Does U.S. monetary policy react to asset prices? Implications of stock market bubbles, volatility and productivity

Indian Journal of Economics and Business, Dec, 2007 by Rajeev Sooreea

[y.sub.t] = [x.sub.t][gamma] [[epsilon].sub.t] (12)

[[]sigma].sup.2.sub.t] = [omega] [alpha][[epsilon].sup.2.sub.t-1] [beta][[sigma].sup.2.sub.t-1]. (13)

Since [[sigma].sup.2.sub.t] is the one-period ahead forecast variance based on past information, it represents the conditional variance. The conditional variance equation specified in (13) is a function of three terms: the mean, [omega]; news about volatility from the previous period, measured as the lag of the squared residual from the mean equation, [[epsilon].sup.2.sub.t-1] (the ARCH term); and the last period's forecast variance, or [[sigma].sup.2.sub.t-1], (the GARCH term), [alpha], [beta] and [gamma], are parameters of the model. However, for equities, it is often observed that downward movements in the market are followed by higher volatilities than upward movements of the same magnitude--the so-called leverage effect. To account for this asymmetry, Nelson (1991) proposed an E-GARCH model. The specification for the conditional variance is:

log([[sigma].sup.2.sub.t]) = [omega] [beta]log([[sigma].sup.2.sub.t-1]) [alpha][absolute value of [[epsilon].sub.t-1] / [[sigma]sub.t-1]] [gamma] [[epsilon].sub.t-1] / [[sigma].sub.t-1]. (14)

Since the left-hand side is the log of the conditional variance, this implies that the leverage effect is exponential, rather than quadratic, and that forecasts of the conditional variance are guaranteed to be nonnegative. Leverage effect exists if [gamma] < 0. The impact is asymmetric if [gamma] [not equal to] 0. Before equation (14) can be estimated, an AR process for the stock returns (log differenced of real stock prices) series is fitted to make it stationary. Monthly stock returns data of the S&P500 index from 1955:01 to 2002:07 are used to capture the maximum information contained in the higher frequency data. The results suggest that an AR(1) process adequately describes the data (the correlogram and the Q-statistics for serial correlation for lags up to 36 indicate that residuals are white noise). Then an AR(1)-E-GARCH (1,1) process is estimated and the results are as follows:

log([[sigma].sup.2.sub.t]) = -1.928 0.729 log([[sigma].sup.2.sub.t-1]) 0.0891 [absolute value of [[epsilon].sub.t-1] / [[sigma]sub.t-1]] - 0.261 [[epsilon].sub.t-1] / [[sigma]sub.t-1] (15)

se (0.375) (0.055) (0.052) (0.047)

The figures in parentheses are the Bollerslev-Wooldridge robust standard errors (se). The estimated value of [gamma] (-0.261) is negative and significant at the 1 percent level, indicating the existence of the leverage effect in stock returns during the sample period. This indicates that bad news ([[epsilon].sub.t-1] / [[sigma].sub.t-1] < 0) and good news ([[epsilon].sub.t-1] / [[sigma].sub.t-1] > 0 have differential effects on the conditional variance, with bad news creating more volatility that good news. The estimated monthly conditional variance series is then converted to a quarterly series using standard aggregation method to match the frequency of other variables in the estimated reaction functions. This conditional variance series [[sigma].sup.2.sub.t] forms the basis of the measure of stock returns volatility that is used in the rest of the paper. Figure 2 shows the volatility in U.S. stock returns as estimated by the E-GARCH model with the shaded bars representing periods of recessions in the U.S. as defined by the National Bureau of Economic Research. Two important facts stand out. First, stock market volatility displays a strong counter-cyclical pattern--peaking just before or during recessions and falling sharply late in recessions or early in recovery periods. Moreover, when volatility increases, investors require a higher risk premium to hold stocks. As a result, stock prices fall. This is evident in the 1987 stock market crash, the 1990 Persian Gulf and S&L crises and recession, the 1994 Mexican Peso Crisis, the 1998 Russian Default and the 2001 recession. Second, the movement in stock returns volatility also appears to be persistent: once volatility rises, it usually stays at high levels for a while. However, it shows no apparent long-run trend. After declining in the early 1990s, volatility started to rise in 1996 and since then has remained at remarkably high levels by postwar standards.