Terminal dates and dynamic properties of National Institute Model 11.

Ireland, Jonathan ; Wren-Lewis, Simon

TERMINAL DATES AND THE DYNAMIC PROPERTIES OF NATIONAL INSTITUTE MODEL 11 Dynamic optimisation problems involving rational expectations require a transversality condition to produce a unique solution (see Hall, Henry and Wren-Lewis, 1986, for an example). These transversality conditions generally amount to a statement about the nature of the endogenous variable in the infinite long run; for example, that the long run involves a stable equilibrium. An analagous problem arises in econometric macro models, where a unique solution requires a 'terminal condition' for any forward-looking variables beyond the solution period of the model. Transversality and terminal conditions are not equivalent, however, because we cannot solve macro models into the infinite future (see Hall and Henry, 1988, pages 192-198).

This poses a potentially serious problem for econometric models that assume consistent expectations, like recent vintages of the National Institute Domestic Model. There is a danger that, by imposing a terminal condition on the model that is really an equilibrium condition before equilibrium is reached, the properties of the model will be distorted. One way to observe this is to change the terminal date (the final period of the solution) for a given simulation, and see if the results change, as in Wallis et al, 1986, pages 52-66.

The severity of this problem will vary from variable to variable depending on the size of the forward-looking root which acts like a discount rate on the future. For example, for most of the forward-looking variables in the Institute's model the forward-looking root is well below one; that is, the current value of the forward-looking variable will only depend on events in the near future. In such cases, rate of growth terminal conditions (that specify an unchanged rate of growth in the period beyond the terminal date) appear to be fairly robust to changes in the terminal date. A key exception is the exchange-rate equation, where the (model-wide) forward root is very close to unity and hence events quite far in the future are likely to affect the exchange rate. Here a rate of growth terminal condition implies no change in the net asset to GDP ratio. Once revaluations have come to an end this is equivalent to the current account being close to balance (see Barrell et al, 1988). Current account balance is essentially a long-run equilibrium property, and as a result it is quite likely that the properties of the whole model will be distorted if the model is not near an equilibrium at the terminal date.

As a result, Wren-Lewis (1988) argues that the latest version of the Institute's domestic model, Model 11, must be solved over at least eight years to avoid distorting its properties. Only after this period will the model be near its equilibrium. In this note we analyse how long the latest version of Model 11 (11.2), estimated using rebased (1985) data, takes to settle to a new equilibrium. The simulation we use is a one period shock to nominal interest rates. Past experience has suggested that changes to interest rates involve some of the longest lags in the model, and so this is quite a demanding test. As the exchange rate is our particular concern, we focus on the initial jump in the exchange rate produced by this simulation. Chart 1 plots the size of this jump as the terminal date is extended. The simulation begins in 1986 Q1, and the horizonal scale gives the terminal date. As a result, the shortest solution period is five years and the longest is twelve years.

As the chart represents the same simulation observed in the same, initial quarter, we would hope to see a horizontal line. For short solution periods (less than eight years) we clearly do not. Only after ten years do the simulation results settle down to an unchanged profile. A similar pattern is observed for the 1980 base data version of this model, Model 11.1, which is deposited at the Warwick Bureau. Even after ten years the initial jump varies slightly, suggesting the model is not quite at equilibrium. Unfortunately the vintage structure of the supply side is likely to lead to extremely long cycles in the model.

Chart 2 plots the time profile of the exchange rate for three simulations, where the terminal dates are 1994 Q2, 1996 Q4 and 1997 Q4 respectively. The latter can be regarded as our best approximation to the 'true' result, that is, the result implied if the model were solved over an infinite horizon. As the higher interest rate in the first period must be associated with a depreciation, the exchange rate jumps upwards. By solving the model over too short a period we distort both the initial jump and the subsequent path of the exchange rate. As Chart 1 suggests this distortion is not monotonic with respect to the length of the solution period.

One other interesting feature of Chart 2 is that the long-run change in the nominal exchange rate does not return to zero, even though the interest rate only changes in the first quarter. This is because, under the assumptions of fixed real government expenditure and nominal interest rates, the price level in the model has hysterisis properties. Its level in the long run depends not only on the long-run value of exogenous variables, but also on the past history of the economy itself. This would not be the case if our policy assumption involved fixing a nominal variable, like money or government debt. Thus while temporary shocks do not appear to alter the long-run solution for real model variables like output or the real exchange rate, they do influence the long-run price level and nominal exchange rate.