## 15. Impulse-Response Functions for Alternate Ordering, United Kingdom

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---

### Introduction
- Focus: bilateral spillovers between the United States and six other industrial countries with inflation-indexed bond markets: Australia, Canada, France (proxy for the Euro Area), Japan, Sweden, and the United Kingdom.
- Motivation: yields on long-term securities have larger impact on activity than short-term policy rates; inflation-indexed bonds permit decomposition into real yields and inflation expectations to identify international spillovers.
- Prior literature summarized: U.S. macro surprises and U.S. interest rates often drive foreign returns and rates; mixed evidence on real interest parity and weak exogeneity of U.S. rates.

### Data and market trading hours
- Data:
  - Daily closing prices for conventional and inflation-indexed bonds for United States, Australia, Canada, France, Italy, Japan, Sweden, United Kingdom.
  - Most data: start in 1997, finish end-2006; Sweden starts June 2000; France from November 1998; Italy and Japan markets opened in 2004.
  - Intraday conventional bond data: December 4, 2006 to April 17, 2007 (50 days).
  - For the United States: government benchmark 10-year maturity; other countries: indexed bond closest to 10 years (typical range 8 to 12 years). Canada indexed bond matures in 2021.
- Market trading hours and overlap with U.S. session:
  - U.S. main session opens at 7 a.m. eastern standard time (EST); fix generally 4.30 p.m. EST. 11/
  - Asian (no overlap): Australia (4.30 p.m. to 12.30 a.m. EST); Japan (7 p.m. to 5 a.m. EST).
  - European (significant overlap): France and Sweden open 1.30 a.m. EST; United Kingdom opens 2.30 a.m. EST; closing quote 1.30 p.m. EST.
  - Canada: essentially synchronous with U.S. trading hours.
- Sample windows:
  - Daily frequency: January 2, 2002 to December 29, 2006.
  - Weekly frequency: January 4, 2002 to December 29, 2006.
  - Intraday: December 4, 2006 to April 17, 2007.
  - Japan sample begins April, 2004 (footnote).

### Theoretical framework and practice of bond yields
- Nominal yield decomposition (annualized, between t and t+k):
  - Yield_{t,t+k} = E_{t}(π_{t,t+k}) + RP_{t,t+k} + r_{t,t+k} + (other terms).
- Inflation-indexed bond yield isolates real return and associated risk premium; nominal minus indexed yields measure expected average inflation over the horizon.
- For foreign investors, exchange-rate appreciation and its real equivalent enter returns; E_{t}(r_{t,t+k}) = E_{t}(r*_{t,t+k} + rs_{t,t+k}).
- International comparisons of indexed yields and nominal-minus-indexed gaps help separate real risk premiums from expected inflation differentials and their risk premiums.

### Descriptive statistics (selected exact figures)
- Table 1. Standard Deviations of Daily Bond Yields and Inflation Expectations (January 2, 2002 to December 29, 2006)
  - Columns (nominal yields level, nominal yields changes, real yields level, real yields changes, inflation expectations level, inflation expectations changes):
    - Australia: 0.35 0.06 0.45 0.03 0.39 0.04
    - Canada: 0.57 0.04 0.72 0.02 0.25 0.03
    - France: 0.71 0.04 0.75 0.03 0.17 0.02
    - Japan 1/: 0.21 0.03 0.23 0.03 0.16 0.02
    - Sweden: 0.81 0.04 0.70 0.03 0.20 0.03
    - United Kingdom: 0.33 0.04 0.33 0.03 0.23 0.02
    - United States: 0.44 0.06 0.47 0.04 0.35 0.03
  - Note: 1/ Data begin in April, 2004.
- Key descriptive observations:
  - Changes in nominal rates are more volatile than either real rates or inflation expectations.
  - Standard deviations of inflation expectations levels are lower than nominal or real yields.
- Table 2. Selected cross-country correlations (January 2, 2002 to December 29, 2006)
  - Nominal yields — Levels (selected entries above diagonal):
    - Australia–Canada: 0.59
    - Australia–France: 0.77
    - Australia–Japan 1/: 0.75
    - Australia–Sweden: 0.66
    - Australia–United Kingdom: 0.89
    - Australia–United States: 0.72
  - Nominal yields — Changes (selected entries below diagonal):
    - Canada–Australia: 0.12
    - France–Australia: 0.27
    - Japan 1/–Australia: 0.40
    - Sweden–Australia: 0.37
    - United Kingdom–Australia: 0.26
    - United States–Australia: 0.06
  - Real yields — Levels (selected entries above diagonal):
    - Australia–Canada: 0.89
    - Australia–France: 0.80
    - Australia–Japan 1/: 0.26
    - Australia–Sweden: 0.90
    - Australia–United Kingdom: 0.86
    - Australia–United States: 0.36
  - Real yields — Changes (selected entries below diagonal):
    - Canada–Australia: 0.14
    - France–Australia: 0.19
    - Japan 1/–Australia: 0.33
    - Sweden–Australia: 0.27
    - United Kingdom–Australia: 0.18
    - United States–Australia: 0.08
  - Inflation expectations — Levels (selected entries above diagonal):
    - Australia–Canada: -0.58
    - Australia–France: 0.29
    - Australia–Japan 1/: 0.00
    - Australia–Sweden: -0.14
    - Australia–United Kingdom: 0.66
    - Australia–United States: 0.58
  - Inflation expectations — Changes (selected entries below diagonal):
    - Canada–Australia: 0.04
    - France–Australia: 0.13
    - Japan 1/–Australia: 0.08
    - Sweden–Australia: 0.24
    - United Kingdom–Australia: 0.14
    - United States–Australia: -0.07
  - Interpretations:
    - Levels of nominal yields are relatively highly correlated (most entries above ½).
    - Correlations of daily changes in nominal yields are generally lower and show regional linkages.
    - Real rates are more correlated than inflation expectations.

### Tests for efficiency and spillovers: empirical strategy
- Market efficiency (weak-form): E_{t}(r_{i,t+1}) = r_{i,t}; equivalently 0 = E_{t}(r_{i,t+1} − r_{i,t}).
- Spillovers defined as contemporaneous correlation between domestic returns r_{t} and foreign returns r*_{t}; testing uses lagged foreign returns r*_{t-1} to distinguish spillovers from contemporaneous correlation.
- Empirical specifications:
  - Daily tests: changes defined from fix to fix; U.S. changes lagged two days relative to foreign markets (except Canada).
  - Weekly tests: first lags used for all series; overlapping-hours issues mitigated.
  - Intraday specification (equation (10)): regressions include domestic and foreign earlier price changes; β2 and γ2 test international spillovers; Canada excluded from intraday analysis due to synchronous trading.
  - VAR reparameterizations with two lags of levels used for inference on short-term inefficiency and long-run error-correction.

### Core empirical findings (direction, strength, type of spillovers)
- Overall summary:
  - Strong evidence of spillovers from U.S. markets to foreign markets across intraday, daily, weekly frequencies.
  - Evidence strongest for real interest rates; weakest for inflation expectations.
  - Little or no evidence that foreign markets consistently Granger-cause U.S. real interest rates.
- Daily-frequency results (January 2, 2002 to December 29, 2006; Table 3):
  - Real interest rates: past U.S. yields Granger-cause current foreign yields in four of six countries (exceptions: Australia—test marginally fails—and Japan—short sample). None of the 36 U.S. entries significant at conventional levels.
  - Inflation expectations: U.S. Granger-cause foreign counterparts in two of six markets; domestic inefficiency present in all foreign markets and in U.S. expectations.
  - Nominal yields: significant U.S. spillovers in four of six foreign markets; spillovers from foreign markets to the United States in two markets.
- Weekly-frequency results (January 4, 2002 to December 29, 2006; Table 4):
  - Similar to daily: real interest rate linkages run entirely from the United States to foreign markets.
  - Somewhat stronger U.S. spillovers to foreign inflation expectations; weaker evidence of spillovers to the United States in nominal bonds and inflation expectations.
- Intraday results (December 4, 2006 to April 17, 2007; Table 5):
  - Strong evidence of U.S. → foreign spillovers; essentially no evidence for spillovers in the other direction.
  - U.S. information significant for foreign markets in every case except Japan.
  - For countries with price data, coefficients insignificantly different from one (U.S. news incorporated one-for-one).
  - Adding U.S. information raised R2 by 5 to 10 percent.

### VAR-based quantification and identification choices
- VAR: Zt = (rt, pt, r*t, p*t) with two lags in levels.
- Identification via Choleski ordering:
  - Base ordering: r t, p t, r t*, p t* (U.S. first, foreign last).
  - Alternate orderings for robustness:
    - European and Canadian alternate: r t, r t*, p t*, p t (U.S. inflation expectations placed last).
    - Asian alternate (no overlap): r t*, p t*, r t, p t (contemporaneous correlation assigned to Asian markets).
  - Baseline generally keeps rUS before r* where trading overlaps.

### Impulse-response and persistence (selected exact values and dynamics)
- U.S. variables:
  - U.S. real interest rates and inflation expectations are extremely close to a random walk.
  - A one standard deviation shock to U.S. real rates moves them up by around 0.045 percent, with slight decay over next 50 days.
- Spillovers from U.S. real shocks to foreign real rates:
  - Domestic shocks in foreign real rates vary between 0.015 and 0.03 percent.
  - Spillovers from U.S. real interest shocks to foreign real rates vary between 0.01 to 0.025 percent.
  - In round figures, between one-quarter and one-half of U.S. real interest rate shocks are transmitted to foreign markets.
  - Changes in U.S. inflation expectations generally have a temporary positive impact on real rates abroad.
- Dynamics of foreign inflation expectations:
  - Foreign inflation expectations have own shocks to foreign real rates of similar magnitude (0.015-0.03 percent).
  - Foreign inflation expectations generally exhibit significant positive spillovers from U.S. inflation expectations and, to a lesser extent, U.S. real rates.
  - Foreign inflation expectations are usually negatively affected by shocks to local real rates.

### Variance decomposition results (exact aggregated entries)
- Table 6 (percent attributed after 50 days; daily data January 2, 2002 to December 29, 2006) — Columns correspond to VAR ordering RUS, PUS, R*, P*:
  - United States average:
    - RUS: 97.6, 0.3, 0.4, 1.8
    - PUS: 1.2, 93.6, 2.2, 3.1
  - Foreign average:
    - R*: 39.7, 3.5, 52.1, 4.6
    - P*: 10.9, 25.2, 4.6, 59.2
- Table 7 (percent attributed after one year; weekly data January 4, 2002 to December 29, 2006):
  - United States average:
    - RUS: 86.2, 3.9, 3.9, 6.0
    - PUS: 7.3, 85.3, 5.3, 2.1
  - Foreign average:
    - R*: 50.3, 9.4, 30.5, 9.8
    - P*: 10.9, 39.5, 14.9, 34.8
- Interpretation:
  - After 50 days, between 20-60 percent of foreign real rate variance determined by U.S. real shocks; foreign real rates mainly driven by domestic and U.S. real shocks.
  - After one year, U.S. factors on average comprise more than half of variation in foreign real interest rates (with exception noted for Australia in the sample).

### Robustness and alternate decompositions
- Alternative Choleski ordering for European markets and Canada leaves U.S. real rates essentially unchanged.
- Under alternative ordering, foreign inflation expectations play a more important role in determining U.S. inflation expectations, but U.S. spillovers—particularly from real rates—to European and Canadian inflation expectations remain.

### Appendix figures (Impulse-Response Functions) — panels and features (selected)
- Figures 7–15 present impulse-response panels for Japan, Sweden, United Kingdom, and alternate-ordering panels for Australia, Canada, France, Japan, Sweden, United Kingdom.
- Common formatting across figures:
  - Horizon ticks printed: 5 10 15 20 25 30 35 40 45 50 (as printed: 5101520253035404550).
  - Y-axis ticks vary by panel (examples printed): -.04 -.02 .00 .02 .04 .06 ; -.02 -.01 .00 .01 .02 .03 ; -.02 .00 .02 .04 .06.
  - Panels include responses of RUS, PUS, and foreign R and P to shocks in each variable (16 panels per figure: Response of RUS to RUS, Response of RUS to PUS, Response of RUS to R*, Response of RUS to P*, etc.).

### Main conclusions and policy-relevant implications
- Real interest rates are substantially more internationally linked than inflation expectations; real rates driven by global factors, inflation expectations by domestic events.
- Real interest rate spillovers flow predominantly from the United States to other countries; U.S. markets efficiently absorb available information relative to foreign markets.
- Tests indicate U.S. factors on average determine about one-half of foreign real interest rates; this proportion tends to rise over longer horizons.
- International spillovers in inflation expectations are smaller; U.S. developments account for roughly one-quarter to one-third of fluctuations in foreign inflation expectations, with some reverse spillovers depending on specification.
- Spillovers to Japan resemble those for other countries despite Japan’s near-zero short-term policy rates in the sample.
- Policy implication: deep and liquid U.S. bond markets play a central role in global price discovery for long-term real rates; U.S. bond yields also reflect international developments (e.g., global “saving glut”), so U.S. financial shocks are an important transmission channel to other industrial countries.

*Source: Content derived from _wp07128 - 15. Impulse-Response Functions for Alternate Ordering, United Kingdom (PDF chapter/section).*

### References .............................................................................................................

### References .................................................................................................................................26

### Figures
- 1. Long-term Interest Rates and Inflation Expectations, 1997–2006 .................................8
- 2. Long-term Interest Rates and Inflation Expectations, 2004–2006 .................................9
- 3. Intraday Price Quote Times ..........................................................................................13

### Tables
- 1. Standard Deviations of Daily Bond Yields and Inflation Expectations .......................10
- 2. Cross-Country Correlations in Bond Yields and Inflation Expectations......................11
- 3. International Bond Market Spillovers at a Daily Frequency ........................................16
- 4. International Bond Market Spillovers at a Weekly Frequency.....................................17
- 5. Intraday Sprillovers in Nominal Bond Markets............................................................20
- 6. Variance Decompositions After 50 Days .....................................................................22
- 7. Variance Decompositions After One Year ...................................................................23

### Appendix Figures
- 4 Impulse-Response Functions, Australia........................................................................29
- 5. Impulse-Response Functions, Canada ..........................................................................30
- 6. Impulse-Response Functions, France ...........................................................................31
- 7. Impulse-Response Functions, Japan .............................................................................32
- 8. Impulse-Response Functions, Sweden..........................................................................33
- 9. Impulse-Response Functions, United Kingdom ...........................................................34
- 10 Impulse-Response Functions for Alternate Ordering, Australia...................................35
- 11. Impulse-Response Functions for Alternate Ordering, Canada .....................................36
- 12. Impulse-Response Functions for Alternate Ordering, France ......................................37
- 13. Impulse-Response Functions for Alternate Ordering, Japan ........................................38
- 14. Impulse-Response Functions for Alternate Ordering, Sweden.....................................39

*Source: _wp07128 - References*

### 15. Impulse-Response Functions for Alternate Ordering, United Kingdom ......................40

### 15. Impulse-Response Functions for Alternate Ordering, United Kingdom

### Introduction
- Rapid financial market globalization increases the likelihood of financial spillovers across countries, particularly in government bond markets.
- Yields on long-term securities are considered to have a larger impact on activity than short-term policy rates.
- Inflation-indexed bonds allow decomposition of nominal yields into real yields and inflation expectations, enabling identification of international spillovers in real interest rates and inflation expectations.
- The paper analyzes bilateral spillovers between the United States and six other industrial countries with inflation-indexed bond markets: Australia, Canada, France, Japan, Sweden, and the United Kingdom. French data are used as a proxy for the Euro Area; Italian data are nearly identical to French series.
- The focus on U.S. bilateral links reflects the dominant position of the United States in global bond markets (almost two-thirds of all private bonds are traded in U.S. markets).
- Prior literature summarized:
  - U.S. macroeconomic surprises affect foreign returns, exchange rates, and interest rates (examples include Faust et al (2007), Andersen et al (2003, 2005), Almeida et al (1998), Fair (2003)).
  - Fewer spillovers from foreign macroeconomic releases to U.S. markets documented (Anderson et al (2003), Ehrmann and Fratzscher (2005), Goldberg and Leonard (2003)).
  - U.S. interest rates drive rates in the euro area, Germany, Canada, and Australia (Ehrmann and Fratzscher, 2005; Bremnes et al, 2001; Gravelle and Moessner, 2001; Kim and Sheen, 2000).
  - Studies of real interest parity (RIP) have mixed results; some evidence that U.S. rates are weakly exogenous for other G7 countries (Breedon et al, 1999) but not universally.

### The Theory and Practice of Bond Yields
- The nominal bond yield decomposition (annualized, between t and t+k):
  - Yield_{t,t+k} = E_{t}(π_{t,t+k}) + RP_{t,t+k} + r_{t,t+k} + (other terms) — formally presented in equation (1).
- Inflation-indexed bond yield isolates the real return and associated risk premium; the difference between nominal and indexed yields measures expected average inflation over the horizon (equation (2) and discussion).
- For foreign investors, exchange rate appreciation and its real equivalent enter returns; E_{t}(r_{t,t+k}) = E_{t}(r*_{t,t+k} + rs_{t,t+k}) implies international nominal yield differences reflect expected inflation, exchange rates, and risk premiums (equations (3) and (4)).
- International comparisons of inflation-indexed yields and nominal-minus-indexed gaps help separate real risk premiums from expected inflation differentials and their risk premiums.

### The Data
- Daily closing prices collected for conventional and inflation-indexed bonds for: United States, Australia, Canada, France, Italy, Japan, Sweden, United Kingdom.
- Most data start in 1997 and finish at end-2006. Sweden series starts in June 2000. French data available from November 1998. Italy and Japan markets opened in 2004.
- Italian data used only to confirm French/Euro Area similarity (correlation coefficient across daily changes in French and Italian real interest rate series is 0.98).
- Japanese results reported despite inflation-indexed issuance only starting in 2004; Japan’s zero interest rate policy through July 2006 noted.
- Intraday conventional bond data available only for December 4, 2006 to April 17, 2007.
- For the United States the series correspond to the government’s benchmark 10-year maturity; for other countries indexed bond chosen closest to 10 years (typical range 8 to 12 years). Canadian indexed bond matures in 2021 and is used throughout the sample.
- Visual and descriptive observations:
  - High correlation of nominal bond yields across countries over 1997–2006; common trends and higher-frequency comovements.
  - Real yields are smoother and less correlated across countries than nominal yields; real yields fell for all countries but earlier in the United Kingdom.
  - Inflation expectations (nominal minus indexed yields) show little evidence of a trend.
  - Since 2004 (Figure 2), Italian data mirrors French series; Japanese yields and inflation expectations are less correlated with others due to unique policy environment.

- Table 1. Standard Deviations of Daily Bond Yields and Inflation Expectations (January 2, 2002 to December 29, 2006)
  - Columns: Level, Changes, Level, Changes, Level, Changes for Nominal yields, Real yields, Inflation expectations respectively.
  - Australia: 0.35 0.06 0.45 0.03 0.39 0.04
  - Canada: 0.57 0.04 0.72 0.02 0.25 0.03
  - France: 0.71 0.04 0.75 0.03 0.17 0.02
  - Japan 1/: 0.21 0.03 0.23 0.03 0.16 0.02
  - Sweden: 0.81 0.04 0.70 0.03 0.20 0.03
  - United Kingdom: 0.33 0.04 0.33 0.03 0.23 0.02
  - United States: 0.44 0.06 0.47 0.04 0.35 0.03
  - Note: 1/ Data begin in April, 2004.
- Key data-based observations:
  - Standard deviations of levels of real returns similar to nominal yields.
  - Changes in nominal rates are more volatile than either real rates or inflation expectations.
  - Standard deviations of inflation expectations levels are lower than nominal or real yields, indicating long-term credibility of monetary authorities.
  - Calculations use data starting in 2002 because U.S. inflation-indexed markets became liquid enough from this period.

- Table 2. Cross-Country Correlations in Bond Yields and Inflation Expectations (January 2, 2002 to December 29, 2006)
  - Note: Entries above diagonal are correlations of levels; entries below diagonal are correlations of daily changes. 1/ Data begin in April, 2004.
  - Nominal Yields — Levels (upper triangle) and Changes (lower triangle):
    - Levels row labels: Australia, Canada, France, Japan 1/, Sweden, United Kingdom, United States
    - Levels (selected entries above diagonal):
      - Australia–Canada: 0.59
      - Australia–France: 0.77
      - Australia–Japan 1/: 0.75
      - Australia–Sweden: 0.66
      - Australia–United Kingdom: 0.89
      - Australia–United States: 0.72
    - Changes (selected entries below diagonal):
      - Canada–Australia: 0.12
      - France–Australia: 0.27
      - Japan 1/–Australia: 0.40
      - Sweden–Australia: 0.37
      - United Kingdom–Australia: 0.26
      - United States–Australia: 0.06
  - Real Yields — Levels and Changes:
    - Levels (selected entries above diagonal):
      - Australia–Canada: 0.89
      - Australia–France: 0.80
      - Australia–Japan 1/: 0.26
      - Australia–Sweden: 0.90
      - Australia–United Kingdom: 0.86
      - Australia–United States: 0.36
    - Changes (selected entries below diagonal):
      - Canada–Australia: 0.14
      - France–Australia: 0.19
      - Japan 1/–Australia: 0.33
      - Sweden–Australia: 0.27
      - United Kingdom–Australia: 0.18
      - United States–Australia: 0.08
  - Inflation Expectations — Levels and Changes:
    - Levels (selected entries above diagonal):
      - Australia–Canada: -0.58
      - Australia–France: 0.29
      - Australia–Japan 1/: 0.00
      - Australia–Sweden: -0.14
      - Australia–United Kingdom: 0.66
      - Australia–United States: 0.58
    - Changes (selected entries below diagonal):
      - Canada–Australia: 0.04
      - France–Australia: 0.13
      - Japan 1/–Australia: 0.08
      - Sweden–Australia: 0.24
      - United Kingdom–Australia: 0.14
      - United States–Australia: -0.07
  - Interpretations:
    - Levels of nominal yields are relatively highly correlated (most entries above ½), though some U.S. correlations are lower in this sample.
    - Correlations of daily changes in nominal yields are generally lower than levels and show regional linkages (European markets; North American markets).
    - Real rates are more correlated than inflation expectations in both levels and changes. Levels of real rates mostly exceed 0.8 except for Japan and the U.S. in this sample.
    - Levels of inflation expectations vary widely between -0.14 and 0.86.
    - Formal tests reject cointegration between U.S. and other countries for nominal yields, real yields, and inflation expectations over this short sample (four years), though results are not conclusive.

- Theoretical alignment:
  - High correlations across real rates consistent with global trends in “world” real rates and potential for significant spillovers.
  - Lower correlations among inflation expectations align with inflation being more influenced by domestic factors than global factors.

### Tests for Efficiency and the Existence of Spillovers
- Market efficiency defined: market prices fully reflect available information. Weak-form efficiency tested where available information is an asset’s own historical returns and those of related markets.
  - Efficiency condition (forecast): E_{t}(r_{i,t+1}) = r_{i,t} (equation (5)); equivalently 0 = E_{t}(r_{i,t+1} − r_{i,t}) (equation (6)).
- Definition of spillovers: contemporaneous correlation between domestic returns r_{t} and foreign returns r*_{t}; if r*_{t} significantly determines r_{t}, there are spillovers. Use impact of lagged foreign returns r*_{t-1} on domestic r_{t} to distinguish spillovers from contemporaneous correlation attribution (equation (7)).
- Empirical strategy: various tests of the impact of r*_{t-1} on r_{t} are used to determine evidence of spillovers and to separate them from domestic market inefficiencies.

*Italic source attribution: Content derived from _wp07128 - 15. Impulse-Response Functions for Alternate Ordering, United Kingdom (PDF chapter/section).*

### conclusions regarding the most likely driver of the correlation between the two markets.

### _wp07128 - conclusions regarding the most likely driver of the correlation between the two markets.

### Market trading hours and data framing
- United States main trading session opens at 7 a.m. eastern standard time (EST) and the fix in the data is generally 4.30 p.m. EST. 11/
- Other markets classified by overlap with U.S. session:
  - Asian markets (no overlap): Australian (4.30 p.m. to 12.30 a.m. EST) and Japanese (7 p.m. to 5 a.m. EST).
  - European markets (significant overlap): France and Sweden open at 1.30 a.m. EST; U.K. opens at 2.30 a.m. EST; closing quote is 1.30 p.m. EST (about two-thirds of the way through the U.S. session). The U.S. market is most active during the overlap with European markets.
  - Canadian market (synchronous): essentially the same trading hours as the United States.
- Intraday dataset limitations and choices:
  - Intraday history available for 50 days.
  - Australia and Japan data are in yields; other countries are in prices.
  - Intraday analysis limited to conventional bonds (higher turnover).
- Data sample windows reported:
  - Daily frequency tests: January 2, 2002 to December 29, 2006.
  - Weekly frequency tests: January 4, 2002 to December 29, 2006.
  - Intraday tests: December 4, 2006 to April 17, 2007.
  - Japan sample begins April, 2004 (footnote).

### Tests of market efficiency and empirical specification
- Daily-data testing approach:
  - Changes defined from the fix on one day to the fix the next day.
  - To avoid overlap due to U.S. later close, change in U.S. yields is lagged two days versus one day for foreign markets (except Canada where first lag can be used).
  - Regression systems (reparameterizations of a VAR with two lags of levels) described in equations (8) and (9), where rt, pt, it represent real interest rate, associated inflation expectations, and nominal interest rates; subscripts t-1 denote prior data.
  - Tests distinguish short-term inefficiency (dependence on past changes) and long-run inefficiency (dependence on past levels / error correction / mean reversion).
- Weekly-data tests:
  - Changes from close of one week to close of next; first lags used for all series.
  - Overlapping-hours issues largely mitigated.
- Intraday specification:
  - Equation (10): it and i*t regressions with coefficients α, β, γ; β1 and γ1 test dependence on earlier domestic price changes; β2 and γ2 test international spillovers.
  - Variable inclusion depends on relative timing of trading (distinct rules for Asian, European, and Canadian cases).
  - Canada excluded from intraday analysis due to synchronous trading.

### Core empirical findings (spillover direction, strength, and type)
- Overall summary:
  - Strong evidence of spillovers from U.S. markets to foreign markets across data frequencies (intraday, daily, weekly).
  - Evidence most convincing for real interest rates; weakest for inflation expectations.
  - Little or no evidence that foreign markets consistently Granger-cause U.S. real interest rates; some evidence of limited reverse effects for nominal bonds and inflation expectations.
- Daily-frequency results (Table 3; p-values reported for Jan 2, 2002 to Dec 29, 2006):
  - Real interest rates: strong evidence that past U.S. yields spillover (Granger-cause) current foreign yields in four of the six other countries (exceptions: Australia—test fails only marginally—and Japan—short sample). None of the 36 entries for the United States are significant at conventional levels.
  - Inflation expectations: weaker and more mixed spillover evidence. U.S. inflation expectations Granger-cause foreign counterparts in two of six markets. All foreign markets show domestic inefficiency with regard to their own past yields; similar domestic inefficiency also appears for U.S. expectations.
  - Nominal yields: significant U.S. spillovers in four of six foreign markets; spillovers from foreign markets to the United States occur in two markets. Several tests indicate domestic inefficiency in the U.S. market.
- Weekly-frequency results (Table 4; Jan 4, 2002 to Dec 29, 2006):
  - Broadly similar to daily results: real interest rate linkages run entirely from the United States to foreign markets.
  - Somewhat stronger evidence of U.S. spillovers to other countries in inflation expectations; somewhat weaker evidence of spillovers to the United States in nominal bonds and inflation expectations.
  - Absence of significance does not rule out contemporaneous links; foreign markets do depend significantly on U.S. developments.
- Intraday results (Dec 4, 2006 to Apr 17, 2007; Table 5):
  - Strong evidence of U.S. spillovers to other markets and essentially no evidence for spillovers in the other direction.
  - U.S. information is significant for foreign markets in every case except Japan.
  - For countries with price data, coefficients are insignificantly different from one (U.S. news incorporated one-for-one into foreign markets).
  - Adding U.S. information to regressions featuring only domestic information raised R2 by 5 to 10 percent.
  - Results robust to use of prices 30 minutes into European sessions (overnight information absorption).

### VAR-based quantification and identification considerations
- VAR specification used to quantify dynamic spillovers (equation (6)):
  - Zt = (rt, pt, r*t, p*t); VAR with two lags in levels.
- Identification (Choleski ordering) choices driven by predominance of U.S. spillovers:
  - Base ordering: r t, p t, r t*, p t* (U.S. variables first, foreign last), assigning contemporaneous correlations to the United States.
  - Alternate orderings to check robustness:
    - For European and Canadian data where feedback in inflation expectations possible: r t, r t*, p t*, p t (U.S. inflation expectations placed last).
    - For Asian markets (no overlap): alternate ordering r t*, p t*, r t, p t, assigning contemporaneous correlation to Asian markets.
  - Where overlapping trading sessions complicate ordering, rUS remains before r* in baseline.

### Interpretation and conclusion
- Consistent interpretation across intraday, daily, and weekly evidence:
  - Real interest rates are significantly determined by events in U.S. markets; U.S. → foreign spillovers dominate.
  - Inflation expectations exhibit more mixed, heterogeneous international interactions.
  - Nominal yields show strong U.S. influence but some limited reverse causation exists.
- Practical implication from intraday explanatory power:
  - U.S. market information materially improves foreign-market return forecasts; R2 increases of 5 to 10 percent when U.S. info included.

*Source: IMF staff calculations and analysis as presented in the provided document.*

### Appendix I, Figures 4-9) and variance decompositions (Table 6). The IRFs indicate that U.S.

### Appendix I, Figures 4–9 and Variance Decompositions (Table 6 and Table 7)

### Impulse-response and persistence of U.S. variables
- U.S. real interest rates and inflation expectations are extremely close to a random walk.
- A one standard deviation shock to U.S. real rates moves them up by around 0.045 percent, with a very slight tendency to fall over the next 50 days.
- U.S. inflation expectations show a similar profile but with a more pronounced decay over time; the French regression finds some significant long-term effects from foreign variables.
- None of the other variables in the daily VARs—U.S. inflation expectations, foreign real rates, and foreign inflation expectations—have any significant impact on U.S. real rates in the base specification.

### Spillovers from U.S. markets to foreign markets
- Domestic shocks in foreign real rates are smaller than in the United States, varying between 0.015 and 0.03 percent.
- Spillovers from U.S. real interest shocks to foreign real rates vary between 0.01 to 0.025 percent.
- In round figures, between one-quarter and one-half of U.S. real interest rate shocks are transmitted to foreign markets.
- These U.S.-origin shocks account for a similar proportion of movements in foreign real rates.
- Changes in U.S. inflation expectations generally have a temporary positive impact on real rates abroad.

### Dynamics of foreign inflation expectations
- Foreign inflation expectations have own shocks to foreign real interest rates of similar magnitude (0.015-0.03 percent).
- Foreign inflation expectations generally exhibit significant positive spillovers from U.S. inflation expectations and, to a somewhat lesser extent, U.S. real rates.
- Foreign inflation expectations are usually negatively affected by shocks to local real rates.
- One interpretation: increases in U.S. real rates are seen as a precursor of global inflation pressures (positive relationship), while higher domestic real rates are seen as monetary tightening (lower expected inflation).

### Variance decomposition results (50 days, Table 6 — daily data from January 2, 2002 to December 29, 2006)
- The base Choleski decomposition confirms outcomes for U.S. real rates and inflation expectations are dominated by local shocks.
- Between 20-60 percent of foreign real rate variances are determined by U.S. real interest rate shocks; most of the remainder reflects domestic real interest rate shocks.
- U.S. and domestic inflation expectations generally play only a minor role in explaining foreign real rate variance over 50 days.

Key aggregated entries from Table 6 (percent attributed to forecasted variable after 50 days):
- United States average:
  - RUS: 97.6, 0.3, 0.4, 1.8
  - PUS: 1.2, 93.6, 2.2, 3.1
- Foreign average:
  - R*: 39.7, 3.5, 52.1, 4.6
  - P*: 10.9, 25.2, 4.6, 59.2

(Columns correspond to VAR ordering RUS, PUS, R*, P* in the left-half specification reported.)

### Variance decomposition results (one year, Table 7 — weekly data from January 4, 2002 to December 29, 2006)
- Using end-week (weekly) data extends the horizon and shows the importance of spillovers rises over longer horizons.
- After one year, U.S. factors on average comprise more than half of the variation in foreign real interest rates (with an anomalous exception for Australia, where the importance declines to 2 percent from 23 percent in the daily data).
- Spillovers from U.S. to foreign inflation expectations also rise over the one-year horizon.

Key aggregated entries from Table 7 (percent attributed to forecasted variable after one year):
- United States average:
  - RUS: 86.2, 3.9, 3.9, 6.0
  - PUS: 7.3, 85.3, 5.3, 2.1
- Foreign average:
  - R*: 50.3, 9.4, 30.5, 9.8
  - P*: 10.9, 39.5, 14.9, 34.8

### Robustness and alternative decompositions
- Results using an alternative Choleski ordering for the three European markets and Canada (right half of Table 6 and Figures 10–15 in Appendix I) leave U.S. real rates essentially unchanged.
- Under the alternative ordering, foreign inflation expectations play a more important role in determining U.S. inflation expectations, but evidence of U.S. spillovers—particularly from real rates—to European and Canadian inflation expectations remains.

### Main conclusions and policy-relevant implications
- Real interest rates are substantially more internationally linked than inflation expectations, consistent with real rates being driven by global factors and inflation expectations by domestic events.
- Real interest rate spillovers flow predominantly from the United States to other countries; U.S. markets efficiently absorb available information relative to foreign markets.
- Tests indicate U.S. factors on average determine about one-half of foreign real interest rates, and this proportion tends to rise over longer horizons.
- International spillovers in inflation expectations are smaller; U.S. developments account for roughly one-quarter to one-third of fluctuations in foreign inflation expectations, with some reverse spillovers depending on specification.
- Spillovers to Japan are similar to those for other countries despite Japan’s period of near-zero short-term policy rates during the sample.
- Given the importance of long-term real interest rates for economic activity and other financial instruments (such as equities), financial spillovers from the United States constitute an important transmission channel to other industrial countries.
- Deep and liquid U.S. bond markets play a central role in global price discovery for long-term real rates; however, U.S. bond yields can also reflect international developments (for example, the global “saving glut”).

*Source: IMF staff calculations (from Appendix I, Figures 4–9 and Tables 6–7 of the provided IMF working paper content).*

### APPENDIX I

### APPENDIX I — Impulse-Response Figures (Figures 7–15)

### Figure 7. Impulse-Response Functions, Japan
- Figure number: 7
- Source note: IMF staff calculations.
- Horizon ticks (as printed): 5101520253035404550
- Y-axis ticks (various panels, as printed): -.04 -.02 .00 .02 .04 .06 ; -.02 -.01 .00 .01 .02 .03
- Panels (responses printed):
  - Response of RUS to RUS
  - Response of RUS to PUS
  - Response of RUS to RJP
  - Response of RUS to PJP
  - Response of PUS to RUS
  - Response of PUS to PUS
  - Response of PUS to RJP
  - Response of PUS to PJP
  - Response of RJP to RUS
  - Response of RJP to PUS
  - Response of RJP to RJP
  - Response of RJP to PJP
  - Response of PJP to RUS
  - Response of PJP to PUS
  - Response of PJP to RJP
  - Response of PJP to PJP

### Figure 8. Impulse-Response Functions, Sweden
- Figure number: 8
- Source note: IMF staff calculations.
- Horizon ticks (as printed): 5101520253035404550
- Y-axis ticks (various panels, as printed): -.02 -.01 .00 .01 .02 .03 .04 .05 ; -.01 .00 .01 .02 .03 .04 ; .00 .01 .02
- Panels (responses printed):
  - Res po n s e   of   RUS  t o  RUS
  - Re s p on s e   o f   RUS  t o   PUS
  - Res po ns e  of   RUS  t o   RSW
  - Res p on s e   of   RUS  t o   PSW
  - Response of PUS to RUS
  - Response of PUS to PUS
  - Response of PUS to RSW
  - Response  of   PUS  to  PSW
  - Res po ns e   of   RSW  t o  RUS
  - Response of RSW to PUS
  - Response of RSW to RSW
  - Response  of   RSW  to  PSW
  - Response of PSW to RUS
  - Response of PSW to PUS
  - Response of PSW to RSW
  - Response  of   PSW  to  PSW

### Figure 9. Impulse-Response Functions, United Kingdom
- Figure number: 9
- Source note: IMF staff calculations.
- Horizon ticks (as printed): 5101520253035404550
- Y-axis ticks (various panels, as printed): -.02 -.01 .00 .01 .02 .03 .04 .05 ; -.010 -.005 .000 .005 .010 .015 .020 ; -.010 -.005 .000 .005 .010 .015 .020
- Panels (responses printed):
  - Response of RUS to RUS
  - Re s p on s e   o f   RUS  t o   PUS
  - Response of RUS to RUK
  - Response of RUS to PUK
  - Response of PUS to RUS
  - Re s p on s e   o f   PUS  t o   PUS
  - Res po ns e  o f   PUS  t o   RUK
  - Response  of   PUS  to  PUK
  - Response of RUK to RUS
  - Re s p on s e   o f   RUK  t o   PUS
  - Response of RUK to RUK
  - Response of RUK to PUK
  - Response of PUK to RUS
  - Re s p on s e   o f   PUK  t o   PUS
  - Res po ns e  o f   PUK  t o   RUK
  - Response  of   PUK  to  PUK

### Figure 10. Impulse-Response Functions for Alternate Ordering, Australia
- Figure number: 10
- Source note: IMF staff calculations.
- Horizon ticks (as printed): 5101520253035404550
- Y-axis ticks (various panels, as printed): -.02 .00 .02 .04 .06 ; -.02 -.01 .00 .01 .02 .03 .04
- Panels (responses printed):
  - Response of RUS to RUS
  - Response of RUS to PUS
  - Re sp o n se   o f   RUS   t o   RA U
  - Response of RUS to PAU
  - Response of PUS to RUS
  - Response of PUS to PUS
  - Re sp o n se   o f   P US   t o   RA U
  - Response of PUS to PAU
  - Response of RAU to RUS
  - Response of RAU to PUS
  - Re sp o n se   o f   RA U  t o   RA U
  - Response of RAU to PAU
  - Response of PAU to RUS
  - Response of PAU to PUS
  - Re sp o n se   o f   P A U  t o   RA U
  - Response of PAU to PAU

### Figure 11. Impulse-Response Functions for Alternate Ordering, Canada
- Figure number: 11
- Source note: IMF staff calculations.
- Horizon ticks (as printed): 5101520253035404550
- Y-axis ticks (various panels, as printed): -.02 .00 .02 .04 .06 ; -.02 -.01 .00 .01 .02 .03 ; -.01 .00 .01 .02 .03
- Panels (responses printed):
  - Response of RUS to RUS
  - Response of RUS to PUS
  - Response of RUS to RCA
  - Response of RUS to PCA
  - Response of PUS to RUS
  - Response of PUS to PUS
  - Response of PUS to RCA
  - Response of PUS to PCA
  - Response of RCA to RUS
  - Response of RCA to PUS
  - Response of RCA to RCA
  - Response of RCA to PCA
  - Response of PCA to RUS
  - Response of PCA to PUS
  - Response of PCA to RCA
  - Response of PCA to PCA

### Figure 12. Impulse-Response Functions for Alternate Ordering, France
- Figure number: 12
- Source note: IMF staff calculations.
- Horizon ticks (as printed): 5101520253035404550
- Y-axis ticks (various panels, as printed): -.02 .00 .02 .04 .06 ; -.01 .00 .01 .02 .03 .04 ; -.004 .000 .004 .008 .012 .016 .020
- Panels (responses printed):
  - Response of RUS to RUS
  - Response of RUS to PUS
  - Response of RUS to RFR
  - Response of RUS to PFR
  - Response of PUS to RUS
  - Response of PUS to PUS
  - Response of PUS to RFR
  - Response of PUS to PFR
  - Response of RFR to RUS
  - Response of RFR to PUS
  - Response of RFR to RFR
  - Response of RFR to PFR
  - Response of PFR to RUS
  - Response of PFR to PUS
  - Response of PFR to RFR
  - Response of PFR to PFR

### Figure 13. Impulse-Response Functions for Alternate Ordering, Japan
- Figure number: 13
- Source note: IMF staff calculations.
- Horizon ticks (as printed): 5101520253035404550
- Y-axis ticks (various panels, as printed): -.04 -.02 .00 .02 .04 .06 ; -.02 -.01 .00 .01 .02 .03 ; -.04 -.02 .00 .02 .04
- Panels (responses printed):
  - Response of RUS to RUS
  - Response of RUS to PUS
  - Response of RUS to RJP
  - Response of RUS to PJP
  - Response of PUS to RUS
  - Response of PUS to PUS
  - Response of PUS to RJP
  - Response of PUS to PJP
  - Response of RJP to RUS
  - Response of RJP to PUS
  - Response of RJP to RJP
  - Response of RJP to PJP
  - Response of PJP to RUS
  - Response of PJP to PUS
  - Response of PJP to RJP
  - Response of PJP to PJP

### Figure 14. Impulse-Response Functions for Alternate Ordering, Sweden
- Figure number: 14
- Source note: IMF staff calculations.
- Horizon ticks (as printed): 5101520253035404550
- Y-axis ticks (various panels, as printed): -.02 .00 .02 .04 .06 ; -.01 .00 .01 .02 .03 .04 ; -.02 -.01 .00 .01 .02 .03
- Panels (responses printed):
  - Response of RUS to RUS
  - Response of RUS to PUS
  - Response of RUS to RSW
  - Response of RUS to PSW
  - Response of PUS to RUS
  - Response of PUS to PUS
  - Response of PUS to RSW
  - Response of PUS to PSW
  - Response of RSW to RUS
  - Response of RSW to PUS
  - Response of RSW to RSW
  - Response of RSW to PSW
  - Response of PSW to RUS
  - Response of PSW to PUS
  - Response of PSW to RSW
  - Response of PSW to PSW

### Figure 15. Impulse-Response Functions for Alternate Ordering, United Kingdom
- Figure number: 15
- Source note: IMF staff calculations.
- Horizon ticks (as printed): 5101520253035404550
- Y-axis ticks (various panels, as printed): -.02 .00 .02 .04 .06 ; -.01 .00 .01 .02 .03 .04 ; -.02 -.01 .00 .01 .02 .03 .04
- Panels (responses printed):
  - Response of RUS to RUS
  - Response of RUS to PUS
  - Response of RUS to RUK
  - Response of RUS to PUK
  - Response of PUS to RUS
  - Response of PUS to PUS
  - Response of PUS to RUK
  - Response of PUS to PUK
  - Response of RUK to RUS
  - Response of RUK to PUS
  - Response of RUK to RUK
  - Response of RUK to PUK
  - Response of PUK to RUS
  - Response of PUK to PUS
  - Response of PUK to RUK
  - Response of PUK to PUK

*Source: _wp07128 - APPENDIX I (figures and panel labels as printed in the source).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp07128.pdf_