Dynamic multiplier plots

An additional benefit of the NARDL model is the computationally simple method to construct graphical illustrations to track the short-run and long-run responses of economic growth to changes in the top income shares. These dynamic mul-tiplier plots map the gradual movement of the growth process from the initial equilibrium to the new one.

Shin et al. (2014) rewrite the NARDL model (equation (4.3)) as

φ(L)yt =_θ⁺(L)Log(Top1t)⁺+_θ⁻(L)Log(Top1t)⁻+et, (4.4) whereφ(L) = 1−_∑^p_i=⁻₁¹_φiLⁱ, θ⁺(L) = _∑^q_i=0θi⁺Lⁱ andθ⁻(L) = _∑^q_i=0θ⁻_i Lⁱ. Multi-plying equation (4.4) by[φ(L)]⁻¹gives

yt =λ⁺(L)Log(Top1t)⁺+λ⁻(L)Log(Top1t)⁻+ [φ(L)]⁻¹et, (4.5) whereλ⁺(L) = _∑^∞_j=0λ⁺_j =φL⁻¹θ⁺(L)andλ⁻(L) = _∑^∞_j=0λ⁻_j =φL⁻¹θ⁻(L). The cumulative dynamic multiplier plots can be constructed using the following:

m⁺_h =

∑

∂y_t+j

∂Log(Top1t)^+,m

−

h =

∑

∂y_t+j

∂Log(Top1t)^−,h=0, 1, 2, . . . , (4.6) whereh is the number of periods (years below) for the dynamic adjustment. By construction, ash −→ _∞, m⁺_h −→ β⁺ andm⁻_h −→ β⁻, i.e. the dynamic multipli-ers converge to the long-run coefficients reported in Table 4.3.

The dynamic multipliers are not only useful in exemplifying the short-run and long-run responses but in fact introduces an additional form of potential asymmetry: the adjustment asymmetry (labeled by the authors (Shin et al., 2014)).

The adjustment asymmetry combines the long-run (reaction,β⁺ 6=β⁻) and short-run (impact, ϕ⁺₀ 6= _ϕ⁻₀) responses with the error correction coefficient ρ. Obvi-ously, the graphical illustrations are conditional on the model specification: for Australia, the adjustments to positive and negative changes in inequality are symmetrical, whereas for the five other countries, the movements toward the new equilibrium associated with positive and negative changes will not be mir-ror images of one another.

Conveniently for an empirical researcher, the ’nardl’ routine of Stata by Marco Sunder includes a plotting option that can be used to construct difference lines, including bootstrapped confidence intervals, between the dynamic adjust-ments to positive and negative shocks. The multipliers are associated with unit changes in the log of top 1 % income share, i.e. percentage changes in the top income share.

Figure 4.4 shows the adjustments of per capita GDP growth to shocks in the top income shares. The illustrations are based on the models reported in Table 4.3 and the converged dynamic multipliers correspond to the coefficients β, β⁺ and β⁻. The green short-dash line corresponds to a positive change in the top income share, the red long-dash line to a negative change while the solid blue line depicts asymmetry as the difference between the dashed lines. The grey area around the solid line is the 95 % confidence interval for asymmetry and it is based on 500 bootstrap replications. The horizontal axis measures time in years after the inequality shock.

The first observation regarding the six individual plots is that the move-ment to new equilibrium takes place between two to seven years depending on the country. Thus, there are no large differences in the adjustment asymmetry between the countries. Moreover, it is unrealistic to assume that mechanisms related to human capital accumulation, legislation of redistributional policies or socio-political movements would be the ones that give rise to the patterns de-picted in Figure 4.4. It seems evident that the ARDL model can capture growth responses that are transmitted through more direct economic mechanisms. This question is revisited below in more detail.

Based on the evidence of Table 4.1, the response in Australia is restricted to be symmetrical. At impact, rise (fall) in inequality supports growth, which is followed by a recoil and eventually the response dies out. In Canada, any change in inequality is bad for growth in the short-run but the cumulative association is minuscule.

In France, the short-run impact is qualitatively similar to Canada, although the response to a negative shock is more pronounced. After the adjustment to the new equilibrium, the illustration portrays the findings discussed above: falling top income shares are associated with lower subsequent growth while the growth process seems to be independent of positive inequality shocks. The long-run case for the United States is similar to France and also already discussed above. In the

-.1-.050.05.1

0 2 4 6 8 10

Years

positive change negative change asymmetry

(a) Australia

-.8-.6-.4-.20.2

0 2 4 6 8 10

Years

positive change negative change

asymmetry CI for asymmetry

Note: 95% bootstrap CI is based on 500 replications

(b) Canada

-.6-.4-.20.2

0 2 4 6 8 10

Years

positive change negative change

asymmetry CI for asymmetry

Note: 95% bootstrap CI is based on 500 replications

(c) France

-.1-.050.05.1

0 2 4 6 8 10

Years

positive change negative change

asymmetry CI for asymmetry

Note: 95% bootstrap CI is based on 500 replications

(d) India

-.1-.050.05.1

0 2 4 6 8 10

Years

positive change negative change

asymmetry CI for asymmetry

Note: 95% bootstrap CI is based on 500 replications

(e) Japan

-.8-.6-.4-.20.2

0 2 4 6 8 10

Years

positive change negative change

asymmetry CI for asymmetry

Note: 95% bootstrap CI is based on 500 replications

(f) United States

FIGURE 4.4 Cumulative responses of per capita GDP growth to 1 % changes in the top income shares

short-run though, the asymmetry is not due to "any change is bad change" type of response. Rather, the negative growth-response to a negative inequality shock is larger in size than the positive response to a positive shock.

For India and Japan, short-run symmetry was imposed based on the results of Table 4.1. In both countries, rising (falling) inequality is associated with higher (lower) growth in the short-run. In India, both responses converge towards zero but the response remains higher in size for the positive shock as discussed above (β⁺ = 0.033,β⁻ = −0.012), i.e. there is long-run asymmetry. In Japan, the posi-tive response crosses the zero line during the adjustment process resulting in an asymmetric long-run response even though the coefficients β⁺ (-0.011) and β⁻ (-0.034) are statistically insignificant.