RDP 2018-02: Affine Endeavour: Estimating a Joint Model of the Nominal and Real Term Structures of Interest Rates in Australia 5. Robustness Checks

5.1 Sample Starting in 1997

The data sample used in Section 4 spans the period before and after the Reserve Bank adopted a formal 2 to 3 per cent inflation target. Therefore, there could be a structural break, or regime shift, for which the model does not adequately account. This would be of particular concern given that the model imposes stationarity. To check the robustness of the results, we estimate the model on a reduced sample beginning in 1997, once inflation expectations had become reasonably anchored.[26]

Nominal and real interest rates, and term premia, follow fairly similar paths to those estimated using the full sample, although expected rates tend to be smoother, especially at longer horizons (Figures E1–E6). In particular, ten-year-ahead nominal and real forward rates are a bit less variable when estimated over the shorter sample, while three- and five-year-ahead real rates show larger declines in recent years. The same is broadly true of inflation expectations, although the smoothness occurs to an even larger degree. Given the results for nominal and real expectations, this last point is perhaps not surprising: expected inflation is calculated as the difference between nominal and real expectations; if these expectations follow similar trends and are relatively smooth, then their difference will tend to be even smoother and flatter still.

More broadly, the smoothness is suggestive of a short-sample problem leading to insufficiently persistent pricing factors. In particular, Guimarães (2016) argues that discarding part of the sample due to changes in the structure of the economy is exactly the opposite of what we should do, as this variation can be extremely useful in separately identifying the P and Q dynamics. This argument could certainly be put forward here. By removing the early period we are potentially removing a period with a large amount of information about the dynamics of inflation expectations, and in particular how they become anchored (and therefore can potentially become unanchored). Nonetheless, both sets of results show broadly similar trends over time for a number of variables, which is reassuring.

5.2 Filtered versus Unfiltered Results

As noted in Section 3.1, we estimate the model in two steps: first we maximise the model's likelihood conditional on the observed factors; second we cast the model in a Kalman filter and re-optimise. The second step allows us to relax the assumption that the factors are priced correctly, and to drop any estimated zero-coupon real yield data that does not have a traded bond with a similar maturity and so is dependent on interpolation. Both of these generalisations are potentially important given the sparsity of inflation-linked bonds. Related to this, by using the Kalman filter and allowing for imperfect pricing in the factors, we allow the surveys to influence these pricing factors, which is also potentially important.

It is interesting to consider what the results would look like if we did not incorporate the second step. Figures F1–F6 contain these results. Again, the estimates of real and nominal interest rates and risk premia are broadly similar, while the estimates of expected inflation show greater differences. In particular, the inflation estimates are generally somewhat smoother, particularly the ten-year-ahead expectations, and there is a larger fall around the onset of the global financial crisis, although the broad trends are still reasonably similar and the results still suggest that inflation expectations are well anchored within the 2 to 3 per cent target band.

The inflation expectation estimates from the first step are also more similar to those from Finlay and Wende (2012). As with the estimates from that paper, the difference seems to be that the Kalman filtered model puts a higher weight on the surveys, as it estimates the variance of the noise associated with the surveys to be lower.[27] This appears to reflect the fact that the Kalman filter approach allows the surveys to affect the estimated pricing factors, rather than constraining the model to use the observed factors. The results are similar whether or not we drop some real yield data, suggesting that fully utilising the information contained in the survey data is the more important generalisation.

The fact that the filtered model places a greater weight on the surveys is particularly evident in Figure F7, which plots the model-implied inflation expectations for both the filtered and unfiltered models alongside the (closest) matching surveys. This also highlights the fact that even in the filtered model, the model-implied expectations do not perfectly coincide with the surveys and that they are taking a substantial signal from the yield data.

Overall, these results suggest that using a Kalman filter, and therefore allowing for pricing factors that diverge from the principal components of the yield data, can lead to a higher weight being put on surveys (though this will not necessarily be a general result). To the extent that we think surveys are good measures of market participants' expectations, this will be preferable. This will be particularly true if we are concerned about the quality of the real yield data, as may be the case in countries with a scarcity of inflation-indexed bonds. However, if for some reason we think that the surveys are a poor measure of expectations, for the full sample or even for some sub-sample, it may be preferable to eschew the Kalman filter or to calibrate the model to place a lower weight on the surveys.


Another approach would be to estimate a model with regime switching. However, the added complexity this would involve was not in keeping with our focus on estimating a usable ‘workhorse’ model. [26]

On the flip side, it estimates the variance of the noise associated with the real yields to be higher. The estimates of the variance of the noise associated with the nominal yields are similar in the filtered and unfiltered models. [27]