RDP 9706: Is the Phillips Curve A Curve? Some Evidence and Implications for Australia 3. Estimating Linear and Non-linear Phillips Curves

In this section, we describe the approach we use to estimate a non-linear Phillips curve for Australia and compare its empirical properties to a similarly estimated linear Phillips curve. The methodology we use is that adopted by Debelle and Laxton (1997) to estimate Phillips curves for Canada, the United Kingdom and the United States. A key feature of the approach is the use of model-consistent estimates of the NAIRU in comparing the linear and non-linear models. Previous attempts to detect the presence of non-linearities in Phillips curves have generally used measures of the NAIRU (or equivalently the output gap) which have been derived in a linear framework, thereby introducing bias into the tests for non-linearities. Laxton, Meredith and Rose (1994) and Clark, Laxton and Rose (1996) document the size of this bias.

3.1 Data

We estimate the linear and non-linear models of the Phillips curve presented in Section 2 (Equations (1) and (3)). Quarterly data from 1959:Q3 until 1997:Q1 is used to estimate the models. We use the four-quarter-ended growth in the (underlying) consumer price index to measure inflation. The unemployment rate is the quarterly average of the monthly seasonally adjusted unemployment rate in the ABS Labour Force Survey^[6] from 1966 onwards, and from the NIF-10 database prior to that.

Inflation expectations are measured in a number of different forms. The backward-looking component is captured by the inclusion of a lagged four-quarter-ended inflation term. For the forward-looking component we use two different measures: an estimate derived from bond-market yields, and the Melbourne Institute measure of consumer inflation expectations. We also estimate a model with only a backward-looking component, where we include four lags of the inflation rate.

The measure of inflation expectations derived from bond-market yields is obtained by subtracting a measure of the equilibrium world real interest rate from the 10-year bond yield. The series for the world real interest rate is based on empirical work that relates the equilibrium world real interest rate to movements in the stock of world government debt (Ford and Laxton 1995; Tanzi and Fanizza 1995). Debelle and Laxton (1997) show that the results are not sensitive to the precise calculation of the world real rate.

3.2 Kalman Filter

As discussed in Section 2, we allow the estimate of the NAIRU to be time-varying. Consequently, we estimate the two models using the Kalman filter. The following provides a short overview of this approach.^[7] Consider the following system,

The parameter (state) vector β_t is time-varying in a manner determined by the transition matrix T. In our estimation we assume that T is such that all parameters are constant except for the NAIRU which follows a random walk.^[8] The Kalman filter produces estimates of this system by minimising the sum of the squared one-step prediction errors of y_t.

In terms of the non-linear model we estimate, y_t = (π_t − π_t−1), and , while in the linear model . Both δ and γ are assumed to be unchanging over time. Estimates of the NAIRU at each point in time can be calculated by taking the negative of the ratio between the second and third elements of β_t.

We need to place some restrictions on the ratio of the two variances in Equations (5) and (6), the signal-to-noise ratio. Multiplying the matrices H, Q and Σ (where Σ is the initial covariance matrix of β) by a scalar leaves the system unchanged. We thus treat the variance of ε_t as the numeraire, setting it equal to 1, and alter the signal-to-noise ratio by changing the magnitude of the non-zero element in Q. In our empirical work, this value is usually found through an optimisation procedure, but is generally around 0.4. The elements of Σ are set to large values,^[9] reflecting lack of knowledge about the ‘true’ value of the NAIRU.

The Kalman filter can be used in two ways, which can broadly be referred to as prediction (one-sided estimation) and smoothing (two-sided estimation).^[10] In prediction mode, the filter computes estimates of the model parameters at time t based on information up to time t. The filter recursively estimates through the entire sample in this manner. In this way, it simulates the process of estimating the NAIRU in real time. In smoothing mode, estimates of the NAIRU at each point are based on information over the entire sample. This two-sided estimate allows the benefit of hindsight in obtaining an estimate of the NAIRU in past time periods. For the last period in the sample, the two approaches will give the same estimate of the NAIRU, because information over the entire sample is available when obtaining the onesided NAIRU estimates.

Footnotes

Labour Force, Australia, ABS Cat. No. 6203.0. [6]

For more information on the Kalman filter, see Chapter 13 of Hamilton (1994). More details on our use of the filter are in Appendix A. The model was estimated using the KALMAN command in RATS version 4.2. The results were replicated using the KFILTER and KSMOOTH procedures available on the ESTIMA homepage (http://www.estima.com/). [7]

This does not mean that we necessarily believe that the NAIRU is indeed a random walk. Rather it is an empirically convenient way to model it. [8]

The diagonal elements are set to a magnitude of 4. [9]

See Kuttner (1992) for a more comprehensive description. [10]