Implied NKPC coefficients

From (7), the Australian NKPC may be expressed in its reduced-form

πt= γ_b,1πt−1+ γ_b,2πt−2+ γ_fEtπt+1+ ˜λmct,

where

γ_b,1 = ρτ − βρ(1 − τ ) 1 + βρτ γ_b,2 = ρ(1 − τ )

1 + βρτ γ_f = β

1 + βρτ .

In the constrained cases, where τ = 1, the coefficients collapse such that γb,1 = ρ/(1 + βρ), γ_f = β/(1 + βρ) and t − 2 inflation does not enter the specification.

Conditioning on the point estimates of the structural parameters, Table 4 displays the implied Table 4

Implied NKPC coefficients Sample: 1960:Q1 - 2007:Q2

γ_b,1 γ_b,2 γ_f

DE_con 0.005 - 0.985

DE_uncon 0.225 0.155 0.615

CF_con 0.284 - 0.711

CF_uncon 0.294 0.097 0.607

Note: the implied NKPC coefficients are conditional on the point estimates of the parameters presented in Table 3.

NKPC coefficients for the full sample period. Unsurprisingly the DEcon specification implies essentially a purely-forward looking NKPC, with the coefficient on expected future inflation near unity. However, all three remaining specifications suggest that there is an important backward-looking component in Australia’s NKPC. The results seem to indicate an enhanced role for backward-looking behaviour in comparison to the conventional GMM estimates of the Australian NKPC presented in Kuttner and Robinson (2010). Kuttner and Robinson (2010) estimate γ_f = 0.806 and γ_b,1 = 0.166 for the same sample period. Comparing the CF estimates to the DE estimates seems to suggest that when all model-consistent restrictions are placed on the evolution of inflation expectations the weight given to lagged inflation tends to increase.

Despite the apparent increased role for lagged inflation, the estimates in Table 4 continue to suggest that the Australian NKPC is predominantly forward-looking, even when one considers the closed form coefficients.

The aforementioned results are largely illustrative but nonetheless display the clear disparity between the DE and CF estimates. The estimates reported for the DE_unconspecification and the closed form suggest the presence of an important backward-looking component in the Australian NKPC. In the next section these issues are explored in greater detail in the context of a refined Calvo model³⁵. Trend inflation is incorporated into the Calvo model yielding an extended NKPC with time-varying coefficients.

35Discussion of the estimates for α and θ are reserved until Section 5.4.

5 The Australian NKPC with Time-Varying Trend Inflation

The traditional NKPC framework, as analysed in the previous section, relies on the assump-tion that inflaassump-tion in the steady-state is zero. This secassump-tion introduces an adapted version of Cogley and Sbordone’s (2008) refined Calvo model, which divorces itself from this conventional assumption, and documents its performance against the Australian inflation data.

In essence, Cogley and Sbordone (2008) develop a dynamic version of the NKPC charac-terised by the inclusion of trend inflation, which is subject to variation over time. It is their contention that variation in the long-run trend component of inflation is the source of the ap-parent persistence in US inflation. In contrast to the simple NKPC framework, the Cogley and Sbordone’s (2008) model takes the log-linearisation of the equilibrium conditions of the refined Calvo model around a steady-state associated with drifting trend inflation. In the baseline NKPC, the log-linearisation is taken around a constant trend (π = 0), which is the same in every period. In the context of this framework trend inflation is assumed to be an exogenous process that is modelled as a random walk. Unlike the traditional framework, with zero trend inflation, the NKPC coefficients in this setup will evolve over time according to the level of trend inflation.

First, the equilibrium conditions of Cogley and Sbordone’s (2008) generalised Calvo model are described. Their model is augmented slightly to capture open economy dimensions and, in the same vein as described in Section 4, also allows for two lags of inflation in the indexation mechanism³⁶.

The primary equilibrium relationship is the restriction between trend inflation and stead-state real marginal costs, which is of the form



1 − αΠ^{(1−ρ)(θ−1)}_t ^1+θω_1−θ

"

1 − α¯q¯g^yΠθ(1+ω)(1−ρ) t

1 − α¯q¯g^yΠ_t^{(1−ρ)(θ−1)}

#

= (1 − α)^1+θω1−θ θ

θ − 1(mc^d_t)^1−φ(mc^m_t )^φ, (22)

where ¯q denotes the steady-state real discount factor, ¯g^yis steady-state output growth, Πtis gross trend inflation at time t. The structural parameters, α, ρ, θ, ω and φ, retain their definitions from the previous section. The above restriction differs in an important respect from the equivalent expression in Cogley and Sbordone (2008)³⁷. In this case, the right hand side includes both

36For the full details of the derivation see Appendix A. Note that this paper follows the version in BGLO (2011), which allows for two lags of inflation in the indexation mechanism.

mc^d_t and mc^m_t , which represent trend real marginal costs of domestic producers and importers, respectively. Both terms enter the restriction according to the respective weights of domestically produced goods and imports in domestic consumption. In this way (22) provides a simple mechanism for capturing the open economy aspects of the inflationary process in Australia.

Following Cogley and Sbordone (2008) it is assumed that the following two inequalities hold

ϕ1,t = α¯q¯g^yΠ^{(1−ρ)(θ−1)}_t < 1 (23)

and

ϕ2,t = α¯q¯g^yΠθ(1−ω)(1−ρ)

t < 1. (24)

These inequalities ensure that the steady-state relationship (22) is well defined. As in the constant trend case, it is assumed that the structural parameters are equivalent across both the domestic and foreign sectors. This simplifying assumption allows for the derivation of a single equation extended NKPC, which is obtained by log-linearising (22) around a steady-state with drifting trend inflation

bπ_t = ρτ (bπ_t−1−bg^π_t) + ρ(1 − τ )(bπ_t−2−bg^π_t−1−bg_t^π) + Ω_tE_t[bπ_t+1− ρτbπ_t− ρ(1 − τ )(πb_t−1−bg_t^π)]

+ λt

h

(1 − φ)mcc^d_t + φmcc^m_t i

+ γtDbt+ uπ,t, (25)

where hatted variables represent log-deviations of stationary variables from their steady-state values³⁸. bD is defined recursively as³⁹

Db_t= ϕ_1,tE_t(bq_t,t+1+bg_t+1^y ) + ϕ_1,t(θ − 1)E_t{bπ_t+1− ρτπb_t− ρ(1 − τ )(bπ_t−1−bg_t^π)} + ϕ_1,tE_tDb_t+1.(26)

For ease of comparison to Cogley and Sbordone (2008), the extended NKPC in its DE form (as

38In particular, πbt = ln(Πt/Πt), mcc^d_t = ln(mc^d_t/mc^d_t), mcc^m_t = ln(mc^m_t /mc^m_t ) andbg^π_t = ln(Πt/Πt−1) is the growth rate of trend inflation; uπ,tis a structural shock.

39Herebg_t^y= ln(g^y_t/g^y) andqbt,t+1= ln(qt,t+1/q_t,t+1). Also note that ϕ0,t=



1−αΠ_t^{(1−ρ)(θ−1)} αΠ_t^{(1−ρ)(θ−1)}

 , and Ωt= ϕ2,t(1 + ϕ0,t).

described by equations (25) and (26)) may be expressed as⁴⁰:

While the extended NKPC in (27) retains a similar form to the traditional NKPC, the two variations differ in some important respects. Firstly, the extended NKPC contains additional variables, such as the growth rate in trend inflation (bg_t^π), as well as terms involving the nominal discount factor ( bQ_t), real output growth (bg_t^y) and higher-order leads of inflation. While these additional terms may suggest the presence of omitted variable bias in the traditional NKPC, perhaps a more important distinguishing feature of the extended NKPC is the fact that the NKPC coefficients in (27) are non-linear functions of trend inflation and the structural param-eters of the model. Consequently these coefficients, which are of interest to policy makers, are subject to variation over time and evolve according to the drift in trend inflation⁴¹.

It is possible to obtain a closed form representation of the NKPC by iterating equations (25) and (26) forward⁴²:

To maintain consistency with the baseline case in Section 4 in the pursuant analysis the same four specifications of the extended NKPC are estimated. In particular, the DE specification (27)

40See Appendix A for complete derivation and definition of coefficients.

41A discussion of the implied NKPC coefficients is undertaken in Section 5.5.

42For a complete derivation of the closed form specification see Appendix B. As stipulated in BGLO (2011) equation (28) is more appropriately referred to as the “quasi-closed form” NKPC as bπt remains a function of higher-order leads of inflation. It is possible to obtain an exact closed form solution, however, BGLO (2011) show that the US estimates of the exact closed form are very similar to the quasi-closed form. Their results indicate that the additional restrictions that the exact closed form solution imposes over (28) are not critical for estimating the structural parameters of the model.

is estimated for the case where τ is constrained to equal one (in this case the NKPC collapses to the corresponding version originally estimated by Cogley and Sbordone, 2008), and for the case where τ is left unconstrained. Similarly, the NKPC in its CF representation (28) is estimated for both the constrained and unconstrained cases.

5.1 Estimating the NKPC Structural Parameters

As in Section 4 the objective is estimate the structural parameters of the NKPC model, ψ = [α, ρ, θ, τ ]. The same two-step estimation approach, as described in Section 4.2, is adopted here.

In the present context the time series vector x_t is extended so as to include not only inflation and real marginal costs, but also a nominal discount factor and output growth, so that n = 5.

The reduced-form VAR is now written as

zt= µ_t+ Atzt−1+ z,t, (29)

where _z,t is a serially uncorrelated error vector. In contrast to the estimation of the baseline NKPC, the reduced-form VAR now has drifting coefficients, captured in µ_t and At. The drift in the VAR coefficients follows from the assumption that trend inflation is time-varying.

As detailed previously, if the extended NKPC is the true data generating process for inflation, then the forecasts from the reduced-form VAR in (29) and the structural forecasts from (27) should be equivalent. Accordingly, the forecasting rule in equation (10) is augmented such that the conditional expectation of a variable yb_t+j∈ x_t+j at time t is now written as

Etybt+j = e⁰_yA^j_tbzt, (30)

where e⁰_y is the selection vector as defined previously. The above condition differs from the original forecasting rule (10) as it defines expectations of yb_t+j, rather than y_t+j. As such bz_t represents the vector of variables expressed in deviations from their time-varying trend levels at time t

bz_t≡ z_t− (I − A_t)⁻¹µ_t. (31)