## _wp08218

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---

### I. Introduction and contribution
- Motivation:
  - Public capital described as the “wheels of economic growth” (World Bank, 1994).
  - Aschauer (1989) estimated a public capital elasticity of 0.39 over 1949-1985: “A 1% increase in the public capital stock was estimated to increase private sector output by 0.39%.”
  - Criticisms of prior work emphasized non-stationarity and endogeneity as potential sources of bias.
- This paper’s contributions:
  - Estimates an aggregate Cobb-Douglas production function for the U.S. postwar period augmented with public capital stock using annual data 1948 to 2004.
  - Uses Johansen’s multivariate cointegration procedure and a Vector Error-Correction Model (VECM) to handle dynamics and endogeneity.
  - Introduces measured knowledge stock (patent-based) and skill-adjusted labor (BLS methodology) into a unified multivariate cointegrating framework.
  - Does not impose a priori returns-to-scale restriction and controls for recessions with energy shocks via dummy variables.

### II. Model specification and estimation approach
- Production function (Cobb-Douglas form estimated):
  - Yt = A_t^(β1) KP_t^(β2) KG_t^(β3) L_t^(β4) ε_t
  - Variables:
    - Y: real private business sector output (billions of chained 2000 dollars; BLS).
    - KP: real net non-residential private capital stock (end of previous year, adjusted by manufacturing capacity utilization).
    - KG: real net non-residential non-military public capital stock (end of previous year; federal, state, local).
    - L: human capital / skill-adjusted labor (BLS methodology).
    - A: stock of knowledge/technology (patent-based).
- Deterministic regressors included in VAR/VECM:
  - Constant.
  - Stepdum86 (equals one after 1985).
  - Impulse74808291 (equals one in years 1974, 1980, 1982, 1991).
  - Impulse97 (equals one in 1997).
- Lag selection and VAR setup:
  - Considered maximum p = 4 lags.
  - Information criteria: AIC minimized at 4 lags; HQ minimized at 3 lags; SC minimized at 2 lags.
  - Finite-sample F-tests indicate proceeding with VAR including 3 lags on each variable.

### III. Data and measurement
- Sample: annual U.S. postwar period, 1948 to 2004.
- Data construction:
  - Yt: Real private business sector production, measured in billions of chained 2000 dollars (BLS).
  - KPt: Real net non-residential private fixed asset stock at end of previous year, multiplied by manufacturing capacity utilization (BEA + Federal Reserve Board).
  - KGt: Real net non-residential non-military public capital stock at end of previous year, measured in billions of chained 2000 dollars (BEA).
  - Lt: Skill-adjusted labor input; Lt = Ht * LCt where Ht = total hours and LCt = labor composition index (BLS 1993 methodology using 1008 worker types).
  - At: Stock of Knowledge; cumulated patent applications using perpetual inventory with 15% depreciation (stock measured end-of-period, lagged one period). Robustness checks with 0%, 5%, and 10% depreciation produce robust results.
- Labor input detail:
  - Hours cross-classified by sex (2), education categories (7), and experience levels (72): 2 * 7 * 72 = 1008 worker types.
  - Labor growth computed using a Tornqvist (Divisia) index with compensation-share weights from a Mincer-type wage regression.

### IV. Initial data features and stationarity
- Average annual growth rates (entire sample):
  - Output: about 3.5% annually.
  - Private capital: about 3.1% annually.
  - Labor input: 1.4% annually.
  - Public capital: 3.1% annually (about 4.4% in the 1950s–late 1960s; slows to 2.4% thereafter).
  - Patent (knowledge) stock: 2.4% annually (about 1.2% prior to mid-1980s; 4.8% thereafter).
- Unit-root testing (ADF):
  - Cannot reject unit root for all variables in levels.
  - Reject unit root for first differences of output, labor, and private capital → these are I(1).
  - First differences of knowledge and public capital initially suggested non-rejection; structural mean shifts (mid-1980s for patents; 1970 for public capital) found and accounted for; conclusion: all series best characterized as I(1) in levels.

### V. VAR diagnostics and misspecification tests
- Residual diagnostics for VAR(3):
  - Individual-equation tests (Breusch-Godfrey LM, Jarque-Bera normality, ARCH, White heteroscedasticity) — none reject at 5% level.
  - Vector tests: no cross-equation autocorrelation up to second lag; vector normality Chi^2(10) = 9.2996, p-value = 0.5039; no heteroscedasticity rejection (Chi^2(450) = 467.87, p-value = 0.2709).
  - Recursive estimation/Chow tests indicate coefficient constancy/stability over time.
- VAR lag-selection statistics (sample 1952–2004):
  - 4 lags: Log-Lik = 1012.5509, SC = -29.220, HQ = -31.966, AIC = -33.681.
  - 3 lags: Log-Lik = 981.92743, SC = -29.937, HQ = -32.111, AIC = -33.469.
  - F-test Unrestricted 4 Lags → Restricted 3 Lags: F = 1.3774 [0.1372] (cannot reject).

### VI. Cointegration testing and identification
- Johansen trace and max-eigenvalue tests (asymptotic):
  - Strongly reject r = 0 at 1% significance.
- Finite-sample adjustments (Reimers, 1992):
  - Adjusted Trace statistic = 41.73 with associated p-value = 0.39.
  - Adjusted Max-Eigenvalue statistic = 24.18 with associated p-value = 0.17.
  - Conclusion after correction: cannot reject at most one cointegrating vector → a single cointegrating relation interpreted as a long-run production function.
- Final reduced-rank standardized coefficients under imposed restrictions (constant returns to private inputs; equal private and public capital elasticities; weak exogeneity for knowledge, private capital, public capital) — sample 1951-2004:
  - LY: Beta = 1, Beta Std Err = 0.000; Alpha = -0.652, Std Err = 0.072
  - LA: Beta = -0.12693, Beta Std Err = 0.016; Alpha = 0.000, Std Err = 0.000
  - LKP: Beta = -0.39253, Beta Std Err = 0.000; Alpha = 0.000, Std Err = 0.000
  - LKG: Beta = -0.39253, Beta Std Err = 0.000; Alpha = 0.000, Std Err = 0.000
  - LL: Beta = -0.60747, Beta Std Err = 0.005; Alpha = -0.280, Std Err = 0.084
  - Constant: 2.43840, Std Err = 0.197

### VII. Long-run elasticities, adjustment, and hypothesis tests
- Long-run elasticities (final estimated log relation):
  - Knowledge Stock elasticity = 0.13
  - Private Capital elasticity = 0.39
  - Public Capital elasticity = 0.39
  - Skill-Adjusted Labor elasticity = 0.61
  - Final estimated relation (natural logs): Output = 0.13 Knowledge Stock + 0.39 Private Capital + 0.39 Public Capital + 0.61 Skill-Adjusted Labor -2.44
- Tests on returns and exogeneity:
  - Test LKP + LL = 1: Chi-square = 1.90; p-value = 0.17 (cannot reject).
  - Equality LKP = LKG: Chi^2(1) = 0.219 [0.6402] (cannot reject).
  - Joint test (LKP + LL = 1; LKP = LKG; LA = LKP = LKG = 0): Chi^2(6) = 16.904 [0.0096] (strongly rejected when adding weak exogeneity of labor).
  - Individual α (speed-of-adjustment) tests:
    - α_output (final) = -0.65 (negative and significant).
    - α_labor = -0.28.
    - Output is the only variable for which weak exogeneity is rejected: Test statistic = 5.58; p-value = 0.018.
  - Joint weak exogeneity:
    - All four inputs jointly weakly exogenous: Chi^2 (4) = 14.00 [0.0073].
    - Three stock series (knowledge, private and public capital) jointly weakly exogenous: Chi^2 (3) = 1.97 [0.58].

### VIII. Dynamics, feedback, and endogeneity
- Feedback tests:
  - Coefficients on once and twice lagged changes in output in the public capital equation: 0.07 [0.016] and 0.006 [0.88].
  - Joint test that lagged changes in output do not explain current changes in public capital: Chi-square (2 df) = 5.89; p-value = 0.053 (some evidence public investment responds positively to private output).
  - Joint test for lagged changes in output not explaining current changes in private capital: Chi-square (2 df) = 22.23; p-value = 0.00 (evidence of feedback to private capital).
  - Joint test for no impact of past changes in output on current change in private and public capital: Chi-square (4 df) = 24.67; significant at 1%.
- Implication:
  - Under cointegration, OLS estimates of public capital elasticity are superconsistent, though finite-sample bias caveat remains.

### IX. Growth accounting for the postwar U.S. economy
- Long-run function rewritten as output per hour (natural logs):
  - Output per hour = 0.13 Knowledge Stock + 0.39 Public Capital + 0.39 Private Capital per hour + 0.61 Labor Composition -2.44
- Average annual growth rates of output per hour:
  - 1949-1973: 3.25 percent
  - 1973-1985: 1.62 percent
  - 1985-2004: 2.27 percent
- Contributions to the 1.63 percentage point decline in labor productivity growth (1949-1973 → 1973-1985) — percent shares of decline:
  - Knowledge stock: 2 percent
  - Public capital: 48 percent
  - Private capital per hour: 42 percent
  - Labor composition: 2 percent
- Contributions to the 0.65 percentage point increase in labor productivity growth (1973-1985 → 1985-2004) — percent shares of increase:
  - Knowledge/patent stock: 70 percent
  - Labor composition: 28 percent
  - Private capital per hour: 1 percent
  - Public capital: 8 percent
  - Residual term: -7 percent
- Subsample quantitative results (selected):
  - 1949-2004 average annual growth of output per hour = 2.54; contributions: Knowledge = 0.31, Public Capital per hour = 1.22, Private Capital per hour = 0.83, Labor Composition = 0.22, Residual = -0.040.
  - 1973-1985: average = 1.62; Knowledge = 0.13, Public Capital per hour = 0.86, Private Capital per hour = 0.54, Labor Composition = 0.14, Residual = -0.048.
  - 1985-2004: average = 2.27; Knowledge = 0.59, Public Capital per hour = 0.91, Private Capital per hour = 0.55, Labor Composition = 0.32, Residual = -0.092.

### X. Conclusions and policy-relevant implications
- Core empirical conclusions:
  - Evidence consistent with an aggregate Cobb-Douglas production function with constant returns to scale with respect to private capital and skill-adjusted labor.
  - Estimated long-run elasticities: Public capital = 0.39; Skill-adjusted labor = 0.61; Knowledge/technology = 0.13; Private capital = 0.39.
  - The estimated long-run elasticity for public capital equals Aschauer (1989)’s estimate of 0.39.
- Growth and policy implications:
  - Public capital accounted for about half of the post-1973 productivity slowdown but only about 8 percent of the subsequent increase in labor productivity growth (1985-2004).
  - The partial recovery since the mid-1980s is mainly due to strong growth in knowledge (patent stock) and human capital / labor composition.
  - Policy inference: skills, technology (knowledge), and both private and public capital are important components for growth; production functions should be estimated in multivariate cointegrating systems to capture long-run relations and to address endogeneity concerns.

*Source PDF filename: _wp08218 - References*

### References                                                                                                            30

### References

### Tables and Figures
- Tables listed:
  - 1. The Information Set:Data Series from 1948 to 2004
  - 2.A. ADF Tests for Variables in Levles, Constant and Trend Included
  - 2.B. ADF Tests for Variables in Levels, Constant Included
  - 3.A. ADF Tests for Variables in First Differences, Constant and Trend Included
  - 3.B. ADF Tests for Variables in First Differences, Constant Included
  - 4.A. Lag Length Analysis: Seclected Statistics
  - 4.B. Lag Length Analysis: F-Tests for Model Reduction
  - 5. Individual Equation and Vector Misspecification Tests for the VAR Model of the Production Function
  - 6. Cointegration Analysis with Johansen’s Test
  - 7. Hypotheses Tests on the Cointegrating Relation
  - 8. Growth Accounting for the Postwar U.S. Economy
- Figures listed:
  - 1. The Variables in Natural Logarithms
  - 2. Recursive System Diagnostics for VAR(3) Model
  - 3. Recursive Likelihood Ratio Test Statistic for Final Restrictions on the Cointegrating Space
  - 4. Output Deviations from the Long Run Aggregate Production Function with Final Restrictions on the Cointegrating Space Imposed

### I. Introduction and Contribution
- Context and motivation:
  - Public capital described as the “wheels of economic growth” by the World Bank (1994).
  - David Aschauer (1989) observed a U.S. post-1973 productivity slowdown “matched or slightly preceded by a precipitous decline in additions to the net stock of public non-military structures and equipment.”
  - Aschauer’s estimation over 1949-1985 using a Cobb-Douglas production function including public capital produced a public capital elasticity of 0.39: “A 1% increase in the public capital stock was estimated to increase private sector output by 0.39%.”
  - Policy implication drawn by Aschauer: increase public investment to boost the economy.
- Criticisms of prior work:
  - Econometric concerns: non-stationarity leading to spurious correlation, and potential endogeneity of public capital.
  - Critics argued these issues could explain the large elasticity (0.39) found by Aschauer (1989).
  - Surveys noted: Munnell (1992), Gramlich (1994), and Romp and de Haan (2007).
- This paper’s approach and contributions:
  - Estimates aggregate production function for the U.S. postwar period augmented with public capital stock.
  - Address spurious correlation and endogeneity by estimating in a cointegrating framework.
  - Uses Johansen’s (1988, 1991) multivariate cointegration procedure and a Vector Error-Correction Model (VECM) to handle dynamics, endogeneity, and multiple cointegrating vectors.
  - Improvements over prior literature:
    - Treats technology and labor inputs differently: uses patent applications to proxy knowledge/technology stocks, and adjusts labor hours for changes in human capital or skill (instead of deterministic time trend for technology and raw hours for labor).
    - First to estimate effects of public capital, human capital/skill-adjusted labor, and measured knowledge stock in a unified multivariate cointegrating framework.
    - Uses longer data span: 1948 to 2004.
    - Does not impose a priori restrictions on returns to scale.
    - Controls for recessions where energy shocks are significant via dummy variables in the cointegrated VAR system.

### Key Empirical Findings
- Production function form:
  - Evidence of an aggregate Cobb-Douglas production function with constant returns to scale with respect to private capital and skill-adjusted labor.
- Long run elasticities (estimated):
  - Public capital: 0.39
  - Skill-adjusted labor: 0.61
  - Technology/knowledge: 0.13
  - Note: “Our estimated long run elasticity for public capital is the same as that of Aschauer (1989).”
- Growth accounting implications:
  - Using these elasticities for a growth accounting exercise for the postwar U.S. economy:
    - Public capital accounts for about half of the post-1973 productivity slowdown.
    - Public capital plays only a minor role in the partial recovery of labor productivity growth since the mid 1980s.
    - The largest contribution to the partial recovery comes from the knowledge stock and human capital.

### Paper Organization (Sections Summary)
- Section II: critical review of the relevant literature.
- Section III: model description.
- Section IV: data set and measurement issues.
- Section V: initial data analysis and reduction of the VAR system.
- Section VI: cointegration analysis — testing for cointegration, estimation of long run aggregate production function, speed of adjustment parameters, tests of weak exogeneity, and hypothesis tests on the aggregate production function.
- Section VII: growth accounting exercise for the U.S. postwar period using estimated production function parameters.
- Section VIII: concluding remarks.

*Source: _wp08218 - References*

### conclusions supporting a strong positive impact of the public capital stock on private sector

### _wp08218 - conclusions supporting a strong positive impact of the public capital stock on private sector

### Criticisms of earlier public capital production-function work
- Two major econometric problems identified in the literature:
  - Non-stationarity / unit roots in levels of inputs and output leading to potential spurious correlation.
  - Endogeneity: feedback (causality) from output into public capital investment causing simultaneous-equation bias.
- First-differencing as a common correction:
  - Removing spurious correlation but destroying the long-run level relationship that theory implies.
  - First-differenced estimates typically yield a very small, often statistically insignificant, effect of public capital.
- Single-equation cointegration (residual-based) studies:
  - Used by several authors; often found smaller public capital elasticities than Aschauer (1989).
  - Limitations: do not fully address endogeneity/feedback and do not allow multiple cointegrating vectors; may be inefficient unless inputs are weakly exogenous.

### Cointegration system (preferred methodology) and advantages of Johansen approach
- Johansen maximum likelihood (system/VAR) approach advantages:
  - Treats all variables as potentially endogenous and explicitly allows testing for feedback from output to public capital.
  - Allows multiple cointegrating vectors and testing for their number.
  - Addresses dynamic interactions, endogeneity, and “causality” within a system.
- Prior uses of Johansen in the literature:
  - Batina (1998), Kamps (2005): found evidence of multiple cointegrating vectors but did not identify vectors or report speeds of adjustment.
  - Pina and Aubyn (2005): one cointegrating vector but concerns about insignificance of skill-adjusted labor and public capital; constant-returns-to-scale restriction imposed rather than tested.
  - Hamilton (1996): single-equation Fully Modified OLS; less powerful than Johansen; used different human capital proxy.

### Model specification (what is estimated)
- Objective: test for and estimate a Cobb-Douglas aggregate production function in a cointegrating VAR framework:
  - Yt = A_t^(β1) KP_t^(β2) KG_t^(β3) L_t^(β4) ε_t   (Cobb-Douglas specification shown as Equation (1) in source)
  - Variables:
    - Y: real private business sector output (billions of chained 2000 dollars; BLS).
    - KP: real net non-residential private capital stock (end of previous year, adjusted by manufacturing capacity utilization).
    - KG: real net non-residential non-military public capital stock (end of previous year; federal, state, local).
    - L: human capital or skill-adjusted labor (BLS methodology; see below).
    - A: stock of knowledge/technology (patent-based).
- Departures from prior literature:
  - Labor input is skill-adjusted (education + work experience) using BLS methodology.
  - Technology/knowledge modeled endogenously using a patent-application-based stock (perpetual inventory, depreciation).
  - Energy-price shocks are not included in the long-run production function; instead, recessions with energy shocks are controlled via dummy variables in the cointegrated VAR.

### Data and measurement (sample and variable construction)
- Sample: annual U.S. postwar period, 1948 to 2004.
- Output, private capital, public capital:
  - Output Yt: real private business sector production, measured in billions of chained 2000 dollars (BLS).
  - KPt: real net non-residential private fixed asset stock (structures, equipment, software) at end of previous year, multiplied by the Federal Reserve Board’s capacity utilization rate (manufacturing sector capacity utilization used).
  - KGt: real net non-residential non-military public capital stock at end of previous year (highways, streets and roads, mass transit and airport facilities, educational buildings, electric/gas/water supply and distribution, wastewater treatment, etc.), measured in billions of chained 2000 dollars (BEA).
- Skill-adjusted labor L_t (BLS 1993 methodology):
  - Hours of work cross-classified by sex, 7 education categories, and 72 levels of experience: 2 * 7 * 72 = 1008 worker types.
  - Labor input growth computed with a Tornqvist (Divisia) index: growth is weighted average of growth rates of hours by compensation-share weights (equations (2)–(6) in source).
  - Labor input can be decomposed: L_t = H_t * LC_t, where H_t is total hours and LC_t is the labor composition index.
- Knowledge/technology stock (A_t):
  - Proxy: cumulated patent applications (invention, designs, plants) from U.S. Patent Office.
  - Patent stock constructed by perpetual inventory method with assumed depreciation rate 15% (stock measured as end-of-period, lagged one period to reflect stock in place).
  - Robustness: stocks also constructed with depreciation rates 0%, 5%, and 10% — model results robust to these alternatives.
  - Rationale: patent applications widely used in literature; applications preferred over grants; patents are imperfect but informative proxies for innovative activity.

### Initial data features and unit-root/integration properties
- Visual/log plots (natural logarithms) and average growth rates (over entire sample):
  - Output grows by an average annual rate of about 3.5%.
  - Private capital grows by about 3.1% annually on average.
  - Labor input grows by 1.4% annually on average.
  - Public capital grows at an average annual rate of 3.1% over the entire sample:
    - About 4.4% in the 1950s through late 1960s.
    - Slows to 2.4% thereafter.
  - Patent (knowledge) stock grows at an average annual rate of 2.4% over the entire sample:
    - About 1.2% prior to the mid 1980s.
    - Increases sharply to 4.8% thereafter (mid 1980s onward).
- Unit-root/ADF testing:
  - Cannot reject the null of a unit root for all variables in levels.
  - Reject null of a unit root for first differences of output, labor, and private capital → these are I(1) in levels.
  - First differences of knowledge stock and public capital initially suggested non-rejection (possible I(2) in levels), but evidence of structural mean shifts:
    - Breaks identified: mid 1980s for patent stock (Stepdum86), 1970 for public capital stock.
    - Perron (1989) structural break tests and recursive regressions indicate correcting for breaks leads to stationarity in first differences.
  - Conclusion: all series best characterized as I(1) in levels (stationary in first differences).

### VAR specification, deterministic terms, and lag selection
- VAR system estimated in levels, cast into VECM for cointegration testing (Equation (7) and transform to (8)/(9) in source).
- Deterministic regressors included:
  - Constant.
  - Stepdum86: equals one after 1985, zero otherwise (captures mid-1980s surge in measured patent activity).
  - Impulse74808291: equals one in years 1974, 1980, 1982, and 1991, zero otherwise (recessions where energy shocks significant).
  - Impulse97: equals one in 1997, zero otherwise (captures institutional changes in U.S. patent policy).
- Lag length:
  - Initial maximum p = 4 considered.
  - Information criteria: AIC minimized at 4 lags; HQ minimized at 3 lags; SC minimized at 2 lags.
  - Finite-sample F-tests for lag reductions:
    - Cannot reject reduction from 4 to 3 lags (F = 1.38, p-value = 0.14).
    - Strongly reject reductions to fewer than 3 lags.
  - Proceed with a VAR that includes 3 lags on each variable.
- Residual diagnostics (VAR with three lags):
  - Individual-equation tests: Breusch-Godfrey LM up to second lag, Jarque-Bera normality, ARCH, White heteroscedasticity — none reject at 5% level.
  - Vector tests: no cross-equation autocorrelation up to second lag; vector normality statistic 9.30 with 10 degrees of freedom, p-value = 0.50; no heteroscedasticity rejection.
  - Recursive estimation/Chow tests indicate coefficient constancy/stability over time.

### Cointegration testing and identification
- Johansen procedure applied to the VECM:
  - Trace and Max-Eigenvalue tests (asymptotic) strongly reject r = 0 (no cointegration) at 1% significance.
  - Using Reimers (1992) finite-sample adjustments for degrees of freedom:
    - Adjusted Trace statistic = 41.73 with associated p-value = 0.39.
    - Adjusted Max-Eigenvalue statistic = 24.18 with associated p-value = 0.17.
    - Conclusion: cannot reject the null of at most one cointegrating vector after finite-sample correction → conclude a single cointegrating relation exists among the variables.
- Interpretation:
  - The single cointegrating relation is interpreted as a long-run production function.
  - The cointegrating β vector contains long-run elasticities of output with respect to knowledge, private capital, public capital, and skill-adjusted labor.
  - α matrix contains speed-of-adjustment coefficients (columns represent speeds of adjustment from equilibrium deviations in each equation).

### Cointegration estimation results (long-run elasticities and adjustment)
- Long-run elasticities (β coefficients; interpreted as elasticities of private output):
  - Knowledge elasticity: about 17% (notated as 0.17 in source).
  - Private capital elasticity: about 0.4.
  - Public capital elasticity: about 0.4.
  - Labor elasticity: 0.49.
- Statistical significance:
  - All four inputs are significant with signs consistent with theory.
  - Johansen multivariate stationarity tests strongly reject stationarity for each variable.
  - Individual β coefficients: all variables strongly significant with p-values of less than 1 percent.
- Speed of adjustment (α coefficients):
  - Output speed of adjustment coefficient estimate: -0.71 (significant) — negative as required for an equilibrium-correcting relation.
  - Speed-of-adjustment estimates for stock variables (knowledge, private capital, public capital): individually less than respective standard errors (intuitive given stock measures are beginning-of-period values).
  - Labor speed of adjustment: barely larger than its standard error; may be weakly exogenous (formal weak exogeneity testing discussed subsequently in source).
- Additional hypothesis tests:
  - Test of equality of private capital and public capital elasticities:
    - Test statistic = 0.22 with p-value = 0.64 → cannot reject equality.

### Measurement, robustness, and methodological notes emphasized in the source
- Labor composition:
  - Constructed from 1008 worker types (2 sexes × 7 education categories × 72 experience levels).
  - Growth of labor input computed by Tornqvist index; wages estimated via a Mincer-type wage regression to compute compensation-share weights.
- Patent stock construction and caveats:
  - Patent applications cumulated with 15% depreciation; robustness checks using 0%, 5%, and 10% depreciation yield robust model results.
  - Patents imperfect proxy for knowledge (not all innovations patentable; heterogeneity in patent quality), but standard in aggregate literature; applications preferred over grants.
- Treatment of energy shocks and recessions:
  - Energy-price shocks not included as long-run regressors; instead, recessions with significant energy components included via impulse dummies (Impulse74808291).
- Estimation strategy:
  - System (Johansen) approach chosen to address spurious-correlation concern while maintaining long-run relationships and to explicitly model endogeneity/feedback among variables.

*Source: _wp08218 - conclusions supporting a strong positive impact of the public capital stock on private sector (PDF chapter/section).*

### references to Cobb-Douglas specifications with constant returns to scale with respect to the

### _wp08218 - references to Cobb-Douglas specifications with constant returns to scale with respect to the

### Tests of Cobb-Douglas specification, returns to scale, and exogeneity
- Test of whether elasticities for private capital stock and skill-adjusted labor sum to unity:
  - Chi-square statistic = 1.90; p-value = 0.17.
- Joint test combining equal capital elasticities and constant returns to private inputs:
  - Test statistic = 2.32; p-value = 0.31.
- Joint significance of knowledge, labor, private and public capital:
  - Test statistic = 37; p-value = 0.000.
- Tests on individual α (speed of adjustment) coefficients:
  - Output is the only variable for which weak exogeneity is rejected:
    - Test statistic = 5.58; p-value = 0.018.
- Weak exogeneity tests for inputs:
  - All four inputs jointly weakly exogenous: Chi^2 (4) = 14.00 [0.0073].
  - Three stock series (knowledge, private and public capital) jointly weakly exogenous: Chi^2 (3) = 1.97 [0.58].
- Joint tests on β and βα vectors:
  - Hypothesis: constant returns to scale w.r.t. private capital and skill-adjusted labor, and three stock variables weakly exogenous (public capital elasticity unrestricted):
    - Chi^2 (4) = 4.10 [0.39] (cannot be rejected).
  - Augmented by restricting public and private capital to have same elasticity:
    - Chi-square with five degrees of freedom; p-value = 0.44 (cannot be rejected).
  - Augmenting the five restrictions with weak exogeneity of skill-adjusted labor:
    - Chi^2 (6) = 16.90 [0.0096] (strongly rejected).
- Final accepted restrictions imposed in the model:
  - Constant returns to scale with respect to private capital and skill-adjusted labor.
  - Equal elasticities for public and private capital stocks.
  - Weak exogeneity of private capital, public capital, and the knowledge stock.

### The final long-run aggregate production function (log form) and estimated parameters
- Final estimated relation in natural logarithms:
  - Output = 0.13 Knowledge Stock + 0.39 Private Capital + 0.39 Public Capital + 0.61 Skill-Adjusted Labor -2.44
- Speeds of adjustment:
  - α_output = -0.65
  - α_labor = -0.28
- Key interpretations and comparisons:
  - Evidence consistent with an aggregate Cobb-Douglas production function with constant returns to scale to the private inputs.
  - Estimated elasticities roughly match shares in national income: skill-adjusted labor ≈ 60 percent, private capital ≈ 40 percent.
  - Public and private capital estimated elasticities are equal (both 0.39).
  - Knowledge/technology stock elasticity = 0.13, consistent with Adams and Coe (1990) and Abdih and Joutz (2006) (0.135 and 0.12 respectively) and at the upper end of firm/industry-level estimates (range 0.01 to 0.1).
  - Public capital elasticity consistent with estimates for the U.S. by Munnell (1990), Ford and Poret (1991), Crowder and Himarios (1997), and Batina (1999) [0.34, 0.30, 0.36 and 0.38 respectively], and identical to Aschauer (1989).

### Dynamics, feedback, and robustness to endogeneity concerns
- Error correction interpretation:
  - Negative and significant speeds of adjustment indicate labor and output jointly correct disequilibrium: if output is below long-run equilibrium, higher growth of labor and output is needed to restore equilibrium.
- Persistence and labor adjustment:
  - Shocks to the economy are less persistent for output than labor.
  - Ratio of speeds of adjustment similar to Okun’s Law relation.
- Feedback from output to public and private capital:
  - Coefficients on once and twice lagged changes in output in the public capital equation:
    - 0.07 [0.016] and 0.006 [0.88] (square-bracketed numbers are p-values for individual coefficient tests).
  - Joint test that lagged changes in output do not explain current changes in public capital:
    - Chi-square (2 df) = 5.89; p-value = 0.053 (some evidence public investment responds positively to private output).
  - Joint test for lagged changes in output not explaining current changes in private capital:
    - Chi-square (2 df) = 22.23; p-value = 0.00 (evidence of feedback to private capital).
  - Joint test for no impact of past changes in output on current change in private and public capital:
    - Chi-square (4 df) = 24.67; significant at one percent.
- Implication on endogeneity:
  - Under cointegration, OLS estimates of public capital elasticity are superconsistent; finite sample bias caveat remains.

### Growth accounting for the postwar U.S. economy (selected quantitative results)
- Rewriting long-run function as output per hour (natural logarithms):
  - Output per hour = 0.13 Knowledge Stock + 0.39 Public Capital + 0.39 Private Capital per hour + 0.61 Labor Composition -2.44
- Average annual growth rates of labor productivity (output per hour):
  - 1949-1973: 3.25 percent
  - 1973-1985: 1.62 percent
  - 1985-2004: 2.27 percent
- Changes in average annual growth rates between 1949-1973 and 1973-1985 (falls by):
  - Knowledge stock: 0.23 percentage points
  - Public capital: 1.98 percentage points
  - Private capital per hour: 1.74 percentage points
  - Labor composition: 0.05 percentage points
- Contributions to the 1.63 percentage point decline in labor productivity growth (percent shares of decline):
  - Knowledge stock: 2 percent
  - Public capital: 48 percent
  - Private capital per hour: 42 percent
  - Labor composition: 2 percent
- Changes in average annual growth rates between 1973-1985 and 1985-2004 (increases by):
  - Knowledge stock: 3.61 percentage points
  - Public capital: 0.13 percentage points
  - Private capital per hour: 0.01 percentage points
  - Labor composition: 0.30 percentage points
- Contributions to the 0.65 percentage point increase in labor productivity growth (percent shares of increase):
  - Knowledge/patent stock: 70 percent
  - Labor composition: 28 percent
  - Private capital per hour: 1 percent
  - Residual term: -7 percent (negative contribution)
- Interpretation:
  - Public capital accounted for about half of the post-1973 productivity slowdown but only about 8 percent of the subsequent increase in labor productivity growth.
  - The partial rebound since the mid-1980s is mostly due to strong growth in knowledge and human capital/labor composition.

### Conclusions and policy-relevant implications
- Strong long-run positive effects on aggregate output from:
  - Public capital (elasticity = 0.39)
  - Human capital / skill-adjusted labor (elasticity = 0.61)
  - Knowledge stock / patents (elasticity = 0.13)
- Methodological implication:
  - Production functions should be estimated as multivariate cointegrating systems to capture long-run relations and avoid biases from single-equation or first-difference approaches.
- Economic implication:
  - Skills, technology (knowledge), and capital—both private and public—are important components in determining and explaining economic growth; reliance on any single type of input is insufficient as the economy evolves.

*Source: content unit _wp08218 - references to Cobb-Douglas specifications with constant returns to scale with respect to the (PDF chapter/section).*

### References

### _wp08218 - References

### Major bibliographic foundations
- Extensive citations on public capital, infrastructure, productivity, and growth models, including but not limited to:
  - Aaron, Henry J., 1990; Aschauer, David Alan, 1989 and 1990; Munnell, Alicia H., 1990 and 1992; Gramlich, Edward M, 1994.
  - Endogenous growth, R&D and patents literature: Aghion and Howitt, 1992; Romer, Paul M., 1990; Griliches, Zvi (1988, 1989, 1990); Kortum, 1997; Jones, 1995 and 2002.
  - Econometric methods and cointegration: Johansen (1988, 1991, 1995), Johansen and Juselius (1990), Perron (1989), Phillips and Hansen (1990), Hendry (1986), Doornik and Hendry (2001).
  - Empirical studies on public capital productivity and cross-country/panel evidence: Evans and Karras, 1994; Everaert and Heylen, 2001; Kamps, 2005; Romp and de Haan, 2007.

### Data definitions and information set (Table 1)
- Variables and descriptions (data series from 1948 to 2004):
  - Yt: Real output in the private business sector measured in billions of (chained/constant) 2000 dollars. Source: Bureau of Labor Statistics (BLS).
  - KPt: Real net stock of nonresidential private fixed assets at the end of the previous year, multiplied by the Federal Reserve Board’s capacity utilization rate. Measured in billions of chained 2000 dollars; consists of structures, equipment and software. Source: Bureau of Economic Analysis (BEA).
  - Capacity utilization: output as a fraction of capacity for the manufacturing sector (data for total private industry available only back to 1967). Source: The Economic Report of the President.
  - KGt: Real net stock of nonresidential, non-defense government (federal, state, local) structures, equipment and software at the end of the previous year. Measured in billions of chained 2000 dollars. Source: BEA.
  - Lt: Skill-adjusted labor input / human capital; Lt = Ht LCt where Ht = total hours worked in the private business economy and LCt = labor composition index. Data for hours, labor composition, skill-adjusted labor input, and private sector output obtained from Larry Rosenblum at the Office of Productivity and Technology, BLS.
  - At: Stock of Knowledge; cumulated total patent applications at the end of the previous year using the perpetual inventory method with a 15% depreciation rate. Patent applications data obtained from the U.S. Patent Office.

### Unit-root (stationarity) testing results (Tables 2.A, 2.B, 3.A, 3.B)
- Augmented Dickey-Fuller (ADF) tests for variables in levels, sample 1953-2004, constant and trend included (Table 2.A):
  - LY: t-ADF = -3.268, π+1 = 0.646, Lags = 1, AIC = -7.277
  - LA: t-ADF = 1.875, π+1 = 1.029, Lags = 2, AIC = -9.615
  - LKP: t-ADF = -2.869, π+1 = 0.691, Lags = 0, AIC = -6.197
  - LKG: t-ADF = -3.440, π+1 = 0.986, Lags = 3, AIC = -12.180
  - LL: t-ADF = -2.530, π+1 = 0.835, Lags = 0, AIC = -7.605
  - Critical Values: 5% = -3.50, 1% = -4.14
- Augmented Dickey-Fuller (ADF) tests for variables in levels, sample 1953-2004, constant included (Table 2.B):
  - LY: t-ADF = -0.199, π+1 = 0.999, Lags = 0, AIC = -7.153
  - LA: t-ADF = 2.261, π+1 = 1.015, Lags = 1, AIC = -9.642
  - LKP: t-ADF = -0.770, π+1 = 0.990, Lags = 0, AIC = -6.088
  - LKG: t-ADF = -2.702, π+1 = 0.997, Lags = 2, AIC = -12.090
  - LL: t-ADF = 0.675, π+1 = 1.009, Lags = 2, AIC = -7.513
  - Critical Values: 5% = -2.92, 1% = -3.56
- ADF tests for first differences, sample 1953-2004, constant and trend included (Table 3.A):
  - DLY: t-ADF = -6.808**, π+1 = 0.028, Lags = 0, AIC = -7.115
  - DLA: t-ADF = -2.549, π+1 = 0.778, Lags = 0, AIC = -9.607
  - DLKP: t-ADF = -5.796**, π+1 = -0.614, Lags = 2, AIC = -6.066
  - DLKG: t-ADF = -2.338, π+1 = 0.904, Lags = 1, AIC = -12.030
  - DLL: t-ADF = -6.031**, π+1 = -0.121, Lags = 1, AIC = -7.529
  - Critical Values: 5% = -3.50, 1% = -4.14
- ADF tests for first differences, sample 1953-2004, constant included (Table 3.B):
  - DLY: t-ADF = -6.878**, π+1 = 0.028, Lags = 0, AIC = -7.153
  - DLA: t-ADF = -1.780, π+1 = 0.901, Lags = 0, AIC = -9.581
  - DLKP: t-ADF = -6.415**, π+1 = -0.339, Lags = 1, AIC = -6.093
  - DLKG: t-ADF = -1.266, π+1 = 0.963, Lags = 1, AIC = -11.990
  - DLL: t-ADF = -5.915**, π+1 = -0.080, Lags = 1, AIC = -7.542
  - Critical Values: 5% = -2.92, 1% = -3.56
- Note: ** denotes rejection of the null hypothesis at the 1% critical value where reported.

### VAR lag selection and model reduction (Tables 4.A and 4.B)
- Lag length selected statistics (Table 4.A):
  - 4 lags: Log-Lik = 1012.5509, SC = -29.220, HQ = -31.966, AIC = -33.681
  - 3 lags: Log-Lik = 981.92743, SC = -29.937, HQ = -32.111, AIC = -33.469
  - 2 lags: Log-Lik = 946.53153, SC = -30.474, HQ = -32.076, AIC = -33.077
  - 1 lag: Log-Lik = 854.30035, SC = -28.867, HQ = -29.896, AIC = -30.54
  - Notes: VARs include variables (LY, LA, LKP, LKG, LL), a constant (restricted), and three dummy variables: Stepdum86, Dum97, Dum74808291. Sample: 1952 through 2004.
- F-tests for model reduction (Table 4.B):
  - Unrestricted 4 Lags → Restricted 3 Lags: F = 1.3774 [0.1372]
  - Unrestricted 4 Lags → Restricted 2 Lags: F = 1.7062 [0.0099]** ; Unrestricted 3 Lags → Restricted 2 Lags: F = 1.9549 [0.0095]**
  - Unrestricted 4 Lags → Restricted 1 Lag: F = 4.0973 [0.0000]** ; Unrestricted 3 Lags → Restricted 1 Lag: F = 5.2565 [0.0000]** ; Unrestricted 2 Lags → Restricted 1 Lag: F = 8.1651 [0.0000]**
  - ** denotes significance at indicated levels.

### VAR misspecification and diagnostic tests (Table 5)
- Individual-equation AR(1-2) F-tests (F(2,33), p-values):
  - LY: 0.4302 (0.654)
  - LA: 2.3728 (0.1089)
  - LKP: 0.8982 (0.417)
  - LKG: 1.7477 (0.1899)
  - LL: 2.9450 (0.0666)
- Normality tests (Chi^2(2), p-values):
  - LY: 0.1467 (0.9293)
  - LA: 5.7030 (0.0578)
  - LKP: 1.5407 (0.4628)
  - LKG: 4.6064 (0.0999)
  - LL: 0.9867 (0.6106)
- ARCH 1-1 tests (F(1,33), p-values):
  - LY: 0.9574 (0.335)
  - LA: 0.6796 (0.4156)
  - LKP: 0.0683 (0.7954)
  - LKG: 0.1107 (0.7414)
  - LL: 0.0024 (0.9609)
- Heteroskedasticity tests (F(30,4), p-values):
  - LY: 0.1676 (0.9988)
  - LA: 0.2749 (0.9844)
  - LKP: 0.2789 (0.9833)
  - LKG: 0.2390 (0.9918)
  - LL: 0.2154 (0.9951)
- Vector diagnostics:
  - Vector AR 1-2 test: F(50,99) = 1.4679 (0.0532)
  - Vector Normality test: Chi^2(10) = 9.2996 (0.5039)
  - Vector hetero test: Chi^2(450) = 467.87 (0.2709)
- Notes: VAR includes three lags on each variable (LY, LA, LKP, LKG, LL), a restricted constant, and the three dummy variables. Sample: 1951-2004.

### Cointegration analysis (Johansen test) and estimated cointegrating relation (Tables 6 and 7)
- Johansen trace and max-eigenvalue test summary (Table 6):
  - Rank 0: Trace test = 122.47 [0.000]**, Max test = 64.68 [0.000]**
  - Rank 1: Trace test = 57.78 [0.021]*, Max test = 33.47 [0.008]**
  - Rank 2: Trace test = 24.31 [0.448], Max test = 13.96 [0.478]
  - Rank 3: Trace test = 10.35 [0.612], Max test = 8.63 [0.486]
  - Rank 4: Trace test = 1.72 [0.825], Max test = 1.72 [0.824]
  - Reduced-rank standardized coefficients (initial):
    - LY: Beta = 1, Std Err = 0; Alpha = -0.706, Std Err = 0.285
    - LA: Beta = -0.168, Std Err = 0.032; Alpha = -0.033, Std Err = 0.085
    - LKP: Beta = -0.438, Std Err = 0.071; Alpha = -0.063, Std Err = 0.493
    - LKG: Beta = -0.370, Std Err = 0.046; Alpha = 0.026, Std Err = 0.030
    - LL: Beta = -0.487, Std Err = 0.091; Alpha = -0.336, Std Err = 0.251
    - Constant: 2.698, Std Err = 0.258
  - Notes: VAR includes three lags on each variable, restricted constant, and the three dummy variables. Sample: 1951-2004.
- Hypothesis tests on the cointegrating relation (Table 7):
  - Beta vector stationarity Chi^2(4) tests (p-values in brackets):
    - LY: 37.006 [0.0000]**
    - LA: 39.851 [0.0000]**
    - LKP: 36.054 [0.0000]**
    - LKG: 34.091 [0.0000]**
    - LL: 39.035 [0.0000]**
    - Constant: 18.352 [0.0000]**
  - Zero beta coefficient Chi^2(1) tests:
    - LA: 13.120 [0.0003]**
    - LKP: 10.283 [0.0013]**
    - LKG: 24.057 [0.0000]**
    - LL: 19.653 [0.0000]**
  - Linear beta restrictions:
    - LKP = LKG: Chi^2(1) = 0.219 [0.6402]
    - LKP + LL = 1: Chi^2(1) = 1.9045 [0.1676]
    - LKP = LKG and LKP + LL = 1: Chi^2(2) = 2.319 [0.3137]
  - Alpha (weak exogeneity) tests (Zero alpha Chi^2(1)):
    - LY: 5.577 [0.0182]*
    - LA: 0.16203 [0.6873]
    - LKP: 0.01627 [0.8985]
    - LKG: 0.933 [0.3340]
    - LL: 1.917 [0.1662]
  - Joint zero alpha coefficients:
    - LA = LKP = LKG = LL = 0: Chi^2(4) = 14.002 [0.0073]**
    - LA = LKP = LKG = 0: Chi^2(3) = 1.971 [0.5784]
  - Joint hypothesis tests (alpha and beta restrictions):
    - (a) Beta: LKP + LL = 1; Alpha: LA = LKP = LKG = 0 → Chi^2(4) = 4.096 [0.3932]
    - (b) Beta: LKP + LL = 1; LKP = LKG; Alpha: LA = LKP = LKG = 0 → Chi^2(5) = 4.781 [0.4432]
    - (c) Beta: LKP + LL = 1; LKP = LKG; Alpha: LA = LKP = LKG = LL = 0 → Chi^2(6) = 16.904 [0.0096]**
  - Final reduced-rank standardized coefficients under restriction (b) (Table 7 final):
    - LY: Beta = 1, Beta Std Err = 0.000; Alpha = -0.652, Std Err = 0.072
    - LA: Beta = -0.12693, Beta Std Err = 0.016; Alpha = 0.000, Std Err = 0.000
    - LKP: Beta = -0.39253, Beta Std Err = 0.000; Alpha = 0.000, Std Err = 0.000
    - LKG: Beta = -0.39253, Beta Std Err = 0.000; Alpha = 0.000, Std Err = 0.000
    - LL: Beta = -0.60747, Beta Std Err = 0.005; Alpha = -0.280, Std Err = 0.084
    - Constant: 2.43840, Std Err = 0.197
  - Notes: final restrictions reflect constant returns to scale for private inputs (labor and private capital), equal elasticities for private and public capital, and weak exogeneity for knowledge, private capital and public capital stocks. Sample: 1951-2004.

### Growth accounting results for the postwar U.S. economy (Table 8)
- Contribution of inputs to average annual growth rate of output per hour (percent per year):
  - Period: 1949-2004
    - Average annual growth rate of output per hour: 2.54
    - Knowledge: 0.31
    - Public Capital per hour: 1.22
    - Private Capital per hour: 0.83
    - Labor Composition: 0.22
    - Residual: -0.040
  - Period: 1949-1973
    - Average annual growth rate of output per hour: 3.25
    - Knowledge: 0.16
    - Public Capital per hour: 1.63
    - Private Capital per hour: 1.23
    - Labor Composition: 0.16
    - Residual: 0.063
  - Period: 1973-1985
    - Average annual growth rate of output per hour: 1.62
    - Knowledge: 0.13
    - Public Capital per hour: 0.86
    - Private Capital per hour: 0.54
    - Labor Composition: 0.14
    - Residual: -0.048
  - Period: 1985-2004
    - Average annual growth rate of output per hour: 2.27
    - Knowledge: 0.59
    - Public Capital per hour: 0.91
    - Private Capital per hour: 0.55
    - Labor Composition: 0.32
    - Residual: -0.092
- Subsample changes (differences between periods) (Table 8):
  - (1973-1985 minus 1949-1973): change in average annual growth = -1.63; component changes: Knowledge -0.03, Public Capital per hour -0.78, Private Capital per hour -0.68, Labor Composition -0.03, Residual -0.11. Percentages reported in table: -2, -48, -42, -2, -7.
  - (1985-2004 minus 1973-1985): change in average annual growth = 0.65; component changes: Knowledge 0.46, Public Capital per hour 0.05, Private Capital per hour 0.003, Labor Composition 0.18, Residual -0.04. Percentages reported in table: 70, 8, 1, 28, -7.
- Notes:
  - Contribution of each input equals the average annual growth rate of the input weighted by its estimated elasticity in the aggregate production function.
  - Contributions may not sum to the growth of output per hour due to rounding.

### Figures and model diagnostic notes (Figures 1–4)
- Figure 1: Variables in natural logarithms plotted for sample period (LY, LKP, LA, LKG, LL). Variable labels:
  - LY: Log of output in the private business sector.
  - LKP: Log of the private capital stock, adjusted for capacity utilization.
  - LKG: Log of the public capital stock.
  - LA: Log of the stock of knowledge.
  - LL: Log of the Labor input.
- Figure 2: Recursive system diagnostics for VAR(3) model with CHOW statistics and 1% critical lines (time axis 1975–2005).
- Figure 3: Recursive Likelihood Ratio Test Statistic for final restrictions on the cointegrating space (restrictions: constant returns to scale for private inputs; equal elasticities for private and public capital; weak exogeneity for knowledge, private capital and public capital stocks). Shows LR(5) and 5% critical line over 1975–2005.
- Figure 4: Output deviations from the long-run aggregate production function with final restrictions imposed (cointegrating relation plotted over 1950–2005).

*Italic: Source PDF filename: _wp08218 - References*

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