Part V Multivariate Time Series Models 429
5.10 Robust hypothesis testing in models with serially correlated/heteroskedastic errors
Kiefer, Vogelsang, and Bunzel (2000) proposed an alternative approach which uses a data trans- formation to deal with the error serial correlation problem and which does not explicitly require a NW type variance matrix estimator. The procedure consists of applying a nonsingular stochas- tic transformation to the OLS estimates, so that the asymptotic distribution of the transformed estimates does not depend on nuisance parameters. Let
Cˆt = 1 T2
T t=1
ˆ stˆst, where
ˆ st=
t j=1
xtuˆt,
anduˆt =yt−βˆxt. Now define Mˆ =
⎛
⎝1 T
T j=1
xtxt
⎞
⎠
−1
Cˆ1/2t ,
whereCˆ1/2t represents a lower triangular Cholesky factor ofCˆt. Kiefer, Vogelsang, and Bunzel (2000) establish that as T → ∞,
ˆ M−1√
T
βˆ −β d
→Z−1k bk(1), (5.66)
where Zkis a lower triangular Cholesky factor of Pkdefined by Pk=
-
[bk(r)−rbk(1)] [bk(r)−rbk(1)]dr, (5.67) and bk(r)denotes a k-dimensional vector of independent standard Wiener processes.5This transformation results in a limiting distribution that does not depend on nuisance parameters.
However, the distribution of Z−1k bk(1)is non-standard, although it only depends on k, the number of regression coefficients being estimated. Critical values have been computed by simu- lation by Kiefer, Vogelsang, and Bunzel (2000). The main advantage of this approach compared
5A brief account of Wiener processes is provided in Section B.13 Appendix B.
116 Introduction to Econometrics
with standard approaches is that estimates of the variance-covariance matrix are not explicitly required to construct the tests. Further, Kiefer, Vogelsang, and Bunzel (2000) show that tests constructed using their procedure can have better finite sample size properties than tests based on consistent NW estimates.
Kiefer and Vogelsang (2002) showed that the above approach is exactly equivalent to using NW standard errors with Bartlett kernel, and without truncation (namely, setting m = T in (5.65)). This result suggests that valid tests can be constructed using kernel based estimators with bandwidth m=T.
Kiefer and Vogelsang (2005) studied the limiting distribution of robust tests based on the NW estimators setting m=bãT, where b∈(0, 1] is a constant, labelling the asymptotics obtained under this framework as ‘fixed-b asymptotics’. The authors showed that the limiting distribution of the F- and t-statistics based on such NW variance estimator are non-standard, and that they depend on the choice of the kernel and on b. Kiefer and Vogelsang (2005) have also analysed the properties of these test statistics via a simulation study. Their results indicate a trade-off between size distortions and power with regard to choice of the bandwidth. Smaller bandwidths lead to tests with higher power but at the cost of greater size distortions, whereas larger bandwidths lead to tests with smaller size distortions but lower power. They also found that, among a group of common choice kernels, the Bartlett kernel leads to tests with highest power in their fixed-b framework.
Phillips, Sun, and Jin (2006) suggested a new class of kernel functions obtained by exponenti- ating a ‘mother’ kernel (such as the Bartlett or Parzen lag window), but without using lag trunca- tion. When the exponent parameter is not too large, the absence of lag truncation influences the variability of the estimate because of the presence of autocovariances at long lags. Such effects can have the advantage of better reflecting finite sample behavior in test statistics that employ NW estimates, and leading to some improvement in test size, as also reported in a simulation study by Phillips, Sun, and Jin (2006).
While this approach works well, Kapetanios and Psaradakis (2007) note that it does not exploit information on the structure of the dependence in the regression errors. However, such information may be used to improve the properties of robust inference procedures. Hence, the authors suggest to employ a feasible GLS estimator where the stochastic process generating disturbances is approximated by an autoregessive model with an order that grows at a slower rate than the sample size (see also Amemiya (1973) on such approximation). Specifically, let ˆ
ut =yt−βˆxt, whereβˆis an initial consistent estimator ofβ. For some positive integer p chosen as a function of T so that p→ ∞and p/T →0 as T→ ∞, letφˆpp=
φˆp,1,φˆp,2,. . .,φˆp,p
be the pthorder OLS estimator of the autoregressive coefficients for
ˆ ut
, obtained as the solu- tion to the minimization of
φp,1,φp,2min,...,φp,p∈Rp
T−p−1 T
t=p+1
uˆt−φp,1uˆt−1−φp,2uˆt−2. . .−φp,puˆt−p 2
.
A feasible GLS estimator ofβmay then be obtained as βˆ =
XˆˆX −1
Xˆˆy, (5.68)
whereˆ is the T−p
×T matrix defined as
ˆ =
⎛
⎜⎜
⎜⎜
⎝
− ˆφp,p − ˆφp,p−1 − ˆφp,p−2 . . . − ˆφp,1 1 . . . 0 0
0 − ˆφp,p − ˆφp,p−1 . . . − ˆφp,2 − ˆφp,1 . . . 0 0
... ... ... ... ... ... ... ... ...
0 0 0 . . . − ˆφp,p − ˆφp,p−1 . . . − ˆφp,1 1
⎞
⎟⎟
⎟⎟
⎠.
Note that (5.68) can be obtained by applying OLS to the regression of ˆy∗t = 1−p
j=1φˆp,jLj yt
onxˆ∗t = 1−p
j=1φˆp,jLj xt.
One drawback of NW-type estimators is that they cannot be employed to obtain valid tests of the significance of OLS estimates when there are lagged dependent variables in the regressors, and errors are serially correlated. The problem is that these procedures require the OLS esti- mator to be consistent. However, as formally proved in Section 14.6, in general OLS estimators will be inconsistent when errors are autocorrelated and there are lagged values of the dependent variable among the regressors. One possible way of dealing with this problem would be to use an instrumental variables (IV) approach for estimating consistently the parameters of the regres- sion model, and then obtain IV-based robust tests. As an alternative, Godfrey (2011) suggested a joint test for misspecification and autocorrelation using the J-test approach by Davidson and MacKinnon (1981) and introduced in Chapter 11 (see Section 11.6). Suppose that the valid- ity of M1is to be tested using information about M2, and that the regressors xtand ztin models (11.22)–(11.23) both contain at least one lagged value of yt. Also suppose that the autoregressive or moving average model of order m is used as the alternative to the assumption of independent errors. The author suggested a heteroskedasticity-robust joint test of the(1+m)restrictions λ=φ1=. . .=φm =0, in the ‘artificial’ OLS regression
yt=β1xt+λ(βˆ2zt)+φ1uˆ1,t−1+. . .+φmuˆ1,t−m+ t, (5.69) whereuˆ1,t−jis a lagged value of the OLS residual from estimation of (11.22) when(t−j) >0 and is set equal to zero when(t−j)≤0. More specifically, Davidson and MacKinnon (1981) pro- posed using the Wald approach, combined with heteroskedasticity-consistent variance (HCV) estimator:
τJ=
λ,ˆ φˆ1,. . .,φˆm
R1CˆJR1
−1
λ,ˆ φˆ1,. . .,φˆm
, (5.70)
where R1=(0, Im+1)is a(1+m)×k1matrix,λ,ˆ φˆ1,. . .,φˆmare OLS estimates ofλ,φ1,. . .,φm
in (5.69), and
ˆ CJ=
T
t=1
ˆ rtˆrt
T t=1
ˆ u21trˆtˆrt
T t=1
ˆ rtrˆt
, (5.71)
118 Introduction to Econometrics
withˆrt =
xt,βˆ2zt,uˆ1,t−1,. . .,uˆ1,t−m
. Note that, under the null hypothesis, the OLS esti- mators of the artificial alternative regression are consistent and asymptotically normal. It follows thatτJ ∼χ2m+1. Recent work has indicated that, when several restrictions are under test, the use of asymptotic critical values with HCV-based test statistics produces estimates of null hypothe- sis rejection probabilities that are too small (see Godfrey and Orme (2004)). To overcome this problem, Davidson and MacKinnon (1981) suggested a bootstrap implementation of the above test, using the wild bootstrap method.