Forty-five years ago when I started to study econometrics seriously I was full of confidence that the application of mathematics and statistical methodology would soon lead to a better understanding of the economic system, and, ultimately, to better public and private policies. Now, with the world economy poised on the edge of a world slump and Eastern Europe and the undeveloped world in turmoil my general hope seems optimistic. Even within econometrics, experience proves that the more we study econometrics the more there is to discover. I will concentrate on a brief survey of the problems in successfully modelling an economy both for understanding the way that it works and for the purpose of deciding on successful government policies.

To some extent there is a basic general philosophical problem which econometrics shares with all sciences: the use of stochastic models to represent both the scientists' ignorance of an immense amount of (hopefully) trivial detail and at a more operational level to allow models to fit data from the real world by introducing errors of various kinds: partly, errors to represent the failures of data-gatherers to collect numbers which represent the ideal economic variables of our economic theories; and partly, errors to represent a very large number of factors which could not be included explicitly in our equations without excessive complication. There is a sense in which, for human behaviour, all activity is determined by some non-random cause; very rarely does a decision taker act by consciously making a random decision. But, we have so little information on the detailed causes of individual decisions that we have to take a treatment in which some part of the causation is assumed to be a random variable. Econometric models are also made much more difficult to use by the problems of size measured simply in terms of the number of individual decision takers, and the number of variables which each decision taker controls. Each model, if it is to represent correctly the true complexity of the economy, should contain millions of variables. Even now when computers can deal with thousands of megabytes we cannot expect to estimate models on this scale if only because we do not collect data in sufficient detail, and even as we accumulate data over time we find that even if we are able to use forty years of quarterly or monthly data, it is unusual to find a way of modelling such a time-series without in effect assuming that the model has fundamentally changed over this period. That is to say, the effective sample size may be limited by the necessity, partly because of discontinuous institutional and governmental changes, to split the sample into shorter sub-periods. This contrasts with other scientific areas, such as earth sciences or meteorology or astronomy/cosmology, where models are large with many variables, but the length of sample time periods used to estimate these models is relatively longer. In practice econometricians must resort to aggregation. Our data is measured for unit time periods which are long compared with some of the decision lags and the production lags in the system. In effect available data restrictions and the computational difficulties of using really short time intervals result in our almost always aggregating to monthly or quarterly data.

The difficulty of working with innumerable micro-variables forces us to use macro-variables which are usually comparatively simple aggregates or index numbers. I am here using micro and macro for convenience of reference rather widely to mean the original variables of the detailed model and the aggregated variables. But, the theory of stochastic aggregation in non-linear dynamic models puts very restrictive conditions on the model before perfect aggregation is possible. Even for a simple linear dynamic model with no a priori constraints on the coefficients, an aggregated model has a much more complicated lag structure than that of the corresponding micro-equations, and the errors on the equations have a complicated moving-average specification (Hendry, Pagan, and Sargan, 1984). The models which we estimate are merely approximations, which can only be expected to work while the data remains in some bounded set or while the economy stays close to some prescribed path. The errors on the equations can then be thought of as defined by approximate macro-equations to be complicated functions of the micro-variables, and so have a very complicated stochastic specification, which on estimation can only be approximated. If we take account of this, what difference does it make to our accustomed methods of estimation and testing? If we estimate as though our macro-model were really generating the macro-variables, setting up the macro-model on the usual basis of treating aggregated variables as if they were perfectly homogeneous commodities, price indices as if they were the corresponding prices, each commodity as if produced by identical perfectly competitive firms, etc., or more generally setting the macro-model up by assuming that we have classes of identical decision takers, so that aggregate variables can be taken as generated at the macro level by the same simple equations as those previously assumed at the micro level, then can we assume that maximum likelihood estimates of the macro-equations will yield forecasting equations for the macro-variables which are at least near optimal in some sense? The commonsense answer is that if our assumptions that aggregation only takes place over homogeneous decision takers and commodities is nearly correct, then we might expect to generate good predictors in this way. But in practice, decision takers vary both in their reaction functions and in their lag behaviour. It seems that if we do not apply too many restrictive constraints we may get models which predict reasonably well, purely from the assumption that after suitable transformations our economic variables can be treated as stationary random variables, and so, that we can use optimal forecasting methods based purely on statistical time series forecasting theory. This is the conclusion drawn by Liu (1960) and by Sims (1980).

In these circumstances we cannot expect our models to yield good forecasts if the variables move outside the range which they covered in the sample estimation period, and if we forecast from a partial model we cannot expect our forecasting equations to be valid if there is a change on the equations required to close the system: for example, a change in the government or a change in public policy may increase the forecasting error of equations describing the private sector of the economy. Clearly there is also some doubt as to the efficacy of using Neyman–Pearson type testing when we do not believe that any of our models are anything but simplified approximations, and there are similar difficulties in applying decision theoretic and Bayesian methods to the choice of model specification in this situation. Having contemplated whether some alternative criterion of choice should be adopted which explicitly takes account of the difficulty in believing in the exact validity of any of our stochastic models, it became clear that there would be great arbritrariness in setting up any other criterion, so that, following in the Popperian tradition it seems that a reasonably robust procedure for simple problems is to estimate and test as if the most general model, which is included in the set of models under consideration, is a valid stochastic model, ensuring that the stochastic processes generating the errors on the equations are rather general and unconstrained. For large scale models with many parameters this may not work out too well, but it is also very difficult to find a reasonable alternative. I also find that the Bayesian or decision theoretic approach is unappealing, since this in general requires us to specify prior probabilities for a large set of alternative models, to then compute posterior probabilities from our sample, and to then give a combined forecast based on a loss function of the errors of forecast which is rather arbitrary.

I have already mentioned that the nature of the omitted variables treated as random errors and the effect of time and variable aggregation make it necessary to treat the random errors as autocorrelated, and to estimate and test fairly complicated stochastic processes to explain these errors. But, a more difficult problem is the treatment of trend. The treatment of trends as random walks ignores the fact that for most economic trends it is possible a priori to determine the sign of the trend, and that the mechanism that ensures this is likely to be rather non-linear. For example one sector of the economy where the random walk form of trend is subjectively most likely to occur is for the price level and related variables which are quoted in current value terms. In this sector both government and private institutions are tolerant of a finite positive rate of price inflation, and costs have a tendency to be rigid in a downward direction. Counteracting this tendency is the effect of technical progress, but its primary effects are only strong in some sectors of industry, and even in these sectors it may have the effect of increasing wages so as to encourage workers to accept new technology, rather than of decreasing prices. Of course technical progress is directly important in determining the level of output, and will only take place if it increases production, leading to an increasing trend in production. It follows that one should expect at least some component of the trend to be non-stochastic. In the last ten years much interesting and original work has been done on testing these different approaches to trend (Phillips and Durlauf, 1986), and it is clear that the power of the tests depends very much on the dynamic properties of the economic variables, but that frequently a rather large sample is required to decide between the hypotheses that there is a linear non-stochastic trend present, or that the trend is of random walk type. Of course, it may be sensible to assume that both types of trend are present. If the sample size is too small to discriminate, then some Monte Carlo results for simple models indicate that the biases in the estimated coefficients of the economic variables are reasonably small compared with their standard errors whichever treatment of trend is used (even if long period forecasts from the two models may be different).

The development of the error correction model (Engle and Granger, 1987; Davidson et al., 1978) gives an interesting compromise where some variables are stationary, apart from a low order non-stochastic trend, and where other variables are first-difference stationary with random walk trends. I will illustrate this by briefly looking at a model of wage-price inflation which I developed in the early 1960s (Sargan, 1964). The model in its simplest form contained two stationary variables, the real wage-rate and the level of unemployment, and a third variable being the first difference or rate of change of money prices. Unemployment only occurred in the equations as a lagged variable and an equation explaining it was not estimated. Prices were explained by a simple first-order homogeneous equation which also included the price of imports as an exogenous variable. The equation for wage-rates was of the error correction type, so that the first difference of money wage-rates is related to the difference between the current real wage-rate and the equilibrium level of real wage-rate. The basic idea of this equation was that workers have memories and that their desired level of income is determined by their recent experience of real wages. If, for any reason, their real wage has fallen, for example, because the price level has risen more rapidly then they expected, then they have the motive to seek a larger wage-rise. This memory of previous real-wage levels might be expected to be rather long term; it is often the case that after a fairly long holding down of some sectional wages the workers will demand an extra large increase quoting the failure of their wages to keep in line with retail prices over the period. In fact, this type of equation performed quite well for a considerable period as an explanation of wage-rates for the British economy.

In so far as this is a valid explanation of the way the labour market works, it reinforces the advantages of an economy where real income and productivity are rising together, as against one where real income is falling. In the former economy, not only will rising income lead to a higher level of savings through lags in the adjustment of consumption, but labour costs will be lower in so far as real wages will lag. This, then, contributes to the problems of an over-mature economy, such as the British, to generate an adequate level of investment, in addition to the problems of these economies in modernising their institutions such as educational and transport systems, and the problem of the depletion of some of the exhaustible resources in mature economies.

One hopeful development which should make the use of aggregated models less necessary, and indeed might make the use of much more detailed and so more easily tested and estimated models possible, is the continued growth in the size and speed of computers, particularly the use of parallel computers. More and more transactions are being computerised, and more and more data is being accumulated on a micro-level individual transaction basis. More sample surveys casting light on the individual expectations of both householders and business are also being carried out. At the same time we are living in interesting times for the world economy with variables taking extreme values and working on unstable regions of the variable space, and this should mean that if the increasing supply of economic data could be properly exploited a far deeper understanding of the complexity of the micro-relationships might be obtained, with possible non-linearities, and individual lag structures, and endogenous public policy variables, and shorter unit-time periods all being sources of possible improvement of the predictive power of the models. Indeed in recent years understanding of the peculiar features of panel data (Chamberlain, 1984) and in particular understanding of the problem of endogenous/exogenous variables, of the peculiar error structures, and of the non-normality of the errors, or of the presence of outliers in these models, has much improved, so that as data accumulates and more complicated models are built better understanding and forecasting should follow.

A further source of confidence is the high quality of young researchers in the field of econometrics. In the 1970s it seemed difficult to attract young scholars with a first rate training in statistical theory into the field. But, it is now clear whenever one reads the current issue of Econometric Theory, The Journal of Econometrics or Econometrica that much exciting and original work is being produced, so that as we discover more problems, at the same time we recruit more quality researchers to solve them. This has the natural consequence that the quality of applied model building is also improving. Applied researchers are no longer so likely to treat estimation as a simple strategy of equation fitting, attempting to diagnose faults in their model only in the most perfunctory way, and are certainly conscious of the problems of data-mining that arise in model specification.

So, there is great hope for the future development of econometrics around the world, and of great benefit to the world economy from this development. Particularly in Spain, there is much interesting new work going forward. I am particularly pleased to be given an honorary degree from the Carlos III University where both statistics and econometrics are so well represented. Thank you.