Post on 30-Mar-2023
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Applied Mathematics & Econometrics Honors Thesis
ARE THE PARAMETERS IN FUNDAMENTAL FACTOR MODELS TIME-INVARIANT?
Joseph Mickel
May 2012
Thesis Advisor
Dr. Russ Wermers
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1. Introduction
Mathematical models are at the heart of investment decisions in quantitative investment
management. Well known mutual funds and hedge funds throughout the United States often employ
such quantitative methodologies, with notable examples including the Vanguard Quantitative Equity
Group, the Fidelity Quantitative Equity Group, and Blackrock Global Funds. Mathematical models can
enhance a fundโs performance by facilitating diversification, optimizing trade flows to reduce
transaction costs, and even identifying investment opportunities arising from equity market anomalies.
In this thesis, I examine the temporal stability of parameters in a specific kind of quantitative model, the
Fundamental Factor Model.
The Fundamental Factor Model is a stock screening and selection tool used by investment firms.
It uses observable and measurable firm characteristics and attributes, the factor exposures, to estimate
factor premia, the marginal stock returns associated with each additional unit of factor exposure. In the
model, factor premia are empirically determined, whereas the factor exposures are exogenously
determined (Connor, 1995). Therefore, a quantitative fund manager using a Fundamental Factor Model
obtains historical equity data and decomposes stock returns into a linear combination of factor
exposures and factor premia. With the model parameters estimated, the manager then predicts future
stock returns from currently available information on the associated factor exposures. These predicted
returns are then used to generate a long-short equity portfolio with the intent to outperform a
benchmark.
The number of funds using quantitative investment management has grown over the years at a
faster pace than qualitative funds. Further, as of 2009, the assets-under-management of quantitative
funds was approximately 25% more than the total amount invested in pure qualitative strategies
(Chincarini, 2010). Hence, a large amount of wealth is made or lost based upon the validity of
assumptions and analysis used to generate predictions of asset returns. For the Fundamental Factor
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Model, a necessary assumption is that the estimated parameters, the factor premia, are time-invariant.
Should these estimated parameters actually vary with time, significant forecasting errors may occur as a
result of model misspecification. Therefore, the existence and frequency of structural breaks is the
primary concern in this analysis.
This paper uses a nested cross-sectional Fundamental Factor Model within a larger pooled OLS
factor model estimated over subsets of twenty years of data. The data consist of corporate accounting
and stock performance metrics contained in the CRSPsift database on US equities listed on US indices.
After constructing factor exposures and a return variable, the model is tested via Wald and Sup-Wald
tests on exogenously determined subsets of the data. All tests in this paper reject the null hypothesis of
parameter stability at the 1% significance level, indicating that structural breaks not only exist, but are
pervasive in the model on this dataset.
2. Review of Literature
As of writing this thesis, I have not been able to find a paper on structural breaks in
Fundamental Factor Models. While some papers acknowledge that factors seem to unexpectedly
change during some estimation periods (Jegadeesh et al., 2004), there is no financial literature that tests
for significant parameter changes over time. However, there is a large body of research involving the
significance of different factor exposures which are commonly used in the Fundamental Factor Model.
Therefore, the selection of factors in this analysis is entirely based upon this established body of
research.
For momentum-based factors, Chan et al. (1999) suggest that earnings momentum related to
analyst recommendations is a unique signal in financial markets that drives cross sectional returns. The
paper examines twenty years of returns from portfolios constructed via price momentum and earnings
momentum factors. The authors constructed zero-investment portfolios based upon individual factors
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and weighted combinations of the factors. They perform in-sample and out-of-sample back-tests using
the model parameters and observe high portfolio returns in each test. In a similar paper by Jegadeesh
et al. (2004), the predictive power of the level and change in consensus recommendations is compared
against twelve previously established factors. The authors conclude that the change in consensus
recommendations is a unique signal which supplements the other factors in predicting cross sectional
returns. The authors demonstrate the effectiveness of this signal through a series of regressions. In
both papers, the authors establish a set of factors as unique signals by comparison to previously studied
factors from the financial literature.
Finally, Campbell, Lo, and Mackinlay (1999, ch. 6) extensively cover factor models in their book.
They discuss factor identification and the issues that arise when determining the source of deviation
from exact factor pricing. They assert a common assumption that omitted variables are the cause of
deviations whenever such issues arise in practice; but, they also state that with finite sample sizes,
additional factors can always be added to reduce residual error. However, the addition of more factors
implicitly assumes that the deviations arise as a result of omitted factors and not another source.
Consequently, this paper employs a comprehensive set of factors from many factor classes, several of
which were discussed in Chan et al. (1999) and Jegadeesh et al. (2004), including: momentum, size,
growth rate, value, industry level, financial risk, operating efficiency, and market liquidity.
3. Models and Tests
a. Models:
For k distinct factors, M distinct industries containing all firms in the sample space, and T distinct
quarters, we have the following two types of factor models:
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The Fundamental Factor Model in pooled OLS, predicting 2 quarters ahead:
๐น๐+๐ = ๏ฟฝ๏ฟฝ๐น๐,๐๐ฐ๐๐ซ๐
๐ด
๐=๐
๐ปโ๐
๐=๐
+ ๏ฟฝ๐ท๐ โ ๐ฟ๐,๐
๐
๐=๐
+ ๐๐+๐
The Fundamental Factor Model using T-2 cross-sections individually, predicting 2 quarters ahead:
๐น๐+๐ = ๏ฟฝ๏ฟฝ๐น๐,๐๐ฐ๐๐ซ๐
๐ด
๐=๐
๐ปโ๐
๐=๐
+ ๏ฟฝ๐ท๐,๐ โ ๐ฟ๐,๐
๐
๐=๐
+ ๐๐+๐
โข ๐ฟ๐,๐ denotes the factor exposure of factor i at quarter t. These factors are observable firm level
characteristics.
โข ๐๐ in the pooled model denotes the factor premium to the factor ๐ฟ๐,๐. It is the return associated
with a unit of exposure to the associated factor, ๐ฟ๐,๐.
โข ๐๐,๐ in the cross sectional model denotes the factor premium to factor i at time t.
โข ๐น๐+๐ denotes the conditional expectation of return two quarters ahead of the observation date.
โข ๐๐+๐ denotes the residual at time t+2.
โข ๐น๐,๐๐ฐ๐๐ซ๐ denotes the Industry-Quarter indicator term in both the cross-sectional and the pooled OLS
model. This allows for variance in return between industries during a given quarter and within a
single industry across time. Hence, I allow for factor premia associated with industry exposures to
vary with time. ๐น๐,๐ is the estimated return to industry j at time t. ๐ฐ๐ is the indicator or characteristic
function for industry j. ๐ซ๐ is the indicator or characteristic function for time t.
The Fundamental Factor Model in pooled OLS with a nested cross-section (used in testing):
๐น๐+๐ = ๏ฟฝ๏ฟฝ๐น๐,๐๐ฐ๐๐ซ๐
๐ด
๐=๐
๐ปโ๐
๐=๐
+ ๏ฟฝ๏ฟฝ๐ท๐ โ ๐ฟ๐,๐๏ฟฝ๐
๐=๐
+ ๏ฟฝ๏ฟฝ๐ท๐,๐โฒ โ ๐ฟ๐,๐โฒ๏ฟฝ๐
๐=๐
+ ๐๐+๐
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โข The inference for all parameters except ๐ท๐,๐โฒ is the same as previously mentioned. The first set of
bracketed terms corresponds to the pooled OLS model. The second bracketed set of terms
corresponds to a single cross section during the specified quarter: tโ.
โข For ๐ท๐,๐โฒ, the new inference is the difference between the factor premia over the tested time
periods and the factor premia at quarter tโ factor premia. Hence, no structural break during quarter
tโ is equivalent to the statement โ๐ โ [๐,๐],๐ท๐,๐โฒ = 0. This inference occurs because I do not
subtract ๐ฟ๐,๐โฒ from ๐ฟ๐,๐ in the pooled model before estimating the factor premia.
โข Note 1: All standard errors in the pooled model are clustered by firm. This is done in order to
mitigate the effect of intra-firm return autocorrelations.
โข Note 2: In the following section on hypothesis tests, ๐ท๐,๐โฒ will be denoted as ๐ท๐,๐ for simplicity.
b. Hypothesis Tests:
The first structural break tests are performed over the entire dataset. Additional tests are
conducted within exogenously determined or pre-specified subsets of the dataset. Analytically, we have
the following hypotheses:
The null hypothesis is that all parameters are time invariant over the observation period. H0: โ๐ โ [๐,๐ป โ ๐], โ๐ โ [๐,๐], ๐ท๐ข,๐ญ = ๐ Define the sets for the hypothesis tests as follow: ๐ป๐โฒ ๐๐๐๐๐ก๐ ๐กโ๐ ๐๐ข๐๐๐ก๐๐ ๐กโ๐๐ก ๐ ๐๐ข๐ ๐๐๐๐ ๐ ๐๐ฆ๐๐๐ ๐๐๐ค๐๐ก๐ข๐๐ ๐๐๐๐๐๐ . ๐ป๐โฒโฒ ๐๐๐๐๐ก๐ ๐กโ๐ ๐๐ข๐๐๐ก๐๐ ๐กโ๐๐ก ๐ ๐๐ข๐ ๐๐๐๐ ๐ ๐๐ฆ๐๐๐ ๐๐๐ค๐๐ก๐ข๐๐ ๐๐๐๐ .
Define a recession interval: ๐๐ = ๏ฟฝ๐ป๐โฒ ,๐ป๐โฒโฒ๏ฟฝ. Denote the recession periods: (โ ๐๐)๐ . De๏ฟฝine the growth periods: ๐ช = [๐,๐ป โ ๐] โ (โ ๐๐)๐ . Thus, ๐ช defines the observation period less the recession intervals. Finally, let ๐(๐๐) will denote the length of a recession period ๐๐.
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I now consider the following alternative hypotheses:
I. A. A structural break exists in the dataset for some parameter. Ha: โ๐ญ โ [๐,๐ โ ๐] โง โ๐ข โ [๐,๐ค] โ ๐๐ข,๐ญ โ ๐
This is the simplest test I will conduct. It will be performed via a Sup-Wald test.
II. A. A structural break exists in any interval about a recession: Ha: โ๐ โ [๐ป๐โฒ โ ๐(๐๐),๐ป๐โฒโฒ + ๐(๐๐)] โง โ๐ โ [๐,๐] โ ๐๐,๐ โ ๐
Rejection of the null indicates that factor model parameters are unstable during
the periods around or within recessions. This test is performed via a Sup-Wald test
on each interval.
III. A. A structural break exists within the growth model. Ha: โ๐ โ ๐ช โง โ๐ โ [๐,๐] โ ๐๐,๐ โ ๐
Rejection of the null in this test indicates that factor models, even after accounting
for recessions, still have unstable parameters. This test will be performed using a
single Sup-Wald test.
IV. A structural break exists on six quarter time intervals. Ha: โ๐ญโฒ โ [๐,๐ โ ๐] โ๐ญ โ [๐ญโฒ, ๐ญโฒ + ๐] โง โ๐ข โ [๐,๐ค] โ ๐๐ข,๐ก โ ๐
Depending on the number of intervals with structural breaks, rejection of the null
in this test might indicate anything from an isolated event to excessive parameter
instability. Without prior knowledge of the data, I would expect that a Sup-Wald
test would fail to reject the null on most, if not all intervals. An excessive number
of statistically significant breaks would suggest this model has unstable parameters
even when estimated over small time scales. This test will be performed via a Sup-
Wald test on each six-quarter interval in the dataset.
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c. Methodology
This paper employs two associated tests for structural breaks. The first is a Wald-type test, a
generalized version of the Chow test. The second, based upon the Wald test, is the Sup-Wald test; this
test compares the supremum of a set of Wald statistics to the asymptotic critical values of its
distribution, which is the supremum of a squared, standardized tied-down Bessel process (Andrews,
1993). The data analysis program used in this analysis, Stata 12.1, computes joint significance tests in
regression models via the Wald test, as opposed to the Chow test, since a Wald-type test accounts for
estimated standard errors. The statistics and distributions follow:
The Wald Test Statistic
๐ญ๐พ๐๐๐ (๐) = ๏ฟฝ๐๏ฟฝ โ ๐๏ฟฝ๐ป๐ฝ๏ฟฝ๐๏ฟฝ๏ฟฝ
โ๐๏ฟฝ๐๏ฟฝ โ ๐๏ฟฝ
Where: ๐ โ โ๐ are parameter values, ๐๏ฟฝ โ โ๐ is an estimator of ๐, and
๐ฝ๏ฟฝ๐๏ฟฝ๏ฟฝ is the variance โ covariance matrix of ๐๏ฟฝ
The Distribution of the Wald Statistic
๐ญ๐พ๐๐๐ (๐)~๐๐(๐) โน for large sample size n,๐ญ๐พ๐๐๐ (๐)
๐~๐ (๐ค,๐ง โ ๐ค)
The Sup-Wald Test Statistic
For ๐ time periods, ๐ โ โ; a trimming value ๐ถ โ ๏ฟฝ๐, ๐๐๏ฟฝ ; and, index ๐ โ [๐ถ, (๐ โ ๐ถ)],
๐ญ๐๐๐ = ๐๐๐{๐ญ๐พ๐๐๐ (๐) | ๐ โ [๐ถ๐, (๐ โ ๐ถ)๐]}
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The Sup-Wald Distribution and Asymptotic Critical Values
๐ท โ โ๐ a ๐ค โ dimensional Wiener process โ โ๐ทโ2~Bessel(k), where (โ๐ท(๐)โ๐)๐ = ๏ฟฝ[๐ท๐(๐)]๐๐
๐=๐
Andrews (1993) laid out the Sup-Wald, Sup-LM, and Sup-LR tests in his original paper. The asymptotic critical values (right) of the Sup-Wald distribution (below) were derived in his corrigendum (Andrews, 2003).
The Sup-Wald Distribution
๐ญ๐๐๐ = ๐๐ฎ๐ฉ๐โ[๐ถ,(๐โ๐ถ)]
๏ฟฝ๐๐๏ฟฝ
๏ฟฝ๐๏ฟฝ๐(๐) โ ๐๐(๐)๏ฟฝ๐
๐
๐
๐=๐
๏ฟฝ
The below table values have not been divided by p,
denoted as โkโ in the Sup-Wald distribution.
This analysis uses the 1% significance level
4. Data
The raw dataset in this analysis came from CRSPsift, a joint research database derived
from the CRSP (Center for Research in Security Prices) and Compustat databases. This raw dataset is an
unbalanced panel on all US equity returns listed on the NYSE, Amex, NASDAQ, and Arca indices with
512,659 firm-quarter observations spanning 14,193 firms and 92 quarters. From this dataset, factor
exposures and the dependent variable, equity returns, are constructed. All unusable observations due
to missing data are then list-wise deleted. The remaining data that is used in the analysis is an
unbalanced panel with 330,090 complete observations on 11,647 firms and 79 quarters. This dataset
covers the quarterly periods from 1989Q4 to 2009Q2. The factors used in this analysis are based upon
those in Jegadeesh (2004), Barra1 Factor models, and Chincarini and Kim (2006). Information on the
1 http://www.msciindia.com/research/articles/barra/Barra_Risk_Models.pdf
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CRSP and Compustat data variables used in constructing equity returns and factor exposures may be
found in the appendix.
Factor Definitions from CRSPsift Data
Auxiliary Variable Constructions
Variable Definition Comments
Recession ๐๐ ๐๐๐๐ ๐ ๐๐๐ = ๏ฟฝ1, ๐๐ ๐๐๐ก๐ โ โ ๐๐๐0, ๐๐ ๐๐กโ๐๐๐ค๐๐ ๐ ๏ฟฝ
The set โ ๐๐๐ is the set of business cycle contractions listed in the
NBER business cycle expansion and contraction dates
IndustryDate ๐๐ผ๐๐๐ข๐ ๐ก๐๐ฆ๐ท๐๐ก๐๐,๐ = ๏ฟฝ1, ๐๐ ๐๐๐๐ข๐ ๐ก๐๐ฆ = ๐ โง ๐๐๐ก๐๐๐ข๐ = ๐ก0, ๐๐ ๐๐กโ๐๐๐ค๐๐ ๐ ๏ฟฝ
Categorical levels of interacted NAICS code and Date number
DateNumber ๐๐ก = ๐ก โ๐ก โ [1,๐ โ 2] An integer valued date number.
T-2 corresponds to the number 86 in this dataset.
The Dependent Variable Construction
Variable Definition Label
๐ถ๐ถ๐ ๐๐๐ฅ๐ก6 Ln๏ฟฝ๐๐๐๐๐๐ก+2 + ๐ด๐๐๐ท๐๐ฃ๐ก+2 + ๐ด๐๐๐ท๐๐ฃ๐ก+1
๐น๐๐๐๐๐๐ก+2๐๐๐๐๐๐ก
๏ฟฝ Continuously Compounded
6-month Return for the next 6 months
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Factor Exposure Constructions
Variable Definition Label
Mom6 Ln๏ฟฝ๐๐๐๐๐๐ก + ๐ด๐๐๐ท๐๐ฃ๐ก + ๐ด๐๐๐ท๐๐ฃ๐กโ1
๐น๐๐๐๐๐๐ก๐๐๐๐๐๐กโ2
๏ฟฝ 6 Month lagged CCRnext6
Mom12 Ln๏ฟฝ๐๐๐๐๐๐กโ2 + ๐ด๐๐๐ท๐๐ฃ๐กโ2 + ๐ด๐๐๐ท๐๐ฃ๐กโ3
๐น๐๐๐๐๐๐กโ2๐๐๐๐๐๐กโ4
๏ฟฝ 12 Month lagged CCRnext6
SGR โ ๐๐๐๐๐ ๐๐ก๐=๐กโ3
โ ๐๐๐๐๐ ๐๐กโ4๐=๐กโ7
Sales Growth Ratio
LnMktCap Ln (1000 โ ๐๐๐๐๐๐ก๐ถ๐๐๐ก) Natural Log of Market Cap
PBratio ๐ถ๐๐๐๐๐๐ก โ ๐ถ๐๐๐๐๐๐โ๐๐๐๐ ๐ก
๐ธ๐๐ข๐๐ก๐ฆ๐ก Price to Book ratio
EPavg โ ๐ธ๐๐๐๐ก๐=๐กโ3
4 โ ๐ถ๐๐๐๐๐๐ก Average Earnings-to-Price ratio
TAT โ ๐๐๐๐๐ ๐๐ก๐=๐กโ3
๏ฟฝ4 โ โ ๐ด๐ ๐ ๐๐ก๐ ๐๐ก๐=๐กโ3 ๏ฟฝ
Totat Asset Turnover
DEratio โ ๐ท๐๐๐ก๐๐ก๐=๐กโ3
โ ๐ธ๐๐ข๐๐ก๐ฆ๐๐ก๐=๐กโ3
Debt to Equity ratio
TURN ๐๐๐๐ข๐๐ ๐ก 90
Average Daily Turnover
Volume
DIV ๏ฟฝ๐ด๐๐๐ท๐๐ฃ๐ก + โ
๐ด๐๐๐ท๐๐ฃ(๐กโ๐)
๏ฟฝโ ๐น๐๐๐๐๐(๐กโ๐+1)๐๐=1 ๏ฟฝ
3๐=1 ๏ฟฝ
๐๐๐๐๐๐ก
Dividend Yield
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Mathematical Definitions of Data Transformations
For ๐ โ ๏ฟฝ0, 12๏ฟฝ, an increasing ordering of X={๐1,๐2, โฆ ,๐๐โ1,๐๐}, an index โjโ and total number โMโ of
industries, an index โhโ and total number โNโ of observations, I define the data transform functions
๐น0:โ โ โ, ๐น1:โ ร ๏ฟฝ0, 12๏ฟฝ โ โ and ๐น2 :โ ร ๏ฟฝ0, 1
2๏ฟฝ โ โ as follows:
โข The identity transform for the โregularโ model, ๐ญ๐:
๐น0(๐ฟ) โ ๐ฟ
โข The winsor transform, ๐ญ๐:
๐น1(๐ฟ,๐) โ ๏ฟฝ๐๐โ๐ ,๐โ ,
๐[๐โ(1โ๐)] , ๏ฟฝ
๐๐ โ < ๐๐๐๐ ๐๐ < โ < ๐(1 โ ๐)
๐๐ ๐(1 โ ๐) < โ๏ฟฝ ๏ฟฝ
โข The normalization transform, ๐ญ๐: (by industry and time levels)
๐น2(๐ฟ,๐) โ๐น1๏ฟฝ๐ฟ๐,๐,๐๏ฟฝ โ ๐๏ฟฝ๐น1๏ฟฝ๐ฟ๐,๐,๐๏ฟฝ๏ฟฝ
๐๏ฟฝ๏ฟฝ๐น1๏ฟฝ๐ฟ๐,๐,๐๏ฟฝ๏ฟฝ๏ฟฝ โ๐ก โ [1,๐ โ 2], โ๐ โ [1,๐], ๐คโ๐๐๐ ๐ฟ = ๏ฟฝ๏ฟฝ๐ฟ๐,๐
๐
๐=1
๐โ2
๐ก=1
The winsor transform is employed to mitigate the effect of outliers in the dataset. A number of
factors contained very large outliers, several magnitudes different from the mean. Plots of all regular
factors are shown in the appendix. Factors with extreme outliers include Sales Growth Ratio (SGR), Debt
to Equity ratio (Deratio), Price to Book ratio (PBratio), and Earnings to Price ratio (EPavg). Results from
using winsorized data appeare similar to those obtained from a median regression; however, due to the
excessive computing time required to run regressions on the median in Stata, this data transform is
applied as an alternative.
The normalization transform is used to mitigate the effects of outliers and the variance among
factor exposures that are correlated with industry, such as price-to-earnings ratio. An observation is
included in a particular industry-date subset if the observed firm belongs to that industry during that
quarter. The inclusion of the time dimension standardizes the transformation method across time
periods and allows for testing within the nested model.
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5. Results:
At the one percent significance level, all tests rejected the null hypothesis of parameter stability.
Graphics and tables of the computed test statistics follow.
Test of alternative I: The Sup-Wald test on the entire dataset, 15% 2-tail trimming
A point represents a Wald statistic.
The suprema of the Wald statistics for regular, winsorized, and normalized models are: 63.03, 75.81,
and 71.34 respectively. The 1% alpha and 15% 2-tail trimming critical value for the Sup-Wald test is
3.23. Based upon the graph, considering the large number of Wald statistics above the 1% Sup-Wald
rejection line for each model, we can reasonably conclude that a structural break exists. However, due
to the excessive number of statistically significant breaks, it is difficult to determine where the structural
break(s) are located or how many breaks exist.
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Test of alternative II: The Sup-Wald test about recessions. Respective 2-tail trim: 10%, 15%, and 15%
Regular Winsor Normalized1990Q1 8.40 6.52 6.14 3.31990Q2 25.41 24.27 25.23 3.31990Q3 6.82 9.42 10.76 3.31990Q4 16.19 16.40 16.26 3.31991Q1 1.56 3.36 2.57 3.31991Q2 7.20 13.04 14.06 3.31991Q3 11.78 11.51 11.99 3.32000Q3 60.20 48.39 41.65 3.232000Q4 26.67 25.21 23.97 3.232001Q1 19.11 11.22 10.08 3.232001Q2 11.43 10.68 9.45 3.232001Q3 4.65 8.88 9.26 3.232001Q4 22.49 21.41 19.19 3.232002Q1 31.03 33.52 34.33 3.232002Q2 5.93 8.19 8.71 3.232006Q3 16.85 6.02 5.60 3.232006Q4 3.89 2.54 2.43 3.232007Q1 7.15 4.92 3.93 3.232007Q2 6.03 7.12 7.46 3.232007Q3 5.64 7.99 7.21 3.232007Q4 6.94 5.38 3.23 3.232008Q1 11.45 14.53 13.53 3.232008Q2 17.79 19.87 19.57 3.232008Q3 7.94 14.52 11.07 3.232008Q4 37.18 47.63 52.36 3.23
1% Critical value
Date
1990Q3 to
1991Q1
2001Q1 to
2001Q4
2007Q4 to
2009Q2
Wald Statistics from Chow TestAssociated Recession
Sup-Wald statistics are displayed in red.
All three recession models have, for all recession intervals, Sup-Wald statistics well over the 1%
Sup-Wald critical values of 3.3, 3.23, and 3.23 respectively. Notably, among all three models, only two
periods have at least one Wald statistic that does not exceed the Sup-Wald critical value; hence, these
statistics indicate a structural break is present in nearly every quarter. The results of this test imply that
Fundamental Factor Models have unstable parameters during transitions between periods of growth
and business cycle downturns. Due to the excessive number of statistics over the Sup-Wald critical
value, it is difficult to determine where a structural break begins and ends.
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Test of alternative III: The Sup-Wald test on the business cycle growth periods, 15% 2-tail trimming
A point represents a Wald statistic.
The suprema of the Wald statistics for regular, winsorized, and normalized models are: 60.39,
80.16, and 71.23 respectively. The associated 1% Sup-Wald critical value is 3.23. From these statistics, it
is clear that a structural break exists during growth periods, as nearly all of the Wald statistics exceed
the Sup-Wald rejection value. This indicates that the each model is highly unstable even over the
growth periods. The excessive number of statistically significant structural breaks makes it difficult to
ascertain exactly where or how any breaks are contained within this time period.
16
Since all three tests rejected the null in favor of the associated alternative hypotheses, I was
interested in testing for structural breaks over much smaller time horizons. With the data excision
requirements of the Sup-Wald test in mind, I considered using one and a half year intervals, or six
quarters, and testing the inner four quarters for the Sup-Wald test. This corresponds to approximately a
15% trimming in each tail. Before performing this test, I fully expected to find that the majority of the
tests over these periods would indicate the parameters were stable by failing to reject the null.
However, quite shockingly, I found the opposite was true. The graph of the computed Wald statistics
follows. Four Wald statistics per winsorized and normalized model are plotted for each period.
Test of alternative IV: Sup-Wald test on all 6 quarter intervals. 15% 2-tail trimming
Regular model statistics omitted due to excessive graphical overlap and similarity of results.
A point represents a Wald statistic.
17
This test rejected the null hypothesis, as every interval contained a Sup-Wald statistic over the
critical value. A list of Wald statistics below the Sup-Wald critical value for this graph may be found in
the appendix. Approximately one out of every six time periods contained a test statistic for either
model below the Sup-Wald value. Hence, five out of every six Wald statistics were in the Sup-Wald
rejection region. This test also leads me to conclude that the Fundamental Factor Model is highly
unstable, even with estimation windows as short as six quarters.
6. Conclusion:
My findings indicate that, with half year holding periods, the factor premia of commonly-
modeled factors in a Fundamental Factor Model are highly unstable with respect to time. In every
model, parameter instability was more the norm than the exception; even in the models with one and a
half year time horizons, the majority of the computed Wald statistics were in the 1% significance Sup-
Wald rejection region. Hence, the factor premia appear non-stationary with any estimation window.
These results suggest that a large number of investment funds could be employing a
misspecified model with questionable predictive power. As Fundamental Factor Models can be used
with a variety of holding periods and any number of unique factor exposures, I cannot generalize my
conclusion to all possible variants of the Fundamental Factor Model; however, my results do suggest
that any model variant which uses commonly modeled factors from financial literature with a similar
holding period is likely subject to significant parameter instability over time.
In spite of this unsettling conclusion, it is possible that high magnitude within-industry cross-
autocorrelations of equity returns are affecting the statistical tests used in this analysis. It is also
possible that factor premia differ between industries, but are constant within industries; hence,
structural breaks observed in this analysis could be driven by changes in the composition of industries
over time. This question may be considered in a future paper.
18
Sources Cited
โข N. Jegadeesh, J. Kim, S. Krische and Charles M. C. Lee. โAnalyzing the Analysts: When Do
Recommendations Add Value?โ The Journal of Finance, Vol. 59, No. 3 (Jun., 2004), pp. 1083-
1124
โข L. Chan, N. Jegadeesh, J. Lakonishok. โThe Profitability of Momentum Strategies.โ Financial
Analysts Journal. 1999. Vol. 55, No. 6, Behavioral Finance (Nov. - Dec., 1999), pp.80-90
โข G Connor. โThe Three Types of Factor Models: A Comparison of Their Explanatory Power.โ
Financial Analysts Journal, Vol. 51, No. 3 (May - Jun., 1995), pp. 42-46
โข John Y. Campbell, Andrew W. Lo, A. Craig MacKinlay. THE ECONOMETRICS OF FINANCIAL
MARKETS. Princeton University Press, 1997
โข L. Chincarini, โA Comparison of Quantitative and Qualitative Hedge Funds.โ Pomona College
Working Paper. Pomona College Working Paper, (Jan. 2010)
โข D. Andrews, โTESTS FOR PARAMETER INSTABILITY AND STRUCTURAL CHANGE WITH UNKNOWN
CHANGE POINTโ, Econometrica, Vol. 63, No. 4 (July, 1993), pp. 821-856
โข D. Andrews, โTESTS FOR PARAMETER INSTABILITY AND STRUCTURAL CHANGE WITH UNKNOWN
CHANGE POINT: A CORRIGENDUMโ, Econometrica, Vol. 71, No. 1 (Jan., 2003), pp. 395-397
19
Appendix
Factor premia estimated over all periods
Factor premia estimated over growth periods
Factor premia estimated over recession periods
20
Alternative IV: Table of Wald statistics below the Sup-Wald critical value
Winsorized Normalized Winsorized Normalized10 15 13 2.79 2.44 63 68 67 2.07 3.2816 21 20 2.89 3.78 64 69 66 2.88 3.2617 22 21 3.06 3.72 64 69 67 2.00 2.9918 23 21 2.74 3.59 65 70 67 2.10 2.6518 23 22 1.60 1.37 66 71 67 1.58 2.2819 24 22 2.60 2.63 66 71 69 2.36 2.5020 25 22 2.21 2.20 66 71 70 2.26 2.9721 26 22 2.50 2.08 67 72 69 2.82 2.7621 26 25 3.50 3.10 67 72 70 2.41 2.8222 27 25 2.45 1.96 67 72 71 2.65 3.4423 28 25 2.64 2.25 69 74 70 2.99 3.2525 30 26 3.03 4.07 70 75 71 3.02 4.0525 30 27 3.10 2.80 70 75 74 3.63 3.1727 32 31 2.93 3.27 71 76 74 2.87 2.3229 34 31 2.97 2.54 71 76 75 2.22 2.3432 37 36 1.99 2.06 72 77 74 1.63 1.1633 38 36 1.98 1.56 72 77 76 1.20 1.8134 39 35 3.02 3.09 73 78 74 2.35 2.3034 39 36 1.88 1.29 73 78 76 1.03 1.7735 40 36 1.15 0.74 74 79 76 1.91 3.0658 63 62 3.22 3.33 75 80 76 1.98 3.6861 66 64 2.85 3.25 75 80 77 3.18 3.3362 67 63 3.56 3.14 76 81 77 2.81 2.7762 67 64 1.57 1.28 76 81 78 2.62 3.1763 68 64 2.45 2.30 77 82 78 2.11 3.1863 68 66 2.96 3.83
Index WaldWald StatisticsTstart Tend Index Tstart Tend
The test intervals in this table are defined by the set of indices in the set [Tstart+1, Tend-1]. As
is evident from the table, no test interval has all four Wald Statistics listed. This occurs only if at least
one index has a corresponding Wald statistic above the 1% significance Sup-Wald critical value.
21
Sources of the quarterly Data Variables used in Factor Construction:
CRSP Variables
Price: โadjprice โ
Facprc: โfacprcโ
Volume: โadjvolโ
MarketCap: โcapโ
AdjDiv: โadjdivโ
Compustat Variables
Sales: โsaleqโ
Debt: โltqโ
Equity: โceqqโ
CommonShares: โcshoqโ
EPS: โepspxqโ
CPrice: โpriceโ
Assets: โatqโ
Naics code: โnaicsโ
Plots of regular factor exposures vs. time The following are plots are taken over the entire dataset. Note the outliers in some graphs.
Returns vs. Time
27
Summary statistics of regular data
within .0759342 -4.692295 15.93388 T-bar = 28.3412 between .0788563 0 4.705118 n = 11647DIV overall .0128227 .0870443 0 19.23077 N = 330090 within 153.9074 -11763.52 63459.03 T-bar = 28.3412 between 129.6485 .0271677 11781.09 n = 11647DEratio overall 3.979764 165.3874 .001056 69231 N = 330090 within .081776 -1.309798 3.488691 T-bar = 28.3412 between .2133991 0 2.963348 n = 11647TAT overall .2625893 .2254904 0 5.306439 N = 330090 within 59.15526 -6757.771 20265.22 T-bar = 28.3412 between 67.54025 0 6765.157 n = 11647SGR overall 1.597242 65.3512 0 27028.78 N = 330090 within .2779586 -115.1175 13.49402 T-bar = 28.3412 between .2692261 -13.85271 1.980056 n = 11647EPavg overall -.0183501 .3001202 -118.3453 15.49242 N = 330090 within 200.813 -3313.132 101069 T-bar = 28.3412 between 44.36919 .0824635 3317.638 n = 11647PBratio overall 4.24708 203.5891 .0189279 102478.1 N = 330090 within .7352949 12.7808 23.35111 T-bar = 28.3412 between 1.883639 12.83233 25.97695 n = 11647LnMktCap overall 19.07941 2.094817 11.68621 27.12424 N = 330090 within 13882.41 -289581.8 3157440 T-bar = 28.3412 between 12861.49 .1590278 487832.1 n = 11647TURN overall 4076.957 23256.7 0 3322031 N = 330090 within .3778661 -3.310242 5.679092 T-bar = 28.3412 between .2447774 -2.807713 2.784478 n = 11647Mom12 overall .0008323 .3950725 -4.153661 5.908147 N = 330090 within .3883649 -4.228867 4.135431 T-bar = 28.3412 between .2568274 -2.685577 2.097141 n = 11647Mom6 overall -.0110231 .4075402 -4.290459 4.190496 N = 330090 within .4540304 -7.819571 4.051639 T-bar = 28.3412 between .3968147 -7.294717 1.642745 n = 11647CCRnext6 overall -.0173998 .4807335 -8.174703 4.190496 N = 330090 Variable Mean Std. Dev. Min Max Observations
28
Summary statistics of winsorized data
within .0110855 -.0885973 .1311436 T-bar = 28.3412 between .0151148 0 .1221053 n = 11647DIV overall .010584 .0187607 0 .1221053 N = 330090 within 2.127588 -19.67986 34.39335 T-bar = 28.3412 between 4.734654 .047489 32.71814 n = 11647DEratio overall 2.902778 4.584336 .047489 32.71814 N = 330090 within .0714359 -.6014877 1.208815 T-bar = 28.3412 between .2041928 .0019785 1.195833 n = 11647TAT overall .2608828 .2129883 .0019785 1.195833 N = 330090 within .4685093 -3.064714 6.30837 T-bar = 28.3412 between .496526 .1701427 6.006501 n = 11647SGR overall 1.17382 .5423698 .1701427 6.006501 N = 330090 within .0774865 -.8766053 .678017 T-bar = 28.3412 between .1106891 -.8545455 .0733333 n = 11647EPavg overall -.0132923 .0963662 -.8545455 .0733333 N = 330090 within 2.904608 -25.91024 37.32603 T-bar = 28.3412 between 3.842162 .2139904 36.50642 n = 11647PBratio overall 2.895137 3.954607 .2139904 36.50642 N = 330090 within .7297016 12.78004 23.35035 T-bar = 28.3412 between 1.870978 14.54193 24.92134 n = 11647LnMktCap overall 19.07865 2.080567 14.54193 24.92134 N = 330090 within 4616.842 -67041.06 89909.69 T-bar = 28.3412 between 7166.603 1.366667 94029.83 n = 11647TURN overall 3395.252 10278.93 1.366667 94029.83 N = 330090 within .3636287 -1.736109 1.89196 T-bar = 28.3412 between .2238177 -1.481605 1.244644 n = 11647Mom12 overall .0010169 .3799221 -1.481605 1.244644 N = 330090 within .3740957 -1.957286 2.066974 T-bar = 28.3412 between .2379111 -1.564986 1.241713 n = 11647Mom6 overall -.0107728 .3923491 -1.564986 1.241713 N = 330090 within .4540304 -7.819571 4.051639 T-bar = 28.3412 between .3968147 -7.294717 1.642745 n = 11647CCRnext6 overall -.0173998 .4807335 -8.174703 4.190496 N = 330090 Variable Mean Std. Dev. Min Max Observations
29
Summary statistics of normalized data
between .7539986 -2.457027 12.19361 n = 11647NDIV overall 8.46e-10 .9972454 -2.55548 14.82217 N = 330090 within .3780099 -5.466204 11.46131 T-bar = 28.3412 between .6859714 -.8216292 10.33942 n = 11647NTURN overall -2.29e-10 .9972454 -1.129668 19.86789 N = 330090 within .722734 -9.278238 18.66465 T-bar = 28.3412 between .9523942 -1.914443 20.00147 n = 11647NPBratio overall 2.72e-10 .9972454 -2.763816 21.10627 N = 330090 within .7939418 -20.38017 9.306662 T-bar = 28.3412 between 1.069475 -13.65239 1.560367 n = 11647NEPavg overall -1.11e-10 .9972454 -23.47325 4.926731 N = 330090 within .6420565 -9.579826 16.00106 T-bar = 28.3412 between 1.002571 -2.185102 11.71153 n = 11647NDEratio overall -1.19e-09 .9972454 -2.185102 17.84907 N = 330090 within .4489305 -5.542224 8.558899 T-bar = 28.3412 between .9457634 -2.64264 9.000388 n = 11647NTAT overall 3.22e-09 .9972454 -2.754076 10.84011 N = 330090 within .8639982 -8.421585 16.0297 T-bar = 28.3412 between .8777673 -3.062694 17.17064 n = 11647NSGR overall 6.52e-09 .9972454 -5.957483 17.71057 N = 330090 within .3209713 -3.042282 1.965051 T-bar = 28.3412 between .8941637 -2.891855 3.35686 n = 11647NLnMkt~p overall -2.17e-10 .9972454 -3.373385 4.358819 N = 330090 within .9529175 -9.075345 9.718304 T-bar = 28.3412 between .5613808 -5.599185 3.621288 n = 11647NMom12 overall 7.76e-11 .9972454 -9.403751 9.21301 N = 330090 within .9492542 -8.38606 8.436458 T-bar = 28.3412 between .6005179 -7.78686 3.919336 n = 11647NMom6 overall -7.77e-10 .9972454 -8.756555 7.67023 N = 330090 within .4540304 -7.819571 4.051639 T-bar = 28.3412 between .3968147 -7.294717 1.642745 n = 11647CCRnext6 overall -.0173998 .4807335 -8.174703 4.190496 N = 330090 Variable Mean Std. Dev. Min Max Observations