Online Appendix (Not intended for Publication): Decomposing the Effects of Monetary Policy Using an External Instruments SVAR

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1 Online Appendix (Not intended for Publication): Decomposing the Effects of Monetary Policy Using an External Instruments SVAR Aeimit Lakdawala Michigan State University December 7 This appendix contains details about identification strategies, factor rotation and the following supplemental tables and figures: Figures: Figure : Path factor constructed controlling for private info Figure : Impulse responses with instruments using 3 min window Figure 3: Impulse responses with instruments using day change Figure 4: Impulse responses for sample: July 979 to November 5 Figure 5: Impulse responses for sample: July 979 to December 8 Figure 6: Impulse responses using Alternate Identification Strategy Figure 7: Impulse responses using the year rate Figure 8: Impulse responses using post 994 futures data Figure 9: Impulse responses using alternative aggregation strategy Figure : Impulse response for specification with commodity prices Figure : Impulse responses with only scheduled FOMC meetings Tables: Table : List of FOMC meeting dates Table : First stage regressions with instruments using 3 min. window and day change Table 3: First stage regressions for sample: July 979 to November 5 Table 4: First stage regressions for sample: July 979 to December 8 Table 5: First stage regressions for sample: July 984 to December Table 6: First stage regressions for specification with excess bond premium Table 7: First stage regressions for specification with commodity prices Table 8: First stage regressions for specification with unemployment Table 9: First stage regressions without non-statement FOMC meetings Table : First stage regressions using post 994 futures data Table First stage regression using alternative aggregation strategy Table : First stage regressions with only one policy tool Table 3: First stage regressions with cleansed path factor Table 4: First stage regressions with only scheduled FOMC meetings Table 5: Private info regressions with unemployment forecasts aeimit@msu.edu

2 Identification with External Instruments Setup for Baseline Identification Strategy The reduced-form covariance matrix in equation. is given by Σ = Σ Σ Σ Σ The instrumental variables estimation gives B B = E[Z tu p t ] E[Z t u q t ] With these two, we can calculate the following matrices B B = [ (B B (B B ) ) + ( Σ B B Σ B B = ( Σ B B Σ B B = Σ + B B B B = Σ B B Z = B B Σ ) Z ( Σ B B Σ ( B B Σ ) ( B B ( B B ) ) ] ) (B B ) ) ( (Σ B B ) ) + B B Σ + Σ The approach in Mertens and Ravn (3) relies on estimating the matrix S which is related to the above estimable matrices in the following manner S S = ( I B B B B ) B B ( I B B B B ) In the case of shock and instrument, we can identify S up to a sign convention from the above equation. With more than shock, S is not identified without further restrictions. As explained in section. in the main draft, the baseline identification strategy involves imposing a triangular structure on S, such that a Cholesky factorization of the above equation gives S. With S in hand, we can get

3 [ the relevant column of the impact matrix B = B B ] from the following two equations B S = ( I B B B B ) B S = B B ( I B B B B ) Setup for Alternative Identification Strategy The alternative identification strategy imposes zero restrictions on the relationship between the structural policy shocks and the instruments. We derive the estimating equations using the approach of Lunsford (5). Recall the relevance and validity conditions of the instrument, E[Z t ε p t ] = φ and E[Z tε q t ] = and that the covariance matrix of the residuals is given by E(u t u t) = BB. Now consider E(Z t u t) E(Z t u t) = E(Z t [Bε t ] ) = E Z t (B B ) εp t ε q t = E Z tε p t (B B ) Z t ε q t = E φ (B B ) = φb Finally consider E(Z t u t) [E(u t u t)] E(u t Z t) E(Z t u t) [ E(u t u t) ] E(ut Z t) = φb (BB ) B φ = φb (B ) B B φ = φφ 3

4 If we have an estimate of φ, we can back out the relevant columns of the impact matrix B, which is B B = E(u t Z t)(φ ) Again, if there is only one shock then φ is a scalar and we can estimate it up to a sign convention. But if there are k > shocks (and instruments) then φ has k unique elements, while E(Z t u t) [E(u t u t)] E(u t Z t) is a symmetric matrix with only k(k+) unique elements. The second strategy involves putting zero restrictions on φ. We assume that φ is triangular and thus a Cholesky factorization of φφ gives φ. The interpretation of a zero restriction on φ is straightforward from the relevance condition of the instruments. A zero restriction on the row i column j element in φ implies that the j th structural policy shock in ε p t is uncorrelated with the ith instrument in Z t. For the application in this paper we will use two instruments, Z t = [Z t Z t ]. We can now re-write the relevance condition as E Z t ε ff t Zt ε ff t E[Z t ε p t ] = φ Z t ε fwd t Z t ε fwd t = φ φ φ φ Thus a triangular identifying assumption that imposes φ = implies that E[Z t ε ff t ] =. This assumption is justified by finding an instrument Z that is uncorrelated with the fed funds rate shock but correlated with the forward guidance shock. Specifically, we will use high frequency futures market data and apply the methodology of GSS. This involves performing a rotation of the principal components to construct a factor that satisfies the above requirement. This factor (labeled the path factor) captures shocks in longer term rates but is uncorrelated to fed funds rate shocks. The construction of the instruments is discussed in more detail next. Target and Path Factor Construction The goal is to construct two new factors Z and Z from the first two principal components F and F by finding an orthogonal matrix U [Z Z ] = [F F ]U (.) 4

5 U matrix has 4 unique elements and requires 4 restrictions for identification U = α β α β The first two come from a simple normalization that imposes the columns of U to have unit length, i.e. α + α = and β + β =. Next, we maintain the orthogonality of the two factors E(Z Z ) =, which gives α β + α β =. Finally we impose the condition required for identification strategy II, that the second factor Z is not related to the current month s futures price change. This condition is given by γ α γ α =. To see this last condition, let γ and γ be the factor loadings on F and F for change in current month s futures contract (given by X()) X() = γ F + γ F (.) From equation. we can write F = F = [β Z α Z ] α β α β [ β Z + α Z ] α β α β Now plug these into equation. and impose the condition that the loading of Z on X() is zero to get the restriction γ α γ α =. 5

6 .3 Path Factors Baseline Pvt Res Difference: (Baseline- Pvt Res) Figure : The top panel shows the path factors from the baseline specification (solid blue line) and the residual after controlling for Fed private info (dashed red line). The bottom panel shows the difference between the two series. 6

7 Shock Forward Guidance Shock Year Rate Year Rate Figure : Tight 3 minute window. The impulse responses to a unit monetary policy shock identified using the external instruments identification strategy I outlined in the text, with 9% confidence intervals. The monetary surprises are constructed using a 3 minute window around FOMC announcements. The first column shows the response to a conventional monetary policy shock (i.e shock to the federal funds rate equation), while the second column shows the response to a forward guidance shock (i.e. shock to the year rate equation) 7

8 Shock Forward Guidance Shock Year Rate Year Rate Figure 3: Two day window. The impulse responses to a unit monetary policy shock identified using the external instruments identification strategy I outlined in the text, with 9% confidence intervals. The monetary surprises are constructed using a two day window around FOMC announcements. The first column shows the response to a conventional monetary policy shock (i.e shock to the federal funds rate equation), while the second column shows the response to a forward guidance shock (i.e. shock to the year rate equation) 8

9 Shock Forward Guidance Shock Year Rate Year Rate Figure 4: Sample: July 979 to November 5 The impulse responses to a unit monetary policy shock identified using the external instruments identification strategy I outlined in the text, with 9% confidence intervals. The first column shows the response to a conventional monetary policy shock (i.e shock to the federal funds rate equation), while the second column shows the response to a forward guidance shock (i.e. shock to the year rate equation) 9

10 Shock Forward Guidance Shock Year Rate Year Rate Figure 5: Sample: July 979 to December 8. The impulse responses to a unit monetary policy shock identified using the external instruments identification strategy I outlined in the text, with 9% confidence intervals. The first column shows the response to a conventional monetary policy shock (i.e shock to the federal funds rate equation), while the second column shows the response to a forward guidance shock (i.e. shock to the year rate equation)

11 Shock: ID II Forward Guidance Shock: ID II Year Rate 3 Year Rate Figure 6: The impulse responses to a unit monetary policy shock identified using the external instruments alternate identification strategy, with 9% confidence intervals. The first column shows the response to a conventional monetary policy shock (i.e shock to the federal funds rate equation), while the second column shows the response to a forward guidance shock (i.e. shock to the year rate equation)

12 Shock Forward Guidance Shock Year Rate Year Rate Figure 7: Impulse responses to a unit monetary policy shock with the year rate as the forward guidance policy tool. The solid blue lines show responses using identification strategy I, with the dashed blue lines showing the 9% confidence intervals. The dashed black lines show the responses using identification strategy II. The first column shows the response to a conventional monetary policy shock (i.e shock to the federal funds rate equation), while the second column shows the response to a forward guidance shock (i.e. shock to the year rate equation).

13 Shock Forward Guidance Shock Year Rate Year Rate Figure 8: Futures Data Sample: February 994 to December. The impulse responses to a unit monetary policy shock identified using the external instruments identification strategy I outlined in the text, with 9% confidence intervals. The first column shows the response to a conventional monetary policy shock (i.e shock to the federal funds rate equation), while the second column shows the response to a forward guidance shock (i.e. shock to the year rate equation) 3

14 Shock Forward Guidance Shock Year Rate.5 Year Rate Figure 9: The impulse responses to a unit monetary policy shock identified using the external instruments identification strategy I outlined in the text, with 9% confidence intervals. The first column shows the response to a conventional monetary policy shock (i.e shock to the federal funds rate equation), while the second column shows the response to a forward guidance shock (i.e. shock to the year rate equation). The daily futures shock series is aggregated to a monthly series by weighing it based on which day of the month the FOMC meeting occurred. If the FOMC meeting occurs in the final 5 days of the month, then the shock is assigned to the next month. 4

15 Shock Forward Guidance Shock Year Rate Year Rate Commodity Prices Commodity Prices Figure : The solid blue lines show the impulse responses to a unit monetary policy shock for the VAR with commodity prices added to the baseline specification, using identification strategy I. The dashed black lines show the responses from the baseline specification. The dashed blue lines show the 9% confidence intervals. The first column shows the response to a conventional monetary policy shock (i.e shock to the federal funds rate equation), while the second column shows the response to a forward guidance shock (i.e. shock to the year rate equation) 5

16 Shock Forward Guidance Shock Year Rate Year Rate Figure : The impulse responses to a unit monetary policy shock with 9% confidence intervals. The solid blue lines show the responses where only the scheduled FOMC meetings are used, while the dashed black line shows the baseline specification with both scheduled and unscheduled meetings. The first column shows the response to a conventional monetary policy shock (i.e shock to the federal funds rate equation), while the second column shows the response to a forward guidance shock (i.e. shock to the year rate equation). 6

17 FOMC Meeting Sched? Stat? FOMC Meeting Sched? Stat? FOMC Meeting Sched? FOMC Meeting Sched? Stat? /8/99 7/3/996 /6/ 3/8/9 //99 8//996 // 4/9/9 /7/99 9/4/996 /9/3 6/4/9 3/8/99 /3/996 3/8/3 8//9 3/7/99 /7/996 5/6/3 9/3/9 4/3/99 /5/997 6/5/3 /4/9 5/5/99 3/5/997 8//3 /6/9 7/5/99 5//997 9/6/3 /7/ 8/6/99 7//997 /8/3 3/6/ 8//99 8/9/997 /9/3 4/8/ 9/3/99 9/3/997 /8/4 6/3/ //99 //997 3/6/4 8// /3/99 /6/997 5/4/4 9// /6/99 /4/998 6/3/4 /3/ /6/99 3/3/998 8//4 /4/ /8/99 5/9/998 9//4 /6/ //99 7//998 //4 3/5/ /6/99 8/8/998 /4/4 4/7/ 4//99 9/9/998 //5 6// 4/9/99 /5/998 3//5 8/9/ 5//99 /7/998 5/3/5 9// 7//99 //998 6/3/5 // 8/9/99 /3/999 8/9/5 /3/ 9/4/99 3/3/999 9//5 /5/ /7/99 5/8/999 //5 3/3/ /8/99 6/3/999 /3/5 4/5/ /3/99 8/4/999 /3/6 6// /4/993 /5/999 3/8/6 8// 3/4/993 /6/999 5//6 9/3/ 5/9/993 //999 6/9/6 /4/ 7/8/993 // 8/8/6 // 8/8/993 3// 9//6 /3/3 9//993 5/6/ /5/6 3//3 /7/993 6/8/ //6 5//3 //993 8// /3/7 6/9/3 /4/994 /3/ 3//7 7/3/3 3//994 /5/ 5/9/7 9/8/3 4/8/994 /9/ 6/8/7 /3/3 5/7/994 /3/ 8/7/7 /8/3 7/6/994 /3/ 8/7/7 /9/4 8/6/994 3// 9/8/7 3/9/4 9/7/994 4/8/ /3/7 4/3/4 /5/994 5/5/ //7 6/8/4 //994 6/7/ //8 7/3/4 //995 8// /3/8 9/7/4 3/8/995 9/7/ 3/8/8 /9/4 5/3/995 // 4/3/8 /7/4 7/6/995 /6/ 6/5/8 /8/5 8//995 // 8/5/8 3/8/5 9/6/995 /3/ 9/6/8 4/9/5 /5/995 3/9/ /8/8 6/7/5 /9/995 5/7/ /9/8 7/9/5 /3/996 6/6/ /5/8 9/7/5 3/6/996 8/3/ /6/8 /8/5 5//996 9/4/ /8/9 /6/5 Table : FOMC meeting dates with a in the Sched? ( Stat? ) column indicating a scheduled meeting (released statement) and a indicating an unscheduled meeting (no released statement). 7

18 (a) (b) (c) (d) VARIABLES FFR residual year residual FFR residual year residual Target Factor.66***.763***.85***.3*** (.34) (.4) (.95) (.35) Path Factor ** (.3) (.6) (.) (.33) Constant (.) (.) (.) (.3) Observations R-squared Adjusted R-squared Robust F-statistic Table : First stage regression of residuals from the reduced form VAR on the target and path factors. Columns (a) and (b) show the results using a day window, while columns (c) and (d) show the results using a 3 minute window. Robust standard errors in parentheses, *** p <., **p <.5, *p <.. () () VARIABLES FFR residual year residual Target Factor.89***.893*** (.8) (.67) Path Factor (.68) (.9) Constant -.6. (.8) (.) Observations R-squared.5.99 Adjusted R-squared.9.93 Robust F-statistic Table 3: Sample: July 979 to November 5 : First stage regression of residuals from the reduced form VAR on the target and path factors, from the baseline specification. Robust standard errors in parentheses, *** p <., **p <.5, *p <.. 8

19 () () VARIABLES FFR residual year residual Target Factor.78***.95*** (.4) (.75) Path Factor (.88) (.5) Constant (.) (.4) Observations 6 6 R-squared.. Adjusted R-squared.3. Robust F-statistic Table 4: Sample: July 979 to December 8. : First stage regression of residuals from the reduced form VAR on the target and path factors, from the baseline specification. Robust standard errors in parentheses, *** p <., **p <.5, *p <.. VARIABLES FFR residual year residual Target Factor.89***.74*** (.) (.7) Path Factor (.3) (.9) Constant (.7) (.) Observations 5 5 R-squared.9.93 Adjusted R-squared.3.86 Robust F-statistic Table 5: Sample: July 984 to December. : First stage regression of residuals from the reduced form VAR on the target and path factors, from the baseline specification. Robust standard errors in parentheses, *** p <., **p <.5, *p <.. 9

20 () () VARIABLES FFR residual year residual Target Factor.73***.79*** (.35) (.58) Path Factor -..6 (.77) (.8) Constant (.) (.) Observations 5 5 R-squared.99.8 Adjusted R-squared Robust F-statistic Table 6: First stage regression of residuals from the reduced form VAR on the target and path factors, from the model with the excess bond premium added to the baseline specification. Robust standard errors in parentheses, *** p <., **p <.5, *p <.. () () VARIABLES FFR residual year residual Target Factor.67***.84*** (.47) (.8) Path Factor (.63) (.33) Constant -..4 (.) (.) Observations 5 5 R-squared Adjusted R-squared Robust F-statistic.67.8 Table 7: First stage regression of residuals from the reduced form VAR on the target and path factors, from the model with commodity prices added to the baseline specification. Robust standard errors in parentheses, *** p <., **p <.5, *p <..

21 () () VARIABLES FFR residual year residual Target Factor.798***.88*** (.46) (.69) Path Factor (.85) (.8) Constant (.) (.3) Observations 5 5 R-squared.6.8 Adjusted R-squared Robust F-statistic Table 8: First stage regression of residuals from the reduced form VAR on the target and path factors, from the model with the unemployment rate added to the baseline specification. Robust standard errors in parentheses, *** p <., **p <.5, *p <.. () () VARIABLES FFR residual year residual Target Factor.75***.857*** (.67) (.99) Path Factor (.87) (.86) Constant -.5. (.) (.3) Observations 5 5 R-squared.69.6 Adjusted R-squared Robust F-statistic Table 9: First stage regression of residuals from the reduced form VAR on the target and path factors where the FOMC meeting dates without an accompanying statement are excluded. Robust standard errors in parentheses, *** p <., **p <.5, *p <..

22 () () VARIABLES FFR residual year residual Target Factor.783***.836*** (.57) (.93) Path Factor (.9) (.76) Constant.4.4 (.) (.4) Observations 6 6 R-squared.5.73 Adjusted R-squared Robust F-statistic Table : Futures Data Sample: February 994 to December. First stage regression of residuals from the reduced form VAR on the target and path factors. Robust standard errors in parentheses, *** p <., **p <.5, *p <..

23 () () VARIABLES FFR residual year residual Target Factor.97***.7*** (.3) (.37) Path Factor.5.4* (.86) (.47) Constant -.5. (.) (.3) Observations 5 5 R-squared Adjusted R-squared.7.89 Robust F-statistic Table : First stage regression of residuals from the reduced form VAR on the target and path factors. The daily futures shock series is aggregated to a monthly series by weighing it based on which day of the month the FOMC meeting occurred. If the FOMC meeting occurs in the final 5 days of the month, then the shock is assigned to the next month. Robust standard errors in parentheses, *** p <., **p <.5, *p <.. 3

24 (a) (b) Variables FFR residual year residual MP.9*** (.3) FF4.38*** (.37) Constant.4.8 (.) (.3) Observations 5 5 R-squared.88. Adjusted R-squared Robust F-statistic Table : First stage regression of residuals from the reduced form VAR with only one policy tool. Panel (a) is the model with only fed funds rate as policy tool and MP as the instrument. Panel (b) is the model with only the year rate as the policy tool with FF4 as the instrument. Robust standard errors in parentheses, *** p <., **p <.5, *p <.. () () VARIABLES FFR residual year residual Target Factor.79***.99*** (.3) (.69) Path Factor (Pvt Res) (.83) (.5) Constant (.) (.3) Observations 4 4 R-squared.8.7 Adjusted R-squared..99 Robust F-statistic Table 3: First stage regression of residuals from the reduced form VAR on the target factor and cleansed path factor (Pvt Res). Robust standard errors in parentheses, *** p <., **p <.5, *p <.. 4

25 (a) (b) Variables FFR residual year residual Target Factor.896***.8** (.87) (.34) Path Factor (.9) (.63) Constant -.6. (.) (.3) Observations 5 5 R-squared.48.7 Adjusted R-squared.4.9 Robust F-statistic Table 4: First stage regression of residuals from the reduced form VAR on the target and path factors with only the scheduled FOMC meeting dates. Robust standard errors in parentheses, *** p <., **p <.5, *p <.. 5

26 () () Target Factor Path Factor GDPt.9 GDPtlag. GDPt.3 GDPtlag -.5 (.3) (.3) (.5) (.) GDPt.5 GDPtlag -. GDPt.5 GDPtlag -.7 (.6) (.) (.4) (.) GDPt3 -.8 GDPt3lag. GDPt3.9 GDPt3lag.8 (.5) (.6) (.5) (.5) GDPt4. GDPt4lag -.4 GDPt4.7 GDPt4lag -.5 (.) (.) (.8) (.) t -. tlag -. t.8 tlag -. (.9) (.7) (.8) (.7) t -. tlag.3 t -.58*** tlag -.5 (.9) (.7) (.) (.3) t3.33 t3lag -.79 t3.3 t3lag.89 (.39) (.37) (.39) (.37) t4 -.3 t4lag.38 t4.77 t4lag -.97 (.4) (.4) (.5) (.5) Ut -.46 Utlag -. Ut -.34 Utlag -. (.8) (.78) (.3) (.) Ut -. Utlag.39 Ut -.9 Utlag.43 (.) (.3) (.35) (.76) Ut3.4 Ut3lag -.73 Ut3.3 Ut3lag -.33 (.3) (.) (.46) (.45) Ut4 -.3 Ut4lag -. Ut4. Ut4lag -.67 (.6) (.7) (.4) (.8) Constant.. (.) (.) Observations R-squared Adjusted R-squared Table 5: Regression results of target and path factor on measure of Federal Reserve private information, which are forecasts differences between Greenbook and Blue Chip forecasts for GDP, and unemployment. Robust standard errors are presented in parentheses. 6