Testing constancy of unconditional variance in volatility models by misspecification and specification tests

3.1 Testing constancy in the GARCH framework

As discussed for example in Amado and Teräsvirta (in press), the first step in generalizing a GARCH model into a TV–GARCH model is to test stability of the unconditional variance over time. This is important because the model defined by (5) and (6) is not identified when the null hypothesis holds, that is, when g_t≡1. In what follows, for simplicity we consider the first-order GARCH model, p=q=1 in (4), assume that r=1 under the alternative in (4) and choose γ₁=0 as our null hypothesis. With this choice of H₀, δ₁ and c₁ are unidentified nuisance parameters under H₀. This being the case, standard asymptotic inference is invalid. We circumvent this identification problem as in Luukkonen, Saikkonen and Teräsvirta (1988) and approximate the transition function by a third-order Taylor expansion around γ₁=0. After merging terms and reparameterising this yields

where R_3t is the remainder and θj=γ1jδ¯j with δ¯j≠0,j=0, 1, 2, 3. The new null hypothesis is thus H′0:θ=0.

Constructing an LM-type test for testing H′0 has the advantage that the model is only estimated under the null hypothesis, and then R_3t=0. This leads to standard asymptotic inference: the LM statistic has an asymptotic χ²-distribution with four degrees of freedom when the null hypothesis holds.

The approximate log-likelihood for observation t has the form

where gt∗=1+θ′t∗. Set α=(α₀, α₁, β₁)′ and denote r^1t=∂ℓ^t/∂α=(ζ^t2−1)h^t−1∂h^t/∂α, where ℓ^t is ℓ_t evaluated under H′0, h^t, equals h_t estimated under H′0, ζ^t2=εt2/h^t, and ∂h^t/∂α=m^t−1+β^1∂h^t−1/∂α is ∂h_t/∂α evaluated under H′0 with m^t=(1, εt2, h^t). Furthermore, let r^2t=∂ℓ^t/∂θ=(ζ^t2−1)g^t∗−1∂gt∗/∂θ=(ζ^t2−1)t∗, where g^t∗=1.

The test can be carried out in stages:

1. Estimate the GARCH(1,1) model, save the squared standardized residuals ζ^t2 and construct the “residual sum of squares” SSR0=∑t=1T(ζ^t2−1)2.

2. Regress ζ^t2−1 on h^t−1∂h^t/∂α and t* and form the residual sum of squares SSR₁.

3. Compute the test statistic

   (9)LM=TSSR0−SSR1SSR0.

[Correction added after online publication (23 April 2016): The stage 2 description in the previous paragraph has been changed from {Regress ζ^t2−1 on r^1t and r^2t and form the residual sum of squares SSR₁.} to the current version]

Under H₀, LM has an asymptotic χ²-distribution with four degrees of freedom.

The value of the test statistic can also be computed from the conventional quadratic form of the χ²-statistic. Our simulations suggest, however, that the TR² form is numerically more stable when the sum α₁+β₁ in (4) assuming p=q=1 is close to (but below) one. The simulation results we report are based on the TR² form of the statistic.

3.2 Testing constancy without specifying the conditional variance component

Another way of testing the constancy of the unconditional variance is to do it before specifying the conditional variance component. The LM-type test would thus serve as a specification tool and not as a misspecification test as it was introduced in Amado and Teräsvirta (in press) and presented in the preceding section. This implies setting h_t=1 in (8). The transition function is defined as

where G₁(t/T; γ₁, c₁) is defined as in (6) and δ₀>0 is a free parameter because h_t=1. The null hypothesis is γ₁=0, and in testing (10) is approximated by (7). Now, since δ₀ is a free parameter, the null hypothesis is H″0:θ1=θ2=θ3=0 in (7). The approximate log-likelihood for observation t equals (8) with h_t=1. The maximum likelihood estimator of the free intercept δ₀ equals δ^0=T−1∑t=1Tεt2. Then r^1t=∂ℓ^t/∂θ0=(εt2/δ^0−1)δ^0−1 and r^2t=∂ℓ^t/∂θ=(εt2/δ^0−1)gt∗−1∂gt∗/∂θ=(εt2/δ^0−1)δ^0−1t∗, where θ=(θ₁, θ₂, θ₃)′ and t^*=(t/T, (t/T)², (t/T)³)′. Stages 1 and 2 of the LM-type test are as follows:

1. Estimate the model (2) and (3) with h_t=1 and g_t=δ₀, save the standardized squared residuals ϕ^t2=εt2/δ^0 and construct the “residual sum of squares” SSR0=∑t=1T(ϕ^t2−1)2.

2. Regress ϕ^t2−1 on δ^0−1 and δ^0−1t* and form the residual sum of squares SSR₁.

[Correction added after online publication (23 April 2016): The stage 2 description in the previous paragraph has been changed from {Regress ϕ^t2−1 on r^2t and form the residual sum of squares SSR₁.} to the current version]

The final stage consists of computing the TR² form of the statistic as in (9). The test statistic has an asymptotic χ²-distribution with three degrees of freedom when the null hypothesis holds. The assumption of independence of ε_t is likely to be violated in practice, however, because {ε_t} typically contains conditional heteroskedasticity. If this is the case, the relevant critical values of the null distribution of the test statistic have to be found by simulation. This will be discussed in Section 4.3.

4.1 Critical values for the LM-type test

In this section we address the size discrepancy of the LM-type test of Amado and Teräsvirta (in press). In the simulations reported in that paper it turned out that the test was somewhat oversized even in relatively large samples. This concerns the test the authors called non-robust, which is the one we are going to consider. In order to correct the size of the test, we compute the relevant critical values for the test statistic by simulation. The standard way of doing this is as follows:

1. Generate T observations from the GARCH(1,1) model we are simulating, estimate the parameters using these observations and compute the value of the test statistic.

2. Draw T variables zt(1), t=1, …, T, with replacement from the population consisting of the estimated residuals z^t and use them and the estimated conditional variances h^t to obtain a new set of observations εt(1)=zt(1)h^t1/2, t=1, …, T.

3. Fit the GARCH(1,1) model to this series and the compute the value of the test statistic. Repeat step 2 and this step B times. This yields one estimate of the critical value(s).

4. Repeat steps 1–3 K times and compute the critical value of interest as the mean of the values resulting from these K replications. We set K=5000.

This method is time-consuming because it requires estimating KB GARCH models. Besides, adjusting the size is just a prelude to power simulations, which are our main object of interest. In order to save time, we adopt the warp-speed bootstrap by Giacomini, Politis, and White (2013). It differs from the ordinary bootstrap in one important respect. Instead of B bootstraps for each replication, only one is performed (B=1). The authors show that the warp-speed bootstrap is practically as efficient as the standard one.

In the ensuing power simulations we generate 1000 replications for T=1000, 2500 and 5000. Because estimation of GARCH models is numerically difficult (estimates unreliable) in small samples, a few realizations have to be discarded when T=1000. The simulated critical values of the LM-type test, a χ²-test with four degrees of freedom, for the three sample sizes and the significance levels α=0.01 and 0.05, are computed separately for each experiment and can be found in Table 1. It is seen that the misspecification LM-type test is indeed size distorted and that the problem becomes worse when the kurtosis of the GARCH process increases. This suggests that distinguishing between GARCH and changes in the unconditional variance can become quite difficult when the GARCH process generates clusters with a large amplitude, when the changes present are rather modest, and when the number of observations is not very large. Results of power simulations in the next section seem to support this conclusion.

4.2 Power simulations 1: the misspecification test

A central assumption of the TV–GARCH model is that the changes in the unconditional variance can be smooth. It is of interest to consider the behavior of tests of constant unconditional variance for different degrees of smoothness measured by the slope parameter γ. It is equally interesting to consider the effect of the size of the switch or switches, measured by δ₁, on the power of the test. In addition, the effect of the GARCH parameters α₁ and β₁ (and thus the kurtosis of ϕ_t) on the power of the test should be investigated as well.

Two sets of parameters in (4) are considered. First, the parameters α₀=0.05, α₁=0.09, and β₁=0.9 form a “big GARCH”: kurtosis of ϕ_t equals 16.1. Second, the parameters α₀=0.05, α₁=0.05, and β₁=0.9 define an intermediate or “mild GARCH”: kurtosis of ϕ_t equals 3.16. The errors are standard normal, and the GARCH(1,1) model is tested against the approximate alternative (7). The true alternative is (5) with r=1 and K₁=1 in (6), i.e.

gt=1+δ1(1+exp{−γ1(t/T−c1)})−1, γ1>0.

The test of H′0:θ=0: in (7) is carried out using the TR² form described in Section 3.1.

The adjusted critical values for the two cases appear in Table 1. It is seen that the size distortion is an increasing function of the kurtosis of the GARCH process. “Big GARCH” five and one per cent critical values are considerably larger than those of “mild GARCH”. This leads one to expect that the size-adjusted power of LM-type test is lower for “big GARCH” than it is for “mild GARCH”.

Since the test appears to be consistent, it is of interest to find out what happens to the power in finite samples when the parameters δ₁ and γ=e^η in the alternative are varied. (We reparameterise γ₁ following Goodwin, Holt, and Prestemon (2011) and Hurn, Silvennoinen, and Teräsvirta (in press) and vary η.) The size-adjusted power of the test at significance levels 0.05 and 0.01 for various values of η and δ₁ when h_t=1 and T=2500, 5000, will be reported for both designs, see Tables 2 and 3. The results for T=1000 can be found at http://creates.au.dk/research/creates-research-papers/supplementary-downloads/.

The results for “big GARCH” for T=2500 and c₁=0.5 in the top panel of Table 2 indicate that when δ₁≤2, the slope parameter η has little effect on the power of the test. This means that for small shifts, smoothness of the shift is not an important factor. Tables 4 and 5 show that for these values of δ₁, the GARCH parameters are well estimated on average. For larger values of δ1, α^1+β^1≈1. (See Hillebrand (2005, Table 1) for similar results when η=∞.) At the same time, for a given δ₁≥2 smooth shifts become easier to detect than abrupt ones. This is particularly clear at α=0.01. In other words, it is more difficult to find evidence for a sudden change in the amplitude of the clusters if the change is abrupt than it is when this change is gradual.

The location of the shift matters. It is seen from the mid-panel of Table 2 that when the mid-point of the positive shift is located early (c₁=0.2), a shift in the conditional variance is easier to detect than if a similar positive shift occurs halfway through the sample or towards the end; results for c₁=0.8 can be found in the bottom panel of Table 2. A smooth change, η=1, constitutes an exception. When the shift has an early location, the power as a function of η is nonmonotonic. It increases from η=1 to η=2 and decays thereafter. This phenomenon cannot be seen in the top or the bottom panel of the table. Furthermore, when c₁=0.2, the decrease in power of the test as a function of η is not fully monotonic. This is particularly clear when δ=2⁵ and α=0.01. There the decay does not begin before η=3.

The bottom panel of Table 2 also shows that a late positive shift (c₁=0.8) is generally difficult to detect unless it happens to be large but at the same time quite smooth (η=1). This may be explained by the fact that a smooth change can begin quite early, although its mid-point is located late in the sample. Evidence about the change thus stretches over a large part of the sample. But then, an analogous argument does not fully apply to the case c₁=0.2. As already pointed out, the results in the mid-panel of Table 2 indicate that the smoothest changes are not as easy to detect as the ones with η>1.

It should be pointed out, however, that these results are not invariant to the direction of the shift. In the reported simulations, the variance component changes from 1 to 1+δ₁, where δ₁>0, so the shift is positive. If the shift is negative and the change is from 1+δ₁ to 1, the conclusions drawn for c₁=0.8 are valid for c₁=0.2 and vice versa. Simulation results not reported here support this claim. In our simulations the shift is a monotonic function of the transition variable t/T. Nonmonotonic shifts are of course possible but are not considered here.

Table 3 contains results for the same designs for T=5000. As may be expected, the powers are higher than for T=2500, but the patterns visible in Table 2 repeat themselves here.

Next consider “mild GARCH”. Results of the power simulations when T=2500 can be found in Table 6. A comparison with the “big GARCH” results indicates that the decrease in kurtosis from 16.1 to 3.18 has a strongly positive effect on power. The power patterns found in the two tables are similar, however, in that even in Table 6 the power of the test decreases with increasing η (decreasing smoothness). The decrease is not very strong at α=0.05 because the power is generally quite high but is more clearly visible at the 1% level of significance.

The results in the mid- and bottom panels of Table 6 show that shifts occurring early (c₁=0.2) are more difficult to find than the ones with their mid-point located late in the sample (c₁=0.8) when the shift is small. When δ₁ is sufficiently large, these differences even out. The differences in power are more visible at α=0.01. An interesting detail in the bottom panel of the table (c₁=0.2) is that the power is nonmonotonic in δ₁ when η=1. This is most clearly seen for α=0.01. Furthermore, for fixed η≥3 and c=0.2, the power first increases, then decreases and then begins to increase again when δ₁ increases. This pattern is also best seen when α=0.01. With few exceptions when α=0.05, the power is highest for δ₁=2⁵ for all three values of c₁.

When T=5000, the power of the test is very close to one for all designs, see Table 7. The previous patterns are now somewhat harder to see because the power of the tests is considerably higher and for large values of δ₁ quite close to one. Differences between the empirical powers reported in this table and the corresponding values in Table 3 are quite remarkable. It seems that volatility in the GARCH component measured by the kurtosis of ϕ_t has a considerable effect on the size-adjusted power of the test. High volatility makes it difficult to detect fluctuations in the unconditional variance.

Table 8 contains the averages of the GARCH parameter estimates for “mild GARCH” for T=2500 and Table 9 reports the same averages for T=5000. While α₁ is reasonably well estimated for small shifts (δ₁≤2), the persistence parameter β₁ is consistently overestimated. The average of the sum α^1+β^1 reaches unity for δ₁≥2³ for c₁=0.2, 0.5, and remains slightly below one for c₁=0.8. An increase in the sample size from 2500 to 5000 has little effect on these values.

In practice, it is not possible to accurately adjust the size because the size distortion varies according to the (unknown) parameters of the null model. Previous experience suggests that after parameterising the unconditional variance and estimating the TV–GARCH model, the sum α^1+β^1 lies clearly below one. A rule of thumb would be to use the critical values determined for “mild GARCH”, but the size correction would in that case be only approximate.

Two solutions to this problem are available. It is possible to use the robustified LM-type test as defined by Amado and Teräsvirta (in press). Simulations in that paper suggest that it is clearly less size distorted than its nonrobust counterpart. Another alternative would be to test constancy of the unconditional variance before estimating the GARCH component. Doing so would prevent the GARCH parameter estimates from absorbing nonstationarity due to nonconstant unconditional variance. This possibility will be considered in the next section.

4.3 Power simulations 2: specification test

We simulate a model in which the conditional heteroskedasticity is generated using the “big GARCH” parameters: α₀=0.05, α₁=0.09 and β₁=0.9. However, in testing constancy of the unconditional variance, the GARCH component is ignored, i.e. it is assumed that h_t=1. The alternative to εt=ztδ01/2 is thus assumed to be εt=ztgt1/2 where g_t is defined in (5) with r=1 and (6) with K=1.

Since the sequence {ε_t} contains (neglected) conditional heteroskedasticity, the test assuming iid observations under H₀ may be oversized. Even here, the size is corrected using the warp-speed bootstrap, so only a single bootstrap replication is performed for each experiment. The following four steps are needed:

1. Generate T observations from the GARCH model we are simulating, estimate the intercept δ₀ using these observations and compute the value of the test statistic.

2. Draw T independent variables εt(1)=zt(1)δ^01/2, t=1, …, T, using the standard normal distribution for zt(1).

3. Estimate the intercept δ₀ from this series and the compute the value of the test statistic. This yields one estimate of the critical value(s).

4. Repeat the steps 1–3 K=5000 times and compute the critical value of interest as the mean of the values resulting from these K replications.

The critical values thus obtained can be found in Table 1. It is seen that the test is hardly size distorted, which means that the asymptotic critical values can be used for the sample sizes considered here.^[1] When conditional heteroskedasticity is present, estimating the GARCH component before testing constancy of the unconditional variance seems to be the single most important cause of the size distortion observed in the misspecification test.

The simulation results for T=2500 and c₁=0.5 show that the power is very close to one for all combinations of η and δ₁, the lowest value being equal to 0.977 for η≥4 and δ₁=1 at the 1% significance level. The situation hardly changes for c₁=0.2 or c₁=0.8. For this reason, no tables for these results are provided. We are able to conclude that it would be preferable to test constancy of the unconditional variance before modeling the GARCH component instead of doing it thereafter. The LM-type test would thus be a specification tool and not a misspecification test. One might even suggest that the whole deterministic unconditional variance component (up to the intercept) defined in (5) should be specified by sequential testing as in Amado and Teräsvirta (in press) but before modeling the conditional variance. This possibility will be investigated elsewhere.

Power simulations of this section suggest that the test could work well in much smaller samples and could therefore be used for testing constancy of the variance in models for the conditional mean. In order to consider this idea we repeated the previous simulations for T=50 but generated the data without any conditional heteroskedasticity. In that case the asymptotic theory is valid and no size correction is necessary. Table 1 shows that this is still true for T=50. The results appear in Table 10. The power is seems largely independent of η for η≥2. It does increase monotonically with δ₁ and c₁. This means that shifts occurring late are easier to detect than ones occurring early in the sample. This is true for a positive shift: for a negative one the conclusion is reversed. In the last two cases, c₁=0.5 and c₁=0.8, the power is already quite reasonable when the shift is sufficiently large. Doubling the length of the series (T=100; results not shown here) considerably increases the power. Summing up, the LM-type test considered here appears a promising tool in testing constancy of the variance in conditional mean models estimated from short time series. A further study of the LM-type test in this context would seem worthwhile but lies beyond the scope of this work.