Measures of the degree of ambiguity

Recent empirical literature uses the disagreement of professional forecasters to proxy the degree of ambiguity in the market (see Anderson, Ghysels, and Juergens (2009) and Drechsler (2013)). Intuitively, if forecasters produce very different and conflict- ing forecasts about the fundamentals (either of a firm or the economy in general), investors are likely to be unsure about the distributions of stock returns as they tend to condition their beliefs on the analysts’ forecasts. Thus, when dispersion among analysts’ opinion regarding the future performance of stock markets is high,

3_{We also use the value-weighted data based on per share numbers, where the per share earnings}

surprisesdEit/Si,t−4are weighted using the number of shares outstanding of quartert. Results are

ambiguity is also likely to be high since investors cannot confidently narrow down the set of their beliefs to a single prior.

At the same time, dispersion of analysts’ forecasts might not necessarily be the ideal proxy for measuring the degree of ambiguity. Barron, Kim, Lim, and Stevens (1998) argue that the ambiguity component of the dispersion of analysts forecasts can be contaminated by the disagreement component that comes from the asymmetric information. This can be a serious issue especially when proxies the degree of ambiguity at individual firm level. Barron, Kim, Lim, and Stevens (1998) propose a decomposition of the forecasts dispersion into uncertainty and disagreement in the following way. Define the consensus measureρ as

ρ= C

V, (3.10)

whereV is overall uncertainty defined as a simple average of individual uncertainty (i.e. variance of forecast errors) overN analysts. C is common uncertainty defined as the average pair-wise covariance among analysts’ beliefs. Thus,ρ measures the degree to which analysts’ beliefs covary relative to the overall uncertainty, in other words, the ratio of common uncertainty to the overall uncertainty.

To compute the consensus among the analysts, we use a special case of above formula where the private information is of equal precision4:

ρ= H

H+Z, (3.11)

where H = (SE−

V N)

(SE−_NV+V)2 measures the precision of common information and Z =

V (SE−V

N+V)2

measures the precision of idiosyncratic information. Here, SE is the mean squared error of forecasts scaled by the absolute value of the actual forecasted variable, V is the variance of forecasts scaled by the absolute value of the actual

4

Please refer to Barron, Kim, Lim, and Stevens (1998) for detailed explanation regarding the relation with common and private information

forecasted variable, andN is the number of forecasts. Thus, the variableρmeasures the uncertainty attributable to experts’ reliance on imprecise common information. We argue that this measure captures the degree of ambiguity embedded in the dispersion of analysts’ forecasts: that is, the more information uncertainty is, the more likely that investors form multiple beliefs about fundamentals of stocks and the economy as a whole. The results using dispersion of analyst forecasts as an alternative measure of ambiguity are similar.

We construct firm-level ambiguity by using analysts’ forecasts of individual firm earnings. They reflect news about individual firms’ news on cash flows. The higher is the uncertainty component in the dispersion of analysts’ earnings forecasts, the more ambiguous is signals about the firm’s cash flow news. Thus, we denote by F Ui the uncertainly measure ρ constructed on the basis of analysts’ earnings forecasts for firm i from IBES data. Specifically, SE will correspond to the mean squaredearningsforecasts error andV corresponds to the variance ofearningsfore- casts. Hence, we argue that F Ui is a good measure for the degree of ambiguity in the market for two reasons. First, because it is based on the dispersion of experts’ forecasts about the fundamentals, it nicely measures the set of reasonable models considered by the representative investor. Secondly, it is free of the impact of asym- metric information component that can possibly contaminate the effect of ambiguity. In fact, Doukas, Kim, and Pantzalis (2006) has demonstrated that uncertainty and asymmetric information component have indeed opposite effects on stock returns.

In order to examine the cross-sectional effect of firm-level ambiguity on the earnings response coefficient we categorize stocks into five different groups based on the degree of firm-level ambiguity. Specifically, every quarter we group stocks into quintiles of firm-level ambiguity variableF Ui. We define dummy variable Ditj,

j = 1, ...,5 to be equal to 1 if stock i falls into firm-level ambiguity quintile j in

quartert. In this way, dummy variableD1 corresponds to the stocks with the least ambiguous cash-flow information andD5 correspond to the most ambiguous stocks.

In order to proxy the degree of market-level ambiguity we use individual analyst’s forecasts for macroeconomic variables, e.g. real GDP growth or inflation growth, that comes from the Survey of Professional Forecasters managed by the Federal Reserve Bank of Philadelphia. Similar to the firm-level ambiguity measure, we obtained the decomposed uncertainty from the dispersion of analysts’ forecasts for the next period real GDP growth rate5. That is, we denote by M U the uncer- tainly measureρ constructed on the basis of experts’ forecasts of the GDP growth. Specifically,SE corresponds now to the mean squared GDP forecasts’ error andV

corresponds to the variance ofGDP growth forecasts. Finally, in order to estimate the differential effect of market-level ambiguity on the earnings response coefficient, we define a dummy variable D_tM that is equal to 1 if the market-level ambiguity

M Ut is above its historical mean value, and zero otherwise.

The choice of the realized value, needed for calculating the mean squared forecasts’ error, depends on the version of data that professional forecasters are trying to predict. Survey of Professional Forecasters database offers five vintages of the realized value, ranging from the initial-release numbers to the values that we understand today. The reliability of the historical values increases in time while the availability decreases in time. We use the latter four vintages on the ground that they suffer less measurement error yet are close enough to what the professionals are try to forecast6. The results are similar for each of the four.