Three essays on estimation uncertainty Open Access
Allena, Rohit (Spring 2021)
Published
Abstract
The dissertation consists of three essays on estimation uncertainty, showing why and how considering estimation uncertainty is important in answering three fundamental asset pricing and market microstructure questions.
The first essay (Confident Risk Premia: Economics and Econometrics of Machine Learning Uncertainties) quantifies ex-ante parameter uncertainty of expected stock return predictions from neural networks by deriving their standard errors or confidence intervals. Considering ex-ante standard errors, the paper provides 1) improved trading strategies known as Confident-high-low portfolios (in contrast to traditional high-low strategies), and 2) ex-post out-of-sample (OOS) inferences by generalizing Diebold-Mariano t-tests to statistically compare OOS returns and Sharpe ratios of any two trading strategies.
The second essay (Comparing Asset Pricing Models with Non-traded Factors and Principal Components) develops a Bayesian methodology to compare asset pricing models containing non-traded factors and principal components. Existing comparison procedures are inadequate when models include such factors due to estimation uncertainties in mimicking portfolios and return covariances. Furthermore, regressions of test assets on such factors are interdependent, rendering comparisons with recently proposed priors sensitive to subsets of the test assets. Thus, the paper derives novel, non-informative priors that deliver invariant inferences. The paper finds that macroeconomic factor models dominate several recent benchmark models with traded factors and principal components.
The third essay (True Liquidity and Fundamental Prices: US Tick Size Pilot) is joint work with Tarun Chordia. This paper develops a big-data methodology to estimate fundamental prices and true liquidity measures, explicitly considering the rounding specification (estimation uncertainty) due to the minimum tick size. Evaluation of the tick size pilot (TSP), which increased the tick size for some randomly chosen stocks, requires estimating the impact of rounding. True liquidity measures capture the TSP-driven decreased inventory costs of market-makers, whereas traditional measures without the rounding adjustment cannot. We find that the TSP increases market-maker profits, but does not improve liquidity and price efficiency. This result contrasts with existing empirical studies but is consistent with recent theoretical studies that account for rounding.
Table of Contents
Confident Risk Premia: Economics and Econometrics of Machine Learning Uncertainties
Contents
1 Introduction 2
A Contribution.............................................................................................................. 9
B Paper Overview........................................................................................................... 10
2 Risk Premium Predictions and Predictive Standard Errors10
A Basics of model-based risk premium predictions...................................................... 11
B Risk Premium Predictions, Standard Errors and Investment Portfolios................. 13
3 NN-based Risk Premia and Standard Errors16
A Neural Networks........................................................................................................ 17
B Parameter Estimation, Regularization, and Dropout.............................................. 19
C Standard Errors of Risk Premium Predictions based on Neural Networks.............. 23
D Dropout Neural Networks and Bayesian Interpretation....................................... 25
E Frequentist Justification for Standard Errors......................................................... 30
4 Ex-ante Estimation Uncertainty and Ex-post OOS Inferences31
A Out-of-Sample Comparisons with theDiebold and Mariano(2002) Tests.............. 31
B Violation of Covariance Stationarity: Empirical Evidence...................................... 33
C Bootstrap Tests for Out-of-Sample Comparisons................................................... 34
D Performance of the Methodology: Monte Carlo Evidence........................................ 37
5 Empirical Results37
A Data, Definitions, and Replication Study.................................................................. 37
B Ex-ante Confidence and Ex-post Out-of-Sample-R2.................................................. 40
C Portfolio Construction............................................................................................... 42
D Economic Gains from Confident-HL Portfolios..................................................... 44
E Reassessing NN-3 and Lewellen Model Comparisons Using Bootstrap Tests......... 47
F Time-Series Variation in Ex-ante Standard Errors................................................... 50
G Cross-sectional Variation in Ex-ante Confidence...................................................... 51
6 Conclusions 52
A Appendix: Proofs53
1 Proof of Proposition-1:............................................................................................... 53
2 Proof of Proposition-2............................................................................................... 55
3 Proof of Proposition-3.............................................................................................. 56
B Appendix: Simulations and Testing theDiebold and Mariano(2002) Assump- tion 58
1 Validity of Standard Errors: Monte Carlo Evidence.................................................. 58
2 Tests of Covariance Stationarity................................................................................. 59
3 Performance of this paper’s OOS Comparison Method: Monte Carlo Evidence....... 60
C Internet Appendix83
C1 Internet Appendix: Simulation Results and Robustness Checks.............................. 83
C2 Internet Appendix: Simulation Details...................................................................... 86
C3 Why Confidence-levels are Better Measures of Precision Relative to Inverse Standard Errors 88
Comparing Asset Pricing Models with Non-traded Factors and Principal Components
Contents 7 Introduction 91 8 Asset Pricing Models with Non-Traded Factors99 9 General Test of a K-Factor Model1019a Prior specification..................................................................................................... 103
9b Bayes Factor............................................................................................................. 108
10 Comparing Asset Pricing Models with Non-Traded Factors112
10a Priors for Comparing Asset Pricing Models............................................................. 115
11 Comparing Models with Principal Components121
11a Priors for Comparing Asset Pricing Models with Principal Components................ 123
12 Marginal Likelihoods and Out-of-Sample Predictions124 13 Simulation Evidence126 14 Empirical Comparison of Prominent Asset Pricing Models130
14a Models with Traded vs Non-Traded Factors............................................................. 131
14b Are these Non-Traded Factors Spurious?................................................................... 134
14c Traded versus Non-Traded versus Principal Components......................................... 135
15 Conclusion 138 16 Appendix 140
16a Bayes Factors for Absolute Tests.............................................................................. 140
16b Invariance to the Choice of Test Assets and Scaling of Mimicking Portfolio Weights.143 16c Bayes Factors for Model Comparisons............................................................................. 144
16d Model Comparisons with Principal Components....................................................... 147
True Liquidity and Fundamental Prices: US Tick Size Pilot
Contents 17 Introduction 162 18 Model 170 19 Methodology: Variational Bayesian Inference17519a Family of Densities for Approximation...................................................................... 178
19b Estimating the Optimal Density Function................................................................ 180
19c Derivations of Updates............................................................................................... 181
20Performance of the Proposed Methodology182
20a Accuracy of the methodology: Monte-Carlo Evidence............................................... 182
21 Data 184 22 Quoted Spreads, True Spreads and Market-Makers’ Profits184
22a True Spreads and Its Components........................................................................... 190
23 Effective Spreads192 24 Price Discovery193
24a Proportion of Price Discovery through Trading and New Information...................... 193
24b Speed of Price Discovery............................................................................................ 196
25 Conclusion 198 A Appendix 199
B Internet Appendix218
Confident Risk Premia: Economics and Econometrics of Machine Learning Uncertainties
List of Tables
1 Calibration of the Confidence Intervals: Monte Carlo Evidence.............................. 58
2 Violation ofDiebold and Mariano(2002) conditions : Non-Stationarities due to Estimation Uncertainty............................................................................................ 59
3 Long-short Portfolios’ Performance on Subsamples with Different Levels of Ex-ante Confidence................................................................................................................ 70
4 Performance of Confident and Low-Confident Long-Short Portfolios: All Stocks..... 71
5 Statistical Comparison of Long-Short Portfolios: All Stocks.................................... 72
6 Performance of Confident and Low-Confident Long-Short Portfolios: Non-Microcap Stocks 73
7 Statistical Comparison of Long-Short Portfolios: Non-Microcap Stocks..................... 74
8 Transaction Costs and Higher-Moment Adjusted Performance of Confident-HL Port- folios 75
9 Statistical Comparison of Long-Short Portfolios: NN-3 versusLewellen(2015).......... 78
10 Aggregate Standard Errors of NN-3-based Risk Premia............................................ 80
11 Cross-sectional Characteristics of Confidence-sorted Deciles...................................... 81
12 Characteristics Distributions of Stocks in the Decile Containing the Most Confident Risk Premium Predictions................................................................................................. 82
A Performance of High-Low and Confident High-Low Portfolios: Simulation Evidence.83 B Performance of Various Long-Short Portfolios: Inverse Standard Errors as Precision
Measures................................................................................................................... 84
C Comparing Confident-HL Portfolios with Double-sorted HL Portfolios...................... 85
Comparing Asset Pricing Models with Non-traded Factors and Principal Components
List of Tables
13 Performance of the Proposed Methodology: Simulation Evidence............................. 153
14 Performance of the Proposed Methodology: Simulation Evidence............................. 154
15 Traded vs Non-Traded: Model Probabilities with 52 Test Assets ofKozak, Nagel, and Santosh(2019)........................................................................................................... 155
16 Traded vs Non-Traded: Model Probabilities with 52 Anomalies + 10 Industry portfolios156
17 Correlations of Macroeconomic Factors with the Cross-Section of Stock Returns..... 157
18 Traded vs Non-Traded : Model Probabilities with Spurious Factors......................... 158
19 Traded vs Non-Traded : Model Probabilities with Spurious Factors......................... 159
20 Traded versus Non-Traded versus Principal Components: Posterior Probabilities with 52 Anomalies.................................................................................................................. 160
21 Traded vs Non-Traded vs Principal Components: Posterior Probabilities with 52 Anomalies + 10 Industry portfolios.............................................................................................. 161
True Liquidity and Fundamental Prices: US Tick Size Pilot
List of Tables31 Performance of the Methodology: Monte Carlo Evidence......................................... 209
32 Quoted Spreads, True Spreads and Market-Maker Profits...................................... 210
33 Aggregated Dollar Value of Quoted Spreads, True Spreads and Market-Maker Profits211 34 Realized Market-Maker Profits................................................................................. 212
35 Components of Bid-Ask Spread................................................................................. 213
36 Inventory Risks of Aggregate Market-Makers.......................................................... 214
37 Effective Spreads with Fundamental Prices, mid-Quotes, and Weighted Mid-quotes.215
38 Proportion of Price Discovery through Market Orders, Limit Orders and New Infor- mation 216
39 Speed of Price Discovery with Fundamental Prices.................................................. 217
40 Market-maker Profits and Depths.............................................................................. 218
Confident Risk Premia: Economics and Econometrics of Machine Learning Uncertainties
List of Figures
1 Example of a 1-layer Neural Network........................................................................ 17
2 NN-1 with Dropout Regularization........................................................................... 21
3 Test Sizes of OOS Comparison Methodologies........................................................ 60
4 Power Curves of OOS Comparison Methodologies.................................................. 61
5 Out-of-Sample (OOS) Performance of Equal-weighted Deciles Based on NN-3 Pre- dictions....................................................................................................................... 66
6 Out-of-Sample (OOS) Performance of Value-weighted Deciles Based on NN-3 Pre- dictions....................................................................................................................... 66
7 Ex-ante Confidence and Ex-post OOS-R2: NN-3-based Predictions and Standard Errors 68
8 Ex-ante Confidence and Ex-post OOS-R2: Lewellen-based Predictions and Standard Errors......................................................................................................................... 69
9 Comparing predictive performance of NN-3 with the benchmarkLewellen(2015) model76
10 Comparing predictive performance of NN-3 with the benchmarkLewellen(2015) model77
11 Time-Series Variation in Standard Errors of NN-based Risk Premia...................... 79
True Liquidity and Fundamental Prices: US Tick Size Pilot
List of Figures
12 ELBO (y-axis) vs Number of Epochs (x-axis).......................................................... 208
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