Testing exogeneity and invariance

Following Engle et al. (1983), the concepts of weak exogeneity and parameter invariance refer to different aspects of ‘exogeneity’, namely the question of valid conditioning in the context of estimation, and valid policy analysis, respectively. In terms of the ‘road-map’ of Figure 9.1, weak exogeneity of the conditional variables for the parameters of the wage-price model Dy (yt | zt, Yt-1, Zt-1) implies that these parameters are free to vary with respect to the parameters of the marginal models for output, productivity, unemployment, and exchange rates DZl (z1t | z2t, z3t, Yt_i, Zt-1). Below we repeat the examination of these issues as in Bardsen et al. (2003): we follow Johansen (1992) and concentrate the testing to the parameters of the cointegration vectors of the wage-price model. Valid policy analysis involves as a necessary condition that the coeffi­cients of the wage-price model are invariant to the interventions occurring in the marginal models. Such invariance, together with weak exogeneity (if that holds), implies super exogeneity.

Following Johansen (1992), weak exogeneity of z1t with respect to the cointegration parameters requires that the equilibrium-correction terms for wages and prices do not enter the marginal models of the conditioning levels variables. Table 9.3 shows the results of testing weak exogeneity of productivity, unemployment, and import prices[97] within the marginal system.

We observe that the weak exogeneity assumptions do not hold (at the 5% critical level) for import prices with respect to the long-run parameters, whereas those assumptions appear to be tenable for productivity and unemployment. Looking at the detailed results, we observe that it is the equilibrium correction term for the price equation ecmp, t that is significant for import prices (through the exchange rate equation). This means that the estimation of the long-run equations is slightly inefficient, whereas the finding of the two long-run rela­tionships (9.3)-(9.4) is likely to be a robust result due to the superconsistency of the cointegrating equations.

Подпись: Table 9.3 Testing weak exogeneity ecmw,t and ecmp,t Apit F (2,105) = 3.67[0.03] Aut F(2, 98) = 1.11[0.33] Aat F(2,105) = 2.45[0.09]

To test for parameter invariance, we need the interventions occurring in the parameterisations of Dzi (z1t | z2t, z3t, Yt-1, Zt-1). Consider therefore the following stacked form of the estimated single equation marginal models (9.7)-(9.13) in Section 9.3:

Подпись: AzMПодпись: + B • EqCM(Zt-1 i image192+ C • Xt + D • INTt + є*м ■


The matrix B contains the coefficients of the equilibrium correction terms (if any) in the marginal models (with the loadings along the diagonal). The matrix C contains the coefficients of the maintained exogenous variables Xt in the marginal models for z1t. Intervention variables affecting the mean of the variables under investigation—significant dummies and non-linear terms— are collected in the INTt matrix, with coefficients D. By definition, the elements in INTt are included because they pick up linear as well as non-linear features of z1t that are left unexplained by the information set underlying the wage-price model.8

To test for parameter invariance in the wage-price model, we test for the significance of all the intervention variables from all the marginal models (9.7)- (9.13) in Section 9.3.9 The results from adding the set of intervention variables to the wage-price model (9.5)-(9.6) are reported in Table 9.4.

The intervention variables are jointly insignificant in the wage-price sys­tem (with p-value = 0.32) as is seen from Table 9.4. As a specification test, this yields support to the empirical model in (9.5)-(9.6). However, we find that three terms in the price equation are significant—the oil-price term and the dummies from the output and productivity equations. Hence, the support for super exo­geneity for the conditioning variables on our sample from 1972(4)-2001(1) is weaker than in Bardsen et al. (2003) on a sample period 1966(4)-1996(4). [98] [99]

Table 9.4
Testing invariance

Awt = ■ ■■ + 0.005 Ydumt + 0.003 Udumt — 0.009 CRdumt (0.011) (0.007) (0.007)

— 0.027 Aoilt x oilSTt — 0.043 sA(euro/dollar)t — 0.003 st (0.027) (0.033) (0.003)

— 0.003 Vdumt + 0.007 Adumt + 0.001 RBOdumt (0.003) (0.013) (0.004)

+ 0.047 sARSt — 0.426 sARWt + 0.0003 RLdumt (0.139) (0.385) (0.0128)

Apt = ■ ■ ■ — 0.008 Ydumt + 0.0026 Udumt — 0.0003 CRdumt (0.004) (0.0023) (0.0021)

+ 0.022 Aoilt x oilSTt — 0.0014 sA(euro/dollar)t + 0.0014 st (0.011) (0.0115) (0.0012)

+ 0.0014 Vdumt + 0.0087 Adumt — 0.0012 RBOdumt (0.0011) (0.0044) (0.0014)

— 0.015 sARSt + 0.100 sARWt + 0.0002 RLdumt (0.049) (0.133) (0.0012)

Note: Testing the invariance with respect to all interventions: X2 (24) = 26.75[0.32].

In the same vein, we have also augmented the wage-price model (9.5)-(9.6) with all equilibrium correction terms in the marginal models (9.7)-(9.13): ecmv, t, ecmy t, ecmu, t, ecma, t, еашсг^, ecmRBo,(, ecm-R-L, t. They are individu­ally and jointly insignificant, with a joint test statistic of x2(14) = 6.82[0.94], providing additional support to the wage-price model specification.

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