Discussion of: Asset Prices with Fading Memory

Transcription

1 Discussion of: Asset Prices with Fading Memory Stefan Nagel and Zhengyang Xu Kent Daniel Columbia Business School & NBER 2018 Fordham Rising Stars Conference May 11, 2018

2 Introduction Summary Model Estimation The paper has two distinct parts: a simple model with some empirical work a more sophisticated Bayesian learning model. I m going to concentrate on the first part. In both models agents learn the underlying cashflow growth rate by observing realized cashflow growth rates. As a result of experiential learning, investors overreact to recent growth rates. Agents get no other information. I m going to argue that you need other shocks to explain market returns. Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 2 / 16

3 Model Estimation Basic Model - Cash flow process The cashflow C t from the endowment (market) follows GBM with constant drift µ: c t = µ + ɛ t where c t = log(c t ) and ɛ t iid N (0, σ 2 c ) However, based on Malmendier and Nagel (2016), the average agent s belief about µ, µ, follows: µ t+1 = µ t + ν ( c t+1 µ t ) MN (2016) estimate ν = 0.018/quarter for inflation data. implying that: µ t = ν(1 ν) j }{{} w j j=0 c t j Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 3 / 16

4 Model Estimation Basic Model - Cash flow process The cashflow C t from the endowment (market) follows GBM with constant drift µ: c t = µ + ɛ t where c t = log(c t ) and ɛ t iid N (0, σ 2 c ) However, based on Malmendier and Nagel (2016), the average agent s belief about µ, µ, follows: µ t+1 = µ t + ν ( c t+1 µ t ) MN (2016) estimate ν = 0.018/quarter for inflation data. implying that: µ t = ν(1 ν) j }{{} w j j=0 c t j Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 3 / 16

5 Model Estimation Basic Model - Cash flow process The cashflow C t from the endowment (market) follows GBM with constant drift µ: c t = µ + ɛ t where c t = log(c t ) and ɛ t iid N (0, σ 2 c ) However, based on Malmendier and Nagel (2016), the average agent s belief about µ, µ, follows: µ t+1 = µ t + ν ( c t+1 µ t ) MN (2016) estimate ν = 0.018/quarter for inflation data. implying that: µ t = ν(1 ν) j }{{} w j j=0 c t j Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 3 / 16

6 Weighting Function, ν = Model Estimation That is, µ t = j=0 w j c t j, where w j looks like: Weight half life is log(0.5) Lag in quarters = 38.2 quarters ( 10 years) log(1 ν) Figure I Weights implied by constant-gain learning ν 0 rationality (i.e., no fading ) Weights on quarterly past observations implied by constant-gain learning with gain = Malmendier and Nagel (2016) suggests that individuals form expectations from data they Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 4 / 16

7 Weighting Function, ν = Model Estimation That is, µ t = j=0 w j c t j, where w j looks like: Weight half life is log(0.5) Lag in quarters = 38.2 quarters ( 10 years) log(1 ν) Figure I Weights implied by constant-gain learning ν 0 rationality (i.e., no fading ) Weights on quarterly past observations implied by constant-gain learning with gain = Malmendier and Nagel (2016) suggests that individuals form expectations from data they Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 4 / 16

8 Basic Model - Pricing Model Estimation Representative agent in model sets price to equal PV of future CFs, using constant discount rate of Ẽ t r t+1 = θ + r f However, agent (mistakenly) extrapolates recent cashflow growth to infer µ. Using a Campbell and Shiller (1988) log-linearization: ( ( ) ) ρ r t+1 Ẽtr t+1 = 1 + ν ( c t+1 µ t ) 1 ρ and ( ( ) ) ρ E t r t+1 Ẽtr t+1 = 1 + ν (µ µ t ) }{{} 1 ρ r f +θ implying a negative relationship between recent cashflow growth and future abnormal returns. Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 5 / 16

9 Basic Model - Pricing Model Estimation Representative agent in model sets price to equal PV of future CFs, using constant discount rate of Ẽ t r t+1 = θ + r f However, agent (mistakenly) extrapolates recent cashflow growth to infer µ. Using a Campbell and Shiller (1988) log-linearization: ( ( ) ) ρ r t+1 Ẽtr t+1 = 1 + ν ( c t+1 µ t ) 1 ρ and ( ( ) ) ρ E t r t+1 Ẽtr t+1 = 1 + ν (µ µ t ) }{{} 1 ρ r f +θ implying a negative relationship between recent cashflow growth and future abnormal returns. Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 5 / 16

10 Basic Model - Pricing Model Estimation Representative agent in model sets price to equal PV of future CFs, using constant discount rate of Ẽ t r t+1 = θ + r f However, agent (mistakenly) extrapolates recent cashflow growth to infer µ. Using a Campbell and Shiller (1988) log-linearization: ( ( ) ) ρ r t+1 Ẽtr t+1 = 1 + ν ( c t+1 µ t ) 1 ρ and ( ( ) ) ρ E t r t+1 Ẽtr t+1 = 1 + ν (µ µ t ) }{{} 1 ρ r f +θ implying a negative relationship between recent cashflow growth and future abnormal returns. Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 5 / 16

11 Model Estimation Estimating µ t The authors don t use cashflows to estimate µ t. They instead use historical returns on the market. Effectively, µ r,t = j=0 w jr t j Reasons: 1 To start in 1926, would need consumption going back to dividends are influenced by shifts in payout policy that can distort estimates of µ constructed from dividend growth rates. 3 The authors simulate µ and µ r (under the null) and show that they are highly correlated. A concern is that price shocks will reflect all information prices/discount rates. How can we confirm the information that is causing E[r]s to change is cashflow innovations? Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 6 / 16

12 Model Estimation Estimating µ t The authors don t use cashflows to estimate µ t. They instead use historical returns on the market. Effectively, µ r,t = j=0 w jr t j Reasons: 1 To start in 1926, would need consumption going back to dividends are influenced by shifts in payout policy that can distort estimates of µ constructed from dividend growth rates. 3 The authors simulate µ and µ r (under the null) and show that they are highly correlated. A concern is that price shocks will reflect all information prices/discount rates. How can we confirm the information that is causing E[r]s to change is cashflow innovations? Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 6 / 16

13 DP decomposition I ll show a set of regressions. Data is from Shiller, over the sample. The dependent variable is always the annual real returns on the S&P 500 (R t+1 ) The forecasting variables I ll use are: 1 dp: log of preceding year s dividend (D t), scaled by this year s price (P t) 2 dpl: dp, lagged 10 years. 3 d: change in the log dividend over the last 10 years. 4 p: change in the log price over the last 10 years. 5 S: Baker and Wurgler (2000) equity share Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 7 / 16

14 An Information Decomposition OLS Regression Results Dep. Variable: R R-squared: Model: OLS Adj. R-squared: No. Observations: 67 AIC: Df Residuals: 65 BIC: Df Model: 1 Covariance Type: HAC coef std err z P> z [ ] const dp Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 8 / 16

15 An Information Decomposition OLS Regression Results Dep. Variable: R R-squared: Model: OLS Adj. R-squared: Method: Least Squares F-statistic: Date: Thu, 10 May 2018 Prob (F-statistic): Time: 09:57:50 Log-Likelihood: No. Observations: 67 AIC: Df Residuals: 65 BIC: Df Model: 1 Covariance Type: HAC coef std err z P> z [ ] const Delta-d the point estimate on the d coefficient is positive, not negative. However, it is statistically insignificant. Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 9 / 16

16 dp decomposition Consider the identity (like that in Daniel and Titman (2006)): dp t = dp t 10 + d t 10,t p t 10,t In words, if the market has a high dp today, there are three possibilities: 1 It was high dp 10 years ago. 2 d was positive. 3 p was negative. At least post-wwii, dp forecasts the market. Which of the three components forecasts the market? Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 10 / 16

17 An Information Decomposition OLS Regression Results Dep. Variable: R R-squared: Model: OLS Adj. R-squared: Method: Least Squares F-statistic: Date: Thu, 10 May 2018 Prob (F-statistic): Time: 09:57:50 Log-Likelihood: No. Observations: 67 AIC: Df Residuals: 63 BIC: Df Model: 3 Covariance Type: HAC coef std err z P> z [ ] const dpl Delta-d Delta-p Note that the coefficient on d is again positive, and now statistically significant. Suggests that d is not just noise w.r.t returns. Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 11 / 16

18 dp decomposition We can also break the market return into the part explained by cashflow changes, and the component that isn t (ɛ). p t 10,t = a dp t 10 + b d t 10,t + ɛ t 10,t ɛ t 10,t is the price change over the last 10 years that can t be explained by the growth rate of dividends. The regression R 2 adj = 51.4% t(b = 0) = 6.7. Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 12 / 16

19 An Information Decomposition OLS Regression Results Dep. Variable: Delta-p R-squared: Model: OLS Adj. R-squared: Method: Least Squares F-statistic: Date: Thu, 10 May 2018 Prob (F-statistic): 4.94e-10 Time: 10:45:20 Log-Likelihood: No. Observations: 67 AIC: Df Residuals: 64 BIC: Df Model: 2 Covariance Type: HAC coef std err z P> z [ ] const dpl Delta-d R 2 adj. = 51.4% ρ 0.7, Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 13 / 16

20 An Information Decomposition OLS Regression Results Dep. Variable: R R-squared: Model: OLS Adj. R-squared: Method: Least Squares F-statistic: Date: Thu, 10 May 2018 Prob (F-statistic): Time: 11:21:31 Log-Likelihood: No. Observations: 67 AIC: Df Residuals: 63 BIC: Df Model: 3 Covariance Type: HAC coef std err z P> z [ ] const dpl Delta-d resid The coefficient on resid is exactly the same as in the previous regression. The coefficients on dp t 10 and d are what they would be were resid not included in the regression. Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 14 / 16

21 An Information Decomposition OLS Regression Results Dep. Variable: R R-squared: Model: OLS Adj. R-squared: Method: Least Squares F-statistic: Date: Thu, 10 May 2018 Prob (F-statistic): Time: 09:57:51 Log-Likelihood: No. Observations: 59 AIC: Df Residuals: 54 BIC: Df Model: 4 Covariance Type: HAC coef std err z P> z [ ] const dpl Delta-d Delta-p S Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 15 / 16

22 References I References Baker, Malcolm, and Jeffrey Wurgler, 2000, The equity share in new issues and aggregate stock returns, Journal of Finance 55, Campbell, John Y., and Robert J. Shiller, 1988, The dividend-price ratio and expectations of future dividends and discount factors, Review of Financial Studies 1, Daniel, Kent D., and Sheridan Titman, 2006, Market reactions to tangible and intangible information, Journal of Finance 61, Kent Daniel Columbia Business School Nagel&Xu Fading Memory 2018 Fordham-RSC 16 / 16