# An Empirical Example

In this section, we use annual UK data to estimate the demand for money (i. e. equation (19.12) extended to include one additional explanatory variable), with and without concomitants, over the long period 1881-1990. The algebraic forms of these two models are given by the sets (19.12)-(19.14) and (19.7) of equations, respectively. Post-sample forecasts are generated over the period 1991-95. The dependent variable is the log of M3 (currency held by the public plus gross deposits at London and country joint stock and private banks divided by the implicit price deflator for net national product). One of the regressors is the log of per capita net national income, deflated by the implicit price deflator. The antilog of rt is a short-term rate – the rate on three month bank bills. Follow­ing Friedman and Schwartz (1982), and others, we use the rate of change of nominal income as a regressor to proxy the nominal yield on physical assets – that is, as an additional opportunity cost variable. Two concomitants are used: (i) a long-term interest rate – the annual yield on consols; and (ii) the inflation rate, as measured by the rate of change of the implicit price deflator. All data are from Friedman and Schwartz (1982) and have been updated by the present

authors.10

Table 19.1 presents the results.11 RCM1 and RCM2 denote the above extended equation (19.12) without and with concomitants, respectively. The coefficient esti­mates are the average values of the individual time-varying coefficients. Point estimates of the elasticities of income and the interest rate in Table 19.1, but not their t-ratios, are within the range of those yielded in previous empirical studies of UK money demand (see, e. g. Hondroyiannis et al., 1999). Also, both RCM1 and RCM2 produce low root mean square errors (RMSEs). In this example, the equation without concomitants yields a lower RMSE over the post-sample period than does the equation with concomitants. An explanation of this result is that over the range of the values of its dependent and independent variables for the period 1991-95 the money demand function without concomitants seems to approximate the true money demand function better than the money demand function with concomitants. The specifications were also used to provide fore­casts over various decades beginning with the 1930s. For this purpose, each specification was re-estimated using data prior to the decade for which it was used to forecast. The equation with concomitants produced lower RMSEs in four out of the six decades. For the sake of brevity, these results are not reported but are available from the authors.

Table 19.1 Long-run elasticities

 RCM1 RCM2 Intercept -3.69 -3.19 (-4.97) (-4.03) Short-term interest rate -0.04 -0.01 (-2.86) (- 0.51) Real per capita income 0.74 0.67 (6.03) (4.98) Nominal income growth -0.41 -0.39 (-4.15) (-3.09) RMSE 0.020 0.041

Estimation period is 1881-1990. Forecast period is 1991-95. Figures in parentheses are f-ratios.

Table 19.2 Long-run elasticities and direct effects from RCM2

Infercepf Inferesf rafe Real per capifa Nominal income

income growfh

 Yet Y1t Direcf effecf Y2t Direcf effecf Yst Direcf effecf -3.19 -0.01 -0.007 0.67 0.68 -0.39 -0.42 (-4.0) (-0.5) (-0.4) (5.0) (5.1) (-3.1) (-3.1)
 Figures in parentheses are f-ratios. Table 19.2 reports the averages of the total and direct effect components of the coefficients from the equation with concomitants (i. e. RCM2). Recall, since this specification includes two concomitants, it is possible to extract the direct effects from the total effects, as shown in Section 3. (For the other specification, the indirect and mismeasurement effects in each coefficient are captured in the cor­responding error term.) As shown in the table, the y (i. e. total) coefficients and the direct-effect coefficients are very close to each other for all variables, as are the corresponding f-ratios. Figures 19.1 and 19.2 show the time profiles of the elasticity of the short-term interest rate in the absence of concomitants and with concomitants, respectively. The figures also include the time profile of the short­term interest rate. Without concomitants, the interest rate coefficients vary within extremely narrow ranges around their average values. To be exact, the elasticity of the short rate varies between -0.0422 (in 1976) and -0.0431 (in 1973). The narrow range of the interest rate elasticities is due to the specification of the coefficients Total effect – ……. Direct effect elasticity of – ……. Short-term elasticity short term due to long term interest rate in the absence of concomitants. Without concomitants the coefficients are equal to a constant mean plus an error term. If the error term has a small variance and does not exhibit serial correlation, the coefficient itself will not vary very much. As shown in Figure 19.2, the coefficients with concomitants exhibit a wider vari­ation than is the case in the specification estimated without concomitants.

The RCM procedure also provides the coefficients of the other regressors. To save space, we report only the time profiles of the interest rate elasticity to give a

flavor of the results obtainable from the procedure. Obviously, a richer specifica­tion of concomitants might well have provided different results. Examples of the use of varying combinations of concomitants can be found in the papers cited at the end of Section 3.

4 Conclusions

This chapter has attempted to provide a basic introduction to the rationale under­lying RCMs. The focus has been on what we characterized as second-generation RCMs, which have been developed to deal with four main problems frequently faced by researchers in applied econometrics. These second-generation RCMs aim to satisfy the conditions for observability of stochastic laws. The use of concomitants – variables that are not included as regressors in the economic relationship to be estimated, but which help deal with correlations between the explanatory variables and their coefficients – allows estimation of both direct and total effects. A set of model validation criteria have also been presented that can be used to discriminate among models and these criteria have been applied to RCMs.

Notes

* Views expressed in this chapter are those of the authors and do not necessarily reflect those of the Office of the Comptroller of the Currency, the Department of the Treasury, International Monetary Fund, or the Bank of Greece. We are grateful to Badi Baltagi for encouragement and guidance, and to four anonymous referees for helpful comments.

1 The suggestion that the coefficients of regression could be random was also made by Klein (1953).

2 An exposition of first-generation RCMs at text-book level has been provided by Judge, Griffiths, Hill, Lutkepohl and Lee (1985, chs 11, 13, and 19). See, also, Chow (1984) and Nicholls and Pagan (1985).

3 The discussion of the real-world sources and interpretations of P, and their implica­tions for its distribution is postponed until the next section.

4 We start with this assumption and, in the next section, detect departures from it that are warranted by the real-world sources and interpretations of the coefficients of equation (19.1).

5 For a derivation of E(\$2|xt), see Judge et al. (1985, p. 435).

6 The above argument is valid even when q(w) is replaced by the Breusch and Pagan (1979) or Judge et al. (1985, p. 436) test statistic.

7 We can write p(mt, rt, yt |zt, 0) = p(mt | rt, yt, zt, 01)p(rt, yt |zt, 02), where p(-) is a probabil­ity density function, zt is a vector of concomitants, and 0, 01, and 02 are the vectors of fixed parameters. Since y0t, y1t, y2t, r, and yt are correlated with one another, the inferences about a1t and a2t can be drawn without violating probability laws by using the density p(yQt + y1trt + y2tyt | rt, yt, zt, 01) if the following three conditions are satisfied: (i) 01 and 02 are independent – a good discussion of parameter independence is given in Basu (1977); (ii) (y0t, y1t, y2t) are independent of (rt, yt), given a value of zt – a good discussion of conditional independence is given in Dawid (1979, pp. 3-4); (iii) pr(rt Є Sr, yt Є Sy |zt, mt) = pr(rt Є Sr, yt Є Sy |zt), where the symbol pr is short­hand for probability, and Sr and Sy are the intervals containing the realized values of rt and y, respectively, to which the observed values of mt are connected by a law – a definition of stochastic law is given in Pratt and Schlaifer (1988). When condition (ii) holds, p(Y0t, Y11, Ya, r t, yt|z t, n) = p(Yot, Y1 t, Y2 t|z t, n 1)p(r„ yt |z t, Є2). This equation provides a formal definition of concomitants.

8 For further discussion of these points, see Schervish (1985).

9 With an inconsistently formulated model, even use of Bayesian methods will prob­ably lead to incoherent forecasts (Pratt and Schlaifer, 1988, p. 49).

10 The data and their sources are available from the authors.

11 The t-ratios were computed by taking account of the correlations among the coefficients.

References

Akhavein, J. D., P. A.V. B. Swamy, S. B. Taubman, and R. N. Singamsetti (1997). A general method of deriving the inefficiencies of banks from a profit function. Journal of Produc­tivity Analysis 8, 71-94.

Basu, D. (1977). On the elimination of nuisance parameters. Journal of the American Statis­tical Association 72, 355-66.

Breusch, T. S., and A. R. Pagan (1979). A simple test for heteroscedasticity and random coefficient variation. Econometrica 47, 1287-94.

Chang, I., P. A.V. B. Swamy, C. Hallahan, and G. S. Tavlas (1999). A computational ap­proach to finding causal economic laws. Computational Economics forthcoming.

Chow, G. C. (1984). Random and changing coefficient models. In Z. Griliches and M. D. Intrilligator (eds.) Handbook of Econometrics, Volume 2, Amsterdam: North-Holland Publishing Company.

Dawid, A. P. (1979). Conditional independence in statistical theory. Journal of the Royal Statistical Society B, 41, 1-31.

de Finetti, B. (1974a). Bayesianism. International Statistical Review 42, 117-30.

de Finetti, B. (1974b). The Theory of Probability, Volume 1. New York: John Wiley & Sons.

Friedman, M., and A. J. Schwartz (1982). Monetary Trends in the United States and the United Kingdom: Their Relation to Income, Prices, and Interest Rates, 1867-1975. Chicago: Univer­sity of Chicago Press.

Friedman, M., and A. J. Schwartz (1991). Alternative approaches to analyzing economic data. American Economic Review 81, 39-49.

Goldberger, A. S. (1991). A Course in Econometrics. Cambridge, MA: Harvard University Press.

Hondroyiannis, G., P. A.V. B. Swamy, and G. S. Tavlas (1999). Modelling the long-run demand for money in the United Kingdom: A random coefficient analysis. Economic Modelling forthcoming.

Judge, G. G., W. E. Griffiths, R. Carter Hill, H. Lutkepohl, and T. Lee (1985). The Theory and Practice of Econometrics, 2nd edn. New York: John Wiley and Sons.

Klein, L. R. (1953). A Textbook of Econometrics. Evanston: Row Peterson and Company.

Leusner, J., J. D. Akhavein, and P. A.V. B. Swamy (1998). Solving an empirical puzzle in the capital asset pricing model. In A. H. Chen (ed.) Research in Finance, Volume 16, Stamford, CT: JAI Press, Inc.

Moggridge, D. (1973). The General Theory and After. New York: Macmillan.

Mosteller, F., and J. W. Tukey (1977). Data Analysis and Regression. Reading, MA: Addison- Wesley Publishing Company.

Nicholls, D. F., and A. R. Pagan (1985). Varying coefficient regression. In E. J. Hannan, P. R. Krishnaiah, and M. M. Rao (eds.) Handbook of Statistics, Volume 5, New York: Elsevier Science Publishers.

Oakes, D. (1985). Self-calibrating priors do not exist. Journal of the American Statistical Association 80, 339.

Phillips, R. J., and P. A.V. B. Swamy (1998). Par clearance in the domestic exchanges: The impact of national bank notes. In A. J. Field, G. Clark, and W. A. Sundstrom (eds.) Research in Economic History. pp. 121-44. Stamford, CT: JAI Press, Inc.

Pratt, J. W., and R. Schlaifer (1984). On the nature and discovery of structure. Journal of the American Statistical Association 79, 9-21, 29-33.

Pratt, J. W., and R. Schlaifer (1988). On the interpretation and observation of laws. Journal of Econometrics 39, 23-52.

Schervish, M. J. (1985). Discussion of "Calibration-based empirical probability" by A. P. Dawid. Annals of Statistics 13, 1274-82.

Stone, M. (1974). Cross-validatory choice and assessment of statistical predictions. Journal of the Royal Statistical Society B, 36, 111-33.

Swamy, P. A.V. B., and J. S. Mehta (1975). Bayesian and non-Bayesian analysis of switching regressions and of random coefficient regression models. Journal of the American Statist­ical Association 70, 593-602.

Swamy, P. A.V. B., and G. S. Tavlas (1995). Random coefficient models: Theory and applica­tions. Journal of Economic Surveys 9, 165-82.

Swamy, P. A.V. B., and P. A. Tinsley (1980). Linear prediction and estimation methods for regression models with stationary stochastic coefficients. Journal of Econometrics 12, 103-42.

Zellner, A. (1969). On the aggregation problem. In K. A. Fox, J. K. Sengupta, and G. V.L. Narasimham (eds.) Economic Models, Estimation and Risk Programming: Essays in Honor of Gerhard Tintner. pp. 365-74. New York: Springer-Verlag.

Zellner, A. (1988). Causality and causal laws in economics. Journal of Econometrics 39, 7-22.