = r 1,, r m ) has been analyzed by Box & Pierce 197) and McLeod 197) Following the proof in McLeod 197, equation 3)), by using a aylor-series expansion of r m around θ = θ, as n, we have: t=1 r m = r m + X m θ θ ) + O p 1/n), 3) where r m = r 1,, r m ), r j = r j {ε t }), X m = X m θ ) is a m p + q) matrix Here each i, j) element of the partitioned matrix of X m is given in 1) of McLeod 197), as follows: [ ] X m = [d 1 θ ),, d m θ )] = αi j βi j ) where d i θ ) is p+q)-vector consists of the ith row of X m and α i α + α1 L + α L +, 1/βL; β ) = β + β1 L + β L + and αi that Iθ ) = d i θ )d i θ ) i=1 s and βi s are defined by 1/αL; α ) = = βi = for i < Note is the Fisher information matrix of the model 1), and n θ θ ) r m) has an approximately normal distribution with a mean vector of m and the following covariance matrix: [ Iθ ) 1 Iθ ) 1 X ] m X m Iθ ) 1, 5) from ) of McLeod 197), where m is the m-dimensional zero vector and I m is an m m identity matrix It follows that r m is approximately normal with a mean vector of m, and a covariance matrix of I m C m )/n with rank m p q Since lim m X mx m = Iθ ), C m = X m Iθ ) 1 X m is an idempotent matrix for moderately large m For this, the portmanteau statistic: Q m = n I m m r j, ) is distributed as χ m p q, where χ m denotes a chi-squared variable with m DF Ljung & Box 197) showed that a better approximation could be achieved using the following modified portmanteau statistic: Q m = nn + ) j=1 m r j/ n j) 7) hese statistics are restricted to large m eg, m ) such that C is an idempotent matrix However, many empirical studies have shown that as m increases, the empirical significance level also increases and the empirical power decreases See Ljung, 19, and references therein) Ljung 19) and Katayama ) propose two solutions for conducting chi-squared test with small values of m s an approximate distribution of Q m in place of χ m p q, Ljung 19) suggested an alternative chi-squared distribution, aχ b, where a and b generally depend on θ and positive real values j=1

3 lternatively, Katayama ) proposed a modified Q m with a correction term Katayama ) rewrote 3) as: I m D m ) m r m = I m D m ) m r m + O p 1/ n ), ) as n, where D m = D m θ ) = X m X mx m ) 1 X m and ) nn + ) nn + ) nn + ) m = diag n 1, n,, n m = ni m + O m/ n ) 9) Since Q m = r m m r m and r m m I m D m ) m r m d χ m p q, we have: Q m B m d χ m p q, as n, 1) where B m = r m m D m m r m herefore, Q m has a positive extra random variable B m for χ m p q approximation Katayama, heorem 1) proved that B m is negligible as m, n increases In addition, Katayama, heorem ) proved that: I m D m ) m r m = I m D m ) m r m + O p 1/ n ), 11) as n, where D m is an consistent estimator of D m given by D m = D m θ) Hence, Katayama ) proposed a modified portmanteau statistic with a correction term: Q m = Q m B m, 1) where B m = r m m Dm m r m Unlike Q m, Q m is not required for large m for the χ m p q approximation Multivariate chi-squared approximation of the portmanteau statistics his section provides the asymptotic joint distribution of the portmanteau statistics Let p + q < m = m1) < m) < < ms) = M < n hen, from ) and 9), we obtain: [ ] Imi) D mi) ) mi) r mi) = ) ne mi) r M + O p 1/ n, 13) M mi) for i = 1,,, s, where E mi) is a M M matrix defined by: [ ] Imi) D E mi) = mi) O, O O and O denotes an appropriate dimensional matrix where all elements are zeros It follows that: Q m1) B m1),, Q ms) B ms)) 1) = n r M E m1) r M,, r M E ms) r M ) + Op 1/ n ) d z M E m1) z M,, z M E ms) z M ), as n, where z M N M M, I M ) he convergence of 1) follows from nr M Similarly, using 9) and 11), yields: d zm and Cramér Wold device Q m1),, Q ms)) d z M E m1) z M,, z M E ms) z M ), as n 15) 3

4 3 Computation of the asymptotic joint distribution his section examines the asymptotic joint probability given by 1) and 15) Unfortunately, computation of the distribution is difficult because all elements are correlated o overcome this difficulty, we provide an independence property of the distribution when m is moderately large he mathematical proofs are given in the ppendix Lemma 1 For i < j k < l, as mi), E mj) E ml) E mk) ) = Oɛ mk)+1 maxp,q) ) and E mj) E mi) )E ml) E mk) ) = Oɛ mi)+1 maxp,q) ), where ɛ, 1) is such that ɛ is larger than the absolute value of any roots of αz 1 ; α )βz 1 ; β ) = From the Craig Sakamoto heorem see, eg, Provost, 199), Lemma 1 indicates the asymptotic independence of z M E mj)z M and z M E ml) E mk) )z M, or z M E mj) E mi) )z M and z M E ml) E mk) )z M, as m increase heorem 1 For j = 1,, s, as m : where ψj) = mj) i=p+q+1 Z i and {Z i } is iid N, 1) z M E mj) z M = ψj) + O p ɛ m+1 maxp,q) ), 1) heorem 1 provides an approximation to evaluate the significance level α, 1) of multiple portmanteau tests by 1) or 15) when m, n are moderately large Suppose that each portmanteau test has the same significance level β, 1), we can approximate α and β by: where d j s are given by: Pr ψj) d j, j = 1,,, s) = 1 α, 17) Pr ψj) d j ) = 1 β, j = 1,,, s 1) We can easily compute 17) and 1) from Hosoya 19, Section ) if m1) p q, mi) mi 1), i =,, s are all even integers However, since Hosoya 19, Section ) contains a few confusing typographical errors, we modify it briefly here ll integrations are easily verified with integration by parts Let d i s be constants such that d 1 d s and let q i, i = 1,, s be nonnegative integers Set for a nonnegative integer j: k, j) = d1 dz 1 d z 1 dz dk z k 1 z q1 1 z z 1 ) q 19) z k 1 z k ) q k 1 z k z k 1 ) j exp z k /)dz k, k =,, s, d1 d dk 1 Bk, j) = dz 1 dz z q1 1 z z 1 ) q ) z 1 z k z k 1 z k ) q k 1 d k z k 1 ) j dz k 1, k =,, s hen those integrals are computed recursively according to the following steps Set p k = q k + q k q s + s k + 1 and then: B, j) = Bk, j) = j q 1! j! q 1 + i + 1)! j i)! dq 1+i+1 1 d d 1 ) j i, j =,, p, 1) i= j q k 1! j! q k 1 + i + 1)! j i)! d k d k 1 ) j i Bk 1, q k 1 + i + 1), i= j =,, p k, k = 3,, s, and then: 1, ) = 1, j) = d1 d1 exp z 1 /)dz 1 = exp d 1 /) +, ) z j 1 exp z 1/)dz 1 = d j 1 exp d 1/) + j1, j 1), j = 1,, p 1, k, ) = exp d k /)Bk, ) + k 1, q k 1 ), k =,, s, k, j) = exp d k /)Bk, j) + jk, j 1), j = 1,, p k, k =,, s

5 Let m1) p q = q 1 + 1), mi) mi 1) = q i + 1), i =,, k We then obtain: )/ k Pr ψj) d j, j = 1,,, k) = k, q k qj+1 q j! 3) and Pr ψj) d j, j = 1,,, k 1 ψk) = d k ) = Bk, q k )/B k, q k ), ) k =,, s, where B k, j) is defined by ) with d i = d k, i = 1,, k s Hosoya 19) pointed out, when the number of DF is odd, the numerical computation of 17) and 1) is barely tractable Hosoya 19) proposed a transformation of test statistics to apply for odd DF However, we do not discuss this as the limitation of an even integer DF is not a serious problem in practical usage of multiple portmanteau tests We now summarize three cases to evaluate significance levels and p-values of multiple portmanteau tests when all values of DF are even integers: I) When we conduct each portmanteau test with the same marginal significance level β by 1), the joint significance level α by 17) is computed from 3) II) When we set the joint significance level α by 17), the critical values d j s and marginal significance level β in 1) are calculated from the Newton-Raphson iteration method in Hosoya 19, Section ) Since d j = d j β)s are defined by 1), the Newton-Raphson iteration method can be conducted as: β i+1) = β i) Pr{ψj) d jβ i) ), j = 1,, s} 1 α) Pr{ψj) d j β i) ), j = 1,, s}/ β i), 5) where β i) denotes the ith-step value of the marginal significance level β Using 3) and ), the derivative term in 5) is computed by: Pr{ψj) d j β), j = 1,, s}/ β ) j=1 = Pr{φ1, j) d j d 1, j =,, s} s 1 Is 3) Pr{ψj) d j, j = 1,, k 1 ψk) = d k } k= Pr{φk, l) d l d k, l = k + 1,, s} Pr{ψj) d j, j = 1,, s 1 ψs) = d s }, where φi, j) = ψj) ψi) = Zmi) Z mj) and Is 3) is the indicator function such that I = 1 if s 3 and I = if s = One reasonable starting value is β ) = α Equations 5) and ) are computed from 3) and ) III) When we compute the joint p-value from the marginal p-values, we also apply Hosoya 199, Section 1) Let Q mj) be a computed portmanteau statistic with mj) p q DF he marginal p-value is then given by p j) = Prψj) > Q mj) ) Putting p min = min{p 1),, p s)}, the critical values d j s are defined by: hen the joint p-value, p s), is given by: where p s) is computed from 3) p min = Prψj) > d j ), j = 1,, s p s) = 1 Prψj) d j, j = 1,, s), 7) able 1 gives Four examples of approximate significance levels and d j s on the multiple portmanteau tests by I) and II) o conduct the Newton-Raphson iteration method given in 5), we chose α as the starting value he iterations continued until β i+1) β i) < 1 5 few iterations were enough for all cases It would seem some monotonicity properties exist between the set of DF and α or β he empirical significance levels corresponding to α = 5 for the R1) model, 1 α L)x t = ε t, are given in Figure 1, where {ε t } is iid N, 1) he solid dashed) line denotes the empirical significance 5

6 able 1: pproximate significance levels of the multiple portmanteau tests given 1α = 1, 5, 1 %) or 1β = 1, 5, 1 %) defined by 17) and 1) and the corresponding critical points, d j s, where DF ) denotes a vector of DF DF,, 1) 1α%) 1β%) d 1 d d DF,, 1) 1α%) 1β%) d 1 d d DF1, 1) 1α%) 1β%) d 1 d DF1, ) 1α%) 1β%) d 1 d level of Q m Q m ) We conducted cases of the set of DF given in able 1 with n = 1 Figure 1 reveals that: i) Q m and Q m perform similarly unless m is very small and α close to 1 Note that approximation of the marginal and joint significance levels depend on ɛ and m, and ɛ = α in the R1) model herefore, departure of Q m an Q m was effected by a correction term B m in this case ii) he empirical marginal joint) significance levels increase as the number of DF increase his is also apparent from previous simulation studies iii) However, the empirical joint significance level of the multiple portmanteau tests performs well if the set of DF contains a small number of DF It would then appear that the empirical marginal significance levels smoothes the empirical joint significance level ogether, these suggest that use of the joint significance level of the multiple portmanteau tests for small to medium values of DF is highly effective because an approximation of the empirical marginal significance levels occasionally performs poorly depending on the values of m and ɛ the case where m is large or small and ɛ is close to 1) From these analysis, and to avoid the arbitrary choice of DF, researchers may conduct multiple portmanteau tests with any possible even DF, mi) p q = i, i = 1,,, s, similarly to the S- PLUS function arimadiag Figure depicts these joint approximate significance levels α by varying s =,,, 1, when β = 1, 5, 1 are given he computations are given by 3) Figure reveals that α increases like a logarithm function and, for s, α is at most 5,, 3 when β = 1, 5, 1, respectively Note that this figure also shows the upper bound of α if we conduct multiple portmanteau tests with even DF, mi) p q, i = 1,, k, k s and mk) p q s he joint empirical significance levels for the case of s = 1 is examined with Figure 3, where we use an R1) model as in Figure 1 he joint empirical significance performs well uniformly in α and β his is consistent with the conjectures obtained by Figure 1

7 R DF,, 1), LPH, BE)=5, 3) R DF1, 1), LPH, BE)=5, 3) R DF,, 1), LPH, BE)=5, 5) R DF1, ), LPH, BE)=5, 3) Figure 1: Percentages of the empirical significance level of the statistics Q m χ m 1 solid line) and Q m χ m 1 dashed line) for the R1) model, 1 α L)x t = ε t ; n = 1, and α =, ±3, ±, ±9, with 1, replications DF ) denotes a vector of DF he significance levels are given in able 1 Lines with denote the empirical joint significance level and lines with a number denote the empirical marginal significance level, where the numbers denote single digit DF For example, in the figure DF,,1), denotes DF and denotes 1 DF 7

8 1 3 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF F FFFF F OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO FF 1 S Figure : Joint significance level 1α% varying s =,,, 1, where O denotes 1β = 1%, F denotes 1β = 5% and denotes 1β = 1% DF,,, ), LPH, BE)=, 1), 17, 5), 9, 1) F F F F F F F O O O O O O O -5 5 R1 Figure 3: Percentages of the joint empirical significance level of the statistics Q m χ m 1 solid line) and Q m χ m 1 dashed line) for the R1) model, 1 α L)x t = ε t ; n = 1, and α =, ±3, ±, ±9, with 1, replications, where DF ) denotes a vector of DF, O denotes 1β = 1% 1α = %), F denotes 1β = 5% 1α = 1%) and denotes 1β = 1% 1α = 91%)

9 PPENDIX Proof of Lemma 1: For simplicity, we shall only prove E m1) E m) E m1) ) = Oɛ m+1 maxp,q) ) However, other cases can be similarly treated First, we prove that: [ ] Dm O D m) = + O ɛ m+1 maxp,q)) 1) O O Let: hen we obtain: X m) = [ ] Xm X 1, where X 1, = [ d m+1 θ ),, d m) θ ) ] [ Xm X D m) = m)x m) ) 1 X m X m X m)x m) ) 1 X ] 1, X 1, X m)x m) ) 1 X m X 1, X m)x m) ) 1 X 1, Since X mx m Iθ ), d m θ ) = O ɛ m maxp,q)) and X 1, = Od m+1 θ )) = O from Katayama, equation 1)), we obtain: X m X m)x m) ) 1 X m D m ɛ m+1 maxp,q)) = X m X m)x m) ) 1{ I p+q X mx m + X 1,X 1, )X mx m ) 1} X m = X m X m)x m) ) 1 X 1,X 1, X mx m ) 1 X m = O X ) 1,X 1, = O i=m+1 ɛ i maxp,q) ) = O ɛ m+1 maxp,q)) Similarly, X m X m)x m) ) 1 X 1, = Oɛ m+1 maxp,q) ) and X 1, X m)x m) ) 1 X 1, = Od m+1 θ ) d m+1 θ )) = O ɛ {m+1 maxp,q)}), ) which proves 1) It follows that: E m) E m = O O O O I m) m O + O ɛ m+1 maxp,q)), O O O which proves E m1) E m) E m1) ) = Oɛ m+1 maxp,q) ) Proof of heorem 1: From Lemma 1 and 3), we obtain: z M E m1)z M 1 z z M E M m)z M = E m1)z M z M E m) E m1) )z M z M E ms)z M 1 1 z M E ms) E ms 1) )z M 1 = 1 1 where the lower triangular matrices are s s matrices m1) Zi i=p+q+1 m) Zi i=m1)+1 ms) Zi i=ms 1)+1 + Oɛ m+1 maxp,q) ), 3) 9

10 CKNOWLEDGMENS he author would like to thank the editor and an anonymous referee for their useful comments, and convey his sincere thanks to Professor Wai Keung Li for valuable suggestions made when the author visited the University of Hong Kong in March he financial support of the Japan Society for the Promotion of Science is gratefully acknowledged REFERENCES BOX, GEP & JENKINS, GM 197) ime Series nalysis: Forecasting and Control San Francisco: Holden-Day BOX, GEP & PIERCE, D 197) Distribution of residual autocorrelations in autoregressiveintegrated moving average time series models Journal of the merican Statistical ssociation 5, 159 HOSOY, Y 19) Simultaneous test in the presence of nested alternative hypotheses Journal of pplied Probability 3, ) Hierarchical statistical models and a generalized likelihood ratio test Journal of the Royal Statistical Society B51, 35 7 KYM, N ) n improvement of the portmanteau statistic Journal of ime Series nalysis 9), LI, WK ) Diagnostic Checks in ime Series New York: Chapman & Hall LJUNG, GM 19) Diagnostic testing of univariate time series models Biometrika 73, 75 3 LJUNG, GM & BOX, GEP 197) On a measure of lack of fit in time series models Biometrika 5, MCLEOD, I 197) On the distribution of residual autocorrelations in Box-Jenkins models Journal of Royal Statistical Society B, 9 3 PROVOS, SB 199) On Craig s theorem and its generalizations Journal of Statistical Planning and Inference 53,

11 OMIED RGUMENS In this section, we present proofs and arguments not included in the main text of the paper and not to be published ypos in Hosoya 19, pp193 19): 1 On equation ), 1st term on RHS p k= should be Ip 3) p 1 On equations 7) and ), these should be defined by k =,, p because the cases of k = 1 are not well defined 3 On equation 9), 1st term on RHS of 1st equation d k 1 exp d 1 /) should be d k 1 exp d 1 /) On equation 9), 3rd equation is not well defined from ) 5 On equation 11), nd term on RHS Bk, m k 1 + j + 1) should be Bk 1, m k 1 + j + 1) Figures showing rates of rejection for the empirical experiments: We present figures that illustrate empirical results for the testing for R1) model, 1 α )x t = ε t ; n = 1, and α =, ±1,, ±9, with 1, replications Notations are the same as used for Figures 1 and 3 k= DF,, 1), LPH, BE)=5, 3) R1 Figure : Empirical significance level %) 11

12 DF,, 1), LPH, BE)=5, 5) R1 Figure 5: Empirical significance level %) DF1, 1), LPH, BE)=5, 3) R1 Figure : Empirical significance level %) 1

13 DF1, ), LPH, BE)=5, 3) R1 Figure 7: Empirical significance level %) DF,,, ), LPH, BE)=, 1), 17, 5), 9, 1) F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F O O O O O O O O O O O O O O O O O O O -5 5 R1 Figure : Empirical significance level %) 13