Testing Jointly for Structural Changes in the Error Variance and Coefficients of a Linear Regression Model (Proof)

“Testing jointly for structural changes in the error variance and coefficients of a linear regression model” Supplementary Material A: Proof of the statement in equation (7): Consider an AR(1) model yt = yt 1 +ut in which the variance of ut has n breaks and j j < 1. Consider the variance adjusted series yt = y t 1 + ut where ut = ut = Then, P s] 0 T 1 [T t=1 zt zt = T = T 1 P[T s] t=1 P[T s] 1 t=1 i0 yt2 yt and yt = 1 2 1 1 + 2T 0 yt with P[T s] t=1 yt 1 (yt 0 = 10 , without loss of generality. yt 1 ) + T 1 1 P[T s] t=1 (yt 1 yt 1 )2 : We here show that the second and the third terms are Op ( T ) and Op ( 2T ) where T ! 0 uniformly in s. For notational simplicity, we show these results for yt and yt instead of yt 1 as the di¤erence is negligible. First, for [T v0 [T v0 k 1] < t k ]: P Pt 1 ut = kl=1 ( l;0 = 0 )vl;t yt = =0 P Pt 1 ut = kl=1 vl;t yt = =0 P[T v0 Pt l ] where vl;t = 0 u for l < k and v = v0 k;t 0 t =[T v0 ]+1 =[T ]+1 and yt 1 yt (yt yt ) = (yt yt )2 = so that for any l < k T 1 P[T v0 l ] v0 l 1 ]+1 t=[T ut . These yield k 1 l 1 yt (yt Pk l1 =1 Pk l1 =1 Pk l2 ;0 l2 =1 l1 ;0 l2 =1 l1 =1 vl1 ;t vl2 ;t 0 Pk Pk yt ) = 0 l2 ;0 0 0 Pk = S1;l = Op ( vl1 ;t vl2 ;t 0 l2 ;0 l2 =1 | 0 0 }| 0 {z =Op ( T) T T) 1 P[T v0 l ] v0 l 1 ]+1 t=[T {z vl1 ;t vl2 ;t =Op (1) and T = 1 P[T v0 l ] v0 l 1 ]+1 t=[T Pk l1 =1 Pk l2 =1 = S2;l = Op ( | 2 T ): (yt yt )2 l1 ;0 0 0 l2 ;0 {z =Op ( 0 0 2) T S-1 T }| 1 P[T v0 l ] t=[T v0 l 1 ]+1 {z =Op (1) vl1 ;t vl2 ;t } } For any l < k; T 1 P[T v0 l ] t=[T v0 l 1 ]+1 vl1 ;t vl2 ;t = Op (1) because vl1 ;t and vl2 ;t are covariance stationary series for any l1 and l2 . We can show that the same property holds uniformly in s for l = k with a minor change of notation. Therefore, uniformly in s, P yt 1 (yt 1 yt 1 ) = 2 kl=1 S1;l = Op ( T ) P s] Pk 2 T 1 [T yt 1 )2 = t=1 (yt 1 l=1 S2;l = Op ( T ) 1 2T P[T s] t=1 B: The choice of ^ . To address what speci…c version of the correction factor to use, we consider the size and power of the sup LR4;T test under the following simple DGP with GARCH(1,1) errors: p yt = 1 + 2 1(t > [:25T ]) + et ; et = ut ht , ut i:i:d: N (0; 1); ht = 1 + 2 1 (t > [:75T ]) + e2t 1 + ht 1 ; with h0 = 1 = (1 ) and 1 = 1. The sample size is T = 100 and " = 0:20. Under H0 , 2 = 2 = 0, while under H1 , one break in mean and one break in variance are allowed ( = 0 under both H0 and H1 ). We consider four versions for the estimate ^ as de…ned by 1 (8): 1) ^ = 2, i.e., no dependence in t is accounted for (labelled “no correction”), 2) using the residuals under H1 to construct the bandwidth bT and to estimate the autocovariances of t (labelled “alternative”); 3) using the residuals under H0 instead (labelled “null”); and, as suggested by Kejriwal (2009), 4) using a hybrid method that constructs the bandwidth bT using the residuals under H1 but uses the residuals under H0 to estimate the autocovariances of t (labelled “hybrid”). Here and elsewhere, we use 1,000 replications. The reason to include the “no correction”option is to assess which cases (i.e., which combinations of values for and ) leads to distortions when serial dependence is not accounted for and how well the various suggested options for corrections improve the size. The results for the exact size of the test (5% nominal size) are presented in Table S.1. The critical values are those of the bound of the limit distribution, hence, a conservative size is expected. The results show that the methods “no correction” and “alternative” exhibit substantial size distortions, that increase with and , which indicates the extent of the correlation in the squared residuals. The method “null”, on the other hand, shows conservative size distortions as expected. The hybrid method shows less conservative size distortions when and are not very large. These results dictate our choice of = 0:2 and = 0:1,..., 0:5 in the subsequent simulations reported in the text since they imply tests that require a correction and using either the “null”and “hybrid”methods yields test with good S-2 …nite samples sizes. The results for power are presented in Table S.2. We only consider the methods “null”and “hybrid”given the high size distortions of the methods “no correction” and “alternative”. The results show that substantial power gains can be achieved using the “hybrid” method as opposed to the “null” method, especially if the GARCH e¤ect is pronounced. Hence, we recommend using the “hybrid” method and all results reported in the main text are based on it. C: Should we always correct? We address the issue of whether it is costly in terms of power to use a correction valid under more general conditions than needed. To that e¤ect we …rst consider the power of the sup LR4;T test under the following DGP with normal errors: yt = 1 + 2 1(t > T1c ) + et ; et i:i:d: N (0; 1 + 1(t > T1v )); where we set 1 = 0 and 2 = . We consider three scenarios for the timing of the breaks: a common break in mean and variance at T1c = T1v = [:5T ], and disjoint breaks at fT1c = [:3T ], T1v = [:6T ]g and fT1c = [:6T ], T1v = [:3T ]g. We use T = 100, 200 and the power, for 5% nominal size tests, is evaluated at values of ranging from 0:25 to 1:5 with " = 0:15. Three versions of the sup LR4;T tests are evaluated: 1) with a full correction based on ^ using the hybrid method (labelled “full”); 2) a correction valid only for i:i:d: errors, though not P P necessarily normal, given by ^ = ^ =^ 4 1, where ^ 2 = T 1 T u^2 and ^ = T 1 T u^4 4 t=1 t 4 t=1 t with u^t the residuals under H0 (labelled “i.i.d.”); 3) no correction, i.e., using ^ = 2, which is the appropriate value with normal errors (labelled “NC”). The results are presented in Table S.3. They show that the power is basically the same using any of the three methods. Hence, there is no cost in using a full correction and we use it throughout for the results reported in the main text. D: Local asymptotic power functions. We consider model (1) focusing on the case of n = m = 1 with the following assumptions. p Assumption L1: Assumptions A1 and A3 hold with 20 = T . We also 10 = P s] P 2 have T 1=2 [T 1] ) W (s) with = limT !1 var(T 1=2 Tt=1 [(ut )2 1]) and t=1 [(ut ) P s] 2 p T 1 [T t=1 (ut ) ! s uniformly in s 2 [0; 1]. p 0 Assumption L2: Assumptions A2 and A3 hold with 02 = T: 1 = In the following, we derive the local asymptotic power of the sup LR2;T and sup LR3;T S-3 tests, allowing for nuisance breaks, i.e., we consider the tests sup LR2;T (n = 1; m = 1; "jn = 0; m = 1) and sup LR3;T (m = 1; n = 1; "jm = 0; n = 1). The results are also valid if the nuisance breaks are not accounted for, in which case the tests reduce to the sup LR1;T and the sup LRT test (Andrews, 1993), respectively. Without loss of generality, we denote by 0 and 0 2 0 by 20 . Lemma S.1 Under Assumption L1 or Assumption L1 with Assumption A3 allowing for 1 = 0, sup LR2;T (n = 1; m = 1; "jn = 0; m = 1) ) sup v 2 ( =2)[J( v )]2 c v; (S.1) where v W (1) W ( v ) p v + v (1 ) 8 q < v0 1 v v q v b( v ) = v0 : (1 ) 1 v 2 p sup LR1;T (n = 1; ") ) sup ( =2)[J( v )]2 : J( v ) = In particular if 1 if if 0 b( v ) v0 v0 v : v > = 0, which is imposed in the construction of the test, v 2 v;" (S.2) Lemma S.2 Under Assumption L2 or Assumption L2 with Assumption A3 allowing for 10 = 0, sup LR3;T (m = 1; n = 1; "jm = 0; n = 1) ) sup c 2 v c;" ( c )0 J( c ) (S.3) where c Wq (1) Wq ( c ) p c + Q1=2 b( c ) c (1 ) 8 q c < c0 1 c c if c0 c q b( ) = c c0 : (1 ) 1 c if c0 > c J( c ) = In particular, if 10 = 0, which is imposed in the construction of the test, sup LRT (m = 1; ") ) sup c 2 c;" J( c )0 J( c ): (S.4) Importantly, the result in (S.4) is the same as Theorem 4(c) in Andrews (1993), if we c0 set (in his notation) (s) = I(s ), S = 20 (T 1 Z 0 Z) and M = (T 1 Z 0 Z). For S-4 comparisons, we also consider the cumulative sum of squares (CU SQ) test when there are no nuisance coe¢ cient breaks. With u^t the OLS regression residuals, the CU SQ test is: i hP v PT T Tv 2 1=2 2 u ^ max T u ^ t=1 t t=1 t T 1 Tv T CU SQ = P T 1 Tt=1 u^2t From Deng and Perron (2008) CU SQ = sup v T 1=2 P[T v u2t ] t=1 2[0;1] 2 0 [T v ] PT t=1 T 1 and we obtain the following result. u2t 2 0 1 + op (1) Lemma S.3 Under Assumption L1, if there is no structural change in the coe¢ cients, then CU SQ ) where p v W ( v) sup v ( )= v 2[0;1] 8 < : v0 v v ) if v0 v0 ) if v0 (1 (1 2 W (1) + p 0 ( v) (S.5) v > v : Lemma S.1 suggests that the local asymptotic power of the sup LR2;T test coincides with that of the sup LR1;T test except for the fact that the set of permissible break dates cv;" becomes smaller than v;" . Lemma S.2 suggests that the local asymptotic power of the sup LR3;T test coincides with that of the standard sup LRT test derived in Theorem 4 of Andrews (1993) except that the set of permissible break dates vc;" is in general smaller than c;" . Hence, when testing for changes in variance allowing for changes in coe¢ cients, we have the same local asymptotic power function as when testing for changes in variance when no change in coe¢ cient is present and none is allowed for. Therefore, we incur no loss in local asymptotic power by adopting our more general approach. We next compare the local asymptotic power functions of the sup LR1;T test given by (S.2), the sup LR2;T test given by (S.1) and the CU SQ test given by (S.5) via Monte Carlo simulations. To this end, the Wiener processes W ( ) are approximated by the partial sums of i.i.d. standard normal random variables with 5,000 discrete steps. The power functions of 5% nominal size tests are computed based on 10,000 Monte Carlo replications with the value of ranging from 0 to 10. We also set the trimming " = 0:15, = 2 and 0 = 1, although these particular choices do not qualitatively a¤ect the results. We use the critical p values of 8:58 for the sup LR1;T and sup LR2;T tests and 2 1:358 for the CU SQ test. S-5 Figure S.1 shows the asymptotic local power functions of the sup LR1;T test and the CU SQ test when a break in variance occurs at v0 = 0:3; 0:5 and 0:7 and no break occurs in the coe¢ cients. They show the local asymptotic power functions to be almost identical. Figure S.2 presents the local asymptotic power functions of the sup LR2;T test when it accounts for a coe¢ cient break at c0 = 0:3; 0:5 or 0:7. It also shows, the local asymptotic power functions of the CU SQ test under the (correct) assumption of no break in the coe¢ cients. Hence, this simulation design gives an advantage to the CU SQ test and some power loss for the sup LR2;T test might be expected. Indeed, the power of the sup LR2;T test is slightly lower when the variance and the coe¢ cient break dates coincide. This is because the permissible break dates around the true break date are not considered due to the concurrent nuisance break. However, the power loss of the sup LR2;T test is very minor even though the sup LR2;T test allows for a coe¢ cient break. The power functions of both tests are almost identical when the two breaks are far apart. i.e., the case of ( v0 ; c0 ) = (0:3; 0:7) and (0:7; 0:3). Proof of Lemma S.1 The sup LR2;T test is: ^ T (Tec ; Tev ) log L eT (T^c )] sup LR2;T = 2[log L = T log e2 (T Tev ) log ^ 22 Tev log ^ 21 P P where e2 = T 1 Tt=1 (yt x0t e zt0et;j )2 , ^ 22 = (T Tev ) 1 Tt=Tev +1 (yt x0t ^ zt0 ^t;j )2 and P ev ^ 2 = Tev 1 T (yt x0 ^ z 0 ^t;j )2 . Applying a Taylor expansion to log e2 , log ^ 2 and log ^ 2 1 around log t t t=1 2 (without loss of 0 2 generality, let 2 0 = 2 20 ), 1 we obtain sup LR2;T = (F1;T + F2;T ) + op (1) where F1;T = ( 20 ) 1 [T e2 Te h X Tev )^ 22 (T v = ( 2 1 0) (yt t=1 +( 20 ) and F2;T = = " 1 T 2 1 (I 2 1 T h X x0t e (yt t=Tev +1 e2 II 2 0 zt0et;j )2 x0t e 2 2 0 III): (T Tev ^ 21 ] Tev ) S-6 x0t ^ (yt zt0et;j )2 ^ 22 (yt 2 0 2 0 zt0 ^t;j )2 x0t ^ 2 Tev i zt0 ^t;j )2 ^ 21 2 0 2 0 i 2 # We …rst show that F1;T = op (1). From Assumption A1, for any partition Tev , we have Xi0 Xi = Op (T ); Zi0 Zi = Op (T ); Xi0 Zi = Op (T ); Xi0 Ui = Op (T 1=2 ) and Zi0 Ui = Op (T 1=2 ) for i = 1 and 2. In addition, under Assumptions A1 and A3 in which the change in the ^ = Op (T 1=2 ), 0 ^j = Op (T 1=2 ), coe¢ cient is assumed to shrink at rate T , we obtain j ^ e = op (T 1=2 ej = op (T ) and ^j 1=2 ) for j = 1 and 2. Hence, F1;T = op (1) is shown by directly following the proof of Theorem 1(b). If there is no break in the coe¢ cient ^ = Op (T 1=2 ), 0 ^j = Op (T 1=2 ), ^ e = op (T 1=2 ) and ( 01 = 02 ), we also obtain j ^j ej = op (T 1=2 ) for j = 1 and 2 no matter where Tec is. Hence, F1;T = op (1). For F2;T , we slightly change the notation and express the change in variance as i0 p ( i = 20 )= T for i = 1; 2. We also denote 20 by 0 without loss of generality so that by construction. Then, i0 = p T 0 = i 2 i0 = 2 0 or = 2 0 1 1=2 = T 1=2 = T ) p II = = = p T T t=1 T X ( u2t ( t=1 ! = 0 )2 = T 1 + 2 pi + O(T T 2 0 2 0 1 1 =0 ) (S.6) ) : # zt0et;j )2 1 1) + T 2 i0 1 T X u2 ( 2t )2 v0 1 1=2 1=2 ; T X ( u2t 2 0 t=Tev +1 T X u2 1=2 ( t2 t=1 t=1 i0 S-7 T1v0 ) + op (1) I(t 1) + op (1) Te X u2 1=2 T ( t2 v 1) 0 T X u2 1=2 ( 2t 1 i0 t=1 8 1=2 < v e T T T : T ! 1=2 8 < v e T T T : T ! x0t e u2t T i 1) + op (1) 2 0 W (1) + 2 Tev ( 1 + 2 pi + O(T T 2 i0 For each of the three terms, we have " T X p (yt I = T 1=2 t=1 T X 2 = 0 + 2 pi + T 1 20 t=1 0 Te X u2 1=2 T ( 2t v 1) t=1 i0 9 = 1) + op (1) ; 1) T X u2 1 ( 2t ) +T t=1 ) = ) Therefore, T v0 ) I(t 1 i0 Tev T ! III = 1=2 8 < T : v0 minf ; v g p 2 1; v 1 9 ! 1=2 8 ev T = < v 2 X u Te ( t2 1) + op (1) T 1=2 ; : T 0 t=1 Te X u2 1 ( 2t )2 1) + T Te X u2 1=2 ( 2t v v i0 t=1 v0 F2;T ) 2 where v b( ) = This yields I(t 1 i0 t=1 v0 v p W ( ) minf ; p v + p v v 9 = T v0 ) + op (1) ; v p W (1) W ( ) p + v 1 p Te X u2 1 ( 2t )2 T " 8 < v v 2 1 : W (1) W ( v ) 2 p v +p v (1 ) v0 : (1 sup LR2;T ) g t=1 q v 1 v v0 sup v 2 c v;" I(t 1 i0 2 q ) 1 if v v if 1 #2 b( v ) v0 v0 9 = T1v0 ) + op (1) ; v v > : [J( v )]2 v W (1) W ( v ) p v J( ) = + v (1 ) 8 q < v0 1 v v q v b( v ) = v0 : (1 ) 1 v v 2 1 p b( v ) if if v0 v0 v > v : The results for the sup LR1;T follow because F1;T = op (1) holds also when there is no break in the coe¢ cients. Proof of Lemma S.2. The sup LR3;T test is: ^ T (Tec ; Tev ) log L eT (T^v )] sup LR3;T = 2[log L = (T T^v ) log e22 + T^v log e21 (T Tev ) log ^ 22 S-8 Tev log ^ 21 P ev x0t e zt0e)2 ; e21 = (Tev ) 1 Tt=1 (yt x0t e zt0e)2 , ^ 22 = t=Tev +1 (yt P ev x0t ^ zt0 ^t;j )2 and ^ 21 = (Tev ) 1 Tt=1 (yt x0t ^ zt0 ^t;j )2 . Applying a t=Tev +1 (yt expansion to log e22 and log ^ 22 around log 220 , and to log e21 and log ^ 21 around log 210 , Tev ) where e22 = (T P (T Tev ) 1 T Taylor 1 PT (S.7) sup LR3;T = (F1;T + F2;T ) + op (1) where 2 e T^v ) 22 F1;T = (T 2 2 v ^2 v e1 e ^ T ) 2 +T 2 (T 20 and " 1 (T 2 " 1 ^v T 2 F2;T = 20 e22 T^v ) e21 2 10 2 ^ 21 ev T 2 20 Tev ) (T 2 20 We …rst show that F2;T = op (1). We have " e22 1 T T^v p T F2;T = 2 2 T 20 " 2 e21 1 T^v p 10 T 2 2 T 10 10 2 10 2 10 2 T T 2 10 2 2 20 2 10 2 v ^1 e T 2 Tev p T T ^ 22 2 # Tev p 2 2 20 2 20 # : ^ 22 T ^ 21 2 20 2 20 2 10 2 2 10 p p 2 2 2 2 Tev )=T ][ T (^ 22 T^v )=T ][ T (e22 20 )= 20 ] and [(T 20 )= p p 2 2 v 2 2 2 v e limit distribution and (T^ =T )[ T (e1 10 )= 10 ] and (T =T )[ T (^ 2 where [(T # 2 # 2 2 20 ] have the same 2 2 2 20 )= 20 ] have the same limit distribution under Assumption A3. These also hold when there is no break in the variance. For F1;T , let 0 = 20 without loss of generality, then h i F1;T = ( 20 ) 1 (T T^v )e22 (T Tev )^ 22 + T^v e21 Tev ^ 21 ( 20 ) The …rst term becomes, 2 T X ( 20 ) 1 4 (yt x0t e t=1 = ( 2 1 r 0 ) [D u D (2) 1 2 10 2 0 2 10 zt0e)2 u Tec X (T^v e21 (yt t=1 D (1)] S-9 x0t e Tev ^ 21 ): zt0e)2 T X t=Tec +1 (yt x0t ^ 3 zt0 ^j+1 )2 5 and the second term is op (1) under Assumption A3. When there is no break in variance, the second term is zero because Dr h = h T T h + T h T h + T h T 2 0 2 10 = 2 20 2 10 = 0. Then, Du (2) Du (1) i PT P 0 1=2 0 c0 0 (u z + I(t T ) z z ) T 1 Tt=1 zt zt0 t t t=1 t t i PT 1=2 0 c0 T )) t=1 (zt ut + zt zt I(t i PTec P ec 0 1=2 0 c0 0 (u z + I(t T ) z z ) T 1 Tt=1 zt zt0 t t t t t=1 i PTec c0 1=2 0 I(t T )) (z u + z z t t t=1 t t i P PT T 0 1=2 0 c0 0 1 0 (u z + I(t T ) z z ) T c t t e t t t=T +1 t=Tec +1 zt zt i PT 1=2 0 c0 (z u + z z I(t T )) + op (1) c t t t t t=Te +1 ) J( c )0 J( c ) where c Wq (1) Wq ( c ) p c J( c ) = c (1 ) 8 q < c0 1 c c c q c b( ) = c0 : (1 ) 1 c ! + Q1=2 b( c ) if c0 if c0 c > c P and Q p limT !1 (T 1 Tt=1 zt zt0 ). Hence, from (S.7) the results of the sup LR3;T test is obtained. The result for the the sup LRT is also obtained since we showed F2;T = op (1) to hold when there is no variance break. Proof of Lemma S.3. By (S.6), we obtain 2 2 i0 = 0 =1+2 i p = T + O(T 1 ) and the test statistic CU SQ is such that: CU SQ = sup v = 1=2 T 1=2 2[0;1] v P[T v ] t=1 2[0;1] sup v T [T ] PT t=1 T P[T v t=1 u2t 2 i0 ] u2t 2 i0 u2t 2 i0 2 i0 2 0 1 [T v ] PT t=1 T 1 + 2 p1 I(t T 1 + 2 p1 I(t T S-10 T v0 ) u2t 2 i0 2 i0 2 0 T v0 ) 1 1 + op (1) 1 + op (1) = sup v 2[0;1] +T ) where T p 1 1=2 P[T t=1 P[T v ] t=1 sup v v u2t 2 i0 [W ( v ) 2[0;1] ] u2t [T v ] PT 1 t=1 2 2 T i0 i0 [T v ] PT u2t 2 1 I(t T v0 ) t=1 2 T i0 2 v W (1)] + p 1 ( v ) u2t 1 v ( )= 8 < v0 : v (1 (1 v ) if v0 v0 ) if v0 2 1 I(t T v0 ) + op (1) v > v : E: Robustness to non-normal errors. Given that our tests are based on a quasi-likelihood framework assuming normal errors, it is useful to assess their size and power under non-normal error distributions. We focus on the tests for the structural changes in variance, i.e. the sup LR1;T and sup LR2;T tests, mostly because these are the most prone to be a¤ected by non-normality; e.g., the test for a single coe¢ cient break coincides with that derived by Andrews (1993) using a GMM-based approach. To investigate the sup LR1;T test, we generate the same data as the experiment pertaining to Table 2: yt = c + yt ht = where h0 = 1 = (1 1 + 1 + et ; et = ut 2 I(t p ht > [:5T ]) + e2t 1 + ht 1 ; ) and ut is drawn from the following well-known non-normal dis- tributions: (a) the t distribution with 5 degrees of freedom (t5 ), (b) a mixture of two normal distributions: v1 I(z 0:5)+v2 I(z > 0:5), where z U [0; 1], v1 N ( 1; 1) and v2 N (1; 1) 2 (c) the distribution with 5 degrees of freedom and (d) an exponential distribution ln(v), v U [0; 1]. These distributions were chosen as empirically relevant examples following Bai and Ng (2005). To facilitate comparisons, the errors are normalized by subtracting the sample mean and dividing by the sample standard deviation of each Monte Carlo repetition. The model parameter values are set at c = 0:5; 1 = 0:1; = 0:2; and " = 0:15. We consider = 0:2; 0:7 and = 0:1; 0:3; 0:5. The sample size is T = 100; 200. Table S.4 presents the exact size and power of the sup LR1;T , U D max1;T and CU SQ tests. The reported values are roughly comparable to those with normal errors in Table 2, i.e., little if any size distortions. In all cases, the power decreases to some extent. Note, however, that this is also the case for S-11 the CU SQ test and the relative advantage of the sup LR1;T and U D max1;T tests over the CU SQ test remains under these non-normal errors. For the sup LR2;T test, we use the same data generating process as that corresponding to Table 4 to assess the size, i.e., yt = 1 + 2 1(t > [0:5T ]) + et ; (S.8) with et drawn from one of the four types of non-normal distributions. Again, the errors are standardized to have mean zero and variance one in each Monte Carlo repetition. We set 1 = 0 and the truncation " = 0:15, although we obtained similar results for other choices of ". Table S.5 shows that the size distortions are minor in all cases. To assess the power of the sup LR2;T test, we use again DGP (S.8). The errors are standardized to have mean zero and variance one when t [0:25T ] and 1 + when t > [0:25T ] so that indicates the magnitude of the break. The results are presented in Table S.6. Relative to the results in Table 5, we have, as expected, some power reductions. The extent of the power losses vary across the di¤erent distributions. Nevertheless, the test remains informative. F: Size and Power of the sup LR1;T test in the case of normal errors. Table S.7 presents results related to Table 2 for the statistic sup LR1;T when testing for a single break in variance assuming no break in regression coe¢ cients but with the static model and normal errors. The DGP is yt = et with et i:i:d:N (0; 1 + I(t > [:5T ]) and varies between 0 and 1.5. The trimming parameter is set to " = 0:15. The results show that using ^ with the full correction yields power and exact size similar to tests with a correction that correctly assumes i:i:d: errors, though here imposing normality can lead to tests with somewhat higher power. This con…rms that using the full correction entails little power loss or size distortions. We also investigate the …ndings that the U D max LR1;T test can have power close to that of sup LR1;T under a single break model even though the former considers a wider range of alternatives by using a simple design with normal errors. We also compare them with the CU SQ test described in the main text. We use the same DGP and the results are presented in Table S.8. They show the exact sizes of the sup LR1;T and U D max LR1;T tests to be close to the nominal 5% size. The CU SQ test is slightly undersized. The power functions of the three tests are very close. S-12 G: Size of the sup LR4;T and U D max tests in the case of normal errors. We present results about the size properties of the sup LR4;T and U D max tests with normal i:i:d: errors, with the DGP set to yt = et i:i:d: N (0; 1). We use three values of the trimming parameter " = 0:1; 0:15 and 0:2. For the U D max test, M = N = 2 and for the sup LR4;T test, we consider the following combinations: a) ma = na = 1, b) ma = 1, na = 2, c) ma = 2, na = 1. Two sample sizes are used, T = 100, 200. The results are presented in Table S.9 and they show the size to be slightly conservative, as expected since the critical values are from a limit distribution that provides an upper bound. Nevertheless, the size is close to the nominal 5% level in every case. H: Split-sample method to select the number of breaks. We present results for a split-sample method to estimate the number of breaks in 2 and . It is based on a speci…c to general sequential procedure which is a modi…cation of the sequential procedure discussed in Qu and Perron (2007). Our problem is, however, more complex since we wish to ascertain what types of break occur at any given selected break date, not only to know whether some kind of break did occur. Hence, the need for some re…nements. The starting point is to consider the testing problem for the number of breaks in the union of the coe¢ cients and variance breaks K. This is implemented by using the SeqT (l + 1jl) test proposed by Qu and Perron (2007). The next step is to decide whether a break in coe¢ cients, in variance or in both has occurred at each of the selected break dates. We then perform standard hypothesis testing for the equality of the parameters across adjacent segments. Since the limit distribution of the estimates of the parameters of the model are the same whether using estimates of the break dates or their true value, standard procedures can be applied. Consider …rst the case of testing whether the regression coe¢ cients are equal across the two regimes (T^k 1 ; T^k ), regime k, and (T^k ; T^k+1 ), regime k+1, separated by the k th break (k = 1; :::; K). Denote the true value of the regression coe¢ cients in regimes k and k + 1 by k and k+1 , respectively. The null and alternative hypotheses are H0 : k = k+1 and H1 : k 6= H0 there must be a change in k+1 . 2 Note that since there is a break in and/or 2 , under . Hence, the test to be applied is a standard Chow-type test allowing for a change in variance across regimes (see Goldfeld and Quandt, 1978). Consider now the testing problem H0 : 2k = 2k+1 versus H1 : 2k 6= 2k+1 , where 2k and 2k+1 are the S-13 error variances in regimes k and k + 1, respectively. The Wald test is Wk = (T^k T^k 1 )(T^k+1 (T^k+1 T^k 1 )(^ 4 where ^ 2k and ^ 2k+1 are the MLE of 2 k and T^k ) 4 ^ ) 2 k+1 ^ 2k+1 ^ 2k 2 ; (constructed allowing E(u4t ), to be di¤erent e.g., ^ 4 = (T^k+1 in regimes k and k + 1), and ^ 4 is a consistent estimate of PT^ T^k 1 ) 1 T^k +1 +1 u^4t , constructed under H1 to maximize power. The simulation design is the k 1 same as stated in Section 6. The results for this split-sample approach are presented in Table S.10. Of note, there are cases for which the probability of making the correct selection is quite low; e.g., when both changes in mean and variance are not large and occur at di¤erent dates, especially when they are far apart. The basic reason for that is that the sequential test of Qu and Perron (2007) jointly tests whether a break in both regression coe¢ cients and variance occur. Hence, if only one type of break occurs the power can be quite low unless the magnitudes of the breaks are large. Unfortunately, this situation is expected to be quite common in practice (see, Perron and Yamamoto, 2019). Hence, though this procedure is valid in large samples, it should not be applied mechanically. Care must be exercised to assess whether we are in a situation where its …nite sample properties are rather poor. References Andrews, D. W. K. (1993), “Tests for parameter instability and structural change with unknown change point.”Econometrica, 61, 821-856. Bai, J. and S. Ng (2005), “Tests for skewness, kurtosis, and normality for time series data.” Journal of Business and Economic Statistics, 23, 49-60. Deng, A. and P. Perron (2008), “The limit distribution of the CUSUM of squares test under general mixing conditions.”Econometric Theory, 24, 809-822. Goldfeld, S. M. and R. E. Quandt (1978), “Asymptotic tests for the constancy of regressions in the heteroskedastic case.” Econometric Research Program Research Memorandum No. 229, Princeton University. Kejriwal, M. (2009), “Tests for a mean shift with good size and monotonic power.”Economics Letters, 102, 78-82. Perron, P. and Y. Yamamoto (2019), “The great moderation: updated evidence from testing jointly for multiple changes in variance and mean.”Unpublished manuscript, Department of Economics, Boston University. Qu, Z. and P. Perron (2007), “Estimating and testing multiple structural changes in multivariate regressions.”Econometrica, 75, 459-502. S-14