RDP 2013-09: Terms of Trade Shocks and Incomplete Information 3. Estimation

I estimate the model using Bayesian methods. This section outlines the estimation strategy, including the choice of priors, and explains how the variables of the theoretical model map into observable time series.

3.1 Measurement

The initial stage of the estimation is to map the model's variables, which are generally unobservable, into observable variables that can be used to estimate the model's parameters. To do this, I first express the log-linear equilibrium conditions, derived in the previous section, in state space form as:

where the state vector collects the model's theoretical variables and the vector collects the observed variables used to estimate the model. Equation (31) governs the transition of the state variables, while Equation (32) maps the state into observable variables. The matrices T, Ξ, V and W are functions of the parameters of the model.

The observable variables I use to estimate the model are the growth rates of real GDP, private consumption, private gross fixed capital formation and the terms of trade as well as the level of the trade balance-to-GDP ratio, . That is:

All variables are expressed in per capita terms and are seasonally adjusted. I estimate the model using quarterly Australian data over the period 1973:Q1–2012:Q2.^[8] The starting point reflects the first quarter for which per capita national accounts data are available for Australia. This is somewhat earlier than the starting date for most Australian DSGE models, which typically use data spanning the period after the adoption of a floating exchange rate in 1983 or inflation targeting in 1993. A later starting date is appropriate for models containing nominal interest rates or inflation, whose behaviour is likely to be affected by changes in the conduct of monetary policy. In contrast, the model in this paper contains no nominal variables. And, given the presence of long-lived trends in the terms of trade, it seems preferable to use a longer time series to estimate the model.

Following Jääskelä and Nimark (2011), the covariance matrix, X, of the vector of measurement errors, ζ_t, in Equation (32) are set to × 0.1 so that 10 per cent of the variance of the data series is assumed to come from measurement errors.

3.2 Bayesian Estimation

I estimate the parameters of the model using Bayesian methods that combine prior information with information from the data. The estimation works in the following way. Denote the vector of parameters to be estimated as Θ. The log posterior distribution of the parameters to be estimated is given by:

where (Θ) is the log of the prior probability of observing a given vector of parameters and is the log likelihood of observing the dataset for a given parameter vector. This likelihood is given by:

where p is the dimension of , Ω is the covariance matrix of the theoretical one-step-ahead forecast errors implied by a given parameterisation of the model and u_t is the vector of actual one-step-ahead forecast errors.

The numerical procedure I use to estimate the posterior distribution follows the methodology outlined in An and Schorfheide (2007). In computing the posterior distribution, I set the number of Metropolis-Hastings draws equal to 500,000, and select these after discarding an initial 250,000 burn-in draws.

3.2.1 Priors

For the AR(1) parameters of the exogenous processes, I assign beta priors with a mean of 0.8 and standard deviation of 0.1. Using the beta distribution for these priors ensures that the estimated parameters lie between 0 and 1, consistent with economic theory. I assign inverse gamma priors with a mean of 5 × 10⁻³ and a standard deviation of 0.01 to the standard deviations of the exogenous processes. Finally, for the capital adjustment cost parameter, ϕ, I assign a truncated normal prior, with a mean of 7.5 and standard deviation of 2.5.^[9]

The theoretical model, of course, contains a number of additional parameters. Many of these are likely to be poorly identified using only the observed data series included in the model but have been estimated many times previously. Rather than rely on imprecise estimates of these parameters, I calibrate them using values determined by previous research or economic theory. In a Bayesian framework, calibration can be thought of as a very tight prior. Table 1 outlines the calibrated parameters.

Table 1: Calibrated Parameters

Parameter	Value	Description
β	0.99	Discount factor
δ	0.02	Depreciation rate
μ	1.0045	Steady-state technology growth rate
	1.10	Steady-state foreign debt level
α	0.33	Capital share of income
η	0.20	Imports share of consumption
φ	1.00	Inverse Frisch labour supply elasticity
ψ	0.001	Portfolio adjustment cost

The parameters for the discount factor, depreciation rate and capital share of income are standard for a model estimated on quarterly data. The parameter for μ broadly conforms to the average quarterly growth rate of GDP per capita over the sample period. The parameter for is set to ensure that the model matches the average net export-to-GDP ratio seen in the data, while that for η matches the import share of consumption. The parameter for ψ is set as a small value that ensures that the model is stationary while having only a minor impact on the dynamics of the model. Finally, the parameter for φ is taken from Jääskelä and Nimark (2011).

3.3 Posterior Distribution

Table 2 shows the main results of estimation. The transitory terms of trade shock is reasonably persistent, with a posterior mean of the AR(1) coefficient ρ_z equal to 0.84. The permanent shock is marginally less so, with ρ_g equal to 0.77. In terms of the magnitude of the shocks, the standard deviation of transitory terms of trade shocks, σ_z is quite large at 1.25 per cent, while the standard deviation of the permanent terms of trade shock, σ_g, is much smaller at just 0.22 per cent. Nonetheless, a shock to ε_g ultimately has a much larger and more lasting impact on the terms of trade. A positive shock to ε_z causes a once-off increase in the terms of trade which then diminishes, although the high value of ρ_z implies that it takes some time for the terms of trade to return to its initial level following the shock. In particular, the half-life of this shock is around six quarters, and the terms of trade does not return to its trend level for several years. In contrast, a positive shock to ε_g increases the terms of trade on impact and then continues to increase the terms of trade further, albeit at a diminishing rate, over time. The accumulation continues over several quarters, and the terms of trade ultimately settles at a level around five times the level of the initial impulse five years after the initial shock.

The standard deviation of the noise shocks, σ_h is also large, at 1.31 per cent. This suggests that agents receive a fairly weak signal about the persistence of terms of trade shocks.

Although the remaining parameter estimates are not the focus of this paper, it is comforting to note that the results seem plausible and are broadly consistent with other empirical estimates. In particular, the magnitude of the transitory productivity shocks are estimated to be larger than those of the permanent productivity shocks, which is consistent with the estimates for Canada in Aguiar and Gopinath (2007), although the persistence of these shocks are slightly larger than in that study. The results also imply large capital adjustment costs. This is a common finding in the open economy literature. In the absence of these adjustment costs, the ability to finance the accumulation of imported capital using foreign borrowing without requiring an accompanying decrease in consumption would lead the model to predict implausibly large investment volatility.

Footnotes

Appendix A outlines the data sources used in the estimation. [8]

The truncation ensures that ϕ is greater than 0. In the estimation, the bulk of the posterior distribution of this parameter lies far away from the truncation point. [9]