Practical Implementation

The following R code allows us to draw a given number of autocorrelations and the significance bands .

 # autocorrelation function
 gamma<-function(x,h){ n<-length(x); h<-abs(h);x<-x-mean(x)
 gamma<-sum(x[1:(n-h)]*x[(h+1):n])/n }
 rho<-function(x,h) rho<-gamma(x,h)/gamma(x,0)
 # acf function with significance bands of a weak white noise
 nl.acf<-function(x,main=NULL,method="NP"){
 n<-length(x); nlag<-as.integer(min(10*log10(n),n-1))
acf.val<-sapply(c(1:nlag),function(h) rho(x,h)); x2<-x^2
var<-1+(sapply(c(1:nlag),function(h) gamma(x2,h)))/gamma(x,0)^2
band<-sqrt(var/n)
minval<-1.2*min(acf.val,-1.96*band,-1.96/sqrt(n))
maxval<-1.2*max(acf.val,1.96*band,1.96/sqrt(n))
acf(x,xlab="Lag",ylab="SACR",ylim=c(minval,maxval),main=main)
lines(c(1:nlag),-1.96*band,lty=1,col="red")
lines(c(1:nlag),1.96*band,lty=1,col="red") }

In Figure 5.1 we have plotted the SACRs and their significance bands for daily series of exchange rates of the dollar, pound, yen and Swiss franc against the euro, for the period from 4 January 1999 to 22 January 2009. It can be seen that the SACRs are often outside the standard significance bands , which leads us to reject the strong white noise assumption for all these series. On the other hand, most of the SACRs are inside the significance bands shown as solid lines, which is in accordance with the hypothesis that the series are realizations of semi‐strong white noises.

Standard Significance Bounds for the SACRs are Not Valid

The right‐hand graph of Figure  5.2 displays the sample correlogram of a simulation of size n = 5000 of the GARCH(1, 1) white noise

5.4

where (η _t ) is a sequence of iid variables. It is seen that the SACRs of order 2 and 4 are sharply outside the 95% significance bands computed under the strong white noise assumption. An inexperienced practitioner could be tempted to reject the hypothesis of white noise, in favour of a more complicated ARMA model whose residual autocorrelations would lie between the significance bounds . To avoid this type of specification error, one has to be conscious that the bounds are not valid for the SACRs of a GARCH white noise. In our simulation, it is possible to compute exact asymptotic bounds at the 95% level (Exercise 5.4). In the right‐hand graph of Figure  5.2, these bounds are drawn in thick dotted lines. All the SACRs are now inside, or very slightly outside, those bounds. If we had been given the data, with no prior information, this graph would have given us no grounds on which to reject the simple hypothesis that the data is a realization of a GARCH white noise.

Estimating the Significance Bounds of the SACRs of a GARCH

Of course, in real situations, the significance bounds depend on unknown parameters, and thus cannot be easily obtained. It is, however, possible to estimate them in a consistent way, as described in Section 5.1.1. For a simulation of model ( 5.4) of size n = 5000, Figure 5.3 shows as thin dotted lines the estimation thus obtained of the significance bounds at the 5% level. The estimated bounds are fairly close to the exact asymptotic bounds.

The SPACs and Their Significance Bounds

Figure 5.4 shows the SPACs of the simulation ( 5.4) and the estimated significance bounds of the , at the 5% level (based on ). By comparing Figures  5.3 and 5.4, it can be seen that the SACRs and SPACs of the GARCH simulation look much alike. This is not surprising in view of Theorem B.4.

Portmanteau Tests of Strong White Noise and of Pure GARCH

Table 5.1 displays p ‐values of white noise tests based on Q _m and the usual Ljung–Box statistics, for the simulation of ( 5.4). Apart from the test with m = 4, the Q _m tests do not reject, at the 5% level, the hypothesis that the data comes from a GARCH process. On the other hand, the Ljung–Box tests clearly reject the strong white noise assumption.

m	1	2	3	4	5	6	7	8	9	10	11	12
Tests based on Q m , for the hypothesis of GARCH white noise
	0.00	0.06	0.03	0.05	0.02	0.00	0.02	0.01	0.02	0.02	0.00	0.01
	0.025	0.028	0.024	0.024	0.021	0.026	0.019	0.023	0.019	0.016	0.017	0.015
Q m	0.00	4.20	5.49	10.19	10.90	10.94	12.12	12.27	13.16	14.61	14.67	15.20
	0.9637	0.1227	0.1391	0.0374	0.0533	0.0902	0.0967	0.1397	0.1555	0.1469	0.1979	0.2306
Usual tests, for the strong white noise hypothesis
	0.00	0.06	0.03	0.05	0.02	0.00	0.02	0.01	0.02	0.02	0.00	0.01
	0.014	0.014	0.014	0.014	0.014	0.014	0.014	0.014	0.014	0.014	0.014	0.014
	0.01	16.78	20.59	34.18	35.74	35.86	38.05	38.44	39.97	41.82	41.91	42.51
	0.9365	0.0002	0.0001	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

Portmanteau Tests Based on Partial Autocorrelations

Table 5.2 is similar to Table  5.1, but presents portmanteau tests based on the SPACs. As expected, the results are very close to those obtained for the SACRs.

m	1	2	3	4	5	6	7	8	9	10	11	12
GARCH white noise tests based on
	0.00	0.06	0.03	0.05	0.02	0.00	0.02	0.01	0.01	0.02	0.00	0.01
	0.025	0.028	0.024	0.024	0.021	0.026	0.019	0.023	0.019	0.016	0.017	0.015
	0.00	4.20	5.49	9.64	10.65	10.650	11.92	12.24	12.77	14.24	14.24	14.67
	0.9637	0.1227	0.1393	0.0470	0.0587	0.0998	0.1032	0.1407	0.1735	0.1623	0.2200	0.2599
Strong white noise tests based on
	0.02	0.01	0.01	0.02	0.00	0.01	0.02	0.01	0.01	0.02	0.00	0.01
	0.014	0.014	0.014	0.014	0.014	0.014	0.014	0.014	0.014	0.014	0.014	0.014
	0.01	16.77	20.56	32.55	34.76	34.76	37.12	37.94	38.84	40.71	40.71	41.20
	0.9366	0.0002	0.0001	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

An Example Showing that Portmanteau Tests Based on the SPACs Can Be More Powerful than Those Based on the SACRs

Consider a simulation of size n = 100 of the strong MA(2) model

5.5

By comparing the top two and bottom two parts of Table 5.3, we note that the hypotheses of strong white noise and pure GARCH are better rejected when the SPACs, rather than the SACRs, are used. This follows from the fact that, for this MA(2), only two theoretical autocorrelations are not equal to 0, whereas many theoretical partial autocorrelations are far from 0. For the same reason, the results would have been inverted if, for instance, an AR(1) alternative had been considered.

m	1	2	3	4	5	6
Tests of GARCH white noise based on autocorrelations
Q m	1.6090	4.5728	5.5495	6.2271	6.2456	6.4654
	0.2046	0.1016	0.1357	0.1828	0.2830	0.3731
Tests of GARCH white noise based on partial autocorrelations
	1.6090	5.8059	9.8926	16.7212	21.5870	25.3162
	0.2046	0.0549	0.0195	0.0022	0.0006	0.0003
Tests of strong white noise based on autocorrelations
	3.4039	8.4085	9.8197	10.6023	10.6241	10.8905
	0.0650	0.0149	0.0202	0.0314	0.0594	0.0918
Tests of strong white noise based on partial autocorrelations
	3.3038	10.1126	15.7276	23.1513	28.4720	32.6397
	0.0691	0.0064	0.0013	0.0001	0.0000	0.0000

Invalidity of the Standard Bartlett Formula and Modified Formula

The validity of the usual Bartlett formula rests on assumptions including the strong white noise hypothesis (Theorem 1.1) which are obviously incompatible with GARCH errors. We shall see that this formula leads to an underestimation of the variances of the SACRs and SPACs, and thus to erroneous ARMA orders. We shall only consider the SACRs because Theorem B.2 shows that the asymptotic behaviour of the SPACs easily follows from that of the SACRs.

We assume throughout that the law of η _t is symmetric. By Theorem B.5, the asymptotic behaviour of the SACRs is determined by the generalised Bartlett formula (B.15). This formula involves the theoretical autocorrelations of (X _t ) and , as well as the ratio . More precisely, using Remark 1 of Theorem 7.2.2 in Brockwell and Davis (1991), the generalised Bartlett formula is written as

where

5.7

and

The following result shows that the standard Bartlett formula always underestimates the asymptotic variances of the sample autocorrelations in presence of GARCH errors.

Consider, by way of illustration, the ARMA(2,1)‐GARCH(1, 1) process defined by

5.8

Figure 5.5 shows the theoretical autocorrelations and partial autocorrelations for this model. The bands shown as solid lines should contain approximately 95% of the SACRs and SPACs, for a realisation of size n = 1000 of this model. These bands are obtained from formula (B.15), the autocorrelations of being computed as in Section 2.5.3. The bands shown as dotted lines correspond to the standard Bartlett formula (still at the 95% level). It can be seen that using this formula, which is erroneous in the presence of GARCH, would lead to identification errors because it systematically underestimates the variability of the sample autocorrelations (Proposition 5.1).

Algorithm for Estimating the Generalised Bands

In practice, the autocorrelations of (X _t ) and , as well as the other theoretical quantities involved in the generalized Bartlett formula (B.15), are obviously unknown. We propose the following algorithm for estimating such quantities:

1. Fit an AR(p ₀) model to the data X ₁,…, X _n using an information criterion for the selection of the order p ₀ .
2. Compute the autocorrelations ρ ₁(h), h = 1, 2,…, of this AR(p ₀) model.
3. Compute the residuals of this estimated AR(p ₀).
4. Fit an AR(p ₁) model to the squared residuals , again using an information criterion for p ₁ .
5. Compute the autocorrelations ρ ₂(h), h = 1, 2,…, of this AR(p ₁) model.
6. Estimate by , where
   
   
   

   and ℓ_max is a truncation parameter, numerically determined so as to have ∣ρ ₁(ℓ)∣ and ∣ρ ₂(ℓ)∣ less than a certain tolerance (for instance, 10⁻⁵ ) for all ℓ > ℓ_max .

This algorithm is fast when the Durbin–Levinson algorithm is used to fit the AR models. Figure 5.6 shows an application of this algorithm (using the BIC information criterion).

The Corner Method

Denote by D(i, j) the j × j Toeplitz matrix

and let Δ(i, j) denote its determinant. Since , for all h > Q , it is clear that D(i, j) is not a full‐rank matrix if i > Q and j > P . More precisely, P and Q are minimal orders (that is, (X _t ) does not admit an ARMA(P ^′, Q ^′) representation with P ^′ < P or Q ^′ < Q ) if and only if

5.10

The minimal orders P and Q are thus characterised by the following table:

where Δ(j, i) is at the intersection of row i and column j , and × denotes a non‐zero element.

The orders P and Q are thus characterised by a corner of zeros in table (T 1), hence the term ‘corner method’. The entries in this table are easily obtained using the recursion on j given by

5.11

and letting Δ(i, 0) = 1, Δ(i, 1) = ρ _X (∣i∣).

Denote by , , , … the items obtained by replacing {ρ _X (h)} by in D(i, j), Δ(i, j), (T1), …. Only a finite number of SACRs are available in practice, which allows to be computed for i ≥ 1, j ≥ 1 and i + j ≤ K + 1. Table is thus triangular. Because consistently estimates Δ(j, i), the orders P and Q are characterised by a corner of small values in table . However, the notion of ‘small value’ in is not precise enough. ³

It is preferable to consider the Studentised statistics defined, for i =  − K,…, K and j = 0,…, K −  ∣ i ∣  + 1, by

5.12

where is a consistent estimator of the asymptotic covariance matrix of the first K SACRs, which can be obtained by the algorithm of Section 5.2.1 or by that of Section 5.2.2, and where the Jacobian is obtained from the differentiation of (5.11):

for k = 1,…, K , i =  − K + j,…, K − j and j = 1,…, K .

When Δ(i, j) = 0, the statistic t(i, j) asymptotically follows a (provided, in particular, that exists). If, in contrast, Δ(i, j) ≠ 0 then when n → ∞. We can reject the hypothesis Δ(i, j) = 0 at level α if ∣t(i, j)∣ is beyond the (1 − α/2)‐quantile of a . We can also automatically detect a corner of small values in the table giving the t(i, j), if no entry in this corner is greater than this (1 − α/2)‐quantile in absolute value. This practice does not correspond to any formal test at level α , but allows a small number of plausible values to be selected for the orders P and Q .

Illustration of the Corner Method

For a simulation of size n = 1000 of the ARMA(2,1)‐GARCH(1, 1) model ( 5.8), we obtain the following table:

 .p.|.q..1....2....3....4....5....6....7....8....9...10...11...12...
 1 | 17.6-31.6-22.6 -1.9 11.5 8.7 -0.1 -6.1 -4.2 0.5 3.5 2.1
 2 | 36.1 20.3 12.2 8.7 6.5 4.9 4.0 3.3 2.5 2.1 1.8
 3 | -7.8 -1.6 -0.2 0.5 0.7 -0.7 0.8 -1.4 1.2 -1.1
 4 |  5.2 0.1 0.4 0.3 0.6 -0.1 -0.3 0.5 -0.2
 5 | -3.7 0.4 -0.1 -0.5 0.4 -0.2 0.2 -0.2
 6 |  2.8 0.6 0.5 0.4 0.2 0.4 0.2
 7 | -2.0 -0.7 0.2 0.0 -0.4 -0.3
 8 |  1.7 0.8 0.0 0.2 0.2
 9 | -0.6 -1.2 -0.5 -0.2
 10 |  1.4 0.9 -0.2
 11 | -0.2 -1.2
 12 |  1.2

A corner of values which can be viewed as plausible realisations of the can be observed. This corner corresponds to the rows 3, 4, … and the columns 2, 3, …, leading us to select the ARMA(2, 1) model. The automatic detection routine for corners of small values gives:

ARMA(P,Q) MODELS FOUND WITH GIVEN SIGNIFICANCE LEVEL
 PROBA   CRIT      MODELS FOUND
 0.200000  1.28  ( 2, 8)  ( 3, 1)  (10, 0)
 0.100000  1.64  ( 2, 1)  ( 8, 0)
 0.050000  1.96  ( 1,10)  ( 2, 1)  ( 7, 0)
 0.020000  2.33  ( 0,11)  ( 1, 9)  ( 2, 1)  ( 6, 0)
 0.010000  2.58  ( 0,11)  ( 1, 8)  ( 2, 1)  ( 6, 0)
 0.005000  2.81  ( 0,11)  ( 1, 8)  ( 2, 1)  ( 5, 0)
 0.002000  3.09  ( 0,11)  ( 1, 8)  ( 2, 1)  ( 5, 0)
 0.001000  3.29  ( 0,11)  ( 1, 8)  ( 2, 1)  ( 5, 0)
 0.000100  3.72  ( 0, 9)  ( 1, 7)  ( 2, 1)  ( 5, 0)
 0.000010  4.26  ( 0, 8)  ( 1, 6)  ( 2, 1)  ( 4, 0)

We retrieve not only the orders (P, Q) = (2, 1) of the simulated model but also other plausible orders. This is not surprising since the ARMA(2, 1) model can be well approximated by other ARMA models, such as an AR(6), an MA(11) or an ARMA(1, 8) (but in practice, the ARMA(2, 1) should be preferred for parsimony reasons).

A Pure GARCH

Consider a simulation of size n = 5000 of the GARCH(2, 1) model

5.13

where (η _t ) is a sequence of iid variables, ω = 1, α = 0.1, β ₁ = 0.05 and β ₂ = 0.8.

The table of Studentised statistics for the corner method is as follows:

.max(p,q).|.p..1....2....3....4....5....6....7....8....9...10...11...12...13...14...15...
    1 |  5.3 2.9 5.1 2.2 5.3 5.9 3.6 3.7 2.9 2.9 3.4 1.4 5.8 2.4 3.0
    2 | -2.4 -3.5 2.4 -4.4 2.2 -0.7 0.6 -0.7 -0.3 0.4 1.1 -2.5 2.8 -0.2
    3 |  4.9 2.4 0.7 1.7 0.7 -0.8 0.2 0.4 0.3 0.3 0.7 1.4 1.4
    4 | -0.4 -4.3 -1.8 -0.6 1.0 -0.6 0.4 -0.4 0.5 -0.6 0.4 -1.1
    5 |  4.6 2.4 0.6 0.9 0.8 0.5 0.3 -0.4 -0.5 0.5 -0.8
    6 | -3.1 -1.7 1.4 -0.8 -0.3 0.3 0.3 -0.5 0.5 0.4
    7 |  3.1 1.2 0.3 0.6 0.3 0.2 0.5 0.1 -0.7
    8 | -1.0 -1.3 -0.7 -0.5 0.8 -0.5 0.3 -0.6
    9 |  1.5 0.3 0.2 0.7 -0.5 0.5 -0.7
    10 | -1.7 0.1 0.3 -0.7 -0.6 0.5
    11 |  1.8 1.2 0.6 0.7 -1.0
    12 |  1.6 -1.3 -1.4 -1.1
    13 |  4.2 2.3 1.4
    14 | -1.2 -0.6
    15 |  1.4

A corner of plausible values is observed starting from the row and the column , which corresponds to GARCH(p, q) models such that (max(p, q), p) = (2, 2), that is, (p, q) = (2, 1) or (p, q) = (2, 2). A small number of other plausible values are detected for (p, q).

GARCH(p,q) MODELS FOUND WITH GIVEN SIGNIFICANCE LEVEL
 PROBA   CRIT      MODELS FOUND
 0.200000  1.28  ( 3, 1)  ( 3, 2)  ( 3, 3)  ( 1,13)
 0.100000  1.64  ( 3, 1)  ( 3, 2)  ( 3, 3)  ( 2, 4)  ( 0,13)
 0.050000  1.96  ( 2, 1)  ( 2, 2)  ( 0,13)
 0.020000  2.33  ( 2, 1)  ( 2, 2)  ( 1, 5)  ( 0,13)
 0.010000  2.58  ( 2, 1)  ( 2, 2)  ( 1, 4)  ( 0,13)
 0.005000  2.81  ( 2, 1)  ( 2, 2)  ( 1, 4)  ( 0,13)
 0.002000  3.09  ( 2, 1)  ( 2, 2)  ( 1, 4)  ( 0,13)
 0.001000  3.29  ( 2, 1)  ( 2, 2)  ( 1, 4)  ( 0,13)
 0.000100  3.72  ( 2, 1)  ( 2, 2)  ( 1, 4)  ( 0,13)
 0.000010  4.26  ( 2, 1)  ( 2, 2)  ( 1, 4)  ( 0, 5)

An ARMA‐GARCH

Let us resume the simulation of size n = 1000 of the ARMA(2, 1)‐GARCH(1, 1) model ( 5.8). The table of Studentised statistics for the corner method, applied to the SACRs of the observed process, was presented in Section 5.2.3. A small number of ARMA models, including the ARMA(2, 1), were selected. Let denote the residuals when an AR(p ₀) is fitted to the observations, the order p ₀ being selected using an information criterion. ⁴ Applying the corner method again, but this time on the SACRs of the squared residuals , and estimating the covariances between the SACRs by the multivariate AR spectral approximation, as described in Section 5.2.2 , we obtain the following table:

.max(p,q).|.p..1....2....3....4....5....6....7....8....9...10...11...12...
    1 |  4.5 4.1 3.5 2.1 1.1 2.1 1.2 1.0 0.7 0.4 -0.2 0.9
    2 | -2.7 0.3 -0.2 0.1 -0.4 0.5 -0.2 0.2 -0.1 0.4 -0.2
    3 |  1.4 -0.2 0.0 -0.2 0.2 0.3 -0.2 0.1 -0.2 0.1
    4 | -0.9 0.1 0.2 0.2 -0.2 0.2 0.0 -0.2 -0.1
    5 |  0.3 -0.4 0.2 -0.2 0.1 0.1 -0.1 0.1
    6 | -0.7 0.4 -0.2 0.2 -0.1 0.1 -0.1
    7 |  0.0 -0.1 -0.2 0.1 -0.1 -0.2
    8 | -0.1 0.1 -0.1 -0.2 -0.1
    9 | -0.3 0.1 -0.1 -0.1
    10 |  0.1 -0.2 -0.1
    11 | -0.4 0.2
    12 | -1.0

A corner of values compatible with the is observed starting from row 2 and column 2, which corresponds to a GARCH(1, 1) model. Another corner can be seen below row 2, which corresponds to a GARCH(0, 2) = ARCH(2) model. In practice, in this identification step, at least these two models would be selected. The next step would be the estimation of the selected models, followed by a validation step involving testing the significance of the coefficients, examining the residuals and comparing the models via information criteria. This validation step allows a final model to be retained which can be used for prediction purposes.

GARCH(p,q) MODELS FOUND WITH GIVEN SIGNIFICANCE LEVEL
 PROBA   CRIT      MODELS FOUND
 0.200000  1.28  ( 1, 1)  ( 0, 3)
 0.100000  1.64  ( 1, 1)  ( 0, 2)
 0.050000  1.96  ( 1, 1)  ( 0, 2)
 0.020000  2.33  ( 1, 1)  ( 0, 2)
 0.010000  2.58  ( 1, 1)  ( 0, 2)
 0.005000  2.81  ( 0, 1)
 0.002000  3.09  ( 0, 1)
 0.001000  3.29  ( 0, 1)
 0.000100  3.72  ( 0, 1)
 0.000010  4.26  ( 0, 1)

The Case Where the LM _n Statistic Takes the Form nR ²

Implementation of an LM test can sometimes be extremely simple. Consider a non‐linear conditionally homoscedastic model in which a dependent variable Y _t is related to its past values and to a vector of exogenous variables X _t by , where ε _t is iid and W _t  = (X _t , Y _t − 1,…). Assume, in addition, that W _t and ε _t are independent. We wish to test the hypothesis

where

To retrieve the framework of the previous section, let R = [0 _{s × (d − s)} : I _s ] and note that

where Σ _λ  = (ℑ ²²)⁻¹ and ℑ ²² = Rℑ ⁻¹ R ^′ is the bottom right‐hand block of ℑ ⁻¹ . Suppose that does not depend on ψ ₀ . With a Gaussian likelihood (Exercise 5.9) we have

where ε _t (θ) = Y _t  − F _θ (W _t ), , , and

Partition ℑ into blocks as

where ℑ ₁₁ and ℑ ₂₂ are square matrices of respective sizes d − s and s . Under the assumption that the information matrix ℑ is block‐diagonal (that is, ℑ ₁₂ = 0), we have where ℑ ₂₂ = RℑR ^′ , which entails Σ _λ  = ℑ ₂₂ . We can then choose

as a consistent estimator of Σ _λ . We end up with

5.18

which is nothing other than n times the uncentred determination coefficient in the regression of on the variables for i = 1,…, s (Exercise 5.10).

LM Test with Auxiliary Regressions

We extend the previous framework by allowing ℑ ₁₂ to be not equal to zero. Assume that σ ² does not depend on θ . In view of Exercise 5.9, we can then estimate Σ _λ by ⁵

where

Suppose the model is linear under the constraint H ₀ , so that

with

up to some negligible terms.

Now consider the linear regression

5.19

Exercise 5.10 shows that, in this auxiliary regression, the LM statistic for testing the hypothesis

is given by

This statistic is precisely the LM test statistic for the hypothesis H ₀ : ψ = 0 in the initial model. From Exercise 5.10, the LM test statistic of the hypothesis in model (5.19) can also be written as

5.20

where , with . We finally obtain the so‐called Breusch–Godfrey form of the LM statistic by interpreting as n times the determination coefficient of the auxiliary regression

5.21

where is the vector of residuals in the regression of Y on the columns of F _β .

Indeed, in the two regressions ( 5.19) and (5.21), the vector of residuals is , because and . Finally, we note that the determination coefficient is centred (in other words, it is the R ² which is provided by standard statistical software) when a column of F _β is constant.

Quasi‐LM Test

When ℓ _n (θ) is no longer supposed to be the log‐likelihood, but only the quasi‐log‐likelihood (a thorough study of the quasi‐likelihood for GARCH models will be made in Chapter 7), the equations can in general be replaced by

5.22

where

It is then recommended that the statistic (5.17) be replaced by the more complex, but more robust, expression

5.23

where and are consistent estimators of I and J . A consistent estimator of J is obviously obtained as a sample mean. Estimating the long‐run variance I requires more involved methods, such as those described in Section 5.2.2 (HAC or other methods).

Equivalence with a Portmanteau Test

Using

it follows from relation (5.25) that

5.26

which shows that the LM test is equivalent to a portmanteau test on the squares.

Expression in Terms of R ²

To establish a connection with the linear model, write

where Y is the n × 1 vector , and X is the n × (q + 1) matrix with first column and (i + 1)th column . Estimating I by , where , we obtain

5.27

which can be interpreted as n times the determination coefficient in the linear regression of Y on the columns of X . Because the determination coefficient is invariant by linear transformation of the variables (Exercise 5.11), we simply have LM _n  = nR ² where R ² is the determination coefficient ⁷ of the regression of on a constant and q lagged variables . Under the null hypothesis of conditional homoscedasticity, LM _n asymptotically follows a . The version of the LM statistic given in expression (5.27) differs from the one given in expression ( 5.25) because relation ( 5.24) is not satisfied when I is replaced by .

5.1 (Asymptotic behaviour of the SACVs of a martingale difference)

Let (ε _t ) denote a martingale difference sequence such that and . By applying Corollary A.1, derive the asymptotic distribution of for h ≠ 0.