The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Online Supplement to “Pooled Bewley Estimator of Long-Run Relationships in Dynamic Heterogenous Panels” Alexander Chudik, M. Hashem Pesaran and Ron P. Smith Globalization Institute Working Paper 409 Supplement June 2021 Research Department https://doi.org/10.24149/gwp409supp Working papers from the Federal Reserve Bank of Dallas are preliminary drafts circulated for professional comment. The views in this paper are those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of Dallas or the Federal Reserve System. Any errors or omissions are the responsibility of the authors. Online Supplement to “Pooled Bewley Estimator of Long Run Relationships in Dynamic Heterogenous Panels” Alexander Chudik Federal Reserve Bank of Dallas M. Hashem Pesaran University of Southern California, USA and Trinity College, Cambridge, UK Ron P. Smith Birkbeck, University of London, United Kingdom May 27, 2021 This online supplement describes implementation of the Pooled Mean Group (PMG) estimator and its bias-corrected versions. S-1 PMG estimator and its bias-corrected versions Consider the same illustrative panel ARDL model as in the paper, namely the model given by equations (1)-(2). PMG estimator of the long-run coe¢ cient , as originally proposed by Pesaran, Shin and Smith (1999), is computed by solving the following equations iteratively: ^ P MG n ^2 X i 0 2 xi Hx;i xi ^ i=1 i = 1 ^ = ^0 Hx;i^ i i i and ^ 2i = T where ^i = yi; 1 1 yi ^^ i i ! 0 1 n X i=1 ^2 i ^ 2i x0i Hx;i yi ^ yi; i 1 ^0i Hx;i yi , i = 1; 2; :::; n, Hx;i yi xi ^ P M G , xi = (xi;1 ; xi;2 ; :::; xi;T )0 , yi; (S.1) (S.2) ^ ^ , i = 1; 2; :::; n, i i yi = yi , 1, (S.3) yi = (yi;1 ; yi;2 ; :::; yi;T )0 , xi ( x0i xi ) 1 x0i , xi = xi xi; 1 , and xi; 1 = (xi;0 ; xi;1 ; :::; xi;T To solve (S.1)-(S.3) iteratively, we set ^ P M G;(0) to the pooled Engle-Granger estimator, and given the initial estimate ^ P M G;(0) , we compute ^i;(0) = yi; 1 xi ^ P M G;(0) , ^ i;(0) and ^ 2i;(0) , for i = 1; 2; :::; n using (S.2)-(S.3). Next we compute ^ using (S.1) and given values ^ and yi; 1 = (yi;0 ; yi;1 ; :::; yi;T 1) 0 , Hx;i = IT P M G;(1) S.1 i;(0) 1) 0 . ^ 2i;(0) . Then we iterate - for a given value of ^ P M G;(`) we compute ^i;(`) , ^ i;(`) and ^ 2i;(`) ; and for given values of ^ i;(`) and ^ 2i;(`) we compute ^ P M G;(`+1) . If convergence is not achieved, we increase ^ ` by one and repeat. We de…ne convergence by ^ < 10 4 .10 P M G;(`+1) P M G;(`) Inference is conducted using equation (17) of Pesaran, Shin and Smith (1999). In particular, p T n ^P M G where n P MG = 1X n i=1 i;0 2 rxi ;xi i;0 ! N (0; 0 P M G) , 1 , and rxi ;xi = plimT !1 T 2 0 xi Hx;i xi . Standard error of ^ P M G , denoted as se ^ P M G , is estimated as se b ^P M G = T where ^ P MG = S-1.1 n 1 X ^ i;0 r^xi ;xi n ^2 i=1 i;0 ! 1 n 1=2 ^ P M G, 1 and r^xi ;xi = T 2 0 xi Hx;i xi . (S.4) Simulation-based bias-corrected PMG Similarly to the simulation-based bias-corrected PB estimator, we consider the following biascorrected PMG estimator ~ P MG = ^P M G ^bP M G , (S.5) where ^bP M G an estimate of the bias of PMG estimator obtained by the following stochastic simulation algorithm, which resembles the algorithm in Subsection 2.2.1. 1. Compute ^ P M G . Given PMG estimate ^ P M G , estimate the remaining unknown coe¢ cients of (1)-(2) by least squares, and compute residuals u ^y;it ; u ^x;it . (r) (r) (r) (r) 2. For each r = 1; 2; :::; R, generate new draws for u ^y;it = ay;it u ^y;it , and u ^x;it = ax;it u ^x;it , where (r) (r) ay;it ,ax;it are randomly drawn from Rademacher distribution (Davidson and Flachaire, 2008) namely (r) ah;it = ( 1, with probability 1/2 1, with probability 1/2 , for h = y; x. Given the estimated parameters of (1)-(2) from Step 1, and initial values yi1 ; xi1 (r) (r) generate simulated data yit ; xit for t = 2; 3; :::; T and i = 1; 2; :::; n. Using the generated (r) data compute ^ P M G . 10 If convergence does not occur within the …rst 500 iterations, we stop and report potential divergence. This event did not happen in any of the simulations in this paper. Convergence of the PMG procedure above is typically fast. S.2 h 3. Compute ^bP M G = R 1 PR r=1 ^ (r) ^ P MG i P MG . The above procedure can be iterated by using the bias-corrected estimate ~ P M G in Step 1, although this is not considered in this paper. ~ ~ ^b ^ We conduct inference by using the 1 con…dence interval C1 P M G = P M G k se P MG = 1 1=2 ~ ^ ^ ^ ^ n k P M G , where k is computed by stochastic simulation. In particular, k is the 1 P MG T n oR (r) (r) (r) (r) (r) (r) percent quantile of tP M G , where tP M G = ~ P M G =se b ^ P M G = T 1 n 1=2 ~ P M G = ^ P M G , r=1 (r) = ^P M G ~ (r) ^bP M G is the bias-corrected PMG estimate of in the r-th draw of the simulated P MG (r) data in the algorithm above, and ^ P M G is computed as in (S.4), but using the simulated data. S-1.2 Jackknife and combined bias-corrected PMG estimators We consider similar jackknife bias correction for PMG estimator as for the PB estimator in Section 2.2. In particular, ~ jk P M G = ~ jk P MG ( ) = ^P M G ^ + ^ P M G;b 2 P M G;a ^ P MG ! , where ^ P M G is the full sample PMG estimator, ^ P M G;a and ^ P M G;b are the …rst and the second half sub-sample PMG estimators, and is suitably chosen weighting parameter. Under our setup ^ with I(1) variables, we need to correct P M G for its O T 2 bias, which gives = 1=3. We also consider a combined, simulation-based adaptive jackknife bias correction where = ^N T is data-dependent and computed by stochastic simulation. Speci…cally, we consider ^P M G = ^bP M G , ^bP M G;a;b ^bP M G (S.6) ^ (r) ^ , and ^bP M G;a;b = ^bP M G;a + ^bP M G;b =2, ^bP M G;a = R 1 PR ^ (r) P MG r=1 P M G;a P (r) R ^ 1 ^ ^ ^ P M G;a , bP M G;b = R P M G;b . r=1 P M G;b ~ ~ We conduct inference by using the 1 con…dence interval C1 jk P M G = jk P M G where ^bP M G = R k^jk se b ^P M G 1 PR = ~ jk r=1 P MG k^jk T simulation. In particular, k^jk is the 1 ~ (r) jk (r) b ^P M G = T P M G =se where k^jk = k^jk ( ) is computed by stochastic n oR (r) (r) , where tjk P M G = percent quantile of tjk P M G 1 n 1=2 ^ P M G, 1 n 1=2 ~ (r) ^ (r) jk P M G = P M G , ~ (r) jk P M G r=1 is the jackknife bias-corrected PMG estimate of using the r-th draw of the simulated data generated using the same algorithm as in (r) Subsection S-1.1, and ^ P M G is computed as in (S.4), but using the simulated data. References Davidson, R. and E. Flachaire (2008). The wild bootstrap, tamed at last. Journal of Econometrics, 146, 162-169. S.3 Pesaran, M. H., Y. Shin, and R. P. Smith (1999). Pooled mean group estimation of dynamic heterogeneous panels. Journal of the American Statistical Association 94, 621–634. S.4