The aggregate productivity shock is one of the key variables in the empirical investigations. To measure it, this study uses the growth accounting framework.
Under the assumptions of competitive markets and constant returns to scale, the growth accounting framework decomposes output growth into two parts:
portions attributed to growth in inputs and changes in aggregate productivity.38 From the production function introduced in Section 2, the conventional Solow residual can be constructed as follows:
∆Aconv= ∆Y − α∆K − (1 − α)∆L, (A.8) where ∆Y is the growth rate of output, α is the factor share distributed to capital, ∆L is the rate of labor hour growth, ∆K is the rate of growth of physical capital, and ∆Aconv is the conventional Solow residual.39
Thus, the adjusted Solow residual can be constructed,
∆Aadj = ∆Y − α∆K − (1 − α)∆L − α∆u
= ∆Aconv− α∆u, (A.9)
where ∆Aadj is the adjusted Solow residual, and ∆u is the growth rate of the utilization rate.
Equation (A.9) shows that as long as the utilization rate is not constant, i.e.,
∆u = 0, the adjusted and the conventional Solow residual are not equivalent, i.e., ∆Aadj= ∆Aconv.
38Hall(1990) said that the following theorem would hold under Solow’s assumptions: the productivity residual is uncorrelated with any variable that is uncorrelated with the rate of growth of true productivity. This is a restatement of Solow’s basic results in which the residual measures the shift of the production function.
39This study constructs the conventional and the adjusted Solow residuals using both the actual and the average shares. The results of the study are not sensitive to the capital share parameter,α
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Table 1: Ng-Perron Unit Root Tests
This table reports the results of the Ng-Perron unit root tests. Significant coefficients at 5% are indicated with **. The sample period is from 1949 to 2001. All variables are in natural logarithm. REALR stands for gross real asset returns, K for capital stock, L for labor hours, and u for utilization rate, SRL for aggregate productivity level, and ADJSRL for adjusted aggregate productivity level.
Ng-Perron test statistics MZa MZt MSB MPT
REALR -1.357 -0.682 0.503 14.549
K -1.925 -0.700 0.363 9.854
L -7.364 -1.709 0.232 4.056
u -20.754** -3.131** 0.151** 1.494**
SRL -1.425 -0.549 0.385 11.23
ADJSRL 1.186 0.965 0.814 50.15
1% -13.8 -2.58 0.174 1.78
Asymptotic Critical Values 5% -8.1 -1.98 0.233 3.17
10% -5.7 -1.62 0.275 4.45
Table 2: Augmented Dickey-Fuller Unit Root Tests
This table reports the results of the Augmented Dickey-Fuller unit root tests.
Augmented Dickey-Fuller test statistics
REALR -1.45
K -2.39
L 0.28
u -4.188**
SRL -1.52
ADJSRL -2.11
1% -3.56
MacKinnon’s critical values 5% -2.92
10% -2.60
Table 3: Descriptive Statistics
GREALR stands for gross real asset return growth, SR for conventional Solow residual, ADJSR for adjusted Solow residual, GY for real GDP growth, GK for capital stock growth, GL for labor-hour growth, and GCU for utilization growth. The sample period is from 1949 to 2001.
GREALR SR ADJSR GY GL GK GCU
Mean 0.0271 0.0158 0.0148 0.0358 0.0139 0.0300 0.0026 Median -0.0434 0.0174 0.0147 0.0375 0.0153 0.0308 0.0023 Max 0.9318 0.0879 0.0499 0.0875 0.0474 0.0473 0.1000 Min -0.3317 -0.0220 -0.0106 -0.0203 -0.0227 0.0138 -0.1151
SD 0.2454 0.0205 0.0126 0.0243 0.0149 0.0071 0.0452
Table 4: Contemporaneous Correlations
GREALR SR ADJSR GY GL GK GCU
GREALR 1.00
SR 0.48 1.00
ADJSR 0.19 0.53 1.00
GY 0.45 0.91 0.23 1.00
GL 0.14 0.17 -0.57 0.55 1.00
GK -0.09 0.02 0.02 0.22 0.24 1.00
GCU 0.42 0.79 -0.09 0.90 0.61 0.01 1.00
Table 5: Pairwise Granger-Causality Tests
This table reports the results of the Granger-Causality tests. GREALR stands for gross real asset return growth, SR for conventional Solow residual, ADJSR for adjusted Solow residual. The sample period is from 1949 to 2001. The number of lagged variable is 1.
Null Hypothesis: F-Statistic Probability
SR does not Granger Cause GREALR 1.699 0.199 GREALR does not Granger Cause SR 1.801 0.186 ADJSR does not Granger Cause GREALR 11.328 0.002**
GREALR does not Granger Cause ADJSR 18.409 0.000**
Table 6: Linear Regression Results
The sample period is from 1950 to 2001. Significant coefficients at 5% and 10%
are indicated with ** and *, respectively. The numbers in the parentheses are standard errors. GREALR stands for gross real asset return growth, SR for conventional Solow residual, GK for capital stock growth, GL for labor-hour growth, ADJSR for adjusted Solow residual, and GCU for utilization growth.
Case I Case II Case III Case IV Conventional Solow Residual
Adj. R-squared 0.215 0.187 0.036 0.292
Adjusted Solow Residual
Adj. R-squared 0.197 0.031 0.067 0.406
Table 7: Ramsey RESET Tests
The table reports results of the Ramsey RESET tests. The tests are based on the results from the regressions (Case I) in Table 6.
Adjusted Solow Residual
F-statistic 1.996 Probability 0.164 Log likelihood ratio 2.214 Probability 0.136
Conventional Solow Residual
F-statistic 1.348 Probability 0.252 Log likelihood ratio 1.442 Probability 0.230
Table 8: Variance Decompositions for Changes in Real Asset Returns due to Adjusted Solow Residual & Conventional Solow Residual
The table reports the results of the variance decompositions. FE stands for forecast error, GREALR for gross real asset return growth, SR for conventional Solow residual, ADJSR for adjusted Solow residual, GK for capital stock growth, GL for labor-hour growth, and GCU for utilization growth.
The orders: Case 1 (Benchmark with adjusted Solow residual): ADJSR, GL, GCU, GK, GREALR; Case 2 (Benchmark with conventional Solow residual):
SR, GL, GK, GREALR; Case 3 (Kiyotaki & Moore with adjusted Solow resid-ual): GREALR, GL, GCU, GK, ADJSR; Case 4 (Kiyotaki & Moore with con-ventional Solow residual): GREALR, GL, GK, SR.
VAR1 Case1 Case2 Case3 Case4
Period F.E. (%) F.E. (%) F.E. (%) F.E. (%)
1 0.175 0.33 0.193 26.85 0.175 0.00 0.193 0.00 2 0.246 13.47 0.254 32.46 0.246 3.35 0.254 0.34 3 0.255 15.76 0.255 32.21 0.255 6.49 0.255 0.70 4 0.259 15.46 0.256 32.45 0.259 6.29 0.256 0.71 5 0.260 15.90 0.256 32.50 0.260 6.51 0.256 0.74 6 0.260 15.89 0.256 32.52 0.260 6.52 0.256 0.75 7 0.261 15.87 0.256 32.53 0.261 6.51 0.256 0.75 8 0.261 15.89 0.256 32.53 0.261 6.53 0.256 0.75 9 0.261 15.89 0.256 32.53 0.261 6.53 0.256 0.75 10 0.261 15.89 0.256 32.53 0.261 6.53 0.256 0.75 VAR6
Period F.E. (%) F.E. (%) F.E. (%) F.E. (%)
1 0.205 16.42 0.220 33.27 0.205 0.00 0.220 0.00 2 0.324 38.75 0.316 32.58 0.324 6.28 0.316 0.23 3 0.351 44.18 0.318 32.42 0.351 7.66 0.318 0.64 4 0.371 43.08 0.329 30.48 0.371 10.54 0.329 3.28 5 0.375 42.37 0.334 30.05 0.375 10.48 0.334 4.08 6 0.386 42.43 0.339 29.40 0.386 11.82 0.339 4.79 7 0.392 41.18 0.340 29.79 0.392 11.48 0.340 5.27 8 0.398 42.04 0.346 28.99 0.398 11.89 0.346 6.38 9 0.401 41.49 0.348 28.65 0.401 11.84 0.348 7.02 10 0.407 42.51 0.350 28.75 0.407 11.96 0.350 7.23
Table 9: Variance Decompositions for Real Asset Returns due to Adjusted Solow Residual & Conventional Solow Residual - Level Specifications
The table reports the results of the variance decompositions. FE stands for forecast error, REALR for gross real asset return, SR for conventional Solow residual, ADJSR for adjusted Solow residual, GK for capital stock growth, GL for labor-hour growth, and GCU for utilization growth.
The orders: Case 1 (Benchmark with adjusted Solow residual): ADJSR, GL, GCU, GK, REALR; Case 2 (Benchmark with conventional Solow residual): SR, GL, GK, REALR.
Figure 1: Changes in Asset Returns vs. the Adjusted Solow Residual & Changes in Asset Returns vs. the Conventional Solow Residual : In the first plot, the effects of other variables (GK, GL) are partialled out in SR. In the second plot, the effects of other variables (GK, GL, GCU) affecting changes in asset returns are partialled out in ADJSR.
Figure 2: Impulse Response Functions, VAR(1): one standard deviation pertur-bation to adjusted Solow residual. GREALR stands for gross real asset return growth, ADJSR for adjusted Solow residual, GK for capital stock growth, GL for labor-hour growth, and GCU for utilization growth. The order - ADJSR, GL, GCU, GK, GREALR.
Figure 3: Impulse Response Functions, VAR(6): one standard deviation per-turbation to adjusted Solow residual. The order - ADJSR, GL, GCU, GK, GREALR.
Figure 4: Impulse Response Functions, VAR(1) - Level: one standard deviation perturbation to adjusted Solow residual. REALR stands for gross real asset return. The order - ADJSR, GL, GCU, GK, REALR.
Figure 5: Impulse Response Functions, VAR(6) - Level: one standard deviation perturbation to adjusted Solow residual. The order - ADJSR, GL, GCU, GK, REALR.
Figure 6: Impulse Response Functions, VAR(1): one standard deviation per-turbation to conventional Solow residual. The order - SR, GL, GK, GREALR.
Figure 7: Impulse Response Functions, VAR(6): one standard deviation per-turbation to conventional Solow residual. The order - SR, GL, GK, GREALR.
Figure 8: Impulse Response Functions, VAR(1) - Level: one standard deviation perturbation to conventional Solow residual. The order - SR, GL, GK, REALR.
Figure 9: Impulse Response Functions, VAR(6) - Level: one standard deviation perturbation to conventional Solow residual. The order - SR, GL, GK, REALR.
Figure 10: Impulse Response Functions, VAR(1): one standard deviation per-turbation to changes in gross real asset returns. The order - GREALR, GL, GCU, GK, ADJSR.
Figure 11: Impulse Response Functions, VAR(6): one standard deviation per-turbation to changes in gross real asset returns. The order - GREALR, GL, GCU, GK, ADJSR.
Figure 12: Impulse Response Functions, VAR(1): one standard deviation per-turbation to changes in gross real asset returns. The order - GREALR, GL, GK, SR.
Figure 13: Impulse Response Functions, VAR(6): one standard deviation per-turbation to changes in gross real asset returns. The order - GREALR, GL, GK, SR.
Figure 14: Capital Utilization Rates: A Comparison between FRB official series vs. capital utilization data used by Burnside, Eichenbaum, and Rebelo (1995):
The sample period runs from 1972 to 1992.