Testing normality for unconditionally heteroscedastic macroeconomic variables
aa r X i v : . [ s t a t . M E ] J un Testing normality for unconditionally heteroscedasticmacroeconomic variables
Hamdi Raïssi ∗ September 4, 2018
Abstract:
In this paper the testing of normality for unconditionally heteroscedastic macroe-conomic time series is studied. It is underlined that the classical Jarque-Bera test (JBhereafter) for normality is inadequate in our framework. On the other hand it is foundthat the approach which consists in correcting the heteroscedasticity by kernel smoothingfor testing normality is justified asymptotically. Nevertheless it appears from Monte Carloexperiments that such methodology can noticeably suffer from size distortion for samplesthat are typical for macroeconomic variables. As a consequence a parametric bootstrapmethodology for correcting the problem is proposed. The innovations distribution of a setof inflation measures for the U.S., Korea and Australia are analyzed.Keywords: Unconditionally heteroscedastic time series; Jarque-Bera test.JEL: C12, C15, C18 ∗ Instituto de Estadística, PUCV. Address: Av. Errazuriz 2734, Valparaíso, V región, Chile. E-mail:[email protected]. This paper was supported by CONICYT-FONDECYT under grant N ◦ . Theauthor gratefully acknowledge Steffen Grønneberg and Genaro Sucarrat for helpful discussions during hisstay at the BI Norwegian Business School. Introduction
In the econometric literature, the Jarque Bera (1980) test is routinely used to assess thenormality of variables. The properties of this test are well documented for stationary con-ditionally heteroscedastic processes. For instance Fiorentini, Sentana and Calzolari (2003),Lee, Park and Lee (2010) and Lee (2012) investigated the JB test in the context of GARCHmodels. However few studies are available on the distributional specification of time series inpresence of unconditional heteroscedasticity. Drees and Stărică (2002), Mikosch and Stărică(2004) and Fryźlewicz (2005) investigated the possibility of modelling financial returns bynonparametric methods. To this end, Drees and Stărică (2002) and Mikosch and Stărică(2004) examined the distribution of S&P500 returns corrected from heteroscedasticity. Onthe other hand Fryźlewicz (2005) pointed out that large sample kurtosis for financial timeseries may be explained by non constant unconditional variance. In general we did not foundreferences that specifically address the problem of assessing the distribution of uncondition-ally heteroscedastic time series. Note that non constant variance constitutes an importantpattern for time series in general, and macroeconomic variables in particular. Reference canbe made to Sensier and van Dijk (2004) who found that most of the 214 U.S. macroeconomictime series they studied have a time-varying variance. In this paper we aim to provide areliable methodology for testing normality for small samples time series with non constantunconditional variance.The structure of the paper is as follows. In Section 2 we first set the dynamics rulingthe observed process. In particular the unconditional heteroscedastic structure of the errorsis given. The inadequacy of the standard JB test in our framework is highlighted. Theapproach consisting in correcting the errors from the heteroscedasticity for building a JBtest is presented. We then introduce a parametric bootstrap procedure that is intendedto improve the normality testing for unconditionally heteroscedastic macroeconomic timeseries. In Section 3 numerical experiments are conducted to shed some light on the finitesample behaviors of the studied tests. In particular it is found that when estimating the non2onstant variance structure by kernel smoothing, a correct bandwidth choice is a necessarycondition for the good implementation of the normality tests based on heteroscedasticitycorrection. We illustrate our outputs examining the distributional properties of the U.S.,Korean and Australian GDP implicit price deflators.
We consider processes ( y t,n ) which can be written as y t,n = ω + x t,n ,x t,n = p X i =1 a i x t − i,n + u t,n , (2.1)where y ,n , . . . , y n,n are available, n being the sample size and E ( x t,n ) = 0 . The conditionalmean of x t,n is driven by the autoregressive parameters θ = ( a , . . . , a p ) ′ . We make thefollowing assumption on the conditional mean. Assumption A0:
The a i ∈ R , ≤ i ≤ p , are such that det( a ( z )) = 0 for all | z | ≤ ,with a ( z ) = 1 − P pi =1 a i z i .In the assumption A1 below, the well known rescaling device introduced by Dahlhaus(1997) is used to specify the errors process ( u t,n ) . For a random variable v we define k v k q = ( E | v | q ) /q , with E | v | q < and q ≥ . Assumption A1:
We assume that u t,n = h t,n ǫ t where:(i) h t,n ≥ c > for some constant c > , and satisfies h t,n = g ( t/n ) , where g ( r ) is ameasurable deterministic function on the interval (0 , , such that sup r ∈ (0 , | g ( r ) | < ∞ .3he function g ( . ) satisfies a Lipschitz condition piecewise on a finite number of somesub-intervals that partition (0 , .(ii) The process ( ǫ t ) is iid and such that E ( ǫ t ) = 0 , E ( ǫ t ) = 1 , and ( E ( k ǫ t k ν ) < ∞ forsome ν > .The non constant variance induced by A1(i) allows for a wide range of non stationaritypatterns commonly faced in practice, as for instance abrupt shifts, smooth changes or cyclicalbehaviors. Note that in the zero mean AR case, tools needed to carry out the Box and Jenkinsspecification-estimation-validation modeling cycle, are provided in Patilea and Raïssi (2013)and Raïssi (2015). For ω = 0 define the estimator ˆ ω = n − P nt =1 y t,n , and x t,n ( ω ) = y t,n − ω for any ω ∈ R . Writing ˆ ω − ω = n − P nt =1 x t,n , it can be shown that √ n (ˆ ω − ω ) = O p (1) , (2.2)using the Beveridge-Nelson decomposition. Now let ˆ θ ( ω ) = (Σ x ( ω )) − Σ x ( ω ) , (2.3)where Σ x ( ω ) = n − n X t =1 x t − ,n ( ω ) x t − ,n ( ω ) ′ and Σ x ( ω ) = n − n X t =1 x t − ,n ( ω ) x t − ,n ( ω ) , with x t − ,n ( ω ) = ( x t − ,n ( ω ) , . . . , x t − p,n ( ω )) ′ . With these notations define the OLS estimator ˆ θ (ˆ ω ) and the unfeasible estimator ˆ θ ( ω ) . Straightforward computations give √ n (ˆ θ (ˆ ω ) − ˆ θ ( ω ))= o p (1) , so that using the results of Patilea and Raïssi (2012) we have √ n (ˆ θ (ˆ ω ) − θ ) = O p (1) . (2.4)Once the conditional mean is filtered in accordance to (2.2) and (2.4), we can proceed tothe test of the following hypotheses: H : ǫ t ∼ N (0 , vs. H : ǫ t has a different distribution , u t,n correspond to those of ǫ t . However in practice E ( u t,n ) = 0 and E ( u t,n ) = 3 is checked using the JB test statistic: Q uJB = n h Q S,uJB + Q K,uJB i , (2.5)where Q S,uJB = ˆ µ µ and Q K,uJB = 124 (cid:18) ˆ µ ˆ µ − (cid:19) , with ˆ µ j = n − P nt =1 (ˆ u t,n − ¯ˆ u ) j and ¯ˆ u = n − P nt =1 ˆ u t,n . The ˆ u t,n ’s are the residuals obtainedfrom the estimation step. Let us denote by ⇒ convergence in distribution. If we supposethe process ( u t ) homoscedastic ( g ( . ) is constant), then the standard result Q uJB ⇒ χ isretrieved (see Yu (2007), Section 2.2). However under A0 and A1 with g ( . ) non constant(the unconditionally heteroscedastic case) we have: Q K,uJB = 124 (cid:2) κ (cid:0) E ( ǫ t )) − (cid:1) + 3 ( κ − (cid:3) + o p (1) , (2.6)where κ = R g ( r ) dr ( R g ( r ) dr ) . Hence if we suppose the errors process unconditionally heteroscedas-tic with E ( ǫ t ) = 3 , we have Q uJB = Cn + o p ( n ) for some strictly positive constant C . As aconsequence, the classical JB test will tend to detect spuriously departures from the null hy-pothesis of a normal distribution in our framework. This argument was used by Fryźlewicz(2004) to underline that unconditionally heteroscedastic specifications can cover financialtime series that typically exhibit an excess of kurtosis.In order to assess the distribution of S&P500 returns, Drees and Stărică (2002) considereddata corrected from heteroscedasticity, using a kernel estimator of the variance. We willfollow this approach in the sequel considering5 h t,n = n X i =1 w ti (ˆ u i,n − ¯ˆ u ) , ≤ t ≤ n, with w ti = (cid:16)P nj =1 K tj (cid:17) − K ti , K ti = K (( t − i ) /nb ) if t = i and K ii = 0 , where K ( · ) is akernel function on the real line and b is the bandwidth. The following assumption is neededfor our variance estimator. Assumption A2: (i) The kernel K ( · ) is a bounded density function defined on the realline such that K ( · ) is nondecreasing on ( −∞ , and decreasing on [0 , ∞ ) and R R v K ( v ) dv < ∞ . The function K ( · ) is differentiable except a finite number of points and the derivative K ′ ( · ) satisfies R R | xK ′ ( x ) | dx < ∞ . Moreover, the Fourier Transform F [ K ]( · ) of K ( · ) satisfies R R | s | τ |F [ K ]( s ) | ds < ∞ for some τ > .(ii) The bandwidth b is taken in the range B n = [ c min b n , c max b n ] with < c min < c max < ∞ and nb − γn + 1 /nb γn → as n → ∞ , for some small γ > .Let ˆ ǫ t = (ˆ u t,n − ¯ˆ u ) / ˆ h t,n . We are now ready to consider the following JB test statistic: Q ǫJB = n h Q S,ǫJB + Q K,ǫJB i , where Q S,ǫJB = ˆ ν ν and Q K,ǫJB = 124 (cid:18) ˆ ν ˆ ν − (cid:19) , with ˆ ν j = n − P nt =1 ˆ ǫ jt . The following proposition gives the asymptotic distribution of Q ǫJB . Proposition 1.
Under the assumptions A0 , A1 and A2 , we have as n → ∞ Q ǫJB ⇒ χ , (2.7) uniformly with respect to b ∈ B n . T cv . The standardtest, that does not take into account the unconditional heteroscedasticity, is denoted by T st .For high frequency time series it is reasonable to suppose that the approximation (2.7)is satisfactory when the bandwidth is carefully chosen. Nevertheless considering the abovesophisticated procedure for small n is questionable. Therefore we propose to apply the fol-lowing parametric bootstrap algorithm inspired from Francq and Zakoïan (2010,p335).1- Generate ǫ ( b ) t ∼ N (0 , , ≤ t ≤ n , build the bootstrap errors u ( b ) t,n = ǫ ( b ) t ˆ h t,n , and thebootstrap series y ( b ) t using (2.1), but with ˆ ω and ˆ θ (ˆ ω ) (see (2.2) and (2.3)).2- Estimate the autoregressive parameters and a constant as in (2.1), but using the y ( b ) t ’s.Build the kernel estimators ˆ h ( b ) t,n from the resulting residuals ˆ u ( b ) t,n .3- Compute ˆ ǫ ( b ) t,n = ˆ u ( b ) t,n / ˆ h ( b ) t,n for t = 1 , . . . , n . Compute Q ǫ, ( b ) JB .4- Repeat the steps 1 to 3 B times for some large B. Use the Q ǫ, ( b ) JB ’s to compute thep-values of the bootstrap JB test.The test obtained using the above parametric bootstrap procedure is denoted by T boot . The finite sample properties of the T st , T cv and T boot tests are first examined by means ofMonte Carlo experiments. The distribution of the U.S., Korean and Australian GDP implicit7rice deflator is then investigated. Throughout this section the asymptotic nominal level ofthe tests is 5%. In the sequel, we fixed B = 499 . We simulate N = 1000 trajectories of AR(1) processes: y t,n = a y t − ,n + u t,n , (3.1)where a = 0 . and u t,n = h t,n ǫ t with ǫ t iid(0,1). Under the null hypothesis we set ǫ t ∼N (0 , . On the other hand under the alternative hypothesis ǫ t = cos( δ ) v t +sin( δ ) w t is taken,with v t ∼ N (0 , , ( √ w t + 1) ∼ χ , < δ ≤ π , v t and w t being mutually independent. Inorder to study the case where the series are actually homoscedastic, we set h t,n = 1 . For theheteroscedastic case, the variance structure is given by h t,n = 1 + 2 exp ( t/n ) + 0 . t/n ) sin (5 πt/n + π/ . (3.2)In such situation the variance structure exhibits a global monotone behavior together with acyclical/seasonal component that is common in macroeconomic data (see e.g. Trimbur, andBell (2010) for seasonal effects in the variance). In all our experiments, the mean in (3.1) istreated as unknown. More precisely the AR parameter in (3.1) is estimated using y t,n − ˆ ω ,where ˆ ω is given in (2.2), and then the resulting centered residuals are used to compute thetest statistics.The outputs obtained under the null hypothesis are first analyzed. The results are givenin Table 1 for the homoscedastic case and in Table 2 for the heteroscedastic case. Notingthat macroeconomic time series with noticeable heteroscedasticity are relatively large butsmaller than n = 400 in general, a special emphasis will be put on interpreting results forsamples n = 100 , , . Since N = 1000 processes are simulated, and under the hypothe-sis that the finite sample size of a given test is , the relative rejection frequencies should be8etween the significant limits 3.65% and 6.35% with probability 0.95. The outputs outsidethese confidence bands will be displayed in bold type.From Table 1, it appears that the T cv is oversized for small samples ( n = 100 and n = 200 ). This could be explained by the fact that this test is too much sophisticated forthe standard case. When the samples are increased the relative rejection frequencies becomeclose to the 5% ( n = 400 and n = 800 ). On the other hand the T st and T boot tests have goodresults for all the samples. Of course if there is no evidence of heteroscedasticity, the simple T st should be used. However Table 1 reveals that in case of doubt, the use of the T boot is agood alternative.In the heteroscedastic case, it is seen from Table 2 that the T st test fails to control thetype I error as n → ∞ . This was expected from (2.6). Next it seems that the relativerejection frequencies of the T cv test are somewhat far from the nominal level 5%, even when n = 800 . From Table 2 it also emerges that the T boot control reasonably well the type Ierror. Therefore we can draw the conclusion that the T boot gives a substantial improvementfor samples that are typical for heteroscedastic macroeconomic variables.Note that the T cv test have better results for larger samples ( n ≫ ). For instanceconducting similar experiences to those of Table 2, we obtained 7.4% rejections for n =1600 and 6.9% rejections for n = 3200 . Hence the potential improvements of the T boot incomparison to the T cv should become slight as n → ∞ . For this reason if high frequency timeseries are analyzed, the T cv should certainly be preferred to the computationally intensive T boot .In general it is important to point out that the bandwidth must be carefully selected toensure a good implementation of the T boot and T cv tests. Its turns out from our experimentsthat selecting the bandwidth by cross-validation leads to relatively correct results. Indeedwe found that the rejection frequencies of the T cv converge to the 5%, and that the rejectionfrequencies of the T boot remain close to the nominal level in such a case. However other choices9an deteriorate the control of the type I errors. For instance let us consider the T f test whichconsists in correcting the heteroscedasticity, but with fixed bandwidth as γ (ˆ σ /n ) . , where ˆ σ is the sample variance and γ is a constant. The corresponding bootstrap test will bedenoted by T f,boot . Here the normal kernel is kept. We only study the heteroscedastic case.The results given in Table 3 show that the rejection frequencies are strongly affected by thisway of selecting the bandwidth.Finally let us point out that when the heteroscedastic structure is relatively easy toestimate (for instance if the sinus part is removed in (3.2)), we found better results (notdisplayed here) for the T boot and T cv tests in comparison to those of Table 2.Now we turn to the analysis of the behavior of the tests under the alternative hypothesis.For a fair comparison we only studied the T st and T boot in the homoscedastic case. Thesample size n = 100 is fixed and recall that the parameter δ defines the departures from thenull hypothesis. The outputs of our simulations, displayed in Figure 1, show that the T boot test does not suffer from a lack of power in comparison to the T st . In conclusion it turns outthat the T boot improves the distribution analysis, in the sense that it ensures a good controlof the type I error, but without entailing noticeable loss of power. The inflation measures data are commonly used to analyze macroeconomic facts. Referencecan be made to the numerous empirical papers studying the relation between price levelsand money supply (see e.g. Jones and Uri (1986)). On the other hand inflation is of greatimportance in finance, as many central banks adjust their interest rates in view of targetinga certain inflation level. Accordingly, constructing valid confidence intervals for inflationforecasts may be often crucial. In such kind of investigations clearly the distributionalanalysis can help to build a model for the data. In a stationary setting, authors aimedto detected ARCH effects assessing asymmetry and/or leptokurticity in inflation variables10ollowing Engle (1982) (see Broto and Ruiz (2008p22) among others). In the same way, it isreasonable to think that a test for normality taking into account the time-varying variance,can help to choose between a deterministic specification, as in A1 , and the case where inaddition to unconditional heteroscedasiticity, second order dynamics are present (as in thecase of spline-GARCH processes introduced by Engle and Rangel (2008)). In other words,once the unconditional heteroscedasiticity is removed from u t = h t ǫ t , the JB tests can helpto decide whether ARCH effects are present or not in ( ǫ t ) .In this part we will study the normality of the log differences of the quarterly GDPimplicit price deflators for the U.S., Korea and Australia from 10/01/1983 to 01/01/2017( n = 132 ). More precisely we use y t,n = 100 log ( GDP t,n /GDP t − ,n ) . The data can bedownloaded from the webpage of the research division of the federal reserve bank of SaintLouis: https://fred.stlouisfed.org. The studied variables plotted in Figure 2 seem to showcyclical heteroscedasticity. In the case of Korea we can suspect a global decreasing behaviorleading to a stabilization after the Asian crisis. The times series are first filtered accordingto (2.1). The non correlation of the residuals is tested using the adaptive portmanteau testof Patilea and Raïssi (2013). On the other hand we applied tests for second order dynamicsdeveloped by Patilea and Raïssi (2014). The outputs (not displayed here) show that thehypothesis of no ARCH effects cannot be rejected. Hence the deterministic specificationof the time-varying variance in A1 seems valid. Once the linear dynamics of the seriesseem captured in an appropriate way, the tests considered in this paper are applied to theresiduals. The results are given in Table 4. When the null hypothesis of normality is rejectedat the 5% level, the p-value is displayed in bold type. It emerges that the outputs of the T boot test are in general clearly different from those of the T cv and T st tests. The p-values of the T cv are all lower than those of the T boot . Note that in the case of the U.S. GDP implicit pricedeflator, the difference between the T boot on one hand, and the T st , T cv tests on the otherhand, lead to different conclusions. In view of the outputs obtained from the simulationsexperiments, it is reasonable to decide that the normality assumption cannot be rejected for11he U.S. data. It is likely that rejecting normality will suggest more sophisticated models,and could entail misspecifications for the confidence intervals of the forecasts by fitting aheavy tailed distribution to the U.S. data. References
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Table 1:
Empirical size (in %) of the studied tests for normality. The homoscedastic case. n
100 200 400 800 T st T cv T boot Empirical size (in %) of the studied tests for normality. The heteroscedastic case. n
100 200 400 800 T st T cv T boot Empirical size (in %) of the T cv and T boot tests for normality with fixed bandwidth. Theheteroscedastic case. γ . n
100 200 100 200 T f T f,boot The p-values (in %) of the tests for normality for GDP implicit price deflators for theU.S., Korea and Australia.
U.S. Korea Australia T st T cv T boot sin ( d ) T st T boot Figure 1:
Empirical power (in %) of the T st and T boot tests in the homoscedastic case. The log differences of the quarterly U.S. (top left panel), Korean (middle left panel) and Aus-tralian (bottom left panel) GDP implicit price deflators from 10/01/1983 to 01/01/2017 ( n = 132 ). Thecorresponding estimations of the innovations variance are on the right. Data source: The research divisionof the federal reserve bank of Saint Louis, fred.stlouisfed.org.). Thecorresponding estimations of the innovations variance are on the right. Data source: The research divisionof the federal reserve bank of Saint Louis, fred.stlouisfed.org.