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

B([D.sub.t]) = c[D.sup.[lambda].sub.t] (7)

where [lambda] is the positive root of the quadratic equation [[lambda].sub.2] [[sigma].sup.2]/2 [lambda][mu] - r = 0, and c is an arbitrary constant. By summing the present value price (6) and the bubble in (7), the basic stock price equation becomes

P([D.sub.t]) = [P.sup.PV.sub.t] B([D.sub.t]) = [kappa][D.sub.t], c[D.sup.[lambda].sub.t]. (8)

Equation (8) says that the price of stocks has two components: the fundamental value, which depends linearly on dividends, and the intrinsic bubble which depends on dividends in a nonlinear fashion. For c>0, stock prices will over-react to changes in dividends:

d[P.sub.t]/d[D.sub.t] = [kappa] [lambda]c[D.sup.[lambda]-1.sub.t] > [kappa] (9)

In order to estimate the fundamental value and the bubble component in (8), the following statistical model is formulated

[P.sub.t]/[D.sub.t] = [kappa] c[D.sup.[lambda]-1.sub.t] [[eta].sub.t] (10)

where [[eta].sub.t] = [[epsilon].sub.t]/ [D.sub.t] are the present value errors. Estimating equation (10) immediately poses a problem because of collinearity among the regressors. So, dividing equation (10) throughout by [D.sub.t] gives

[P.sub.t]/[D.sub.t] = [kappa] c[D.sup.[lambda]-1.sub.t] [[eta].sub.t] (11)

where [[eta].sub.t]= [[epsilon].sub.t]/[D.sub.t]. The focus in equation (11) is on the intercept term [kappa]. Once the value of k is obtained, the fundamental value of stocks from equation (6) can be estimated as [P.sup.PV.sub.t] = [kappa][D.sub.t]. Then an estimate of the bubble would be B(D.sub.t]) = [P.sub.t] - [P.sup.PV.sub.t] or c[D.sup.[lambda].sub.t].

To empirically estimate the fundamental values and stock market bubble, this study uses data on dividends and stock prices (Standard & Poor's 500) for the period 1955:1 to 2002:2 available from Shiller's (2000) updated dataset. The data are deflated using the Consumer Price Index with 1996 as the base year. Equation (11) is estimated using the non-linear least squares method. The objective is to come up with an optimal value of [lambda]-1 that minimizes the sum of squared residuals (SSR) from equation (11). Because of the non-linear property of the equation, the issue of convergence is resolved using a grid search (see Table 1).

The value off [lambda]-1 of 3 that Froot and Obstfeld (1991) estimated is used as a baseline value. Experiments are then conducted with extreme values of [lambda]-1 ranging from 1 to 50 with increments of i unit at a time. The criterion to include incremental points is whether each increment is adding to the efficiency of the SSR. After pinning down an initial estimated value of [lambda]-1 of 13 that gives a local minimum in a first round (see Table 1), a finer grid search with increments of 0.1 unit is used. In this second round of the grid search the optimal value of [lambda]-1 extracted turns out to be 12.7. Corresponding to this value of [lambda]-1, the estimate of [kappa] is 25.48 and significant with a t-ratio of 38. Using the estimated [kappa] value of 25.48, the fundamental value of stocks is then given by [P.sup.PV.sub.t] = 25.48 [D.sub.t]. Figure 1 shows the log of the estimated fundamental value of stocks versus the log of actual real stock prices (S&P 500). Froot and Obstfeld's (1991) estimate of [kappa] is 14 and is lower than the one reported here. This may be due to the fact that their sample consists of annual data and for the 1900-1988 period only which excludes the recent periods of asset price booms. Balke and Wolhar (2001), however, extract a fundamental value parameter of 24.91 for a very long historical sample period (from 1881 to 1999), by using the Gordon's (1962) stock valuation model. For the 100-year period before 1983, they extract a price-dividend ratio which is not very dissimilar.