RDP 9111: Monthly Movements in the Australian Dollar and Real ShortTerm Interest Differentials: An Application of the Kalman Filter 4. Econometric Methodology
November 1991
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A state space representation of the structural model is derived to avoid the computational problems arising from the inclusion of unobserved variables. The resulting representation can then be estimated using the Kalman filter. Harvey (1988)^{[29]} describes this method of model specification and estimation as being “… in many ways preferable to the more conventional approach based on ARIMA processes”. This section briefly reviews Kalman filter estimation and its application in our paper (see Harvey (1981, 1988, and 1989) for a detailed discussion).
4.1 Motivation for Choice of Estimation Technique
The Kalman filter was derived for applications in control engineering (Kalman (1960)). In this paper, its importance for maximum likelihood estimation is exploited. Depending on the form of the function to be optimised, maximum likelihood problems can be solved either analytically or numerically. The analytical solution involves equating the vector of first derivatives to zero and solving for the parameters. More complex functions are solved numerically. Two approaches exist for numerical optimisation:
 Direct evaluation is used for functions which can be calculated but not differentiated.
 Kalman filter evaluation is used either because the likelihood function cannot be evaluated directly, or because the system contains unobservable variables.
Our paper uses the Kalman filter because it permits a more flexible and rigorous treatment of the unobservable variables, the ex ante real interest differential, E_{t}(d_{t,t+1}) the longrun real exchange rate, w_{t}, and the forecast error, e_{t}.^{[30]} The alternative approach, selecting proxy variables before estimation, is subject to measurement error and may not be optimal in any welldefined sense.
The more standard method of estimating models with unobservable components is that of “instrumental variables” (IV). This technique relies on choosing a proxy (or “instrument”) which, while highly correlated with its unobservable equivalent, is not correlated with the model's disturbance terms. The parameter estimates obtained in this way are sensitive to the choice of instrument and may be biased. The IV technique would also restrict the dynamics of Campbell and Clarida's (1987) structural model:
 The longrun real equilibrium exchange rate and the risk premium are allowed to vary over time. Determining an instrument for either of these variables is problematic. With respect to wt, a separate model could be estimated, but this exercise lies beyond the scope of our paper. The temptation would be to assume wt constant as in Shafer and Loopesko (1983). Moreover, these assumptions reduce the explanatory power of the model.

We suggest that d_{t−1} would be an appropriate instrument for E_{t}(d_{t,t+1}).^{[31]} Using IV,
equation [10] could be estimated as:
where:
 w is a constant term representing the longrun real exchange rate;
 ω_{t} is a composite error term incorporating u_{1,t}, u_{2,t} and u_{3,t}.
Estimation of [10'] reveals strong serial correlation. This can be corrected by using an appropriate estimation technique (for example simple CochraneOrcutt AR(1) modification). Alternatively, the model can be estimated in first differences. Although this approach would yield coefficient estimates, it would eliminate evaluation of the proportion of the variation in q_{t} explained by E_{t}(d_{t,t+1}). Rather, it would enable us to evaluate the proportion of the variation in the change in q_{t} explained by the change in the proxy for E_{t}(d_{t,t+1}).
 As well, the Kalman filter allows solution of the model regardless of stationary/nonstationary data considerations.
4.2 Application of the Kalman Filter
The filter can only be applied once the structural model is cast in a framework which enables the treatment of its unobservable components. A linear dynamic system can be written in the socalled state space form.^{[32]} A key feature of this representation is the presence of an unobservable vector, α_{t} called the state vector. In the context of this paper, α_{t} is [E_{t}(d_{t,t+1}) e_{t} w_{t}]'.
The state space form consists of two equations: the measurement equation and the transition equation. The measurement equation relates the vector of unobservable, explanatory variables, α_{t}, to a vector of observable variables, y_{t}, such that:
For our model this takes the form:
β ≠ 1 : Linear risk premium model;
β = 1 : Uncovered interest parity model;
β and ρ are assumed constant for these two models and, in keeping with our theorybased relationships, an error term – generally added onto the righthandside of equation [11] – is not included.
The second equation in the state space formulation is the transition equation. This describes the evolution of the state vector over time such that:
For our model:
u_{1t} is the innovation, unrelated to inflation surprise, associated with the ex ante real interest differential. u_{2t} is the inflation forecast error; rational expectations dictates that this is uncorrelated contemporaneously with either u_{1t} or u_{3t}. u_{3t} is the innovation to the longrun real exchange rate. All errors are serially uncorrelated and have standard deviations σ_{1}, σ_{2} and σ_{3} respectively. σ_{13}, the contemporaneous covariance between u_{1t} and u_{3t}, is not restricted.
Initialised with a set of priors, the Kalman filter estimates (that is, it “predicts” and “updates”) the vector of unobservable variables, α_{t} period by period. “Predicting” means predicting α_{t} given the information set of period t1. “Updating” means using a onestepahead forecast error of y_{t} (necessarily calculated by the filter at each execution) to update the predicted value of α_{t}. This two stage procedure, carried out at each new time period, defines the Kalman filter. The linear estimate of α_{t} denoted , is optimal in the minimum mean square error sense. The filter also provides the error covariance matrix of , denoted P_{t}. Starting values for and P_{0} must be provided. At time period t, observations y_{1}, y_{2} … y_{t} are available.
Apart from its ability to generate optimal estimates (as opposed to proxies) for the vector of unobservable variables, α_{t} the Kalman filter may be used in maximum likelihood estimation. Its execution generates a time series for the onestepahead prediction errors made in estimating the vector of observable, dependent variables, y_{t}. Harvey (1985) notes that the likelihood can be calculated from these normally distributed errors and their variances. Estimation of our model yields a bivariate normal log likelihood function which can be expressed as:
where:
 υ_{t} is the set of onestepahead prediction errors made in estimating the vector of observable, explanatory variables, y_{t} (the real exchange rate, q_{t}, and the ex post real interest differential, d_{t}, in our model). That is, υ_{t} = y_{t} – E_{t−1}[y_{t}]. The onestepahead prediction of the observation y_{t} is calculated as: where is the optimal predictor of the state vector, α_{t}, given observations up to t−1 and Z is the matrix of coefficients on the state vector.
 F_{t} is the estimated covariance matrix of these onestepahead prediction errors.
Hence, the Kalman filter can be run in conjunction with an optimisation routine^{[33]} – calculating the likelihood where required by the routine. This is what it means to “estimate” a model using the Kalman filter. For each run through the filter, the coefficients on the model are fixed. However, the likelihood function [15] can be calculated for each run.^{[34]}
Equations [12] and [14] are estimated with the Kalman filter. The model contains seven unknown parameters, β, ρ, λ, σ_{1}, σ_{2}, σ_{3} and σ_{13}. Where equality between the expected real exchange rate change and the real interest differential is imposed, β is set equal to one. We estimate models for Australia's real exchange rate with the United States, Japan, West Germany and the United Kingdom, as well as a tradeweighted index. In each case, models are estimated with β unrestricted and with β restricted to one.
4.3 Variance Decomposition
Innovation relationships were investigated by expanding the oneperiod ahead variance of the actual dependent variable (namely, the observable real exchange rate) in the following manner.
Equation [10] implies:
where the subscripts of σ carry their obvious meaning (refer equation [10]).
The variance T periods ahead (i.e., E_{t}(q_{t+T}−Et(q_{t+T}))^{2}) is
as T →∞, the first and third terms of the RHS converge, i.e.:
Clearly, a variance decomposition calculated over a long time horizon will always attribute a very large proportion of the variation in q_{T} to w_{t} a priori. This is a direct result of the assumption that w_{t} is a random walk and that E_{t}(d_{t}) is stationary. In the limit, w_{t} explains 100 per cent of the variation in q_{T}. However, what is not imposed is how much of the variability in q_{T} is explained by w_{t} in the short run. In principle, ρ could be very close to one, in which case E_{t}(d_{t,t+1}) could explain a large proportion of the variability of q_{t} for a small value of T. Using the parameter estimates obtained from the Kalman filter and the optimisation algorithm, the appropriate expression for the variance (equation [17] with T=1) is used to calculate estimated variance decompositions. That is, each of the three terms on the right hand side of equation [17] are expressed as a percentage of σ_{q(1)}^{2} and reported in Tables 4 and 5.
Footnotes
Harvey (1988), Chapter 8, p.285. [29]
Clearly, the specification of the unobservable components in the state space model must depend, to some extent, on a priori considerations. However, estimates of the unobservable variables generated by the Kalman filter are optimal in the minimum MSE sense. [30]
Nevertheless, given the IV approach to estimation, ρ = 1, and d_{t−1} becomes a suitable instrument for E_{t}(d_{t,t+1}). [31]
Such models, driven by innovations of some macroeconomic time series, are sometimes referred to as “innovation models” (see Aoki and Havenner (1986)). [32]
Since it is explicitly designed for optimisation of likelihood functions, the BerndtHallHallHausman (1974) algorithm (BHHH) is used on the final iteration for all models, with the exception of the United Kingdom. BHHH uses a modified method of scoring (see Berndt et al. (1974)) to optimise the function. On the final iteration of the United Kingdom's uncovered interest parity model, a more general purpose algorithm is applied. [33]
See Appendix 2 for a flow chart of our estimation procedure. [34]