We begin by writing the price system in an error correction form:
where Axt - i is a (2 x 1) vector of first differences lagged i periods and st~N(0,Q), assumed independent over time. The matrices ni and Q are both (2 x 2), and the latter is assumed positive definite. П is the (2 x 2) long-run impact matrix that contains the cointegrating vector. Inclusion of this term in this vector autoregressive regression (VAR) follows from the Granger Representation Theorem. PTs rank is equal to the number ofcointegrating relations which in our simple case is just one. Following earlier literature, we parameterize П as follows:
where both a and b are (2 x 1) vectors. The vector b contains the cointegrating coefficients, and for identification we normalize its first element to - 1, that is, b=( - 1,b). For example, if, on average, analysts forecast that target prices are 20 percent higher than current market prices, then b = [- 1 1.2]. The vector a is the vector of weights in the VAR regression. This vector can be interpreted as a vector of adjustment coefficients; that is, the elements in a allow us to quantify how target prices and market prices react to past deviations from the long-term relation. By obtaining estimates of the elements in a, we can quantify the way in which the two time series contribute to the correction of the system back to the long-term relation.
B. Cointegration Results for IBM
We present the cointegration results in Panel A of Table VI. We focus our attention on b, the parameter capturing the long-run ratio of target prices relative to market prices, as well as on the (2 x 1) vector a, capturing the response coei-cients of each price variable to deviations from the long-run relation. The long-term relation for IBM is 1.23, indicating that, ex ante, analysts expected that IBMs annual return would be 23 percent.
The estimates of the vector of response coeicients, a, provide interesting evidence on the manner with which target and market prices adjust to the long-term relation. The target price response, atp, is large and positive (0.17, with a t-statistic = 8.15), while the market price response, amarket, is statistically indistinguishable from zero. This inding can be interpreted as follows. Suppose market and target prices are currently 100 and 123 dollars, respectively, and that a 4-dol-lar revision in the consensus target price leads to a market price adjustment of1.6 dollars. In this case, the new ratio (1.25) differs from its long-term value of 1.23. The regression analysis indicates that the consensus target price is adjusted by 17 percent of (127 -1.23 n 101.6), or 35 cents in the first week. Since amarket is insig-niicantly diferent from zero, it can be seen that in IBMs case, once the system of prices has been shocked away from its long-term relation, it is predominantly the analysts, rather than market participants, who tend to revise their target prices toward the long-run relation.
П = ab,
Cointegration Regression Results
The table provides regression results of the cointegration analysis. Panel A presents results for IBMs 151 weekly observations of target and market prices spanning the period January 1997 through December 1999. The target price series is formed by equal-weighting all outstanding target prices that were issued within the previous 90 days. The regression setup is given in Section III and we estimate the parameter b, which captures the long-run ratio of target prices-to-market prices, and the vector a, which contains the response coefficients of each price variable to deviations from the long-run equilibrium.The first row contains the parameter estimates and the second contains the corresponding p-values. We do not report standard errors associated with b, as these are based on asymptotic theory whose inite sample properties are undetermined. Panel B provides results for the full sample of 900 irms that have at least one year of continuous record of weekly target and market prices. For each parameter we calculate the mean, median, 25th, 50th, and 75th percentile, as well as the standard deviation across the 900 firm regression estimate. Panel C provides regression results for subsamples of firms sorted by market capitalization (size). Size terciles are formed on the basis of NYSE capitalization cutoffs and are adjusted quarterly. For each size sort, we report the average of the regression estimates. For example, for the smallest 300 irms in this sample the average long-run ratio b is 1.37.
C. Full-Sample Implementation of the Cointegration Analysis
In this subsection, we implement the cointegration analysis for the full sample of 900 firms and estimate, for each firm, the parameters that capture the long-run ratio of target prices relative to market prices, b,aswellasthe(2x 1) vector of response coefficients, a, that captures how each price variable responds to deviations from the long-run relation. We report the results in Panel B of Table VI.
Since the regression analysis results in 900 sets of parameter estimates, we re-
port summary statistics only.
Consider irst the full sample results of the long-run ratio of target-to-market prices, b, given in the irst row. The irst column indicates that the grand-average (median) of the 900 estimates equals 1.28 (1.26). That is, conditional on at least two years of continuous consensus coverage, the average irm in this sample is expected to earn 28 percent annually. Furthermore, with the 25th and 75th per-centiles equal to 1.20 and 1.33, we learn that the distribution ofthese estimates is quite disperse, with a slight skew to the right.24
Next, consider the full-sample estimates of the response coefficients, atp and amarket. The second column provides the grand average (median) of atp,which equal 9.0 (9.0) percent. From this we learn that for the average irm, the analysts weekly response to a one-dollar shock in either target price or stock price that causes a deviation from the long-term relation is to revise the target price by nine percent of the resulting deviation. As discussed earlier, a positive response co-eicient is consistent with analysts revising their target prices toward the long-term relation once the system has been perturbed away from it. The market price response coeicient, amarket, is smaller by one order of magnitude, as can be seen from the third column. The grand-average (median) ofthe markets response coef-icient is only - 0.02 ( - 0.01) percent, suggesting a two percent weekly correction in response to a one-dollar deviation from the price systems long-term ratio. The above evidence is consistent with the interpretation that analysts revise their targets toward the long-term relation once the system has been shocked away from it. The same statistics for market prices, amarket, indicate a much smaller reaction to deviations from the long-term relation by market participants.
The evidence regarding the estimates of the response coefficients may seem inconsistent with the results reported in Section II.C, in which we detect abnormal return drifts subsequent to the target price revision. We argue, however, that the two empirical indings are, in fact, not inconsistent with each other. While the event-study approach allows us to isolate investor reactions to extreme target price revisions by conditioning both on the magnitude of the target price revision and the type of recommendation change, the cointegration approach provides lower frequency evidence in which unconditional estimates of target and market price are calculated. Hence, abrupt target price revisions are averaged with less
23 Because we estimate the regressions on a irm-by-irm basis, the results do not account for possible cross-correlation in the regression errors. While it is beyond the scope of this paper to estimate a large variance-covariance matrix or price errors, we note that the individual irm parameter estimates are, however, consistent and, in our setup, estimated quite precisely.
24 Our analysis leaves open the question of whether the estimates of ex ante returns are consistent with those elicited from an asset-pricing model such as the CAPM or the Fama and French three-factor model. We note here that, much like the high estimates that we report, the geometric average annual market return over our sample period is quite high at 19.9 percent. Furthermore, in unreported analysis we have contrasted the cointegration estimates of b with expected returns from the Fama and French model, allowing for the possibility of mispricing as in Pastor and Stambaugh (1999). We ind that allowing for mispricing uncertainty regarding the Fama and French three-factor model can indeed account for the cross-sectional dispersion in the reported estimates of b.
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