Appendix 1: Tests of the Lag Structure | RDP 8806: Employment, Output and Real Wages

RDP 8806: Employment, Output and Real Wages Appendix 1: Tests of the Lag Structure

Bill Russell and Warren J. Tease

September 1988

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The Koyck specification was tested against two less restrictive equations. The first had a general lag structure of the form:

The second was an error-correction model of the form:

The error correction model (2″) is nested in the more general model (2′) and imposes the restrictions b₂ = c₄−c₁, b₄ = c₅−c₂, and b₇=0. The Koyck, in turn, is nested in both (2′) and (2″). To get from the general specification to the Koyck, the following restrictions are imposed; b₂ = 0, b₄ = 0 and b₇ = 0, while the following restrictions are imposed on the error correction model; c₄−c₁ = 0 and c₅−c₂ = 0. These restrictions are tested with a likelihood ratio test. Before presenting the test statistics, the estimates of 2′ and 2″ are reported in Tables 1 and 2, respectively.

Table 1
lnE_t = b₀ + b₁ lnY_t + b₂ lnY_t−1 + b₃ lnW_t + b₄ lnW_t−1 + b₅ T + b₆ lnE_t−1 + b₇ lnE_t−2
Variable	Coefficient	Standard Error
b₀	0.91	0.35
lnY_t	0.12	0.05
InY_t−1	0.003	0.05
lnW_t	−0.08	0.03
InW_t−1	−0.03	0.03
T	−0.0005	0.0002
lnE_t−1	0.85	0.13
lnE_t−2	−0.41	0.37
RBAR² = 0.99	SEE = 0.005	SSR = 0.0013
DW^[5] = 1.98	Q(24) = 25.51

Table 2
ΔlnE_t = c₀ + c₁ ΔlnY_t + c₂ ΔlnW_t + c₃ lnE_t−1 + c₄ lnY_t−1 + c₅ lnW_t−1 + c₆ T
Variable	Coefficient	Standard Error
c_o	0.95	0.32
WlnY_t	0.12	0.05
WlnW_t	−0.09	0.03
lnE_t−1	−0.19	0.05
lnY_t−1	0.12	0.03
T	−0.0005	0.0002
lnW_t−1	−0.12	0.02
RBAR² = 0.47	SEE = 0.004	SSR = 0.0013
DW = 1.98	Q(24) = 26.23

The results reported above suggest that the restrictions imposed by the Koyck specification are reasonable; the coefficients b₂, b₄ and b₇ are not significantly different from zero.

As in the Koyck specification, all the variables are significant and of the right sign. The results are very close to those of the Koyck. The short-run output and real wage elasticities are 0.12 and −0.09 respectively (compared with 0.12 and −0.11 for the Koyck). The long-run output and real wage elasticities are 0.63 and −0.63 respectively (compared with 0.65 and −0.61 in the Koyck).

Table 3 reports the results of testing the restricted equations.

Table 3
Likelihood Ratio Test of a Restricted Equation Against a More General Specification.
Restriction^[6]	Test Statistic^[7]	Critical Value 95%
General – Error Correction	0.90	7.82
General – Koyck	0.49	7.82
Error Correction – Koyck	0.54	5.99

Neither the restrictions imposed by the error-correction specification nor those imposed by Koyck on the more general model can be rejected. Furthermore, the restrictions imposed by the Koyck on the error-correction specification could not be rejected. For this reason, the results reported in the body of the paper employ the Koyck specification.

Footnotes

Durbin's h could not be calculated since T.Var(b₆)>1. [5]

This column refers to the restriction tested. The first (General – Error Correction) refers to the restrictions imposed by the error correction specification on the General model. [6]

The test statistic is −2(Lr-Lu), where Lr is the log likelihood value of the specification which imposes the restriction while Lu is the log likelihood value of the unrestricted alternative. It is distributed Χ²(k) where k is the number of restrictions. [7]