Introduction Model Conclusion Discussion

The Dog That Did Not Bark: A Defense ofReturn Predictability

John CochraneOctober 23, 2006

Presented By: Michelle ZemelSeptember 25, 2007

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Motivation

Are stock returns predictable?

What is the (statistically) correct way to answer this question?

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Motivation

Are stock returns predictable?

What is the (statistically) correct way to answer this question?

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Motivation

Regression b t R2(%) σ(%)

Rt+1 = a + b(Dt/Pt) + εt+1 3.39 2.28 5.8 4.9Rt+1 − R f

t = a + b(Dt/Pt) + εt+1 3.83 2.61 7.4 5.6Dt+1/Dt = a + b(Dt/Pt) + εt+1 0.07 0.06 0.0001 0.001

rt+1 = ar + br (dt − pt) + εrt+1 0.097 1.92 4.0 4.0

4dt+1 = ad + bd(dt − pt) + εdpt+1 0.008 0.18 0.00 0.003

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Main Findings

The relationship between the dividend yield, dividend growthand return variables can be used to form more powerfulstatistical tests to evaluate return predictability.

The dividend price ratio does predict returns, while it does notpredict dividend growth

Long-horizon return regressions provide stronger evidence forreturn predictability than regressions on the one-period return.

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Main Findings

The relationship between the dividend yield, dividend growthand return variables can be used to form more powerfulstatistical tests to evaluate return predictability.

The dividend price ratio does predict returns, while it does notpredict dividend growth

Long-horizon return regressions provide stronger evidence forreturn predictability than regressions on the one-period return.

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Main Findings

The relationship between the dividend yield, dividend growthand return variables can be used to form more powerfulstatistical tests to evaluate return predictability.

The dividend price ratio does predict returns, while it does notpredict dividend growth

Long-horizon return regressions provide stronger evidence forreturn predictability than regressions on the one-period return.

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Predictability - What are we testing?

1st order VAR system

rt+1 = ar + br (dt − pt) + εrt+1 (1)

4dt+1 = ad + bd(dt − pt) + εdt+1 (2)

dt+1 − pt+1 = adp + φ(dt − pt) + εdpt+1 (3)

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Estimation

Estimates Error Correlation and S.E.

b̂, φ̂ σ(b̂) t-stat r 4d dp

r 0.097 0.050 1.92 .196 .66 -.704d 0.008 0.044 .182 .66 .140 .075dp .941 0.047 20.02 -.70 .075 .153

Table: Forecasting Regressions. The data was taken from the CRSPdatabase, the sample period covers 1927-2004. The length of the periodis 1 year. Coefficients were estimated using OLS. The standard errorsinclude a GMM correction for heteroskedasticity.

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Dividend yields,Dividend Growth and Returns

By the definition of a return ⇒

Rt+1 =Pt+1 + Dt+1

Pt=

(1 + Pt+1

Dt+1)

PtDt

Dt+1

Dt

Log linearizating as in Cambell and Shiller (1988) ⇒

rt+1 = log [1 + e(pt+1−dt+1)] +4dt+1 − (pt − dt)

rt+1 ≈ log [1+P̄/D̄]+P̄/D̄

1 + P̄/D̄(pt+1−dt+1−(p̄−d̄))+4dt+1−(pt−dt)

rt+1 ≈ k + ρ(pt+1 − dt+1) +4dt+1 − (pt − dt)

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Dividend yields,Dividend Growth and Returns

By the definition of a return ⇒

Rt+1 =Pt+1 + Dt+1

Pt=

(1 + Pt+1

Dt+1)

PtDt

Dt+1

Dt

Log linearizating as in Cambell and Shiller (1988) ⇒

rt+1 = log [1 + e(pt+1−dt+1)] +4dt+1 − (pt − dt)

rt+1 ≈ log [1+P̄/D̄]+P̄/D̄

1 + P̄/D̄(pt+1−dt+1−(p̄−d̄))+4dt+1−(pt−dt)

rt+1 ≈ k + ρ(pt+1 − dt+1) +4dt+1 − (pt − dt)

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Dividend yields,Dividend Growth and Returns

By the definition of a return ⇒

Rt+1 =Pt+1 + Dt+1

Pt=

(1 + Pt+1

Dt+1)

PtDt

Dt+1

Dt

Log linearizating as in Cambell and Shiller (1988) ⇒

rt+1 = log [1 + e(pt+1−dt+1)] +4dt+1 − (pt − dt)

rt+1 ≈ log [1+P̄/D̄]+P̄/D̄

1 + P̄/D̄(pt+1−dt+1−(p̄−d̄))+4dt+1−(pt−dt)

rt+1 ≈ k + ρ(pt+1 − dt+1) +4dt+1 − (pt − dt)

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Dividend yields,Dividend Growth and Returns

By the definition of a return ⇒

Rt+1 =Pt+1 + Dt+1

Pt=

(1 + Pt+1

Dt+1)

PtDt

Dt+1

Dt

Log linearizating as in Cambell and Shiller (1988) ⇒

rt+1 = log [1 + e(pt+1−dt+1)] +4dt+1 − (pt − dt)

rt+1 ≈ log [1+P̄/D̄]+P̄/D̄

1 + P̄/D̄(pt+1−dt+1−(p̄−d̄))+4dt+1−(pt−dt)

rt+1 ≈ k + ρ(pt+1 − dt+1) +4dt+1 − (pt − dt)

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Dividend yields,Dividend Growth and Returns

By the definition of a return ⇒

Rt+1 =Pt+1 + Dt+1

Pt=

(1 + Pt+1

Dt+1)

PtDt

Dt+1

Dt

Log linearizating as in Cambell and Shiller (1988) ⇒

rt+1 = log [1 + e(pt+1−dt+1)] +4dt+1 − (pt − dt)

rt+1 ≈ log [1+P̄/D̄]+P̄/D̄

1 + P̄/D̄(pt+1−dt+1−(p̄−d̄))+4dt+1−(pt−dt)

rt+1 ≈ k + ρ(pt+1 − dt+1) +4dt+1 − (pt − dt)

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Dividend yields,Dividend Growth and Returns

By the definition of a return ⇒

Rt+1 =Pt+1 + Dt+1

Pt=

(1 + Pt+1

Dt+1)

PtDt

Dt+1

Dt

Log linearizating as in Cambell and Shiller (1988) ⇒

rt+1 = log [1 + e(pt+1−dt+1)] +4dt+1 − (pt − dt)

rt+1 ≈ log [1+P̄/D̄]+P̄/D̄

1 + P̄/D̄(pt+1−dt+1−(p̄−d̄))+4dt+1−(pt−dt)

rt+1 ≈ k + ρ(pt+1 − dt+1) +4dt+1 − (pt − dt)

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Important Identities

rt+1 ≈ k + ρ(pt+1 − dt+1) +4dt+1 − (pt − dt) (4)

Iterate Forward ....

dt − pt = Et

∞∑j=1

ρj−1rt+j − Et

∞∑j=1

ρj−14dt+j (5)

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Important Identities

rt+1 ≈ k + ρ(pt+1 − dt+1) +4dt+1 − (pt − dt) (4)

Iterate Forward ....

dt − pt = Et

∞∑j=1

ρj−1rt+j − Et

∞∑j=1

ρj−14dt+j (5)

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Important Identities

rt+1 ≈ k + ρ(pt+1 − dt+1) +4dt+1 − (pt − dt) (4)

Iterate Forward ....

dt − pt = Et

∞∑j=1

ρj−1rt+j − Et

∞∑j=1

ρj−14dt+j (5)

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Relationship between the regression coefficients

Use (4) to link the regressions (1)-(3):

rt+1 = k − ρ(dt+1 − pt+1) +4dt+1 + (dt − pt)

= k−ρ∗(adp +φ(dt−pt)+εdpt+1)+ad +bd(dt−pt)+εdt+1+(dt−pt)

rt+1 = (k−ρadp +ad)+(−ρφ+bd +1)(dt −pt)+(−ρεdpt+1 + εdt+1)

= ar + br (dt − pt) + εrt+1

⇒

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Relationship between the regression coefficients

Use (4) to link the regressions (1)-(3):

rt+1 = k − ρ(dt+1 − pt+1) +4dt+1 + (dt − pt)

= k−ρ∗(adp +φ(dt−pt)+εdpt+1)+ad +bd(dt−pt)+εdt+1+(dt−pt)

rt+1 = (k−ρadp +ad)+(−ρφ+bd +1)(dt −pt)+(−ρεdpt+1 + εdt+1)

= ar + br (dt − pt) + εrt+1

⇒

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Relationship between the regression coefficients

Use (4) to link the regressions (1)-(3):

rt+1 = k − ρ(dt+1 − pt+1) +4dt+1 + (dt − pt)

= k−ρ∗(adp +φ(dt−pt)+εdpt+1)+ad +bd(dt−pt)+εdt+1+(dt−pt)

rt+1 = (k−ρadp +ad)+(−ρφ+bd +1)(dt −pt)+(−ρεdpt+1 + εdt+1)

= ar + br (dt − pt) + εrt+1

⇒

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Relationship between the regression coefficients

Use (4) to link the regressions (1)-(3):

rt+1 = k − ρ(dt+1 − pt+1) +4dt+1 + (dt − pt)

= k−ρ∗(adp +φ(dt−pt)+εdpt+1)+ad +bd(dt−pt)+εdt+1+(dt−pt)

rt+1 = (k−ρadp +ad)+(−ρφ+bd +1)(dt −pt)+(−ρεdpt+1 + εdt+1)

= ar + br (dt − pt) + εrt+1

⇒

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Relationship between the regression coefficients

Use (4) to link the regressions (1)-(3):

rt+1 = k − ρ(dt+1 − pt+1) +4dt+1 + (dt − pt)

= k−ρ∗(adp +φ(dt−pt)+εdpt+1)+ad +bd(dt−pt)+εdt+1+(dt−pt)

rt+1 = (k−ρadp +ad)+(−ρφ+bd +1)(dt −pt)+(−ρεdpt+1 + εdt+1)

= ar + br (dt − pt) + εrt+1

⇒

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Relationship between the regression coefficients

Use (4) to link the regressions (1)-(3):

rt+1 = k − ρ(dt+1 − pt+1) +4dt+1 + (dt − pt)

= k−ρ∗(adp +φ(dt−pt)+εdpt+1)+ad +bd(dt−pt)+εdt+1+(dt−pt)

rt+1 = (k−ρadp +ad)+(−ρφ+bd +1)(dt −pt)+(−ρεdpt+1 + εdt+1)

= ar + br (dt − pt) + εrt+1

⇒

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Relationship between the regression coefficients

br = 1− ρφ + bd (6)

and

εrt+1 = εdt+1 − ρεdpt+1 (7)

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Forming the Null Hypothesis

Any two coefficients in the system br , bd , φ uniquelydetermine the third coefficient

We would like to say something about the return predictabilitycoefficient br

Let’s ask about br using what we know about φ

br = 0 and φ = 0.941 ⇒ bd = br + ρφ− 1 = −0.1

The evaluation of the statement br = 0 is equivalent in thisframework to an evalutaion of the null hypothesis H0 : br = 0and bd = −0.1

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Forming the Null Hypothesis

Any two coefficients in the system br , bd , φ uniquelydetermine the third coefficient

We would like to say something about the return predictabilitycoefficient br

Let’s ask about br using what we know about φ

br = 0 and φ = 0.941 ⇒ bd = br + ρφ− 1 = −0.1

The evaluation of the statement br = 0 is equivalent in thisframework to an evalutaion of the null hypothesis H0 : br = 0and bd = −0.1

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Forming the Null Hypothesis

Any two coefficients in the system br , bd , φ uniquelydetermine the third coefficient

We would like to say something about the return predictabilitycoefficient br

Let’s ask about br using what we know about φ

br = 0 and φ = 0.941 ⇒ bd = br + ρφ− 1 = −0.1

The evaluation of the statement br = 0 is equivalent in thisframework to an evalutaion of the null hypothesis H0 : br = 0and bd = −0.1

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Forming the Null Hypothesis

Any two coefficients in the system br , bd , φ uniquelydetermine the third coefficient

We would like to say something about the return predictabilitycoefficient br

Let’s ask about br using what we know about φ

br = 0 and φ = 0.941 ⇒ bd = br + ρφ− 1 = −0.1

The evaluation of the statement br = 0 is equivalent in thisframework to an evalutaion of the null hypothesis H0 : br = 0and bd = −0.1

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Forming the Null Hypothesis

Any two coefficients in the system br , bd , φ uniquelydetermine the third coefficient

We would like to say something about the return predictabilitycoefficient br

Let’s ask about br using what we know about φ

br = 0 and φ = 0.941 ⇒ bd = br + ρφ− 1 = −0.1

The evaluation of the statement br = 0 is equivalent in thisframework to an evalutaion of the null hypothesis H0 : br = 0and bd = −0.1

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Forming the Null Hypothesis

Any two coefficients in the system br , bd , φ uniquelydetermine the third coefficient

We would like to say something about the return predictabilitycoefficient br

Let’s ask about br using what we know about φ

br = 0 and φ = 0.941 ⇒ bd = br + ρφ− 1 = −0.1

The evaluation of the statement br = 0 is equivalent in thisframework to an evalutaion of the null hypothesis H0 : br = 0and bd = −0.1

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Testing the Null Hypothesis

We would like to test the hypothesisH0 : br = 0 and bd = −0.1

What is the chance of seeing the sample coefficients if ourdata came from the null distribution?

The hypothesis will be tested by simulation

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Testing the Null Hypothesis

We would like to test the hypothesisH0 : br = 0 and bd = −0.1

What is the chance of seeing the sample coefficients if ourdata came from the null distribution?

The hypothesis will be tested by simulation

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Testing the Null Hypothesis

We would like to test the hypothesisH0 : br = 0 and bd = −0.1

What is the chance of seeing the sample coefficients if ourdata came from the null distribution?

The hypothesis will be tested by simulation

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Testing the Null Hypothesis

Simulation Procedure:

Simulate dividend growth and dividend yield and let the returnfollow from the identity

To create one simulated data set:

Step 1: Draw initial observation of d0 − p0. For φ < 1, drawd0 − p0 from the unconditional distributiond0− p0 ∼ N[0, σ2(εdp/(1−φ2)]. If φ ≥ 1 start at d0− p0 = 0.

Step 2:Sample εdp and εd using the estimated covariancematrix of the VAR system

Step 3: Using dt − pt and the random errors calculatedividend yield and dividend growth

Step 4: Using the dividend yield and dividend growthcalculate the return

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Testing the Null Hypothesis

Simulation Procedure:

Simulate dividend growth and dividend yield and let the returnfollow from the identity

To create one simulated data set:

Step 1: Draw initial observation of d0 − p0. For φ < 1, drawd0 − p0 from the unconditional distributiond0− p0 ∼ N[0, σ2(εdp/(1−φ2)]. If φ ≥ 1 start at d0− p0 = 0.

Step 2:Sample εdp and εd using the estimated covariancematrix of the VAR system

Step 3: Using dt − pt and the random errors calculatedividend yield and dividend growth

Step 4: Using the dividend yield and dividend growthcalculate the return

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Testing the Null Hypothesis

Simulation Procedure:

Simulate dividend growth and dividend yield and let the returnfollow from the identity

To create one simulated data set:

Step 1: Draw initial observation of d0 − p0. For φ < 1, drawd0 − p0 from the unconditional distributiond0− p0 ∼ N[0, σ2(εdp/(1−φ2)]. If φ ≥ 1 start at d0− p0 = 0.

Step 2:Sample εdp and εd using the estimated covariancematrix of the VAR system

Step 3: Using dt − pt and the random errors calculatedividend yield and dividend growth

Step 4: Using the dividend yield and dividend growthcalculate the return

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Testing the Null Hypothesis

Simulation Procedure:

Simulate dividend growth and dividend yield and let the returnfollow from the identity

To create one simulated data set:

Step 1: Draw initial observation of d0 − p0. For φ < 1, drawd0 − p0 from the unconditional distributiond0− p0 ∼ N[0, σ2(εdp/(1−φ2)]. If φ ≥ 1 start at d0− p0 = 0.

Step 2:Sample εdp and εd using the estimated covariancematrix of the VAR system

Step 3: Using dt − pt and the random errors calculatedividend yield and dividend growth

Step 4: Using the dividend yield and dividend growthcalculate the return

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Testing the Null Hypothesis

Simulation Procedure:

Simulate dividend growth and dividend yield and let the returnfollow from the identity

To create one simulated data set:

Step 1: Draw initial observation of d0 − p0. For φ < 1, drawd0 − p0 from the unconditional distributiond0− p0 ∼ N[0, σ2(εdp/(1−φ2)]. If φ ≥ 1 start at d0− p0 = 0.

Step 2:Sample εdp and εd using the estimated covariancematrix of the VAR system

Step 3: Using dt − pt and the random errors calculatedividend yield and dividend growth

Step 4: Using the dividend yield and dividend growthcalculate the return

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Testing the Null Hypothesis

Simulation Procedure:

Simulate dividend growth and dividend yield and let the returnfollow from the identity

To create one simulated data set:

Step 1: Draw initial observation of d0 − p0. For φ < 1, drawd0 − p0 from the unconditional distributiond0− p0 ∼ N[0, σ2(εdp/(1−φ2)]. If φ ≥ 1 start at d0− p0 = 0.

Step 2:Sample εdp and εd using the estimated covariancematrix of the VAR system

Step 3: Using dt − pt and the random errors calculatedividend yield and dividend growth

Step 4: Using the dividend yield and dividend growthcalculate the return

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Testing the Null Hypothesis

Simulation Procedure:

Simulate dividend growth and dividend yield and let the returnfollow from the identity

To create one simulated data set:

Step 1: Draw initial observation of d0 − p0. For φ < 1, drawd0 − p0 from the unconditional distributiond0− p0 ∼ N[0, σ2(εdp/(1−φ2)]. If φ ≥ 1 start at d0− p0 = 0.

Step 2:Sample εdp and εd using the estimated covariancematrix of the VAR system

Step 3: Using dt − pt and the random errors calculatedividend yield and dividend growth

Step 4: Using the dividend yield and dividend growthcalculate the return

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Testing the Null Hypothesis

Simulate 50,000 data sets

Run regressions (1)-(3) on each of the simulated samples,calculate coefficient estimates and t statistics

Using the sample distribution of coefficient estimates and tstatistics, what is the probability of observing bd , br estimatepairs and t statistics more extreme than our original sampleestimates?

br tr bd tdReal Returns 22.3% 10.3% 1.77% 1.67%

Excess Returns 17.4% 6.32% 1.11% 0.87%

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Testing the Null Hypothesis

Simulate 50,000 data sets

Run regressions (1)-(3) on each of the simulated samples,calculate coefficient estimates and t statistics

Using the sample distribution of coefficient estimates and tstatistics, what is the probability of observing bd , br estimatepairs and t statistics more extreme than our original sampleestimates?

br tr bd tdReal Returns 22.3% 10.3% 1.77% 1.67%

Excess Returns 17.4% 6.32% 1.11% 0.87%

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Testing the Null Hypothesis

Simulate 50,000 data sets

Run regressions (1)-(3) on each of the simulated samples,calculate coefficient estimates and t statistics

Using the sample distribution of coefficient estimates and tstatistics, what is the probability of observing bd , br estimatepairs and t statistics more extreme than our original sampleestimates?

br tr bd tdReal Returns 22.3% 10.3% 1.77% 1.67%

Excess Returns 17.4% 6.32% 1.11% 0.87%

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Testing the Null Hypothesis

Simulate 50,000 data sets

Run regressions (1)-(3) on each of the simulated samples,calculate coefficient estimates and t statistics

Using the sample distribution of coefficient estimates and tstatistics, what is the probability of observing bd , br estimatepairs and t statistics more extreme than our original sampleestimates?

br tr bd tdReal Returns 22.3% 10.3% 1.77% 1.67%

Excess Returns 17.4% 6.32% 1.11% 0.87%

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Testing the Null Hypothesis

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coeffients

Recall identity (7) : br = 1− ρφ + bd

Dividing through by 1− ρφ yields

br

1− ρφ− bd

1− ρφ= 1

blrr − blr

d = 1

blrr has two interpretations:

1)Regression coefficient on long run returns on dividend yields

2)Fraction of variance of dividend yields that can beattributed to time-varying expected returns and time-varyingexpected dividend growth

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coeffients

Recall identity (7) : br = 1− ρφ + bd

Dividing through by 1− ρφ yields

br

1− ρφ− bd

1− ρφ= 1

blrr − blr

d = 1

blrr has two interpretations:

1)Regression coefficient on long run returns on dividend yields

2)Fraction of variance of dividend yields that can beattributed to time-varying expected returns and time-varyingexpected dividend growth

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coeffients

Recall identity (7) : br = 1− ρφ + bd

Dividing through by 1− ρφ yields

br

1− ρφ− bd

1− ρφ= 1

blrr − blr

d = 1

blrr has two interpretations:

1)Regression coefficient on long run returns on dividend yields

2)Fraction of variance of dividend yields that can beattributed to time-varying expected returns and time-varyingexpected dividend growth

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coeffients

Recall identity (7) : br = 1− ρφ + bd

Dividing through by 1− ρφ yields

br

1− ρφ− bd

1− ρφ= 1

blrr − blr

d = 1

blrr has two interpretations:

1)Regression coefficient on long run returns on dividend yields

2)Fraction of variance of dividend yields that can beattributed to time-varying expected returns and time-varyingexpected dividend growth

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coeffients

Recall identity (7) : br = 1− ρφ + bd

Dividing through by 1− ρφ yields

br

1− ρφ− bd

1− ρφ= 1

blrr − blr

d = 1

blrr has two interpretations:

1)Regression coefficient on long run returns on dividend yields

2)Fraction of variance of dividend yields that can beattributed to time-varying expected returns and time-varyingexpected dividend growth

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coeffients

Recall identity (7) : br = 1− ρφ + bd

Dividing through by 1− ρφ yields

br

1− ρφ− bd

1− ρφ= 1

blrr − blr

d = 1

blrr has two interpretations:

1)Regression coefficient on long run returns on dividend yields

2)Fraction of variance of dividend yields that can beattributed to time-varying expected returns and time-varyingexpected dividend growth

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coeffients

Recall identity (7) : br = 1− ρφ + bd

Dividing through by 1− ρφ yields

br

1− ρφ− bd

1− ρφ= 1

blrr − blr

d = 1

blrr has two interpretations:

1)Regression coefficient on long run returns on dividend yields

2)Fraction of variance of dividend yields that can beattributed to time-varying expected returns and time-varyingexpected dividend growth

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coefficients

dt − pt = Et

∞∑j=1

ρj−1rt+j − Et

∞∑j=1

ρj−14dt+j (5)

After some manipulation ...

var(dt−pt) = cov(∞∑j=1

ρj−1rt+j , dt−pt)−cov(∞∑j=1

ρj−14dt+j , dt−pt)

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coefficients

dt − pt = Et

∞∑j=1

ρj−1rt+j − Et

∞∑j=1

ρj−14dt+j (5)

After some manipulation ...

var(dt−pt) = cov(∞∑j=1

ρj−1rt+j , dt−pt)−cov(∞∑j=1

ρj−14dt+j , dt−pt)

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coefficients

dt − pt = Et

∞∑j=1

ρj−1rt+j − Et

∞∑j=1

ρj−14dt+j (5)

After some manipulation ...

var(dt−pt) = cov(∞∑j=1

ρj−1rt+j , dt−pt)−cov(∞∑j=1

ρj−14dt+j , dt−pt)

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coefficients

dt − pt = Et

∞∑j=1

ρj−1rt+j − Et

∞∑j=1

ρj−14dt+j (5)

After some manipulation ...

var(dt−pt) = cov(∞∑j=1

ρj−1rt+j , dt−pt)−cov(∞∑j=1

ρj−14dt+j , dt−pt)

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coefficients

var(dt−pt) = cov(∞∑j=1

ρj−1rt+j , dt−pt)−cov(∞∑j=1

ρj−14dt+j , dt−pt)

β(∞∑j=1

ρj−1rt+j , dt − pt)− β(∞∑j=1

ρj−14dt+j , dt − pt) = 1

β(∞∑j=1

ρj−1rt+j , dt−pt) =∞∑j=1

ρj−1β(rt+j , dt−pt) =∞∑j=1

ρj−1φj−1br

=br

1− ρφ= blr

r

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coefficients

var(dt−pt) = cov(∞∑j=1

ρj−1rt+j , dt−pt)−cov(∞∑j=1

ρj−14dt+j , dt−pt)

β(∞∑j=1

ρj−1rt+j , dt − pt)− β(∞∑j=1

ρj−14dt+j , dt − pt) = 1

β(∞∑j=1

ρj−1rt+j , dt−pt) =∞∑j=1

ρj−1β(rt+j , dt−pt) =∞∑j=1

ρj−1φj−1br

=br

1− ρφ= blr

r

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coefficients

var(dt−pt) = cov(∞∑j=1

ρj−1rt+j , dt−pt)−cov(∞∑j=1

ρj−14dt+j , dt−pt)

β(∞∑j=1

ρj−1rt+j , dt − pt)− β(∞∑j=1

ρj−14dt+j , dt − pt) = 1

β(∞∑j=1

ρj−1rt+j , dt−pt) =∞∑j=1

ρj−1β(rt+j , dt−pt) =∞∑j=1

ρj−1φj−1br

=br

1− ρφ= blr

r

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coefficients

var(dt−pt) = cov(∞∑j=1

ρj−1rt+j , dt−pt)−cov(∞∑j=1

ρj−14dt+j , dt−pt)

β(∞∑j=1

ρj−1rt+j , dt − pt)− β(∞∑j=1

ρj−14dt+j , dt − pt) = 1

β(∞∑j=1

ρj−1rt+j , dt−pt) =∞∑j=1

ρj−1β(rt+j , dt−pt) =∞∑j=1

ρj−1φj−1br

=br

1− ρφ= blr

r

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coefficients

var(dt−pt) = cov(∞∑j=1

ρj−1rt+j , dt−pt)−cov(∞∑j=1

ρj−14dt+j , dt−pt)

β(∞∑j=1

ρj−1rt+j , dt − pt)− β(∞∑j=1

ρj−14dt+j , dt − pt) = 1

β(∞∑j=1

ρj−1rt+j , dt−pt) =∞∑j=1

ρj−1β(rt+j , dt−pt) =∞∑j=1

ρj−1φj−1br

=br

1− ρφ= blr

r

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run estimates and tests

Variable b̂lr s.e. t % p-value

r 1.09 0.44 2.48 1.39-1.834d 0.09 .44 2.48 1.39-1.83

Excess r 1.23 0.47 2.62 0.47 - 0.69

Table: The long run coefficients are calculated from the one periodcoefficients. The standard errors are calculated by the delta method

using the standard errors for b̂lrr , b̂lr

d , φ̂. The t statistic for 4d is the

statistic for the hypothesis b̂lrd = −1. Percent probability values were

generated by Monte Carlo under the φ = 0.941 null.

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coefficients

Using either of the long run coefficients, it is easy to rejectthe null of no return predictability

The tests on blrr and blr

d give the same results, and there is noneed to choose between return and dividend growth tests

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coefficients

Using either of the long run coefficients, it is easy to rejectthe null of no return predictability

The tests on blrr and blr

d give the same results, and there is noneed to choose between return and dividend growth tests

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Long-run Coefficients

Using either of the long run coefficients, it is easy to rejectthe null of no return predictability

The tests on blrr and blr

d give the same results, and there is noneed to choose between return and dividend growth tests

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power, Φ and negative correlation

Whyare the dividend growth and long run coefficient tests more powerful?

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power, Φ and negative correlation

b̂r and φ̂ are highly negatively correlated

High blrr requires high b̂r and φ̂, which is hard under the null

Negative correlation of coefficient estimates comes from thecorrelation structure of the error terms

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power, Φ and negative correlation

b̂r and φ̂ are highly negatively correlated

High blrr requires high b̂r and φ̂, which is hard under the null

Negative correlation of coefficient estimates comes from thecorrelation structure of the error terms

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power, Φ and negative correlation

b̂r and φ̂ are highly negatively correlated

High blrr requires high b̂r and φ̂, which is hard under the null

Negative correlation of coefficient estimates comes from thecorrelation structure of the error terms

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power, Φ and negative correlation

b̂r and φ̂ are highly negatively correlated

High blrr requires high b̂r and φ̂, which is hard under the null

Negative correlation of coefficient estimates comes from thecorrelation structure of the error terms

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power, Φ and negative correlation

From identity (7), the error terms in this system are related asfollows:

εrt+1 = εdt+1 − ρεdpt+1

cov(εrt+1, εdpt+1) = cov(εdp

t+1, εdt+1)− ρσ2(εdp

t+1)

So,cov(εdp

t+1, εdt+1) = 0⇒

cov(εrt+1, εdpt+1) = −ρσ2(εdp

t+1)

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power, Φ and negative correlation

From identity (7), the error terms in this system are related asfollows:

εrt+1 = εdt+1 − ρεdpt+1

cov(εrt+1, εdpt+1) = cov(εdp

t+1, εdt+1)− ρσ2(εdp

t+1)

So,cov(εdp

t+1, εdt+1) = 0⇒

cov(εrt+1, εdpt+1) = −ρσ2(εdp

t+1)

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power, Φ and negative correlation

From identity (7), the error terms in this system are related asfollows:

εrt+1 = εdt+1 − ρεdpt+1

cov(εrt+1, εdpt+1) = cov(εdp

t+1, εdt+1)− ρσ2(εdp

t+1)

So,cov(εdp

t+1, εdt+1) = 0⇒

cov(εrt+1, εdpt+1) = −ρσ2(εdp

t+1)

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

It all comes down to φ

The value of φ can be bounded by theoretical arguments

φ >1

ρ⇒ infinite dividend yield

dt−pt = Et

∞∑j=1

ρj−1rt+j−Et

∞∑j=1

ρj−14dt+j+ limk→∞

ρkEt(pt+k−dt+k)

ρkφk(pt+k − dt+k) will explode if φ > 1ρ

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

It all comes down to φ

The value of φ can be bounded by theoretical arguments

φ >1

ρ⇒ infinite dividend yield

dt−pt = Et

∞∑j=1

ρj−1rt+j−Et

∞∑j=1

ρj−14dt+j+ limk→∞

ρkEt(pt+k−dt+k)

ρkφk(pt+k − dt+k) will explode if φ > 1ρ

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

It all comes down to φ

The value of φ can be bounded by theoretical arguments

φ >1

ρ⇒ infinite dividend yield

dt−pt = Et

∞∑j=1

ρj−1rt+j−Et

∞∑j=1

ρj−14dt+j+ limk→∞

ρkEt(pt+k−dt+k)

ρkφk(pt+k − dt+k) will explode if φ > 1ρ

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

It all comes down to φ

The value of φ can be bounded by theoretical arguments

φ >1

ρ⇒ infinite dividend yield

dt−pt = Et

∞∑j=1

ρj−1rt+j−Et

∞∑j=1

ρj−14dt+j+ limk→∞

ρkEt(pt+k−dt+k)

ρkφk(pt+k − dt+k) will explode if φ > 1ρ

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

It all comes down to φ

φ =1

ρ⇒ Rational Bubble

Et

∞∑j=1

ρj−14dt+j =∞∑j=1

ρj−1φj−1bd(pt − dt)

For these terms to converge,

br = 0 and bd = 0

Recalling that,

dt − pt = Et

∞∑j=1

ρj−1rt+j − Et

∞∑j=1

ρj−14dt+j + limk→∞

ρkEt(pt − dt)

The dividend yield in this case will depend only on the last term

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

It all comes down to φ

φ =1

ρ⇒ Rational Bubble

Et

∞∑j=1

ρj−14dt+j =∞∑j=1

ρj−1φj−1bd(pt − dt)

For these terms to converge,

br = 0 and bd = 0

Recalling that,

dt − pt = Et

∞∑j=1

ρj−1rt+j − Et

∞∑j=1

ρj−14dt+j + limk→∞

ρkEt(pt − dt)

The dividend yield in this case will depend only on the last term

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

It all comes down to φ

φ =1

ρ⇒ Rational Bubble

Et

∞∑j=1

ρj−14dt+j =∞∑j=1

ρj−1φj−1bd(pt − dt)

For these terms to converge,

br = 0 and bd = 0

Recalling that,

dt − pt = Et

∞∑j=1

ρj−1rt+j − Et

∞∑j=1

ρj−14dt+j + limk→∞

ρkEt(pt − dt)

The dividend yield in this case will depend only on the last term

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

It all comes down to φ

φ =1

ρ⇒ Rational Bubble

Et

∞∑j=1

ρj−14dt+j =∞∑j=1

ρj−1φj−1bd(pt − dt)

For these terms to converge,

br = 0 and bd = 0

Recalling that,

dt − pt = Et

∞∑j=1

ρj−1rt+j − Et

∞∑j=1

ρj−14dt+j + limk→∞

ρkEt(pt − dt)

The dividend yield in this case will depend only on the last term

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

It all comes down to φ

φ =1

ρ⇒ Rational Bubble

Et

∞∑j=1

ρj−14dt+j =∞∑j=1

ρj−1φj−1bd(pt − dt)

For these terms to converge,

br = 0 and bd = 0

Recalling that,

dt − pt = Et

∞∑j=1

ρj−1rt+j − Et

∞∑j=1

ρj−14dt+j + limk→∞

ρkEt(pt − dt)

The dividend yield in this case will depend only on the last termJohn Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

It all comes down to φ

φ = 1⇒ dividend yield follows a random walk

The dividend yield does pass standard unit root tests, butstatistical evidence is not compelling

A random walk in dividend yields generates far more variationthan we have seen over the history of stock trading

If the dividend yield is not stationnary, than either returns ordividend growth are not stationnary

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

It all comes down to φ

φ = 1⇒ dividend yield follows a random walk

The dividend yield does pass standard unit root tests, butstatistical evidence is not compelling

A random walk in dividend yields generates far more variationthan we have seen over the history of stock trading

If the dividend yield is not stationnary, than either returns ordividend growth are not stationnary

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

It all comes down to φ

φ = 1⇒ dividend yield follows a random walk

The dividend yield does pass standard unit root tests, butstatistical evidence is not compelling

A random walk in dividend yields generates far more variationthan we have seen over the history of stock trading

If the dividend yield is not stationnary, than either returns ordividend growth are not stationnary

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

It all comes down to φ

φ = 1⇒ dividend yield follows a random walk

The dividend yield does pass standard unit root tests, butstatistical evidence is not compelling

A random walk in dividend yields generates far more variationthan we have seen over the history of stock trading

If the dividend yield is not stationnary, than either returns ordividend growth are not stationnary

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

It all comes down to φ

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power in long-run regression coefficients

How do we reconcile these findings with the literature on longhorizon returns?

Three main differences between these and typical long horizonreturns:

Infinite vs. Finite long run returns

∞∑j=1

ρj−1rt+j on dt − pt

Implied vs. Directly Calculated

Weighted vs. Unweighted Returns

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power in long-run regression coefficients

How do we reconcile these findings with the literature on longhorizon returns?

Three main differences between these and typical long horizonreturns:

Infinite vs. Finite long run returns

∞∑j=1

ρj−1rt+j on dt − pt

Implied vs. Directly Calculated

Weighted vs. Unweighted Returns

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power in long-run regression coefficients

How do we reconcile these findings with the literature on longhorizon returns?

Three main differences between these and typical long horizonreturns:

Infinite vs. Finite long run returns

∞∑j=1

ρj−1rt+j on dt − pt

Implied vs. Directly Calculated

Weighted vs. Unweighted Returns

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power in long-run regression coefficients

How do we reconcile these findings with the literature on longhorizon returns?

Three main differences between these and typical long horizonreturns:

Infinite vs. Finite long run returns

∞∑j=1

ρj−1rt+j on dt − pt

Implied vs. Directly Calculated

Weighted vs. Unweighted Returns

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power in long-run regression coefficients

How do we reconcile these findings with the literature on longhorizon returns?

Three main differences between these and typical long horizonreturns:

Infinite vs. Finite long run returns

∞∑j=1

ρj−1rt+j on dt − pt

Implied vs. Directly Calculated

Weighted vs. Unweighted Returns

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power in long-run Regression Coeficients

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power in long-run Regression Coeficients

The key factor is horizon

A short finite horizon does not capture all the power of φ

To see why, consider a short finite return

bkr = (1 + φρ + φ2ρ2 + ... + φkρk)

Longer horizons give more weight to φ, generating more power

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power in long-run Regression Coeficients

The key factor is horizon

A short finite horizon does not capture all the power of φ

To see why, consider a short finite return

bkr = (1 + φρ + φ2ρ2 + ... + φkρk)

Longer horizons give more weight to φ, generating more power

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power in long-run Regression Coeficients

The key factor is horizon

A short finite horizon does not capture all the power of φ

To see why, consider a short finite return

bkr = (1 + φρ + φ2ρ2 + ... + φkρk)

Longer horizons give more weight to φ, generating more power

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power in long-run Regression Coeficients

The key factor is horizon

A short finite horizon does not capture all the power of φ

To see why, consider a short finite return

bkr = (1 + φρ + φ2ρ2 + ... + φkρk)

Longer horizons give more weight to φ, generating more power

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Power in long-run Regression Coeficients

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Out of Sample R2

Goyal and Welch (2005) show that the dividend yield does notdo well forecasting out of sample

The Goyal and Welch statistic is calculated as follows:

Run a regression from time 1 to time T, estimate the returnpredictability coefficient

Use the coefficients estimated on time 1 to T to forecast thereturn in period T + 1

Compute the sample mean return from time 1 to T

Use the sample mean calculated on time 1 to T to forecastthe return in period T + 1

Compare the MSE of the two strategies

Goyal and Welch find that the out of sample MSE is higherfor the return forecast than for the sample mean

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Out of Sample R2

Goyal and Welch (2005) show that the dividend yield does notdo well forecasting out of sample

The Goyal and Welch statistic is calculated as follows:

Run a regression from time 1 to time T, estimate the returnpredictability coefficient

Use the coefficients estimated on time 1 to T to forecast thereturn in period T + 1

Compute the sample mean return from time 1 to T

Use the sample mean calculated on time 1 to T to forecastthe return in period T + 1

Compare the MSE of the two strategies

Goyal and Welch find that the out of sample MSE is higherfor the return forecast than for the sample mean

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Out of Sample R2

Goyal and Welch (2005) show that the dividend yield does notdo well forecasting out of sample

The Goyal and Welch statistic is calculated as follows:

Run a regression from time 1 to time T, estimate the returnpredictability coefficient

Use the coefficients estimated on time 1 to T to forecast thereturn in period T + 1

Compute the sample mean return from time 1 to T

Use the sample mean calculated on time 1 to T to forecastthe return in period T + 1

Compare the MSE of the two strategies

Goyal and Welch find that the out of sample MSE is higherfor the return forecast than for the sample mean

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Out of Sample R2

Goyal and Welch (2005) show that the dividend yield does notdo well forecasting out of sample

The Goyal and Welch statistic is calculated as follows:

Run a regression from time 1 to time T, estimate the returnpredictability coefficient

Use the coefficients estimated on time 1 to T to forecast thereturn in period T + 1

Compute the sample mean return from time 1 to T

Use the sample mean calculated on time 1 to T to forecastthe return in period T + 1

Compare the MSE of the two strategies

Goyal and Welch find that the out of sample MSE is higherfor the return forecast than for the sample mean

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Out of Sample R2

Goyal and Welch (2005) show that the dividend yield does notdo well forecasting out of sample

The Goyal and Welch statistic is calculated as follows:

Run a regression from time 1 to time T, estimate the returnpredictability coefficient

Use the coefficients estimated on time 1 to T to forecast thereturn in period T + 1

Compute the sample mean return from time 1 to T

Use the sample mean calculated on time 1 to T to forecastthe return in period T + 1

Compare the MSE of the two strategies

Goyal and Welch find that the out of sample MSE is higherfor the return forecast than for the sample mean

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Out of Sample R2

Goyal and Welch (2005) show that the dividend yield does notdo well forecasting out of sample

The Goyal and Welch statistic is calculated as follows:

Run a regression from time 1 to time T, estimate the returnpredictability coefficient

Use the coefficients estimated on time 1 to T to forecast thereturn in period T + 1

Compute the sample mean return from time 1 to T

Use the sample mean calculated on time 1 to T to forecastthe return in period T + 1

Compare the MSE of the two strategies

Goyal and Welch find that the out of sample MSE is higherfor the return forecast than for the sample mean

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Out of Sample R2

Goyal and Welch (2005) show that the dividend yield does notdo well forecasting out of sample

The Goyal and Welch statistic is calculated as follows:

Run a regression from time 1 to time T, estimate the returnpredictability coefficient

Use the coefficients estimated on time 1 to T to forecast thereturn in period T + 1

Compute the sample mean return from time 1 to T

Use the sample mean calculated on time 1 to T to forecastthe return in period T + 1

Compare the MSE of the two strategies

Goyal and Welch find that the out of sample MSE is higherfor the return forecast than for the sample mean

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Out of Sample R2

Goyal and Welch (2005) show that the dividend yield does notdo well forecasting out of sample

The Goyal and Welch statistic is calculated as follows:

Run a regression from time 1 to time T, estimate the returnpredictability coefficient

Use the coefficients estimated on time 1 to T to forecast thereturn in period T + 1

Compute the sample mean return from time 1 to T

Use the sample mean calculated on time 1 to T to forecastthe return in period T + 1

Compare the MSE of the two strategies

Goyal and Welch find that the out of sample MSE is higherfor the return forecast than for the sample mean

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Out of Sample R2

What is the chance of seeing such forecasting results if returnsreally are predictable?

Simulate the data with br = 1− ρφ and bd = 0

Run the out of sample forecasts and calculate the Goyal andWelch statistic

30 - 40% of the draws show even worse results than thesample results

Poor R2 cannot be used to reject the null hypothesis thatreturns are predictable

However, these regressions are not likely to be useful informing real-time forecasts because of the difficulty inestimating accurate coefficients in our short sample

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Out of Sample R2

What is the chance of seeing such forecasting results if returnsreally are predictable?

Simulate the data with br = 1− ρφ and bd = 0

Run the out of sample forecasts and calculate the Goyal andWelch statistic

30 - 40% of the draws show even worse results than thesample results

Poor R2 cannot be used to reject the null hypothesis thatreturns are predictable

However, these regressions are not likely to be useful informing real-time forecasts because of the difficulty inestimating accurate coefficients in our short sample

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Out of Sample R2

What is the chance of seeing such forecasting results if returnsreally are predictable?

Simulate the data with br = 1− ρφ and bd = 0

Run the out of sample forecasts and calculate the Goyal andWelch statistic

30 - 40% of the draws show even worse results than thesample results

Poor R2 cannot be used to reject the null hypothesis thatreturns are predictable

However, these regressions are not likely to be useful informing real-time forecasts because of the difficulty inestimating accurate coefficients in our short sample

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Out of Sample R2

What is the chance of seeing such forecasting results if returnsreally are predictable?

Simulate the data with br = 1− ρφ and bd = 0

Run the out of sample forecasts and calculate the Goyal andWelch statistic

30 - 40% of the draws show even worse results than thesample results

Poor R2 cannot be used to reject the null hypothesis thatreturns are predictable

However, these regressions are not likely to be useful informing real-time forecasts because of the difficulty inestimating accurate coefficients in our short sample

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Out of Sample R2

What is the chance of seeing such forecasting results if returnsreally are predictable?

Simulate the data with br = 1− ρφ and bd = 0

Run the out of sample forecasts and calculate the Goyal andWelch statistic

30 - 40% of the draws show even worse results than thesample results

Poor R2 cannot be used to reject the null hypothesis thatreturns are predictable

However, these regressions are not likely to be useful informing real-time forecasts because of the difficulty inestimating accurate coefficients in our short sample

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Out of Sample R2

What is the chance of seeing such forecasting results if returnsreally are predictable?

Simulate the data with br = 1− ρφ and bd = 0

Run the out of sample forecasts and calculate the Goyal andWelch statistic

30 - 40% of the draws show even worse results than thesample results

Poor R2 cannot be used to reject the null hypothesis thatreturns are predictable

However, these regressions are not likely to be useful informing real-time forecasts because of the difficulty inestimating accurate coefficients in our short sample

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Conclusion

The absence of dividend growth predictability provides strongevidence that returns are predictable

There is even stronger evidence for predictability when longhorizon returns are considered

The long run coefficient tests capture the three variables inone number and provide a powerful test

This testing framework addresses the issue of how to evaulatethe question of return predictability; it does not suggest thatthis is the best model for forecasting

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Conclusion

The absence of dividend growth predictability provides strongevidence that returns are predictable

There is even stronger evidence for predictability when longhorizon returns are considered

The long run coefficient tests capture the three variables inone number and provide a powerful test

This testing framework addresses the issue of how to evaulatethe question of return predictability; it does not suggest thatthis is the best model for forecasting

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Conclusion

The absence of dividend growth predictability provides strongevidence that returns are predictable

There is even stronger evidence for predictability when longhorizon returns are considered

The long run coefficient tests capture the three variables inone number and provide a powerful test

This testing framework addresses the issue of how to evaulatethe question of return predictability; it does not suggest thatthis is the best model for forecasting

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Conclusion

The absence of dividend growth predictability provides strongevidence that returns are predictable

There is even stronger evidence for predictability when longhorizon returns are considered

The long run coefficient tests capture the three variables inone number and provide a powerful test

This testing framework addresses the issue of how to evaulatethe question of return predictability; it does not suggest thatthis is the best model for forecasting

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Discussion

Approximation Error:

Formulation of Null Hypothesis is based on the approximatelinear relationship between dividend yields, dividend growthand returns

If the approximation does not hold then the implication thatbr = 0⇒ bd = c breaks down

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Discussion

Approximation Error:

Formulation of Null Hypothesis is based on the approximatelinear relationship between dividend yields, dividend growthand returns

If the approximation does not hold then the implication thatbr = 0⇒ bd = c breaks down

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Discussion

Approximation Error:

Formulation of Null Hypothesis is based on the approximatelinear relationship between dividend yields, dividend growthand returns

If the approximation does not hold then the implication thatbr = 0⇒ bd = c breaks down

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Discussion

Approximation Error:

Formulation of Null Hypothesis is based on the approximatelinear relationship between dividend yields, dividend growthand returns

If the approximation does not hold then the implication thatbr = 0⇒ bd = c breaks down

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Discussion

Evaluation of Approximation Error

Using the data of the study, I created a series of returnapproximations using identity (4).

I regressed the actual returns on the approximate returns toevaluate the accuracy of the approximation

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Discussion

Evaluation of Approximation Error

Using the data of the study, I created a series of returnapproximations using identity (4).

I regressed the actual returns on the approximate returns toevaluate the accuracy of the approximation

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Discussion

Evaluation of Approximation Error

Using the data of the study, I created a series of returnapproximations using identity (4).

I regressed the actual returns on the approximate returns toevaluate the accuracy of the approximation

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Discussion

Correlation is .99.John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Discussion

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Discussion

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Discussion

The role of φ and stationnarity

Results are sensitive to the value of φ and the estimate of φmay be biased

Cochrane claims that the stationnarity of dividend yieldsimplies predictability in either dividend growth or returns

However, if the dividend yield is not stationnary, thisrelationship breaks down

If for example, φ = 1 then the stationnarity argument breaksdown

The implications are the you cannot estimate the covariancematrix from which you generate the parameters for thesimulation

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Discussion

The role of φ and stationnarity

Results are sensitive to the value of φ and the estimate of φmay be biased

Cochrane claims that the stationnarity of dividend yieldsimplies predictability in either dividend growth or returns

However, if the dividend yield is not stationnary, thisrelationship breaks down

If for example, φ = 1 then the stationnarity argument breaksdown

The implications are the you cannot estimate the covariancematrix from which you generate the parameters for thesimulation

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Discussion

The role of φ and stationnarity

Results are sensitive to the value of φ and the estimate of φmay be biased

Cochrane claims that the stationnarity of dividend yieldsimplies predictability in either dividend growth or returns

However, if the dividend yield is not stationnary, thisrelationship breaks down

If for example, φ = 1 then the stationnarity argument breaksdown

The implications are the you cannot estimate the covariancematrix from which you generate the parameters for thesimulation

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Discussion

The role of φ and stationnarity

Results are sensitive to the value of φ and the estimate of φmay be biased

Cochrane claims that the stationnarity of dividend yieldsimplies predictability in either dividend growth or returns

However, if the dividend yield is not stationnary, thisrelationship breaks down

If for example, φ = 1 then the stationnarity argument breaksdown

The implications are the you cannot estimate the covariancematrix from which you generate the parameters for thesimulation

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Discussion

The role of φ and stationnarity

Results are sensitive to the value of φ and the estimate of φmay be biased

Cochrane claims that the stationnarity of dividend yieldsimplies predictability in either dividend growth or returns

However, if the dividend yield is not stationnary, thisrelationship breaks down

If for example, φ = 1 then the stationnarity argument breaksdown

The implications are the you cannot estimate the covariancematrix from which you generate the parameters for thesimulation

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability

Introduction Model Conclusion Discussion

Discussion

The role of φ and stationnarity

Results are sensitive to the value of φ and the estimate of φmay be biased

Cochrane claims that the stationnarity of dividend yieldsimplies predictability in either dividend growth or returns

However, if the dividend yield is not stationnary, thisrelationship breaks down

If for example, φ = 1 then the stationnarity argument breaksdown

The implications are the you cannot estimate the covariancematrix from which you generate the parameters for thesimulation

John Cochrane October 23, 2006

The Dog That Did Not Bark: A Defense of Return Predictability