Liderazgo situacional de Hersey y Blanchard

In this section we investigate another implication of the model. Namely, we assess whether the price of market risk is also a function of investor beliefs about the state of the economy. As discussed above, the model predicts that, as investor uncertainty increases, the price of one unit of market risk, the so-called market premium, should also increase and reach a maximum price around the point of maximum uncertainty.

As opposed to the estimation of betas above, the task of estimating the price of market risk is more complicated, particularly because it requires the joint estimation of betas and prices of risk. The traditional approach of Fama and MacBeth (1973) involves a two-step procedure, where in the first step time-series regressions are used to estimate betas, and in the second, cross-section regressions with the estimated betas as regressands are used to obtain an estimate of the price of risk. Despite the computational simplicity, this procedure requires, for correct statistical inference, the adjustment of t-statistics to account for the existence of error-in-variables problem.

To impose fewer restrictions on the distribution of returns and avoid the problems associated with the two-step approach, we jointly estimate the parameters by GMM. The GMM framework is also flexible enough to allow us investigate time-varying functional forms for the price of market risk. However, as pointed out by Shanken and Zhou (2007), the moment restrictions typically imposed on the cross-section of stock returns, such as those in chapter 12 of Cochrane

(2005), are difficult to solve numerically, and the convergence, when possible, depends on the initial values chosen. These computational complications arise from the need of joint estimation of the constants on the time-series factor regressions and the constants on the cross-section pricing restriction. To avoid such difficulties, we follow Shanken and Zhou (2007) and estimate the parameters sequentially, imposing the following set of moments suggested by Harvey and Kirby (1996): E          ri,t−µi rm,t−µm (rm,t−µm)2−σ2m ri,t−λ0−λm,t(ri,t−µi)(r_σ2m,t−µm) m          = 0 (3.6)

where i = 1, ..., N. The first N + 2 set of moment restrictions exactly identify the N + 2

parametersµi,i= 1, .., N,µm andσ2m and are estimated separately, in the initial step. In the second step, the remainingN moment restrictions are used to estimate λ0 and the parameters ofλm,t. Here, we allow the price of market risk to be time-varying with the suggested functional forms, λm,t =b1+b2πˆt+b3πˆt2 and λm,t =b4 +b5U Ct. The proxy of beliefs, ˆπt, is the same as before, and also the two proxies for uncertainty, U Ct = ˆπt(1−πˆt) and U Ct = V IXt. The GMM estimators, λˆ0,ˆbj, j = 1, ...,5, are obtained analytically. For more details on this sequential procedure, the reader is referred to Shanken and Zhou (2007).

Table 3.6 shows the GMM estimated parameters λˆ0,ˆbj,j = 1,2,3, implied by the same 40 portfolios used in previously. To simplify our analysis, however, the betas of such portfolios are assumed static here. Model (1) is the classic CAPM cross-sectional regression. The intercept is positive and significantly different than zero and the price of market risk is negative and insignificant. This economically inconsistent result usually arises when portfolios sorted on book-to-market and size are used in the regressions. The specifications on Model (2) and Model (3) allow the price of market risk to be time-varying, by projecting λm,t on beliefs, πˆt, and squared beliefs, πˆ_t2. The significance of the coefficients indicates that time-variation is a statistically relevant characteristic of market premium. The point estimates, however, indicate a price dynamics that is at odds with our theory. The parameters imply that the price of

market risk is actually lower during periods of high uncertainty. Model (5) controls for other two commonly used risk factors, the HML and SMB factors of the Fama and French 3 factor model. In this case, all the coefficients are insignificant, which suggests an over-specification of the dynamics of price of risk.

Table 3.7 shows the GMM estimated parameters ˆλ0,ˆbj, j = 4,5, with a more restrictive functional form on the price of market risk. Two different sample sizes and proxies for uncer- tainty are considered. The coefficients in Model (6) confirm the results on Table 3.6 that the price of market risk is time-varying and also decreasing on levels of uncertainty. The coeffi- cientˆb5 is negative,−0.1799, and statistically significant with a t-statistic well below−3when

ˆ

πt(1−πˆt)is the uncertainty proxy, and on the monthly sample from 1956 to 2010. The inclu- sion of the High-Minus-Low (HML) and Small-Minus-Big (SMB) risk factors of the three-factor Fama-French model does not change its negative sign, but reduces its statistical significance. In Model (8) a different proxy,V IXt, as well as a different sample period, from January 1990 to December 2010, are used. The point estimates cannot be compared because of the different scales of the proxies, but the qualitative results are the same, as the negative sign remains. Thus, the evidence also points to a decreasing equity premium on uncertainty in this restricted sample periods and for the model-free proxy of uncertainty, although this time not statistically different than zero.

The dynamics of market risk price revealed by the data does not match our model pre- dictions. This inconsistent result is typically found in similar empirical investigations. For instance, a large body of literature has long debated on what is the appropriate econometric approach to assess the risk versus return trade-off on the market portfolio. Depending on the chosen approach, a negative relation between return and market variance can be found (Whitelaw (1994) and Brandt and Kang (2004)). The task of assessing the price of market risk from the cross-section of returns, which is essentially the same as finding a positive risk-return trade-off on the market portfolio, has also presented with contradictory results, as evidenced by the murderings and resurrectings of the CAPM (Fama and French (1996) and Lettau and Ludvigson (2001)).