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Capital Shallowness: A Problem for New Zealand? - WP 05/05

Appendix 2

We test the sensitivity of our results by also running the relationship in first differences (see equation 14 below), from which we can obtain an estimate of the short-run elasticity of substitution (). The results are presented below, and give estimates for the short-run elasticities which are similar to the estimates obtained using equation (11), confirming the results found by equation (11) (although for Australia the short-run elasticity when the mining sector is excluded drops from 0.57 to 0.20 and from 0.33 to 0.09 when both mining and agriculture, forestry and fishing are excluded).

(14)    

Appendix Table 2: Results of Regression in First Differences
Dependent Variable:
Base Case Excluding Housing Excluding Mining
Australia New Zealand Australia New Zealand Australia New Zealand
First difference of the Log of relative prices (α1) 0.21*
(1.98)
0.25***
(3.43)
0.27*
(2.14)
0.19***
(3.58)
0.20
(0.31)
0.28***
(3.90)
Adjusted R-squared 0.16 0.65 0.20 0.68 -0.09 0.66

*= sig. at 10% level, **=sig. at 5% level, ***=sig. at 1% level

All of the New Zealand estimated equations have been corrected for autocorrelation, as well as the Australian estimated equation excluding mining.

Appendix Table 3: Results of Regression in First Differences
Dependent Variable:
Excluding Mining and Agriculture, Forestry and Fishing Using new OECD capital stock
Australia New Zealand Australia New Zealand
First difference of the Log of relative prices (α1) 0.09
(0.46)
0.24***
(3.37)
0.42***
(3.35)
0.36***
(4.47)
Adjusted R-squared -0.15 0.56 0.42 0.62

The Canadian equation has been corrected for Autocorrelation, as well as the Australian and New Zealand equations excluding Mining and Agriculture, Forestry and Fishing, and the New Zealand equation using the new OECD capital stock data.

We also estimated the short-run and long-run elasticities using an error correction framework. This framework requires that the variables be cointegrated. The unit root tests (see below) indicate that the variables cannot be cointegrated for both New Zealand and Australia. However, the unit root test has been found to have very low power in small samples. Thus we estimated the error correction models as a further robustness check (see equation 15 below). The results are presented below, and again give similar results for the short-run elasticities. The long-run elasticities are also very similar. Exceptions are the model for Australia when housing is excluded, and the model for New Zealand using the new OECD capital stock. The long-run elasticity for Australia when housing is excluded was 0.94 and not statistically different from one; in the error correction model this decreases to 0.58 and becomes significantly smaller than unity. When the error correction model is run for New Zealand using the new OECD capital stock data, the long-run elasticity estimate becomes statistically smaller than unity (previously the estimate was 0.93 and not significantly different from one).

(15)     

Appendix Table 4: Error Correction Estimation Results
Dependent Variable:
Base Case Excluding Housing Excluding Mining
Australia New Zealand Australia New Zealand Australia New Zealand
First difference of the Log of relative prices (α1) (short-run elasticity) 0.30
(1.45)
0.33***
(5.80)
0.28
(1.55)
0.27***
(4.70)
0.54**
(3.08)
0.30***
(4.87)
Lagged log of the capital-labour ratio (λ) -0.38
(-1.54)
-0.56***
(-5.87)
-0.31
(-1.31)
-0.63***
(-4.34)
-0.52**
(-2.76)
-0.66***
(-4.43)
Lagged log of relative prices (λβ1) 0.29
(1.41)
0.35***
(5.61)
0.18
(0.68)
0.28***
(3.55)
1.14**
(4.37)
0.39***
(5.49)
Time Trend (λβ2) 0.003
(0.53)
0.008***
(4.71)
0.004
(0.71)
0.01****
(3.57)
-0.01*
(-2.21)
0.007***
(4.15)
Long-run elasticity () 0.76*
(-1.46)a
0.63***
(-15.14)a
0.58**
(-1.93)a
0.44***
(-25.92)a
2.19***
(5.22)a
0.59***
(-18.94)a
Adjusted R-squared 0.28 0.87 0.30 0.78 0.66 0.87

a: These t-ratios were calculated using the Delta method and indicate whether the long-run elasticities are statistically different from one.

Appendix Table 5: Error Correction Estimation Results

Appendix Table 5: Error Correction Estimation Results
Dependent Variable:
Excluding Mining and Agriculture, Forestry and Fishing Using the new OECD capital stock
Australia New Zealand Australia New Zealand(smoothed)
First difference of the Log of relative prices (α1) (short-run elasticity) 0.28*
(1.90)
0.30***
(4.80)
0.38**
(2.71)
0.38***
(6.10)
Lagged log of the capital-labour ratio (λ) -0.50**
(-2.36)
-0.68***
(-4.30)
-0.76**
(-2.72)
-0.58***
(-6.40)
Lagged log of relative prices (λβ1) 0.79***
(3.65)
0.38***
(4.89)
0.44**
(2.59)
0.44***
(6.09)
Time Trend (λβ2) -0.008
(-0.97)
0.007***
(3.86)
0.007
(1.32)
0.01***
(5.29)
Long-run elasticity () 1.58***
(2.83)a
0.56***
(-19.93)a
0.58***
(-7.44)a
0.76***
(-9.80)a
Adjusted R-squared 0.58 0.84 0.63 0.89

a: These t-ratios were calculated using the Delta method and indicate whether the long-run elasticities are statistically different from one.

The unit root test used in this study is the Augmented Dickey-Fuller t test. The optimal lag length has been chosen using Schwarz criterion.

All variables were tested first to ascertain whether a trend should be included in the unit root test. If we did not reject the unit root null hypothesis, we took the first difference of the series and reran the test excluding a time trend.

New Zealand’s capital-labour ratio and capital-labour ratio excluding mining were both stationary, as well as Australia’s adjusted relative price series excluding housing. All other series were found to be I(1), except Australia’s adjusted relative price series and New Zealand’s unadjusted relative price series excluding mining, which were both I(2).

Appendix Table 6: Unit Root Tests (levels)
  Australia New Zealand
  Lag Order t-statistic Lag Order t-statistic
Log(K/L) 0 4.44 1 -3.37*
Log(w/r) (adjusted) 4 -3.07 0 -1.13
Log(K/L) excluding housing 0 4.66 1 1.74
Log(w/r) (adjusted, excluding housing OS) 4 -4.78*** 2 -2.92
Log(K/L) excluding mining 4 0.86 3 -4.75***
Log(w/r) (unadjusted, excluding mining OS and COE) 0 -0.07 4 -1.25
Log(K/L) excluding mining and agriculture 4 2.10 4 -0.24
Log(w/r) (unadjusted, excluding mining and agriculture OS and COE) 0 0.16 4 -1.45
Log(K/L) using new OECD capital stock 4 1.11 0 -1.50
Log(w/r) (adjusted, using new OECD capital stock) 1 -1.69 0 -0.24

*= sig. at 10% level, **=sig. at 5% level, ***=sig. at 1% level

Appendix Table 7: Unit Root Tests (first differences)
  Australia New Zealand
  Lag Order t-statistic Lag Order t-statistic
Log(K/L) 0 -5.92*** - -
Log(w/r) (adjusted) 4 -1.54 0 -2.84*
Log(K/L) excluding housing 0 -3.44** 0 -2.92*
Log(w/r) (adjusted, excluding housing OS) - - 0 -2.74*
Log(K/L) excluding mining 0 -2.93* - -
Log(w/r) (unadjusted, excluding mining OS and COE) 0 -4.84*** 0 -2.39
Log(K/L) excluding mining and agriculture 0 -3.01** 3 -4.37***
Log(w/r) (unadjusted, excluding mining and agriculture OS and COE) 0 -4.12*** 0 -3.01**
Log(K/L) using new OECD capital stock 3 -5.19*** 0 -3.56**
Log(w/r) (adjusted, using new OECD capital stock) 0 -4.69*** 0 -3.38**

*= sig. at 10% level, **=sig. at 5% level, ***=sig. at 1% level

Appendix Table 8: Unit Root Tests (second differences)
  Lag Order t-statistic
Australia Log(w/r) (adjusted) 3 -3.24**
NZ Log(w/r) (unadjusted, excluding mining OS and COE) 4 -4.79***

*= sig. at 10% level, **=sig. at 5% level, ***=sig. at 1% level

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