## wp17268 - 0.13  percent, while US unconventional monetary policy increased Canadian output by 1.2   percent

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### Introduction and objectives
- Study period: 1994–2016 (monthly data from August 1994 to December 2016).
- Focus: quantify effects of unconventional monetary policy (UMP) in a small open economy (Canada) and spillovers from foreign UMP (US).
- Key methodological choices:
  - Capture international spillovers mainly via US variables given strong response of small open economies to foreign shocks.
  - Use shadow interest rates to measure policy stance uninterrupted by the ZLB (Zero lower bound defined as 25 bps).
- Counterfactual interpretation: movements of the shadow rate below 25 bps are interpreted as unconventional monetary policy; counterfactuals holding shadow rate at 25 bps are interpreted as an upper bound on the effect of unconventional monetary policy.

### Data, model, and estimation
- Shadow rates:
  - Canadian shadow rate estimated via a shadow rate term structure model (SRTSM) following Wu and Xia (2016) methodology; US shadow rate taken from Wu and Xia (2016).
  - Assumed ZLB at 25 bps for both Canada and the US.
  - Estimated minima:
    - Canadian shadow rate low: approximately -0.34 percent in November 2009.
    - US shadow rate low: -2.99 percent (decreasing from 2008 through 2014).
- Model: two-country Bayesian structural VAR (B-SVAR) with block-exogeneity (Canada cannot influence US block).
  - Canadian block (main specification): policy rate r; exchange rate s (CAD/USD); seasonally-adjusted real Canadian industrial production y; Canadian CPI p; real Canadian exports ex; real Canadian imports im. Net exports computed post-estimation.
  - US block: US policy rate r*; VIX; commodity export prices wpx; US CPI p*; US industrial production y*.
- Policy rate construction: splice target rates with shadow rates whenever shadow rates are below 25 bps.
- Estimation details:
  - Lags: 12.
  - Gibbs sampler: 5000 draws, discarding first 1000.
  - Prior: Sims and Zha (1998) with μi = 10 for i = 1,...,6 for lag decay.
  - Contemporaneous restrictions: 111 total restrictions; model over-identified.
  - Joint likelihood-ratio test for 385 restrictions (over-identification and block exogeneity) fails to reject block exogeneity; likelihood-ratio statistic = 419.01, 5 percent critical value = 431.75.

### Impulse response findings (25 bps expansionary shocks)
- Canadian 25 bps expansionary shock:
  - Raises industrial production and prices significantly on impact and for approximately 18 to 24 months.
  - Currency depreciates on impact and for about one year (significant at the 68 percent level).
  - Temporary rise in exports (about 6 months after shock) and imports; net exports impulse response slightly negative and significant at the 68 percent level.
- US 25 bps expansionary shock and spillovers to Canada:
  - US industrial production rises with a delay of approximately 1–2 years.
  - US prices fall on impact then rise slightly after about 6 months; commodity prices rise on impact; VIX falls significantly.
  - Strong spillovers to Canada:
    - Canadian policy rate responds on impact and for over two years.
    - Canadian industrial production rises after about 6 months and again following US dynamics.
    - Canadian prices fall in the first month and then rise after about 5 months.
    - Net exports show alternating exchange rate and income effects.

### Counterfactual experiments and quantified effects of UMP (ZLB imposed at 25 bps)
- Counterfactual method: replace realized monetary policy shocks with shocks that constrain the shadow rate to 25 bps during ZLB episodes (separately for Canada and for the US).
- Main historical-decomposition quantified results:
  - Canadian unconventional monetary policy (Canadian ZLB Imposed):
    - Canadian industrial production average percent difference: -0.131 (68% Conf. Bound Upper: -0.064; Lower: -0.198).
    - Cumulative percent difference for Canadian industrial production: -1.828 (68% Conf. Bound Upper: -0.902; Lower: -2.772).
    - Alternative reporting: industrial production (level terms) would have been 0.13 percent lower on average during the Canadian ZLB period under the counterfactual; cumulative effect over the ZLB period: 1.828 percent lower; annualized phrasing in source: "In annualized terms, the decline in output is 1.572 percent percent."
  - US unconventional monetary policy (US ZLB Imposed):
    - Canadian industrial production average percent difference: -1.194 (68% Conf. Bound Upper: -0.215; Lower: -2.283).
    - US industrial production average percent difference under US ZLB Imposed: -1.535 (68% Conf. Bound Upper: -0.009; Lower: -3.337).
  - Government Expenditure Model (selected):
    - Canadian industrial production average percent difference under Canadian ZLB Imposed: -0.132 (68% Conf. Bound Upper: -0.062; Lower: -0.203).
    - Canadian industrial production average percent difference under US ZLB Imposed: -1.054 (68% Conf. Bound Upper: -0.167; Lower: -1.993).
  - Current Account Model (selected):
    - Canadian industrial production average percent difference under Canadian ZLB Imposed: -0.147 (68% Conf. Bound Upper: -0.077; Lower: -0.219).
    - Canadian industrial production average percent difference under US ZLB Imposed: -1.054 (68% Conf. Bound Upper: -0.184; Lower: -2.02).
- Selected large cumulative and unit differences reported (preserved from source tables):
  - Table 2 cumulative percent difference for Canadian industrial production under US ZLB Imposed: -101.51 (68% Conf. Bound Upper: -18.278; Lower: -194.095).
  - Table 4 cumulative unit difference for Canadian industrial production under Canadian ZLB Imposed: -20.281 (68% Conf. Bound Upper: -10.606; Lower: -30.117).
  - Table 4 cumulative unit difference for US industrial production under US ZLB Imposed: -1484.171 (68% Conf. Bound Upper: -8.913; Lower: -3190.650).
- Robustness: Canadian 0.131 percent result is robust across several specifications; combined counterfactual constraining both Canadian and US shadow rates at 25 bps produces results essentially unchanged from the US-only counterfactual because of the large relative impact of US UMP.

### Alternative specifications and robustness checks
- Current account specification:
  - Replacing exports and imports with a current account series (monthly via linear interpolation) yields Canadian industrial production average percent difference under Canadian ZLB Imposed: -0.147 and under US ZLB Imposed: -1.054.
  - Current account rises marginally in response to Canadian and US expansionary policy.
- Unrestricted Bank of Canada contemporaneous reaction function:
  - Impulse responses for industrial production and prices remain robust.
  - Counterfactuals show Canadian industrial production would have been significantly lower without US and Canadian UMP.
- Taylor Rule specification:
  - Taylor Rule constructed with π* = 2%, β = 0.5, ρ = 0, natural rate from Holston, Laubach, and Williams (2017); quarterly rates linearly interpolated to monthly.
  - Results broadly consistent with main model; Canadian industrial production average percent difference under Canadian ZLB Imposed: -0.194 (68% Conf. Bound Max 0.011; Min -0.400).
  - Table E.1 reports large cumulative percent and unit differences (examples preserved verbatim in Appendix E).

### Interpretation, policy implications, and caveats
- Interpretation:
  - Domestic UMP in Canada had measurable positive effects on industrial production and prices during the ZLB episode, but magnitudes are relatively small compared with US UMP spillovers.
  - US unconventional monetary policy generated much larger favorable spillovers to Canada over the July 2007–December 2015 period.
- Policy implications:
  - Unconventional monetary policy in a small open economy is effective, and favorable foreign monetary policy spillovers can be important for recovery.
  - International coordination of monetary policy merits consideration given cross-border spillovers of UMP.
- Caveats and limitations:
  - Shadow-rate-based estimates cannot disentangle specific policy channels (e.g., LSAPs versus forward guidance).
  - Shadow-rate methodology may understate UMP effects if long-term asset holdings after December 2015 generated additional stimulus.
  - The brevity of Canada’s ZLB period limits the extent to which UMP could affect the real economy, making some effects transitory.

### Appendix highlights (data, SRTSM, confidence bands)
- Data sources and construction (selected):
  - r: Bank of Canada, author’s calculations — Bank of Canada target rate spliced with shadow rate at ZLB when shadow rate < target rate = 25 bps, Monthly.
  - r_US: FRED, Wu and Xia (2016) — Federal funds target rate spliced with Wu and Xia (2016) shadow rate at ZLB when SSR < FFR = 25 bp, Monthly.
  - y, y_US: IFS — Industrial Production, Seasonally adjusted, Index, Monthly.
  - current account: CANSIM, quarterly series linearly interpolated to monthly; log not taken because current account can be negative.
- Shadow-rate term structure model (SRTSM) implementation (Canada):
  - Observed nominal rate r_t = max{sr_t, r} = sr_t + max{0, r − sr_t}; sr_t modeled affine in state variables; three-factor model estimated via extended Kalman filter using forward-rate observations.
- Confidence bands:
  - 90 percent confidence bands constructed from 5000 Gibbs sampling draws (Figures B.1, B.2; Table B.1).
  - Example 90% percent-difference entries:
    - Canadian industrial production: Mean -0.132, 90% Conf. Bound Upper -0.024, Lower -0.243.
    - US industrial production: Mean -1.531, 90% Conf. Bound Upper 0.983, Lower -4.727.

*Source: https://www.imf.org/-/media/files/publications/wp/2017/wp17268.pdf*

### 0.13  percent, while US unconventional monetary policy increased Canadian output by 1.2   percent

### wp17268 - 0.13  percent, while US unconventional monetary policy increased Canadian output by 1.2   percent

### Introduction and objectives
- Study period: 1994–2016 (monthly data from August 1994 to December 2016).
- Focus: quantify effects of unconventional monetary policy (UMP) in a small open economy (Canada) and spillovers from foreign UMP (US).
- Key challenges addressed:
  - Small open economies respond strongly to foreign shocks → capture international spillovers mostly via US variables.
  - No variation in the short-term policy rate at the ZLB → use shadow interest rates to measure policy stance uninterrupted by the ZLB.

### Definitions and key parameters
- Zero lower bound (ZLB) defined as 25 bps.
- Shadow rate framework: observed nominal rate rt = max{sr t, r} = sr t + max{0, r − sr t} (option value of currency).
- Interpretation: movements of the shadow rate below 25 bps are interpreted as unconventional monetary policy; counterfactuals holding shadow rate at 25 bps attribute below-ZLB movements to UMP.
- Counterfactual interpretation: results interpreted as an upper bound on the effect of unconventional monetary policy.

### Measuring unconventional monetary policy / Shadow rates
- Canadian shadow rate estimated using a shadow rate term structure model (SRTSM) following Wu and Xia (2016) methodology; US shadow rate taken from Wu and Xia (2016).
- Assumed ZLB at 25 bps for both Canada and the US.
- Estimated minima:
  - Canadian shadow rate reached a low of approximately -0.34 percent in November 2009.
  - US shadow rate reached a low of -2.99 percent (decreasing from 2008 through 2014).
- Fed policy context: federal funds rate at the ZLB (0–25 bps) from November 2008; Fed moved to 50 bps on December 16, 2015.
- Bank of Canada policy context: target rate lowered from 50 bps to 25 bps on April 21, 2009 (conditional commitment), conditional commitment removed April 20, 2010, and target rate raised to 50 bps on June 1, 2010.

### Data and model specification (Small open economy B-SVAR)
- Model: two-country Bayesian structural VAR (B-SVAR) with block-exogeneity restrictions (Canada cannot influence US block).
- Canadian block variables (main specification): policy rate r; exchange rate s (CAD/USD); seasonally-adjusted real Canadian industrial production y; Canadian CPI p; seasonally-adjusted Canadian real exports ex; seasonally-adjusted Canadian real imports im. Net exports computed post-estimation.
- US block variables: US policy rate r*; VIX; commodity export prices wpx; US CPI p*; seasonally-adjusted real US industrial production y*.
- Policy rate construction: splice target rates with shadow rates whenever shadow rates are below 25 bps.
- Estimation details:
  - Lags: 12.
  - Gibbs sampler: 5000 draws, discarding first 1000.
  - Prior: Sims and Zha (1998) with μi = 10 for i = 1,...,6 for lag decay.
  - Contemporaneous restrictions: 111 total restrictions; exact identification would require 55 (n(n−1)/2), hence model is over-identified.
  - Joint likelihood-ratio test for 385 restrictions (over-identification and block exogeneity) fails to reject block exogeneity; likelihood-ratio statistic = 419.01, 5 percent critical value = 431.75.

### Impulse response findings (monetary policy shocks)
- Canadian 25 bps expansionary shock:
  - Raises industrial production and prices significantly on impact and for approximately 18 to 24 months.
  - Currency depreciates on impact and for about one year (significant at the 68 percent level).
  - Temporary rise in exports (about 6 months after shock) and imports → income effect; net exports impulse response slightly negative and significant at the 68 percent level.
- US 25 bps expansionary shock:
  - US industrial production rises with a delay of approximately 1–2 years.
  - US prices fall on impact then rise slightly after about 6 months.
  - Commodity prices rise on impact; VIX falls significantly.
  - Strong spillovers to Canada:
    - Canadian policy rate responds on impact and for over two years.
    - Canadian industrial production rises after about 6 months and again following US dynamics.
    - Canadian prices fall in the first month and then rise after about 5 months.
    - Net exports show alternating exchange rate and income effects.

### Counterfactual experiments and quantified effects of UMP
- Counterfactual method: replace realized monetary policy shocks with shocks that constrain the shadow rate to 25 bps during ZLB episodes (separately for Canada and for the US).
- Main quantified results (historical decomposition):
  - Canadian unconventional monetary policy: boosted Canadian output (industrial production) by approximately 0.131 percent on average during the ZLB episode.
  - US unconventional monetary policy: without US UMP, Canadian output (industrial production) would have been approximately 1.194 percent lower on average during the ZLB episode.
- Robustness: Canadian 0.131 percent result is robust to several different specifications; a combined counterfactual constraining both Canadian and US shadow rates at 25 bps produces results essentially unchanged from the US-only counterfactual due to the large relative impact of US UMP.

### Interpretation and implications
- Results demonstrate:
  - Effectiveness of domestic unconventional monetary policy in a small open economy (Canada).
  - Strong spillovers from foreign (US) unconventional monetary policy to Canada.
- Policy implication:
  - Under certain circumstances, and in particular when international monetary policy is coordinated, international spillovers from unconventional policies can be favorable for small open economies.
- Caveat:
  - Shadow-rate-based estimates cannot disentangle specific policy channels (e.g., LSAPs versus forward guidance) and may understate UMP effects if long-term asset holdings after December 2015 generated additional stimulus.

*Source: https://www.imf.org/-/media/files/publications/wp/2017/wp17268.pdf*

### 1.   Restricted monetary policy in Canada

### 1.   Restricted monetary policy in Canada

### Counterfactual construction and experiment
- Counterfactual scenario: restrict policy to be bound by 25 bps when the shadow rate was below 25 bps (2009–2010), interpreting the counterfactual path as an upper bound of the economy had the Bank of Canada not provided any additional stimulus beyond lowering the target rate to 25 bps.
- Visualization: Figure 4a shows the counterfactual path (red line) when the shadow rate is forced to be at least 25 bps; Figure 4e shows ZLB period comparisons (counterfactual vs observed).

### Estimated effects on industrial production and prices (ZLB period)
- Industrial production (level terms): would have been 0.13 percent lower on average during the Canadian ZLB period under the counterfactual.
- Prices (level terms): would have been 0.10 percent lower on average during the Canadian ZLB period under the counterfactual.
- Statistical significance: both values are significantly different than zero on average through the entire period based on 68 percent confidence bands.
- Cumulative effect on industrial production over the entire ZLB period: 1.828 percent lower.
- Annualized phrasing in source: "In annualized terms, the decline in output is 1.572 percent percent."

### Timing and persistence
- Recovery timing: industrial production and prices would have returned to their observed path within two years under the counterfactual.
- Interpretation: unconventional monetary policy operations conducted by the Bank of Canada sped up the recovery significantly but produced relatively short-term or transitory effects.
- Limitation noted: the brevity of the ZLB period in Canada limits the extent to which unconventional monetary policy could affect the real economy, though observed and counterfactual paths are nonetheless significantly different.

### Confidence-band robustness notes
- Results hold with 90 percent confidence bands; these are displayed in Figure B.1 and Table B.1 of Appendix B (as reported in the source). 

*Source: wp17268 - 1.   Restricted monetary policy in Canada*

### 0.132 percent lower on average during the ZLB without Canadian unconventional monetary

### wp17268 - 0.132 percent lower on average during the ZLB without Canadian unconventional monetary policy

### B. Current Account
- Motivation:
  - Replace import and export variables with estimates of the current account to capture trade balance and international receipts and payments of income.
  - In Canada from 1994 to 2016, receipts of income from abroad accounted for about 9 percent of all current account receipts, and payments of income from abroad account for about 15 percent of current account payments, on average.
- Implementation:
  - Export and import variables in the main model are replaced with a single current account variable, ordered first in the vector of Canada variables, Y_CAN_t.
  - A monthly series is created by linearly interpolating quarterly values and the log is not taken because the current account can take negative values.
- Counterfactual experiment results (Table 4):
  - Canadian industrial production would have been about -0.147 percent lower on average during the Canadian ZLB without unconventional monetary policies from the Bank of Canada.
  - Canadian industrial production would have been 1.054 percent lower on average had the Fed not enacted any unconventional monetary policies.
  - The current account rises marginally in response to Canadian and US expansionary policy, consistent with evidence that unconventional monetary policy is associated with modest positive increases in current account balances.

### C. Alternative Specifications
- Two alternative specifications tested:
  - Unrestricted Bank of Canada contemporaneous reaction function (leave Bank of Canada contemporaneous reaction function completely unconstrained).
  - Use of derived Taylor Rule rates instead of US and Canadian shadow rates as proxies for monetary policy stance.
- Findings:
  - In the unrestricted Bank of Canada reaction function model the estimated impulse response functions are robust for main variables of interest (industrial production and prices).
  - Counterfactual exercises in the unrestricted model show Canadian industrial production would have been significantly lower without US and Canadian unconventional monetary policies.
  - Results from the Taylor Rule model are also highly robust to the main model.

### VI. Conclusion
- Methodology:
  - Use shadow rates (Wu and Xia, 2016) as a proxy for monetary policy at the ZLB.
  - Construct a block-exogenous B-SVAR model to allow for international policy spillovers, studying Canada as a small open economy with the US as the adequate control for the outside world.
- Main empirical findings (1994–2016):
  - Both Canadian and US expansionary monetary policy shocks were associated with increased industrial production and prices in Canada.
  - Without the Bank of Canada’s unconventional policies, industrial production would have been 0.131 percent lower on average during the Canadian ZLB period.
  - Without the Fed’s unconventional monetary policies, over the July 2007– December 2015 period:
    - US industrial production would have been approximately 1.535 percent lower on average in the US.
    - Canadian industrial production would have been approximately 1.194 percent lower on average in Canada.
  - The relatively small impact of Bank of Canada unconventional policies is consistent with the limited unconventional intervention by the Bank of Canada during the ZLB period (no quantitative easing).
  - Without the Fed’s unconventional monetary policies, both Canadian and US industrial production would have been on different paths at the end of 2016.
- Policy implications and suggestions for future research:
  - Unconventional monetary policy in a small open economy is effective, but favorable foreign monetary policy spillovers are important.
  - Discussion on international coordination of monetary policy is rekindled by concerns about cross-border spillovers of unconventional monetary policies.
  - In some cases spillovers may have beneficial effects on economic recovery; the Bank of Canada ceased UMP operation long before the Fed yet still benefited from spillovers.
  - Future research should quantify the costs and benefits of spillovers from monetary policy normalization or reversal, which may not have symmetric impacts on small open economies.

### Key quantitative results and robustness checks (selected exact figures from counterfactual tables and model variants)
- Main model counterfactual averages (Table 2):
  - Canadian industrial production average percent difference under Canadian ZLB Imposed: -0.131 (68% Conf. Bound Upper: -0.064; Lower: -0.198).
  - Canadian industrial production average percent difference under US ZLB Imposed: -1.194 (68% Conf. Bound Upper: -0.215; Lower: -2.283).
  - US industrial production average percent difference under US ZLB Imposed: -1.535 (68% Conf. Bound Upper: -0.009; Lower: -3.337).
- Government Expenditure Model (Table 3):
  - Canadian industrial production average percent difference under Canadian ZLB Imposed: -0.132 (68% Conf. Bound Upper: -0.062; Lower: -0.203).
  - Canadian industrial production average percent difference under US ZLB Imposed: -1.054 (68% Conf. Bound Upper: -0.167; Lower: -1.993).
  - US industrial production average percent difference under US ZLB Imposed: -1.169 (68% Conf. Bound Upper: 0.255; Lower: -2.774).
- Current Account Model (Table 4):
  - Canadian industrial production average percent difference under Canadian ZLB Imposed: -0.147 (68% Conf. Bound Upper: -0.077; Lower: -0.219).
  - Canadian industrial production average percent difference under US ZLB Imposed: -1.054 (68% Conf. Bound Upper: -0.184; Lower: -2.02).
  - US industrial production average percent difference under US ZLB Imposed: -1.591 (68% Conf. Bound Upper: 0.003; Lower: -3.432).
- Selected unit and cumulative differences (preserved as reported in Tables 2–4):
  - Table 2 cumulative percent difference for Canadian industrial production under Canadian ZLB Imposed: -1.828 (68% Conf. Bound Upper: -0.902; Lower: -2.772).
  - Table 2 cumulative percent difference for Canadian industrial production under US ZLB Imposed: -101.51 (68% Conf. Bound Upper: -18.278; Lower: -194.095).
  - Table 4 cumulative unit difference for Canadian industrial production under Canadian ZLB Imposed: -20.281 (68% Conf. Bound Upper: -10.606; Lower: -30.117).
  - Table 4 cumulative unit difference for US industrial production under US ZLB Imposed: -1484.171 (68% Conf. Bound Upper: -8.913; Lower: -3190.650).

*Source: wp17268 - 0.132 percent lower on average during the ZLB without Canadian unconventional monetary policy (PDF).*

### REFERENCES

### REFERENCES

### Bibliography (selection)
- Lists academic and policy literature on unconventional monetary policy, shadow rates, international spillovers, and estimation methods. Key cited authors and works include:
  - Alpanda, Sami, and Serdar Kabaca, 2015, “International spillovers of large-scale asset purchases,” Journal of the European Economic Association.
  - Anzuini, Alessio, Marco J Lombardi, and Patrizio Pagano, 2013, “The impact of monetary policy shocks on commodity prices,” International Journal of Central Banking, Vol. 9, No. 3, pp. 119–144.
  - Bauer, Michael D, and Christopher J Neely, 2014, “International channels of the Fed’s unconventional monetary policy,” Journal of International Money and Finance, Vol. 44, pp. 24–46.
  - Bernanke, Ben S, and Vincent R Reinhart, 2004, “Conducting monetary policy at very low short-term interest rates,” The American Economic Review, Vol. 94, No. 2, p. 85.
  - Wu and Xia (2016) as the source for US shadow rate estimates and methodology.
  - Holston, Kathryn, Thomas Laubach, and John C Williams, 2017, “Measuring the natural rate of interest: International trends and determinants,” Journal of International Economics.
  - Many other working papers, Bank of Canada and Federal Reserve research staff reports, and Journal articles addressing LSAPs, term structure modeling, VAR identification, and international spillovers.

### Methodology and Estimation References
- Term structure and shadow-rate literature:
  - Black (1995), Krippner (2012, 2013), Christensen and Rudebusch (2014), Wu and Xia (2016).
- Structural VAR, Bayesian methods, and identification:
  - Sims and Zha (1998, 1999), Waggoner and Zha (2003).
- Taylor Rule and policy-rule literature:
  - Taylor (1993, 1999, 2013); Federal Reserve Bank of Atlanta’s Taylor Rule Utility referenced for prescribed Taylor Rule rates.
- Empirical evidence on international spillovers and quantitative easing:
  - Gagnon et al. (2010), Fratzscher, Lo Duca and Straub (2013, 2014), Neely (2015), Bauer and Rudebusch (2013), Krishnamurthy and Vissing-Jorgensen (2011, 2013), Eichengreen and Gupta (2014).

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### APPENDIX A. DATA

- Table A.1 lists data sources and definitions used in the analysis:
  - ca: CANSIM, author’s calculation — Current account balance, seasonally adjusted. Quarterly series linearly interpolated to monthly frequency.
  - g: CANSIM, author’s calculations — General federal governments final consumption expenditure, seasonally adjusted at annual rates, chained 2007 dollars. Quarterly series linearly interpolated to monthly frequency.
  - g_US: Federal Reserve of St. Louis Economic Data (FRED) — Federal government final consumption expenditures, seasonally adjusted at annual rates, chained 2009 dollars. Quarterly series linearly interpolated to monthly frequency.
  - p: CANSIM — Consumer Price Index, All items, Index, Monthly, seasonally adjusted.
  - p_US: FRED — Consumer Price Index, All items, Index, Monthly, seasonally adjusted.
  - r: Bank of Canada, author’s calculations — Bank of Canada target rate spliced with shadow rate at ZLB when shadow rate < target rate = 25 bps, Monthly.
  - r_US: FRED, Wu and Xia (2016) — Federal funds target rate spliced with Wu and Xia (2016) shadow rate at ZLB when SSR < FFR = 25 bp, Monthly.
  - s: IFS — National Currency per US Dollar, Rate, Monthly average.
  - vix: CBOE — Chicago Board Options Exchange (CBOE) VIX index measuring market’s expectation of 30-day volatility. Constructed using the implied volatilities of a range of S&P 500 index options, Monthly average.
  - wxp: IFS — Export Price, All Commodities, Index, Monthly average.
  - y: IFS — Industrial Production, Seasonally adjusted, Index, Monthly.
  - y_US: IFS — Industrial Production, Seasonally adjusted, Index, Monthly.
  - ex,im: CANSIM — International merchandise trade, by Standard International Trade Classification, customs-based price index, monthly.

---

### APPENDIX B. ALTERNATIVE CONFIDENCE BANDS

- Counterfactual experiments imposing ZLB (25 bps) for Canada and US:
  - Figures B.1 and B.2 plot observed series and counterfactual paths with 90 percent confidence bands constructed from 5000 Gibbs sampling draws from the posterior distribution at the 90 percent confidence level.
  - Variables presented include interest rates, exchange rates ($ CDN/US), imports, exports, industrial production, price levels, VIX, commodity prices, and net exports.
- Table B.1. Average Counterfactual Path for ZLB (90 % confidence band) reports:
  - Percent Difference panel (selected entries):
    - Canadian interest rate: Mean 0.24, 90% Conf. Bound Upper 0.24, Lower 0.24.
    - Canadian CPI: Mean -0.1, 90% Conf. Bound Upper -0.017, Lower -0.19.
    - Canadian industrial production: Mean -0.132, 90% Conf. Bound Upper -0.024, Lower -0.243.
    - US interest rate: Mean 1.48, 90% Conf. Bound Upper 1.48, Lower 1.48.
    - US CPI: Mean 0.257, 90% Conf. Bound Upper 1.726, Lower -1.004.
    - US industrial production: Mean -1.531, 90% Conf. Bound Upper 0.983, Lower -4.727.
  - Unit Difference panel (selected entries):
    - Canadian CPI: Mean -1.156, 90% Conf. Bound Upper -0.201, Lower -2.190.
    - Canadian industrial production: Mean -1.293, 90% Conf. Bound Upper -0.235, Lower -2.386.
    - US CPI: Mean 5.989, 90% Conf. Bound Upper 40.396, Lower -23.526.
    - US industrial production: Mean -16.810, 90% Conf. Bound Upper 10.567, Lower -51.668.
- Note: Calculations use formulas:
  - percent difference = (1 / ˜T) ∑_{t ∈ ZLB} (y_t − y^CF_t) / y_t
  - unit difference = (1 / ˜T) ∑_{t ∈ ZLB} (y_t − y^CF_t)
  - ZLB denotes the time period for the experiment and ˜T the number of observations during the ZLB.
  - When max and min bounds include 0, paths may still be significantly different from zero for individual months.

---

### APPENDIX C. GOVERNMENT EXPENDITURE MODEL

- Impulse response findings (Figures C.1–C.4; 90% and 68% confidence bands reported):
  - Canadian expansionary shock (25 bps):
    - Industrial production and prices rise.
    - Canadian dollar depreciates on impact.
    - Government spending shows no significant change.
    - Net exports responses plotted (calculated ex-post estimation).
  - US expansionary shock:
    - Canadian and US industrial production rise significantly after a short delay.
    - Canadian government spending doesn’t change significantly.
    - US government spending shows a significant temporary drop as monetary policy expands.
- Counterfactual paths imposing ZLB shown for both Canadian and US ZLB experiments (Figures C.5 and C.6) for variables including interest rates, exchange rates ($ CDN/US), imports, exports, price levels, industrial production, government expenditures ($ Mil for gov’t expenditures series), and net exports.

---

### APPENDIX D. CURRENT ACCOUNT MODEL

- Impulse responses for 25 bps expansionary monetary policy shocks:
  - Canadian monetary policy shock (Figure D.1):
    - Price level and industrial production rise.
    - Initial exchange rate depreciation is no longer statistically significant.
    - No significant impact on the Canadian current account.
  - US monetary policy shock (Figure D.2):
    - Prices fall significantly on impact very temporarily, then rise after about 5 months.
    - US industrial production rises; Canadian industrial production rises after approximately a two year lag.
    - Canadian currency depreciates as Canadian policy rate falls (not significantly).
    - Significant rise in the Canadian current account balance, suggesting a spillover via an income effect from rising US income receipts flowing to Canada.
- Counterfactual paths imposing ZLB for Canadian and US experiments reported in Figures D.3 and D.4 for interest rates, exchange rates ($ CDN/US), current account, price levels, and industrial production.

---

### APPENDIX E. ALTERNATIVE SPECIFICATIONS

E.1 Alternative Bank of Canada Reaction Function
- Results for specification with Bank of Canada’s reaction function unconstrained:
  - Impulse responses and counterfactuals reported (Figures E.1–E.4).
  - Variables include interest rates, exchange rate, price level, industrial production, imports, exports, net exports, VIX, commodity prices, and US variables where applicable.

E.2 Using a Taylor Rule
- Taylor Rule construction:
  - Prescribed FFR: ˆFFR_t = ρ FFR_{t−1} + (1−ρ)[(r*_t + π*_t) + 1.5(π_t − π*_t) + β gap_t].
  - π* set to 2%.
  - β (weight on output gap) assumed 0.5.
  - ρ (interest rate smoothing) assumed 0.
  - Natural rate r*_t taken from Holston, Laubach, and Williams (2017).
  - Output gap: Congressional Budget Office’s real GDP gap for US; Bank of Canada’s published output gap for Canada.
  - Taylor Rule rates calculated quarterly then linearly interpolated to monthly.
- Comparisons and results:
  - Figure E.5 shows Taylor Rule rates and actual policy rates for Canada and US (1994–2016); Taylor Rule rates do not follow observed rates closely, particularly for Canada, but do fall below ZLB during parts of unconventional regimes.
  - Policy rate constructed using the prescribed Taylor Rule rate when the official rate is at ZLB and prescribed Taylor Rule rate is below ZLB; otherwise set to 25 bps where prescribed > 25 bps but official = 25 bps.
  - Impulse responses using Taylor Rule (Figures E.6–E.7) broadly consistent with main model:
    - Canadian monetary policy shock: significant rise in price level and industrial production, depreciation of currency, slight fall in net exports.
    - US monetary policy shock: strong spillovers to Canada; Canadian prices rise after about six months and industrial production after about 25 months.
  - Counterfactuals and averages reported (Figures E.8–E.9; Table E.1).
- Table E.1. Average Counterfactual Path for ZLB, Taylor Rule (selected entries):
  - Percent Difference panel:
    - Canadian Interest rate: Mean 0.39, 68% Conf. Bound Max 0.39, Min 0.39.
    - Canadian CPI: Mean -0.204, 68% Conf. Bound Max -0.038, Min -0.373.
    - Canadian industrial production: Mean -0.194, 68% Conf. Bound Max 0.011, Min -0.400.
    - US interest rate: Mean 0.771, 68% Conf. Bound Max 0.771, Min 0.771.
    - US CPI: Mean -0.420, 68% Conf. Bound Max 0.532, Min -1.487.
    - US industrial production: Mean -1.193, 68% Conf. Bound Max 0.424, Min -3.709.
  - Cumulative Percent Difference and Unit Difference panels contain large cumulative effects for industrial production (e.g., cumulative percent and unit differences reported, including values like -2.721, -145.558, -101.376, -315.301, and unit figures such as -26.672, -1575.004, -1116.936).
- Note: Table E.1 references “See Table 3 of paper.”

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### APPENDIX F. SHADOW RATES

- Shadow rate term structure model (SRTSM) overview:
  - Observed nominal rate r_t = max{sr_t, r} = sr_t + max{0, r − sr_t} (equation (4)), where sr_t is the unobserved shadow rate and r the effective lower bound.
  - Non-linearity from the max operator implies no analytical solution in general; Wu and Xia (2016) provide an approximate tractable solution based on the yield curve and the probability shadow rate falls below the ELB.
- Canadian SRTSM implementation:
  - Canadian shadow rate modeled as affine in state variables: sr_t = δ_0 + δ'_1 X_t (equation (5)).
  - State variables X_t are forward rates derived from Bank of Canada zero-coupon bond yield data for 3 month, 6 month, 1, 2, 5, 7, and 10 year maturities.
  - Transition under P: X_{t+1} = μ + ρ X_t + Σ ε_{t+1}, ε_{t+1} ∼ N(0, I) (equation (6)).
  - Stochastic discount factor and price of risk λ_t are affine in factors (equation (7)); risk-neutral dynamics under Q follow VAR(1) (equation (8)) with parameter relations μ − μ^Q = Σ λ_0 and ρ − ρ^Q = Σ λ_1.
  - Approximate forward rate in SRTSM: f^{SRTSM}_{n,n+t,t} = r + σ^Q_n g(a_n + b'_n X_t − r / σ^Q_n) (equation (10)), where g(z) = z Φ(z) + φ(z).
  - Definitions for a_n, b'_n, and (σ^Q_n)^2 given explicitly in the appendix (see equations following (10)).
  - Measurement equation: f^o_{n,n+t,t} = r + σ^Q_n g(a_n + b'_n X_t − r / σ^Q_n) + η_{nt}, η_{nt} ∼ N(0, ω) (equation (11)).
  - Identification and estimation:
    - Three-factor model assumed.
    - Normalizing restrictions: δ = [1, 1, 0]′; μ^Q = 0; Σ lower triangular.
    - ρ^Q assumed real Jordan form with eigenvalues in descending order.
    - Model estimated with the extended Kalman filter using forward-rate observations.
- Notes on alternatives:
  - Some models use quadratic or square-root short-rate diffusions to avoid negative rates, but these treat ZLB as a reflecting barrier; SRTSM treats ZLB as absorbing with option-value term.
  - Wu and Xia (2016) method chosen for US shadow rate; analogous approach applied to Canadian data.

*Source: wp17268 - REFERENCES (IMF Working Paper Appendices and References).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2017/wp17268.pdf_