RDP 2019-07: MARTIN Has Its Place: A Macroeconometric Model of the Australian Economy Appendix A: Estimation of Time-varying Constants in Error Correction Equations
August 2019
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Several of the error correction equations in MARTIN feature time-varying constants. We include these to account for slow-moving events, such as structural reforms or technological change. These can lead to a gradual change in the steady state of a cointegrating relationship, without altering the dynamic relationship between the cointegrating variables.
A leading example of where these types of developments occur is in MARTIN's trade block. Decreases in trade costs and the dismantling of global tariff barriers meant that export volumes grew faster than the GDP of Australia's major trading partners, particularly during the 1980s and 1990s. To the extent that these changes represent one-off developments, we would expect them to cause a permanent increase in the ratio between exports and GDP (both foreign and domestic). However, the response of Australian exports to a given change in foreign GDP may not be affected. Because the expansion of export volumes following a permanent decrease in trade costs or tariff barriers is likely to occur gradually and at an uneven rate, it may not adequately be accounted for by including a trend or one-off break in the trade equations. In contrast, a time-varying constant can account for gradual changes in the empirical relationships, albeit at the cost of potentially overfitting the data.
To estimate the time-varying constants, we first set up each of the equations in a state-space system. The first part of the system is the signal equation. This shows the relationship between variables that we can observe in the data and the unobservable variable – the time-varying constant – whose value we have to estimate. The typical form for this type of equation in MARTIN is:
where y_{t} is the dependent variable, x_{1,t} and x_{2,t} are explanatory variables, ${\beta}_{1}$ and ${\beta}_{2}$ are coefficients whose value we estimate and y_{c,t} is the time-varying constant. The error term ${\epsilon}_{sig,t}\sim N\left(0,{\sigma}_{sig}^{2}\right)$.
The second part of the system is the transition equation. This shows how the unobserved state variable evolves over time. We typically specify this equation as a random walk, that is the value of the constant in the current period is equal to its value in the previous period plus a random disturbance:
where error term ${\epsilon}_{trans,t}\sim N\left(0,{\sigma}_{trans}^{2}\right)$.
We estimate the parameters of the system by maximum likelihood. In some cases, we restrict the variance of the error in the transition equation to be small relative to the variance of the signal equation. This ensures that the time-varying constant captures only low-frequency changes in the relationship between the observable variables. Given the estimated parameter values, we use the Kalman filter to infer the value of y_{c}_{,t}.