Statistical Models of Stocks and Bonds
Zachary D Easterling: Department of Economics
The University of Akron
Abstract
One of the key ideas in monetary economics is that the prices of investments tend to move in the
opposite direction as interest rates. Is there a correlation between stocks, bonds and housing and do
they move in the same direction? Can an economist predict the movement of one by looking at the
aggregate of the others in relation to interest rates? Using Simple and Multiple Linear Regression, ChiSquare, One-Sample, and Two-Sample T-tests, this short work will focus on any positive or negative
correlation between the Federal Funds Rate and investment media.
Simple Linear Regression: Using the Federal Reserve Federal Funds Rate to predict the yield to
maturity on a One-Year Treasury Security.
H0: The Federal Funds Rate is not a useful predictor of the One-Year Treasury Security yield.
Ha: The Federal Funds Rate is a useful predictor of the One-Year Treasury Security yield.
Figure 1: MiniTab Output for Simple Linear Regression
The regression equation is
One Year Security = 1.38 + 0.758 Federal Funds Rate
Predictor
Constant
Fed Funds Rate
S = 1.29798
Coef SE Coef
1.3772
0.2483
0.75839 0.05592
R-Sq = 49.2%
T
P
5.55 0.000
13.56 0.000
R-Sq(adj) = 48.9%
MiniTab was used to
predict the yield to maturity on a
One-Year Treasury Security
using the Federal Funds Rate;
which is set daily by the Federal
Reserve Bank.
The output
generated by MiniTab in
conjunction with the hypotheses
made in a simple linear
regression leads me to accept
the idea that the Federal Funds
rate is indeed a useful predictor
of the One-Year Treasury
Security
because
of
the
incredibly
small
P-value
indicated in the output. The R2
values indicate however, that
only 48.9% of the variability in
the One Year Treasury is
explained by the movement of
the Federal Funds Rate.
Unfortunately,
the
residual plot information for the
Simple Linear Regression is
somewhat suspect. While the
Versus Fits scatter plot does
indicate a random cloud of
residuals, the Histogram does
not resemble a clear Normal
Analysis of Variance
Curve.
In this case the
Histogram appears more right
Source
DF
SS
MS
F
P
skewed, and may therefore lead
Regression
1 309.92 309.92 183.96 0.000
Residual Error 190 320.10
1.68
to the questioning of the viability
Total
191 630.03
of this model as an overall
useful predictor. And while the
ANOVA information does lead to the conclusion that this model is useful1 to make Treasury Bill Security
rate predictions using the Federal Funds Rate, as indicated by the low P-value; the overall ability of the
Federal Funds Rate to predict the One Year Treasury Security rate requires further study.
H0: The Model is not useful and contains no linear relation.
Ha: The Model is useful and has linear relation.
Multiple Linear Regression: Using the Federal Funds Rate, Standard and Poor’s 500, and the Average
Selling Price of Housing in the United States to predict the yield to maturity on a One Year Treasury
Security.
H0: ß1: The Federal Funds Rate does not contribute to the model consisting of all other predictors.
H0: ß2: The Standard and Poor’s 500 does not contribute to the model consisting of all other
predictors.
H0: ß3: Average Selling Price of a US Home does not contribute to the model consisting of all other
predictors.
Figure 2: MiniTab Output for Multiple Linear Regression.
MiniTab was next used to
predict the monthly percentage
change in the One Year
Treasury Security using the
percentage changes in the
Federal Funds Rate, the
Predictor
Coef
SE Coef
T
P
Standard and Poor’s 500, and
Constant
-0.000197 0.008724 -0.02 0.982
the Average Selling Price (ASP)
Fed Funds Rate
0.09633
0.05022
1.92 0.057
S & P 500
-0.000237 0.001153 -0.21 0.838
of a US Home.
Output
Avg. Selling Price
-0.00046
0.01753 -0.03 0.979
confirmed the previous simple
linear regression; that there is
S = 0.0623407
R-Sq = 2.0%
R-Sq(adj) = 0.4%
indeed
some
relationship
between the Federal Funds Rate and the One Year Treasury Security. In this regression the relatively
low P-value for the Federal
Funds Rate indicates that it
does contribute to the model
which includes the S&P 500
and the ASP of a US Home.
However the exceptionally
high P-values of both the S &
P 500 and the ASP of a US
Home indicate they do not
contribute to the overall
model. The coefficients for
the slope of the lines of the S
& P 500 and the ASP of a US
Home are in direct conflict
with the slope of the Federal
Funds Rate.
The regression equation
Tbill_MnthlyPctChng = -
is
0.00020 + 0.0963 FFR_Pctchng
0.00024 SNP_PctChng_Monthly
0.0005 Housing_AvgPctChng
Residuals
for
the
Multiple
Linear
Regression
confirm
(despite
the
one outlier
Source
DF
SS
MS
F
P
Regression
3 0.014777 0.004926 1.27 0.287
in the Versus Fits scatter plot)
Residual Error 188 0.730636 0.003886
that this model is not useful;
Total
191 0.745413
ANOVA further solidifies this
confirmation. The large P-value
stemming from the ANOVA F-test indicates weak evidence against the model null hypothesis1; implying
that there is little to no linear relationship between the One Year Treasury Security and the combined
variables: Federal Funds Rate, S & P 500, and ASP of a US Home.
Analysis of Variance
Chi-Square Test: Comparing the proportions of increasing and decreasing months of the Federal
Reserve Federal Funds Rate and the Standard and Poor’s 500.
H0: The direction of movement of the Federal Funds Rate has no association with the direction of
movement in the Standard and Poor’s 500.
Ha: The direction of movement of the Federal Funds Rate has an association with the direction of
movement of the Standard and Poor’s 500.
Since the previous Multiple Linear
Regression analysis indicated that there
seemed to be no linear relationship between the
One Year Treasury Security and both the S & P
500 and the ASP of a US Home; MiniTab was
used to perform a Chi-Square test to determine
whether or not there was any association
between the movement (increasing or
decreasing) of the Federal Funds Rate and the
movement of the S & P 500. For the Federal
Funds Rate, all the months that had a positive
rate change were labeled as FFR Inc, and
subsequently all months that had a negative rate
change were labeled as FFR Dec. The same
process was then applied to the S & P 500 to
create categorical variables associated with a
numerical change.
Figure 3: MiniTab Output for Chi-Square Test.
Rows: S & P 500
Columns: Fed Funds Rate
FFR Dec
FFR Inc
All
SNP Dec
34
33.50
33
33.50
67
67.00
SNP Inc
62
62.50
63
62.50
125
125.00
All
96
96.00
96
96.00
192
192.00
Cell Contents:
Count
Expected count
Pearson Chi-Square = 0.023, DF = 1,
P-Value = 0.880
Likelihood Ratio Chi-Square = 0.023,
DF = 1, P-Value = 0.880
The Chi-Squared Test led to the
conclusion that there is no association between the movement of the Federal Funds Rate and the S & P
500. The P-value returned by the Chi-Square Test indicates very weak evidence against the null
hypothesis, leading to its acceptance.
Figure 4: MiniTab Output for two One Sample T-tests.
One-Sample T – 1990’s (January 1991 to December 1999)
One Sample and Two
Sample T-tests: Testing
the movement of the
average Federal Funds
Rate.
Test of mu = 4.11224 vs not = 4.11224
N
108
Mean
4.817
StDev
3.674
SE Mean
0.354
95% CI
(4.116, 5.518)
T
1.99
P
0.049
One-Sample T – 2000’s (January 2000 to December 2006)
Test of mu = 4.11224 vs not = 4.11224
N
84
Mean
3.206
StDev
4.121
SE Mean
0.450
95% CI
(2.312, 4.101)
T
-2.01
P
0.047
would be periods where the return on One Year Treasury Securities
as compared to the overall average.
Since the Federal
Funds Rate can be used to
estimate the One Year
Treasury Security, it is
important
to
determine
whether certain snippets of
time have an average rate
that is different from the
overall Federal Funds Rate.
If the mean Federal Funds
Rate moves, then there
would yield higher or lower returns
H01: The Federal Funds Rate of the 1990’s is no different than the overall Federal Funds Rate.
Ha1: The Federal Funds Rate of the 1990’s is different from the overall Federal Funds Rate.
H02: The Federal Funds Rate of the 2000’s is no different than the overall Federal Funds Rate.
Ha2: The Federal Funds Rate of the 2000’s is different from the overall Federal Funds Rate.
As the One Sample T-tests show, in both cases the sample means are different from the overall
averages. The P-value of 0.049 as returned by the 1990’s One Sample T-test indicates strong evidence
against H01. In this case the T-test confirms that the Federal Funds Rate in the 1990’s is not the same as
the overall Federal Funds Rate from January 1991 to December 2006. In fact, with 95% certainty, the
mean can be found somewhere between 4.116% and 5.518% as compared to the overall average of
4.112%.
The second One Sample T-test indicates that as well, the mean Federal Funds Rate during
January 2000 and December 2006 is not the same as the overall Federal Funds Rate. The P-value of
0.047 is strong evidence against H02 leading to the conclusion that this mean rate has also moved. Again,
with 95% certainty, the mean can be found somewhere between 2.312% and 4.101%.
Figure 5: MiniTab Output for a Two Sample T-test.
Two-Sample T-Test and CI – 1990’s vs 2000’s
Sample
1
2
N
108
84
Mean
4.82
3.21
StDev
3.67
4.12
SE Mean
0.35
0.45
Finally, since there was no
overlapping confidence interval
between the 1990’s and the 2000’s,
a Two Sample T-test was run to
determine if there was a difference
in the two means.
The test confirmed that yes
there
is
a difference in the two
Difference = mu (1) - mu (2)
means, as indicated by the small PEstimate for difference: 1.611
95% CI for difference: (0.481, 2.740)
value.
This is strong evidence
T-Test of difference = 0 (vs not =):
against the null hypothesis, leading
T-Value = 2.82 DF = 167
to a rejection of the null, and a
P-Value = 0.005
conclusion that the means are
indeed different. A 95% confidence interval puts the difference somewhere between 0.481% and
2.740%.
H0: The mean Federal Funds Rate of the 1990’s is not different from the mean Federal Funds Rate
in the 2000’s.
Ha: The mean Federal Funds Rate of the 1990’s is different from the mean Federal Funds Rate in
the 2000’s.
Data Sources
Bureau of Labor Statistics. Consumer Price Index - All Urban Consumers. US Department of Labor.
http://data.bls.gov/servlet/SurveyOutputServlet?data_tool=latest_numbers&series_id=CUSR0000SA0&ou
tput_view=pct_1mth (10 November 2008).
Federal Reserve Board of Governors. Monthly Federal Funds Rate.
http://www.federalreserve.gov/releases/h15/data/Monthly/H15_FF_O.txt (1 December 2008).
Federal Reserve Bank. One Year Government Securities by Month.
http://federalreserve.gov/releases/h15/data/Monthly/H15_TB_Y1.txt (4 December 2008).
Office of Federal Housing Enterprise Oversight. Housing Prices Indexes.
http://www.ofheo.gov/media/hpi/MonthlyIndex_to_1991.xls (8 November 2008).
The National Data Book. Banking, Finance, & Insurance: Money Stock, Interest Rates, Bond Yields. US
Census Bureau. http://www.census.gov/compendia/statab/tables/08s1167.xls (10 November 2008).
The Federal Reserve Board. H.15 Selected Interest Rates . Federal Reserve Bank.
https://www.federalreserve.gov/datadownload/Download.aspx?rel=H15&series=40afb80a445c5903ca2c4
888e40f3f1f&filetype=csv&label=include&layout=seriescolumn&from=01/01/1990&to=12/31/2008 (10
November 2008).
Yahoo! Finance. S&P 500 Historical Index.
http://finance.yahoo.com/q/hp?s=%5EGSPC&a=00&b=1&c=1990&d=11&e=31&f=2006&g=m (16
November 2008).