This document presents a study analyzing the time variation paths of factors affecting bank stock returns using a flexible least squares (FLS) method. It discusses motivations, describes the problem and data collection/processing steps. It then provides details on the FLS method, including an example application. The document analyzes a three-index FLS model fit to the data and discusses the role of a parameter μ in identifying time-sensitive FLS coefficients. It outlines a procedure to estimate an appropriate μ value by building ordinary least squares models over different sub-intervals of the time period.

1. The Time Variation Paths of Factors Effecting Bank
Stock Returns
Kaiyi Chen
Department of Mathematics
University of Central Arkansas
August 5, 2015
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 1 / 43

2. Outline
1 Motivation
2 Problem Description
3 Data
4 Flexible Least Squares (FLS) Method
5 Analysis & Discussion
6 Conclusion
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 2 / 43

3. Outline
1 Motivation
2 Problem Description
3 Data
4 Flexible Least Squares (FLS) Method
5 Analysis & Discussion
6 Conclusion
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 3 / 43

4. Motivation
In my undergraduate, my second major was in finance, and I always
interested in bank performance.
According to the previous study of Dr.Ling He, bank stock returns
depend on adjusted monthly percentage changes in stock market ,
interest rates on long-term government bond, and monthly changes in
median sales price of new house during 1972-1995. Hence, I am
curious if these factors still affect bank stock returns now. In this
study we fitted model with more recent data.
In order to investigate the effect of the time-varying independent
variables, we use the FLS method developed by Kalaba and Tesfatsion
to fit model. In this study, we implemented the FLS method in R.
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 4 / 43

5. Outline
1 Motivation
2 Problem Description
3 Data
4 Flexible Least Squares (FLS) Method
5 Analysis & Discussion
6 Conclusion
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 5 / 43

6. Problem Description
Investors have long been interested in both risk and performance of
commercial banks.
Upon reviewing the empirical literature, equity market factors are
useful in predicting future bank holding company performance.
More recently, a number of empirical studies show the potential
effectiveness of using bond prices and spread in predicting bank risk.
Real estate yields could be another factor affects bank stock returns,
as commercial banks have crucial holdings of both residential and
commercial real estate mortgages.
To quantify the performance of commercial banks, a three index model
was fitted by using the variables mentioned above.
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 6 / 43

7. Outline
1 Motivation
2 Problem Description
3 Data
4 Flexible Least Squares (FLS) Method
5 Analysis & Discussion
6 Conclusion
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8. Data
Collection
We collected data from Nov.1990 to Nov. 2014 for the following variables:
NASDAQ bank index
S&P 500 index
Interest rates on long-term government bonds
median sales price of new houses
3-month treasury bill
The first two sets of data are collected from Yahoo Finance. The rest of
datas are collected from Federal Reserve Bank of St. Louis.
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9. Data
Collection
Figure 1: Original Data of S&P 500
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10. Data
Processing
In order to minimize the effect of the magnitude of data of the
variables, we used monthly changes of NASDAQ bank index, S&P
500 index and median sales price for new houses
In order to get risk premium model, all these 4 variables are further
adjusted by subtracting corresponding 3-month treasury bills (T-bill)
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11. Data
Processing
Figure 2: Monthly data substract corresponding 3-month treasury bills
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12. Data
Descriptive Analysis
We denote these processed variables as follows:
y = monthly changes in NASDAQ bank stock index adjusted for T-bills
x1 = monthly percentage changes in S&P 500 adjusted for T-bills
x2 = interest rates on long-term government bond adjusted for T-bills
x3 = monthly changes in median sales price of new houses sold in the U.S.
adjusted for T-bills
Variables Mean Std.Dev Minimun Maximum
y -0.019 0.050 -0.249 0.140
x1 -0.021 0.046 -0.195 0.108
x2 0.017 0.015 -0.012 0.050
x3 -0.025 0.047 -0.134 0.111
Table 1: Summary Statistics from 1990 to 2014
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13. Data
Descriptive Analysis
Figure 3: Time series of 4 variables
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14. Data
Description Analysis
Figure 4: Correlation scatterplot matrix
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15. Outline
1 Motivation
2 Problem Description
3 Data
4 Flexible Least Squares (FLS) Method
5 Analysis & Discussion
6 Conclusion
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16. FLS
Ordinary Least Squares (OLS) Model
The general form of Ordinary Least Squares model is estimated as:
y = b1x1 + b2x2 + ... + bkxk + (1)
where y is the dependent variable. xi , i = 1, 2, ...k, are n independent
variables. bj , j = 1, 2, .., k, are parameters of the model.
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 16 / 43

17. FLS
Flexible Least Squares Method
Suppose noisy scalar observations y1, y2, ..., yT obtained on a process
over a time-span 1, 2, ..., T are assumed to have been generated by a
linear regression model with coefficients which evolve only slowly over
time, if at all. More precisely, suppose these prior theoretical beliefs
take the following form and it is FLS model:
yt = xT
t bt + , t = 1, 2, ..., T, (2)
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 17 / 43

18. FLS
Error Terms
An incompatibility cost:
= µr2
D + r2
M (3)
consisting of the µ-weighted average of the associated dynamic error and
measurement error sums.
Where
r2
M = (y − ˆy)2
r2
D = (bt+1 − bt)T
(bt+1 − bt)
The coefficients of the model will be estimated by minimizing the
incompatibility cost.
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19. FLS
Role of µ
The predetermined positive constant µ of the FLS model plays a crucial
role in identifying suitable FLS coefficients. The value of µ is inversely
proportional to the time sensitivity of FLS coefficients.
For i = 1, 2, 3,
bit → bi as µ → ∞ (4)
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20. FLS
Algorithm
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21. FLS
Example
We have taken this example from Kalaba and Tesfatsion[p.1228].
In this example, the yn, n = 1, 2, ..., T, values are computed as follows:
yt = xT
t bt + , t = 1, 2, ..., T, (5)
where
xt=
sin(10 + t) + 0.01
cos(10 + t)
, t = 1,2,...,T
bt=
sin(4πt/10)
sin(2πt/10)
, t = 1,2,...,T
We used the computed values of yt and xt for different values of µ & T
using this equations as inputs for the R code we developed, and computed
bt.
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 21 / 43

22. FLS
Example
0 200 400 600 800 1000
−1.0−0.50.00.51.01.52.0
n
ExactvsEstimated
Exact b1
Estimated b1
Exact b2
Estimated b2
Figure 5: Estimated & exact coefficients when µ = 1000 and T = 1000
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 22 / 43

23. FLS
Example
0 200 400 600 800 1000
−1.0−0.50.00.51.01.52.0
n
ExactvsEstimated
Exact b1
Estimated b1
Exact b2
Estimated b2
Figure 5: Estimated & exact coefficients when µ = 10 and T = 1000
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 22 / 43

24. FLS
Example
0 200 400 600 800 1000
−1.0−0.50.00.51.01.52.0
n
ExactvsEstimated
Exact b1
Estimated b1
Exact b2
Estimated b2
Figure 5: Estimated & exact coefficients when µ = 0.1 and T = 1000
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 22 / 43

25. FLS
Example
When µ value is fixed, the coefficients get better fit as T increases.
0 5 10 15 20 25 30
−1.0−0.50.00.51.01.52.0
n
ExactvsEstimated
Exact b1
Estimated b1
Exact b2
Estimated b2
Figure 6: Estimated & exact coefficients when µ = 1 and T = 30
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26. FLS
Example
When µ value is fixed, the coefficients get better fit as T increases.
0 20 40 60 80 100
−1.0−0.50.00.51.01.52.0
n
ExactvsEstimated
Exact b1
Estimated b1
Exact b2
Estimated b2
Figure 6: Estimated & exact coefficients when µ = 1 and T = 100
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 23 / 43

27. FLS
Example
When µ value is fixed, the coefficients get better fit as T increases.
0 200 400 600 800 1000
−1.0−0.50.00.51.01.52.0
n
ExactvsEstimated
Exact b1
Estimated b1
Exact b2
Estimated b2
Figure 6: Estimated & exact coefficients when µ = 1 and T = 1000
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 23 / 43

28. Outline
1 Motivation
2 Problem Description
3 Data
4 Flexible Least Squares (FLS) Method
5 Analysis & Discussion
6 Conclusion
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 24 / 43

29. Analysis & Discussion
Building FLS Model
According to Dr.He’s the previous study, the three-index FLS model
was built here:
yt = b1tx1t + b2tx2t + b3tx3t + , n = 1, 2, ..., T (6)
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 25 / 43

30. Analysis & Discussion
Building FLS Model
According to Dr.He’s the previous study, the three-index FLS model
was built here:
yt = b1tx1t + b2tx2t + b3tx3t + , n = 1, 2, ..., T (6)
The corresponding OLS model will be:
y = b1x1 + b2x2 + b3x3 + (7)
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 25 / 43

31. Analysis & Discussion
Role of µ
−0.50.00.51.01.52.0
Time
Estimatedcoefficients
MTK BOND PRICE
1990 1993 1996 1999 2002 2005 2008 2011 2014
Figure 7: FLS estimated coefficients when µ = 0.1
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32. Analysis & Discussion
Role of µ
−0.50.00.51.01.52.0
Time
Estimatedcoefficients
MTK BOND PRICE
1990 1993 1996 1999 2002 2005 2008 2011 2014
Figure 7: FLS estimated coefficients when µ = 0.5
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 26 / 43

33. Analysis & Discussion
Role of µ
−0.50.00.51.01.52.0
Time
Estimatedcoefficients
MTK BOND PRICE
1990 1993 1996 1999 2002 2005 2008 2011 2014
Figure 7: FLS estimated coefficients when µ = 0.9
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 26 / 43

34. Analysis & Discussion
Procedure to estimate µ
How to estimate a proper µ value?
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 27 / 43

35. Analysis & Discussion
Procedure to estimate µ
How to estimate a proper µ value?
We answer this question by building various OLS models for different
non-overlapping sub-intervals of entire time region.
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 27 / 43

36. Analysis & Discussion
Procedure to estimate µ
How to estimate a proper µ value?
We answer this question by building various OLS models for different
non-overlapping sub-intervals of entire time region.
We identify sub-interval by using the procedure discussed in the next
slide.
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 27 / 43

37. Analysis & Discussion
Procedure to estimate µ
1 START by setting a value for µ (choose from (0,0.9)).
2 Build a FLS model for the entire time period [1, T].
3 Identify time-varying paths (time-series plots) of FLS coefficients.
4 Identify FLS coefficients whose signs of the time-varying paths do not
change. If either one of the conditions following conditions fails, go to
Step 1 to repeat Steps 1 through 4 for different value for µ; else go to
Step 5:
a. whether the corresponding OLS coefficients are statistically
significant, and
b. whether the signs of the time-varying paths of FLS coefficients are
the same as the signs of the corresponding OLS coefficients.
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 28 / 43

38. Analysis & Discussion
Procedure to estimate µ
5 Identify a FLS coefficient satisfying the following conditions: Over
interval [1, T],
a. at least there is one sign change, and
b. minimum number of sign changes when compared to other FLS
coefficients.
6 Identify the time points at which the FLS coefficient identified in Step
3 change signs.
7 Divide the interval [1, T] into subintervals with endpoints
corresponding to the points identified in Step 5 including the initial
point 1, and the end point T.
8 Construct as many OLS models as the subintervals identified in Step
7 using the corresponding datasets of these intervals.
9 Check a. whether the coefficients of the two OLS models are
statistically significant, and
b. whether the signs of the OLS coefficients are the same as the signs
of the corresponding FLS coefficient identified in Step 4.
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 29 / 43

39. Analysis & Discussion
Procedure to estimate µ
10 If both conditions of Step 9 are satisfied, then the path of the FLS
coefficient identified in Step 4 is a reliable path to analyze the
dependencies between y and the corresponding independent variable.
11 Identify the next FLS coefficient satisfying conditions in Step 4, and
repeat Steps 5 through 9. If there are no more coefficients, then
EXIT the procedure.
12 If either one of the conditions of Step 8 is not satisfied, then adjust
time points found in Step 5 (which also includes the scenario where
some of the subintervals in Step 6 are either merged or split) and
repeat Steps 8 through 11. If either one of the conditions of Step 8 is
still not satisfied, go to Step 1 to repeat all steps for another value of
µ.
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 30 / 43

40. Analysis & Discussion
Estimate µ value for x1 (stock market)
For x1, the coefficient remained positive throughout the entire time
period no matter how µ value changed. The fitted OLS model whose
results are shown below:
Variables Est.Coef P − value
x1 0.7572 < 0.001
x2 0.1554 0.1116
x3 0.0765 < 0.1
Table 2: Time period from Nov.1990 to Nov.2014
Hence, the value of µ could as small as we want. Then we can say
the µ value for the FLS model doesn’t depend on x1
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 31 / 43

41. Analysis & Discussion
Estimate µ value for x2 (goverment bond)
We identified three sub-intervals (Nov.1990 to May 2006, Jun.2006 to
Oct.2008, and Nov.2008 to Nov.2014) with the FLS model when
µ = 0.5 .
The following two slides show results of OLS models and a figure for
FLS & OLS estimations.
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 32 / 43

42. Analysis & Discussion
Estimate µ value for x2 (goverment bond)
Variables Est.Coef P − value
x1 0.67527 < 0.001
x2 0.2315 < 0.05
x3 0.08048 0.1292
Table 3: Time period from Nov.1990 to May 2006
Variables Est.Coef P − value
x1 0.8194 < 0.001
x2 0.7675 0.2221
x3 0.1402 0.388138
Table 4: Time Period from Jun.2006 to Oct.2008
Variables Est.Coef P − value
x1 1.11081 < 0.001
x2 -0.4773 < 0.01
x3 0.13168 0.07335
Table 5: Time Period from Nov.2008 to Nov.2014
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 33 / 43

43. Analysis & Discussion
Estimate µ value for x2 (goverment bond)
−0.50.00.51.0
time
OLS&FLSresultforGB
Estimated FLS
Estimated OLS
1990 1993 1996 1999 2002 2005 2008 2011 2014
Figure 8: FLS & OLS estimation for BOND
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 34 / 43

44. Analysis & Discussion
Estimate µ value for x3 (sales price of new houses)
Variables Est.Coef P − value
x1 0.7362 0.001
x2 0.1789 0.395
x3 -0.1118 0.319
Table 6: Time Period from Nov.1990 to Jan.1994
Variables Est.Coef P − value
x1 0.75452 < 0.001
x2 0.03971 0.7412
x3 0.1099 < 0.05
Table 7: Time Period from Feb.1994 to Nov.2014
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 35 / 43

45. Analysis & Discussion
Estimate µ value for x3 (sales price of new houses)
−0.50.00.51.0
time
OLS&FLSresultforPrice
Estimated FLS
Estimated OLS
1990 1993 1996 1999 2002 2005 2008 2011 2014
Figure 9: FLS & OLS estimation for PRICE
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 36 / 43

46. Analysis & Discussion
Summary
The market performance, based on S&P 500 index, had significant
positive impact during the entire period Nov.1990 through Nov.2014.
The long-term government bond rates had significant positive impact
during Nov.1990 through May 2006 and had no significant impact
during Jun.2006 through Oct.2008 and had significant negative
impact during Nov.2008 through Nov.2014.
The sales price of new houses had a positive significant impact after
Jan.1994 and had a decline during 2007 through 2011.
These mixed impacts of the independent variables reflect the impact of the
events, which occurred during the period that covered the entire financial
meltdown caused by massive subprime mortgages, on the bank stock
returns.
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 37 / 43

47. Outline
1 Motivation
2 Problem Description
3 Data
4 Flexible Least Squares (FLS) Method
5 Analysis & Discussion
6 Conclusion
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 38 / 43

48. Conclusion
The objectives of this study are to implement the Flexible
least-squares (FLS) method in R, validate it, and use it to compute
the time varying coefficients of S&P 500 index, interest rate on
long-term government bond, and median sale price of new house in
order to analyze the effect of these variables on the NASDAQ bank
stock index for the period Nov 1990 to Nov 2014.
Major findings of this study include: long-term government bond
rates had no impact on banks’ benefit during time period Jun.2006 to
Oct.2008, and there is a decline trend for coefficients of sales price of
new house during 2007 to 2011.
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 39 / 43

49. Possible extension
One possible extension of the present work is to automate this
procedure adaptively based on the OLS coefficients and their
statistical significance. This would significantly reduce the time in
identifying reliable time varying paths of FLS coefficients more
objectively.
The second possible extension of this study would be to replace S&P
500 index data with New York Stock Exchange (NYSE) composite
index and study the effect of time varying coefficient of this data on
bank stock returns.
As another possible extension, more interesting and relevant
independent variables can be added to the model to study their effect
on bank stock returns.
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 40 / 43

50. Reference I
Ling T.He and Alan K. Reichert
Time variation paths of factors affecting financial institutions and
stock returns.
Atlantic Economic Journal,31(1):71-86, 2003.
R.Kalaba and L.Tesfatsion
Time-varing linear regression via Flexible Least Squares.
Computers Math.Applic,17(8/9):1215-1245, 1989
YAHOO FINANCE
NASDAQ Bank Index
http://finance.yahoo.com/q/hp?s=%5EBANK+Historical+Prices
Federal Reserve Bank of ST. Louis
Long-Term Government Bond Yields
https://research.stlouisfed.org/fred2/series/IRLTLT01DEM156N
Kaiyi Chen (University of Central Arkansas) FLS Model for Bank Stock Returns August 5, 2015 41 / 43

51. Acknowledgement
Dr.R.B.Lenin (committee chairperson)
Dr.Ling He and Dr.Patrick Carmack (committee member)
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52. End
Thank You!
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