DEPARTMENT OF STATISTICS

A Tes for Mulivariae ARCH Effecs R. Sco Hacker and Abdulnasser Haemi-J 004: DEPARTMENT OF STATISTICS S-0 07 LUND SWEDEN

A Tes for Mulivariae ARCH Effecs R. Sco Hacker Jönköping Inernaional Business School P.O. Box 106, SE-551 11, Jönköping, Sweden E-mail: Sco.Hacker@ihh.hj.se Telephone: +46 36 15 75 14, Fax + 46 36 1 18 3 and Abdulnasser Haemi-J Lund Universiy and Universiy of Skövde P.O. Box 408, SE-541 8, Skövde, Sweden. E-mail: abdulnasser.haemi-j@ish.his.se Tel: +46 500 44 87 31, Fax: +46 500 44 87 99 Absrac This paper exends Engle s LM es for ARCH affecs o mulivariae cases. The size and power properies of his mulivariae es for ARCH effecs in VAR models are invesigaed based on asympoic and boosrap disribuions. Using he asympoic disribuion, deviaions of acual size from nominal size do no appear o be very excessive. Neverheless, here is a endency for he acual size o overrejec he null hypohesis when he nominal size is 1% and underrejec he null when he nominal size is 5% or 10%. We find ha using a boosrap disribuion for he mulivariae LM es is generally superior in achieving he appropriae size o using he asympoic disribuion when (1) he nominal size is 5%, () he sample size is small (40 observaions) and/or he VAR sysem is sable. Wih he small sample, he power of he es using he boosrap disribuion appears also beer a he 5% nominal size. Key words: Engle Tes, VAR model, Sabiliy, Mulivariae ARCH, Boosrap JEL classificaion: C3, C15 Running ile: Tes for Mulivariae ARCH Effecs 1

A Tes for Mulivariae ARCH Effecs Absrac This paper exends Engle s LM es for ARCH affecs o mulivariae cases. The size and power properies of his mulivariae es for ARCH effecs in VAR models are invesigaed based on asympoic and boosrap disribuions. Using he asympoic disribuion, deviaions of acual size from nominal size do no appear o be very excessive. Neverheless, here is a endency for he acual size o overrejec he null hypohesis when he nominal size is 1% and underrejec he null when he nominal size is 5% or 10%. We find ha using a boosrap disribuion for he mulivariae LM es is generally superior in achieving he appropriae size o using he asympoic disribuion when (1) he nominal size is 5%, () he sample size is small (40 observaions) and/or he VAR sysem is sable. Wih he small sample, he power of he es using he boosrap disribuion appears also beer a he 5% nominal size. Key words: Engle Tes, VAR model, Sabiliy, Mulivariae ARCH, Boosrap JEL classificaion: C3, C15 Running ile: Tes for Mulivariae ARCH Effecs 1. Inroducion Since he seminal paper by Engle (198) auoregressive condiional heeroscedasiciy (ARCH) models have been exensively used in applied research. Engle also showed ha he ordinary leas squares (OLS) esimaor is no consisen if ARCH effecs are presen. He suggesed a Lagrange Muliplier (LM) es for esing for ARCH effecs, which is regularly used as a diagnosic es in regression analyses. This es was inroduced for single equaion cases. In his paper we exend his es mehod o es for mulivariae ARCH effecs in he vecor auoregressive (VAR) model. Since he inroducion of he VAR model by Sims (1980) his model has been exensively used in applied research. Thus, performing ess for mulivariae ARCH effecs can be of ineres o he praciioners. The aim of he presen sudy is o invesigae he size and power properies of a mulivariae version of Engle s es for ARCH effecs under differen siuaions of sabiliy and insabiliy and of small and moderae sample sizes. The properies of he es will be invesigaed using asympoic and boosrap disribuions.

We invesigae he size and power properies for 1%, 5% and 10% significance levels. A wide number of parameers will be used in he VAR model o make he resuls as represenaive as possible. This paper is organized as he follows. Secion describes he VAR model and he LM es for esing ARCH effecs in a mulivariae perspecive. Secion 3 inroduces our simulaion design. Secion 4 presens he simulaion resuls. The conclusions are provided in he las secion.. The VAR Model and Mulivariae ARCH Effecs Consider he following vecor auoregressive of order p, VAR(p), process: y = c + A1 y 1 + Κ + Ap y p + ε, (1) where y = a vecor of n variables, c = a vecor of n inerceps, ε = a vecor of n error erms, and A r = an n n marix of coefficien marix for lag order r. Suppose he vecor of error erms is disribued wih zero covariance among he error erms and a corresponding n 1 variance vecor, σ. The variance vecor is assumed o follow he process: σ v Κ + u, () = + ρ1ε 1 + + ρ mε m where u is an error erm vecor, each elemen of which is assumed o be whie noise. To es for ARCH effecs, we consider an equaion in which he equaion () erms are replaced wih heir residual counerpars, i.e. ˆ ε = v + ρ ˆ 1ε 1 + Κ + ρ ˆ mε m + u, (3) where ˆ ε is he vecor of squared residuals a ime. The null hypohesis of no mulivariae ARCH effecs of degree m is : ρ ρ = Λ = ρ 0 and he alernaive hypohesis of H 0 1 = m = mulivariae ARCH effecs of degree m is a leas one of he ρ i marixes (i = 1,..., m) is no a zero marix. The following mulivariae LaGrange Muliplier (MLM) es saisic can be used o es he null hypohesis: 3

Ω R ( ) ( ) MLM m = T log (4) ΩU where he denoaion is an adjusmen for parameers, 1 and T is he sample size. Ω R is he esimaed variance-covariance marix for he error erm vecor in equaion (3) when he null hypohesis of no mulivariae ARCH(m) is imposed. Ω U is he esimaed variance-covariance marix for he error erm vecor in equaion (3) when he null hypohesis is no imposed. Ω represens he deerminan of marix Ω. Under he null hypohesis of no mulivariae ARCH effecs of order m, he LM saisic is asympoically disribued as χ wih m n degrees of freedom. 3. Mone Carlo Simulaion and Boosrapping For he purpose of simulaions in his sudy we will concenrae on a bivariae VAR(1) model wih mulivariae ARCH effecs of firs degree as he following: y1 1.0 α 1,11 α1,1 y1 1 ε1 = + +, (5) y 1.0 α 1,1 α1, y 1 ε where each ε i (i = 1, ) is independenly drawn from N(0, σ ), and σ 1 ρ 1,11 ρ1,11 0 u 1 ε1 1 1 = + +. (6) 3 σ 1 ρ 1, 0 ρ1, u ε 1 The error erms u i (i = 1, ) are drawn independenly from a sandard normal disribuion. To allow us o use he same number of observaions regardless of number of observaions and o remove effecs from sar up values, we generae 100 presample observaions. We use a variey of combinaions of parameer values in he rue model. Table 1 displays he various values for he parameers ha we use in equaion (5). Enumeraing all he combinaions of hese parameers, we come up wih 94 (6 7 7) combinaions. i 1 The adjusmen is done by making use of an Edgeworh expansion suggesed by Anderson (1958), which is generalized by Haemi-J (004). In our case, = T - m n + 0.5(n(m-1)-1). Johansen suggesed he es saisic in (4) for esing mulivariae auocorrelaion, and indicaes i is asympoically disribued as χ wih m n degrees of freedom. We use his es saisic for esing anoher consrain on he mulivariae sysem, so i should have he same asympoic disribuion. 3 For more deails regarding he derivaion of he mulivariae ARCH effecs as described by equaion (6) see Haemi-J (004). 4

Table 1. Parameer values for VAR model of equaion (4) α 1,11-1 -0.6-0. 0. 0.6 1.0 α 1, -0.8-0.5 0.1 0.0 0.1 0.5 0.8 α 1,1 = α 1,1-0.8-0.5 0.1 0.0 0.1 0.5 0.8 We run simulaions separaely for wo assumpions on he ARCH effecs. One, o check he size properies, is o assume he rue model has no ARCH effecs, i.e., ρ 1,11 = ρ 1, =0. The oher, o consider power properies, is o assume he rue model has firs-degree ARCH effecs wih ρ 1,11 = ρ 1, =0.5. In general, having he inercep vecor in he mulivariae ARCH process as ( 1 ρ, ρ ) 1,11 1 1,, wih he ARCH coefficien erms beween 0 and 1 inclusive as we have, guaranees he uncondiional variance for each error erm is equal o one. We also run our simulaions separaely for hose parameers values resuling in sabiliy of he model and hose ha do no. The disincion beween sable and unsable cases is relevan, as sabiliy of he model is imporan for sandard asympoic heory o apply, bu insabiliy is common in economic ime series. By looking a he modulus of he companion marix for α, B = α, 111 1 1 α α 11, 1,, (7) we can deermine he sabiliy of he model (he modulus is he square roo of he summed squares of he real and imaginary eigenvalue elemens). Sabiliy is indicaed when he modulus of each eigenvalue is less han uniy. Oherwise insabiliy is indicaed. Parameer value combinaions ha resul in explosive processes, wih a maximum absolue-value modulus greaer han 1, are dropped. All bu 150 of he parameer-value combinaions are dropped for his reason. Table indicaes he disribuion of he remaining parameer cases for various levels of he maximum absolue-value modulus. Table. Disribuion of parameer cases wih respec o he modulus STABILITY STATUS NUMBER OF CASES PERCENT OF CASES WITH MAXIMUM ABSOLUTE-VALUE MODULUS FALLING WITHIN RANGE. <0.5 [0.5,0.5) [0.5, 0.75) [0.75,0.95) [0.95,1) 1 Sable 136 10.9.94 39.71 41.18 5.88 0.00 Unsable 14 0.00 0.00 0.00 0.00 0.00 100 5

In addiion o disinguishing beween sable and unsable models and beween models wih and wihou mulivariae ARCH effecs our simulaions, we also disinguish beween wo sample sizes, T, of 40 and 100. This leads o eigh scenarios consising of he eigh combinaions of he wo differen sabiliy sauses, he wo differen sample sizes, and he wo differen assumpions abou ARCH effecs. For each case in each of he eigh scenarios we perform 1000 simulaions. In each of hese simulaions we esimae each of he wo scalar equaions implied in (5) using OLS, resuling in residuals for each equaion εˆ i (i = 1, and = 1,..., T) and he esimaed squared residual vecor ˆ ε ( ˆ ε, ˆ ε ). Wih he esimaed squared residual vecor we esimae (3) = 1 wih and wihou he resricion : ρ ρ = Λ = ρ 0 using OLS, find he esimaed H 0 1 = m = variance-covariance marix under each siuaion, and compue he MLM es saisic as shown in equaion (4). In each of hese simulaions we es for ARCH effecs by deermining wheher he MLM saisic is significanly large according o he χ as described earlier and according o quaniles from a boosrap disribuion o be described below. The size and power properies of hese ess are invesigaed a he nominal levels of 1%, 5%, and 10% for m = 1,..., 5. The procedure ha we use o deermine he boosrap disribuion consiss of he following seps: 1. For each scalar equaion implied in (5), T values are drawn randomly and independenly wih replacemen from he residuals for ha equaion, εˆ i, and he mean of hose seleced values is subraced from each value o give he boosrap residuals, ε. i. The boosrap values for y, denoed by y ˆ ˆ, = c + A1 y 1 + ε y, are compued by using he equaion where a circumflexed parameer represens he OLS esimae for ha parameer and ε = ( 1, ) ε ε. 3. The boosrap MLM (BMLM) is calculaed using he y series ( = 1,..., T), calculaing he saisic he same way as we do using he rue simulaed daa. 4. Seps 1-3 are repeaed 500 imes o generae he boosrap disribuion of he es saisic. 6

4. The Resuls of he Mone Carlo Experimens The simulaion resuls for size properies of MLM es and he BMLM es are presened in Tables 3-6 and he resuls for power properies of hese wo ess are presened in Tables 7-8. In his sudy wo measures are used o check he size properies of he ess. The firs measure is he esimaed acual size. The second measure is absolue deviaions from he nominal size. The means over he parameer combinaions in he paricular scenario are presened for boh measures. Based on he simulaion resuls he following can be noed. Using he asympoic disribuion, he acual size ends o be higher han he nominal size when he nominal size is 1%, bu he reverse is generally rue when he nominal size is 5% or 10% and he sample size is small (40 observaions) and/or he VAR model is sable. Also, when he sample size is small and/or he VAR model is sable, and he nominal size is 5% or 10%, using he boosrap disribuion raher han he asympoic one generally resuls in he acual size being closer o he nominal size. Similar conclusions can be drawn when one looks a he average absolue deviaion from he nominal size for he asympoic and boosrap mulivariae LM ess. I is of course a subjecive issue, bu he average absolue deviaions do no seem very excessive for he asympoic LM ess. For he moderae sample size (T = 100) wih unsable VAR models for example, hey are 0.4 o 0.7 percenage poins a he 5% nominal size, and 0.8 o 1.6 percenage poins a he 10% nominal size. However, hey are 0.4 o 0.7 percenage poins a he 1% nominal size in his example, which seems somewha high a his level. A he 5% and 10% nominal size level, he power of he boosrapped LM es is superior o he asympoic one when he sample size is small and/or he mulivariae ARCH lag order of he es is equivalen o ha of he rue model (we es for power only in he siuaion where he rue model has a 1-lag ARCH srucure, however). A he 1% nominal size level, he power of he asympoic MLM es is superior of he boosrap version, and his is likely due o he acual size ending o be higher han he acual size when using he asympoic es a his nominal size level. Anoher resul ha can be drawn from he simulaions is ha, unsurprisingly, larger sample sizes produce higher power irrespecive of wheher he boosrap disribuion or he asympoic disribuion is used. Also in unsable VAR models he power seems o be higher compared o sable VAR models for he same sample size. 5. Conclusions This sudy has exended Engle s LM es for ARCH effecs o a mulivariae es. The size and power properies of his es are invesigaed for esing mulivariae ARCH effecs in he 7

sable and unsable VAR models. We have shown ha for he ypical nominal size used in business and economics of 5% ha he boosrapped version of he mulivariae LM es has relaively beer size and power properies in small samples and/or sable VAR models. However he gains from using he boosrap version of he mulivariae LM es are no clear, especially when using 1% nominal size, where boosrapping leads o lile if any gain in achieving he appropriae size and leads o slighly worse power performance, and when dealing wih unsable VAR cases wih moderae sample sizes. To our bes knowledge any es for mulivariae ARCH effecs is no available auomaically in he common economeric packages in he marke, unlike oher mulivariae diagnosic ess, e.g. ess for auocorrelaion and normaliy. Given wha we see in our simulaion resuls, we hink ha making a leas he asympoic mulivariae LM es available in sofware packages would be useful for praciioners. We find his imporan because OLS esimaion for VAR, which is he sandard VAR esimaion mehod, resuls in esimaes ha are no consisen if ARCH effecs are presen. Therefore, i is imporan o es for mulivariae ARCH effecs when esimaing a VAR model. 8

Table 3: Acual sizes and average absolue deviaions from he nominal size when T = 40 in sable VAR models ABSOLUTE DEVIATION FROM ACTUAL SIZES FOR EACH TEST NOMINAL SIZE ARCH order 1 3 4 5 1 3 4 5 Nominal size 1% Nominal size 1% MLM 1.1% 1.% 1.% 1.3% 1.% 0.3% 0.3% 0.3% 0.3% 0.3% BMLM 1.3% 1.% 1.% 1.% 1.% 0.4% 0.3% 0.3% 0.3% 0.3% Nominal size 5% Nominal size 5% MLM 3.6% 4.% 4.3% 4.4% 4.4% 1.4% 0.9% 0.8% 0.7% 0.7% BMLM 5.0% 5.1% 5.% 5.% 5.3% 0.7% 0.5% 0.6% 0.6% 0.6% Nominal size 10% Nominal size 10% MLM 6.7% 7.7% 7.7% 8.0% 8.0% 3.3%.3%.3%.0%.0% BMLM 9.5% 9.7% 9.9% 10.1% 10.% 1.0% 0.8% 0.7% 0.7% 0.8% MLM is he mulivariae LaGrange Muliplier presened in equaion (3) and BMLM is he boosrapped MLM. Bold values indicae bes relaive performance. The means over he parameer combinaions in he paricular scenario are presened for boh measures. Table 4: Acual sizes and average absolue deviaions from he nominal size when T = 40 in unsable VAR models ACTUAL SIZES FOR EACH TEST ABSOLUTE DEVIATION FROM NOMINAL SIZE ARCH order 1 3 4 5 1 3 4 5 Nominal size 1% Nominal size 1% MLM 1.3% 1.4% 1.4% 1.4% 1.4% 0.4% 0.4% 0.4% 0.4% 0.4% BMLM 1.3% 1.% 1.% 1.% 1.3% 0.3% 0.3% 0.3% 0.3% 0.4% Nominal size 5% Nominal size 5% MLM 4.% 4.7% 4.6% 4.6% 4.5% 0.9% 0.4% 0.5% 0.7% 0.7% BMLM 5.1% 5.3% 5.1% 5.1% 5.% 0.7% 0.5% 0.5% 0.5% 0.6% Nominal size 10% Nominal size 10% MLM 7.4% 8.% 8.3% 8.3% 8.4%.6% 1.8% 1.8% 1.7% 1.6% BMLM 9.7% 10.0% 10.% 9.9% 10.3% 1.0% 0.6% 0.9% 0.9% 0.9% MLM is he mulivariae LaGrange Muliplier presened in equaion (3) and BMLM is he boosrapped MLM. Bold values indicae bes relaive performance. The means over he parameer combinaions in he paricular scenario are presened for boh measures. 9

Table 5: Acual sizes and average absolue deviaions from he nominal size when T = 100 in sable VAR models ABSOLUTE DEVIATION FROM ACTUAL SIZES FOR EACH TEST NOMINAL SIZE ARCH order 1 3 4 5 1 3 4 5 Nominal size 1% Nominal size 1% MLM 1.3% 1.4% 1.4% 1.4% 1.3% 0.3% 0.5% 0.4% 0.5% 0.4% BMLM 1.3% 1.3% 1.3% 1.3% 1.3% 0.4% 0.4% 0.4% 0.4% 0.4% Nominal size 5% Nominal size 5% MLM 4.3% 4.7% 4.8% 4.8% 4.6% 0.8% 0.6% 0.6% 0.5% 0.6% BMLM 5.% 5.% 5.3% 5.% 5.3% 0.5% 0.6% 0.6% 0.6% 0.6% Nominal size 10% Nominal size 10% MLM 7.9% 8.6% 8.6% 8.7% 8.5%.1% 1.5% 1.4% 1.3% 1.5% 10.1 10.1 BMLM 9.9% 10.1% 10.1% 0.7% 0.8% 0.7% 0.7% 0.8% % % MLM is he mulivariae LaGrange Muliplier presened in equaion (3) and BMLM is he boosrapped MLM. Bold values indicae bes relaive performance. The means over he parameer combinaions in he paricular scenario are presened for boh measures. Table 6: Acual sizes and average absolue deviaions from he nominal size when T = 100 in unsable VAR models ABSOLUTE DEVIATION FROM ACTUAL SIZES FOR EACH TEST NOMINAL SIZE ARCH order 1 3 4 5 1 3 4 5 Nominal size 1% Nominal size 1% MLM 1.5% 1.6% 1.6% 1.7% 1.4% 0.5% 0.6% 0.6% 0.7% 0.4% BMLM 1.4% 1.4% 1.4% 1.4% 1.3% 0.4% 0.4% 0.4% 0.4% 0.4% Nominal size 5% Nominal size 5% MLM 4.8% 5.3% 5.4% 5.3% 4.8% 0.6% 0.4% 0.7% 0.6% 0.6% BMLM 5.4% 5.6% 5.6% 5.6% 5.4% 0.7% 0.7% 0.8% 0.8% 0.6% Nominal size 10% Nominal size 10% MLM 8.4% 9.3% 9.5% 9.% 8.7% 1.6% 0.9% 0.8% 1.0% 1.3% 10.6 10.5 10. BMLM 10.1% 10.7% 0.9% 0.8% 0.8% 0.6% 0.7% % % % MLM is he mulivariae LaGrange Muliplier presened in equaion (3) and BMLM is he boosrapped MLM. Bold values indicae bes relaive performance. The means over he parameer combinaions in he paricular scenario are presened for boh measures. 10

ARCH order Table 7: Power properies when T = 40 STABLE VAR MODEL UNSTABLE VAR MODEL 1 3 4 5 1 3 4 5 Nominal size 1% Nominal size 1% MLM 17.4% 14.0% 11.5% 10.0% 8.8% 19.6% 16.1% 13.1% 11.8% 10.% BMLM 16.7% 1.3% 10.0% 8.5% 7.5% 17.0% 1.8% 10.5% 9.% 8.% Nominal size 5% Nominal size 5% MLM 30.0% 5.3% 1.3% 19.0% 17.1% 3.9% 7.6% 3.6% 1.4% 19.1% BMLM 34.0% 6.9%.7% 0.0% 18.1% 35.0% 8.1% 4.1% 1.4% 19.4% Nominal size 10% Nominal size 10% MLM 38.4% 3.9% 8.5% 5.9% 3.5% 41.4% 35.6% 31.% 8.4% 5.8% BMLM 44.6% 36.9% 3.1% 8.9% 6.6% 46.4% 38.5% 34.0% 30.7% 8.% MLM is he mulivariae LaGrange Muliplier presened in equaion (3) and BMLM is he boosrapped MLM. Bold values indicae bes relaive performance. The means over he parameer combinaions in he paricular scenario are presened for boh measures. ARCH order Table 8: Power properies when T = 100 STABLE VAR MODEL UNSTABLE VAR MODEL 1 3 4 5 1 3 4 5 Nominal size 1% Nominal size 1% MLM 71.7% 63.8% 57.3% 5.% 47.5% 73.1% 65.% 58.8% 53.6% 48.8% BMLM 66.6% 56.3% 49.6% 44.4% 40.3% 66.1% 56.5% 49.8% 44.4% 40.4% Nominal size 5% Nominal size 5% MLM 83.1% 76.5% 70.8% 66.1% 61.7% 84.% 77.6% 7.1% 67.4% 63.0% BMLM 84.1% 75.8% 69.6% 64.5% 60.4% 84.3% 76.% 70.0% 65.1% 61.3% Nominal size 10% Nominal size 10% MLM 88.0% 8.4% 77.4% 73.1% 69.3% 88.9% 83.4% 78.7% 74.% 70.5% BMLM 89.9% 83.6% 78.6% 74.1% 70.4% 90.% 84.3% 79.% 74.6% 71.0% MLM is he mulivariae LaGrange Muliplier presened in equaion (3) and BMLM is he boosrapped MLM. Bold values indicae bes relaive performance. The means over he parameer combinaions in he paricular scenario are presened for boh measures. References Anderson T. W. (1958) An Inroducion o Mulivariae Saisical Analysis. New York, Wiley. Davison, A. C., and Hinkley, D.V. (1999) Boosrap Mehods and Their Applicaion. Cambridge Universiy Press. Cambridge, UK. Efron, B. (1979) Boosrap Mehods: Anoher Look a he Jackknife, Annals of Saisics 7, 1-6. Engle, R. (198) Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion, Economerica, 50(4), 987-1007. Haemi-J (004) Mulivariae Tess for Auocorrelaion in Sable and Unsable VAR Models, Economic Modelling, Vol 1(4), 661-683. 11

Johansen, S. (1996) Likelihood-Based Inference in Coinegraed Vecor Auoregressive Models. Advanced Texs in Economerics. Oxford Universiy Press. New York. Sims, C. A., (1980) Macroeconomics and Realiy. Economerica, 48, 1-8. 1