Part V Multivariate Time Series Models 429
4.5 Diagnostic checks and tests of homoskedasticity
There exist three general methods that can be used to check the validity of the homoskedasticity assumption:
(i) Graphical methods
(ii) General non-parametric methods, such as the Goldfeld–Quandt test (iii) Parametric tests.
4.5.1 Graphical methods
The graphical approach simply involves plotting the squares of the OLS residuals against the square of fitted values (i.e.,ˆy2i), or against other variables thought to be important in explaining heteroskedasticity of the error variances. Identification of systematic patterns in such graphical displays can be viewed as casual evidence of heteroskedasticity.
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4.5.2 The Goldfeld–Quandt test
This test involves grouping the observations into m (m ≥ 2) categories and then testing the hypothesis that error variances are equal across these m groups. The test assumes that the error variances, Var(ui), are homoskedastic within each observation group. In practice, the observa- tions are placed into three categories and the test of the equality of error-variances is carried out across the first and the third category, thus ignoring the observations in the middle category. This is done to reduce the possibility of dependence between the estimates of Var(ui)over the first group of observations and over the third group of observations. Application of the Goldfeld–
Quandt test to the simple regression model comprises the following steps.
Step I: Split the sample of observations into three groups, so that Group 1: yi=αI+βIxi+uIi i=1, 2,. . .T1,
Group 2: yi=αII+βIIxi+uIIi i=T1+1,. . ., T1+T2, Group 3: yi=αIII+βIIIxi+uIIIi i=T1+T2+1,. . ., T, where T=T1+T2+T3.
Step II: Run the OLS regressions of yion xifor the first and the third groups separately. Obtain the sums of squares of residuals for these two regressions, and denote them by SSRI and SSRIII, respectively.
Step III: Construct the statistic
F= SSRI/ (T1−2) SSRIII/ (T3−2) = σˆ2I
ˆ σ2III, whereσˆ2I andσˆ2IIIare the unbiased estimates of Var
uIi
and Var uIIIi
, computed using the observations in groups I and III, respectively. It is convenient to compute the above F statistic such that it is larger than unity (by putting the larger estimate of the variance in the numera- tor), so that the test statistic is more directly comparable to the critical values in F Tables.
Under the null hypothesis of homoskedasticity, the above F-statistic has an F-distribution with T1−2 and T3−2 degrees of freedom. Large values of F are associated with the rejec- tion of the homoskedasticity assumption, and possible evidence of the heteroskedasticity. The Goldfeld–Quandt test readily generalizes to multivariate regressions and to more than three observation groups.
4.5.3 Parametric tests of homoskedasticity
The starting point of these tests is a parametric formulation of the heteroskedasticity, such as those specified by (4.11), (4.12), and (4.13). In the case of these specifications, the homoskedas- ticity hypothesis can be formulated in terms of the following null hypotheses:
1. Multiplicative specification:
H0:γ1=γ2,. . .,γp =0,
2. Additive specification:
H0:λ1=λ2= ã ã ã =λp=0, 3. Mean-variance specification:
H0:δ=0.
Any one of the three likelihood-based approaches discussed in Section 9.7 can be used to implement the tests. The simplest procedure to compute is the Lagrange multiplier (LM) method, since this method does not require the estimation of the regression model under heteroskedasticity. One popular LM procedure for testing the homoskedasticity assumption is based on an additive version of the mean-variance model, (4.13). The LM statistic is computed as the t-ratio of the slope coefficient in the regression ofuˆ2i on ˆ
y2i (including an intercept), whereuˆiare the OLS residuals andyˆiare the fitted values.
Under the null hypothesis of homoskedastic variances, this t-ratio is asymptotically dis- tributed as a standard normal variable. In small samples, however, it is more advisable to use critical values from the t-distribution rather than the critical values from the normal distribution.
An LM test of the homoskedasticity assumption based on the additive specification, (4.12), involves running the following OLS regression:
ˆ
u2i =α+λ1zi1+λ2zi2+ ã ã ã +λpzip+error, and then testing the hypothesis
H0:λ1=λ2= ã ã ã =λp =0, against
H1:λi=0, λ2=0,ã ã ã,λp=0,
using the F-test or other asymptotically equivalent procedures. For example, denoting the mul- tiple correlation coefficient of the regression ofuˆ2i on zi1, zi2,. . ., zipby R, it is easily seen that under H0 : λ1 = λ2 = ã ã ã = λp = 0, the statistic Tã R2is asymptotically distributed as aχ2with p degrees of freedom. This test is also asymptotically equivalent to the test pro- posed by Breusch and Pagan (1980), which tests H0against the more general alternative spec- ification:σ2i = f
α0+λ1zi1+ ã ã ã +λpzip
, where f(ã)could be any general function. The White (1980) test of homoskedasticity is a particular application of the above test where zij’s are chosen to be equal to the regressors, their squares and their cross products. For example, in the case of the regression equation
yi=α+β1xi1+β2xi2+ui, White’s test set, p=5 and
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zi1 =xi1, zi2 =xi2, zi3=x2i1, zi4=x2i2, zi5=xi1xi2.
A particularly simple example of the above testing procedure involves running the auxiliary regression
ˆ
u2i =constant+αˆy2i, (4.21)
and then testingα=0, againstα=0, using the standard t-test.