Excerpt

## TABLE OF CONTENTS

1. Introduction

2. Linear Predictive Regression Model

3. Econometric Methodology and Hypothesis ofInterest

4. Empirical Results

5. Economic Interpretation

6. Conclusions

References

Appendix

Table I, II, III

Table IV, V, VI

Table VII, VIII

Table IX, X

Figure I

## 1 Introduction

A vast recent literature aimed to explore the concept of predictive regressions and highlighted their application in forecasting particularly economic and financial variables (Fama and French (1988), Bekaert and Hodrick (1992), Stambaugh (1999), Jansson and Moreira (2006), Campbell and Yogo (2006), Wei and Wright (2009), among others). An adoption of predictive regression has been used to determine the predictive power of lagged explanatory variables with a high degree of persistence, such as interest rates, dividend yields, earnings or other possibly non-stationary variables, to forecast the future value of generally more volatile variable, for example, stock market returns. Torous and Valkanov (2000) in their studies suggest that building future stock returns prediction upon their realized past values is not a reliable procedure, since returns typically follow a random walk or 1(0) process (Fama 1965). Nonetheless, there is a belief that some of the variation of stock returns can be predicted using broadly available data, which has provided an incentive for econometricians to develop the predictive regression approach in finance. The evidence provided in researches have documented that financial predictors, such as dividend yields, may have some forecasting power in predicting the equity market stock returns in particular time spans (Bekaert and Hodrick (1992) and references therein).

A number of studies have examined the performance of predictive regressions and paid special attention to issues that may arise from statistical estimation of such models. Apparently, the predictive regressions can capture some predictive power of specific instruments used in the model, but they have been found to be subject to many statistical problems at the same time. Many variables, that are widely thought to have some predictive power in forecasting market returns, are likely to suffer from endogeneity or persistence. The persistence of the explanatory variable leads the standard OLS estimates to be biased as well as to poor finite sample approximation (Stambaugh (1999)). Lack of consistency (if regressors happen to be nonstationary) and explosive conventional t-statistics may result in inferences being misleading, if not meaningless. Campbell and Yogo (2006) argue that the conventional t-statistics derived on the coefficients of financial variables provide solid evidence for the predictability of stock returns, which, in reality, might not be the case. Thus, special care must be undertaken while analyzing the predictive power of financial instruments.

A simple linear estimation of stock returns using past values of persistent regressors casts doubts not only on the methodology and quality of the forecast, but also rises a question of structural stability of the model, which was predominantly ignored in the 20th century literature, when the predictive ability of financial instruments was frequently claimed to be negligible. There are number of reasons to believe that the predictive power of financial instruments changes across time due to either business cycle fluctuations, economic events, the evolution of the economy, speculations or simply exogenous factors that affect the market. Hence, if the data covering several decades were used in linear estimation to conduct the analysis, some may expect estimates to be unstable. Identification of such instability is extremely crucial, as the consequences of omitting the presence of structural breaks can be perilous and may lead to untrue conclusions, e.g. not reveal the predictive ability. Allowing for multiple breaks, assuming that their number is bounded asymptotically, may be a source of remedy and enhancement in terms of provision of efficient inferences. These arguments were considered in contemporary applied works inter alia by Rapach and Wohar (2006) and Lettau and Nieuwerburgh (2007). Pitarakis and Gonzalo (2009) in their paper argue that the linear predictability of financial instruments may change together with economic regimes, that is, principally during economic downturns a highly persistent dividend yield can partially forecast the dynamics of stock returns, whereas during 'booms' the predictive power is negligible. Moreover, when the linear prediction is implemented and no structural breaks are allowed, then results revealing disappointing performance of predictive regressions are consistent with the findings presented by Goyal and Welch (2002) or Torous and Valkanov (2000), where no-bubble models are applied.

A special attention that predictive regressions have received in the past 20 years aimed to answer two key questions. Firstly, taking into account econometric issues, how do regressors perform in terms of predictive power. Secondly, assuming that the predictive power exists, is the forecasting stable over time. The purpose of this work is to provide a discussion to the latter question addressed above as the previous literature has put more weight to deal with the first one. Moreover, I also concentrate on a predictive ability of dividend yields on excess returns at a one-month horizon. Certainly, testing for a structural break can be done using simple econometric tool, such as the classical Chow test (I960), assuming the location of instability is known. However, more efficient approach is to test for a break of unknown location due to uncertainty, since expectations based on economic theory or arbitrarily chosen break-point may not provide an exact clue where to search for instability. What is more, the evidence of instability found, inter alia, by Dangl and Halling (2006) together with arguments described in the previous paragraph suggest that the coefficients are likely to change over time, so that it is rational to allow for a structural break in both, intercept and slope coefficients, in the model that combines stock returns and dividend yields. In order to proceed with the analysis I construct the Supremum test (SupWald) of Andrews (1993), that is a Wald test for each possible location k. Since the model suffers from high persistence in explanatory variable and contemporaneous correlation between innovations from the linear predictive model and autoregressive dividend yields regression, the size and power of the test are affected. However, the dispersion between tabulated critical values and outcomes obtained from Monte Carlo simulation closes as the number of observations increases. Similarly, the power of the test seems to converge to one as the sample size goes to infinity, even for a N1(1) process. In my dissertation I find that Andrews test is likely to over-reject the null hypothesis of no structural instability. Although there is no strong statistical evidence of a break based on maximum SupWald statistic, I provide arguments supporting the claim that the predictive power may be uncovered when the sample was split into two subsamples according to the potential break-date provided by Andrews coefficient instability test.

The plan for the remainder of this dissertation is as follows. Chapter 2 describes the linear predictive regression model and briefly states some econometric issues. Chapter 3 introduces hypothesis of interests and presents arguments for the methodology applied in empirical analysis. The Monte Carlo simulation is conducted in order to verify the appropriateness of Andrews' approach when the lagged explanatory variable is persistent and when the contemporaneous correlation between disturbances is nonzero. Chapter 4 implements described methodology in empirical analysis. Chapter 5 provides description of results from economic point of view and Chapter 6 concludes.

## 2 Linear Predictive Regression Model

The linear predictive regression model associates the noisy variable, sometimes characterized by clusters of volatility, such as stock returns, with a lagged variable that may have some predictive power. It can be said that the variable x_{t} has the ability to predict returns if β≠0. In the framework I use, the explanatory variable follows an AR(1) process. The classical set-up of a linear predictive regression is as follows:

[illustration not visible in this excerpt](2.1)

where y_{t+1} is the variable that the model aims to forecast, in this scenario stock returns, a is the intercept (if assume that y_{t+1} has mean zero, then ⍺ does not appear in the model), u_{t} and v_{t} are assumed to be independently and identically distributed (i.i.d.) bivariate Normal with zero mean N(0,Σ), where the covariance matrix is:

[illustration not visible in this excerpt](2.2)

One of the standard statistical issues described by Stambaugh (1999) is that the innovations u_{t} and v_{t} may be correlated, σ_{uv}≠0 (from (2.2)), which violates the initial assumption and raises a problem of endogeneity. This in turns may result in the finite- sample standard OLS estimate of β being still [illustration not visible in this excerpt]-consistent but imprecisely estimated (upward biased) as [illustration not visible in this excerpt]. Despite Hjalmarsson (2OO7) suggestion, that the Stambaugh bias in the time series case is less stern when no intercept is included in the model, I do not exclude the intercept in order to test for a break in both coefficients. Hence, the sampling distribution is typically different, and analysis conducted under the standard setting could provide misleading results. To be more specific, the conventional t-statistics may be over-rejecting the hypothesis that β=0. To control for serial correlation researchers commonly apply Newey-West of Hansen-Hodrick standard errors. As an addition, Faust and Wright (2OO8) suggest that augmenting a standard linear predictive regression model with variables that are correlated with the innovations ut and uncorrelated with the regressor x_{t} can substantially increase the asymptotic efficiency of an estimate of β and improve forecasting. Alternatively, to deal with this problem the instrumental variable (IV) approach could be implemented, as long as the instrumental validity is satisfied, i.e. the instrument is highly correlated with the financial variable xt and uncorrelated with the innovations ut in (2.1). Nevertheless, such valid instruments are problematic to obtain since the new potential IV is likely to display similar problems as the original regressor.

Forecasting a highly volatile variable with a regressor, that is characterized by the significant persistence, creates complications in analysis and statistical inferences. Numerous explanatory variables considered to be reasonable predictors fail to reject the unit-root hypothesis. At first sight, it is difficult to believe that such persistent explanatory variable could have a notable predictive power in explaining much noisier variable. This educated guess has been justified in many research papers, as the coefficients of determination, R^{2}, were typically found to be low, when financial instruments were used as predictors. What is more, if the persistent regressor turns out to have some predictive power, then it may be the case, that the explained variable may have some persistent component. Furthermore, Torous and Valkanov (2000) argue that increasing the frequency of observations may not necessarily improve the precision of estimates, but can contribute to further growth of the noise keeping the signal in the data nonessential. In the past literature, variables that have been suggested to have some predictive power in explaining the variation of the regressand were primarily stationary, [illustration not visible in this excerpt], e.g. Dangl and Halling (2006), among others. However, less attention has been paid to estimation using regressors xt that can be described as a local-to-unity or nearly integrated, N1(1), process (see Campbell and Yogo (2006) or Gonzalo and Pitarakis (2009)). In my work I define the independent variable as a local-to-unity or nearly nonstationary process, that is:

[illustration not visible in this excerpt] (2.3)

where c is a fixed constant. The key complication in the linear predictive regression model I target to emphasize is the possible presence of instability in the predictability of forecasted variable. If we recall the arguments provided in the third paragraph of Chapter 1, it is likely that the model predicting returns with generally smooth financial variables, such as dividend yields or interest rates, will exhibit some inconsistency across time. More precisely, the unstable predictability results in variation of coefficients, what may have been stimulated by economic factors. Park (2005) in his paper conclude that most of

researches implied the disappearing predictive power of dividend-price ratios due to a temporary instability in the late 1990s. Hence, to efficiently capture the variation of excess returns, it is sensible to take into account the possibility of structural break in the model and possibly increase the predictability of returns. Although, Rapach and Wohar (2006) report the presence of multiple breaks, in my work I assume the circumstance under which only one break may have occurred in the model, say at time k:

[illustration not visible in this excerpt] (2.4)

Allowing for a structural break in the linear predictive regression model may increase the forecasting ability of the lagged regressor in a specific time span. In other words, if this phenomenon was ignored, the presence of some dynamics may have been omitted by the model. It is worth a mention that this procedure is likely to reinforce the goodness of fit, R^{2}, whose value was typically observed to be modest when no structural break was permitted- Moreover, the explanatory power of dividend yields, as a regressor in returns model, has been observed to vary, i.e. over distinct four-year horizons, R^{2} 's deviated from 19% to 64% (Goetzmann and Jorion (1993) and references therein) or 34% in 1982 to 0% at the end of the 1990s (Lettau and Nieuwerburgh (2007)). Number of studies have stressed the importance of detecting structural breaks in the predictive regression model to determine how the strength of predictability changes over time, seasonally or across different economic regimes. Therefore, the conjecture of possible instability in returns model has to be reconsidered in order to assess the actual predictive power of financial instruments.

## 3 Econometric Methodology and Hypothesis of Interest

Lettau and Nieuwerburgh (2007) in their studies describe the incompatible outcomes of stock predictability shown in the recent literature. There have been various opinions, some supporting the claim that the stock returns may be partially forecastable with financial data, and others declaring the nonexistence of predictability. Certainly, findings were not identical because test measures, assumptions and methodologies presented in these paper were not unified and not all statistical problems were taken into account. For example, Fama and French (1988) report a high predictive ability of dividend-price ratio on the future stock returns, but they relied mainly on asymptotic theory. Hence, due to the inference problem that may appear as a result of structural instability it would be sensible to test the null hypothesis of linearity:

[illustration not visible in this excerpt](3.1)

against the general model presented in equations (2.4). Hence, the intuition is to test the structural stability of both, intercept and] slope coefficient, in the linear predictive regression model (parameters α and β from equation (2.1)). The implication of the presence of a break is that it may alter the conditional expected stock return E(y_{t+1}|x_{t}). As a consequence, the predictive power of the lagged explanatory variable can be inappropriately estimated. If the null hypothesis cannot be rejected, this result could justify the methodology previously described in the literature by, exempli gratia, Campbell and Yogo (2006). What is more, in the case where the result suggests the constant mean specification, that is α_{1}=α_{2}=0 and β_{1}=β_{2}=0, the predictive power of the lagged independent variable used in the regression is said to be nonexistent. On the other hand, the evidence of the structural break would support the claim presented by Rapach and Wohar (2006). In order to verify which of the confronting claims may be correct, I proceed to testing for the structural instability of the coefficients.

**[...]**

- Quote paper
- Lukasz Prochownik (Author), 2011, Linear predictive regression framework, Munich, GRIN Verlag, https://www.grin.com/document/182493

Comments

Great!!