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  • 1. LONG-RUN RELATIONSHIPS BETWEEN GDP, RISK, EARNINGS AND THE STOCK MARKET C. Barry Pfitzner and David A. Brat, Department of Economics and Business Randolph-Macon College, Ashland, VA 23005, 804-752-7353 bpfitzne@rmc.edu,dbrat@rmc.edu ABSTRACT In this paper we test for cointegration among the stock market, nominal GDP, and risk, the latter measured as the sum of squared monthly returns measured at quarterly intervals. Employing the Engle- Granger technique, cointegration is not found in the trivariate framework. Since the measure of risk we use does not contain a unit root, we also test for cointegration in a bivariate framework with nominal GDP and the stock market. Here we find in favor of cointegration, suggesting that total returns for the S&P 500 and nominal GDP share common trends. The results of the second stage estimation of the error correction form, however, do not suggest that deviations from the long-run relationship necessarily result in movements back to the common trend. INTRODUCTION The growth of the stock market in the 1990’s (and thereafter) has been the subject of great debate as it relates to what many analysts see as market fundamentals. Authors such as Shiller [7] argued that the market was grossly overvalued at the beginning of the 2000s—referencing among other factors, the divergence of the price earnings ratio from its historical averages. Other authors, including Glassman and Hassett [3], believed that the stock market was not overvalued arguing that stocks were no more risky than bonds, so that much higher stock prices were justified in order to equalize the returns on stocks and bonds. The risk associated with the stock market is not directly observable. However researchers have often used some measure of stock market volatility as a proxy for risk. Generally, these measures are the sum of the monthly (or daily) stock market returns over, say, a quarter or some longer time period. It is clear that the growth in the stock market is related in theory to general economic activity. It is also clear from many prior research efforts that the stock market is forward looking; such that most research finds that the stock market leads economic activity, and not vice-versa. Notable exceptions to these findings are those Guo [4, 5] and Lettau and Ludvigson [6]. One avenue of research that may prove fruitful is to test for long-run relationships between the stock market (measured by the S&P 500 index) and measures of economic activity. To be clear, we do not wish to establish that economic activity precedes movements in the stock market, but rather we will test to see if the market shares a long-run relationship with economic activity. Consider Figure 1, which shows GDP, S&P 500 nominal price, nominal earnings, and nominal dividend normalized to a value of 1 on the left axis. Note that the S&P 500 index and economic activity measured by nominal GDP increase at similar rates and that the series return to a similar trend after the market fall at the beginning of the 2000s. This research will focus on total returns on the S&P 500 index, that is, nominal returns plus dividends. We will assume that the Modigliani-Miller result that dividend policy does matter in total returns holds for the S&P 500.
  • 2. GDP and the S&P 500 600 Log of GD P Log of S&P5 00 500 400 300 200 100 0 1952 1955 1958 1961 1964 1967 1970 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 Time Figure 1: Nominal GDP and the S&P 500, Normalized to 1 at Start DATA AND METHOD Data Since we wish to estimate long-run relationships between the total return on the S&P 500 and GPD growth, all of our data will be in quarterly form. Specifically we will collect the following variables for use in the analysis. S&P 500 = the index of the S&P 500, measured quarterly DIV = the dividend rate on the S&P 500 VOL= sum of the squared monthly stock market returns over a given quarter, a proxy for risk on the S&P 500 NGDP = quarterly nominal GDP The data period is second quarter, 1952 through third quarter, 2002. The data period is constrained by the length of the series on the dividend rate. The measure of the stock market we employ is the total return on the S&P 500, that is, with re-invested dividends computed by the authors. All series are in log form. Method We will employ the Engle-Granger method to test for long-term common trends. The method is called co-integration analysis. In general, the series believed to be co-integrated are tested for the order of integration—that is a test for unit roots. The augmented Dickey-Fuller test for a series yt implemented by a regression of the form: n ∆y t = α 0 + α1 y t −1 + ∑ ∆y t −i + et (1) i =1 2
  • 3. The null hypothesis of a unit root (non-stationarity) is a t-test of α1 = 0. If the null is not rejected the series is differenced and re-tested for a second root. The number of lags (n) in chosen by the Akiake Information Criterion (AIC). Assuming the order of integration is the same for the series under consideration, cointegration analysis proceeds to estimate the long run equilibrium relationship (co-integrating vector) between y and z as follows: yt = β 0 + β1 z t + ε t (2) Next the error-correction model is estimated: n n ∆y t = α 1 + α y ( y t −1 − β 1 z t −1 ) + ∑ α 11 (i )∆y t −i + ∑ α 12 ∆z t −i + ε yt (3) i =1 i =1 n n ∆zt = α 2 + α z ( yt −1 − β1 zt −1 ) + ∑ α 21 (i )∆yt −i + ∑ α 22 ∆zt −i + ε zt (4) i =1 i =1 The error correction forms can be examined for significance of the terms αy, αz . If for example, αy ,is not different from zero, then deviations from the long term relationship in period t-1 do not affect yt. Similarly, if αz,is not different from zero, then deviations from the long term relationship in period t-1 do not affect zt. The extension of the method to three variables is straight forward. RESULTS In the trivariate framework, the first step is to test each series for a unit root. The null hypothesis of a unit root cannot be rejected (as expected) for either log of the S&P 500 total returns or the log of nominal GDP. However, the volatility series, the proxy for risk, does not contain a unit root. Table I: Results of the ADF Unit Roots Tests Series Calculated t Critical t (5%) Critical t (10%) Lags Log of GDP -1.71 -3.13 -3.43 16 Log of S&P 500 -1.91 -3.13 -3.43 1 Log of Vol -6.82 -2.57 -2.58 1 Despite the rejection of the null of a unit root in the volatility measure, we estimated the long-run relationship for the three variables. In no formulation was the measure of risk statistically significant, nor was the error term significant for the error-correction equation for the ΔVol equation. For these reasons, the volatility measure was dropped from the analysis. We proceed to estimate the long-run relationship between the GDP and the S&P 500 series. The estimated equation is as follows: LogGDP = 3.32 + 0.673logS&P500 + et (5) (t = 54.50) R 2 = .937 The next step is to test the residuals from (5) for a unit root the calculated value of t for the ADF is -2.65. The critical values are -2.57 (α = .10) and -2.88 (α = .05). Thus at α = .10, we can conclude that the 3
  • 4. residuals from (5) do not contain a unit root. The GDP series and the S&P 500 series are cointegrated over this time period. Next the error correction form is estimated with t-scores in parentheses: ΔlogS&P = –.045 –.044et-1 + .224 ΔlogS&Pt-1 + .022 ΔlogS&Pt-2 – .280ΔGDPt-1 – 1.20ΔGDPt-2 + espt (6) (-2.69) (3.21) (0.30) (-0.58) (-2.53) ΔlogGDP = –.008 –.003et-1 + .027 ΔlogS&Pt-1 + .012 ΔlogS&Pt-2 + .317ΔGDPt-1 + .129ΔGDPt-2 + egdpt (7) (-1.39) (2.49) (1.14) (4.48) (1.82) Equations 6 and 7 represent a first-order vector autoregression (VAR) augmented by a single error correction term, et-1. The estimated coefficients on the et-1 terms in each equation are of importance in the process of determining that the variables do not drift apart. The coefficient on the error term is statistically significant in (6) but not in (7). Further, in theory the sign of the coefficient in (6) should be positive. Generally, et-1 is positive when the growth of GDP exceeds that of the S&P 500 as estimated by the equilibrium relationship in (5). Thus a positive coefficient on et-1 would help to move the S&P 500 back to the common trend. Similarly, a negative coefficient is to be anticipated on et-1 in equation (7). That coefficient is negative, and weakly significant (α = .10, one-tailed test). Another standard diagnostic tool is the decomposition of the variance for the variables in the VAR system. Tables II and III are the tabular representation of the decomposition. At one-step ahead, the dependent variable normally accounts for most if not all of the variance in itself. Notably here, at three years ahead (12 steps or quarters) GDP accounts for almost 8% of the variance in the S&P 500 index (see Table II), whereas in Table III, the S&P 500 accounts for a smaller percentage (5%) of the variance in nominal GDP at the three year horizon. Table II: Decomposition of Variance for Series Log of the S&P 500 Step Std Error Log of the S&P 500 Log of GDP 1 0.059969669 100.000 0.000 2 0.093599355 99.949 0.051 3 0.119380704 98.687 1.313 4 0.139226605 97.138 2.862 5 0.154977572 95.642 4.358 6 0.167948550 94.465 5.535 7 0.178991751 93.624 6.376 8 0.188655813 93.059 6.941 9 0.197279033 92.696 7.304 10 0.205070396 92.472 7.528 11 0.212163457 92.345 7.655 12 0.218650397 92.284 7.716 Further notice that the effect of the S&P 500 on the variance of nominal GDP diminishes somewhat at the longer time horizon, perhaps suggesting that the lead of the stock market for GDP is greatest at less than two years. CONCLUSIONS This paper tests for the Engle-Granger [1] form of cointegration on a quarterly basis among nominal GDP, risk, and total returns to the S&P 500 index. We find no evidence that our measure of risk, proxied for by the quarterly sum of squared monthly returns on the S&P 500, is useful as a variable in the 4
  • 5. cointegration analysis. We find in favor of cointegration for in the bivariate framework of nominal GDP and total returns to the S&P 500. We find a perverse sign in the second stage estimations that may be the result of the persistent rise in stock prices in excess of GDP over the period from the early 1980’s until 2000. It will be of interest to extend the data period in both directions to test the robustness of these findings. Table III: Decomposition of Variance for Series Log of GDP Step Std Error Log of the S&P 500 Log of GDP 1 0.008945037 0.242 99.758 2 0.014947932 1.945 98.055 3 0.020677329 3.895 96.105 4 0.025668099 5.207 94.793 5 0.030012619 5.938 94.062 6 0.033848336 6.224 93.776 7 0.037313858 6.221 93.779 8 0.040517931 6.047 93.953 9 0.043535872 5.779 94.221 10 0.046417160 5.466 94.534 11 0.049193716 5.134 94.866 12 0.051886616 4.803 95.197 REFERENCES [1] Engel, R.E. and C.W.J. Granger. “Cointegration and Error Correction: Representation, Estimation, and Testing.” Econometrica 55, March 1987, pp. 251-76. [2] Granger, C.W.J. “Investigating causal relations by econometric models and cross-spectral models.” Econometrica, Volume 46, 1969, pp. 424-438. [3] Glassman, James and Kevin Hassett. Dow 36,000. New York: Times Business, 1999. [4] Guo, Hui. “Does stock market volatility forecast returns?” Monetary Trends, The Federal Reserve Bank of St. Louis, February 2003 [5] Guo, Hui, “Time-varying risk premia and the cross section of stock returns,” Working Paper Series, The Federal Reserve Bank of St. Louis, 2002 http://research.stlouisfed.org/wp/2002/2002-013.pdf [6] Lettau, Martin and Sydney Ludvigson, “Consumption, aggregate wealth, and expected stock returns,” Journal of Finance, Vol. LVI, No. 3, June 2001 [7] Shiller, Robert. Irrational Exuberance. Princeton: Princeton University Press, 2000. [8] Studenmund, A. H. Using econometrics: a practical guide. Addison-Wesley Publishing; 4th edition (July 24, 2000) 5

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