Forecasting UK stock prices

Applied Financial Economics, 1996, 6, 279 286 Forecasting UK stock prices CHULHO JUNG and ROY BOYD Department of Economics, Ohio ºniversity, Athens, OH 45701, ºSA The Vector Autoregressive (VAR) model, the Error Correction Model (ECM), and the Kalman Filter Model (KFM) are used to forecast UK stock prices. The forecasting performance of the three models is compared using out of sample forecasting. The results show that the forecasting performance of the ECM is better than that of the VAR and the KFM, and that the VAR performs a forecasting better than the KFM. It seems that the ECM outperforms the VAR and the KFM, since the ECM allows for dynamic updating via an error correction term. I. INTRODUCTION Many economists believe that the overall performance of a country s economy is strongly related to the performance of its stock market. Some empirical studies, however, show that this relationship does not necessarily hold. In the sense that stock market performance is highly unpredictable, the movement of stock prices has sometimes been expressed as a random walk process. Shiller (1989) and DeBondt (1991) have also shown that stock prices are both unpredictable and volatile in the short run. Other empirical studies show that overall economic performance has an influence on the performance of stock prices. Studies by Umstead (1977) and Fama (1981) show that a positive correlation exists between real economic growth and stock prices. Spiro (1990) and Cochrane (1991) find that economic fluctuations influence stock prices and that macroeconomic variables such as real output and the interest rate can explain stock market movement significantly. For over 100 years, the UK has been a major investor in international ventures and British stocks play an important part in the portfolios of most major international investors. Concern over the vulnerability of oil prices and the future role of the UK in a united Europe have recently led to concern over potential volatility in UK stock market indices. While most studies of movements in stock markets have been limited to the US stock market, the long-term trends in the markets of such countries as the UK, however, should be of equal importance to corporations and private investors. The presence of large numbers of multinational corporations, floating exchange rates and instant communication have contributed to a global market in stock and securities trade. This means that understanding markets of other countries has become a necessity rather than mere curiosity. Although observed volatility of the UK stock market pales in comparison with that of other markets, for example the Tokyo Stock Exchange, the selection of an accurate long-run price forecasting model is a topic of considerable interest. This study compares three such models and determines which makes the most accurate out-of-sample forecast. Section I describes the data on interest rates, GNP, and inflation to be used as explanatory variables. A basic theoretical model is presented which subsequently will be developed into, arguably, the three most popular empirical models used for forecasting purposes; namely the Vector Autoregressive (VAR) Model, the Error Correction Model (ECM), and the Kalman Filter (i.e., Stochastic Parameter) Model (SPM). While all three models are similar in structure, there is a fundamental difference between the classic VAR and the other two. Whereas the VAR yields estimates which are unchanging over time, ECM s and SPM s allow for dynamic updating via either an error correction term or time-varying coefficients. Indeed, it is these features of ECM s and SPM s which give them a certain appeal as forecasting instruments. Section II empirically estimates all three models and contrasts their forecasting performance. See, for example, Fama (1970), DeBondt and Thaler (1989), Randolph (1991), and McQueen (1992). 0960 3107 1996 Chapman & Hall 279

280 C. Jung and R. Boyd II. DATA USED Besides the positive impact of general economic activity, noted above, it is also generally thought that changes in interest and inflation rates have a significant impact on stock prices. As pointed out by Sease and Steiner [1991], lower interest rates cut short-term CD rates. This makes stocks more attractive to investors. At higher interest rates, more investors would wish to avoid stock-related risks by selling stock and driving down share prices. Furthermore, by eroding corporate profits and, concomitantly, lowering stock prices as risk to firms increases, higher interest rates have a negative effect on share prices. Since many investors buy shares on loans one direct impact that rising interest rates have on stock prices is the so-called margin shake-out. As interest rates rise, the marginal cost of carrying shares will increase. In these circumstances many investors will choose to liquidate their positions, thus lowering share prices. The lower value of stock, in turn, will force more investors to meet margin calls, promoting further sales (Geisst, 1989). The effect of inflation on stock prices is not as clear as that of the interest rate. Most economists have observed a negative relationship between inflation and stock prices. Various hypotheses have been proposed to explain this phenomenon (Fama, 1981). A common interpretation is that an increase in inflation raises the nominal rate of return required by investors. This can only be satisfied by a drop in the share price (Asikoglu and Ercan, 1992). Many economists and financial analysts believe that there should be a positive relationship between inflation and stock prices, since the stock market provides investors with a reasonably good opportunity to hedge against inflation. There are, however, a number of empirical studies which indicate that, on the contrary, stock markets provide a rather poor hedge against inflation. For all intents and purposes then, the sign of the coefficient on inflation is indeterminate. The dependent variable employed here is the real London Stock Exchange (LSE) price index. This price index was chosen because the LSE composite index is a broad market indicator and, as a result, provides a superior measure of overall market movements. Monthly data from January 1964 December 1987 (288 months) are used for estimation and data from January 1988 June 1989 (18 months) data are used to perform out-of-sample predictions. The monthly series of LSE indices are divided by monthly series of consumer price index. These two variables fare obtained from Citibank Database. The real level of output is used as a proxy for aggregate economic activity. There is an implicit assumption here that, overall, investors have perfect knowledge of the current performance of the economy. Monthly data of Index of Industrial Production are obtained from Citibank Database and used for the output variable. Real GDP or real GNP could also be used for the output variable, but monthly data are not available for these variables. The UK Treasury 3-month bill new issues rate is used as a proxy for the short-term interest rate. The monthly data for this series are obtained from International Monetary Fund CD Rom. Pretesting revealed that the rate of inflation had a statistically insignificant coefficient and, consequently, it was dropped from the model as an explanatory variable. III. BASIC MODEL The general functional form of the model to be considered is SP "#β GNP#y ¹R #ε t"1, 2, ¹ (1) or, ln SP"#β lngnpt #y ln¹r#ε (2) where SP Real London Stock Exchange Composite Index GNP Index of Industrial Production (1987"100) ¹R UK Government Treasury 3-month bill new issues rate A specification test is conducted to find the most appropriate functional form. While a PE test would be appropriate here, it offers no guidance between the linear and the log-linear form. Therefore, a Box Cox test is performed to test for linearity. The Box Cox test results show that the null hypothesis of linearity is rejected at the 5% level of significance. Next, the variables used for the presence of a unit root are tested. When variables follow a unit root process, it can lead to spurious results when the level of the variables is used for estimation purposes. More specifically, the variance of a unit root process becomes infinite and OLS estimation might be inappropriate. Traditionally, this problem has been handled by differencing the data, but this can lead to a significant loss of information. As Granger (1981) points out, however, a set of variables, all of which are stationary only after differencing, may have linear combinations which are stationary without differencing. In such a case, those variables are said to be cointegrated. Engle and Granger For a concise but complete presentation of all relevant theories on this issue, see Spiro (1990). See Mackinnon, White, and Davidson (1991) for details of the PE test. See Box and Cox (1964) for general discussion of the testing procedure.

Forecasting ºK stock prices 281 (1987) find a cointegration testing procedure using the Dickey Fuller unit root testing procedure. However, their cointegration testing procedure is constrained by the limitation that it requires prior knowledge about cointegrating vectors. Among many other cointegration testing procedures, Johansen (1988) and Johansen and Juselius (1990) consider multivariate cointegration tests which can resolve this problem. Johansen s maximal eigenvalue testing procedure is used in the present study. To perform a cointegration test, we need to test if given variables are stationary. Dickey Fuller and Augmented Dickey Fuller tests are applied to determine the order of integration of each variable and the results are shown in Table 1. The Akaike s information criterion (AIC) and final prediction error (FPE) are used to determine the lag order for the Augmented Dickey Fuller tests. The unit root test is conducted for both levels and first differences of each series. According to the results in Table 1, the null hypothesis of a unit root is accepted for the level series, but rejected for the first differenced series at the 5% level of significance. These results lead to the conclusion that each variable is nonstationary and integrated of the order of one. The multivariate cointegration test can, hence, be conducted. Johansen s maximal eigenvalue test is applied to three variables. The null and alternative hypotheses for the maximal eigenvalue test are H : r)k and H : r'k where k is some integer from 0 to 2 and r is the number of cointegrating vectors. The results in Table 2 show that the null hypothesis is rejected for r"0 and accepted for r)1 and 2. That is, Table 1. Dickey Fuller unit-root testing results Augmented Dickey Fuller Dickey Fuller Lag Variables test statistic test statistic order** evels ln SP!1.0631!1.4238 2 ln GNP!1.8819!1.6929 1 ln ¹R!5.2749*!2.7537 4 First differenced ln SP!13.0577*!12.4377* 1 ln GNP!22.1691*!15.7856* 1 ln ¹R!27.3603*!12.4864* 3 *We reject the null-hypothesis of unit root at both the 1% and 5% levels. The critical value is!2.8628 at the 5% level and!3.4269 at the 1% level. **The lag order for the augmented term was determined using the minimum value of Akaike s Final Prediction Error (FPE) and Akaike s Information Criterion (AIC). They lead to the same results. Table 2. Maximal eigenvalue testing result Null Co-integration Critical hypothesis test statistic value* Ho: r"0 26.827** 21.144 Ho: r)1 7.841 14.839 Ho: r)2 1.234 8.106 *Critical values are for the 95% Quantile and are taken from Johansen and Juselius (1990). **Significant at the 5% level. there is a single cointegrating vector. These results show that the long-run movements of the three variables are determined by some common driving fundamentals, which make the model stationary without differencing. IV. VECTOR AUTOREGRESSIVE MODEL (VAR) The VAR model has proven to be a successful technique for forecasting systems of interrelated time-series variables. VARs are also frequently used for analysing the dynamic impact of different types of random disturbances and controls on systems of variables. The model makes the implicit assumption that the coefficients are fixed and that no adjustment is being made towards some long-term equilibrium. Hence, it provides a good benchmark from which to test the predictive power of both the Error Correction and Stochastic Parameter Models described below. The basic relations are estimated using the VAR model to incorporate the dynamics of lagged variables. The following system of three equations are estimated: ln SP " # ln SP # ln GNP # ln ¹B #ξ (3) ln GNP"β # β ln SP # β ln GNP # β ln ¹B #ξ (4) ln ¹B "y # y ln SP # y ln GNP # y ln ¹B #ξ (5) See Akaike (1974) for details of the AIC. See Akaike (1970) for details of the FPE.

282 C. Jung and R. Boyd Table 3. Estimation of vector autoregressive model ln SP ln GNP ln ¹B Variables (Equation 1) (Equation 2) (Equation 3) Constant 0.098 0.064!1.432* (0.158) (1.274) (!2.546) ln SP 0.985** 0.006!0.132** (90.514) (1.676) (!3.410) ln GNP!0.003 0.978** 0.616** (!0.090) (80.377) (4.519) ln ¹B!0.007 0.005 0.656** (!0.546) (1.196) (14.428) R 0.976 0.975 0.731 Note: Numbers in parentheses are t-statistics. *Significant at the 5% level of significance. **Significant at the 1% level of significance. where ξ, ξ, and ξ are the error terms. The three equations are estimated with different lag length, and the lag length is determined by Akaike s AIC and FPE measures. They are minimized at p"q"r"1. Therefore, the model is estimated with p"q"r"1, and the estimation results are shown in Table 3. The VAR model is also estimated with different lag length such as 4, 8, 12, and 16 for the purpose of forecasting. V. ERROR CORRECTION MODEL (ECM) Engle and Granger (1987) show that an error-correction model should be estimated for cointegrating variables. The cointegrating linear combination of variables is interpreted as an equilibrium relationship, since it can be shown that variables in the error-correction term in an ECM must be cointegrated, and that cointegrated variables must have an ECM representation. Cointegration provides a formal framework for testing for, and estimating, long-run equilibrium relationships among economic variables. The ECM is a dynamic system in which an error-correction term represents deviations from a long-run equilibrium relationship, while short-run dynamics are represented by lagged difference terms. In this section, the following system of three error-correction models are estimated: ln SP " # e # ln SP # ln GNP # ln ¹B #η (6) ln GNP "β #β e # β ln SP # β ln GNP # β ln ¹B #η (7) ln ¹B "y #y e # y ln SP # y ln GNP # y ln ¹B #η (8) where e "ln SP!δK lngnp!δk ln¹b with δk and δ being the least squares estimates of an equation, ln SP"δ ln GNP #δ ln ¹B #ζ, η, η, η, and ζ are the error terms, and, as before, is a difference operator. The system of these three equations is estimated (i.e., full ECM) for different lag length, and the optimal lag length determined by Akaike s AIC and FPE. Various lag lengths were tried, p)16, but it was found that p"1 gives the minimum of AIC and FPE. Therefore, the model is estimated with p"1, and the results are shown in Table 4. For the purpose of forecasting, however, the model is estimated with different lag lengths such as 4, 8, 12, and 16. VI. STOCHASTIC PARAMETER MODEL (SPM) In fixed parameter models, the relationship between the dependent and independent variables is assumed to remain Table 4. Estimation of error correction model ln SP ln GNP ln ¹B Variables (Equation 1) (Equation 2) (Equation 3) Constant 0.0004 0.0016 0.0028 (0.1642) (1.5910) (0.2375) e!0.0169* 0.0063!0.0789* (!1.6426) (1.8125) (!2.0003) ln SP 0.3634** 0.0034!0.4712* (6.4898) (0.1800) (!2.1914) ln GNP!0.0282!0.1533**!0.1906 (!0.0473) (!2.6238) (!0.2866) ln ¹B 0.0065 0.0043!0.3690** (0.4431) (0.8729) (!6.5494) R 0.14 0.14 0.17 Note: Numbers in parentheses are t-statistics. *Significant at the 5% level of significance. **Significant at the 1% level of significance. Phillips (1991) compares the statistical properties of the ECM and the VAR, and concludes that the ECM is preferred over the VAR.

Forecasting ºK stock prices 283 fixed. Over long periods of time, however, these relationships may change due to outside shocks or the expectations of economic agents within the market itself. Hence, in the stochastic parameter regression model, the impact of real output and the interest rate on the movement of the real stock price is allowed to vary over time. The actual functional form of this model is similar to that of the fixed coefficient model above, except that, now, the coefficients are random variables which are allowed to evolve over time. The SPM is used in the context of the Vector Autoregressive model. More formally: ln SP " # ln SP # ln GNP # ln ¹R #ε (9) where the coefficients follow a first-order autoregressive process, as in Equation 10:! and where " 0 0 0 0 0 0 0 0 0 0 0 0! # μ μ (10) μ μ,,, : Diagonal transition factors μ, μ, μ, μ : White noise,,, : Fixed means of,,, and Similarly, for the other two variables, ln GNP "β #β ln SP #β ln GNP #β ln ¹R #ε (11) ln ¹B "½ #½ ln SP #½ ln GNP #½ ln ¹R #ε (12) The coefficients of Equations 11 and 12 also follow a firstorder autoregressive process as in the transition Equation 10. The model is estimated by means of a maximum likelihood procedure using the Kalman filter algorithm (see Kalman, 1960). The details of this algorithm are straightforward but lengthy and, hence, for expositional purposes are relegated to the appendix. To make use of the model Table 5. Estimation of stochastic parameter model ln SP ln GNP ln ¹B Variables (Equation 1) (Equation 2) (Equation 3) Constant 0.4165* 0.0639* 0.5321* (0.2924) (0.3206) (!0.6084) ln SP 0.9775* 0.0058*!0.5603* (0.3726) (0.5188) (!0.5255) ln GNP!0.0771* 0.9780* 0.8728* (0.3229) (0.4254) (!0.3279) ln ¹B 0.0143 0.0049* 0.0318* (0.3851) (0.2567) (0.8771) R 0.9999 0.9749 0.9999 Note: Numbers in parentheses are diagonal transition factors. *Mean value of stochastic parameter. a set of starting values are chosen somewhat arbitrarily, because as computations proceed, the influence of the choice is negligible. These estimation results are given in Table 5. A comparison A Likelihood Ratio Test is conducted here to test the null hypothesis that there is no significant difference between the VAR model and the stochastic parameter model (SPM). The alternative hypothesis is that the SPM is better. The likelihood ratio test statistic is: R"2(ln!ln ) (13) which in large samples is distributed as X with K"8 degrees of freedom where K is the number of restrictions imposed by assuming constant coefficients. The test statistic for the first equation is 63.68, that for the second equation is 0.50, and that for the third equation is 130.03. Since the critical value of the X is 15.507, the null hypothesis can be rejected for Equations 1 and 3, but it cannot be rejected for Equation 2 at the 5% level of significance. Hence, the test provides evidence that only the SPM estimates of Equations 1 and 3 are statistically superior to its fixed coefficient counterpart (VAR). It should be noted that, while the likelihood ratio statistic above tests for the overall stability of the model, it does not test for the relative stability of individual coefficients in the model. In this sense, it is similar to the CUSUM and CUSUM squares tests on recursive residuals (see Brown, Durbin and Evans, 1975). One of the advantages of the Kalman filter, however, is that it allows the value of each While, theoretically, off-diagonal elements need not be restricted to equal zero, the Φ matrix was assumed to be diagonal to reduce the computational burden. For additional information on state-space models in general and the Kalman Filter in particular, the interested reader should consult Newbold and Bos (1985), Diderrich (1985), Harvey (1990), and Mills (1991). When estimating a SPM one must obtain estimates of the first-order autocorrelation coefficients (i.e., the Φ s), the variance of the stochastic coefficients of the measurement equation (i.e., var( ), var( ), var( ), var( )) as well as the variance of ε in the measurement equation.

284 C. Jung and R. Boyd stochastic coefficient to be recovered for each observation. This situation seems quite plausible when one considers that agent s expectations with regard to short-term interest rates are generally considered to be highly volatile, and the stock market s response to interest rate changes has varied considerably over the period studied. VII. A COMPARISON OF FORECASTING PERFORMANCE Even if the estimation results indicate that the SPM is not preferred over the VAR model for all three equations, this does not necessarily mean that one should discard the SPM in favour of the VAR. As both Pindyck and Rubinfeld (1991) and Swamy and Tinsley (1980) point out, models should be judged in terms of their intended use, and the intended use of these models is accurately to forecast fluctuations in the stock market. With this point in mind, out-of-sample predictions were made for the period January 1988 June 1989. Although later data was available for such forecasts, this period was chosen because it predates the 1989 stock market crash, the inclusion of which would have entailed making use of non-representative data. Tables 6 and 7 show a comparison between the ECM, the VAR and the SPM forecasts. To measure the relative predictive accuracy of the three models, the root mean squared error (RMSE) and the mean absolute deviation (MAD) for the forecasts of the three models were calculated. As Tables 6 and 7 show, the ECM performs better than both the VAR and the SPM for all 18 periods following the sampling period. The VAR, however, performs better than the SPM. The models were also estimated with different lag lengths such as 4, 8, 12, and 16. The performance of the ECM is consistently better than the VAR or the SPM. In Tables 6 and 7 the sample period is divided into three periods: January 1988 June 1988; July 1988 December 1988; and January 1989 June 1989. Although the error measurement for all three models increases considerably in the third period (reflecting the increased volatility of the market in early 1989, just before the market crash), the ECM far out-performs the SPM and VAR in all three periods. The ECM has an error-correction term that adjusts the deviation from a long-run equilibrium, while the lagged difference terms of the ECM represent short-run dynamics. The SPM adjusts in a manner so that it can only update the information of the previous period, (see the Appendix) while the VAR has no ability to adjust to a deviation from a long-run equilibrium, or to update information from the previous period. Thus, the ECM was able to account for the change in the long run equilibrium of the market Table 6. RMSE measurement Time period ECM VAR SPM January June, 1988 7.09 10.68 15.53 July December, 1988 5.95 12.59 25.55 January June, 1989 17.40 28.37 47.39 Total January 1988 June 1989 11.38 18.95 32.35 Table 7. MAD measurement Time period ECM VAR SPM January June, 1988 6.91 10.46 14.81 July December, 1988 5.04 12.24 25.47 January June, 1989 16.72 27.82 46.91 Total January 1988 June 1989 9.56 16.84 29.07 over the period January 1988 June 1989 and out-perform both the VAR and SPM models. VIII. SUMMARY AND CONCLUSIONS In this study, three models were built using macroeconomic indicators as independent variables and a broad, real stockmarket index as the independent variable. These three models have the same basic functional form, but the stochastic parameter regression model assumes that the parameters are not fixed. Rather, they follow a first-order autoregressive process. The fixed coefficient VAR model, then, can be viewed as a special case of the SPM. That is to say, when the Φ matrix in the transition equation (Equation A2 in the Appendix) is a zero matrix, and the vector has a variance of zero. Even though the ECM is, technically, a fixed parameter model, it can be seen as a dynamic system in which an error-correction term represents deviations from a long-run equilibrium relationship while short-run dynamics are represented by lagged difference terms. Overall, the forecasting performance of all three models is reasonably good. However, the ECM performs much better than both the VAR and SPM, while the VAR out-performed the SPM. The ECM captures market movement much better than the others, especially in the 6 12 months that immediately followed the sampling period. The ECM has an error-correction term that adjusts the deviation from a long-run equilibrium. The lagged difference terms of the ECM represent short-run dynamics. This enables it to update information from previous periods in a manner that the One can calculate asymptotic t statistics for each of the random coefficients. Their statistical properties are not well understood, however, and it is best to use them only as a rough measure of the variability of the corresponding random coefficient.

Forecasting ºK stock prices 285 other models cannot. Even so, it would seem that since the SPM automatically updates its estimates from the immediately proceeding period it would outperform the VAR. The results here, however, suggest otherwise. One explanation is that the varying parameters are picking up white noise in the estimation process that, normally, would be picked up by the error term. This, in turn, results in high R-squared values but in biased coefficients which forecast poorly. Forecasting over the intermediate run is especially important to economic agents interested in long-run investments. This case study clearly indicates that, when used properly, error correction models can be extremely important tools for this purpose. REFERENCES Akaike, H. (1970) A fundamental relation between predictor identification and power spectrum estimation, Annals of the Institute of Statistical Mathematics, 22, 219 23. Akaike, H. (1974) A new look at the statistical model identification, IEEE ¹ransactions on Automatic Control, 19, 716 23. Ansley, C. F. and Kohn, R. (1982) A geometrical derivation of the fixed interval smoothing algorithm, Biometrika, 69, 486 7. Asikoglu, Y. and Ercan, M. (1992) Inflation flow-through and stock prices, ¹he Journal of Portfolio Management, Spring, 63 68. Box, G. E. P. and Cox, D. R. (1964) An analysis of transformations, Journal of the Royal Statistical Society, 26, 211 43. Brown, R. L., Durbin, J. and Evans, J. M. (1975) Techniques for testing the constancy of regression relationships over time, Journal of the Royal Statistical Society, 37, 149 92. Chen, N. F. (1991) Financial investment opportunities and the macroeconomy, Journal of Finance, 46, 529 54. Cochrane, J. H. (1991) Production-based asset pricing and the link between stock returns and the economic fluctuations, Journal of Finance, 46, 209 37. De Bondt, W. F. M (1991) What do economists know about the stock market? ¹he Journal of Portfolio Management, Winter, 84 91. DeBondt, W. F. M. and Thaler, R. H. (1989) A mean-reverting walk down Wall Street, Journal of Economic Perspectives, 3, 189 202. Dickey, D. A. and Fuller, W. A. (1981) Likelihood ratio statistics for autoregressive time series with a unit root, Econometrica, 49, 1057 72. Diderrich, G. T. (1985) The Kalman filter from the perspective of Goldberger-Theil estimators, ¹he American Statistician, 39. Engle, R. F. and Granger, C. W. J. (1987) Co-integration error correction: representation, estimation, and testing, Econometrica, 55, 251 76. Fama, E. F. (1970) Efficient capital markets: a review of theory and empirical work, ¹he Journal of Finance, 25, 383 416. Fama, E. F. (1981) Stock returns, real activity, inflation, and money, American Economic Review, 71, 545 65. Geisst, C. R. (1989) A Guide to the Financial Markets, St. Martin s Press, New York. Granger, C. W. J. (1981) Some properties of time series data and their use in econometric model specification, Journal of Econometrics, 16, 121 30. Harvey, A. C. (1990) Forecasting, Structural ¹ime Series Models and the Kalman Filter, Cambridge University Press, New York. Harvey, A. C. and Phillips, G. D. A. (1982) The estimation of regression models with time-varying parameters, in M. Deistler, E. Fust and G. Schwodiaur (eds.) Games, Economic Dynamics and ¹ime Series, Physica-Verlog, Wurzburg. Johansen, S. (1988) Statistical analysis of cointegration vectors, Journal of Economic Dynamics and Control, 12, 231 54. Johansen, S. and Juselius, K. (1990) The full information likelihood procedure for inference on cointegration with applications to the demand for money, Oxford Bulletin of Economics and Statistics, 52, 169 210. Judge, G. G., Griffiths, W. E., Hill, R. C., Lu tkepohl, H. and Lee, T. C. (1985) ¹he ¹heory and Practice of Econometrics, Wiley, New York. Kalman, R. E. (1960) A new approach to linear filtering and prediction problems, Journal of Basic Engineering, 82, 35 45. Kalman, R. E. and Bucy, R. (1961) New results in linear filtering and prediction theory, Journal of Basic Engineering, 83, 95 108. Mackinnon, J. G., White, H. and Davidson, R. (1983) Tests for model specification in the presence of alternative hypotheses: some further results. Journal of Econometrics, 21, 53 70. McQueen, G. (1992) Long-horizon mean-reverting stock prices revisited, Journal of Financial and Quantitative Analysis, 27, 1 18. Meinhold, R. J. and Singpurwalla, N. D. (1983) Understanding the Kalman filter, ¹he American Statistician, 37, 123 7. Mills, T. L. (1991) ¹ime Series ¹echniques for Economists, Cambridge University Press, Cambridge. Newbold, P. and Bos, T. (1985) Stochastic Parameter Regression Models, Sage Publications: Beverly Hills. Phillips, P. C. B. (1991) Optimal inference cointegrated systems, Econometrica, 59, 283 306. Pindyck, R. S. and Rubinfeld, D. L. (1991) Econometric Models and Economic Forecasts, McGraw-Hill, New York. Randolph, W. L. (1991) Use of the mean reversion model in predicting stock market volatility, ¹he Journal of Portfolio Management, Spring, 22 26. Sease, D. R. and Steiner, R. (1991) Stock soar, bonds extend rally on rate cut, ¹he ¼all Street Journal, December 24. Shiller, R. J. (1989) Market»olatility, The MIT Press, Cambridge, Massachusetts. Smith, R. and Steiner, R. (1992) Fed move ignites explosive stock, bond rallies, ¹he ¼all Street Journal, April 10. Spiro, P. S. (1990) The impact of interest rate changes on the stock price volatility, ¹he Journal of Portfolio Management, Winter, 63 68. Swamy, P. A. V. B. and Tinsley, P. A. (1980) Linear prediction and estimation methods for regression models with stationary coefficients, Journal of Econometrics, 12, 103 42. Touhey, J. C. (1980) Stock Market Forecasting for Alert Investors, AMACOM, New York. Umstead, D. A. (1977) Forecasting stock market prices, ¹he Journal of Finance, 32, 427 41. APPENDIX. A PRESENTATION OF THE GENERAL STOCHASTIC PARAMETER MODEL Description of the Kalman Filter state-space model can be found in Kalman (1960), Kalman and Bucy (1961), Harvey and Phillips (1982), Newbold and Bos (1985) and Judge et al. (1985). A detailed discussion of the generality of the Kalman

286 C. Jung and R. Boyd filter approach is available in Harvey (1990). Hence, only a simple outline of the Kalman algorithm will be given here. In general, the equation to be estimated may be written as: ½ "X b #ε (A1) where X is a row of vector of predetermined variables, denotes the transpose operator, b is a column vector of coefficients, and ε is the random error and is assumed to have a mean of zero, a fixed variance of σ, and to be free of serial correlation. The b parameters, as the subscript implies, are free to vary over time. This distinguishes the approach from the usual fixed-coefficients regression model. It is assumed that b has a mean vector β and evolves through time according to the first-order autoregressive process given by b!β"φ(b!β)# (A2) where Φ is a square matrix and is a column vector of random error terms (associated with the autoregressive process) where each element has mean zero and has covariance matrix Ω. These error terms are assumed to be uncorrelated with the regression error term ε. Equations A1 and A2 comprise the state-space model to be estimated. Equation A1 is the measurement equation which relates the observed variable to the unobservable coefficients b while Equation A2 is the transition equation which describes the evolution of the unobservable coefficients over time (Newbold and Bos (1985)). Following Diderrich (1985), the conditional expectation of b given previous realizations of y is E(b y, y, 2, y )"β#φ E(b y, y, 2, y )!Φβ #E( y, y, 2, y ), where E is the expectation operator. By assumption, is uncorrelated with ε and the y, S"1, 2, t!1. Then, since is assumed to have a mean vector of zeros, we can rewrite this equation as β(tt!1)"φβ(t!1t!1)#(i!φ)β (A3) where β(st!1) denotes E(b y, y, 2, y ). Using Equation A3, and the fact that the mean vector β has zero variance, we can denote the covariance matrix of b conditional on the history of y as P(tt!1)"ΦP(t!1t!1)Φ#Ω (A4) where P(st!1) is the covariance matrix of β(st!1). The mean and variance of y given its history, are derived from Equation A1. In particular, the variance of y, given its history, is» "X»ar(b y, y, 2, y )X #σ (A5) where»"»ar (y y, y, 2, y ) and, as before P(tt!1)"»ar(b y, y, 2, y ). As Diderrich (1985) points out, Equations A3, A4 and A5 comprise the first part of the Kalman algorithm which predicts the mean and variance of the coefficients, given the information at time t!1. The second part of the Kalman algorithm updates the estimate using current information, given the current value y as well as its history in Bayesian fashion (see Meinhold and Singpurwalla, 1983). From a general result of normal distributions (Newbold and Bos, 1985, p. 125) the mean of the β vector conditional on the current value of y is β(tt)"φβ(t!1 t!1)#(i!φ)β #P(t t!1)x»[½!x β (tt!1)] (A6) while its covariance matrix is P(tt)"P(tt!1)!P(t t!1)x» X P(tt!1) (A7) Thus, Equations A6 and A7 use the information from Equations A3 A5 in the Bayesian updates of the mean vector and covariance matrix given the new information at time t. A first run of the Kalman algorithm is initiated by choosing starting values for the fixed elements in Ω, Φ, β, andσ. This is typically done by using estimates from ordinary least squares regressions to compute estimates for β(10), P(10),», β(11), and P(11) which, in turn, are used for calculating β(21), P(21),», P(22), and P(22). The process is then continued through to the last observation. This is the procedure we follow in our estimation of the SPM. The first run gives estimated values for the fixed parameters of interest, but we do not assume they are true values. Instead, we vary the values of these parameters in order to maximize the log-likelihood function, ln "! ¹ 2 ln (2π)!1 2 ln»#[y!x β(tt!1)]», or, equivalently, to minimize the value of the associated function (see Newbold and Bos, 1985, p. 33): ln»#[y!x β(tt!1)]» We then recover the values of the stochastic parameters for each period t by employing the fixed-interval smoothing algorithm (see Ansley and Kohn, 1982).. The independent variables used in Section VI are lagged values of SP, GNP, and TB. However, it does not change the basic properties of the model. The programme we use is a modification of a published Kalman Filter program (Aptech Systems Inc. (1988)). The published programme restricted all coefficients to lie in the interval (!1, 1). We modified it to allow higher and lower coefficient values.