Security price responses to unexpected earnings: a ... - Springer Link
Stat Methods Appl (2011) 20:241–258 DOI 10.1007/s10260-010-0159-3
Security price responses to unexpected earnings: a nonparametric investigation Lei Gao · Li Wang
Accepted: 22 December 2010 / Published online: 6 January 2011 © Springer-Verlag 2011
Abstract The widely used linear model of the unexpected earnings/returns relationship has been challenged. In this article we propose a flexible nonparametric approach to study this relationship in which splines are used to approximate the unknown regression function. Spline confidence bands are constructed based on wild bootstrap to examine the adequacy of certain linear/nonlinear specifications. Monte Carlo results show that the proposed bands have excellent coverage of the true regression function with little computing load. These properties make the procedure highly recommended for extracting information from large and complicated datasets. The proposed approach has also been applied to the real world financial data from the unexpected earnings/returns study, and we find significant evidence of nonlinearity. The nonlinearity persists when we control the measurement errors of the earning surprises and firm size. Keywords Confidence bands · Earnings response coefficient · Measurement error · Spline · Wild bootstrap 1 Introduction Financial analysts make earning forecast for public firms. The financial analyst forecast error is called the unexpected earning. Since the stock market responds to the
L. Gao Department of Banking and Finance, University of Georgia, Athens, GA 30602, USA e-mail: [email protected] L. Wang (B) Department of Statistics, University of Georgia, Athens, GA 30602, USA e-mail: [email protected]
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information quickly and accurately, the stock price should have integrated all available information including the financial analyst forecast reports. The expected return of a single stock is defined as the risk-adjusted market index return, such as the S & P 500 return or the Dow Jones Industrial Average return. When there is a deviation between the firm’s earning announcement and the analysts’ forecast, the stock price will pick up the newly available information, so we would expect a corresponding unexpected stock return with respect to the unexpected earning information during the short period before and after the announcement event. For investors and researchers, it is very important to understand the relationship between the unexpected returns and earnings, and this article contributes to explaining the mystery. For the past three decades, the studies on the relationship of unexpected earnings and returns have occupied much of the accounting literature; see Collins and Kothari (1989), Freeman and Tse (1992) and Stephen and Zarowin (2003), for example, to name one article representative of each decade; and see Kothari et al. (2005), Anilowskia et al. (2007), and Easton and Sommers (2008) for some of the more recent research. Regression analysis has been widely used in learning about this relationship. Classic linear regression concerns primarily the inference on some parameters, for instance, the earnings response coefficient, referred as ERC in the literature. Many researchers have challenged the linearity and nonlinear models have been employed to study this relationship. For example, Cheng et al. (1992) and Subramanyam (1996) both find evidence that linear models of the earnings/returns relationship are misspecified. Freeman and Tse (1992) propose an arctan earnings/returns model that provides higher explanatory power and a richer explanation for differences between ERCs and the price-earnings ratio. Some other studies, including Das and Lev (1994) and Beneish and Harvey (1998), used the quadratic and broken stick models. However, it is difficult to choose the best regression curve in the absence of a comprehensive equity valuation model. These specific function forms are called “parametric” because they are explicitly determined by those fixed number of parameters. As discussed in Das and Lev (1994), such specifications are ad hoc, and if misspecified, can lead to substantial estimation bias problems. Another issue in the prior studies of earnings/returns relationship is that positive and negative surprises are treated symmetrically. However, Abdel-khalik (1990) and Basu (1997) show that the earnings/returns relationship is asymmetric. Collins et al. (1999) and Skinner and Sloan (2002) also find that the valuation effect of earnings is asymmetric between positive and negative earnings. The reason for this asymmetry is that bad news is reported more timely than good news, and hence anticipated losses are recognized immediately whereas unanticipated gains are spread out into the future. The earlier defined models are not be able to study this “leverage” effect on security returns. The main objectives of this article are (a) to introduce a flexible method, which is robust to the model misspecification, to estimate the unknown earnings/returns relationship and (b) to test the adequacy and validity of a specific model such as the linear and arctan models. For objective (a), we propose using a nonparametric model to study the relationship between unexpected earnings/returns. The nonparametric approach increases the
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flexibility of the class of models we use without restricting it a priori to have or not have certain shapes. The curve can be asymmetric and it allows one to study the different “leverage” of good news and bad news. In addition, the nonparametric models are more robust to model misspecification than the parametric ones, at least asymptotically, when dealing with the regression problem related with the “generated regressors” and “dependent variables”. The variables involved in the study of earnings/returns relationship are forecast errors or generated variables, i.e., the variation between the actual value of earnings and the value forecasted. The extra variability added by the estimation of these quantities can be smoothed out by the nonparametric estimators by averaging the irrelevant noise in the data. Details of nonparametric regression can be found in Fan and Gijbels (1996) and Ruppert et al. (2003). As in any estimation procedure, confidence regions provide more information than a single estimate. For objective (b), we make use of simultaneous confidence intervals (confidence bands) which give us insightful information of the unexpected earnings/returns relationship. This method allows us to visually examine the bands to test how well a curve captures the relationship of the unexpected earnings/returns. The early studies of Hall and Titterington (1988) and Härdle (1989) establish an asymptotic confidence band of nonparametric regression function based on Nadaraya-Watson kernel smoothing. The more efficient local polynomial smoothing has been investigated in parallel; see Xia (1998) and Claeskens and Van Keilegom (2003). All of the aforementioned methods involve local averaging by kernel weights. One drawback of the localized kernel estimator is its instability near the boundaries of the interval, which will lead to a bad assessment of the tail behavior on the regression curve. As an alternative, Proietti and Luati (2008) evaluate a family of asymmetric filters that can improve the estimation of the underlying function at the end of the sample period. However, kernel method is in general computationally expensive since it requires solving a linear least squares problem for every design point. The majority of financial data are large and complicated, for example, the sample size of the data used in our earnings/returns study is 13,314; see Sect. 2 for details. The kernel method is computationally too expensive for such large datasets. In fact for large datasets like that, it is useful and required that estimation and inference methods are efficient and computationally easily implemented. In this article, we propose approximating the regression function by spline smoothing, which is practically as easy to implement and fast as a simple linear regression with slowly increasing number of parameters (Ruppert et al. 2003). In addition, the implementation can be easily achieved in a commonly used computational environment like R. The rest of our article is organized as follows. Section 2 contains a detailed description of the financial data. Section 3 first introduces a flexible nonparametric heteroscedastic model and our hypothesis. Section 3.2 presents a class of spline estimators and an asymptotic confidence band for the regression function. We find the asymptotic band to be a little conservative and unstable, so a wild bootstrap band is developed in Sect. 3.3. Monte Carlo simulation studies are given in Sect. 4. Section 5 reports our statistical analysis and empirical findings for the data introduced in Sect. 2. We close with conclusions and discussions
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Table 1 Descriptive statistics of earnings forecast errors Sample size Philbrick and Ricks (1991) Freeman and Tse (1992) Beneish and Harvey (1998) This article
Mean
Standard deviation
Median 0.0024
4, 770
0.0087
0.0501
12, 381
0.0159
–
–
2, 929
0.0051
0.0042
0.0039
13, 314
0.0047
0.0269
0.0011
2 Data The dataset used to investigate the unexpected earnings/returns relationship comprises four resources: (i) Price per share of a firm at the end of the quarter and quarterly earnings announcement date on the 2007 quarterly industrial file obtained from Standard & Poor’s Compustat, a database of delivering financial, statistical and market information on active and inactive companies throughout the world; (ii) Daily stock returns on the 2007 CRSP daily returns file; (iii) Actual earnings quarterly data on the 2007 detail file from Institutional Brokers’ Estimate System (I/B/E/S), which provides both summary and individual analyst forecasts of company earnings, cash flows, and other important financial items, as well as buy-sell-hold recommendations; (iv) Forecast earnings quarterly data on the 2007 I/B/E/S detail file, in which financial models have been built to estimate the prospective revenues and costs and to predict the earnings; (v) Common shares outstanding (CSHO) of a firm on the 2007 Compustat fundamentals annual file. It has been conjectured in Beneish and Harvey (1998) that the nonlinear relationship could be an artifact phenomenon due to the measurement errors. So to examine whether the control for errors eliminates the nonlinearity, we removed 3 observations whose absolute per-share forecast error exceeds the price per share. The final selected sample contains a total of 1,647 firms and 13,314 observations from the first quarter of 2004 through the fourth quarter of 2006. Table 1 presents the comparison of our absolute forecast errors with those in the prior studies. One key variable of the study, unexpected earnings of a fiscal quarter, is measured by the ratio of earnings forecast errors to the stock price per share at the end of the previous quarter. The earnings forecast errors are defined as the difference between the actual quarterly earnings per share and the median I/B/E/S analyst forecast in the last month of the corresponding quarter. The other key variable, unexpected return, is measured based on the daily abnormal return of a firm, which is the return of a specific firm less the equally weighted mean return in its beta-matched portfolio for each trading date. Long and short time windows are selected to capture quick or slow stock price reaction to the unexpected earnings. For the long period window, the unexpected return is defined as the sum of daily abnormal returns for the period beginning two days after the previous quarterly
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15 10 5
0.01+variance*E-3
20
Fig. 1 Plot of the estimated variance of the unexpected earnings based on the pooled samples
245
-4
-2
0
2
4
unexpected earnings*E-2
earnings announcement and ending one day after the current announcement; for the short period window, it is calculated as the the sum of daily abnormal returns from day −2 to day +1 relative to the current announcement date. 3 Methodology Motivated by the dataset in Sect. 2, we propose using a flexible nonparametric model to study the unexpected earnings/returns relationship. In this section, we describe the model, hypothesis and general methodology. 3.1 Model and hypothesis n satisfying We formulate the problem broadly for any random sample {X i , Yi }i=1
Yi = m (X i ) + εi , i = 1, ..., n,
(1)
where m (x) = E (Y |X = x ) is some smooth but unknown conditional mean function [a, b]. The errors εi ’s are of mean zero and variance σ 2 (x) = 2on an interval E ε |X = x , not necessarily independent or normal. Model (1) is superior as it essentially gives one the flexibility of having a smooth function m, not necessarily in linear, quadratic or any other specific parametric forms. Another advantage is that it allows heteroscedastic errors, for example, the variance of the unexpected return (Y ) can be set to be a function of the earning surprises (X ), instead of a constant as assumed in the earlier studies. The asymmetry of the variance has been confirmed by our empirical analysis; see the empirically estimated variance function of unexpected earnings in Fig. 1.
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One of the major objectives of this article is to examine the validity of certain parametric forms of the regression function m (·). This problem can be formulated in terms of the hypothesis of m (·), i.e., H0 : m (x) = m θ (x), for any point x ∈ [a, b], where θ ∈ , and is a parameter space. For example, one can verify the generally assumed linearity of the unexpected earnings/returns by testing the hypothesis H0 : m (x) = β0 + β1 x, for any point x ∈ [a, b].
3.2 Spline estimators and confidence bands Next we introduce a class of estimators of function m(·) with confidence bands. To estimate m, one of the most common nonparametric methods is smoothing, in which observational errors are reduced by different types of averaging of the data. Splines, kernels and orthogonal series expansions are all examples of smoothing estimators. As pointed out in the introduction, one drawback of the localized kernel estimator is its instability near the boundaries of the interval, which will lead to poor assessment of the tail behavior of effect from the unexpected earnings to unexpected returns. Kernel smoothing is also computationally intensive since it requires to solve an optimization problem at every design point. In contrast, spline estimator is more stable and very popular forits simple implementation and fast computation (Ruppert et al. 2003). p p Denote 1, x, ..., x p , (x − κ1 )+ , ..., (x − κ N )+ the truncated power spline bases of order p, where (ν)+ = ν if ν > 0 and 0 otherwise, and κ j , j = 1, ..., N , are interior points in [a, b], called knots. combination of the p above spline bases pAny linear m x; β0 , ..., β p , γ1 , ..., γ N = k=0 βk x k + Nj=1 γ j x − κ j + is a piecewise poly nomial function, called a spline, where β0 , ..., β p , γ1 , ..., γ N are the coefficients and N is the number of interior knots. If N is sufficiently large, m x; β0 , ..., β p , γ1 , ..., γ N can approximate m (x) with a high degree of accuracy. We define the spline estimator for the regression function m(x) as
mˆ (x) =
p k=0
βˆk x k +
N
p γˆ j x − κ j + ,
(2)
j=1
βˆ0 , ..., βˆ p , γˆ1 , ..., γˆN are the least square estimators of β0 , ..., β p , n γ1 , ..., γ N based on sample {(X i , Yi )}i=1 . The shape of the basis functions is determined by the position of the knots κ1 < · · · < κ N , which can, for example, be uniformly spread sample quantiles of the X i values. To test the hypotheses in Sect. 3.1, we make use of the confidence bands, which allow us to visually examine and summarize how the regression curve changes along the whole range of predictor values. Under the null hypothesis, m θ would have to lie completely inside the confidence band with probability 1 − α. If this is violated, the tested curve will be rejected at the significance level α. For example, for the aforementioned linear hypothesis, we can obtain a weighted OLS estimator (βˆ0 , βˆ1 ) of n , then visually examine the confidence band to (β0 , β1 ) from the data {(X i , Yi )}i=1
where
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check whether the band covers βˆ0 + βˆ1 x for all x. If so, then we accept at level α the null hypothesis that m is linear. Otherwise the linearity is rejected. n , Huang (2003) obtains asymptotic 100(1 − α)% Given any sample {(X i , Yi )}i=1 pointwise confidence intervals in the form of mˆ (x) ± z 1−α/2 σˆ mˆ (x) ,
(3)
where z 1−α/2 is the (1 − α/2) quantile of the standard normal distribution, mˆ (x) is the spline estimator in (2) and σˆ mˆ (x) is its standard error, whose explicit formula is contained in Huang (2003). Wang and Yang (2009) derive an asymptotic 100(1 − α)% confidence band of m (x) over interval [a, b] in the form of mˆ (x) ± σˆ mˆ (x) {2 log (N + 1) − 2 log α}1/2
(4)
by using the piecewise linear spline basis. The spline bands in (4) have the same width order as the Nadaraya-Watson bands of Härdle (1989), and the local polynomial bands of Xia (1998) and Claeskens and Van Keilegom (2003). However, a plug-in method has to be implemented first to estimate the standard error σˆ mˆ . However, the plug-in method requires additional smoothing steps which may lead to less accurate and stable results. Hence in this article we develop a better method to construct the confidence bands using the so-called “wild bootstrap” by Wu (1986), which preserves the conditional heteroscedasticity in the original residuals. 3.3 Bootstrap confidence band We define the residuals εˆ i = Yi − mˆ (X i ) , 1 ≤ i ≤ n, and denote a predetermined large integer by n B . 1≤b≤n Step 1 Let δi,b 1≤i≤n B be i.i.d. samples from a Bernoulli distribution, and
δi,b =
√ √ √ −(√ 5 − 1)/2, with probability( √5 + 1)/(2 √5); ( 5 + 1)/2, with probability( 5 − 1)/(2 5).
It is easily verified that E δi,b ≡ 0 and V ar δi,b ≡ 1, for all 1 ≤ i ≤ n, 1 ≤ b ≤ n B . Step 2 For any 1 ≤ b ≤ n B , define the b-th wild bootstrap sample Yi,b = mˆ (X i ) + estimator of m (x) is the spline δi,b εˆ i , 1 ≤ i ≤ n. Then the b-th bootstrap n estimator mˆ (b) (x) based on sample X i , Yi,b i=1 . Step 3 Denote mˆ L ,α/2 (x) and mˆ U,α/2 (x) the lower and upper 100(1 − α/2)% quantiles of the set mˆ (b) (x) 1≤b≤n . The wild bootstrap 100(1 − α)% B pointwise confidence interval for function value m (x) at one point x is mˆ L ,α/2 (x) , mˆ U,α/2 (x) . Step 4 According to Wang and Yang (2009), when localized at any point x, the uniform confidence band in (4) is wider than the pointwise confidence inter−1 {2 log (N + 1) − 2 log α}1/2 . val in (3) by a common factor of K = z 1−α/2
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Hence we define the wild bootstrap 100(1 − α)% confidence band for m (x) over [a, b] as m L ,α/2 (x) = mˆ (x) + mˆ L ,α/2 (x) − mˆ (x) K , m U,α/2 (x) = mˆ (x) + mˆ U,α/2 (x) − mˆ (x) K . Härdle and Mammen (1993) studied several different bootstrap procedures and concluded that the wild bootstrap is the most pertinent method for testing the regression structure. Neumann and Kreiss (1998) have extended the result to a time series context and shown that the wild bootstrap still consistent even if the errors are not independent. The justification for making use of the wild bootstrap sample is also given in Sperlich et al. (2002). An important aspect for spline smoothing is the choice of the appropriate number of knots. Splines with few knots are generally smoother than splines with many knots; however, increasing the number of knots usually increases the fit of the spline function to the data. Wang and Yang (2009) show that the bias of the spline estimator can be killed if N ∼ n 1/5 . The number of knots used to construct the band is selected to minimize the following bayesian information criterion (BIC)
n 2 1 log n . Yi − mˆ (X i ) BIC(N ) = log + (N + 1) n n
(5)
i=1
For the issue of stability, here the selection is just made on the original series, not on the bootstrap replicates. 4 Simulation studies To illustrate the finite-sample behavior of the spline bootstrap bands, we conduct some simulation studies. For comparison purposes we also include the kernel-type bands from Hardle 1990, and the spline-type bands in (4) based on the asymptotic distribution theory. For the kernel-type bands, we use the Nadaraya-Watson smoother with the quartic kernel and rule-of-thumb bandwidth. For the spline-type bands, we use the piecewise linear spline smoothing. The number of interior knots for the asymptotic bands is taken to be [5n 1/5 ] + 1 as suggested in Wang and Yang (2009), and the number of knots for the bootstrap bands is selected via the BIC given in (5). We generate samples from the following heteroscedastic models Yi = m (X i ) + σ0 |X i |1/5 /5 + 1.2 εi , where Model 1 : m(x) = arctan (2.5x) ; Model 2 : m(x) = 1.5 sin (π x) ; Model 3 : m(x) = 3 (x + 1) (x − 1) + 1.5.
(6)
The X i ’s are i.i.d from uni f or m(−1, 1). The noise level σ0 = 0.5 and the error εi ’s are generated independently from N (0, 1).
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Table 2 Monte Carlo coverage probabilities and areas of the asymptotic kernel-type, asymptotic splinetype and bootstrap spline-type confidence bands Model Size
1
2
3
Confidence level 0.99 Kernel Spline Asymptotic
Bootstrap
Confidence level 0.95 Kernel Spline Asymptotic
Bootstrap
200
0.992(1.534) 0.968 (3.126) 0.962(1.996)
0.934(1.180) 0.908(2.764)
0.928(1.826)
500
1.000(1.044) 0.988(2.241)
0.944(0.815) 0.952(1.988)
0.950(1.344)
0.978(1.479)
1000 0.994(0.783) 0.984(1.690)
0.980 (1.067) 0.938(0.617) 0.946 (1.502) 0.958(0.964)
2000 0.994(0.585) 0.994(1.294)
0.992(0.872)
0.938(0.466) 0.970(1.152)
0.970(0.790)
5000 0.992(0.502) 0.984(0.910)
0.992(0.619)
0.942(0.402) 0.962(0.813)
0.978(0.561)
200
0.868(2.037) 0.966(3.126)
0.956 (2.004) 0.684(1.635) 0.910 (2.764) 0.906(1.831)
500
0.858(1.404) 0.988(1.870)
0.968(2.241)
0.592(1.139) 0.950(1.988)
0.942(1.346)
1000 0.798(1.060) 0.984(1.690)
0.974(1.068)
0.510(1.065) 0.946(1.502)
0.966(0.946)
2000 0.648(0.797) 0.994(1.294)
0.982(0.872)
0.330(0.656) 0.964(0.813)
0.960(0.790)
5000 0.388(0.550) 0.984(0.910)
0.980(0.986)
0.134(0.456) 0.964(0.813)
0.972(0.561)
200
0.610(1.909) 0.968(3.126)
0.956(2.002)
0.286(1.512) 0.910(2.764)
0.924(1.828)
500
0.354(1.298) 0.990(2.241)
0.974(1.480)
0.100(1.040) 0.952(1.988)
0.946(1.346)
1000 0.120(0.971) 0.980(1.690)
0.978(1.067)
0.034(0.784) 0.944(1.502)
0.958(0.964)
2000 0.016(0.727) 0.994(1.294)
0.990(0.872)
0.000(0.591) 0.972(1.152)
0.972(0.790)
5000 0.000(0.498) 0.984(0.910)
0.996(0.620)
0.000(0.409) 0.962(0.813)
0.972(0.561)
The result is based on 500 replications. The values outside ( ) are the coverage probabilities of the bands; the values inside ( ) are the areas of the bands
We calculate the percentage of coverage of the true function by the confidence bands constructed from the above three different methods. Two nominal confidence levels 0.99 and 0.95 are chosen for sample sizes 200, 500, 1000, 2000 and 5000. We carry out 500 simulation replications, and for each replication, 1000 bootstrap samples are generated for the bootstrap bands. Table 2 contains the Monte Carlo coverage probabilities and the enclosed areas of three types of bands. From Table 2, one sees that the spline-type bands have much higher coverage probability than the kernel-type bands, especially for Models 2 and 3. On the other hand, the spline-type bands are more conservative than the kerneltype bands. In terms of the width of the confidence bands, the kernel-type band is the narrowest, followed by the spline bootstrap band and the spline asymptotic band. In addition, the enclosed area decreases when sample size increases or confidence level decreases. The spline asymptotic bands seem to be too wide for some small sample size, for example, n = 200. Overall, the spline bootstrap band is a useful compromise between the kernel-type band and the spline asymptotic band. The enclosed area of the spline bootstrap bands is on average comparable to the kernel ones, while at the same time the bands achieve very good coverage probability. Figure 2 shows the spline fits for regression functions in (6) and the corresponding bootstrap bands. Here the graphs are created based on one random sample of size 200, each with four types of symbols: points (data), center smooth curve (true curve), center piecewise linear line (the piecewise-linear spline estimator), upper and lower
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y
0
1
true spline conf. band
−1
0 −2
−2
−1
y
2
(b)
true spline conf. band
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250
−1.0
−0.5
0.0
0.5
1.0
−1.0
−0.5
0.0
0.5
1.0
x
true spline conf. band
y
−2
−1
0
1
(c)
2
x
−1.0
−0.5
0.0
0.5
1.0
x Fig. 2 Spline estimates and 95% spline bootstrap confidence bands based on models 1, 2 and 3 in (6), n = 200, σ0 = 0.5
curves (confidence band). The computing burden is extremely light by borrowing the strength of spline smoothing. Remarkably, it takes merely 3 seconds in average when n = 10000 on a regular PC. 5 Unexpected earnings/return study revisited The proposed procedure has been applied to the financial data described in Sect. 2. We compare the performance of the following five estimators: Linear:mˆ 1 (x) = αˆ 1 + βˆ1 x; Broken stick (Beneish and Harvey 1998):mˆ 2 (x) = αˆ 2 + βˆ2 x + γˆ2 I{x $200 millions) sectors by the CSHO values. We conduct a cross sectional analysis for each sector separately, and each sector data are truncated at the 98th percentile of absolute value of the earning surprises. Table 7 summarizes the average MSE, R 2 and adjusted R 2 for each sector. It again shows the superiority of the spline smoothing to all the parametric models regardless the size of the firms. But here we do observe a little improvement of the parametric models for large firms compared with the results for the small firms. We also construct the confidence bands to test the specifications of the earnings/returns relation for different firm sizes. Table 8 provides a coverage report of
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−0.1
0.0
0.1
0.2
linear broken stick quadratic arctan spline confidence band
−0.2
unexpected returns
255
−0.02
−0.01
0.00
0.01
0.02
unexpected earnings Fig. 4 95% confidence band: the third quarter of 2005 Table 7 Cross-sectional results by firm size Firm size Small
Medium
Large
Model
Mean squared error
R2
Adjusted R 2
Linear
0.0177
0.0233
0.0231
Broken stick
0.0176
0.0250
0.0248
Quadratic
0.0177
0.0240
0.0238
Arctan
0.0171
0.0547
0.0545
Spline
0.0170
0.0595
0.0593
Linear
0.0129
0.0262
0.0260
Broken stick
0.0129
0.0263
0.0261
Quadratic
0.0129
0.0262
0.0260
Arctan
0.0127
0.0434
0.0432
Spline
0.0127
0.0448
0.0446
Linear
0.0097
0.0074
0.0071
Broken stick
0.0097
0.0099
0.0097
Quadratic
0.0097
0.0078
0.0076
Arctan
0.0096
0.0189
0.0187
Spline
0.0096
0.0261
0.0258
all estimated curves for small, medium and large firms. From Table 8, we find that for small firms, none of the hypothesized curves can be fully covered by the 95% bands. Another finding is that the coverage percentages of all tested curves are much bigger for large firms than for small firms. Figure 5a provides the 95 confidence band for small firms. The fitted spline curve presents strong deviation from strict linearity. For medium and large firms, the spline
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Table 8 Coverage percentage of hypothesized specification by firm size Firm size
Linear (%)
Broken stick (%)
Quadratic (%)
Arctan (%)
27.33
26.88
26.88
100.00
53.62
53.27
53.50
100.00
Large
50.17
64.74
51.46
95.02
−0.04
−0.02
0.02
0.04
0.1 0.0
0.02
linear broken stick quadratic arctan spline confidence band
−0.1
0.0
0.1
0.2
0.01
−0.2
unexpected returns
−0.02 −0.01 0.00
unexpected earnings
unexpected earnings
(c)
−0.1
unexpected returns 0.00
linear broken stick quadratic arctan spline confidence band
−0.2
−0.1
0.0
0.1
0.2
(b)
linear broken stick quadratic arctan spline confidence band
−0.2
unexpected returns
(a)
0.2
Small Medium
−0.02 −0.01 0.00
0.01
0.02
unexpected earnings Fig. 5 95% confidence bands: a small firms b medium firms; c large firms
estimated curves become much smoother due to the fact that large firms are better followed, hence increases the likelihood that analysts’ forecasts approximate market expectations of earnings. However, the 95% confidence bands in (b) and (c) of Fig. 5 still portray the nonlinearity. We also find that the arctan captures the shape of the regression curve quite well with the firm size controlled. Hence, if all we are interested in are some qualitative features of the underlying regression curves like derivatives, the arctan model seems to work sufficiently well if the firm size is controlled.
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6 Conclusions and discussions Motivated by the study of the unexpected earnings/returns, we propose a flexible and effective method to analyze the structure relationship of large and complex data using spline smoothing techniques. In addition, we develop an informative confidence band to explore the adequacy of certain linear/nonlinear parametric specifications as the underlying true model. There are three principal advantages of our method over the published ones: (i) the proposed approach provides an ability of infer some implications about the structural relationship for complex data (ii) the proposed band has excellent coverage of the true function and its width rapidly shrinks to zero with increasing sample size; (iii) the approach avoids the computational challenges that occur when one uses kernel smoothing methods for large datasets. We believe that the proposed method can be used for testing for structural change in the study of large datasets from financial research and many other areas. Our major findings of the unexpected earnings/returns study using the proposed method include (i) the measurement errors of the unexpected earnings account for some nonlinearity, however, they do not fully explain this phenomenon and the nonlinearity persists even the measurement errors are controlled; (ii) the removal of the firm size factor does not eliminate the nonlinearity, either; (iii) the degree of nonlinearity varies from small to large firms and not all the nonlinear parametric models fit the data well; (iv) for some type of firms, it appears difficult to use a single parametric equation to describe the complicated earnings/returns relationship, and effective tools for extracting information from such complex data have to be nonparametric (no particular formulae are imposed on the regression structures); (v) the arctan model seems to capture the relationship of the unexpected earnings/returns very well if we control the size of the firms. The proposed nonparametric approach reveals in parts the reason why various studies found different regression curves in the unexpected earnings/returns study. The more flexible model, however, suggests that there were some factors that the traditional parametric earnings/return studies did not account for. In addition to the magnitude of earning surprises and firm size difference, there might be a number of other factors responsible for the nonlinear earnings/returns relationship, such as the risk-free interstate rate, growth opportunity and other macroeconomic variables. It is thus interesting to search for other factors that would explain observed nonlinearity. In the future work, we would like to extend the proposed approach to nonparametric multivariate regression to further explore the underlying cause of nonlinearity. Acknowledgments The authors are grateful to James H. Albert, Stephen P. Baginski, Paul Irvine, Harold Mulherin and two referees for helpful comments that greatly improved the readability of the manuscript. The authors also would like to thank Lijian Yang and Jing Wang for helpful discussion on confidence bands.
References Abdel-khalik AR (1990) Specification problems with information content of earnings: revisions and rationality of expectations and self-selection bias. Contemp Account Res 7:142–172
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Anilowskia C, Feng M, Skinner DJ (2007) Does earnings guidance affect market returns? The nature and information content of aggregate earnings guidance. J Account Econ 44:36–63 Basu S (1997) The conservatism principle and the asymmetric timeliness of earnings. J Account Econ 24:1–37 Beneish MD, Harvey CR (1998) Measurement error and nonlinearity in the earnings–returns relation. Rev Quant Finance Account 11:219–247 Cheng SA, Hopwood WS, McKeown JC (1992) Nonlinearity and specification problems in unexpected earnings response regression model. Account Rev 67:379–398 Claeskens G, Van Keilegom I (2003) Bootstrap confidence bands for regression curves and their derivatives. Ann Stat 31:1852–1884 Collins DW, Kothari SP (1989) An analysis of intertemporal and cross-sectional determinants of earnings response coefficients. J Account Econ 11:143–181 Collins DW, Pincus M, Xie H (1999) Equity valuation and negative earnings: the role of book value of equity. Account Rev 74:29–62 Das S, Lev B (1994) Nonlinearity in the returns–earnings relation: tests of alternative specification and explanations. Contemp Account Rev 11:353–379 Easton PD, Sommers GA (2008) Effect of analysts’ optimism on estimates of the expected rate of return implied by earnings forecasts. J Account Res 45:983–1015 Fan J, Gijbels I (1996) Local polynomial modelling and its applications. Chapman and Hall, London Freeman RN, Tse S (1992) A nonlinear model of security price responses to unexpected earnings. J Account Res 30:185–209 Hall P, Titterington DM (1988) On confidence bands in nonparametric density estimation and regression. J Multivar Anal 27:228–254 Härdle W (1989) Asymptotic maximal deviation of M–smoothers. J Multivar Anal 29:163–179 Härdle W, Mammen E (1993) Comparing nonparametric versus parametric regression fits. Ann Stat 21:1926–1947 Huang JZ (2003) Local asymptotics for polynomial spline regression. Ann Stat 31:1600–1635 Kothari SP, Lewellen JW, Warner JB (2005) Stock returns, aggregate earning surprises, and behavioral finance. J Finance Econ 79:537–568 Lev B, Thiagarajan SR (1993) Fundamental information analysis. J Account Res 31:190–215 Neumann MH, Kreiss JP (1998) Regression-type inference in nonparametric autoregression. Ann Stat 26:1570–1613 Philbrick DR, Ricks WE (1991) Using value line and IBES analysts forecasts in accounting research. J Accont Res 29:397–415 Proietti T, Luati A (2008) Real time estimation in local polynomial regression, with application to trendcycle analysis. Ann Appl Stat 2:1523–1553 Ruppert D, Wand MP, Carroll RJ (2003) Semiparametric regression. Cambridge, New York Skinner DJ, Sloan RG (2002) Earnings expectations, growth expectations, and stock returns, or don’t let an earnings torpedo ink your portfolio. Rev Accont Stud 7:289–312 Sperlich S, Tjøtheim D, Yang L (2002) Nonparametric estimation and testing of interaction in additive models. Econ Theory 18:197–251 Stephen R, Zarowin P (2003) Why has the contemporaneous linear returns–earnings relation declined?. Account Rev 78:523–553 Subramanyam KR (1996) The pricing of discretionary accruals. J Account Econ 22:249–281 Wang J, Yang L (2009) Polynomial spline confidence bands for regression curves. Stat Sinica 19:325–342 Wu CFJ (1986) Jackknife, bootstrap and other resampling methods in regression analysis (with discussion). Ann Stat 14:1261–1295 Xia Y (1998) Bias-corrected confidence bands in nonparametric regression. J R Stat Soc B 60:797–811
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Security price responses to unexpected earnings: a nonparametric investigation Lei Gao · Li Wang
Accepted: 22 December 2010 / Published online: 6 January 2011 © Springer-Verlag 2011
Abstract The widely used linear model of the unexpected earnings/returns relationship has been challenged. In this article we propose a flexible nonparametric approach to study this relationship in which splines are used to approximate the unknown regression function. Spline confidence bands are constructed based on wild bootstrap to examine the adequacy of certain linear/nonlinear specifications. Monte Carlo results show that the proposed bands have excellent coverage of the true regression function with little computing load. These properties make the procedure highly recommended for extracting information from large and complicated datasets. The proposed approach has also been applied to the real world financial data from the unexpected earnings/returns study, and we find significant evidence of nonlinearity. The nonlinearity persists when we control the measurement errors of the earning surprises and firm size. Keywords Confidence bands · Earnings response coefficient · Measurement error · Spline · Wild bootstrap 1 Introduction Financial analysts make earning forecast for public firms. The financial analyst forecast error is called the unexpected earning. Since the stock market responds to the
L. Gao Department of Banking and Finance, University of Georgia, Athens, GA 30602, USA e-mail: [email protected] L. Wang (B) Department of Statistics, University of Georgia, Athens, GA 30602, USA e-mail: [email protected]
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information quickly and accurately, the stock price should have integrated all available information including the financial analyst forecast reports. The expected return of a single stock is defined as the risk-adjusted market index return, such as the S & P 500 return or the Dow Jones Industrial Average return. When there is a deviation between the firm’s earning announcement and the analysts’ forecast, the stock price will pick up the newly available information, so we would expect a corresponding unexpected stock return with respect to the unexpected earning information during the short period before and after the announcement event. For investors and researchers, it is very important to understand the relationship between the unexpected returns and earnings, and this article contributes to explaining the mystery. For the past three decades, the studies on the relationship of unexpected earnings and returns have occupied much of the accounting literature; see Collins and Kothari (1989), Freeman and Tse (1992) and Stephen and Zarowin (2003), for example, to name one article representative of each decade; and see Kothari et al. (2005), Anilowskia et al. (2007), and Easton and Sommers (2008) for some of the more recent research. Regression analysis has been widely used in learning about this relationship. Classic linear regression concerns primarily the inference on some parameters, for instance, the earnings response coefficient, referred as ERC in the literature. Many researchers have challenged the linearity and nonlinear models have been employed to study this relationship. For example, Cheng et al. (1992) and Subramanyam (1996) both find evidence that linear models of the earnings/returns relationship are misspecified. Freeman and Tse (1992) propose an arctan earnings/returns model that provides higher explanatory power and a richer explanation for differences between ERCs and the price-earnings ratio. Some other studies, including Das and Lev (1994) and Beneish and Harvey (1998), used the quadratic and broken stick models. However, it is difficult to choose the best regression curve in the absence of a comprehensive equity valuation model. These specific function forms are called “parametric” because they are explicitly determined by those fixed number of parameters. As discussed in Das and Lev (1994), such specifications are ad hoc, and if misspecified, can lead to substantial estimation bias problems. Another issue in the prior studies of earnings/returns relationship is that positive and negative surprises are treated symmetrically. However, Abdel-khalik (1990) and Basu (1997) show that the earnings/returns relationship is asymmetric. Collins et al. (1999) and Skinner and Sloan (2002) also find that the valuation effect of earnings is asymmetric between positive and negative earnings. The reason for this asymmetry is that bad news is reported more timely than good news, and hence anticipated losses are recognized immediately whereas unanticipated gains are spread out into the future. The earlier defined models are not be able to study this “leverage” effect on security returns. The main objectives of this article are (a) to introduce a flexible method, which is robust to the model misspecification, to estimate the unknown earnings/returns relationship and (b) to test the adequacy and validity of a specific model such as the linear and arctan models. For objective (a), we propose using a nonparametric model to study the relationship between unexpected earnings/returns. The nonparametric approach increases the
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flexibility of the class of models we use without restricting it a priori to have or not have certain shapes. The curve can be asymmetric and it allows one to study the different “leverage” of good news and bad news. In addition, the nonparametric models are more robust to model misspecification than the parametric ones, at least asymptotically, when dealing with the regression problem related with the “generated regressors” and “dependent variables”. The variables involved in the study of earnings/returns relationship are forecast errors or generated variables, i.e., the variation between the actual value of earnings and the value forecasted. The extra variability added by the estimation of these quantities can be smoothed out by the nonparametric estimators by averaging the irrelevant noise in the data. Details of nonparametric regression can be found in Fan and Gijbels (1996) and Ruppert et al. (2003). As in any estimation procedure, confidence regions provide more information than a single estimate. For objective (b), we make use of simultaneous confidence intervals (confidence bands) which give us insightful information of the unexpected earnings/returns relationship. This method allows us to visually examine the bands to test how well a curve captures the relationship of the unexpected earnings/returns. The early studies of Hall and Titterington (1988) and Härdle (1989) establish an asymptotic confidence band of nonparametric regression function based on Nadaraya-Watson kernel smoothing. The more efficient local polynomial smoothing has been investigated in parallel; see Xia (1998) and Claeskens and Van Keilegom (2003). All of the aforementioned methods involve local averaging by kernel weights. One drawback of the localized kernel estimator is its instability near the boundaries of the interval, which will lead to a bad assessment of the tail behavior on the regression curve. As an alternative, Proietti and Luati (2008) evaluate a family of asymmetric filters that can improve the estimation of the underlying function at the end of the sample period. However, kernel method is in general computationally expensive since it requires solving a linear least squares problem for every design point. The majority of financial data are large and complicated, for example, the sample size of the data used in our earnings/returns study is 13,314; see Sect. 2 for details. The kernel method is computationally too expensive for such large datasets. In fact for large datasets like that, it is useful and required that estimation and inference methods are efficient and computationally easily implemented. In this article, we propose approximating the regression function by spline smoothing, which is practically as easy to implement and fast as a simple linear regression with slowly increasing number of parameters (Ruppert et al. 2003). In addition, the implementation can be easily achieved in a commonly used computational environment like R. The rest of our article is organized as follows. Section 2 contains a detailed description of the financial data. Section 3 first introduces a flexible nonparametric heteroscedastic model and our hypothesis. Section 3.2 presents a class of spline estimators and an asymptotic confidence band for the regression function. We find the asymptotic band to be a little conservative and unstable, so a wild bootstrap band is developed in Sect. 3.3. Monte Carlo simulation studies are given in Sect. 4. Section 5 reports our statistical analysis and empirical findings for the data introduced in Sect. 2. We close with conclusions and discussions
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Table 1 Descriptive statistics of earnings forecast errors Sample size Philbrick and Ricks (1991) Freeman and Tse (1992) Beneish and Harvey (1998) This article
Mean
Standard deviation
Median 0.0024
4, 770
0.0087
0.0501
12, 381
0.0159
–
–
2, 929
0.0051
0.0042
0.0039
13, 314
0.0047
0.0269
0.0011
2 Data The dataset used to investigate the unexpected earnings/returns relationship comprises four resources: (i) Price per share of a firm at the end of the quarter and quarterly earnings announcement date on the 2007 quarterly industrial file obtained from Standard & Poor’s Compustat, a database of delivering financial, statistical and market information on active and inactive companies throughout the world; (ii) Daily stock returns on the 2007 CRSP daily returns file; (iii) Actual earnings quarterly data on the 2007 detail file from Institutional Brokers’ Estimate System (I/B/E/S), which provides both summary and individual analyst forecasts of company earnings, cash flows, and other important financial items, as well as buy-sell-hold recommendations; (iv) Forecast earnings quarterly data on the 2007 I/B/E/S detail file, in which financial models have been built to estimate the prospective revenues and costs and to predict the earnings; (v) Common shares outstanding (CSHO) of a firm on the 2007 Compustat fundamentals annual file. It has been conjectured in Beneish and Harvey (1998) that the nonlinear relationship could be an artifact phenomenon due to the measurement errors. So to examine whether the control for errors eliminates the nonlinearity, we removed 3 observations whose absolute per-share forecast error exceeds the price per share. The final selected sample contains a total of 1,647 firms and 13,314 observations from the first quarter of 2004 through the fourth quarter of 2006. Table 1 presents the comparison of our absolute forecast errors with those in the prior studies. One key variable of the study, unexpected earnings of a fiscal quarter, is measured by the ratio of earnings forecast errors to the stock price per share at the end of the previous quarter. The earnings forecast errors are defined as the difference between the actual quarterly earnings per share and the median I/B/E/S analyst forecast in the last month of the corresponding quarter. The other key variable, unexpected return, is measured based on the daily abnormal return of a firm, which is the return of a specific firm less the equally weighted mean return in its beta-matched portfolio for each trading date. Long and short time windows are selected to capture quick or slow stock price reaction to the unexpected earnings. For the long period window, the unexpected return is defined as the sum of daily abnormal returns for the period beginning two days after the previous quarterly
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Fig. 1 Plot of the estimated variance of the unexpected earnings based on the pooled samples
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earnings announcement and ending one day after the current announcement; for the short period window, it is calculated as the the sum of daily abnormal returns from day −2 to day +1 relative to the current announcement date. 3 Methodology Motivated by the dataset in Sect. 2, we propose using a flexible nonparametric model to study the unexpected earnings/returns relationship. In this section, we describe the model, hypothesis and general methodology. 3.1 Model and hypothesis n satisfying We formulate the problem broadly for any random sample {X i , Yi }i=1
Yi = m (X i ) + εi , i = 1, ..., n,
(1)
where m (x) = E (Y |X = x ) is some smooth but unknown conditional mean function [a, b]. The errors εi ’s are of mean zero and variance σ 2 (x) = 2on an interval E ε |X = x , not necessarily independent or normal. Model (1) is superior as it essentially gives one the flexibility of having a smooth function m, not necessarily in linear, quadratic or any other specific parametric forms. Another advantage is that it allows heteroscedastic errors, for example, the variance of the unexpected return (Y ) can be set to be a function of the earning surprises (X ), instead of a constant as assumed in the earlier studies. The asymmetry of the variance has been confirmed by our empirical analysis; see the empirically estimated variance function of unexpected earnings in Fig. 1.
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One of the major objectives of this article is to examine the validity of certain parametric forms of the regression function m (·). This problem can be formulated in terms of the hypothesis of m (·), i.e., H0 : m (x) = m θ (x), for any point x ∈ [a, b], where θ ∈ , and is a parameter space. For example, one can verify the generally assumed linearity of the unexpected earnings/returns by testing the hypothesis H0 : m (x) = β0 + β1 x, for any point x ∈ [a, b].
3.2 Spline estimators and confidence bands Next we introduce a class of estimators of function m(·) with confidence bands. To estimate m, one of the most common nonparametric methods is smoothing, in which observational errors are reduced by different types of averaging of the data. Splines, kernels and orthogonal series expansions are all examples of smoothing estimators. As pointed out in the introduction, one drawback of the localized kernel estimator is its instability near the boundaries of the interval, which will lead to poor assessment of the tail behavior of effect from the unexpected earnings to unexpected returns. Kernel smoothing is also computationally intensive since it requires to solve an optimization problem at every design point. In contrast, spline estimator is more stable and very popular forits simple implementation and fast computation (Ruppert et al. 2003). p p Denote 1, x, ..., x p , (x − κ1 )+ , ..., (x − κ N )+ the truncated power spline bases of order p, where (ν)+ = ν if ν > 0 and 0 otherwise, and κ j , j = 1, ..., N , are interior points in [a, b], called knots. combination of the p above spline bases pAny linear m x; β0 , ..., β p , γ1 , ..., γ N = k=0 βk x k + Nj=1 γ j x − κ j + is a piecewise poly nomial function, called a spline, where β0 , ..., β p , γ1 , ..., γ N are the coefficients and N is the number of interior knots. If N is sufficiently large, m x; β0 , ..., β p , γ1 , ..., γ N can approximate m (x) with a high degree of accuracy. We define the spline estimator for the regression function m(x) as
mˆ (x) =
p k=0
βˆk x k +
N
p γˆ j x − κ j + ,
(2)
j=1
βˆ0 , ..., βˆ p , γˆ1 , ..., γˆN are the least square estimators of β0 , ..., β p , n γ1 , ..., γ N based on sample {(X i , Yi )}i=1 . The shape of the basis functions is determined by the position of the knots κ1 < · · · < κ N , which can, for example, be uniformly spread sample quantiles of the X i values. To test the hypotheses in Sect. 3.1, we make use of the confidence bands, which allow us to visually examine and summarize how the regression curve changes along the whole range of predictor values. Under the null hypothesis, m θ would have to lie completely inside the confidence band with probability 1 − α. If this is violated, the tested curve will be rejected at the significance level α. For example, for the aforementioned linear hypothesis, we can obtain a weighted OLS estimator (βˆ0 , βˆ1 ) of n , then visually examine the confidence band to (β0 , β1 ) from the data {(X i , Yi )}i=1
where
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check whether the band covers βˆ0 + βˆ1 x for all x. If so, then we accept at level α the null hypothesis that m is linear. Otherwise the linearity is rejected. n , Huang (2003) obtains asymptotic 100(1 − α)% Given any sample {(X i , Yi )}i=1 pointwise confidence intervals in the form of mˆ (x) ± z 1−α/2 σˆ mˆ (x) ,
(3)
where z 1−α/2 is the (1 − α/2) quantile of the standard normal distribution, mˆ (x) is the spline estimator in (2) and σˆ mˆ (x) is its standard error, whose explicit formula is contained in Huang (2003). Wang and Yang (2009) derive an asymptotic 100(1 − α)% confidence band of m (x) over interval [a, b] in the form of mˆ (x) ± σˆ mˆ (x) {2 log (N + 1) − 2 log α}1/2
(4)
by using the piecewise linear spline basis. The spline bands in (4) have the same width order as the Nadaraya-Watson bands of Härdle (1989), and the local polynomial bands of Xia (1998) and Claeskens and Van Keilegom (2003). However, a plug-in method has to be implemented first to estimate the standard error σˆ mˆ . However, the plug-in method requires additional smoothing steps which may lead to less accurate and stable results. Hence in this article we develop a better method to construct the confidence bands using the so-called “wild bootstrap” by Wu (1986), which preserves the conditional heteroscedasticity in the original residuals. 3.3 Bootstrap confidence band We define the residuals εˆ i = Yi − mˆ (X i ) , 1 ≤ i ≤ n, and denote a predetermined large integer by n B . 1≤b≤n Step 1 Let δi,b 1≤i≤n B be i.i.d. samples from a Bernoulli distribution, and
δi,b =
√ √ √ −(√ 5 − 1)/2, with probability( √5 + 1)/(2 √5); ( 5 + 1)/2, with probability( 5 − 1)/(2 5).
It is easily verified that E δi,b ≡ 0 and V ar δi,b ≡ 1, for all 1 ≤ i ≤ n, 1 ≤ b ≤ n B . Step 2 For any 1 ≤ b ≤ n B , define the b-th wild bootstrap sample Yi,b = mˆ (X i ) + estimator of m (x) is the spline δi,b εˆ i , 1 ≤ i ≤ n. Then the b-th bootstrap n estimator mˆ (b) (x) based on sample X i , Yi,b i=1 . Step 3 Denote mˆ L ,α/2 (x) and mˆ U,α/2 (x) the lower and upper 100(1 − α/2)% quantiles of the set mˆ (b) (x) 1≤b≤n . The wild bootstrap 100(1 − α)% B pointwise confidence interval for function value m (x) at one point x is mˆ L ,α/2 (x) , mˆ U,α/2 (x) . Step 4 According to Wang and Yang (2009), when localized at any point x, the uniform confidence band in (4) is wider than the pointwise confidence inter−1 {2 log (N + 1) − 2 log α}1/2 . val in (3) by a common factor of K = z 1−α/2
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Hence we define the wild bootstrap 100(1 − α)% confidence band for m (x) over [a, b] as m L ,α/2 (x) = mˆ (x) + mˆ L ,α/2 (x) − mˆ (x) K , m U,α/2 (x) = mˆ (x) + mˆ U,α/2 (x) − mˆ (x) K . Härdle and Mammen (1993) studied several different bootstrap procedures and concluded that the wild bootstrap is the most pertinent method for testing the regression structure. Neumann and Kreiss (1998) have extended the result to a time series context and shown that the wild bootstrap still consistent even if the errors are not independent. The justification for making use of the wild bootstrap sample is also given in Sperlich et al. (2002). An important aspect for spline smoothing is the choice of the appropriate number of knots. Splines with few knots are generally smoother than splines with many knots; however, increasing the number of knots usually increases the fit of the spline function to the data. Wang and Yang (2009) show that the bias of the spline estimator can be killed if N ∼ n 1/5 . The number of knots used to construct the band is selected to minimize the following bayesian information criterion (BIC)
n 2 1 log n . Yi − mˆ (X i ) BIC(N ) = log + (N + 1) n n
(5)
i=1
For the issue of stability, here the selection is just made on the original series, not on the bootstrap replicates. 4 Simulation studies To illustrate the finite-sample behavior of the spline bootstrap bands, we conduct some simulation studies. For comparison purposes we also include the kernel-type bands from Hardle 1990, and the spline-type bands in (4) based on the asymptotic distribution theory. For the kernel-type bands, we use the Nadaraya-Watson smoother with the quartic kernel and rule-of-thumb bandwidth. For the spline-type bands, we use the piecewise linear spline smoothing. The number of interior knots for the asymptotic bands is taken to be [5n 1/5 ] + 1 as suggested in Wang and Yang (2009), and the number of knots for the bootstrap bands is selected via the BIC given in (5). We generate samples from the following heteroscedastic models Yi = m (X i ) + σ0 |X i |1/5 /5 + 1.2 εi , where Model 1 : m(x) = arctan (2.5x) ; Model 2 : m(x) = 1.5 sin (π x) ; Model 3 : m(x) = 3 (x + 1) (x − 1) + 1.5.
(6)
The X i ’s are i.i.d from uni f or m(−1, 1). The noise level σ0 = 0.5 and the error εi ’s are generated independently from N (0, 1).
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Table 2 Monte Carlo coverage probabilities and areas of the asymptotic kernel-type, asymptotic splinetype and bootstrap spline-type confidence bands Model Size
1
2
3
Confidence level 0.99 Kernel Spline Asymptotic
Bootstrap
Confidence level 0.95 Kernel Spline Asymptotic
Bootstrap
200
0.992(1.534) 0.968 (3.126) 0.962(1.996)
0.934(1.180) 0.908(2.764)
0.928(1.826)
500
1.000(1.044) 0.988(2.241)
0.944(0.815) 0.952(1.988)
0.950(1.344)
0.978(1.479)
1000 0.994(0.783) 0.984(1.690)
0.980 (1.067) 0.938(0.617) 0.946 (1.502) 0.958(0.964)
2000 0.994(0.585) 0.994(1.294)
0.992(0.872)
0.938(0.466) 0.970(1.152)
0.970(0.790)
5000 0.992(0.502) 0.984(0.910)
0.992(0.619)
0.942(0.402) 0.962(0.813)
0.978(0.561)
200
0.868(2.037) 0.966(3.126)
0.956 (2.004) 0.684(1.635) 0.910 (2.764) 0.906(1.831)
500
0.858(1.404) 0.988(1.870)
0.968(2.241)
0.592(1.139) 0.950(1.988)
0.942(1.346)
1000 0.798(1.060) 0.984(1.690)
0.974(1.068)
0.510(1.065) 0.946(1.502)
0.966(0.946)
2000 0.648(0.797) 0.994(1.294)
0.982(0.872)
0.330(0.656) 0.964(0.813)
0.960(0.790)
5000 0.388(0.550) 0.984(0.910)
0.980(0.986)
0.134(0.456) 0.964(0.813)
0.972(0.561)
200
0.610(1.909) 0.968(3.126)
0.956(2.002)
0.286(1.512) 0.910(2.764)
0.924(1.828)
500
0.354(1.298) 0.990(2.241)
0.974(1.480)
0.100(1.040) 0.952(1.988)
0.946(1.346)
1000 0.120(0.971) 0.980(1.690)
0.978(1.067)
0.034(0.784) 0.944(1.502)
0.958(0.964)
2000 0.016(0.727) 0.994(1.294)
0.990(0.872)
0.000(0.591) 0.972(1.152)
0.972(0.790)
5000 0.000(0.498) 0.984(0.910)
0.996(0.620)
0.000(0.409) 0.962(0.813)
0.972(0.561)
The result is based on 500 replications. The values outside ( ) are the coverage probabilities of the bands; the values inside ( ) are the areas of the bands
We calculate the percentage of coverage of the true function by the confidence bands constructed from the above three different methods. Two nominal confidence levels 0.99 and 0.95 are chosen for sample sizes 200, 500, 1000, 2000 and 5000. We carry out 500 simulation replications, and for each replication, 1000 bootstrap samples are generated for the bootstrap bands. Table 2 contains the Monte Carlo coverage probabilities and the enclosed areas of three types of bands. From Table 2, one sees that the spline-type bands have much higher coverage probability than the kernel-type bands, especially for Models 2 and 3. On the other hand, the spline-type bands are more conservative than the kerneltype bands. In terms of the width of the confidence bands, the kernel-type band is the narrowest, followed by the spline bootstrap band and the spline asymptotic band. In addition, the enclosed area decreases when sample size increases or confidence level decreases. The spline asymptotic bands seem to be too wide for some small sample size, for example, n = 200. Overall, the spline bootstrap band is a useful compromise between the kernel-type band and the spline asymptotic band. The enclosed area of the spline bootstrap bands is on average comparable to the kernel ones, while at the same time the bands achieve very good coverage probability. Figure 2 shows the spline fits for regression functions in (6) and the corresponding bootstrap bands. Here the graphs are created based on one random sample of size 200, each with four types of symbols: points (data), center smooth curve (true curve), center piecewise linear line (the piecewise-linear spline estimator), upper and lower
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0.5
1.0
x Fig. 2 Spline estimates and 95% spline bootstrap confidence bands based on models 1, 2 and 3 in (6), n = 200, σ0 = 0.5
curves (confidence band). The computing burden is extremely light by borrowing the strength of spline smoothing. Remarkably, it takes merely 3 seconds in average when n = 10000 on a regular PC. 5 Unexpected earnings/return study revisited The proposed procedure has been applied to the financial data described in Sect. 2. We compare the performance of the following five estimators: Linear:mˆ 1 (x) = αˆ 1 + βˆ1 x; Broken stick (Beneish and Harvey 1998):mˆ 2 (x) = αˆ 2 + βˆ2 x + γˆ2 I{x $200 millions) sectors by the CSHO values. We conduct a cross sectional analysis for each sector separately, and each sector data are truncated at the 98th percentile of absolute value of the earning surprises. Table 7 summarizes the average MSE, R 2 and adjusted R 2 for each sector. It again shows the superiority of the spline smoothing to all the parametric models regardless the size of the firms. But here we do observe a little improvement of the parametric models for large firms compared with the results for the small firms. We also construct the confidence bands to test the specifications of the earnings/returns relation for different firm sizes. Table 8 provides a coverage report of
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−0.1
0.0
0.1
0.2
linear broken stick quadratic arctan spline confidence band
−0.2
unexpected returns
255
−0.02
−0.01
0.00
0.01
0.02
unexpected earnings Fig. 4 95% confidence band: the third quarter of 2005 Table 7 Cross-sectional results by firm size Firm size Small
Medium
Large
Model
Mean squared error
R2
Adjusted R 2
Linear
0.0177
0.0233
0.0231
Broken stick
0.0176
0.0250
0.0248
Quadratic
0.0177
0.0240
0.0238
Arctan
0.0171
0.0547
0.0545
Spline
0.0170
0.0595
0.0593
Linear
0.0129
0.0262
0.0260
Broken stick
0.0129
0.0263
0.0261
Quadratic
0.0129
0.0262
0.0260
Arctan
0.0127
0.0434
0.0432
Spline
0.0127
0.0448
0.0446
Linear
0.0097
0.0074
0.0071
Broken stick
0.0097
0.0099
0.0097
Quadratic
0.0097
0.0078
0.0076
Arctan
0.0096
0.0189
0.0187
Spline
0.0096
0.0261
0.0258
all estimated curves for small, medium and large firms. From Table 8, we find that for small firms, none of the hypothesized curves can be fully covered by the 95% bands. Another finding is that the coverage percentages of all tested curves are much bigger for large firms than for small firms. Figure 5a provides the 95 confidence band for small firms. The fitted spline curve presents strong deviation from strict linearity. For medium and large firms, the spline
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Table 8 Coverage percentage of hypothesized specification by firm size Firm size
Linear (%)
Broken stick (%)
Quadratic (%)
Arctan (%)
27.33
26.88
26.88
100.00
53.62
53.27
53.50
100.00
Large
50.17
64.74
51.46
95.02
−0.04
−0.02
0.02
0.04
0.1 0.0
0.02
linear broken stick quadratic arctan spline confidence band
−0.1
0.0
0.1
0.2
0.01
−0.2
unexpected returns
−0.02 −0.01 0.00
unexpected earnings
unexpected earnings
(c)
−0.1
unexpected returns 0.00
linear broken stick quadratic arctan spline confidence band
−0.2
−0.1
0.0
0.1
0.2
(b)
linear broken stick quadratic arctan spline confidence band
−0.2
unexpected returns
(a)
0.2
Small Medium
−0.02 −0.01 0.00
0.01
0.02
unexpected earnings Fig. 5 95% confidence bands: a small firms b medium firms; c large firms
estimated curves become much smoother due to the fact that large firms are better followed, hence increases the likelihood that analysts’ forecasts approximate market expectations of earnings. However, the 95% confidence bands in (b) and (c) of Fig. 5 still portray the nonlinearity. We also find that the arctan captures the shape of the regression curve quite well with the firm size controlled. Hence, if all we are interested in are some qualitative features of the underlying regression curves like derivatives, the arctan model seems to work sufficiently well if the firm size is controlled.
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6 Conclusions and discussions Motivated by the study of the unexpected earnings/returns, we propose a flexible and effective method to analyze the structure relationship of large and complex data using spline smoothing techniques. In addition, we develop an informative confidence band to explore the adequacy of certain linear/nonlinear parametric specifications as the underlying true model. There are three principal advantages of our method over the published ones: (i) the proposed approach provides an ability of infer some implications about the structural relationship for complex data (ii) the proposed band has excellent coverage of the true function and its width rapidly shrinks to zero with increasing sample size; (iii) the approach avoids the computational challenges that occur when one uses kernel smoothing methods for large datasets. We believe that the proposed method can be used for testing for structural change in the study of large datasets from financial research and many other areas. Our major findings of the unexpected earnings/returns study using the proposed method include (i) the measurement errors of the unexpected earnings account for some nonlinearity, however, they do not fully explain this phenomenon and the nonlinearity persists even the measurement errors are controlled; (ii) the removal of the firm size factor does not eliminate the nonlinearity, either; (iii) the degree of nonlinearity varies from small to large firms and not all the nonlinear parametric models fit the data well; (iv) for some type of firms, it appears difficult to use a single parametric equation to describe the complicated earnings/returns relationship, and effective tools for extracting information from such complex data have to be nonparametric (no particular formulae are imposed on the regression structures); (v) the arctan model seems to capture the relationship of the unexpected earnings/returns very well if we control the size of the firms. The proposed nonparametric approach reveals in parts the reason why various studies found different regression curves in the unexpected earnings/returns study. The more flexible model, however, suggests that there were some factors that the traditional parametric earnings/return studies did not account for. In addition to the magnitude of earning surprises and firm size difference, there might be a number of other factors responsible for the nonlinear earnings/returns relationship, such as the risk-free interstate rate, growth opportunity and other macroeconomic variables. It is thus interesting to search for other factors that would explain observed nonlinearity. In the future work, we would like to extend the proposed approach to nonparametric multivariate regression to further explore the underlying cause of nonlinearity. Acknowledgments The authors are grateful to James H. Albert, Stephen P. Baginski, Paul Irvine, Harold Mulherin and two referees for helpful comments that greatly improved the readability of the manuscript. The authors also would like to thank Lijian Yang and Jing Wang for helpful discussion on confidence bands.
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