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### I. Introduction and Research Objective
- Research question: Do corporate income tax cuts stimulate business investment, job creation, and economic growth?
- Goal: Estimate the long-run user cost elasticity of demand for capital (UCE) using an error correction model (ECM) applied to Canadian panel data varying by industry, year, and province.
- Identification advantages:
  - Canada treated as a small open economy; real interest rate and price of machinery largely determined in the U.S.
  - Exploits intertemporal (2001–2004 and later tax reforms), inter-industry, and inter-provincial variation in tax policies and capital stocks.
- Data dimensions: 10 provinces × 7 industries × 17 years (1997–2013) → cross-section of 70 industry-province pairs and 17 years.
- Estimation method: two-step Arellano-Bond System-GMM with collapsed instrument matrix and Windmeijer (2005) robustness correction.

### II. Theory and Key Definitions
- User cost of capital (C) identity:
  - C = q(1 + t_S)(r_f + δ)[(1 − φ) + τ(1 − u)/(δ + r_f + π)]/(1 − u)
- Definitions and decompositions:
  - q = p_K/p; δ = economic depreciation rate; r_f = real cost of corporate funds; t_S = provincial sales tax on capital goods; τ = capital-based tax rate; u = statutory corporate income tax rate; φ = reduction from tax shields (capital cost allowances and investment tax credits).
  - φ = uZ + κ
  - Z = (1 − κ) × α(1 + r_f + π)/(α + r_f + π)
  - κ = investment tax credit rate; α = tax depreciation rate.
  - r_f = βi(1 − u) + (1 − β)η − π
  - Net-of-depreciation user cost: C_n = C − qδ
  - METR: METR = (C_n − q r*)/C_n, where r* = βi + (1 − β)η − π (world real net rate in small open economy)
  - User cost in METR terms: C = q(r* + δ) + q r* (METR/(1 − METR))
  - Percentage change of user cost with respect to METR (c ≡ ln C): dc/dMETR = 1/(1 − METR) · 1/(1 + (1 − METR)(δ/r*))
- Benchmark structural relationship for desired capital:
  - K* = γ Y^ζ C^−UCE
- Bewley-transformed ECM (level form used for estimation):
  - k_i,j,t = ψ_0 − ψ_1 ∆k_i,j,t + ψ_2 y_i,j,t − ψ_3 c_i,j,t − ∑_{h=0}^{H_y−1} ψ_4h ∆y_i,j,t−h + ∑_{h=0}^{H_c−1} ψ_5h ∆c_i,j,t−h + υ_i,j,t
  - UCE = −ψ_3
  - METR semi-elasticity (MSE) = −ψ_3 × (dc/dMETR)

### III. Data and Measurement
- Sample: 1997–2013, 10 provinces, 7 industries (manufacturing, construction, communication, transportation, utilities, wholesale and retail trade, other services — forestry and other services excluded from sample construction reasons).
- Capital decomposition: 15 asset classes of machinery and equipment (M&E); 3 asset classes of non-residential construction (NRC).
- METR construction: Duanjie Chen and Jack Mintz METR model; capital weights from Department of Finance; provincial sales tax shares by industry-province; asset-specific economic depreciation rates from Statistics Canada; federal and provincial investment tax credits from KPMG Tax Factsbooklets and Canadian Tax Foundation sources.
- Price and rate inputs:
  - r* value held fixed at 0.0349 in METR construction available to authors.
  - Alternative robustness: imported M&E price index (varies by industry, not province).
- Missing data: 108 missing observations (communications or utilities, mostly Atlantic provinces).
- Representative shares: M&E represents 28 percent and NRC 72 percent of combined M&E and NRC capital stock in sample construction context (descriptive proportions used within text).

### IV. Time Series Properties and Cointegration
- Unit root testing applied to log capital stocks, user cost, and GDP:
  - Tests: Harris-Tzavalis (1999), Im-Pesaran-Shin (2003), Hadri (2000).
  - Results indicate non-stationarity (unit roots) in variables.
- Panel cointegration (Westerlund, 2007): mixed evidence
  - For M&E: group and panel tests provide evidence of panel cointegration (G_α and P_α marginally significant at about the 12 percent level).
  - For NRC: weaker evidence; some tests insignificant except P_τ suggesting cointegration in some panels.
- Implication: cointegration supports using an ECM to estimate long-run relationships.

### V. Empirical Estimation Strategy
- Challenge: dynamic panel bias from lagged dependent variable and fixed effects (Nickell, 1981).
- Estimator: System-GMM (Arellano-Bover/Blundell-Bond).
  - Instruments: lagged levels for differenced equations; differences for level equations; instrument matrix collapsed; minimal instrument count; Windmeijer (2005) correction.
- Diagnostic tests used:
  - Hansen (1982) J-test for instrument validity.
  - AR(2) test for second-order serial correlation.
  - Difference-in-Hansen test for additional system-GMM instruments.

### VI. Main Empirical Findings — Machinery & Equipment (M&E)
- ECM (Bewley-transformed) baseline results (Table 5):
  - UCE estimates range from −1.078 to −1.312 (columns 1–3).
  - GDP elasticities range from 0.568 to 0.621.
  - Dynamic adjustment terms: differenced GDP terms mostly negative (theoretically expected); differenced UCC terms wrong-signed but statistically insignificant.
- Robustness using imported M&E price index (columns 4–6):
  - UCE magnitudes increase (in absolute value) to between about 1.6 and 1.8.
  - GDP elasticities decrease to between about 0.3 and 0.5.
- Diagnostics:
  - Observations: 960, 896, 832 (depending on column).
  - Number of Cross-Sections: 64.
  - Number of Instruments: 40, 39, 38.
  - AR(1) P-Values range listed (e.g., 0.787, 0.357, 0.324).
  - AR(2) P-Values range listed (e.g., 0.844, 0.170, 0.339).
  - Hansen Test P-Values: 0.819, 0.728, 0.414.
  - Difference-in-Hansen Test P-Values: 0.952, 0.916, 0.284.
- METR semi-elasticity (MSE) calculations (Table 6):
  - Using UCE = −1.312.
  - METR values for M&E: 27.42 percent (sample mean) and 10.95 percent (most recent year in sample).
  - Resulting MSEs: −0.324 and −0.223, respectively.
  - Interpretation: a 1 percentage point increase in METR reduces capital stock by 100×MSE percent.
  - Example: a 5 percentage point decrease in METR (e.g., 15 percent → 10 percent) would in the long run increase the M&E stock by about 1.0 percent when evaluated at an METR of 11 percent (text example uses 100×(−0.05)×(−0.22) = 1.1 percent).
  - Example mapping statutory rate to METR: a hypothetical five percentage point reduction in federal statutory CIT rate in 2009 (19 percent → 14 percent) would translate into a 3.9 percentage point reduction in the aggregate METR.

### VII. Alternative Estimates — Distributed Lag Model (DLM) in First Differences
- Motivation: differencing addresses non-stationarity but may omit long-run information.
- DLM results for M&E (Table 7):
  - UCE estimates (sum of coefficients on ∆UCC) between −0.098 and −0.424 across specifications.
  - Column 2 best-fit: UCE ≈ −0.4 (significant at 10 percent).
  - Robustness with imported price series yields similar conclusions.
  - Conclusion: DLM in first differences underestimates UCE compared to ECM estimates (corroborates Dwenger (2014)).

### VIII. Non-Residential Construction (NRC) — Results and Interpretation
- Empirical outcome:
  - UCE estimates for NRC are statistically insignificantly different from zero across ECM and DLM specifications (Table A1).
  - Some specifications show coefficients with wrong signs and poor fit.
- Possible explanations:
  - Lack of secular decline in price of NRC investments (unlike IT/M&E) reduces identification power in cointegration framework.
  - Less inter-provincial and inter-industry variation in METR for NRC compared to M&E.
  - METR construction limitations for NRC:
    - Time-to-build for owner-built structures not incorporated.
    - Long-lived nature of NRC increases sensitivity to uncertainty; METR under irreversibility differs from conventional METR.
- Recommendation for future research: modify METR calculations for NRC to incorporate time-to-build and uncertainty; importance of disaggregating M&E vs NRC in policy impact analysis.

### IX. Key Quantitative Results and Summary Statistics
- Sample period: 1997–2013.
- Cross-section: 70 industry-province pairs (7 industries × 10 provinces).
- UCE for M&E (ECM baseline): range −1.078 to −1.312.
- UCE for M&E (robustness, imported price): about −1.6 to −1.8 (in absolute value).
- UCE for M&E (DLM in first differences): ≈ −0.4 (lower magnitude).
- METR values used in MSE examples:
  - 27.42 percent (sample mean for M&E).
  - 10.95 percent (most recent year in sample for M&E).
  - METR used in METR construction had r* fixed at 0.0349.
- Implied METR semi-elasticities for M&E:
  - MSE ≈ −0.324 at METR = 27.42 percent.
  - MSE ≈ −0.223 at METR = 10.95 percent.
- Aggregate implications (illustrative):
  - Using elasticity −1.3 for M&E and zero for NRC implies weighted-average UCE ≈ −0.4 for combined M&E and NRC, or overall semi-elasticity ≈ −0.06.
  - If applied to all physical capital, a reduction in aggregate METR from 28.0 percent in 2009 to 20.0 percent in 2015 may have increased the national capital stock by 0.5 percent in the long run, holding other influences constant (calculation presented in text).
- Summary statistics (selected; time period 1997–2013):
  - Building and Engineering (NRC) Capital Stock: Obs: 1088; Mean: 6074.79; Std Dev.: 708.51; Min: 9.75; Max: 7683.6.
  - Machinery and Equipment (MEq) Capital Stock: Obs: 1088; Mean: 2397.14; Std Dev.: 422.91; Min: 0.23; Max: 2195.8.
  - METR for NRC: Obs: 1190; Mean: 0.35; Std Dev.: 0.13; Min: -0.05; Max: 0.56.
  - METR for MEq: Obs: 1190; Mean: 0.35; Std Dev.: 1.08; Min: -4.70; Max: 14.39.
  - Relative input/output Price for Building Engineering: Obs: 1190; Mean: 1.10; Std Dev.: 0.14; Min: 0.65; Max: 1.68.
  - Relative input/output Price for Machinery and Equipment: Obs: 1190; Mean: 0.87; Std Dev.: 0.21; Min: 0.40; Max: 1.44.
  - Relative imported input/output Price for Machinery and Equipment: Obs: 1190; Mean: 0.81; Std Dev.: 0.18; Min: 0.45; Max: 1.24.
  - Real Interest rate: Obs: 1190; Mean: 2.43; Std Dev.: 1.19; Min: 0.42; Max: 4.13.
  - Real GDP: Obs: 1190; Mean: 7682129242697938.

### X. Policy-Relevant Conclusions and Recommendations
- Main conclusion: Machinery and equipment investment is sensitive to business tax rates; ECM estimates indicate a UCE for M&E of about −1.3 and an METR semi-elasticity of about −0.2 at a weighted-average METR of 11 percent.
- Interpretation for tax policy:
  - Corporate tax reductions and METR reductions can generate positive, long-run increases in physical capital, particularly M&E.
  - Estimated magnitudes imply modest but non-negligible long-run effects (e.g., a 5 percentage point METR cut ~ 1.0–1.1 percent rise in M&E stock in examples).
- Caveats and policy considerations:
  - NRC responses to tax changes are empirically weak or ambiguous; policy effects may differ substantially across asset types.
  - Efficient allocation of capital depends on both the level of METR and its dispersion across industries, provinces, and asset types.
- Research agenda:
  - Improve METR calculations for NRC to incorporate time-to-build and uncertainty.
  - Further disaggregation of capital types to better capture heterogeneous tax responsiveness.

*Italic: Source: wpiea2020077-print-pdf - References (Content extracted from the specified PDF chapter).*

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

### wpiea2020077-print-pdf - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### I. Introduction and Research Objective
- Research question: Do corporate income tax cuts stimulate business investment, job creation, and economic growth?
- Goal: Estimate the long-run user cost elasticity of demand for capital (UCE) using an error correction model (ECM) applied to Canadian panel data varying by industry, year, and province.
- Identification advantages:
  - Canada treated as a small open economy; real interest rate and price of machinery largely determined in the U.S.
  - Exploits intertemporal (2001–2004 and later tax reforms), inter-industry, and inter-provincial variation in tax policies and capital stocks.
- Data dimensions: 10 provinces × 7 industries × 17 years (1997–2013) → cross-section of 70 industry-province pairs and 17 years.
- Estimation method: two-step Arellano-Bond System-GMM with collapsed instrument matrix and Windmeijer (2005) robustness correction.

### II. Theory and Key Definitions
- User cost of capital (C) identity:
  - C = q(1 + t_S)(r_f + δ)[(1 − φ) + τ(1 − u)/(δ + r_f + π)]/(1 − u)
  - q = p_K/p; δ = economic depreciation rate; r_f = real cost of corporate funds; t_S = provincial sales tax on capital goods; τ = capital-based tax rate; u = statutory corporate income tax rate; φ = reduction from tax shields (capital cost allowances and investment tax credits).
- φ decomposition:
  - φ = uZ + κ
  - Z = (1 − κ) × α(1 + r_f + π)/(α + r_f + π)
  - κ = investment tax credit rate; α = tax depreciation rate.
- r_f definition:
  - r_f = βi(1 − u) + (1 − β)η − π
  - β = debt share; i = nominal interest rate on debt; η = required nominal return on equity; π = inflation rate.
- Net-of-depreciation user cost:
  - C_n = C − qδ
- Marginal Effective Tax Rate (METR):
  - METR = (C_n − q r*)/C_n
  - r* = βi + (1 − β)η − π (world real net rate in small open economy)
- User cost expressed in terms of METR:
  - C = q(r* + δ) + q r* (METR/(1 − METR))
- Percentage change of user cost with respect to METR (with c ≡ ln C):
  - dc/dMETR = 1/(1 − METR) · 1/(1 + (1 − METR)(δ/r*))
- Benchmark structural relationship for desired capital:
  - K* = γ Y^ζ C^−UCE
- Bewley-transformed ECM (level form used for estimation):
  - k_i,j,t = ψ_0 − ψ_1 ∆k_i,j,t + ψ_2 y_i,j,t − ψ_3 c_i,j,t − ∑_{h=0}^{H_y−1} ψ_4h ∆y_i,j,t−h + ∑_{h=0}^{H_c−1} ψ_5h ∆c_i,j,t−h + υ_i,j,t
  - UCE = −ψ_3
  - METR semi-elasticity (MSE) = −ψ_3 × (dc/dMETR)

### III. Data and Measurement
- Sample: 1997–2013, 10 provinces, 7 industries (manufacturing, construction, communication, transportation, utilities, wholesale and retail trade, other services — note forestry and other services were excluded from sample construction reasons).
- Capital decomposition: 15 asset classes of machinery and equipment (M&E); 3 asset classes of non-residential construction (NRC).
- METR construction: Duanjie Chen and Jack Mintz METR model; capital weights from Department of Finance; provincial sales tax shares by industry-province; asset-specific economic depreciation rates from Statistics Canada; federal and provincial investment tax credits from KPMG Tax Factsbooklets and Canadian Tax Foundation sources.
- Price and rate inputs:
  - r* value held fixed at 0.0349 in METR construction available to authors.
  - Alternative robustness: imported M&E price index (varies by industry, not province).
- Missing data: 108 missing observations (communications or utilities, mostly Atlantic provinces).
- Representative shares: M&E represents 28 percent and NRC 72 percent of combined M&E and NRC capital stock in sample construction context (note these proportions used descriptively within the text).

### IV. Time Series Properties and Cointegration
- Unit root testing: Harris-Tzavalis (1999), Im-Pesaran-Shin (2003), Hadri (2000) applied to log capital stocks, user cost, and GDP; results indicate non-stationarity (unit roots) in variables.
- Panel cointegration (Westerlund, 2007): mixed evidence
  - For M&E: group and panel tests provide evidence of panel cointegration (G_α and P_α marginally significant at about the 12 percent level).
  - For NRC: weaker evidence; some tests insignificant except P_τ suggesting cointegration in some panels.
- Implication: cointegration supports using an ECM to estimate long-run relationships.

### V. Empirical Estimation Strategy
- Challenge: dynamic panel bias from lagged dependent variable and fixed effects (Nickell, 1981).
- Estimator: System-GMM (Arellano-Bover/Blundell-Bond), instruments:
  - Lagged levels as instruments for first-differenced equations; differences as instruments for level equations.
  - Instrument matrix collapsed; minimal instrument count; Windmeijer (2005) correction.
- Diagnostic tests used:
  - Hansen (1982) J-test for instrument validity.
  - AR(2) test for second-order serial correlation.
  - Difference-in-Hansen test for additional system-GMM instruments.

### VI. Main Empirical Findings — Machinery & Equipment (M&E)
- ECM (Bewley-transformed) baseline results (Table 5):
  - UCE estimates range from −1.078 to −1.312 (columns 1–3).
  - GDP elasticities range from 0.568 to 0.621.
  - Dynamic adjustment terms: differenced GDP terms mostly negative (theoretically expected); differenced UCC terms wrong-signed but statistically insignificant.
  - Robustness using imported M&E price index (columns 4–6):
    - UCE magnitudes increase (in absolute value) to between about 1.6 and 1.8.
    - GDP elasticities decrease to between about 0.3 and 0.5.
  - Diagnostic tests (Hansen, AR(2), difference-in-Hansen) reported as satisfactory.
- METR semi-elasticity (MSE) calculations (Table 6):
  - Using UCE = −1.31 (from column 2 of Table 5).
  - METR values for M&E: 27.42 percent (sample mean) and 10.95 percent (most recent year in sample).
  - Resulting MSEs: −0.32 and −0.22, respectively.
  - Interpretation: a 1 percentage point increase in METR reduces capital stock by 100×MSE percent.
  - Example: a 5 percentage point decrease in METR (e.g., 15 percent → 10 percent) would in the long run increase the M&E stock by about 1.0 percent when evaluated at an METR of 11 percent (description in text uses 100×(−0.05)×(−0.22) = 1.1 percent as the calculation).
  - Example mapping statutory rate to METR: a hypothetical five percentage point reduction in federal statutory CIT rate in 2009 (19 percent → 14 percent) would translate into a 3.9 percentage point reduction in the aggregate METR.

### VII. Alternative Estimates — Distributed Lag Model (DLM) in First Differences
- Motivation: common alternative; differencing addresses non-stationarity but may omit long-run information.
- DLM results for M&E (Table 7):
  - UCE estimates (sum of coefficients on ∆UCC) between −0.098 and −0.424 across specifications.
  - Column 2 best-fit: UCE ≈ −0.4 (significant at 10 percent).
  - Robustness with imported price series yields similar conclusions.
  - Findings corroborate Dwenger (2014): DLM in first differences underestimates UCE compared to ECM estimates.

### VIII. Non-Residential Construction (NRC) — Results and Interpretation
- Empirical outcome:
  - UCE estimates for NRC are statistically insignificantly different from zero across ECM and DLM specifications (see Table A1).
  - Some specifications show coefficients with wrong signs and poor fit.
- Possible explanations:
  - Lack of secular decline in price of NRC investments (unlike IT/M&E) reduces identification power in cointegration framework.
  - Less inter-provincial and inter-industry variation in METR for NRC compared to M&E.
  - METR construction limitations for NRC:
    - Time-to-build for owner-built structures not incorporated.
    - Long-lived nature of NRC increases sensitivity to uncertainty; METR under irreversibility differs from conventional METR.
  - Recommendation for future research: modify METR calculations for NRC to incorporate time-to-build and uncertainty; importance of disaggregating M&E vs NRC in policy impact analysis.

### IX. Key Quantitative Results and Policy-Relevant Metrics
- Sample period: 1997–2013.
- Cross-section: 70 industry-province pairs (7 industries × 10 provinces).
- UCE for M&E (ECM baseline): range −1.078 to −1.312.
- UCE for M&E (robustness, imported price): about −1.6 to −1.8 (in absolute value).
- UCE for M&E (DLM in first differences): ≈ −0.4 (lower magnitude).
- METR values used in MSE examples:
  - 27.42 percent (sample mean for M&E).
  - 10.95 percent (most recent year in sample for M&E).
  - METR used in METR construction had r* fixed at 0.0349.
- Implied METR semi-elasticities for M&E:
  - MSE ≈ −0.32 at METR = 27.42 percent.
  - MSE ≈ −0.22 at METR = 10.95 percent.
- Aggregate implications (illustrative):
  - Using elasticity −1.3 for M&E and zero for NRC implies weighted-average UCE ≈ −0.4 for combined M&E and NRC, or overall semi-elasticity ≈ −0.06.
  - If applied to all physical capital, a reduction in aggregate METR from 28.0 percent in 2009 to 20.0 percent in 2015 may have increased the national capital stock by 0.5 percent in the long run, holding other influences constant (calculation presented in text).

### X. Policy-Relevant Conclusions and Recommendations
- Main conclusion: Machinery and equipment investment is sensitive to business tax rates; ECM estimates indicate a UCE for M&E of about −1.3 and an METR semi-elasticity of about −0.2 at a weighted-average METR of 11 percent.
- Interpretation for tax policy:
  - Corporate tax reductions and METR reductions can generate positive, long-run increases in physical capital, particularly M&E.
  - Estimated magnitudes imply modest but non-negligible long-run effects (e.g., a 5 percentage point METR cut ~ 1.0–1.1 percent rise in M&E stock in examples).
- Caveats and policy considerations:
  - NRC responses to tax changes are empirically weak or ambiguous; policy effects may differ substantially across asset types.
  - Efficient allocation of capital depends on both the level of METR and its dispersion across industries, provinces, and asset types.
- Research agenda:
  - Improve METR calculations for NRC to incorporate time-to-build and uncertainty.
  - Further disaggregation of capital types to better capture heterogeneous tax responsiveness.

*Italic: Source: wpiea2020077-print-pdf - References (Content extracted from the specified PDF chapter).*

### REFERENCES

### REFERENCES (wpiea2020077-print-pdf)

### Data sources and variables
- Marginal Effective Tax Rates (METR): percent — School of Public Policy of University of Calgary.
- Real Capital Stock: 2007 CAD — CANSIM 310002.
- Capital Input Prices: index (2002) — CANSIM 3840039.
- Machinery and Equipment Import Price: index (2002) — CANSIM 18100107.
- Output price components (indices, 2002) — CANSIM 3260021 and CANSIM 3290056 and CANSIM 810009 as listed for Utilities, Transportation, Construction, Retail, Communication, Manufacturing, Wholesale.
- Real GDP: Chained 2007 Dollars — CANSIM 3790030.
- Real Return on Long-Term Canadian Bonds: percent — CANSIM 1760043.
- Note: Wholesale data is available for the period 2002-2009 and the rest of the years were extrapolated using growth of CPI in the relevant years.

### Summary statistics (Table 2; time period 1997–2013)
- Building and Engineering (NRC) Capital Stock
  - Obs: 1088
  - Mean: 6074.79
  - Std Dev.: 708.51
  - Min: 9.75
  - Max: 7683.6
- Machinery and Equipment (MEq) Capital Stock
  - Obs: 1088
  - Mean: 2397.14
  - Std Dev.: 422.91
  - Min: 0.23
  - Max: 2195.8
- METR for NRC
  - Obs: 1190
  - Mean: 0.35
  - Std Dev.: 0.13
  - Min: -0.05
  - Max: 0.56
- METR for MEq
  - Obs: 1190
  - Mean: 0.35
  - Std Dev.: 1.08
  - Min: -4.70
  - Max: 14.39
- Relative input/output Price for Building Engineering
  - Obs: 1190
  - Mean: 1.10
  - Std Dev.: 0.14
  - Min: 0.65
  - Max: 1.68
- Relative input/output Price for Machinery and Equipment
  - Obs: 1190
  - Mean: 0.87
  - Std Dev.: 0.21
  - Min: 0.40
  - Max: 1.44
- Relative imported input/output Price for Machinery and Equipment
  - Obs: 1190
  - Mean: 0.81
  - Std Dev.: 0.18
  - Min: 0.45
  - Max: 1.24
- Real Interest rate
  - Obs: 1190
  - Mean: 2.43
  - Std Dev.: 1.19
  - Min: 0.42
  - Max: 4.13
- Real GDP
  - Obs: 1190
  - Mean: 7682129242697938
  - (Note: Time period is 1997–2013. All the variables, except real interest rate, vary by province and industry. The capital stock is defined as geometric (infinite) end-year net stock and lagged for one period, Kt=Kt-1.)

### Panel unit root tests (Table 3; p-values)
- Harris-Tzavalis (Panels contain unit roots): 1.00 for all listed series.
- Im-Pesaran-Shin (All panels contain unit roots):
  - values: 0.46, 0.87, 0.13, 0.02, 0.00 (as shown in the table).
- Hadri LM test (All panels are stationary): 0.000 for all listed series.
- Note: The table shows p-values. In all the tests, the mean of variables are subtracted and a time trend considered. All of the variables are in logs.

### Panel cointegration tests (Table 4; Capital Stocks, UCC and GDP)
- Group statistics (M&E and NRC reported separately in table layout):
  - G τ: value = -1.79, z-value = -3.10, p-value = 0.01 (first column context)
  - G α: value = -1.785, z-value = 5.91, p-value = 0.12
  - Panels P τ: value = -7.37, z-value = 0.09, p-value = 0.00
  - Panels P α: value = -1.09, z-value = 2.30, p-value = 0.11
- Note: Robust (bootstrapped ten thousands times) p-values are reported. One lead and one lag is included in all of the tests. The results are robust to adding a constant term and a time trend. All of the variables are in logs. Null hypothesis is “no co-integration exists.”

### Error Correction Model for Machinery and Equipment Capital Stock (Table 5; System GMM Arellano-Bond)
- Dependent variable: ∆log(Capital Stock for MEq). Clustered (by Province X Industry) standard errors in brackets. Capital stock and GDP treated as endogenous. Instruments constructed using up to seven lags following the last lag included.
- Key coefficient estimates (selected; Baseline and Robustness Check columns shown as columns 1–6)
  - ∆log(Capital Stock for MEq) t: -0.332 (col1), -2.419 (col2), 0.404 (col3), -0.501 (col4), -2.667 (col5), 0.301 (col6). Standard errors: [0.711], [2.071], [1.225], [0.583], [2.110], [1.140].
  - Log of UCC t: -1.078* (col1) with [0.592]; -1.312*** (col2) [0.345]; -1.206*** (col3) [0.416]; -1.686** (col4) [0.828]; -1.561*** (col5) [0.436]; -1.756*** (col6) [0.532].
  - Log of GDP t: 0.589* (col1) [0.300]; 0.621*** (col2) [0.189]; 0.568** (col3) [0.245]; 0.308 (col4) [0.338]; 0.540** (col5) [0.212]; 0.482* (col6) [0.285].
- Model diagnostics (selected)
  - Observations: 960, 896, 832, 960, 896, 832.
  - Number of Cross-Sections: 64 in all columns.
  - Number of Instruments: 40, 39, 38, 40, 39, 38.
  - AR(1) P-Value: 0.787, 0.357, 0.324, 0.953, 0.245, 0.670.
  - AR(2) P-Value: 0.844, 0.170, 0.339, 0.745, 0.175, 0.308.
  - Hansen Test (Joint Validity) P-Value: 0.819, 0.728, 0.414, 0.973, 0.868, 0.791.
  - Difference-in-Hansen Test of Exogeneity of Instrument Subsets (P-Value) / Hansen Test Excluding Group: 0.952, 0.916, 0.284, 0.983, 0.996, 0.771.
  - Difference (null H = exogenous): 0.241, 0.301, 0.486, 0.575, 0.316, 0.63.
- Significance notation:
  - ∗∗∗ p<0.01, ∗∗ p<0.05, ∗ p<0.1.
- Baseline Results display estimates for UCC computed with the main price index for machinery and equipment in Canada; Robustness Check displays estimates using a price index of imported machinery and equipment.

### METR semi-elasticities for Machinery & Equipment (Table 6)
- UCE = −1.312
- Average METR (27.42%)
  - Semi-elasticity (MSE): -0.324
- Recent METR (10.95%)
  - Semi-elasticity (MSE): -0.223

### Distributed Lag Model for Machinery and Equipment Capital Stock (Table 7; System GMM)
- Key coefficients (selected; columns 1–6)
  - ∆log(UCC for MEq) t: -0.0209 [0.0654]; -0.0771 [0.0574]; -0.0355 [0.0832]; -0.0387 [0.0675]; -0.109 [0.0726]; -0.0909 [0.0922].
  - ∆log(UCC for MEq) t−1: -0.0770 [0.0864]; -0.224* [0.121]; -0.135 [0.126]; -0.112 [0.0791]; -0.284* [0.146]; -0.245 [0.153].
  - Sum of ∆log(UCC): -0.0980, -0.424*, -0.0799, -0.151, -0.583*, -0.464 (as reported).
- F-Test (Joint Significance) P-Value: 0.496, 0.0912, 0.817, 0.270, 0.0782, 0.287.
- Diagnostic statistics (selected)
  - Observations: 960, 896, 832, 960, 896, 832.
  - Number of Cross-Sections: 64 in all columns.
  - Number of Instruments: 26, 27, 28, 26, 27, 28.
  - AR(1) P-Value: 0.00924, 0.0280, 0.0218, 0.00760, 0.0386, 0.0190.
  - AR(2) P-Value: 0.03640, 0.578, 0.348, 0.03630, 0.618, 0.300.
  - Hansen Test (Joint Validity) P-Value: 0.104, 0.523, 0.149, 0.106, 0.488, 0.156.
- Notes: Same estimation and instrument construction conventions as for the Error Correction Model. Baseline uses main price index; Robustness Check uses imported machinery and equipment price index.

### Appendix A — Distributed Lag Model in First Differences (DLM)
- Estimating equation (in first differences):
  - ∆k_{i,j,t} = ρ ∆k_{i,j,t−1} + Σ_{h=0}^{H_y} β_h ∆y_{i,j,t−h} − Σ_{h=0}^{H_c} θ_h ∆c_{i,j,t−h} + ξ_{i,j,t}
- Growth rates defined as ∆x_{i,j,t−h} ≡ ln X_{i,j,t−h} − ln X_{i,j,t−h−1}.
- Permanent change in capital stock (in percentage) from a sustained percentage increase in user cost over H_c periods:
  - UCE = − Σ_{h=0}^{H_c} θ_h
- Note: Chirinko et al. (1999) estimate the equation with ρ = 0 in their specification after approximating the left-hand side with an investment-to-capital ratio.

### Main results for Non-Residential Construction (Table A1; System GMM)
- Error Correction Model and Distributed Lag Model results summarized (selected)
  - ∆log(Capital Stock for NRC) t: 0.408 [2.261], 0.571 [3.216], 0.336 [6.008] (columns as shown).
  - Log of UCC t: 1.555 [2.108], 1.272 [1.903], 0.084 [2.702].
  - Log of GDP t: 0.659* [0.374], 0.469* [0.260], 0.913*** [0.241].
  - Sum of ∆log(UCC): -0.0584, -0.0342, -0.0755 (Prob >F: 0.206, 0.823, 0.586).
  - Sum of ∆log(GDP): 0.0428, -0.349, 0.801 (Prob >F: 0.874, 0.609, 0.191).
- Diagnostics (selected)
  - Observations: 615, 574, 533, 960, 896, 832 (across columns).
  - Number of new_id: 41, 41, 41, 64, 64, 64.
  - Number of Instruments: 40, 39, 38, 26, 27, 28.
  - AR(1) P-Value: 0.454, 0.502, 0.329, 0.00678, 0.144, 0.0786.
  - AR(2) P-Value: 0.738, 0.246, 0.953, 0.813, 0.951, 0.389.
  - Hansen Test (Joint Validity) P-Value: 0.980, 0.174, 0.119, 0.190, 0.0579, 0.0568.

*Italicized source: wpiea2020077-print-pdf - REFERENCES*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020077-print-pdf.pdf_