Asymptotic Properties of White s Test for Heteroskedasticity and the Jarque-Bera Test for Normality

Transcription

1 Asymptotic Properties of White s est for Heteroskedasticity and the Jarque-Bera est for Normality Carlos Caceres Nuffield College, University of Oxford Bent Nielsen Nuffield College, University of Oxford March 006 Abstract he aim of this paper is to analyse the asymptotic properties of White s test for heteroskedasticity and those of the Jarque-Bera test for normality from a time series perspective. For this purpose, we study the characteristics of these tests when applied to a first order autoregressive model with independent, identically and normally distributed errors. It will be proved here that White s test for heteroskedasticity is theoretically valid, and thus certainly applicable, in both the case of a stationary autoregressive process and in that of an autoregressive process with a unit root i.e. a marginally stable autoregressive process or random walk). However, it will be shown that White s test for heteroskedasticity is not valid when applied to an explosive autoregressive process. On the other hand, it is proved here that the Jarque-Bera test for normality is parameter independent. his means that this normality test can be applied to any first order autoregressive process, regardless of whether the latter contain a stationary, a unit or an explosive root. i

2 Contents Introduction. Notation White s test for Heteroskedasticity Jarque-Bera s test for Normality Assumptions Unit Root Case: α = 0 3 Explosive Case: α > 8 4 Conclusion 34 References 35 ii

3 Introduction In Halbert White s words 980), the presence of heteroskedasticity in an otherwise properly specified linear model leads to consistent but inefficient parameter estimates and faulty inference when testing statistical hypotheses. In other words, heteroskedasticity represents a serious problem in statistics and has to be taken into consideration when performing any econometric application that could be affected by the latter. herefore, many statisticians have put a lot of effort into the elaboration of diagnostic tests enabling accurate detection of heteroskedasticity. In particular, a widely and commonly used test for heteroskedasticity is that proposed by White himself 980). Nowadays, several econometrics software packages, such as PCGive, Stata and EViews, include this test - or an adaptation of it - amongst their key features. Nonetheless, White seems to have proposed this test without having had time series applications directly in mind. He only considered the case where the regressors were a sequence of independent random variables. hus this test was probably inspired by cross-section data analysis instead. he same applies to Amemiya s work 977) whose results also assumed the absence of lagged dependent variables. In fact, only few results are available regarding the theoretical validity of White s test for heteroskedasticity in a model including lagged endogenous variables. Kelejian 98) only proved that the test is applicable in the stable stationary) case, but did not consider a more general case. A similar result was presented by Godfrey and Orme 994). In addition the latter argue that White s test is ineffective in the presence of omitted variables. Doornik 996) suggested that under simplifying assumptions Kelejian s result reduces to a multivariate extension of White s test. He seemed to assert the legitimacy of the latter but did not explicitly specify any restriction on the parameters of interest. his also applies to the paper by Ali and Giaccotto 984). Wooldridge 999) mentioned that this and other heteroskedasticity tests have not yet been truly analysed in the unit root case for which the use of the Functional Central Limit heorem would be required. Another important mis-specification test is the normality test due to Jarque and Bera 980). Many econometric models and results are based on the fact that some variables follow a normal distribution. In particular the maximum likelihood estimator coincides with the OLS estimator when normality is assumed. hus the importance of having a theoretical test for normality that can be used in time series analysis. In the case of the Jarque-Bera test for normality, some studies in the past have analysed its validity when dealing with stationary autoregressive processes. Notably the contribution of Lutkepohl and Shneider 989). Other authors have expanded this issue to accommodate for more general models. Kilian and Demiroglu 000) proved the validity of the Jarque-Bera s normality test for Vector-Error Correction VEC) models and for unrestricted Vector Autoregression VAR) models with possibly integrated or cointegrated variables. Yet, theoretical results regarding the validity of the Jarque-Bera test for normality in the general case are not available. In this paper, the asymptotic properties of White s test for heteroskedasticity and that of the Jarque-Bera test for normality are analysed from a time series econometrics

4 perspective. Unfortunately, it is found that the validity of White s test for heteroskedasticity depends upon the value of the parameters of interest. In more concrete words, this test may work well for some values of these parameters and yet it may not work for others. It will be shown here that White s test for heteroskedasticity is indeed valid in the unit root case, but no longer valid in the explosive case. his is a significant theoretical result. It implies that in order to do statistical inference on the parameters of interest we need to know if heteroskedasticity is present in the model. But, to test for heteroskedasticity we need to know the value of these parameters! As a consequence, any applied statistician should not use this test unless he/she is completely certain of dealing with data from a stationary or marginally stable process. It is important to mention that this drawback is not a general feature of all misspecification tests. Remarkably, the results obtained here regarding White s heteroskedasticity test are, for instance, in contrast to the theoretical outcome obtained also in this paper for the Jarque-Bera normality test. he latter is in fact valid in all three, stationary, unit root and explosive cases. Another example would be the findings of Nielsen 00) regarding the tests for lag length. Nielsen proved that the methods used for order determination in general vector autoregressions are valid regardless of the values of the characteristic roots. In other words, the validity of these lag length tests does not depend upon the values of the parameters of interest. Again, this is not the case in White s heteroskedasticity test. Before presenting White s test for heteroskedasticity and the Jarque-Bera s test for normality in a formal way, some notation should be introduced.. Notation he following notation is used throughout this paper: N 0 denotes the set of all positive integers including zero) N denotes the set of all strictly positive integers Z denotes the set of all integers R denotes the set of all real numbers n N, [[, n]] = [, n] N R n n denotes the field of square matrices of dimension n and with general term in R R n denotes the set of vectors of dimension n and with all terms in R M R n n and V R n, M and V denote the transpose of M and V respectively

5 i.i.d. stands for independent and identically distributed Having introduced the above notation, we are now in the position to formally present White s test for heteroskedasticity in a time series context.. White s test for Heteroskedasticity As we said earlier, our workhorse in this paper is a first order autoregressive model or AR) model). hus, lets start by defining the latter in a more formal way. Let X 0, X,..., X ) be a one-dimensional time series satisfying the first order autoregressive equation: X t = αx t + ε t for t =,..., ) where ε t ) is a sequence of i.i.d. normally distributed random shocks and α R represents the parameter of interest i.e. the parameter space). In order to test for heteroskedasticity in the AR) model presented above, White s idea is basically equivalent to the use of the following auxiliary model : ˆε t = β 0 + β X t + β X t + u t for t =,..., ) where ˆε t ) are the least square residuals from regression ). In this sense, one needs to consider the constant-adjusted squared multiple correlation coefficient R from the regression exhibited in equation ). his is given by: Where: R = S 00.S 0.S.S 0 3) S 00 R and S 00 = t= R 0t S 0 R and S 0 = t= R 0t.R t S 0 R and S 0 = S 0 And: S R and S = t= R t.r t R 0t R and R 0t = ˆε t ˆε ) X t X R t R and R t = Xt X where the bar over a random variable denotes the sample mean of that random variable, i.e. X = t= X t or the second sample moment in the case of ˆε = t= ˆε t 3

6 and X = t= X t. herefore, the main components of the significant statistic S 00, S 0 and S presented above can be decomposed in the following way: And: S = t= = = S 00 = t= ˆε t ˆε ) = ) t= ˆε4 t t= ˆε t S 0 = Xt X t= ˆε t ˆε ) Xt X = Xt X X t X t= ˆε t ˆε )X t X) t= ˆε t ˆε )Xt X ) ) Xt X Xt X t= X t X) t= X t X)Xt X ) t= X t X)Xt X ) t= X t X ) ) t= X t X t= X3 t X X t= X3 t X X t= X4 t X ) And, assuming that model ) is true we have ˆε t = ε t ˆα α)x t, with ˆα α) given by: ˆα α) = t= X t ε t t= X t 4) Hence the above expressions for S 00, S 0 and S can be further expanded using the binomial rule to obtain: 4

7 S 00 = ) t= ε4 t 4ˆα α) 4ˆα α) 3 S 0 = And: t= X t ε 3 t ) ) t= X3 t ε t + ˆα α) 4 t= X4 t t= ε t P t= X t ε t ) P t= X t ) t= X t X)ε t ) + ˆα α) ) ) + 6ˆα α) t= X t ε t t= X3 t t= X t X )ε t ) + ˆα α) t= X4 t ˆα α) ˆα α) S = ) t= X t ε t ˆα α) X.X + ˆα α) ) t= X3 t ε t ˆα α) X + ˆα α) ) ) X X t= X t X t= X3 t X X t= X3 t X X t= X4 t X t= X t ε t ) t= X t ε t ) 5) 6) 7) Note that despite the presence of several terms in the expanded form of components S 00 and S 0, it will be shown that only few of these terms play and important role in the asymptotic distribution of S 00 and S 0 i.e. many of these terms become negligible as increases). In fact, it will be shown that a key feature of the explosive case is that terms that would have been otherwise negligible will in that case have an effect upon the asymptotic distribution of S 0 and thus upon that of R. Finally, under the null i.e. β = β = 0), White s test for heteroskedasticity should provide the following result: R d χ ) 8) In other words, if the null hypothesis is true, and thus heteroskedasticity is rejected, then the R above should have a chi-square as a limiting distribution. It is worthwhile mentioning here that several different significant statistics have been analysed in the time series literature. Most statisticians and econometricians have been particularly concerned with the asymptotic properties of the estimators of the parameter α i.e. Maximum Likelihood Estimator MLE), Least Squares Estimator). hese are driven by: 5

8 ˆα α) = t= X t ε t t= X t for which the asymptotic results in the stationary case are universally known and are a direct consequences of the Law of Large Numbers LLN) for serially dependent processes and the Central Limit heorem CL) for martingale differences c.f. Hamilton 994), Nielsen 004) among others). Nonetheless, the asymptotic properties of these estimators have been largely studied for unstable autoregressive processes as well. Notably, the contributions of Rubin 950), Anderson 959), Lai and Wei 983), Chan and Wei 988), Jeganathan 988) and Nielsen 005). On the other hand, it can be seen upon examination of the components S 00, S 0 and S presented in equations 5), 6) and 7) that a particular feature of the significant statistic of White s test is the presence of higher powers, for instance: t= X4 t, t= X3 t, t= X3 t ε t and t= X t ε t ) among others. hese kinds of terms have not been analysed in the general case in the literature. More precisely, most of the techniques used in this paper are known but have not been applied to this particular problem before. his represents a key attribute of the analysis presented here, both for White s test for heteroskedasticity and for the Jarque-Bera normality test to be presented in the next sub-section..3 Jarque-Bera s test for Normality Consider again the one-dimensional time series satisfying the first order autoregressive equation ). And provided that this is the true model the residuals are given by ˆε t = ε t ˆα α)x t, with ˆα α) defined as in equation 4). he Jarque-Bera normality test is based on the idea of analyzing the asymptotic properties of the following two statistics: And, ˆK 3 = [ t= ˆε t ˆε) 3 ] 3/ = S 3 t= ˆε t ˆε) S 3/ 9) Where: ˆK 4 = [ t= ˆε t ˆε) 4 ] 3 = S 4 3S t= ˆε t ˆε) 0) S 6

9 S = t= ˆε t ˆε) = t= ˆε t ) t= ˆε t S 3 = t= ˆε t ˆε) 3 = t= ˆε3 t 3 t= ˆε t ) ) t= ˆε t + ) 3 t= ˆε t S 4 = t= ˆε t ˆε) 4 And S 4 3S = t= ˆε4 t 4 6 t= ˆε t t= ˆε t ) 4 3 ) t= ˆε t ) t= ˆε3 t ) + ) ) t= ˆε t t= ˆε t hen, under the null i.e. results: the ε t s are normally distributed) we have the following ˆK 3 6 d χ ) ˆK 4 4 d χ ) ) ˆK 3 + ˆK 4 d χ 6 4 ) his means that when we fail to reject the null hypothesis of normality, the above three statistics should have a chi-square as limiting distribution. Now, as mentioned previously, it will be shown in this paper that White s method for testing for heteroskedasticity works well in the stationary and unit root cases, but not in the general case. Here the general case denotes all three stationary, unit root and explosive cases together. Additionally, it will be proved that the Jarque-Bera normality test works well in all three cases. hus this paper is organised in the following way: the validity of White s test for Heteroskedasticity and that of the Jarque-Bera test for Normality will be analsed when dealing with a marginally stable process i.e. unit root case). his is going to be shown explicitly in heorems. and. in section. hen, heorem 3. in section 3 will reveal that White s test for heteroskedasticity is not 7

10 valid in the explosive case. Whilst, heorem 3. will demonstrate that the Jarque-Bera normality test is nevertheless valid in the explosive case. Following the outline just described, it is worth emphasizing here that for each of the two cases studied in this paper i.e. unit root and explosive cases) a different distribution theory is required. he stationary case as shown by previous authors) is based on the application of the Law of Large Numbers LLN) and the Central limit heorem CL) to stationary ergodic mixing) autoregressive processes and ergodic martingale differences respectively. In the unit root case, heorems. and. are consequences of the Functional Central Limit heorem FCL), the Continuous Mapping heorem and a result provided by Ibragimov and Phillips 004). Last but not least, heorems 3. and 3. which represent the main results in the explosive case are based on a different theory from the previous two. As pointed out by Nielsen 005), this theory is based on the use of strong consistency arguments rather than weak consistency and weak convergence arguments used in non-explosive time series. Furthermore, it will be pointed out that the order of magnitude of the process X t ) t N varies according to the case under consideration i.e. according to the values of the parameter α). Before proceeding to the presentation of the mentioned theorems, it is important at this point to introduce the main assumptions that are used in the rest of the paper unless stated otherwise..4 Assumptions hroughout the entire paper, the following two assumption are used: Assumption.. ε t ) t is a sequence of i.i.d. normally distributed random variables with mean zero and variance one. Assumption.. X 0 = 0 Note that assumption. could be modified so that variance equal to one could be replaced by constant variance σ. his is due to the scale invariance property of this particular problem and simplifies the presentation thereafter. In other words, the distribution of the significant statistic R is unaffected when the variance σ is scaled to one. Additionally, it is worth mentioning that most of the results presented in this paper still hold if we further relax assumption.. Notably if i.i.d.-ness is replaced by letting ε t ) t N be a martingale difference sequence with respect to an increasing sequence of σ-fields F t ) t N where F t denotes the so called natural filtration). his is due to the fact that as mentioned earlier) most of the theorems used here: Law of Large Numbers, CL s, Functional CL s FCL s), etc, are applicable to martingale differences. However, the fact that White s test for heteroskedasticity is not valid in the explosive case using assumption. is a sufficient condition to prove that this test is not generally valid when ε t ) t N is a martingale difference sequence. hus assumption. is apposite here. 8

11 Assumption. is used for simplicity only. he latter could be effectively replaced by the assumption that X 0 is fixed and the main results would be unchanged. However the presentation would become more cumbersome due to the extra term. It is therefore convenient to use assumption. henceforth. his means that X t ) is taken to be an autoregressive model without an intercept. 9

12 Unit Root Case: α = he characteristics of this test for heteroskedasticity have been considered previously by a number of researchers in the stationary case. Kelejian 98) has shown that White s test for heteroskedasticity is valid in this case using a stationary p-dimensional vector autoregressive process a VAR) model). A similar result was presented by Godfrey and Orme 994). In this section, the asymptotic properties of White s test for heteroskedasticity in the unit root or marginally stable) case are analysed. herefore the findings presented in this section are ground-breaking in the sense that, as mentioned by Wooldridge 999), the properties of White s test have not been studied formally in the unit root case before. One of the main results of this section, which is the validity of White s test for heteroskedasticity when applied to a first order autoregressive process with a unit root i.e. a random walk), is presented in the following theorem: heorem.. Let R be defined as in equation 3). If assumptions. and. are satisfied and if α =, then R d χ ) ) he second main result of this section is that of the validity of the Jarque-Bera normality test when dealing with a first order autoregressive process with a unit root as in the previous theorem. his point is presented in the following theorem: heorem.. Let ˆK 3 and ˆK 4 be defined as in equation 3). If assumptions. and. are satisfied and if α =, then ˆK 3 d χ ), ˆK 4 d ˆK χ ) and ˆK ) 4 d χ ) ) 4 In order to establish the proofs for the above theorems, the following three lemmas are required. Lemma.. Let ε t ) t N be a sequence of i.i.d. random variables satisfying assumption.. herefore, on the space D[0, ] of right continuous functions with left limits, we have: [ ] t= [ ] ε t, ε t ) t= d B, B ) 3) Also: [ ] [ ] ε t ), t= t= ε 3 t d B, B 3 ) 4) 0

13 where B, B and B 3 are three Brownian motions, and B is independent of both B and B 3. he above lemma is a consequence of the Functional Central Limit heorem FCL) for martingale difference sequence. his theorem can be found in Nielsen 004). Proof of lemma.. : Let ε t ) t N be a sequence of i.i.d. normally distributed random variables with mean zero and variance one. Also, let u t ) t N be a sequence of i.i.d. random variables with mean zero and variance K0 K 0 constant). Let us define S = t= E [u t ] = K0. Note that u t ) t N can in turn be equal to the sequences ε t ) t N, ε t ) t N and ε 3 t ) t N. In that case u t ) t N is clearly a martingale difference and satisfies the following three conditions: i) ii) iii) t= u t S = K0 t= u t P u max t t [[, ]] S = P K 0 max t [[, ]] u t 0 since u t = O p ) ) [ r] t= E u t = [ r] Eu t) S t= K0 = [ r] = r frac r) r where, r) R + [0, ], [ r] is the integer value of r and frac r) is the mantissa of r defined as frac r) = r [ r]. herefore, from the functional central limit theorem on the space D[0, ] of right continuous functions with left limits or CadLag functions), we have: S [ ] t= u t = K 0 [ ] t= u t d B 5) where B is a Brownian motion. Now, setting u t ) t N equal to the sequences ε t ) t N, ε t ) t N and ε 3 t ) t N respectively and using the result presented in 5) we obtain that: [.] t= t= ε t d B. 6) [.] ε d t ) B. 7) [.] t= ε 3 t d B 3. 8)

14 where B., B. and B 3. are three Brownian motions. Having stated the three individual marginal results presented in equations 6), 7) and 8) we can provide a stronger result by combining the latter. In fact, the processes ε t ) t N and ε t ) t N are uncorrelated. Likewise the processes ε t ) t N and ε 3 t ) t N are also uncorrelated. In other words: cov [ ε t, ε t ) ] = cov [ ε 3 t, ε t ) ] = 0 Hence, using the result presented by Helland 98, heorem 3.3) based on the Cramer-Wold device, we conclude that: [ ] [ ] ε t, ε t ) d B, B ) Similarly: t= t= t= [ ] [ ] ε t ), t= ε 3 t d B, B 3 ) where B u, B u and B 3u are three Brownian motions. B u is independent of both B u andb 3u. his concludes the proof of Lemma.. Lemma.. Let X t ) t N be the process satisfying equation ). If assumptions. and. are satisfied and if α =, then k [[, 4]], t= ) k Xt d 0 B k u du 9) where B u represents the standard Brownian motion. he above lemma is simply a corollary of Lemma. and its proof makes use of the Continuous Mapping heorem which can also be found in Nielsen 004). Proof of lemma.. : Let ε t ) t N be a sequence of i.i.d. normally distributed random variables with mean zero and variance one and let X.) N be a sequence of random elements taking values in D[0, ] defined as: u [0, ], X u) = [ u] t= ε t Also, let f : R R be a continuous function and B u) = B u denotes a standard

15 Brownian motion. And we have already shown in equation 6) that: X u) = [ u] t= Let G : D[0, ] R be the function defined by: Z ) D[0, ], G ) Z ) = f Z u ) du ε t d B u 0) G is a continuous mapping as the composed of the continuous mappings: f : R R above and x x 0 u du. herefore, applying the Continuous Mapping heorem to the result presented in equation 0) we obtain that: In other words, we obtain that: f t= G X u)) ) Xt 0 d G B u ) d 0 f B u ) du he proof of Lemma. is completed by choosing the continuous function f : R R such that: x, k) R [[, 4]], fx) = x k Lemma.3. Let ε t ) t N be a sequence of i.i.d. random variables satisfying assumption. and let X t ) t N be the autoregressive process satisfying equation ) and X 0 = 0. If α =, then ) k k [[, 3]], Xt d t= εt 0 Bk u db u ) k k [[, ]], Xt t= ε d t ) 0 Bk u db u And: t= Xt ) ε 3 t d 0 B u db 3u where B u, B u and B 3u are three Brownian motions as in Lemma.. Proof of lemma.3. : Lemma.3 is an immediate consequence of heorem 0.3 presented by Caceres and Nielsen 006). 3

16 Additionally, a second consequence of Caceres and Nielsen s heorem 0.3 is that the convergence results in Lemma. and Lemma.3 also hold jointly. Another important point to mention regarding these two lemmas above is the weights imposed to each sum. hese clearly differ from the stationary case where all sums are equally weighted by a coefficient for the convergence in probability and a coefficient for the convergence in distribution). However, in the unit root case a polynomial weight is in order. Now, having established Lemmas. and.3, we are in the position of formally presenting the proofs of heorem. and heorem.. Proof of theorem.. : Lets consider R defined in equation 3) as R = S00.S 0.S.S 0. However, here we use the fact that R can also be written in the following way: R = S 00.S 0.S.S 0 = S 00.S 0.A ).A.S.A ).A.S 0 ) = S 00.S 0.S.S 0 ) where A is a positive-definite diagonal matrix defined by: A = 0 ) 0 Note that the component S 00 remains unchanged by the transformation presented in equation ), and it expanded form is given as in equation 5). However, the terms S 0 and S above are now given by: S0 = 3 t= X ) t X)ε t ) + ˆα α) 3 t= X3 t t= X t X )ε t ) + ˆα α) ) t= X4 t ˆα α) ˆα α) 3 t= X t ε t ) ) ˆα α) X.X + ˆα α) 3 X t= X t ε t t= X3 t ε t ) ˆα α) X + ˆα α) X t= X t ε t ) S = t= X t X) 5 t= X t X)X t X ) 5 t= X t X)X t X ) 3 t= X t X ) First of all, using the results from Lemmas. and.3 presented above for α =, we obtain that: ˆα α) = t= X t ε t t= X t 4 = O p )

17 Now, applying Lemma. to each of the terms of S presented above we obtain the following result: S d 0 B u.g u du 0 B u.g u du 0 B u.g u du 0 B u.g u du = [ Gu 0 G u ] [ Gu G u ] du 3) where B u represents the standard Brownian motion and G u and G u are given by: G u = B u G u = B u 0 0 B v dv B v dv Similarly, we can apply both Lemma. and Lemma.3 to each of the terms of S 00 and S 0 to obtain the following results: Furthermore, we have that: ) S 00 = ε 4 t ε P t + o p ) 4) t= t= t= X t X)ε t ) + o p ) S 0 = 5) 3 t= X t X )ε t ) + o p ) S 0 d [ 0 G u db u 0 G u db u ] = [ Gu 0 G u ] db u 6) Finally, combining the three results presented in equations 3), 4) and 6) we obtain that the convergence of R =.S00.S0.S.S0 is given by: R d 0 db u [ Gu G u ] 0 [ Gu G u ] [ Gu G u ] ) du [ Gu 0 G u ] dbu Furthermore, based on Johansen s 996, chapter 3) exposition, we find that, conditioning on the vector G u = [G u, G u ], the integral 0 db u )G u is Gaussian: N [ 0 0 ), G u G u du 0 because we have that G u and B u are independent from the independence of B u and B u ). herefore we conclude that: )] 5

18 R d χ ) 7) conditionally on G u. But since the conditional distribution of R does not depend on G u, the above result also holds marginally. his completes the proof of heorem.. Proof of theorem.. : Lets consider again the terms ˆK 3 and ˆK 4 defined in equations 9) and 0). hen, using once more the results presented in Lemma. and Lemma.3, for α =, combined with the LLN and the Lindberg-Levy CL, we obtain that: And t= ˆα α) = X t ε t = O p ) t= X t S = ) ε P t + O p 8) S = S3 = t= t= t= ε t ) ) P + O p 9) ) ε 3 d t 3ε t + op ) N0, 6) 30) S4 3S ) = herefore, combining the above results we now obtain that: t= ε 4 t 6ε t + 3 ) d N0, 4) 3) ˆK ) 3 6 = S3 d 3 χ ) 3) S 6 ˆK 4 4 = [ S4 3S ] ) d 4 χ ) 33) S 4 Furthermore, let Z t = 6 ε 3 t 3ε t ) and Z t = 4 ε 4 t 6ε t + 3). hen, we have: EZ t ) = 0, EZ t) = 0, varz t ) = 0, varz t) = 0 and covz t, Z t) = 0 And, from the Lindberg-Levy CL we obtain that: 6

19 t= [ Zt Z t ] [ d 0 N 0 ) 0, 0 hus, from the above results and using the Kramer-Wold device, we conclude that: ˆK ˆK ) 4 d χ ) 35) 4 his completes the proof. Hence, the results presented in this section prove that White s test for heteroskedasticity is valid in the unit root case. his also proves the validity of the Jarque-Bera normality test in the same case. In other words, both tests are applicable to a marginally stable first order autoregressive process or random walk. his is quite an important and innovative result since several commonly encountered processes in economics, finance, biology and thermodynamics are believed to be represented by random walks. Based on the findings presented here, such processes can be submitted legitimately and accurately to these diagnostic tests. )] 34) 7

20 3 Explosive Case: α > In this section, the asymptotic properties of White s test for heteroskedasticity and that of the Jarque-Bera test for normality in the explosive case are examined. he main result is presented in the following theorem, which states that White s test for heteroskedasticity is not valid when applied to a first order autoregressive process with an explosive root. heorem 3.. Let R be defined as in equation 3). If assumptions. and. are satisfied and if α >, then it is not true that R d χ ) 36) Similarly, the corresponding result for the Jarque-Bera normality test when dealing with an explosive autoregressive process is presented in the following theorem. heorem 3.. Let ˆK 3 and ˆK 4 be defined as in equation 3). If assumptions. and. are satisfied and if α >, then ˆK 3 d χ ), ˆK 4 d ˆK χ ) and ˆK ) 4 d χ ) 37) 4 Following again the structure of the previous section, we are going to introduce the next two Lemmas which are essential for the proof of heorem 3.. Nevertheless, it is important to mention at this point that the standard Law of Large Numbers LLN s), the Central Limit heorems CL s) and/or the Functional CL s FCL s) do not apply in this case and a different asymptotic theory is therefore required. Lemma 3.. Let X t ) t N be the process defined by equation ). If assumptions. and. are satisfied and if α >, then k [[, 4]], Xt k a.s. = O α ) k 38) t= And: Where: F k) = α k t= X k t F k) a.s. 0 39) α ki Z k And: Z = α i= α j ε j j= It is worth mentioning here that the weights used for the almost sure a.s.) convergence in the explosive case clearly differ from those used in the stationary and 8

21 unit root cases. Additionally, as proved by e.g. Lai and Wei 983), the order of the process itself is given by: X a.s. = O α ). his is quite interesting as the term t= X t a.s. = O α ) as well. his refers to the fact that the sum of exponentials gives an exponential. Proof of lemma 3.. : In order to prove Lemma 3. we follow Anderson 959) in that he defines a random variable Z t such that: t t N, Z t = α α j ε j j= In other words: t N, X t = α t ) Z t Additionally, let the Z, F k, ) and F k) be defined by: Z = α j= α j ε j F k, ) = i= α ki Z k F k) = i= α ki Z k From the martingale convergence theorem and the Marcinkiewicz-Zygmund result a.s. Lai and Wei, 983): Z Z. Now, Xt k [ ] F k, ) = α t ) k Z t Fk, ) α k t= = = = α k t= α k α k t) Zt k α ki Z k t= i= α k α ki+k Z k i+ α ki Z k i= i= α ) ki Z k i+ Z k And, using the triangular inequality: Xt k F k, ) α ki Z k i+ Z k α k t= i= i= α ki k Z i+ Z i= j=0 Z j i+ Zk j 9

22 And we know that Z n ) n N0 is a converging sequence Z n Z), therefore this can be also written as: c > 0, n 0 N 0 / n > n 0, Z n Z < c let: c > 0/ i [[, n 0 ]], Z i < c i.e. c > max i [[,n0 ]] Z i then, V So, now we can write: α k + j N 0, Z j < K 0 = maxc, c + Z ) Xt k F k, ) t= α ki k Z i+ Z i= n 0 i= j=0 Z j i+ Zk j α ki k Z i+ Z) Z Z) j=0 α ki k Z i+ Z i= n 0 + j=0 Z j i+ Zk j Z j i+ Zk j n 0 i= α ki k c K0 k j=0 + α ki k K 0 i= n 0 + K0 k j=0 k c K0 k j=0 n 0 i= α ki k + K 0 j=0 K k 0 i= n 0 + α ki k c K0 k j=0 α ki + o) i= and c can be taken to be as small as possible and the rest are simply constants. hus, Xt k a.s. F k, ) 0 40) Now, consider: α k t= 0

23 F k) F k, ) = = = α ki Z k α ki Z k i= i= α ki Z k α ki Z k α ki Z k i= i= i= + α ki Z k Z k) α ki Z k i= i= + And, using the triangular inequality: Fk) F k, ) = α ki Z k i= α ki Z k i= α ki Z k Z k + i= Z k Z k Z k Z k i= + α ki Z k α ki + Z k α k α k α k i= α ki + o) i= and: Z k Z k a.s. 0 Hence: Fk) F k, ) a.s. 0 4) Finally, combining the results presented in equations 40) and 4) we obtain: Xt k a.s. F k) 0 4) where: α k t= F k) = α ki Z k 43) i= herefore this concludes with the proof of Lemma 3. by simply letting k to take values in [[, 4]] accordingly. Note that another proof for the above result, in the case where k =, was provided by Nielsen 005). He also presents a proof for the case k = following the lines of Lai and Wei 983). he above exposition is a generalisation of these results where k can take any value in the integer interval [[,4]]. In fact the proof presented above is still

24 valid for any positive finite integer k. Lemma 3.. Let ε t ) t N be a sequence of i.i.d. random variables satisfying assumption. and let X t ) t N be the autoregressive process defined by equation ) and X 0 = 0. If α >, then k [[, 3]], k [[, ]], And: α k t= Xk t ε t a.s. α k Z k Y k α k t= Xk t ε t ) a.s. α k Z k Y k α t= X t ε 3 t a.s. α Z Y 3 where Z is defined as in Lemma 3. and k [[, 3]], Y k, Y k, Y 3 are random variables defined by: Y k = lim [ j= αkj ) ε j ] [ ] Y k = lim j= αkj ) ε j ) Y 3 = lim [ j= αj ) ε 3 j ] Proof of lemma 3.. : Let ε t ) t N be a sequence of i.i.d. random variables satisfying assumption. and let u t ) t N be a sequence of i.i.d. random variables satisfying the following assumptions: i) Eu t ) = 0 ii) Eu t ) = K i.e. a positive constant iii) Eu t u s ) = 0 for all t s Note that all three processes ε t ) t N, ε t ) t N and ε 3 t ) t N satisfy the above three conditions and can therefore replace u t ) t N in what follows. Again, following Anderson s 959) notation, let: Z = α j= α j ε j = α ) X Z = α j= α j ε j Y u, = i= αi ) u i

25 Consider: α ) X t u t = α +t ) u t α t+ X t t= = = t= α +t u t Z t t= α +t u t [Z t Z ) + Z ] t= = Z Y u, + α +t u t Z t Z ) t= = Z Y u, + α j u j Z j Z ) j= herefore: E α ) X t u t Z Y u, = E α j u j Z j Z ) t= j= α j E u j Z j Z ) j= [ α j Eu j ) E [ Z j Z ) ]] j= α j K [E [ Z j Z ) ]] j= And: hen, E [ Z j Z ) ] = α ) [ ] α j α 44) 3

26 E α ) X t u t Z Y u, t= α j K α ) j= K K K j= j= [ ] α j α ) α α α j ) α α α ) α α a.s. 0 Hence, E t= α ) X t u t Z Y u, a.s. 0 hus, using Anderson s theorems.3,.4 and.6 Anderson, 959) given that u t ) t N is i.i.d we can say that Y u,, Z ) has a limiting distribution given by Y u, Z). Furthermore: ) α t= X t u t a.s. Z Y u In other words, we obtain three of the results presented in Lemma 3. by simply letting the process u t ) t N to be equal to processes ε t ) t N, ε t ) t N and ε 3 t ) t N respectively: α t= X t ε t a.s. α Z Y α t= X t ε t ) a.s. α Z Y α t= X t ε 3 t a.s. α Z Y 3 Where: Y = lim [ j= αj ) ε j ] [ ] Y = lim j= αj ) ε j ) Y 3 = lim [ j= αj ) ε 3 j ] Note that a similar result corresponding to the case u t = ε t above was presented by Jeganathan 988). In fact he included a result regarding the joint convergence of α t= X t, α t= X t ε t ). he exposition presented here is therefore a gen- 4

27 eralisation of that result where u t ) t N can be any i.i.d. process satisfying the required three conditions. he other proofs for the other terms presented in Lemma 3. are also based on the technique presented above. hese go as follows: Lets consider Z and Z as presented above. Lets also define Y u, by: Y u, = α i ) u i i= Now, consider: α ) Xt u t = α +t ) u t α t ) Xt t= = = t= α t ) u t Zt t= α t ) u t [Z ] t Z ) + Z t= = Z Y u, + α t ) u t Zt Z ) t= = Z Y u, + α j u j Z j Z ) herefore: E α ) X t u t Z Y u, = E α j u j Z j Z ) t= And: E [ Z j Z ) ] = α8 α j= j= α j E u j Z j Z ) j= [ α j Eu j ) E [ Z j Z ) ]] j= α j K [E [ Z j Z ) ]] j= α ) α j ) 4α 3α α j)) 45) 5

28 hen, E α ) X t u t Z Y u, t= j= j= j= K α j α4 α α ) α j ) 4α 3α α j)) K α j α4 +) α α j) 34 α ) α ) 4 ) α j ) K α4 +) α α ) K α α ) ) α a.s. 0 Hence, E t= α ) X t u t Z Y u, a.s. 0 hus, using Anderson s theorems.3,.4 and.6 Anderson, 959) given that u t ) t N is i.i.d we can say that Y u,, Z ) has a limiting distribution given by Y u, Z). Furthermore: ) α t= X t u t a.s. Z Y u In other words, we obtain two other results presented in Lemma 3.: α t= X t ε t a.s. α 4 Z Y α t= X t ε t ) a.s. α 4 Z Y where: Y = lim [ j= αj ) ε j ] [ ] Y = lim j= αj ) ε j ) he last result presented in Lemma 3. is proved in exactly the same way, and we therefore obtain: where: α 3 t= X3 t ε t a.s. α 6 Z 3 Y 3 6

29 his completes the proof of Lemma 3. Y 3 = lim [ j= α3j ) ε j ] Having established the above two lemmas, we can now state the following heorem regarding the asymptotic distribution of R in the explosive case. heorem 3.3. Let R be defined as in equation 3). If assumptions. and. are satisfied and if α >, then Where Y R is given by: Y = R d Y Λ z Y 46) α Z Y + α )α+) α 3 α +α+) Z Y α ) α 4 Z Y Y Z Y α 4 + α ) α 4 α +) Z Y α ) α 6 Z Y Y 3 47) and Λ z R is given by: Λ z = z α α ) z 3 α 3 α 3 ) z 3 α 3 α 3 ) z 4 α 4 α 4 ) 48) where i, j) [[, ]] [[, 3]], Y ij and Z given as in Lemma 3.. Proof of theorem 3.3. : Lets consider once more the constant-adjusted squared correlation coefficient R defined in equation 3) as R = S00.S 0.S.S 0. However, here we use the fact that R can also be written in the following form: R = S 00.S 0.S.S 0 = S 00.S 0.A α ).A α.s.a α ).A α.s 0 ) = S 00.S 0.S.S 0 49) where A α is a diagonal matrix defined by: A α = [ 0 α 0 α ] 50) herefore the terms S 0 and S in equation 49) are given by: 7

30 S 0 = α t= X t X)ε t ) + ˆα α) α α t= X t X )ε t ) + ˆα α) α t= X3 t ) t= X4 t ˆα α) α ) ) t= X t ε t ˆα α) X X.X α + ˆα α) α t= X t ε t ˆα α) α ) t= X3 t ε t ˆα α) X X α + ˆα α) ) α t= X t ε t ) S = α t= X t X α α 3 t= X3 t α 3 X.X α 3 t= X3 t α 3 X.X α 4 t= X4 t X α 4 Note that the term S 00 remains unchanged after this transformation and it expanded form was presented in equation 5). Now applying Lemma 3. to the different terms of S presented above we get: S = α t= X t + o) α 3 t= X3 t + o) α 3 t= X3 t + o) α 4 t= X4 t + o) a.s. F ) F 3) F 3) F 4) = Λ z 5) where: k [[, 4]], F k) = α ki Z k And: Z = α i= α j ε j Similarly, combining both Lemma 3. and Lemma 3. we obtain the corresponding results for S 00 and S 0: S 00 = ε 4 t t= t= ε t j= ) + o) a.s. 5) S 0 = α t= X t ε t ) + ˆα α) α t= X t ε t ) + ˆα α) α t= X3 t α t= X4 t ) ˆα α) ) ˆα α) ) α t= X t ε t + o) ) α t= X3 t ε t + o) 8

31 Furthermore, we obtain that: 53) S 0 a.s. α Z Y + α 4 Z [Y ] F 3) α 6 Z3 Y Y [F )] F ) α 4 Z Y + α 4 Z [Y ] F 4) α 8 Z4 Y Y 3 [F )] F ) = Y 54) where Y, Y, Y 3, Y and Y are defined as in Lemma 3.. Anderson s 959) theorem.6 implies ) that the distributions of Z and Y are normal with mean zero and α variance. hese two variables have zero correlation and are in fact independent. α Furthermore, the same theorem implies ) that Y ) and Y 3 have normal distributions with mean zero and variances and respectively. Y and Y 3 are both α 4 α 4 α 6 α 6 independent of Z. Simplifying the above expression of Y according to the definition of F k) given by equation 43), we obtain that: Y = α Z Y + α )α+) α 3 α +α+) Z Y α ) α 4 Z Y Y Z Y α 4 + α ) α 4 α +) Z Y α ) α 6 Z Y Y 3 55) Finally, combining the results presented in equations 5), 5) and 54), we conclude that: R = S00.S0.S.S0 d Y Λ z Y 56) his terminates the proof of heorem 3.3. At this point a few remarks are in order. First of all, as mentioned in the introduction, several terms of the components S 00 and S 0 equivalently S0) are asymptotically negligible i.e. these become negligible as increases). In the case of S 00, whose expanded form is presented in equation 5), all the terms disappear asymptotically except for the two terms t= ε t and t= ε4 t. Furthermore, and more importantly, this is true in all three stationary, unit root and explosive cases. In fact, the asymptotic distribution of S 00 is the same in all three cases and is independent of the value of the parameter α. On the other hand, the asymptotic results obtained for S 0 are very different in the explosive case and in the non-explosive cases i.e. α ). Precisely, it has been shown in the previous two sections that all the terms in S 0 are asymptotically negligible when α except for the two terms t= X t X)ε t ) and t= X t X )ε t ). In other words, these two are the leading terms of S 0 in both the stationary and 9

32 the unit root cases. hese results are explicitly presented in equations??) and 5) respectively. However in the explosive case, as shown by equation 53) in the above proof, several other terms in S 0 do not become negligible when increases, and thus play an important role in the asymptotic distribution of R. Finally, the proof of heorem 3. is presented below. Proof of theorem 3.. : o begin with, recall that from heorem 3.3 we have that R d Y Λ z Y. However, note that it would be quite difficult to determine explicitly the distribution of Y given by equation 55) and therefore that of Y Λ z Y ) as well. Nevertheless, it is important to mention at this point that a necessary condition for R =.S00.S0.S.S0 to converge to a χ is that the moments of Y Λ z Y ) must be the same as those of the χ distribution. In particular, the first moment of a χ with k degrees of freedom is equal to k. In this case k =. herefore, a sufficient condition for Y Λ z Y ) not to have a χ distribution is that its first moment is different than for all α R such that α >. In fact Y Λ z Y ) can be written as: Y Λ z Y = X Z X Z z α α ) z 3 α 3 α 3 ) z 3 α 3 α 3 ) z 4 α 4 α 4 ) X Z X Z 57) where X and X are given by: X = α Y + α )α + ) α 3 α + α + ) Y α ) α 4 Y Y X = α 4 Y + α ) α 4 α + ) Y α ) α 6 Y Y 3 hus Y Λ z Y Y ) becomes: Λ z Y = [ X α 4 α 4 ) X ] X α 3 α 3 ) + X α α ) 58) where = α 4 α )α )α + α + ) α 3 + α + α + ). Note that Z does not appear in the simplified expression of Y Λ z Y ) presented in equation 58) above. Additionally, we have that the first moment of Y Λ z Y ) is given by: E [ Y Λ z Y ] = 3α 8 + 3α 5 + 8α 4 + 5α 3 + 6α + 5 α + )α α + )α 4 + α 3 + α + α + ) 59) which is different than for α >. 30

33 3.5 Y * Λz Y* α Figure : E [ Y Λ z Y ] as a function of α herefore, Y Λ z Y ) does not have a χ distribution when α >. We can conclude that R does not converge to a χ distribution in the explosive case. hus heorem 3.. In fact, note that the expression for E [ ] Y Λ z Y presented in equation 59) varies with α. Figure exhibits E [ ] Y Λ z Y as a function of α. Interestingly, at the limit when α =, E [ ] Y Λ z Y =, indicating a continuity when moving from the unit root value into the explosive region and viceversa. Additionally, E [ ] Y Λ z Y tends to 3/ when α tends to infinity. In conclusion to this section, the proof of heorem 3. suggests that White s test for heteroskedasticity is not valid when applied to an autoregressive process with an explosive root. his result emphasises a clear handicap of this heteroskedasticity test. It implies that the latter cannot be used when working with data originated from an explosive process. Among others, examples of such cases include hyperinflation and unstable vibrating systems. In other words, an important consequence of the above theorem is that White s test 3

34 for heteroskedasticity is parameter dependent. It is not valid in the general case, and therefore, only applicable for a certain range of values of the parameter α i.e. valid for α ). We now proceed to the proof of the result concerning the validity of the Jarque-Bera normality test in the explosive case. Proof of theorem 3.. : In a similar way to the proof of heorem., but applying this time the results presented in Lemmas 3. and 3., we obtain that: And α ˆα α) = α t= X t ε t α = O p ) t= X t S = ) ε P t + O p 60) S = S3 = t= t= t= ε t ) ) P + O p 6) ) ε 3 d t 3ε t + op ) N0, 6) 6) S4 3S ) = herefore, combining the above results we now obtain that: t= ε 4 t 6ε t + 3 ) d N0, 4) 63) ˆK ) 3 6 = S3 d 3 χ ) 64) S 6 ˆK 4 4 = [ S4 3S ] ) d 4 χ ) 65) S 4 And, following the same argument as in the proof of heorem. we show that the above two convergence results also hold jointly, that is: ˆK ˆK ) 4 d χ ) 66) 4 which concludes the proof of heorem 3.. he above result proves that the Jarque-Bera normality test is thus theoretically 3

35 valid in the general case. hat is all three, the stationary, the unit root and the explosive case. his is also a very important result, in the sense that it comes as a relief given that this normality test is widely used in various standard statistical and econometric software packages. herefore, this paper shows that such test can be used when dealing with a first order autoregressive processes without the need to worry about the value of the root present in that model. 33

36 4 Conclusion he objective of this paper was to analyse the asymptotic properties of White s test for heteroskedasticity and that of the Jarque-Bera test for normality from a time series perspective. In particular, we were interested in studying whether these tests are valid in the general case when applied to a first order autoregressive process such as X t ) t N0 defined in equation ). In fact, it has been shown that the validity of White s test for heteroskedasticity is parameter dependent. In other words, the latter is valid for some values of the parameter of interest α but it is not valid for others. It has been proved in Section that this test works well in the unit root case when α = ). he validity of this test in the stationary case when α < ) was already established by several authors c.f. Kelejian 98), Godfrey and Orme 994) among others). Nevertheless, section 3 proved that this test is not valid in the explosive case when α > ). his is a very important theoretical result. It implies that in order to do statistical inference on the parameter α we need to know if heteroskedasticity is present in the model. But, to test for heteroskedasticity we need to know the value of α! Paraphrasing, this means that any applied statistician should not use this test unless he/she is absolutely sure of dealing with data from a stationary or marginally stable process. Regarding the Jarque-Bera normality test, it was proved that this test is theoretically valid in the general case when applied to a first order autoregressive process. his means that the Jarque-Bera test for normality can be used when dealing with a first order autoregressive process independently of whether the latter contains a stationary, an unit or an explosive root. It is worthwhile mentioning that the results presented in this paper are based on a purely theoretical framework. his applies in particular to the results obtained for White s test for heteroskedasticity. In order to know in practice how different the distribution of the sufficient statistic R in the explosive case is from that of a chi-squared distribution, some computer simulations could be very valuable. From a practical point of view, this test for heteroskedasticity could still be applicable for a certain range of values of the parameter α that could be less restrictive than the theoretical one. his is certainly an area that needs further investigation. Alternatively, a new heteroskedasticity test could be designed altogether. his should essentially have solid basis upon the corresponding probability and measure theory. Ideally, a heteroskedasticity test should not be parameter dependent. Also it should be applicable to a multivariate model i.e. VARk) models), and furthermore, should be able to accommodate for different types of deterministic terms in the model. 34

37 References [] Ali, M.M. and Giaccotto, C. 984). A study of several new and existing tests for heteroscedasticity in the general linear model. Journal of Econometrics 6, [] Amemiya,. 977). A note on a heteroskedasticity model. Journal of Econometrics 6, [3] Anderson,.W. 959). On asymptotic distributions of estimates of parameters of stochastic difference equations. Annals of Mathematical Statistics 30, [4] Caceres, C. and Nielsen, B. 006). Limiting Distributions of Non-Stationary Processes. Working paper, Nuffield College, Oxford. [5] Chan, N.H. and Wei, C.Z. 988). Limiting Distributions of Least Squares Estimates of Unstable Autoregressive Processes. Annals of Statistics 6, [6] Doornik, J.A. 996). esting Vector Autocorrelation and Heteroscedasticity in Dynamic Models. Econometric Society 7th World Congress, okyo. [7] Godfrey, L.G. and Orme, C.D. 994). he sensitivity of some general checks to omitted variables in the linear model. International Economic Review 35, [8] Hamilton, J.D. 994). ime Series Analysis. Princeton University Press. [9] Helland, I. S. 98). Central Limit heorems for Martingales with Discrete or Continuous ime. Scandinavian Journal of Statistics 9, [0] Ibragimov, R. and Phillips, P.C.B. 004). Regression Asymptotics Using Martingale Convergence Methods. Cowles Foundation Discussion paper No [] Jarque, C.M. and Bera, A.K. 980). Efficient tests for normality, homoscedasticity and serial independence of regression residuals. Economic Letters 6, [] Jeganathan, P. 988). On the strong approximation of the distributions of estimators in linear stochastic models, I and II: stationary and explosive AR models. Annals of Statistics 6, [3] Johansen, S. 996). Likelihood-based inference in cointegrated vector autoregressive models, nd print, Oxford University Press. [4] Kelejian, H.H. 98). An Extension of a Standard est for Heteroskedasticity to a System Framework. Journal of Econometrics 0, [5] Kilian, L. and Demiroglu, U. 000). Residual-Based ests for Normality in Autoregressions: Asymptotic heory and Simulation Evidence. Journal of Business and Economic Statistics 8,

38 [6] Lai,.L. and Wei, C.Z. 983). Asymptotic properties of general autoregressive models and strong consistency of least-squares estimates of their parameters. Journal of Multivariate Analysis 3, -3. [7] Lai,.L. and Wei, C.Z. 985). Asymptotic properties of multivariate weighted sums with applications to stochastic regression in linear dynamic systems. In P.R. Krishnaiah, ed., Multivariate Analysis VI, Elsevier Science Publishers, [8] Lutkepohl, H. and Schneider, W. 989). esting for Nonnormality of Autoregressive ime Series. computational Statistics Quarterly, [9] Nielsen, B. 00). Order determination in general vector autoregressions. Discussion paper, Nuffield College, Oxford. [0] Nielsen, B. 004). Unit Roots. Advanced Econometrics lecture notes, MPhil in Economics, Oxford University. [] Nielsen, B. 005). Strong consistency results for least squares estimators in general vector autoregressions with deterministic terms. Econometric heory. [] Rubin, H. 950). Consistency of maximum-likelihood estimates in the explosive case. Statistical Inference in Dynamic Economic Models ed..c. Koopmans), , New York: Wiley. [3] White, H. 980). A Heteroskedastic-Consistent Covariance Matrix and a Direct est for Heteroskedasticity. Econometrica 48, [4] White, J. 958). he Limimiting Distribution of the Serial Correlation Coefficient in the Explosive Case. Annals of Mathematical Statistics 9, [5] Wooldridge, J.M. 999). Asymptotic properties of some specification tests in linear models with integrated processes. In R.F. Engle and H. White, Cointegration, Causality and Forecasting: Festschrift in Honour of Clive W.J.Granger. Oxford University Press. 36