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## Maximum likelihood estimation of the equity premium (2013)

Citations: | 1 - 0 self |

### Citations

3868 |
Time Series Analysis
- Hamilton
- 1994
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Citation Context ...(Box and Tiao, 1973; Poirier, 1978), assuming that it is a draw from the stationary distribution. In our case, the stationary distribution of x0 is normal with mean µx and variance σ2x = σ2v 1− θ2 , (=-=Hamilton, 1994-=-). The resulting likelihood function is p (r1, . . . , rT ;x0, . . . , xT |µr, µx, β, θ,Σ) =( 2piσ2x )− 1 2 exp { −1 2 ( x0 − µx σx )2} × |2piΣ|−T2 exp { −1 2 ( σ2v |Σ| T∑ t=1 u2t − 2 σuv |Σ| T∑ t=1 u... |

2406 | Generalized Autoregressive Conditional Heteroskedasticity." Journal of Econometrics 31: 307-327. http://dx.doi.org/10.1016/0304-4076(86)90063-1 Bouye, Eric. 2002. "Multivariate Extremes at Work for Portfolio Risk Measurement." Working paper - Bollerslev - 1986 |

1751 | The equity premium: a puzzle. - Mehra, Prescott - 1985 |

1453 | By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior."
- Campbell, Cochrane
- 1995
(Show Context)
Citation Context ...s necessary, e.g. understanding long-horizon returns or the statistical properties of estimators for β.3 Indeed, most leading economic models imply that xt is stationary (e.g. Bansal and Yaron, 2004; =-=Campbell and Cochrane, 1999-=-). A large and sophisticated literature uses this setting to explore the bias and size distortions in estimation of β, treating other parameters, including µr, as “nuisance” parameters. 4 Our work dif... |

1067 |
Advanced Econometrics
- Amemiya
- 1985
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Citation Context ... maximum likelihood, assuming that the specification is correct, provides the most efficient estimates of the parameters, that is, the estimates with the (weakly) smallest asymptotic standard errors (=-=Amemiya, 1985-=-). Furthermore, in large samples, and assuming no mis-specification, introducing more data 6We could extend our results to multiple predictor variables (Kelly and Pruitt (2013), for example, allow mul... |

847 | Do stock prices move too much to be justified by subsequent changes in dividends?,
- Shiller, J
- 1981
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Citation Context ...dge result in which the variance of the log dividend-price ratio xt and the covariance of xt with log price changes cancel out. However, it is well-known that prices are more volatile than dividends (=-=Shiller, 1981-=-). 15This point is also made by Constantinides (2002), who suggests adjusting the mean return by the difference in the valuation ratio between the first and last observation. Constantinides derives co... |

766 |
ARCH Modeling in Finance: A Review of the Theory and Empirical Evidence,”
- Bollerslev, Chou, et al.
- 1992
(Show Context)
Citation Context ...nces of structural breaks for our results. 5.1 Conditional Heteroskedasticity It is well known that stock returns exhibit time-varying volatility (French, Schwert, and Stambaugh, 1987; Schwert, 1989; =-=Bollerslev, Chou, and Kroner, 1992-=-). In this section we generalize our estimation method to take this into account. Because of our focus on maximum likelihood, a natural approach is to use the GARCH model of Bollerslev (1986). We will... |

761 | Risks For the Long Run: A Potential Resolution of Asset Pricing Puzzles. - Bansal, Yaron - 2004 |

726 | The Dividend-price Ratio and Expectations of Future Dividends and Discount Factors. - Campbell, Shiller - 1988 |

716 | Expected Stock Returns and Volatility,”
- French, Schwert, et al.
- 1987
(Show Context)
Citation Context ... does not. Finally, Section 5.3 analysis the consequences of structural breaks for our results. 5.1 Conditional Heteroskedasticity It is well known that stock returns exhibit time-varying volatility (=-=French, Schwert, and Stambaugh, 1987-=-; Schwert, 1989; Bollerslev, Chou, and Kroner, 1992). In this section we generalize our estimation method to take this into account. Because of our focus on maximum likelihood, a natural approach is t... |

674 | Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model - Bollerslev - 1990 |

662 |
Why does stock market volatility change over time?
- Schwert
- 1989
(Show Context)
Citation Context ...is the consequences of structural breaks for our results. 5.1 Conditional Heteroskedasticity It is well known that stock returns exhibit time-varying volatility (French, Schwert, and Stambaugh, 1987; =-=Schwert, 1989-=-; Bollerslev, Chou, and Kroner, 1992). In this section we generalize our estimation method to take this into account. Because of our focus on maximum likelihood, a natural approach is to use the GARCH... |

612 |
Business conditions and expected returns on stocks and bonds
- Fama, French
- 1989
(Show Context)
Citation Context .... Indeed, the first equation is equivalent to the standard ordinary least squares regression that has been a focus of measuring predictability in stock returns for 30 years (Keim and Stambaugh, 1986; =-=Fama and French, 1989-=-). We have simply rearranged the parameters so that the mean excess return µr appears explicitly as a parameter. In a time-series setting in which the shocks are generally correlated with future value... |

490 | On estimating the expected return on the market: An exploratory investigation.
- Merton
- 1980
(Show Context)
Citation Context ...conomy. 1 The equity premium is usually estimated by taking the sample mean of stock returns and subtracting a measure of the riskfree rate such as the average Treasury Bill return. As is well known (=-=Merton, 1980-=-), it is difficult to estimate the mean of a stochastic process. If one is computing the sample average, a tighter estimate can be obtained only by extending the data series in time which has the disa... |

476 | The Equity Premium: It's Still a Puzzle," - Kocherlakota - 1996 |

466 | Predictive regressions.
- Stambaugh
- 1999
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Citation Context ...able 4 shows nearly identical results from setting the parameters equal to their sample means and OLS estimates. It is well known that OLS estimates of predictive coefficients can be severely biased (=-=Stambaugh, 1999-=-). Tables 3 and 4 replicate this result. For example, in the simulation in Table 3, the “true” value of the predictive coefficient β in the simulated data is 0.69, however, the median OLS value from t... |

444 | Investing for the long run when returns are predictable, - Barberis - 2000 |

396 | Household finance, - Campbell - 2006 |

365 |
Predicting returns in the stocks and the bond markets
- Keim, Stambaugh
- 1986
(Show Context)
Citation Context ...standard in the literature. Indeed, the first equation is equivalent to the standard ordinary least squares regression that has been a focus of measuring predictability in stock returns for 30 years (=-=Keim and Stambaugh, 1986-=-; Fama and French, 1989). We have simply rearranged the parameters so that the mean excess return µr appears explicitly as a parameter. In a time-series setting in which the shocks are generally corre... |

341 | Consumption and Portfolio Decisions When Expected Returns Are Time Varying. - Campbell, Viceira - 1999 |

325 | On the Predictability of Stock Returns: An Asset Allocation Perspective,” - Kandel, Stambaugh - 1996 |

247 | The Equity Premium. - Fama, French - 2002 |

246 | Consumption strikes back? Measuring long-run risk. - Hansen, Heaton, et al. - 2008 |

243 | A Random Walk Down Wall - Malkiel - 1996 |

187 | Predictable stock returns: The role of small sample bias. - Nelson, Kim - 1993 |

169 | The Dog That Did Not Bark: A Defense of Return Predictability, - Cochrane - 2005 |

166 | Consumption-based asset pricing,” - Campbell - 2003 |

163 | Predicting returns with financial ratios. - Lewellen - 2004 |

143 | On biases in tests of the expectations hypothesis of the term structure of interest rates. - Bekaert, Hodrick, et al. - 1997 |

140 |
Exactly Median-unbiased Estimation of First-order Autoregressive/ Unitroot Models.
- Andrews
- 1993
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Citation Context ...nsbergen and Koijen (2010). 5 Estimation of the mean and of the predictive coefficient are related, in that the bias in β arises from the bias in θ, which in turn arises from the need to estimate µx (=-=Andrews, 1993-=-). 5This likelihood function, however, ignores the information contained in the initial draw x0. For this reason, studies have proposed a likelihood function that incorporates the first observation (... |

127 | Equity premia as low as three percent? Evidence from analysts’ earnings forecasts for domestic and international stock markets
- Claus, Thomas
- 2001
(Show Context)
Citation Context ...enges, it is not surprising that a number of studies investigate how to estimate the equity premium using techniques other than taking the sample average. These include making use of survey evidence (=-=Claus and Thomas, 2001-=-; Graham and Harvey, 2005; Welch, 2000), as well as data on the cross section (Polk, Thompson, and Vuolteenaho, 2006). The branch of the literature most closely related to our work uses the accounting... |

112 | On predicting stock returns with nearly integrated explanantory variables. - Torous, Valkanov, et al. - 2005 |

109 | Inference in models with nearly integrated regressors. - Cavanagh, Elliott, et al. - 1995 |

76 | The equity premium and structural breaks. - Pastor, Stambaugh - 2001 |

71 | Inference in time series regression when the order of integration of a regressor is unknown. - Elliott, Stock - 1994 |

68 | The Equity Premium in Retrospect, - Mehra, Prescott - 2003 |

67 | Predictive regressions: A present-value approach, working paper, - Binsbergen, Koijen - 2008 |

64 |
E"cient tests for a unit root when the initial observation is drawn from its unconditional distribution.
- Elliott
- 1999
(Show Context)
Citation Context ...the effect of incorporating this 5Wachter and Warusawitharana (2009, 2012) use Bayesian methods rather than maximum likelihood. 5 first term (referred to as the initial condition) on unit root tests (=-=Elliott, 1999-=-; Müller and Elliott, 2003).6 We derive the values of µr, µx, β, θ, σ 2 u, σ 2 v and σuv that maximize the likelihood (3) by solving a set of first-order conditions. We give closed-form expressions f... |

63 |
Analyzing Investments Whose Histories Differ in Length,
- Stambaugh
- 1997
(Show Context)
Citation Context ...is not related to predictability.18 17This point is related to the result that longer time series can help estimate parameters determined by shorter time series, as long as the shocks are correlated (=-=Stambaugh, 1997-=-; Singleton, 2006; Lynch and Wachter, 2013). Here, the time series for the predictor is slightly longer than the time series of the return. Despite the small difference in the lengths of the data, the... |

63 | Spurious regressions in financial economics, - Ferson, Sarkissian, et al. - 2003 |

56 | Optimal inference in regression models with nearly integrated regressors. Preliminary Draft. - Jansson, Moreira - 2004 |

56 | Reconciling the return predictability evidence. - Lettau, Nieuwerburgh - 2008 |

56 | Predictive systems: Living with imperfect predictors,
- Pastor, Stambaugh
- 2008
(Show Context)
Citation Context ...on overwhelms the gains from introducing data on 21However, if the equity premium were indeed varying over time, one would expect return innovations to be negatively correlated with realized returns (=-=Pastor and Stambaugh, 2009-=-). 22Though the data generating process assumes bias-corrected estimates, MLE will still find values of β that are high relative to the values specified in the simulation. This will hurt its finite-sa... |

48 | Rational Asset Prices. - Constantinides - 2002 |

35 | Motohiro Yogo, 2006, Efficient tests of stock return predictability - Campbell |

34 |
Empirical dynamic asset pricing: Model specification and econometric assessment.
- Singleton
- 2006
(Show Context)
Citation Context ... predictability.18 17This point is related to the result that longer time series can help estimate parameters determined by shorter time series, as long as the shocks are correlated (Stambaugh, 1997; =-=Singleton, 2006-=-; Lynch and Wachter, 2013). Here, the time series for the predictor is slightly longer than the time series of the return. Despite the small difference in the lengths of the data, the structure of the... |

24 | Stocks for the Long Run: A Guide to Selecting Markets for Long Term Growth. - Siegel - 1994 |

22 | The long-run equity risk premium,”
- Graham, Harvey
- 2005
(Show Context)
Citation Context ...ing that a number of studies investigate how to estimate the equity premium using techniques other than taking the sample average. These include making use of survey evidence (Claus and Thomas, 2001; =-=Graham and Harvey, 2005-=-; Welch, 2000), as well as data on the cross section (Polk, Thompson, and Vuolteenaho, 2006). The branch of the literature most closely related to our work uses the accounting identity that links pric... |

22 | Predictability of stock returns and asset allocation under structural breaks, - Pettenuzzo, Timmermann - 2011 |

16 | Vuolteenaho (2006): “Cross-Sectional Forecasts of the Equity Premium - Polk, Thompson, et al. |

16 | Tests for Unit - Müller, Elliott - 2003 |

13 | Long-run stock returns: Participating in the real economy, - Ibbotson, Chen - 2003 |

12 |
The Effect of the First Observation in Regression Models With First-Order Autoregressive Disturbances
- Poirier
- 1978
(Show Context)
Citation Context ... function, however, ignores the information contained in the initial draw x0. For this reason, studies have proposed a likelihood function that incorporates the first observation (Box and Tiao, 1973; =-=Poirier, 1978-=-), assuming that it is a draw from the stationary distribution. In our case, the stationary distribution of x0 is normal with mean µx and variance σ 2 x = σ2 v , 1 − θ2 see Hamilton (1994). The result... |

11 | Using samples of unequal length in generalized method of moments estimation,
- Lynch, Wachter
- 2004
(Show Context)
Citation Context ...8 17This point is related to the result that longer time series can help estimate parameters determined by shorter time series, as long as the shocks are correlated (Stambaugh, 1997; Singleton, 2006; =-=Lynch and Wachter, 2013-=-). Here, the time series for the predictor is slightly longer than the time series of the return. Despite the small difference in the lengths of the data, the structure of the problem implies that the... |

9 | Perspectives on the Equity Risk Premium - Siegel - 2005 |

6 | 2013): “Market Expectations in the Cross-Section of Present Values - Kelly, Pruitt |

5 | 2007, What is the chance that the equity premium varies over time? evidence from predictive regressions, working paper - Wachter, Warusawitharana |

5 | 2009): “Predictable Returns and Asset Allocation: Should a Skeptical Investor Time the Market - Wachter, Warusawitharana |

4 | Exact maximum likelihood estimation of observation-driven econometric models, NBER technical working paper 194
- Diebold, Shuermann
- 1996
(Show Context)
Citation Context ...d thus is the GARCH analogue of the conditional maximum likelihood function (2). However, unlike in the homoskedastic case, there is no analytical expression for the unconditional distribution of x0 (=-=Diebold and Schuermann, 2000-=-). 24 For this reason, we adopt 23Applying the law of iterated expectations, we find Eu2t = E[Et−1u 2 t ] = Eσ 2 u,t. The result for σu follows under stationarity by taking the expectation of the left... |

3 | The U.S. Equity Return Premium - DeLong, Magin |

3 | Estimating the equity premium - Donaldson, Kamstra, et al. - 2010 |

1 | Missaka Warusawitharana, 2009, Predictable returns and asset allocation: Should a skeptical investor time the market - Jessica |

1 | Estimating the equity premium - Donaldson, Kamstra, et al. - 2010 |