## _wp0574 - 2. Log of Domestic Patent Applications .........................................................................27

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---

### I. Introduction and contribution
- Uses postwar U.S. time-series to directly estimate parameters of the knowledge production function.
- Constructs knowledge flows and stocks from historical time series of patent filings.
- Employs Johansen’s (1988,1991) maximum-likelihood cointegration procedure to estimate the knowledge production function, addressing nonstationarity and potential endogeneity of inputs.
- Motivated by the Romer–Jones debate on the magnitude of intertemporal knowledge spillovers and implications for long-run growth and policy effectiveness.

### II. Theoretical framework and empirical specification
- Knowledge production function (generic): Ȧ = δ A^φ L_A^λ (equation (4)).
  - φ interprets intertemporal knowledge spillovers: φ>0 = “standing on shoulders”; φ<0 = “fishing out”; φ=0 = spillover offsets fishing out.
- Romer (1990) specialization: φ = 1 and λ = 1 → Ȧ = δ A L_A; steady-state growth g_A = δ L_A (equations (6)–(7)); implies permanent long-run effects of R&D scale on growth.
- Jones (1995a,1995b) critique: smaller φ removes policy-driven long-run growth effects; steady-state results summarized:
  - Modified steady-state (equation (9)): Ȧ/A = λ (L_A)^{φ−1}
  - Steady-state growth (equation (10)): g_A = λ (L_A)^{φ−1}
  - If L̇_A/L_A = n (steady state), then (equation (11)): g_A = λ n^{1−φ}

### III. Data and measurement
- Sample frequency and period: Annual data available from 1948 to 1997 (many series referenced 1948–97; estimation sample 1953 to 1997 in VAR).
- Flow of new knowledge: domestic patent applications (dp) filed at the USPTO used to measure new knowledge.
- Stock of knowledge:
  - stp: stock of total patent applications (domestic + foreign) constructed via perpetual inventory with depreciation rate 15 percent; robustness checks with 0, 5, and 10 percent depreciation produced very little difference.
  - sdpl1 / sdpl1 notation: lagged stock of domestic patents constructed similarly.
- R&D input: number of scientists and engineers engaged in R&D (s&e).
- Productivity: total factor productivity (tfp) for the private business sector.
- Key data sources: U.S. Patent and Trademark Office counts; National Science Foundation (Science and Engineering Indicators-2000) and earlier NSF publications; Larry Rosenblum, Office of Productivity and Technology, Bureau of Labor Statistics.

### IV. Empirical patterns and stylized facts
- Domestic patent applications (dp):
  - Average annual growth rate 1.7 percent between 1948 and 1997.
  - 1948–1965: average annual growth 1.6 percent.
  - Mid 1960s to mid 1980s: relative constancy.
  - Since about 1985: average annual growth 5.1 percent.
- Stock of total patent applications (stp):
  - Prior to 1985: stock grew at average annual rate 1.2 percent.
  - After 1985: stock growth about 4 percent (more than tripled).
- Scientists and engineers engaged in R&D (s&e):
  - Grew at an average annual rate of 4.3 percent over 1948–97.
- Total factor productivity (tfp):
  - Grew at average annual rate about 1.4 percent between 1948 and 1997.
  - Prior to 1973: 2.1 percent.
  - After 1973: about 0.7 percent.

### V. Time-series properties and estimation approach
- Nonstationarity: inputs and outputs plausibly I(1); ADF tests in levels cannot reject unit root for all four variables; first differences reject unit root for dp, tfp, s&e but not for stp (stp shows structural mean shift around 1985).
- Structural break: apparent permanent shift in mean of patent stock starting mid 1980s; Perron (1989) procedure indicates I(1) after correcting for structural mean shift.
- Modeling approach:
  - General-to-specific modeling (Hendry (1986)).
  - Form VAR system including dp, tfp, stpl1, s&e, constant, trend, and dummies: Stepdum86 (one after 1985), Impulse9495 (one in 1994 and 1995), Impulse96 (one in 1996).
  - Lag selection: sequential testing from VAR(4) to VAR(1); Schwarz criterion and Hannan-Quinn minimized at VAR(1) → VAR(1) adopted.
  - VAR(1) residual diagnostics: tests do not reject no serial correlation, joint normality, or homoskedasticity; recursive estimation and Chow tests indicate model stability at 1 percent significance.

### VI. Cointegration evidence and long-run relationships
- Johansen trace statistics:
  - 0.937 r = 0 183.03** [0.000]
  - 0.687 r ≤ 1 58.87 ** [0.000]
  - 0.120 r ≤ 2 6.65 [0.993]
  - 0.019 r ≤ 3 0.88 [0.997]
- Two cointegrating relationships identified and interpreted as:
  - (1) Knowledge production function (flow of new knowledge depends on lagged stock and s&e).
  - (2) Determinants of total factor productivity (tfp related to lagged stock).
- Unrestricted cointegrating vectors (selected display):
  - Vector 1 (unrestricted): dp = 1, tfp = -0.255, stpl1 = -1.415, s&e = -0.195, Trend = 0.025
  - Vector 2 (unrestricted): dp = 0.170, tfp = 1, stpl1 = -0.418, s&e = -0.106, Trend = -0.002
- Restricted (just-identified) specifications impose β12 = 0 and β21 = 0 to identify vectors normalized on dp and tfp respectively.

### VII. Main empirical findings (cointegrated parameter estimates)
- First cointegrating vector — knowledge production (restricted estimates, Table 5):
  - dp normalized to 1; stpl1 coefficient: -1.436** (0.0619) → interpreted as long-run elasticity of new knowledge with respect to existing stock = 1.4361 (significantly greater than unity; likelihood ratio test χ2(1)= 29.12 [0.00]).
  - s&e coefficient: -0.208** (0.020) → interpreted as long-run elasticity of new knowledge with respect to researchers ≈ 0.21 (positive, highly significant, < 1).
  - Trend coefficient: 0.023** (0.001) reported (negative trend reported elsewhere as -2.3 percent when sign conventions applied; interpretation: faster growth in s&e than dp).
- Feedback (error-correction) for knowledge relation (Panel B, Table 5):
  - ∆dp feedback α11: -0.449** (SE 0.121) → positive deviation of dp above long-run path declines next period.
  - ∆stpl1 feedback α31: 0.135** (SE 0.008) → dp above equilibrium raises next-period growth of patent stock.
  - ∆s&e feedback α41: -0.190** (SE 0.068) → dp above equilibrium slows growth of s&e next period.
  - Conclusion: dp, stpl1, s&e are not weakly exogenous for the knowledge relation; significant feedback mechanisms operate.
- Second cointegrating vector — tfp relation (restricted estimates, Table 5):
  - tfp normalized to 1; stpl1 coefficient: -0.108 (0.141) → implied long-run elasticity of tfp with respect to stock of knowledge ≈ 0.1081 (positive but small and not statistically significant; likelihood ratio test χ2(1) = 0.62 [0.43]).
  - Trend: reported implying trend growth of tfp about 1 percent (highly significant), matching observed average.
  - Feedback in ∆tfp equation from tfp cointegrating relation is significant and relatively quick; stpl1 appears weakly exogenous with respect to tfp relation.

### VIII. Robustness: domestic patent stock model (Table 6)
- Re-estimation using sdpl1 (domestic-only stock) yields two cointegrating vectors.
- Key differences:
  - Long-run impact of lagged domestic patent stock on dp: coefficient -1.000** (0.009) → elasticity of unity (sdpl1 → dp).
  - Long-run impact of sdpl1 on tfp: -0.134 (0.127) → implied elasticity ≈ 0.134 (small, statistically insignificant).
  - Feedback coefficients remain significant for dp and sdpl1 (e.g., sdpl1 α1: 0.162** (SE 0.001)).
- Interpretation: using domestic-only stock produces spillover coefficient of unity; using total (domestic+foreign) stock yields larger spillover estimate (1.4361), consistent with access to global knowledge.

### IX. Quantitative summaries and stylized elasticities
- Long-run elasticity of new knowledge with respect to existing stock:
  - Using domestic patent stock: unity (1.000** (0.009) in Table 6).
  - Using total patent stock: 1.4361 (reported as -1.436 in normalized presentation with sign conventions; magnitude 1.4361; highly significant).
- Long-run elasticity of new knowledge with respect to s&e: about 0.21 (0.208** (0.020)).
- Long-run elasticity of tfp with respect to stock of knowledge: about 0.1081 (0.1081 reported; small and imprecise; example alternative 0.1 with standard error 0.1 noted).

### X. Interpretation, reconciliation, and policy implications
- Spillovers:
  - Evidence indicates strong intertemporal knowledge spillovers in the knowledge production function (φ≥1 in estimates using patent-based measures).
  - Romer–type scale effects (φ = 1) are consistent with some specifications; total stock measure suggests even larger spillovers.
- TFP response and diffusion:
  - Despite strong spillovers, measured long-run impact of patent stock on TFP is small: doubling the stock of knowledge is estimated to increase TFP by only 10 percent in the long run.
  - Explains Jones (1995a) finding of weak relationship between R&D workforce and TFP: knowledge generation may be substantial while diffusion and application into measured productivity are slow and incomplete.
- Policy implications:
  - If φ≥1, policies that permanently increase labor devoted to R&D can have permanent effects on long-run growth (endogenous growth logic).
  - However, modest measured TFP gains from patent accumulation alone imply policymakers should complement R&D-promoting policies with policies that speed diffusion and application of knowledge into productive sectors to realize aggregate productivity gains.
- Suggested further research:
  - Examine transmission mechanisms from knowledge production to productivity, timing and diffusion of knowledge into production, and measurement issues.

*Source: _wp0574 - 2. Log of Domestic Patent Applications .........................................................................27*

### 2. Log of Domestic Patent Applications .........................................................................27

### _wp0574 - 2. Log of Domestic Patent Applications .........................................................................27

### I. Introduction and contribution
- Uses postwar U.S. time-series to directly estimate parameters of the knowledge production function.
- Exploits historical time series of patent filings to construct knowledge flows and stocks.
- Employs Johansen’s (1988,1991) maximum-likelihood cointegration procedure to estimate the knowledge production function, addressing nonstationarity and potential endogeneity of inputs.
- Motivated by the Romer–Jones debate on the magnitude of intertemporal knowledge spillovers and their implications for long-run growth and policy effectiveness.

### II. Theoretical framework and empirical specification
- Knowledge production function framework:
  - New knowledge (Ȧ) depends on research labor (L_A) and existing knowledge stock (A).
  - Generic specification: Ȧ = δ A^φ L_A^λ (equation (4)).
  - φ captures intertemporal knowledge spillovers: φ>0 = “standing on shoulders”; φ<0 = “fishing out”; φ=0 = spillover offsets fishing out.
  - Romer (1990) imposes φ=1 and λ=1, yielding Ȧ = δ A L_A and steady-state growth g_A = δ L_A (equations (6)–(7)), implying permanent long-run effects of R&D scale on growth.
  - Jones (1995a,1995b) critiques Romer’s strong spillover assumption by documenting weak empirical relationship between number of R&D scientists and engineers and TFP growth; argues for smaller φ which removes policy-driven long-run growth effects.

### III. Data and variables considered
- Inputs and outputs used in empirical model:
  - Flow of new knowledge measured by flow of new patents.
  - Stock of existing knowledge measured by patent stock.
  - R&D input measured by number of scientists and engineers engaged in R&D.
  - Total Factor Productivity (TFP) included to:
    - Capture relation between patent-based knowledge measure and Jones’s TFP measure.
    - Reflect that long-run economic growth depends on TFP as application/embodiment of knowledge.
    - Shed light on the weak observed relationship between R&D workforce and TFP growth.
- Appendix and tables listed include: Augmented Dickey-Fuller tests (1953–97), cointegration analyses, VAR diagnostics, and data construction and sources.

### IV. Estimation approach and time-series issues
- Time-series properties addressed:
  - Inputs and output plausibly nonstationary and I(1); conventional OLS would risk spurious correlations.
  - Johansen cointegration used to correct for spurious correlations and explicitly account for endogeneity.
- Model stability and diagnostic testing:
  - Model stability tests include recursive residuals and Chow tests.
  - VAR(1) residual diagnostics and restricted cointegration analyses are reported (tables referenced).

### V. Main empirical findings
- Two long-run cointegrating relationships are found:
  - A long-run knowledge production function: flow of new knowledge depends positively on the existing stock of knowledge and the number of R&D scientists and engineers.
  - A long-run positive relationship between TFP and the stock of knowledge (patents).
- Evidence on spillovers and elasticities:
  - Results indicate the presence of strong intertemporal knowledge spillovers, consistent with Romer (1990).
  - The long-run elasticity of new knowledge creation with respect to the existing stock is at least as large as unity.
- Impact of patent stock on measured productivity:
  - The long-run impact of the knowledge (patent) stock on TFP is small: doubling the stock of knowledge (patents) is estimated to increase TFP by only 10 percent in the long run.
- Interpretation regarding Jones (1995a):
  - The weak relationship documented by Jones between R&D workforce and TFP growth is not necessarily evidence of weak intertemporal spillovers.
  - Knowledge generation is an intermediate step; diffusion and application of new knowledge into productive sectors appear slow.
  - Paper provides evidence that the rate of diffusion of new knowledge into the productive sector of the U.S. economy has been slow over the past 20 years.

### VI. Growth and policy implications
- If φ≥1 (strong spillovers): Policies that permanently increase labor devoted to R&D (e.g., subsidies encouraging research) can have permanent effects on the long-run growth rate, consistent with endogenous growth models (Romer).
- Despite strong spillovers in knowledge production, measured TFP responds modestly to increases in patent stock (doubling patents → TFP +10 percent), implying limited measured productivity gains from patent accumulation alone.
- Policy interpretation:
  - Increasing R&D inputs likely raises knowledge production substantially due to strong spillovers.
  - However, policymakers should account for slow diffusion/application of knowledge into TFP; complementary policies to speed diffusion and embodiment of knowledge into production may be necessary to realize aggregate productivity gains.

### VII. Organization of the paper (structure)
- Section II: Simple R&D-based growth model and Romer–Jones debate.
- Section III: Data description for inputs and output of the knowledge production function.
- Section IV: Univariate and multivariate time-series analysis and estimation of the knowledge production function.
- Section V: Concluding remarks.

*Source: _wp0574 - 2. Log of Domestic Patent Applications .........................................................................27*

### 1.4 percent per year over the postwar period. Therefore, L

### _wp0574 - 1.4 percent per year over the postwar period. Therefore, L

### Critique of Romer (1990) knowledge production function
- Romer’s knowledge production function (equation (6) in the source) implies scale effects: long-run growth should increase with the level of research inputs.
- Time series evidence contradicts this implication:
  - Labor in research (LA), measured by number of scientists and engineers engaged in R&D for the U.S., rises from about 100,000 in 1950 to about 1 million by 1997.
  - Because LA is not relatively constant and exhibits a strong upward trend, Romer’s assumption (φ = 1) and resulting scale effects are inconsistent with the observed time series.

### Jones (1995b) alternative specification and steady state logic
- Jones proposes relaxing the assumption φ = 1 (i.e., φ < 1) in a modified knowledge production function to reconcile the model with data.
- Key algebraic relationships from the source:
  - Modified knowledge production steady-state condition (equation (9)): Ȧ/A = λ (L_A)^{φ−1}
  - Steady-state growth of A solved (equation (10)): g_A = λ (L_A)^{φ−1}
  - Under steady state where growth of researchers equals population growth n (i.e., L̇_A/L_A = n), Jones obtains (equation (11)): g_A = λ n^{1−φ}
- Interpretations and implications:
  - If φ < 1, a constant steady-state growth of A can coexist with a rising L_A; the long-run growth rate of the stock of knowledge depends on the growth rate of L_A rather than its level.
  - Positive knowledge spillovers (φ > 0) are not ruled out, but the magnitude must be less than unity to match time series evidence.
  - Long-run growth in Jones’s model depends on φ, λ, and n—parameters typically treated as exogenous—implying long-run growth is invariant to policy changes such as subsidies to R&D (policy changes affect transitional dynamics and long-run levels but not long-run growth rate).

### Empirical data, variables, and measurement
- Sample frequency and period:
  - Annual data available from 1948 to 1997.
- Variables used (natural logs for levels):
  - Patent applications (used to construct knowledge flows).
  - Stock of patents (constructed from cumulated patent filings).
  - Number of scientists and engineers engaged in R&D (LA).
  - Total factor productivity (TFP) as proxy for Ȧ/A.
- Choice of patent measure:
  - Domestic patent applications (DP) filed at the U.S. Patent and Trademark Office (USPTO) used to measure new knowledge.
  - Rationale: patent applications are viewed as outputs of innovative activity and better approximate knowledge flows than R&D expenditures.
- Stock construction:
  - Stock of knowledge (STP) derived from cumulated total patents applied for by U.S. and foreign inventors using perpetual inventory method with a depreciation rate of 15 percent.
  - Robustness: experiments with 0, 5, and 10 percent depreciation rates made very little difference; results are not sensitive to depreciation choice.

### Empirical patterns and stylized facts on patenting and knowledge
- Domestic patent applications (dp):
  - Overall upward trend over 1948–1997.
  - Average annual growth rate of domestic patent applications: 1.7 percent between 1948 and 1997.
  - Sub-period behavior:
    - 1948–1965 average annual growth rate: 1.6 percent.
    - Mid 1960s to mid 1980s: relative constancy.
    - Since about 1985: dramatic increase at an average annual rate of 5.1 percent.
- Potential explanations for the mid-1980s surge in patenting:
  - Institutional change: establishment of the Court of Appeals of the Federal Circuit (CAFC) in 1982, which unified patent appeals and is widely perceived as “pro-patent,” possibly strengthening patent protection and raising patenting incentives.
  - Real increase in discovery and innovation: evidence of “explosion of formation of new firms and innovation in the high-tech industries” in the 1980s and 1990s.
  - Productivity improvements in research: use of information technology in discovery boosting research productivity.
  - Reallocation of R&D toward applied activities and improved R&D management increasing patentable discoveries.
- Stock of total patent applications (stp):
  - Strong upward trend over 1948–1997.
  - Stock constructed via perpetual inventory with 15 percent depreciation.

### Policy and modeling implications
- The degree of positive knowledge spillovers assumed in R&D-based endogenous growth models is empirically important and cannot be arbitrarily large if models are to match time series behavior.
- With φ < 1:
  - Scale effects postulated by Romer (1990) are removed.
  - Long-run growth becomes tied to parameters φ, λ, and n—not directly to policy instruments like R&D subsidies—so policy shifts alter levels and transitional growth but not the long-run growth rate.
- Jones’s conclusion cited in the source: “Nothing in the U.S. experience appears to have had a permanent effect on growth. In light of this evidence, the invariance result maybe exactly what the data requires.”

*Source: _wp0574 PDF chapter content (pages excerpted).*

### 1.9 percent. There also appears to be a substantially stronger trend since the mid 1980s:

### _wp0574 - 1.9 percent. There also appears to be a substantially stronger trend since the mid 1980s:

### Empirical trends in patent stocks, R&D inputs, and productivity
- Stock of total patent applications:
  - Prior to 1985, the stock grew at an average annual rate of 1.2 percent.
  - After 1985, the average annual growth rate of the stock more than tripled to about 4 percent.
  - The more rapid increase in the stock since the mid 1980s captures the more rapid increase in (the number of) domestic patent applications that occurred over that period.
- Domestic vs. foreign patent applications:
  - By definition, the stock of total patent applications includes (1) the cumulated numbers of domestic patent applications (i.e., patent applications by U.S. inventors at the U.S. Patent Office), and (2) the cumulated numbers of foreign patent applications (i.e., patent applications by foreign inventors at the U.S. Patent Office).
  - The data on the number of foreign patent applications is characterized by a sustained smooth rise over the period 1948–97. There is no evidence of a more rapid increase in the series beginning in the mid 1980s. Thus, the more rapid increase in the stock of total patent applications since the mid 1980s indeed captures the more rapid increase in the numbers of domestic patent applications.
- Scientists and engineers engaged in R&D (s&e):
  - Series represents scientists and engineers employed in industry, the federal government, educational institutions, and nonprofit organizations.
  - The number of R&D scientists and engineers grew at an average annual rate of 4.3 percent over the period 1948–97.
  - The series exhibits a very strong upward trend over the past 50 years, with a decline in employment in the late 1960s through the early 1970s associated with reduced federal and business funding for certain research programs (e.g., space-related R&D).
- Total factor productivity (tfp) for the private business sector:
  - Grew at an average annual rate of about 1.4 percent between 1948 and 1997.
  - Average annual growth rate prior to 1973: 2.1 percent.
  - Average annual growth rate after 1973: about 0.7 percent (the productivity slowdown).

### Data sources and series construction
- s&e data:
  - Period 1979–97 obtained from National Science Foundation, Science and Engineering Indicators-2000.
  - Years prior to 1979 taken from Jones (2002) and Machlup (1962), who obtain data from NSF (1993, 1962, 1961, and 1955).
- tfp data:
  - Obtained from Larry Rosenblum at the Office of Productivity and Technology, the Bureau of Labor Statistics, U.S. Department of Labor.
- Patent stock measures:
  - stp: log of the stock of total patent applications (domestic and foreign).
  - sdp: stock of domestic patents constructed from the number of domestic patent applications using the perpetual inventory method (statistical arguments for stp apply to sdp).
- Time span referenced: 1948–97 for many series; specific events noted in 1985, 1994, 1995, 1996.

### Theoretical framework and long-run relationships
- Two long-run relationships specified in log terms:
  - Knowledge production function: dp = F(stp, s&e) where dp is log of number of domestic patent applications, stp is log of stock of total patent applications, s&e is log of number of scientists and engineers engaged in R&D.
  - Productivity relation: tfp = G(stp, dp, s&e) where tfp is log of total factor productivity.
- Assumptions:
  - Function F(.) is assumed to be linear in log-levels of variables.
  - R&D scientists and engineers produce U.S. patents while drawing upon the “world” stock of knowledge.
  - Total stock of patents represents cumulative R&D output that leads to higher productivity.

### Econometric approach and steps
- Modeling approach: general-to-specific modeling (Hendry (1986)).
- Steps:
  1. Examine time series properties, patterns, trends, stationarity, and order of integration.
  2. Form a Vector Autoregressive Regression (VAR) system; test lag length, residual diagnostics, and stability.
  3. Test the system for potential cointegration relationships using Johansen procedure.
  4. Interpret cointegrating relations and test for weak exogeneity; conditional error correction model would follow (not performed here).
- VAR system: includes dp (log domestic patent applications), tfp (log total factor productivity), stpl1 (lagged log stock of total patent applications), s&e (log number of scientists and engineers engaged in R&D), plus a constant, trend, and three dummy variables:
  - Stepdum86: one after 1985, zero otherwise (capture dramatic increase in patenting since mid 1980s).
  - Impulse9495: one in 1994 and 1995, zero otherwise (captures institutional changes in U.S. patent policy and twelve-year patent renewal fees collected in 1994/95).
  - Impulse96: one in 1996, zero otherwise (captures instantaneous negative response to increased cost of patent applications).

### Integration (unit root) findings
- Augmented ADF tests in levels and first differences with constant and trend:
  - Cannot reject the null hypothesis of a unit root for all four variables in levels.
  - In first differences:
    - Domestic patents, total factor productivity, and scientists and engineers engaged in R&D reject the null of a unit root.
    - The stock of patents (stp) does not reject; recursive analysis indicates non-constant coefficient and a break around 1985.
  - Plot of first difference of log patent stock shows apparent permanent shift in mean starting in the mid 1980s.
  - Perron (1989) structural break procedure: could not reject the hypothesis of an I(1) process after correcting for the structural mean shift.

### VAR specification, lag selection, and diagnostics
- Lag selection:
  - Sequential testing from VAR(4) down to VAR(1).
  - None of the F statistics for simplifications are significant at 1 percent, 5 percent, or 10 percent critical values.
  - Schwarz criterion (SC) and Hannan-Quinn criterion (HQ) minimized at VAR(1).
  - Conclusion: VAR(1) is statistically valid.
- Residual diagnostics for VAR(1):
  - F-test for no residual vector serial correlation: not rejected.
  - Chi-square test for joint normality of residuals: not rejected.
  - F-tests for no residual vector heteroskedasticity: not rejected.
  - Conclusion: VAR(1) residuals appear normal, homoskedastic, and serially uncorrelated.
- Stability and constancy:
  - Recursive estimation and 1-step ahead residuals: model fairly stable.
  - 1-step ahead Chow tests and Break-point Chow tests indicate VAR stable at the 1 percent significance level.

### Cointegration analysis and main empirical findings
- Johansen cointegration tests (Table 4 summary):
  - Panel A:
    - Test of r=0: Trace statistic = 184.03; null soundly rejected (no measurable p-value).
    - Test of r≤1 vs >1: Trace statistic = 58.87; p-value = [0.00]. Suggests at least two cointegrating vectors.
    - Cannot reject the null hypothesis that there is at most two cointegrating relations.
  - Panel B:
    - Two cointegrating vectors presented, normalized on (domestic) patent filings and total factor productivity respectively.
    - Interpreted as: (1) knowledge production function, and (2) determinants of total factor productivity.
  - Panel C:
    - Reports feedback coefficients (α) and standard errors associated with each long-run equation for the variables in first differences.
- Identification and restrictions:
  - Unrestricted long-run (cointegrating) solutions implied as:
    - dp = β11 + β12 tfp + β13 stpl1 + β14 s&e + β15 Trend
    - tfp = β21 + β22 dp + β23 stpl1 + β24 s&e + β25 Trend
  - Two restrictions required for just-identification:
    - β12 = 0 (no direct effect from total factor productivity in the knowledge production relation).
    - β21 = 0 (current patent applications should not determine productivity).
  - Over-identifying restrictions tested:
    - Excluding s&e from the second cointegrating vector by testing β24 = 0:
      - Likelihood ratio test: χ2(1) = 1.43 [0.23] (p-value in square brackets).
      - The restriction β24 = 0 cannot be rejected.
- Weak exogeneity tests (feedback coefficients):
  - Hypotheses:
    - α21 = 0: cointegrating relation for new patent flows (dp) does not help explain changes in tfp (tfp∆).
    - α32 = 0: total factor productivity does not provide information about changes in the lagged stock of knowledge (stpl1∆); stpl1 may be weakly exogenous with respect to the second cointegrating vector.
  - The feedback coefficients α21 and α32 appear numerically small in Panel C of Table 4 (full statistical inference and exact values reported in Table 4).

### Interpretation and economic implications emphasized in the content
- Empirical evidence supports:
  - A substantive role for R&D scientists and engineers in generating patents (dp) drawing on the accumulated stock of knowledge (stp).
  - A positive dependence of total factor productivity on the stock of patents (consistent with micro productivity literature: Jaffe, Henderson, and Trajtenberg (1993) and Thompson and Kean (2004)).
  - A structural shift in patenting behavior beginning in the mid 1980s, captured by Stepdum86 and reflected in the growth of the patent stock.
- Econometric implications:
  - Nonstationarity of series calls for cointegration methods to retain long-run information.
  - VAR(1) with specified dummies yields a stable system suitable for cointegration analysis.
  - Two cointegrating relations identified and interpretable as knowledge production and productivity determinants; over-identifying restrictions and weak exogeneity tests guide model reduction and inference.

*Italic: Content based only on the supplied PDF excerpt._wp0574 - 1.9 percent. There also appears to be a substantially stronger trend since the mid 1980s:*

### 4. If all three over-identifying restrictions are imposed, the joint hypothesis cannot be

### _wp0574 - 4. If all three over-identifying restrictions are imposed, the joint hypothesis cannot be rejected: χ2(3) = 4.05 [0.26].

### Restricted Cointegration Model — overview
- Table 5 reports results from three restrictions on the just identified model, split into:
  - Panel A: restricted (and identified) estimates for the cointegrating vectors, the β’s, with standard errors.
  - Panel B: feedback coefficients estimates, α’s, with standard errors.
- Two cointegrating relationships are identified: the first interpreted as a long-run knowledge or “idea” production function; the second links total factor productivity (tfp) to the stock of knowledge.

### First cointegrating vector — knowledge (idea) production function
- Implied long run / cointegrating relationship as reported (equation 18):
  - 1.4361   0.208&0.023dpstplseTrend=+−                                        (18)
- Key coefficients and interpretations:
  - Coefficient on lagged stock of knowledge, stpl1: 1.4361 (reported as highly significant).
    - Interpreted as positive spillovers or “standing on shoulders effect”.
    - Magnitude is “significantly greater than unity,” implying stronger spillovers than in Romer (1990) and Jones (1995b).
    - Likelihood ratio test rejects the null that that coefficient is unity: χ2(1)= 29.12 [0.00].
  - Coefficient on s&e (productivity of researchers), reported as 0.21.
    - Described as positive, highly significant and less than unity.
    - Matches micro-literature range 0.1 to 0.6 (Kortum (1993)).
  - Coefficient on time trend: reported as -2.3 percent (negative).
    - Negative trend reflects faster growth in R&D scientists and engineers (4.3 percent) than growth in domestic patent applications (1.7 percent); their difference roughly matches the trend coefficient (-2.3 percent).
    - Consistent with Griliches (1990) interpretation of a declining propensity to patent.

### Feedback coefficients for the knowledge production relation (panel B)
- All feedback coefficients for the first cointegrating vector are significant from zero, implying dp, stpl1, and s&e are not weakly exogenous and jointly correct deviations from long-run equilibrium.
- Reported feedback coefficients and interpretations:
  - ∆dp equation feedback coefficient (α11 in equation 15): –0.45 (significant).
    - Implies stability of the error correction mechanism; a positive deviation of dp from its long-run path declines next period.
  - ∆stpl1 equation feedback coefficient (α31 in equation 15): positive and significant.
    - If dp is above its long-run equilibrium path, growth of the stock of knowledge increases next period.
  - ∆s&e equation feedback coefficient (α41 in equation 15): negative and significant.
    - If dp is above its long-run equilibrium path, growth of R&D scientists and engineers slows next period, consistent with balanced growth implications.

### Second cointegrating vector — total factor productivity (tfp)
- Implied long run relationship reported (equation 19):
  - **0.1081   0.009tfpstplTrend=+                                                                    (19)
- Key coefficients and interpretations:
  - Coefficient on time trend: reported as implying trend growth of total factor productivity of about 1 percent (highly significant), matching average annual growth rate over 1948–97.
  - Coefficient on lagged stock of knowledge (stpl1): 0.1081 (positive but small and reported as not significant).
    - Doubling the stock of knowledge increases tfp by about 10 percent in the long run (interpretation given).
    - Likelihood ratio test under null that the coefficient is zero: χ2(1) = 0.62 [0.43].
  - Interpretation: weak/significant diffusion of new knowledge into measured productivity may explain the insignificant coefficient; diffusion may have been slow and incomplete in OECD countries over the past 20 years, with U.S. data suggesting benefits of knowledge were not fully realized in measured productivity until possibly the late 1990s (beyond the sample).

### Feedback coefficients for the tfp relation
- The cointegrating relation for tfp enters the ∆tfp equation with the appropriate sign and is significant; feedback is relatively quick.
- Changes in scientists and engineers, ∆s&e, are not weakly exogenous:
  - If tfp is above its long-run level, it reduces s&e growth over several years (less “demand” for knowledge-producing workers).
  - This reduction in s&e has a negative effect on ∆dp, observed in a marginally significant feedback coefficient.
- The stock of knowledge (stpl1) is weakly exogenous with respect to the tfp cointegrating relation.
- Caution noted: interpretation of feedback coefficients from the tfp relation is less confident due to complexity and less theoretical guidance.

### Robustness check: using domestic stock of patents (sdpl1)
- Re-estimated VAR with dp, tfp, sdpl1 (lagged stock of domestic patents), and s&e; Johansen methodology yields 2 cointegrating vectors (reported in Table 6).
- Results relevant for spillovers:
  - Long-run impact of lagged stock of domestic patents on flow of new domestic patents (dp): unity and significant.
  - Long-run impact of lagged stock of domestic patents on tfp: coefficient 0.1 with standard error 0.1 (small and insignificant).
- Interpretation:
  - Using domestic-only patent stock yields a spillover coefficient of unity (consistent with Porter and Stern (2000) and Romer (1990)).
  - Using total (domestic + foreign) patent stock produces a larger spillover effect (coefficient 1.4361 reported earlier), consistent with researchers having access to a larger global pool of knowledge.

### Reconciling with prior time-series evidence (Jones 1995a)
- Romer (1990) implies spillover parameter of unity and proportional growth of knowledge to R&D scientists and engineers.
- Jones (1995a) measured knowledge via tfp and found a weak relationship between TFP growth and number of R&D scientists and engineers, concluding weak spillovers.
- Current paper’s findings:
  - When knowledge is measured via patents and the knowledge production function is directly estimated, the spillover parameter is unity or larger.
  - Thus, large knowledge spillovers can coexist with a weak observed relationship between TFP growth and R&D scientists/engineers if the produced knowledge has limited measured impact on TFP.
- Possible explanations for weak observed TFP relationship:
  - Small impact of produced knowledge on measured TFP (slow/complex diffusion and embodiment).
  - Measurement problems and errors-in-variables attenuating estimated relationships; Johansen dynamic approach may mitigate some measurement issues by using lagged variables as instruments.

### Main empirical conclusions (V. Reconciliation and Conclusion)
- Two potential long-run relationships identified:
  - Knowledge production function (new knowledge depends on existing stock and number of research workers) — evidence of large knowledge spillovers.
  - Positive but small and imprecise long-run dependence of tfp on the stock of knowledge.
- Quantitative summary of key elasticities:
  - Long–run elasticity of new knowledge with respect to existing stock of knowledge:
    - Unity when using domestic patent stock proxy.
    - Greater than unity (1.4361 reported) when using global patent stock (domestic + foreign).
  - Long–run elasticity of tfp with respect to stock of knowledge: positive, small, and imprecise (examples: 0.1081; alternative estimate 0.1 with standard error 0.1).
- Policy implication emphasized:
  - Large knowledge spillovers imply potential sensitivity of long-run growth to policy variables (e.g., subsidies to R&D per Romer (1990)); small spillovers would imply long-run growth invariant to such policy (per Jones (1995b)).
- Suggested direction for further research:
  - Examine transmission mechanisms between the knowledge production function and productivity, timing and diffusion of knowledge into productive sectors, and measurement issues.

*Italic: Source — _wp0574 - 4. If all three over-identifying restrictions are imposed, the joint hypothesis cannot be rejected: χ2(3) = 4.05 [0.26].*

### 0.937 r = 0 183.03** [0.000]

### _wp0574 - 0.937 r = 0 183.03** [0.000]

### Cointegration test summary and trace statistics
- Trace test statistics and associated probabilities:
  - 0.937 r = 0 183.03** [0.000]
  - 0.687 r ≤ 1 58.87 ** [0.000]
  - 0.120 r ≤ 2 6.65 [0.993]
  - 0.019 r ≤ 3 0.88 [0.997]

### Estimated cointegrating vectors (unrestricted and restricted presentations)
- (A) Estimated Cointegrating Vectors β′ (unrestricted display)
  - Vector 1 coefficients:
    - dp: 1
    - tfp: -0.255
    - stpl1: -1.415
    - s&e: -0.195
    - Trend: 0.025
  - Vector 2 coefficients:
    - dp: 0.170
    - tfp: 1
    - stpl1: -0.418
    - s&e: -0.106
    - Trend: -0.002

- (B) Restricted Cointegration Analysis — Table 5 (selected entries)
  - Vector 1 (β′) reported elements and standard errors (in parentheses):
    - dp: 1
    - tfp: 0.0
    - stpl1: -1.436** (0.0619)
    - s&e: -0.208** (0.020)
    - Trend: 0.023** (0.001)
  - Vector 2 (β′) reported elements and standard errors:
    - dp: 0.0
    - tfp: 1
    - stpl1: -0.108 (0.141)
    - s&e: 0.0
    - Trend: -0.009** (0.002)

- (A) Restricted Cointegration Analysis — Table 6 (Domestic Patent Stock Model)
  - Vector 1 (β′) reported elements and standard errors:
    - dp: 1
    - tfp: 0.0
    - sdpl1: -1.000** (0.009)
    - s&e: -0.011* (0.004)
    - Trend: 0.00046* (0.00016)
  - Vector 2 (β′):
    - dp: 0.0
    - tfp: 1
    - sdpl1: -0.134 (0.127)
    - s&e: 0.0
    - Trend: -0.0088** (0.0013)

### Feedback coefficients α and standard errors
- Unrestricted α (Table header alignment corresponds to columns 1 and 2; SE(α) columns correspond to 1 and 2)
  - dp:
    - α1: -0.398
    - α2: -0.342
    - SE(α)1: 0.120
    - SE(α)2: 0.121
  - tfp:
    - α1: 0.062
    - α2: -0.109
    - SE(α)1: 0.050
    - SE(α)2: 0.050
  - stpl1:
    - α1: 0.132
    - α2: 0.022
    - SE(α)1: 0.008
    - SE(α)2: 0.008
  - s&e:
    - α1: -0.162
    - α2: -0.402
    - SE(α)1: 0.071
    - SE(α)2: 0.072

- Restricted α (Table 5 — reported significance indicators)
  - dp:
    - α1: -0.449** (SE 0.121)
    - α2: -0.184 (SE 0.111)
  - tfp:
    - α1: ----- (not estimated)
    - α2: -0.108* (SE 0.050)
  - stpl1:
    - α1: 0.135** (SE 0.008)
    - α2: ----- (not estimated)
  - s&e:
    - α1: -0.190** (SE 0.068)
    - α2: -0.405** (SE 0.053)

- Restricted α (Table 6 — Domestic Patent Stock Model)
  - dp:
    - α1: -0.341** (SE 0.109)
    - α2: -0.356** (SE 0.116)
  - tfp:
    - α1: ----- (not estimated)
    - α2: -0.102* (SE 0.048)
  - sdpl1:
    - α1: 0.162** (SE 0.001)
    - α2: ----- (not estimated)
  - s&e:
    - α1: -0.166** (SE 0.060)
    - α2: -0.413** (SE 0.069)

- Significance notation:
  - * and ** indicate rejection (at the 5 percent and 1 percent critical values) of the null hypothesis that a particular coefficient is zero. Tests are likelihood ratio statistics distributed under the null as χ2 with 1 degree of freedom.

### Model specification and sample
- VAR specification:
  - Includes a single lag on each variable (dp, tfp, sdpl1, s&e), a constant, trend, and three dummy variables: Stepdum86, Impluse9495, and Impulse96.
  - Estimation sample: 1953 to 1997.

### Data construction and sources (key points)
- Stock of total patent applications:
  - Source: U.S. Patent and Trademark Office counts of patent applications (invention, designs, plants).
  - Patent stock constructed via perpetual inventory:
    - Stock_t = (total number of patent applications)_t + Stock_{t-1} (1-d)  (equation (A1))
    - d denotes depreciation rate.
  - Initial stock alternatives:
    - Coe and Helpman (1995) method: Stock_0 = (number of patent applications)_0 / (g + d), where g is average annual growth rate over 1840–1999 and (number of patent applications)_0 is for 1840.
    - Or assume initial stock in 1840 equals number of patent applications in 1840 (i.e., no prior knowledge).
  - Typical assumed depreciation rate: 15 percent [see Griliches (1989, 1990)].
  - Alternative depreciation rates used in robustness checks: 0, 5, 10 percent. Results consistent and robust across this range.

- Domestic patent applications:
  - Data 1940–1999 from multiple USPTO tabulations and archival sources (tabulations available at http://www.uspto.gov/web/offices/ac/ido/oeip/taf/us_stat.pdf and several unpublished or printed historical sources).
  - For 1840–1939, domestic patent applications proxied by multiplying total patent applications by the fraction of total patent grants issued to U.S. inventors (following Kortum and Lerner (1998)).
  - Domestic patent stock cumulated via perpetual inventory with depreciation rate 15 percent.

- Scientists and Engineers engaged in R&D:
  - Data 1979–97 from National Science Foundation, Science and Engineering Indicators–2000 (http://www.nsf.gov/sbe/srs/seind00/start.htm).
  - Prior to 1979, data taken from Jones (2002) and Machlup (1962) who obtained data from NSF (1993, 1962, 1961, and 1955).

- Total Factor Productivity (TFP):
  - Data for private business sector obtained from Larry Rosenblum at the Office of Productivity and Technology, Bureau of Labor Statistics, U.S. Department of Labor.
  - TFP measures output per combined unit of labor and capital inputs; labor input constructed from hours-at-work cross-classified by education, experience, and gender and aggregated using wage-compensation shares.

*Source: _wp0574 - 0.937 r = 0 183.03** [0.000] (PDF chapter content).*

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