Lena Boneva and Oliver Linton. January 2017

Transcription

1 Appendx to Staff Workng Paper No. 640 A dscrete choce model for large heterogeneous panels wth nteractve fxed effects wth an applcaton to the determnants of corporate bond ssuance Lena Boneva and Olver Lnton January 207 Staff Workng Papers descrbe research n progress by the authors and are publshed to elct comments and to further debate. Any vews expressed are solely those of the authors and so cannot be taken to represent those of the Bank of England or to state Bank of England polcy. Ths paper should therefore not be reported as representng the vews of the Bank of England or members of the Monetary Polcy Commttee, Fnancal Polcy Commttee or Prudental Regulaton Authorty Board.

2 Appendx to Staff Workng Paper No. 640 A dscrete choce model for large heterogeneous panels wth nteractve fxed effects wth an applcaton to the determnants of corporate bond ssuance Lena Boneva and Olver Lnton 2 Bank of England. Emal: lena.boneva@bankofengland.co.uk 2 Unversty of Cambrdge. Emal: obl20@cam.ac.uk Informaton on the Bank s workng paper seres can be found at Publcatons and Desgn Team, Bank of England, Threadneedle Street, London, EC2R 8AH Telephone emal publcatons@bankofengland.co.uk Bank of England 207 ISSN on-lne

3 Supplementary materal for: A Dscrete Choce Model For Large Heterogeneous Panels wth Interactve Fxed Effects wth an Applcaton to the Determnants of Corporate Bond Issuance Lena Boneva Bank of England and CEPR Olver Lnton Unversty of Cambrdge January 3, 207 Proofs Proof of Lemma 4. We have Defne the event max t T ĥt h t = max t T u t = max t T N u t. B = { u t τ N,T for all N, t T }, = where τ N,T < s to be determned below. We have for any x > 0 { } N N Pr max t T ĥt h t > x Pr max t T ĥt h t > x B + PrB c We would lke to thank Peter Robnson, our edtor Therry Magnac and 3 anonymous referees for ther helpful comments and Hashem Pesaran, Martn Wedner, Lu Lu, Menno Mddeldorp and Magda Rutkowska for many useful dscussons. The vews expressed n ths paper are those of the authors and do not necessarly reflect the vews of the Bank of England, the Monetary Polcy Commttee, the Fnancal Polcy Commttee or the Prudental Regulaton Authorty. Bank of England, Threadneedle St, London, EC2R 8AH. Emal: Lena.Boneva@bankofengland.co.uk. Fnancal support from Cusanuswerk and the ESRC grant no.: K67783X s gratefully acknowledged. Faculty of Economcs, Austn Robnson Buldng, Sdgwck Avenue, Cambrdge, CB3 9DD. Emal: obl20@cam.ac.uk. Fnancal support from the Keynes Fund s gratefully acknowledged. Appendx to Staff Workng Paper No. 640 January 207

4 Then by Bonferron and Bernsten s nequalty Van der Vaart, 998, p285 { } N Pr max t T ĥt h t > x B 2T exp 2 x 2 σ 2 u + xτ N,T / N where we use that u t are..d wth mean zero and fnte varance σ 2 u. Then, takng x = log T we have { } N Pr max t T ĥt h t > log T B 2T exp 2 provded τ N,T log T/ N 0. Furthermore, we note that wth τ N,T have PrB = F u τ N,T NT NT c, NT πα log 2 T σ 2 u + τ N,T log T/ N, = o = NT π for some π > 0, we where F u denotes the c.d.f. of the random varable u t. The moment condtons mply that F u x cx α for x large for α 4. If πα >, then PrB. Therefore, provded π > /α and N π /2 T π log T 0 the result s establshed. Proof of Theorem 4. Because the nfeasble estmator θ s consstent, t suffces to show that estmatng the unobserved factors does not affect the crteron functon. We have Pr sup Q T θ Q T θ ɛ Pr 0, sup sup h h H δ T θ Θ T θ Θ q t θ, h t qtθ, h t ɛ + Pr ĥ h H > δ T 2 3 Proof of Theorem 4.2 To show that θ s asymptotcally normal, t suffces to show that estmatng the unobserved factors does not affect the lmtng dstrbuton, that s, T θ θ = o P. 4 By the Mean Value Theorem, we have 0 = T q t θ, ĥt 2 Appendx to Staff Workng Paper No. 640 January 207

5 = T = T = T q t θ, ĥt q t θ, h t + T + T 2 q t θ, h t h ĥt h t + T 2 q t θ, h t θ θ 2 q t θ, h t h ĥt h t + T 2 q t θ, h t θ θ, 2 q t θ, h t θ θ where h t and h t are ntermedate values. Then, provded T lm nf N,T T 2 q t θ, h t > 0 2 qt θ, h t ĥt h h t = o P T /2. the result 4 follows. These propertes follow from the unform convergence of ĥt h t, the Cauchy- Schwarz nequalty and condtons D. For example, wth probablty tendng to one T 2 qt θ, h t ĥt h h t Proof of Theorem T 2 q t θ, h 2 t h sup θ θ 0 δ T T = o P T sup h h H <δ T = O P log 2 T N ĥ h 2 H 2 q tθ, h t 2 ĥ h 2 It suffces to show that the feasble objectve functons are unformly close to the nfeasble ones, see CJL 206. Thus Pr max Pr Pr 0. sup N θ Θ max N max N Q T θ Q T θ ɛ sup sup q h h H δ T θ Θ T tθ, h t qtθ, h t ɛ + Pr ĥ h H > δ T sup sup q h h H δ T θ Θ T t θ, h t qtθ, h t ɛ + Pr ĥ h H > δ T H 5 3 Appendx to Staff Workng Paper No. 640 January 207

6 Proof of Theorem 4.4 We show that θ θ = o P N /2. 6 The estmators θ and θ, =,..., N satsfy the frst order condtons T q t θ, h t = 0 = T qt θ, ĥt. We frst work wth a lnear approxmaton to θ. Defne L T θ = T q t θ, ĥt + M θ θ 7 from whch we obtan for θ such that L T θ = 0, θ θ = M T qt θ, ĥt. We frst establsh the result for ths lnear approxmaton. By the Mean-Value Theorem, we have for r =,..., p T q t θ, ĥt r = T + 2T q tθ 0, h t r + T 2 q tθ 0, h t r h ĥt h t + T ĥt h t 3 qt r h h θ, h t ĥt h t + θ θ 0 + θ θ 0 = 6 J rk;, k= 2T 3 q tθ, h t r θ θ 0 2T 2 q tθ 0, h t r θ θ 0 3 q tθ, h t r h ĥt h t where h t and θ are ntermedate values. It follows that N = θ θ = 6 k= N = M J k; 6 R rk, 8 k= where J k; denotes the vector wth r th element J rk;. We consder n sequence the vector random varables R R 6. 4 Appendx to Staff Workng Paper No. 640 January 207

7 We have ER = 0 and q tθ 0, h t / s..d. across and t condtonal on X, d, f, so that R = N = M T q tθ 0, h t = N T = M qtθ 0, h t = O P N /2 T /2. Consder R 2 = T N = M 2 qt h θ 0, h t ĥt h t. 9 We have Φ t0 / h t = φ t0 κ and φ t0 / h t = θ 0z t φ t0 κ, whence 2 qt h θ 0, h t = φ 2 t0z t κ T Φ t0 Φ t0 Y t Φ t0 T Φ t0 Φ t0 2Φ t0 φ 2 t0z 2 t κ Y t Φ t0 θ T Φ t0 Φ t0 0z t φ 2 t0 κ z t. We decompose 9 nto three terms: the second and thrd terms are just lnear combnatons of the random varables Y t Φ t0, whch are..d. mean zero condtonal on the factors; the frst term s dfferent and we treat ths more carefully. Ths term can be wrtten as W NT = N 2 T r d t, f t, u t, u t u lt = l= for some functon r. Wrte for each l =,..., N r d t, f t, u t, u t = r d t, f t, u l t for some ntermedate value u l t, where u l t l, E r ;3 d t, f t, u l t, u t + r ;3 d t, f t, u l t, u t u lt N r ;33d t, f t, u l t = j l u jt/n so that u t u l t, u t u2 lt N 2 = u lt /N. We have for, u t u lt = 0 and E r;33 d t, f t, u l t, u t u 2 lt <. From ths we obtan that EW NT = ON. By smlar arguments we obtan E W 2 NT = N 4 T 2 = l= = l = t = E [ r d t, f t, u t, u t r d t, f t, u t, u ] t u lt u l t = ON T, because whenever ether t t or all four ndces n {,, l, l } are dstnct, then the expectaton s zero for the leave out case, or small otherwse. Therefore, W NT = O P N + O P N /2 T / Appendx to Staff Workng Paper No. 640 January 207

8 In concluson, R 2 = O P N + O P N /2 T /2. We have R 3 = M 2 qtθ 0, h t θ θ N T 0 = = M 2 qtθ 0, h t θ θ N T 0 = = θ θ 0 + M 2 qtθ 0, h t I N N T p θ θ 0 = = = o P N /2 + O P T = o P N /2, by Cauchy-Schwarz and the assumpton that T 2 /N 0. The remanng terms, R 4 R 6 are also treated by crude boundng. For example, wth probablty tendng to one M ĥt h t 3 q t N 2T = r h h θ 0, h t ĥt h t ĥ h 2 H M sup 3 qt 2NT = h h <δ T r h h θ 0, h t H log 2 T = O P = o P N /2. N max N N = M θ θ 0 2T 3 qtθ, h t θ θ r 0 2 max θ θ 0 N 2NT = log 2 N = O P = o P N /2. T M sup h h H <δ T 2 sup θ θ 0 δ T 3 q tθ, h t r Fnally, we show that the lnear approxmaton s very close to the actual score functon of the feasble 2 estmator N max N = N max N sup θ θ 0 δ N N max N sup θ θ 0 δ T T T q tθ, ĥt θ θ 0 2 max N L T θ θ θ 2 q tθ, ĥt θ θ sup θ θ 0 δ T sup h h H <δ T T 2 q tθ, h t 6 Appendx to Staff Workng Paper No. 640 January 207

9 = o P. The argument follows as n CJL 206, p68. 2 Addtonal tables for the smulaton study Tables -8 report the results for experments 2-5 that are dscussed n Secton 5 of the man paper. 3 Tables for the robustness checks Tables 9 to 2 report the results of our robustness checks that are dscussed n Secton 6.3 of the man paper. 7 Appendx to Staff Workng Paper No. 640 January 207

10 Table : Small sample propertes of the mean group estmator β: Experment 2 T/N Bas 000 RMSE 000 Power Sze Coverage probablty Infeasble estmator CCEMG estmator Nave estmator Notes: The data generatng process s defned n 7-2 n the man paper except that β = 0.5, for all. The nomnal sze s 5% and power s computed under the alternatve β = The number of replcatons s set to 2000.

11 Table 2: Small sample propertes of the mean group estmator β: Experment 3 T/N Bas 000 RMSE 000 Power Sze Coverage probablty Infeasble estmator CCEMG estmator Nave estmator Notes: The data generatng process s defned n 7-2 n the man paper except that kj2 N ID0, 0.. The nomnal sze s 5% and power s computed under the alternatve β = The number of replcatons s set to 2000.

12 Table 3: Small sample propertes of the mean group estmator β: Experment 4 T/N Bas 000 RMSE 000 Power Sze Coverage probablty Infeasble estmator CCEMG estmator Nave estmator Notes: The data generatng process s defned n 7-2 and 22 n the man paper. The nomnal sze s 5% and power s computed under the alternatve β = The number of replcatons s set to 2000.

13 Table 4: Small sample propertes of the mean group estmator β: Experment 5 T/N Bas 000 RMSE 000 Power Sze Coverage probablty Infeasble estmator CCEMG estmator Nave estmator e Notes: The data generatng process s defned n 7-2 n the man paper except that kj2 = 0, j and κ2 = 0. The nomnal sze s 5% and power s computed under the alternatve β = The number of replcatons s set to 2000.

14 Table 5: Small sample propertes of the margnal effect ME: Experment 4 T/N Bas 000 RMSE Infeasble estmator CCEMG estmator Nave estmator Lnear probablty estmator Notes: The mean group estmator of the average margnal effect s s reported. The data generatng process s defned n 7-2 and 22 n the man paper. The number of replcatons s set to Appendx to Staff Workng Paper No. 640 January 207

15 Table 6: Small sample propertes of the margnal effect ME: Experment 5 T/N Bas 000 RMSE Infeasble estmator CCEMG estmator Nave estmator Lnear probablty estmator Notes: The mean group estmator of the average margnal effect s s reported. The data generatng process s defned n 7-2 n the man paper except that k j2 = 0, j and κ 2 = 0. The number of replcatons s set to Appendx to Staff Workng Paper No. 640 January 207

16 Table 7: Small sample propertes of the margnal effect ME: Experment 2 T/N Bas 000 RMSE Infeasble estmator CCEMG estmator Nave estmator Lnear probablty estmator Notes: The mean group estmator of the average margnal effect s reported. The data generatng process s defned n 7-2 n the man paper except that β = 0.5, for all. The number of replcatons s set to Appendx to Staff Workng Paper No. 640 January 207

17 Table 8: Small sample propertes of the margnal effect ME: Experment 3 T/N Bas 000 RMSE Infeasble estmator CCEMG estmator Nave estmator Lnear probablty estmator Notes: The mean group estmator of the average margnal effect s reported. The data generatng process s defned n 7-2 n the man paper except that k j2 NID0, 0.. The number of replcatons s set to Appendx to Staff Workng Paper No. 640 January 207

18 Table 9: The effect of yelds on bond ssuance for US frms: controllng for lqudty Pre-crss Post-crss All Fnancal Other All Fnancal Other Coeffcent estmates Yeld Sze Lqudty Margnal effects Yeld Sze Lqudty Observatons Notes: The dependent varable s f a frm ssues a bond n a partcular month and zero otherwse. Yeld s the frm-specfc corporate bond yeld, sze s measured by assets/000 and lqudty s the share of current debt among total debt. All specfcatons nclude a measure of credt supply leverage n the broker-dealer market, the federal funds rate pre-crss only and the change n Federal Reserve Holdngs of Treasury Notes post-crss only as common factors. Columns and 4 use all frms, 2 and 5 use fnancal sector frms and 3 and 6 use all other frms excludng mnng and agrculture. t-statstcs are shown n parenthess. 6 Appendx to Staff Workng Paper No. 640 January 207

19 Table 0: The effect of yelds on bond ssuance for US frms: corporate spreads nstead of yelds Pre-crss Post-crss All Fnancal Other All Fnancal Other Coeffcent estmates Spread Sze Margnal effects Spread Sze Observatons Notes: The dependent varable s f a frm ssues a bond n a partcular month and zero otherwse. Spread s the frm-specfc corporate bond yeld mnus the federal funds rate and sze s measured by assets/000. All specfcatons nclude a measure of credt supply leverage n the broker-dealer market and the change n Federal Reserve Holdngs of Treasury Notes post-crss only as common factors. Columns and 4 use all frms, 2 and 5 use fnancal sector frms and 3 and 6 use all other frms excludng mnng and agrculture. t-statstcs are shown n parenthess. 7 Appendx to Staff Workng Paper No. 640 January 207

20 Table : The effect of yelds on bond ssuance for US frms: alternatve sample choces a Frms wth at least 20 tme seres observatons Pre-crss Post-crss All Fnancal Other All Fnancal Other Coeffcent estmates Yeld Sze Margnal effects Yeld Sze Observatons b Frms wth at least 40 tme seres observatons Pre-crss Post-crss All Fnancal Other All Fnancal Other Coeffcent estmates Yeld Sze Margnal effects Yeld Sze Observatons Notes: The dependent varable s f a frm ssues a bond n a partcular month and zero otherwse. Yeld s the frm-specfc corporate bond yeld and sze s measured by assets/000. All specfcatons nclude a measure of credt supply leverage n the broker-dealer market, the federal funds rate pre-crss only and the change n Federal Reserve Holdngs of Treasury Notes post-crss only as common factors. Columns and 4 use all frms, 2 and 5 use fnancal sector frms and 3 and 6 use all other frms excludng mnng and agrculture. t-statstcs are shown n parenthess. 8 Appendx to Staff Workng Paper No. 640 January 207

21 Table 2: The effect of yelds on bond ssuance for US frms: frms wth at least 2 ssuances Pre-crss Post-crss All Fnancal Other All Fnancal Other Coeffcent estmates Yeld Sze Margnal effects Yeld Sze Observatons Notes: The dependent varable s f a frm ssues a bond n a partcular month and zero otherwse. Yeld s the frm-specfc corporate bond yeld and sze s measured by assets/000. All specfcatons nclude a measure of credt supply leverage n the broker-dealer market, the federal funds rate pre-crss only and the change n Federal Reserve Holdngs of Treasury Notes post-crss only as common factors. Columns and 4 use all frms, 2 and 5 use fnancal sector frms and 3 and 6 use all other frms excludng mnng and agrculture. t-statstcs are shown n parenthess. 9 Appendx to Staff Workng Paper No. 640 January 207