An Independent Research Project by
Transcript of An Independent Research Project by
YIELD CURVE INVERSION TRADING STRATEGIES
An Independent Research Project by:
Mary Rachide & Erik Schneider
Under the Direction of Faculty Advisor:
Dr. Campbell Harvey
The Fuqua School of Business
Duke University
April 28, 2003
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Hypothesis:
The ability to forecast the economic cycle will allow an informed investor to outperform the
widely discussed long-term strategies for investors – automatically rebalancing between a
diversified portfolio of equities and either a diversified portfolio of bonds or cash; or simply
buying and holding a diversified portfolio of equities.
Genesis:
Common wisdom holds that one of these scenarios must be superior over specific time
horizons. If different asset classes demonstrate mean reversion and typically outperform
during different phases of the business cycle, then an automatic rebalancing strategy will
systematically outperform a buy and hold strategy by causing an investor to sell relatively
more dear assets while acquiring ones that are relatively cheaper. Conversely, some pundits
hold that simply buying and holding equities is the superior solution over time. While
seeking to determine which of these competing camps is correct, we also sought an
opportunity to outperform either of these low-thought investment options by applying a
simple, recognizable signal that would require minimal monitoring or trading effort and
therefore be suitable for the average investor. As we believe that different asset classes do
perform in a differentiated manner across the economic cycle, we sought a signal for which
phase of the economic cycle we were experiencing (recession or expansion).
Data:
To benchmark our base case investment rules of automatic rebalancing and buy and hold we
utilized market benchmarks for the total returns of each asset class – Equity (S&P 500),
Bonds (U.S. Long-Term Government) and Cash (U.S. 30-day Treasury Bill). Additionally
we looked at an opportunity to subdivide Equity into Growth (Fama-French Large Company
Growth) and Value (Fama-French Large Company Value) components. All data for these
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total returns was provided by Ibbotson and data was utilized from January 1954 through June
2002.
To determine which phase of the economic cycle we were experiencing in the U.S. at any
given time we sought multiple signals. As a benchmark we used the National Bureau of
Economic Research official designations of economic peaks and troughs. Additionally we
utilized monthly data from the U.S. Treasury Department to identify 3-month, 5-year and 10-
year Treasury yields, from which we calculated yield spreads to determine periods of yield
curve inversion.
One Variable Data Analysis:
First we individually examined each of the asset classes to determine if the returns conform
to the normality assumption. A summary is tabulated below (full statistical analysis in
Appendix 1):
Asset Class Average Monthly
Total Return
Standard Deviation
in Monthly Total
Return
Skewness Kurtosis
Cash (T-bills) .00438379 .0028226 10.6368 8.51173
Bonds (LT Gov’t) .00509972 .0261816 4.85855 13.3507
Equity (S&P 500) .00922077 .0421526 -5.7003 12.3417
Growth .00997271 .0468033 -3.61399 9.60853
Value .0133739 .0450135 0.468686 11.6595
As presented in the summary table, none of our five asset classes pass the standard normality
tests of both skewness and kurtosis. None of the asset classes met the requirements of
kurtosis, and only one, Value, was normal by measure of skewness.
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T-bills, our proxy for holding cash, performed as expected with the lowest average total
monthly returns and a very tight standard deviation around the mean. T-bill total returns are
very positively skewed (resulting from the impossibility of negative returns) and also have
non-normal kurtosis readings.
Long-term government bonds have a slightly higher mean total return, but a much greater
standard deviation than T-bills. While this asset class is less positively skewed (some
negative returns are possible), it is still non-normal by the tests of skewness and even further
from normality as described by kurtosis. As compared to the T-bill, the long-term bond
returns a mere 7 basis point improvement in average total monthly return while the standard
deviation of returns is multiplied by about 9.3 times.
The S&P 500 is our proxy for equity, and average total returns and standard deviation are
both much higher than for either cash or long term government bonds. Interestingly, the S&P
500 is too negatively skewed to meet the requirements of a normal distribution. The negative
skewness implies that while an investor will make more on average by investing in equities
than in bonds, the investor will also be more likely to suffer from large negative returns in
equities than in bonds.
Growth equities, as defined by Fama & French, returned slightly more than all equities on an
average monthly return basis, involved a slightly higher standard deviation, and were
somewhat less negatively skewed.
Value equities, as defined by Fama & French were far and away the best performer of our
non-cash asset classes. Total average returns were significantly higher than Cash, Bonds,
S&P 500, and Growth equities. At the same time this was the equity class with the lowest
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standard deviation on monthly returns and was the only positively skewed asset class. In
terms of relating this to a trading strategy, investors always prefer positive skewness which
results in a greater likelihood of strong positive returns.
As we are attempting to determine a superior investment system over time, we also sought to
determine if it were possible to simply predict short-term returns of each of the asset classes
through time-series forecasting techniques. Unfortunately for us, but fortunately for the
proponents of efficient market theories, only Cash exhibited strong autocorrelations of
monthly returns. Assuring ourselves by this simple analysis that we would have to bring to
bear more information than a chart and a ruler, we also attempted to uncover any ability to
combine asset classes to create a portfolio with superior risk-adjusted returns.
Multiple Variable Data Analysis :
To evaluate the potential to create portfolios with superior risk-adjusted returns among our
asset classes, we first sought to uncover any clear relationships among the asset classes
selected. The degree and statistical certainty of correlation among the asset classes revealed
some interesting information. Not surprisingly, since the Growth and Value assets were
subsets of Equity, there was a very high and statistically significant correlation among these
three asset classes. This suggests that they could not be well combined to create lower risk
portfolios. Furthermore, we were surprised to discover that, while the degree of correlation
was not nearly as high as among the various subsets of equity asset classes, the correlation of
monthly bond returns with each of the other asset classes was statistically significant. Only
the T-bill, our proxy for cash, was not correlated with any of the other asset classes at the
95% confidence level. This suggests, and our later analysis confirmed, that Cash will be
superior to Bonds as an asset class to combine with any of the subsets of equity to create a
lower risk portfolio at any given level of return. A complete correlation matrix is presented in
Appendix 2.
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Correlation Matrix
LTBond
SP500
TBill
Growth
Value
ANOVA:
After creating a binomial indicator of the status of the yield curve, we applied a one-way
analysis of variance asset pairs to determine if the relative returns acted differently based on
the economic regime implied by the yield curve. Here we were testing to determine if the
yield curve determined by the 5-year bond would be more or less useful than that implied by
the 10-year bond. Using the 5-year bond determined yield curve, produced meaningfully
different relative returns for both the Bond – Cash and Cash – S&P 500 asset pairs. As we do
not believe that combining these two assets is a useful scenario, we hoped for more
interesting results from the 10-year bond determined yield curve. Asset pairs showing
statistically significant differences given both applied tests were Cash – S&P 500 and Cash –
Growth. Asset pairs showing statistically significant differences given only one of the
applied tests were Cash – Value and Bond – Growth. Details of the ANOVA analysis for
each of the 5-year and 10-year inversion indicators are included in Appendix 4 and Appendix
5, respectively.
Benchmark Returns:
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Since the ultimate measure of a superior investment system must be the level of risk-adjusted
returns, we developed benchmarks which combined each of the investment classes against
which we could compare our expected returns from implementing a predictive strategy. As
outlined above, we consider portfolios invested, 100% in Equity, 100% in Bonds or 100% in
Cash and combinations thereof. A graphical representation of the monthly returns and
standard deviation of those returns is below:
1954 - 1998
10% bonds
20% bonds
30% bonds
40% bonds
50% bonds
60% bonds
70% bonds
80% bonds
90% bonds
100% bonds
20% cash
30% cash
40% cash
50% cash
60% cash
70% cash
80% cash
90% cash
100% cash
100% equities
10% cash
0.0040
0.0050
0.0060
0.0070
0.0080
0.0090
0.0100
0.0110
0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200 0.0225 0.0250 0.0275 0.0300 0.0325 0.0350 0.0375 0.0400 0.0425 0.0450
Standard Deviation
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The buy-and-hold strategy demonstrated by the extreme points on the graph is clearly
implemented. A strategy involving a combination of the asset classes assumes monthly
rebalancing to the preferred level. Therefore, for an investor holding two asset classes, he
would sell some of the asset that performed better to buy some of the asset that performed
worse. The greater the disparity in relative performance, the greater the dollar volume of
assets shifted in a given month. Note that the shift of assets to cash always decreases the
standard deviation of monthly returns while that to bonds initially decreases variability, but
eventually serves to increase it. Additionally, there is a more complex set of portfolios that
could be created by combining each of the three asset classes, whose returns and standard
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deviation of returns would be contained in the space between the two sets of portfolios
outlined above.
Additionally, we benchmarked the potential to receive superior risk-adjusted returns by
subdividing the Equity asset class into Growth and Value components:
1954 - 1998
10% growth20% growth
30% growth40% growth50% growth60% growth70% growth
80% growth90% growth
100% growth
20% cash
30% cash
40% cash
50% cash
60% cash
70% cash
80% cash
90% cash
100% cash
20% bond
30% bond
40% bond
50% bond
60% bond
70% bond
80% bond
90% bond
100% bond
100% value
10% cash10% bond
0.0040
0.0060
0.0080
0.0100
0.0120
0.0140
0.0160
0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200 0.0225 0.0250 0.0275 0.0300 0.0325 0.0350 0.0375 0.0400 0.0425 0.0450 0.0475 0.0500
Standard Deviation
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As was clear from the Data Analysis above, the high correlation between Growth and Value
suggests that there would be little benefit to combining them in a single portfolio. This is
born out by the tight cluster of highly variable returns shown above. Since the 100% Value
asset provided superior returns and lower standard deviation, relative to the 100% Growth
asset, we subsequently combined the Value asset with the Bond and Cash assets as described
above.
Analysis:
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In selecting the forecast of the economic cycle, the question is raised as to whether prior
knowledge of the NBER’s assignment of the peaks and troughs of each economic cycle
would enable an investor to achieve excess returns over the aforementioned trading
strategies. Because American financial markets are highly liquid and well informed, the
effects of an economic downturn are probably foreseen and discounted into the equity
markets well in advance of the actual beginning of the contraction. We tested this belief over
the period from 1954 to 2001 and found, in fact, that prior knowledge of the NBER
assignments of economic peaks and troughs would be useful in creating an effective trading
strategy.
The strategies we examined were 1) investing in bonds in NBER defined recessions and
equities in expansion periods and 2) holding cash in NBER defined recessions and equities in
expansions. While the cash/equity strategy barely beat the benchmark, the bond/equity
strategy was clearly optimal to a simple buy and hold strategy. The problem, however, is that
this trading strategy relies upon knowing ahead of time whether you are in an expansion or
contraction period. As of now, there is no reliable way to predict the NBER economic cycles.
Thus we turn next to an examination of the yield curve as a means of identifying a superior
trading strategy. Building on the research of Dr. Campbell Harvey and others suggesting that
the yield curve is a predictor of the economic cycle, we tested the five and ten-year yield
spreads to the three-month as predictive variables. We also examined the Resnick Shoemaker
Probit Model. Prior to any examination of the data, we expected the ten-year yield spread to
be a stronger indicator because a more extreme expected economic downturn is necessary to
cause an inversion here than in the five-year yield spread. A preliminary examination of the
data reveals, however, that the ten-year is actually inverted more frequently than the five-
year. Even so, the ten-year spread over the three-month is still a better predictor of regime
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changes in the market. The probit model developed by Resnick and Shoemaker was not
useful, particularly over the period from 1998 – 2002 period.
The yield curve model was first examined relative to the S&P500 portfolios created by
rebalancing with Bonds or Cash. Then it was compared to that available from the
combination of Growth and Value with Bonds or Cash. The results for each case were similar
and are demonstrated graphically below. Note that the returns for the Value and Cash
strategy are superior to the Equity and Cash strategy, the standard deviation is also higher.
1954 - 1998
5 YR Inv - Bond
10 YR Inv - Bond5 YR Inv - Cash
10 YR Inv - Cash
5 YR Inv - Short
10 YR Inv - Short
0.0040
0.0050
0.0060
0.0070
0.0080
0.0090
0.0100
0.0110
0.0120
0.0300 0.0325 0.0350 0.0375 0.0400 0.0425 0.0450
Standard Deviation
Mo
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1954 - 1998
5 YR Inv - G V Momentum10 YR Inv - G V Momentum
5 YR Inv - G V Contrarian10 YR Inv - G V Contrarian5 YR Inv - Value Cash
10 YR Inv - Value Cash
0.0040
0.0050
0.0060
0.0070
0.0080
0.0090
0.0100
0.0110
0.0120
0.0130
0.0140
0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350 0.0400 0.0450 0.0500
Standard Deviation
Mo
nth
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rn
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The implication here is that the ten-year yield spread should be used to identify inversions.
Our analysis focused on only a one month lag of this variable because the information about
future economic prospects conveyed by such an inversion is immediately communicated to
the financial markets.
Mechanics:
Each trading strategy was developed in detail using Microsoft Excel. At the beginning of
each monthly period, current funds were calculated be multiplying the funds in each
investment option from the previous month by the total monthly returns for that vehicle. In
the case of a simple buy and hold strategy, this became the investment for the next month and
the process was repeated. For an automatic rebalancing strategy, the funds were reallocated
to match the desired allocation ratio after the previous monthly returns had been applied. By
this method, a small amount of whichever vehicle had returned the greatest amount was sold
and the proceeds were invested in the laggard. The return series described by these
mechanics became the benchmarks against which trading strategies were tested.
To apply each trading strategy, we inserted a check at the point of monthly reinvestment. If
the binomial indicator being examined indicated recession, all funds were switched to the
alternate investment strategy (e.g., 100% cash). Funds were maintained in the alternate
investment until the point where the binomial indicator returned to its normal state. At this
point, funds were divided again according to our benchmark strategy.
Testing a Variety of Strategies:
The buy and hold strategies we tested included: 100% growth, 100% value, 100% bonds,
100% cash. Simple, monthly rebalancing strategies included S&P to cash, bonds to S&P,
Value to Growth, Value to Bonds and cash to Value.
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We found that switching from a buy-and-hold or an auto-rebalancing strategy to allocating
monthly based on the binomial inversion factor yield excess returns. Because the cash to
Value was the most difficult of our benchmarks, we focused on this strategy in evaluating our
binomial yield curve inversion variable. Specifically, using a Cash to Value strategy based
on the binomial inversion factor outperforms holding cash, holding value, or rebalancing
among cash and Value on a monthly basis.
Out of Sample Testing:
To determine if the backward looking information provided by examining historical data
could be applied as a forward looking trading strategy, requires performing a test on data not
included in the original sample. In this case, a complicating factor is that we are seeking a
relatively rare, event-driven strategy, so there are not a large number of useful historical data
points. Additionally, each event will be driven by its own unique dynamics which may differ
radically from those of previous events. For our out of sample test, we held out data from
January 1998 through June 2002.
The result of examining Cash or Bonds versus the S&P 500 provides a much different picture
from the analysis period. This was due the collapse of long term bond rates to historic lows
during this time frame.
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1998 - 2002
10% bonds
20% bonds
30% bonds
40% bonds50% bonds
60% bonds80% bonds90% bonds
100% bonds
20% cash30% cash40% cash50% cash
60% cash
70% cash
80% cash
90% cash
100% cash
100% equities
70% bonds
10% cash
0.0035
0.0037
0.0039
0.0041
0.0043
0.0045
0.0047
0.0049
0.0051
0.0053
0.0055
0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350 0.0400 0.0450 0.0500 0.0550 0.0600
Standard Deviation
Mo
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A similar dynamic carried over to the examination of Growth versus Value and the
combination of value with Cash or Bonds. 100% Value was not the best strategy for equity
ownership, nor was further diversifying with Cash relative to Bonds. Over our out of sample
period, there is a return penalty for Cash, but it did serve to lower monthly standard deviation
of returns.
1998 - 2002
10% growth20% growth
60% growth
90% growth100% growth
20% cash30% cash
40% cash
50% cash
60% cash
70% cash
80% cash
100% cash
20% bond30% bond40% bond50% bond
60% bond70% bond
80% bond
90% bond
100% bond
100% value
30% growth40% growth
50% growth 70% growth80% growth
10% cash
90% cash
10% bond
0.0030
0.0035
0.0040
0.0045
0.0050
0.0055
0.0060
0.0065
0.0070
0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350 0.0400 0.0450 0.0500 0.0550 0.0600 0.0650
Standard Deviation
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Although there was a radical change in the shape of the return profile of (and indeed, the
choice of) superior underlying strategy, the value of trading on the yield curve inversion to
lower standard deviation of returns is maintained. In this case, although standard deviation of
monthly return would be less than that of any of the all equity portfolios the penalty to
returns is severe.
1998 - 2002
5 YR Inv - G V Momentum
5 YR Inv - G V Contrarian
10 YR Inv - G V Momentum
10 YR Inv - G V Contrarian
5 YR Inv - Value Cash 10 YR Inv - Value Cash
(0.0020)
0.0000
0.0020
0.0040
0.0060
0.0080
0.0100
0.0120
0.0140
0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350 0.0400 0.0450 0.0500 0.0550 0.0600 0.0650
Standard Deviation
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Conclusion:
Our analysis of the value of predicting the economic cycle and trading accordingly, showed a
number of interesting results. Within our benchmark period, we were surprised to discover
that Cash was superior to Bonds to combine with an equity portfolio. Further, even when we
had strong signals from our preferred yield curve inversion model, Cash was the better asset
to trade into versus Bonds or shorting the selected equity portfolio.
Across our benchmark period from 1954 to 1998, using an inverted yield curve as a trading
signal to convert from the S&P 500 to Cash served to increase monthly returns from 0.99%
to 1.09% while decreasing standard deviation of returns from 4.09% to 3.54%. Over the same
time period, we determined that a Value portfolio would have been a superior equity
Yield Curve Inversion Trading Strategies Mary Rachide Erik Schneider
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investment with a monthly return of 1.39% and a standard deviation of 4.39%. Using the
yield curve inversion as a trading signal did not return a similar increase in returns with
decreased standard deviation relative to the Value only equity portfolio. Instead, monthly
returns dropped to 1.30% while standard deviation dropped to 3.87%. It was not surprising to
discover that, compared to the particular equity class with the best ex post performance over
the time period, we would lose some return by trading into Cash. More importantly to us,
was the usefulness of the trading strategy when utilized ex ante. We therefore performed a
test on the period from 1998 to 2002 which had been held out of our benchmarking data.
Within this period, a radically inflating then deflating equity market, combined with a
simultaneous collapse in long-term interest rates, played havoc our expected buy and hold
strategies, but clearly demonstrated the value of utilizing the yield curve inversion trading
strategy. The S&P 500 returned 0.45% monthly with a standard deviation of 5.38% while
trading into Cash yield a nearly identical 0.44% monthly return, but with a standard deviation
of only 4.61%. We would however, based on our benchmark data, have been using the Value
portfolio which returned 0.63% monthly with a standard deviation of 5.57%. By applying the
yield curve inversion trading strategy, monthly returns would have dropped to 0.48% with a
commensurate decrease in standard deviation to 4.68%.
Collectively, the results point to the conclusion that compared to simply investing in the S&P
500, applying a yield curve inversion signal to trade into and out of Cash can be expected to
significantly reduce standard deviation of returns, without negatively impacting returns.
However, compared to a Value only equity investment, trading based on the yield curve
inversion will decrease standard deviation of returns, but at the cost of decreased monthly
returns.
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Bibliography Federal Reserve Statistical Release, “Selected Interest Rates.” December 2002.
http://www.federalreserve.gov/releases/h15/data.htm.
French, Kenneth R. “Description of Fama-French Benchmark Factors.”
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library
Harvey, Campbell R., “Term Structure and the Economic Growth: The Recession of 2001.”
Duke University; Fall 2002.
Ibbotson Financial Database. January 1926 – June 2002
Liu, Wei, Bruce G. Resnick and Gary L. Shoesmith, Market Timing of International Stock
Markets Using the Yield Spread. Indiana University and Wake Forest University;
2002.
Resnick, Bruce G. and Gary Shoesmith, “Using the Yield Curve to Time the Stock Market.”
Financial Analysts Journal. Charlottesville; May/Jun 2002.
Appendix 1 – One Variable Data Analysis
Page 16 of 58
Scatterplot
LTBond-0.09 -0.05 -0.01 0.03 0.07 0.11 0.15
Normal Probability Plot
LTBond
perc
enta
ge
-0.09 -0.05 -0.01 0.03 0.07 0.11 0.150.1
15
2050809599
99.9
Symmetry Plot
distance below median
dist
ance
abo
ve m
edia
n
0 0.03 0.06 0.09 0.12 0.150
0.03
0.06
0.09
0.12
0.15
Appendix 1 – One Variable Data Analysis Descriptive statistics (measures of central tendency, variability, and distribution shape) for each asset class are detailed within this appendix. None of the asset classes passes both the skewness and kurtosis tests of normality. Descriptive Statistics – Long Term Government Bonds Summary Statistics for LTBond Count = 582 Average = 0.00509972 Variance = 0.000685477 Standard deviation = 0.0261816 Minimum = -0.0878044 Maximum = 0.141801 Range = 0.229605 Stnd. skewness = 4.85855 Stnd. kurtosis = 13.3507
Appendix 1 – One Variable Data Analysis
Page 17 of 58
Scatterplot
TBill0 3 6 9 12 15
(X 0.001)
Symmetry Plot
distance below median
dist
ance
abo
ve m
edia
n
0 2 4 6 8 10(X 0.001)
0
2
4
6
8
10(X 0.001)
Normal Probability Plot
TBill
perc
enta
ge
0 3 6 9 12 15(X 0.001)
0.115
2050809599
99.9
Descriptive Statistics – Treasury Bills
Summary Statistics for TBill Count = 582 Average = 0.00438379 Variance = 0.00000520872 Standard deviation = 0.00228226 Minimum = 0.000296956 Maximum = 0.013385 Range = 0.0130881 Stnd. skewness = 10.6368 Stnd. kurtosis = 8.51173
Appendix 1 – One Variable Data Analysis
Page 18 of 58
Scatterplot
SP500-0.25 -0.15 -0.05 0.05 0.15 0.25
Normal Probability Plot
SP500
perc
enta
ge
-0.25 -0.15 -0.05 0.05 0.15 0.250.1
15
2050809599
99.9
Symmetry Plot
distance below median
dist
ance
abo
ve m
edia
n
0 0.05 0.1 0.15 0.2 0.25 0.30
0.05
0.1
0.15
0.2
0.25
0.3
Descriptive Statistics – S&P 500
Summary Statistics for SP500 Count = 582 Average = 0.00922077 Variance = 0.00177684 Standard deviation = 0.0421526 Minimum = -0.242326 Maximum = 0.153341 Range = 0.395667 Stnd. skewness = -5.7003 Stnd. kurtosis = 12.3417
Appendix 1 – One Variable Data Analysis
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Scatterplot
Growth-0.24 -0.14 -0.04 0.06 0.16 0.26
Symmetry Plot
distance below median
dist
ance
abo
ve m
edia
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0 0.05 0.1 0.15 0.2 0.25 0.30
0.05
0.1
0.15
0.2
0.25
0.3
Descriptive Statistics – Fama & French Growth Equities
Summary Statistics for Growth Count = 576 Average = 0.00997371 Variance = 0.00219054 Standard deviation = 0.0468033 Minimum = -0.239951 Maximum = 0.210817 Range = 0.450768 Stnd. skewness = -3.61399 Stnd. kurtosis = 9.60853
Normal Probability Plot
Growth
perc
enta
ge
-0.24 -0.14 -0.04 0.06 0.16 0.260.1
15
2050809599
99.9
Appendix 1 – One Variable Data Analysis
Page 20 of 58
Scatterplot
Value-0.2 -0.1 0 0.1 0.2 0.3
Normal Probability Plot
Value
perc
enta
ge
-0.2 -0.1 0 0.1 0.2 0.30.1
15
2050809599
99.9
Symmetry Plot
distance below median
dist
ance
abo
ve m
edia
n
0 0.04 0.08 0.12 0.16 0.2 0.240
0.04
0.08
0.12
0.16
0.2
0.24
Descriptive Statistics – Fama & French Value Equities
Summary Statistics for Value Count = 576 Average = 0.0133739 Variance = 0.00202622 Standard deviation = 0.0450135 Minimum = -0.198438 Maximum = 0.230568 Range = 0.429006 Stnd. skewness = 0.468686 Stnd. kurtosis = 11.6595
Appendix 2 – Autocorrelation Analysis
Page 21 of 58
Estimated Autocorrelations for LTBond
lag
Aut
ocor
rela
tions
0 2 4 6 8-1
-0.6
-0.2
0.2
0.6
1
Time Series Plot for LTBond
LTB
ond
1/26 1/35 1/44 1/53 1/62 1/71 1/80-0.09
-0.05
-0.01
0.03
0.07
0.11
0.15
Appendix 2 – Autocorrelation Analysis This appendix illustrates the autocorrelation analysis conducted for each asset class. Tables show the estimated autocorrelations between values of each asset at various lags. Autocorrelation Analysis – Long Term Government Bonds Data variable: LTBond Number of observations = 582 Start index = 1/26 Sampling interval = 1.0 month(s)
Estimated Autocorrelations for LTBond
Lower 95.0% Upper 95.0% Lag Autocorrelation Stnd. Error Prob. Limit Prob. Limit ---------------------------------------------------------------------------------- 1 0.0625393 0.0414513 -0.0812433 0.0812433 2 -0.00336936 0.0416131 -0.0815604 0.0815604 3 -0.0997163 0.0416136 -0.0815613 0.0815613 4 0.0541813 0.0420222 -0.0823621 0.0823621 5 0.0341596 0.042142 -0.082597 0.082597 6 0.0373293 0.0421896 -0.0826902 0.0826902 In this case, one of the 6 autocorrelation coefficients is statistically significant at the 95.0% confidence level, implying that the time series is probably completely random.
Appendix 2 – Autocorrelation Analysis
Page 22 of 58
Estimated Autocorrelations for TBill
lag
Aut
ocor
rela
tions
0 2 4 6 8-1
-0.6
-0.2
0.2
0.6
1
Autocorrelation Analysis – Treasury Bills Data variable: TBill Number of observations = 582 Start index = 1/26 Sampling interval = 1.0 month(s)
Estimated Autocorrelations for TBill Lower 95.0% Upper 95.0% Lag Autocorrelation Stnd. Error Prob. Limit Prob. Limit ---------------------------------------------------------------------------------- 1 0.95238 0.0414513 -0.0812433 0.0812433 2 0.921242 0.0695352 -0.136287 0.136287 3 0.899639 0.0880431 -0.172562 0.172562 4 0.872016 0.10263 -0.201151 0.201151 5 0.854027 0.114656 -0.224722 0.224722 6 0.837928 0.125109 -0.24521 0.24521 In this case, 6 of the 6 autocorrelation coefficients are statistically significant at the 95.0% confidence level, implying that the time series is probably completely random.
Time Series Plot for TBill
TB
ill
1/26 1/35 1/44 1/53 1/62 1/71 1/800
3
6
9
12
15(X 0.001)
Appendix 2 – Autocorrelation Analysis
Page 23 of 58
Time Series Plot for SP500
SP
500
1/26 1/35 1/44 1/53 1/62 1/71 1/80-0.25
-0.15
-0.05
0.05
0.15
0.25
Estimated Autocorrelations for SP500
lag
Aut
ocor
rela
tions
0 2 4 6 8-1
-0.6
-0.2
0.2
0.6
1
Autocorrelation Analysis – S&P 500
Data variable: SP500 Number of observations = 582 Start index = 1/26 Sampling interval = 1.0 month(s)
Estimated Autocorrelations for SP500 Lower 95.0% Upper 95.0% Lag Autocorrelation Stnd. Error Prob. Limit Prob. Limit ---------------------------------------------------------------------------------- 1 0.0206921 0.0414513 -0.0812433 0.0812433 2 -0.0319949 0.0414691 -0.0812781 0.0812781 3 0.00936037 0.0415115 -0.0813611 0.0813611 4 0.0242488 0.0415151 -0.0813683 0.0813683 5 0.0892807 0.0415394 -0.0814159 0.0814159 6 -0.0456714 0.0418678 -0.0820596 0.0820596 In this case, one of the 6 autocorrelation coefficients is statistically significant at the 95.0% confidence level, implying that the time series is probably completely random.
Appendix 2 – Autocorrelation Analysis
Page 24 of 58
Time Series Plot for Growth
Gro
wth
1/26 1/35 1/44 1/53 1/62 1/71 1/80-0.24
-0.14
-0.04
0.06
0.16
0.26
Estimated Autocorrelations for Growth
lag
Aut
ocor
rela
tions
0 2 4 6 8-1
-0.6
-0.2
0.2
0.6
1
Autocorrelation Analysis – Fama & French Growth Equities Data variable: Growth Number of observations = 576 Start index = 1/26 Sampling interval = 1.0 month(s)
Estimated Autocorrelations for Growth
Lower 95.0% Upper 95.0% Lag Autocorrelation Stnd. Error Prob. Limit Prob. Limit ---------------------------------------------------------------------------------- 1 0.0618676 0.0416667 -0.0816653 0.0816653 2 -0.0378888 0.0418258 -0.0819773 0.0819773 3 0.00111439 0.0418854 -0.082094 0.082094 4 0.0091741 0.0418854 -0.0820941 0.0820941 5 0.0363455 0.0418889 -0.082101 0.082101 6 -0.0299562 0.0419436 -0.0822082 0.0822082 In this case, none of the 6 autocorrelations coefficients are statistically significant, implying that the time series may well be completely random.
Appendix 2 – Autocorrelation Analysis
Page 25 of 58
Estimated Autocorrelations for Value
lag
Aut
ocor
rela
tions
0 2 4 6 8-1
-0.6
-0.2
0.2
0.6
1
Time Series Plot for Value
Val
ue
1/26 1/35 1/44 1/53 1/62 1/71 1/80-0.2
-0.1
0
0.1
0.2
0.3
Autocorrelation Analysis – Fama & French Value Equities Data variable: Value Number of observations = 576 Start index = 1/26 Sampling interval = 1.0 month(s)
Estimated Autocorrelations for Value Lower 95.0% Upper 95.0% Lag Autocorrelation Stnd. Error Prob. Limit Prob. Limit ---------------------------------------------------------------------------------- 1 0.0840571 0.0416667 -0.0816653 0.0816653 2 -0.0263515 0.04196 -0.0822403 0.0822403 3 -0.00753506 0.0419888 -0.0822966 0.0822966 4 -0.0106556 0.0419911 -0.0823012 0.0823012 5 0.0562525 0.0419958 -0.0823104 0.0823104 6 -0.0202184 0.0421264 -0.0825664 0.0825664 In this case, one of the 6 autocorrelation coefficients is statistically significant at the 95.0% confidence level, implying that the time series is probably completely random.
Appendix 3: Multiple Variable Analysis
Page 26 of 58
Appendix 4: Multiple Variable Analysis Data variables: LTBond, SP500, TBill, Growth, Value There are 576 complete cases for use in the correlation calculations. Correlation Matrix
LTBond SP500 TBill Growth Value ----------------------------------------------------------------------------------------------------------------------- LTBond 0.2378 0.1086 0.2223 0.2063
0.0000 0.0091 0.0000 0.0000 SP500 0.2378 -0.0783 0.9680 0.8558
0.0000 0.0604 0.0000 0.0000 TBill 0.1086 -0.0783 -0.0802 -0.0645
0.0091 0.0604 0.0545 0.1219 Growth 0.2223 0.9680 -0.0802 0.7795
0.0000 0.0000 0.0545 0.0000 Value 0.2063 0.8558 -0.0645 0.7795
0.0000 0.0000 0.1219 0.0000 ------------------------------------------------------------------------------------------------------------------------ Correlation P-Value This table shows Pearson product moment correlations between each pair of variables to measure the strength of the linear relationship between the variables. The third number in each location of the table is a P-value which tests the statistical significance of the estimated correlations. P-values below 0.05 indicate statistically significant non-zero correlations at the 95% confidence level. The following pairs of variables have P-values below 0.05, indicating that they are correlated. LTBond and SP500 LTBond and TBill LTBond and Growth LTBond and Value SP500 and Growth SP500 and Value Growth and Value SP500 and Value Growth and Value
Appendix 4: ANOVA on the 5-Year Inversion
Page 27 of 58
Box-and-Whisker Plot
BondSPDelta
lag(
Inv_
5_yr
,1)
0
1
-0.14 -0.04 0.06 0.16 0.26 0.36
Means and 95.0 Percent LSD Intervals
lag(Inv_5_yr,1)
Bon
dSP
Del
ta
0 1-9
-6
-3
0
3
6(X 0.001)
Appendix 4: ANOVA on the 5-Year Inversion Within this appendix, one-way analysis of variance tests for the deltas between monthly returns of sets of two asset classes are presented based on our binomial classification of yield curve inversions on a one month lag. This analysis allows the comparison of the relative performance of two asset classes in and out of inverted yield curves. The F-test in the ANOVA tables test whether there are any significant differences amongst the means. Because of the non-normality of the distributions of returns for our asset classes, the Kruskal-Wallis test provides a good secondary test of differences in relative asset class performance in and out of inverted yield curves. The Kruskal-Wallis test evaluates the null hypothesis that the medians of the relative performance metric within each of the lagged binomial yield curve classification are statistically the same. Use of the median instead of the mean as the test criteria eliminates the impact of outliers in the data set. To conduct the test, the data from both periods of inversion and normal yield curves are combined and ranked from smallest to largest. The average rank is then computed for the data for each of the binomially classified sets. If the P-value from this test is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.
One-Way ANOVA – Relative Performance of LT Bonds and the S&P 500 Dependent variable: BondSPDelta Factor: lag(Inv_5_yr,1) Number of observations: 581 Number of levels: 2
Appendix 4: ANOVA on the 5-Year Inversion
Page 28 of 58
One-Way ANOVA – Relative Performance of LT Bonds and the S&P 500 cont. Summary Statistics for BondSPDelta lag(Inv_5_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 494 -0.00446731 0.0018014 0.0424429 -0.122186 1 87 -0.00170421 0.00290541 0.0539018 -0.136218 ---------------------------------------------------------------------------------------------------------------- Total 581 -0.00405356 0.00196297 0.0443054 -0.136218 lag(Inv_5_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ---------------------------------------------------------------------------------------------------------------- 0 0.302746 0.424932 9.1155 27.391 1 0.148834 0.285053 0.919685 0.27451 ---------------------------------------------------------------------------------------------------------------- Total 0.302746 0.438964 8.13383 21.8447 ANOVA Table for BondSPDelta by lag(Inv_5_yr,1)
------------------------------------------------------------------------------------------------------------ Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------ Between groups 0.000564763 1 0.000564763 0.29 0.5921 Within groups 1.13796 579 0.00196538 ------------------------------------------------------------------------------------------------------------ Total (Corr.) 1.13852 580 The F-ratio, which in this case equals 0.287355, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean BondSPDelta from one level of lag(Inv_5_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BondSPDelta by lag(Inv_5_yr,1) lag(Inv_5_yr,1) Sample Size Average Rank ------------------------------------------------------------ 0 494 290.237 1 87 295.333 ------------------------------------------------------------ Test statistic = 0.0681859 P-Value = 0.793997 Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.
Appendix 4: ANOVA on the 5-Year Inversion
Page 29 of 58
Means and 95.0 Percent LSD Intervals
lag(Inv_5_yr,1)
Bon
dBill
Del
ta
0 1-10
-7
-4
-1
2
5(X 0.001)
Box-and-Whisker Plot
BondBillDelta
lag(
Inv_
5_yr
,1)
0
1
-0.1 -0.06 -0.02 0.02 0.06 0.1 0.14
One-Way ANOVA - Relative Performance of LT Bonds and T-Bills Dependent variable: BondBillDelta Factor: lag(Inv_5_yr,1) Number of observations: 581 Number of levels: 2
Summary Statistics for BondBillDelta lag(Inv_5_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 494 0.0017383 0.000590337 0.0242969 -0.0826588 1 87 -0.00517067 0.00115448 0.0339776 -0.0964916 ---------------------------------------------------------------------------------------------------------------- Total 581 0.000703737 0.000679055 0.0260587 -0.0964916 lag(Inv_5_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.121288 0.203947 3.47795 9.34901 1 0.129323 0.225814 1.46602 4.32526 ---------------------------------------------------------------------------------------------------------------- Total 0.129323 0.225814 2.98716 11.948
Appendix 4: ANOVA on the 5-Year Inversion
Page 30 of 58
One-Way ANOVA - Relative Performance of LT Bonds and T-Bills cont.
ANOVA Table for BondBillDelta by lag(Inv_5_yr,1) Analysis of Variance --------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value --------------------------------------------------------------------------------------------------------- Between groups 0.00353098 1 0.00353098 5.24 0.0225 Within groups 0.390321 579 0.00067413 --------------------------------------------------------------------------------------------------------- Total (Corr.) 0.393852 580 The F-ratio, which in this case equals 5.23784, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is less than 0.05, there is a statistically significant difference between the mean BondBillDelta from one level of lag(Inv_5_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BondBillDelta by lag(Inv_5_yr,1) lag(Inv_5_yr,1) Sample Size Average Rank ------------------------------------------------------------ 0 494 297.067 1 87 256.552 ------------------------------------------------------------ Test statistic = 4.30909 P-Value = 0.0379059 The Kruskal-Wallis test tests the null hypothesis that the medians of BondBillDelta within each of the 2 levels of lag(Inv_5_yr,1) are the same. Since the P-value is less than 0.05, there is a statistically significant difference amongst the medians at the 95.0% confidence level.
Appendix 4: ANOVA on the 5-Year Inversion
Page 31 of 58
Box-and-Whisker Plot
BillSPDelta
lag(
Inv_
5_yr
,1)
0
1
-0.15 -0.05 0.05 0.15 0.25
Means and 95.0 Percent LSD Intervals
lag(Inv_5_yr,1)
Bill
SP
Del
ta
0 1-9
-5
-1
3
7
11(X 0.001)
One-Way ANOVA - Relative Performance of T-Bills and the S&P 500 Analysis Summary Dependent variable: BillSPDelta Factor: lag(Inv_5_yr,1) Number of observations: 581 Number of levels: 2
Summary Statistics for BillSPDelta lag(Inv_5_yr,1) Count Average Variance Standard dev. Minimum ----------------------------------------------------------------------------------------------------------- 0 494 -0.00620561 0.00162372 0.0402954 -0.121873 1 87 0.00346646 0.00271202 0.0520771 -0.1483 ------------------------------------------------------------------------------------------------------------Total 581 -0.0047573 0.00179422 0.0423583 -0.1483 lag(Inv_5_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ------------------------------------------------------------------------------------------------------------0 0.248282 0.370155 6.61718 15.2549 1 0.132443 0.280743 0.225281 0.613454 ------------------------------------------------------------------------------------------------------------ Total 0.248282 0.396582 5.92577 12.3445
Appendix 4: ANOVA on the 5-Year Inversion
Page 32 of 58
One-Way ANOVA - Relative Performance of T-Bills and the S&P 500 cont. ANOVA Table for BillSPDelta by lag(Inv_5_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00692005 1 0.00692005 3.88 0.0495 Within groups 1.03373 579 0.00178537 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.04065 580 The F-ratio, which in this case equals 3.87598, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is less than 0.05, there is a statistically significant difference between the mean BillSPDelta from one level of lag(Inv_5_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BillSPDelta by lag(Inv_5_yr,1) lag(Inv_5_yr,1) Sample Size Average Rank ------------------------------------------------------------ 0 494 285.856 1 87 320.207 ------------------------------------------------------------ Test statistic = 3.09757 P-Value = 0.0784055 Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.
Appendix 4: ANOVA on the 5-Year Inversion
Page 33 of 58
Means and 95.0 Percent LSD Intervals
lag(Inv_5_yr,1)
GV
Del
ta
0 1-14
-11
-8
-5
-2
1(X 0.001)
Box-and-Whisker Plot
GVDelta
lag(
Inv_
5_yr
,1)
0
1
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
One-Way ANOVA - Relative Performance of Growth and Value Dependent variable: GVDelta Factor: lag(Inv_5_yr,1) Number of observations: 575 Number of levels: 2
Summary Statistics for GVDelta lag(Inv_5_yr,1) Count Average Variance Standard dev. Minimum ----------------------------------------------------------------------------------------------------------------0 488 -0.0023697 0.000801739 0.028315 -0.10753 1 87 -0.00877434 0.00164484 0.0405567 -0.14001 ---------------------------------------------------------------------------------------------------------------- Total 575 -0.00333875 0.000931937 0.0305276 -0.14001 lag(Inv_5_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.12503 0.23256 -0.746931 9.11199 1 0.138661 0.278671 0.189348 4.62528 ---------------------------------------------------------------------------------------------------------------- Total 0.138661 0.278671 -1.09382 12.7949
Appendix 4: ANOVA on the 5-Year Inversion
Page 34 of 58
One-Way ANOVA - Relative Performance of Growth and Value cont. ANOVA Table for GVDelta by lag(Inv_5_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------ Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------ Between groups 0.00302873 1 0.00302873 3.26 0.0714 Within groups 0.531903 573 0.000928278 ------------------------------------------------------------------------------------------------------------ Total (Corr.) 0.534932 574 The F-ratio, which in this case equals 3.26274, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean GVDelta from one level of lag(Inv_5_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for GVDelta by lag(Inv_5_yr,1) lag(Inv_5_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 488 293.214 1 87 258.753 -------------------------------------------------------------- Test statistic = 3.17705 P-Value = 0.0746756 The Kruskal-Wallis test tests the null hypothesis that the medians of GVDelta within each of the 2 levels of lag(Inv_5_yr,1) are the same. Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.
Appendix 4: ANOVA on the 5-Year Inversion
Page 35 of 58
Means and 95.0 Percent LSD Intervals
lag(Inv_5_yr,1)
Bon
dGD
elta
0 1-9
-6
-3
0
3
6(X 0.001)
Box-and-Whisker Plot
BondGDelta
lag(
Inv_
5_yr
,1)
0
1
-0.17 -0.07 0.03 0.13 0.23 0.33
One-Way ANOVA - Relative Performance of LT Bonds and Growth
Dependent variable: BondGDelta Factor: lag(Inv_5_yr,1) Number of observations: 575 Number of levels: 2
Summary Statistics for BondGDelta lag(Inv_5_yr,1) Count Average Variance Standard dev. Minimum ----------------------------------------------------------------------------------------------------------------0 488 -0.00541221 0.00209873 0.0458119 -0.136733 1 87 -0.00164752 0.00368114 0.0606724 -0.163053 ---------------------------------------------------------------------------------------------------------------- Total 575 -0.0048426 0.00233398 0.0483113 -0.163053 lag(Inv_5_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.30037 0.437104 6.88741 19.2342 1 0.162041 0.325094 0.423122 0.510276 ---------------------------------------------------------------------------------------------------------------- Total 0.30037 0.463424 5.91763 15.4029
Appendix 4: ANOVA on the 5-Year Inversion
Page 36 of 58
One-Way ANOVA - Relative Performance of LT Bonds and Growth cont.
ANOVA Table for BondGDelta by lag(Inv_5_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00104648 1 0.00104648 0.45 0.5036 Within groups 1.33866 573 0.00233623 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.3397 574 The F-ratio, which in this case equals 0.447935, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean BondGDelta from one level of lag(Inv_5_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BondGDelta by lag(Inv_5_yr,1) lag(Inv_5_yr,1) Sample Size Average Rank ------------------------------------------------------------ 0 488 286.299 1 87 297.54 ------------------------------------------------------------ Test statistic = 0.338046 P-Value = 0.560959 The Kruskal-Wallis test tests the null hypothesis that the medians of BondGDelta within each of the 2 levels of lag(Inv_5_yr,1) are the same. Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.
Appendix 4: ANOVA on the 5-Year Inversion
Page 37 of 58
Means and 95.0 Percent LSD Intervals
lag(Inv_5_yr,1)
Bon
dVD
elta
0 1-18
-15
-12
-9
-6
-3(X 0.001)
Box-and-Whisker Plot
BondVDelta
lag(
Inv_
5_yr
,1)
0
1
-0.21 -0.11 -0.01 0.09 0.19 0.29
One-Way ANOVA - Relative Performance of LT Bonds and Value Dependent variable: BondVDelta Factor: lag(Inv_5_yr,1) Number of observations: 575 Number of levels: 2
Summary Statistics for BondVDelta lag(Inv_5_yr,1) Count Average Variance Standard dev. Minimum ----------------------------------------------------------------------------------------------------------------0 488 -0.00778192 0.00203464 0.045107 -0.202883 1 87 -0.0104219 0.00330588 0.0574968 -0.208345 ---------------------------------------------------------------------------------------------------------------- Total 575 -0.00818135 0.00222246 0.0471429 -0.208345 lag(Inv_5_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.258857 0.461741 2.62287 14.5755 1 0.105941 0.314285 -2.18773 2.57142 ---------------------------------------------------------------------------------------------------------------- Total 0.258857 0.467202 0.497432 13.9685
Appendix 4: ANOVA on the 5-Year Inversion
Page 38 of 58
One-Way ANOVA - Relative Performance of LT Bonds and Value cont.
ANOVA Table for BondVDelta by lag(Inv_5_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.000514591 1 0.000514591 0.23 0.6308 Within groups 1.27518 573 0.00222544 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.27569 574 The F-ratio, which in this case equals 0.231231, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a\ statistically significant difference between the mean BondVDelta from one level of lag(Inv_5_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BondVDelta by lag(Inv_5_yr,1) lag(Inv_5_yr,1) Sample Size Average Rank ------------------------------------------------------------ 0 488 288.631 1 87 284.46 ------------------------------------------------------------ Test statistic = 0.0465502 P-Value = 0.829179 The Kruskal-Wallis test tests the null hypothesis that the medians of BondVDelta within each of the 2 levels of lag(Inv_5_yr,1) are the same. Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.
Appendix 4: ANOVA on the 5-Year Inversion
Page 39 of 58
Means and 95.0 Percent LSD Intervals
lag(Inv_5_yr,1)
BillG
Del
ta
0 1-11
-7
-3
1
5
9
13(X 0.001)
Box-and-Whisker Plot
BillGDelta
lag(
Inv_
5_yr
,1)
0
1
-0.21 -0.11 -0.01 0.09 0.19 0.29
One-Way ANOVA - Relative Performance of T-Bills and Growth
Dependent variable: BillGDelta Factor: lag(Inv_5_yr,1) Number of observations: 575 Number of levels: 2
Summary Statistics for BillGDelta lag(Inv_5_yr,1) Count Average Variance Standard dev. Minimum ----------------------------------------------------------------------------------------------------------------0 488 -0.0071054 0.00196346 0.0443109 -0.137027 1 87 0.00352315 0.00356451 0.0597036 -0.205777 ---------------------------------------------------------------------------------------------------------------- Total 575 -0.00549725 0.00221445 0.0470579 -0.205777 lag(Inv_5_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.245906 0.382933 4.99172 10.1325 1 0.14565 0.351426 -0.697078 1.85611 ---------------------------------------------------------------------------------------------------------------- Total 0.245906 0.451683 3.91312 9.52679
Appendix 4: ANOVA on the 5-Year Inversion
Page 40 of 58
One-Way ANOVA - Relative Performance of T-Bills and Growth cont.
ANOVA Table for BillGDelta by lag(Inv_5_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00834101 1 0.00834101 3.78 0.0522 Within groups 1.26275 573 0.00220376 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.27109 574 The F-ratio, which in this case equals 3.7849, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean BillGDelta from one level of lag(Inv_5_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BillGDelta by lag(Inv_5_yr,1) lag(Inv_5_yr,1) Sample Size Average Rank --------------------------------------------------------------- 0 488 282.561 1 87 318.506 --------------------------------------------------------------- Test statistic = 3.45638 P-Value = 0.0630045 The Kruskal-Wallis test tests the null hypothesis that the medians of BillGDelta within each of the 2 levels of lag(Inv_5_yr,1) are the same. Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.
Appendix 4: ANOVA on the 5-Year Inversion
Page 41 of 58
Means and 95.0 Percent LSD Intervals
lag(Inv_5_yr,1)
BillV
Del
ta
0 1-13
-10
-7
-4
-1
2(X 0.001)
Box-and-Whisker Plot
BillVDelta
lag(
Inv_
5_yr
,1)
0
1
-0.23 -0.13 -0.03 0.07 0.17 0.27
One-Way ANOVA - Relative Performance of T-Bills and Value
Dependent variable: BillVDelta Factor: lag(Inv_5_yr,1) Number of observations: 575 Number of levels: 2
Summary Statistics for BillVDelta lag(Inv_5_yr,1) Count Average Variance Standard dev. Minimum ----------------------------------------------------------------------------------------------------------------0 488 -0.0094751 0.00182446 0.0427137 -0.207198 1 87 -0.0052512 0.00326529 0.0571427 -0.224761 ---------------------------------------------------------------------------------------------------------------- Total 575 -0.00883601 0.00203945 0.0451602 -0.224761 lag(Inv_5_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.204393 0.411591 1.27693 9.91818 1 0.149888 0.374649 -2.05064 4.2917 ---------------------------------------------------------------------------------------------------------------- Total 0.204393 0.429154 -0.329068 11.8175
Appendix 4: ANOVA on the 5-Year Inversion
Page 42 of 58
One-Way ANOVA - Relative Performance of T-Bills and Value cont.
ANOVA Table for BillVDelta by lag(Inv_5_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00131734 1 0.00131734 0.65 0.4220 Within groups 1.16933 573 0.00204071 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.17064 574 The F-ratio, which in this case equals 0.645533, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean BillVDelta from one level of lag(Inv_5_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BillVDelta by lag(Inv_5_yr,1) lag(Inv_5_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 488 284.9 1 87 305.391 -------------------------------------------------------------- Test statistic = 1.1233 P-Value = 0.289206 The Kruskal-Wallis test tests the null hypothesis that the medians of BillVDelta within each of the 2 levels of lag(Inv_5_yr,1) are the same. Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.
Appendix 5: ANOVA on 10-Year Inversion
Page 43 of 58
Means and 95.0 Percent LSD Intervals
lag(Inv_10_yr,1)
Bon
dSP
Del
ta
0 1-9
-5
-1
3
7
11(X 0.001)
Box-and-Whisker Plot
BondSPDelta
lag(
Inv_
10_y
r,1)
0
1
-0.14 -0.04 0.06 0.16 0.26 0.36
Appendix 5: ANOVA on 10-Year Inversion One-Way ANOVA – Relative Performance of LT Bonds and the S&P 500 Analysis Summary Dependent variable: BondSPDelta Factor: lag(Inv_10_yr,1) Number of observations: 581 Number of levels: 2
Summary Statistics for BondSPDelta lag(Inv_10_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 479 -0.00564214 0.00180792 0.0425196 -0.122186 1 102 0.0034065 0.00264803 0.051459 -0.136218 ---------------------------------------------------------------------------------------------------------------- Total 581 -0.00405356 0.00196297 0.0443054 -0.136218 lag(Inv_10_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.302746 0.424932 9.4434 28.1298 1 0.148834 0.285053 0.296377 0.521339 ---------------------------------------------------------------------------------------------------------------- Total 0.302746 0.438964 8.13383 21.8447
Appendix 5: ANOVA on 10-Year Inversion
Page 44 of 58
One-Way ANOVA – Relative Performance of LT Bonds and the S&P 500 cont. ANOVA Table for BondSPDelta by lag(Inv_10_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00688535 1 0.00688535 3.52 0.0610 Within groups 1.13164 579 0.00195447 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.13852 580 The F-ratio, which in this case equals 3.52288, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean BondSPDelta from one level of lag(Inv_10_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BondSPDelta by lag(Inv_10_yr,1) lag(Inv_10_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 479 285.015 1 102 319.108 -------------------------------------------------------------- Test statistic = 3.46879 P-Value = 0.0625334 Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.
Appendix 5: ANOVA on 10-Year Inversion
Page 45 of 58
Means and 95.0 Percent LSD Intervals
lag(Inv_10_yr,1)
Bon
dBill
Del
ta
0 1-68
-48
-28
-8
12
32(X 0.0001)
Box-and-Whisker Plot
BondBillDelta
lag(
Inv_
10_y
r,1)
0
1
-0.1 -0.06 -0.02 0.02 0.06 0.1 0.14
One-Way ANOVA - Relative Performance of LT Bonds and T-Bills Analysis Summary Dependent variable: BondBillDelta Factor: lag(Inv_10_yr,1) Number of observations: 581 Number of levels: 2
Summary Statistics for BondBillDelta lag(Inv_10_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 479 0.00152147 0.000590955 0.0243096 -0.0826588 1 102 -0.0031364 0.00108467 0.0329343 -0.0964916 ---------------------------------------------------------------------------------------------------------------- Total 581 0.000703737 0.000679055 0.0260587 -0.0964916 lag(Inv_10_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.121288 0.203947 3.69659 9.75244 1 0.129323 0.225814 0.991146 4.23979 ---------------------------------------------------------------------------------------------------------------- Total 0.129323 0.225814 2.98716 11.948
Appendix 5: ANOVA on 10-Year Inversion
Page 46 of 58
One-Way ANOVA - Relative Performance of LT Bonds and T-Bills cont. ANOVA Table for BondBillDelta by lag(Inv_10_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00182446 1 0.00182446 2.69 0.1012 Within groups 0.392028 579 0.000677077 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 0.393852 580 The F-ratio, which in this case equals 2.6946, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean BondBillDelta from one level of lag(Inv_10_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BondBillDelta by lag(Inv_10_yr,1) lag(Inv_10_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 479 295.136 1 102 271.578 ------------------------------------------------------------ Test statistic = 1.65612 P-Value = 0.198125 Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.
Appendix 5: ANOVA on 10-Year Inversion
Page 47 of 58
Means and 95.0 Percent LSD Intervals
lag(Inv_10_yr,1)
BillS
PD
elta
0 1-10
-6
-2
2
6
10
14(X 0.001)
Box-and-Whisker Plot
BillSPDelta
lag(
Inv_
10_y
r,1)
0
1
-0.15 -0.05 0.05 0.15 0.25
One-Way ANOVA - Relative Performance of T-Bills and the S&P 500 Analysis Summary Dependent variable: BillSPDelta Factor: lag(Inv_10_yr,1) Number of observations: 581 Number of levels: 2
Summary Statistics for BillSPDelta lag(Inv_10_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 479 -0.0071636 0.00161614 0.0402013 -0.121873 1 102 0.0065429 0.00249836 0.0499836 -0.1483 ---------------------------------------------------------------------------------------------------------------- Total 581 -0.0047573 0.00179422 0.0423583 -0.1483 lag(Inv_10_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.248282 0.370155 6.83975 15.9861 1 0.132443 0.280743 -0.113135 0.902562 ---------------------------------------------------------------------------------------------------------------- Total 0.248282 0.396582 5.92577 12.3445
Appendix 5: ANOVA on 10-Year Inversion
Page 48 of 58
One-Way ANOVA - Relative Performance of T-Bills and the S&P 500 cont. ANOVA Table for BillSPDelta by lag(Inv_10_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.0157984 1 0.0157984 8.93 0.0029 Within groups 1.02485 579 0.00177004 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.04065 580 The F-ratio, which in this case equals 8.92547, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is less than 0.05, there is a statistically significant difference between the mean BillSPDelta from one level of lag(Inv_10_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BillSPDelta by lag(Inv_10_yr,1) lag(Inv_10_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 479 281.66 1 102 334.863 ------------------------------------------------------------ Test statistic = 8.44723 P-Value = 0.00365483 Since the P-value is less than 0.05, there is a statistically significant difference amongst the medians at the 95.0% confidence level.
Appendix 5: ANOVA on 10-Year Inversion
Page 49 of 58
Box-and-Whisker Plot
GVDelta
lag(
Inv_
10_y
r,1)
0
1
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
One-Way ANOVA - Relative Performance of Growth and Value Analysis Summary Dependent variable: GVDelta Factor: lag(Inv_10_yr,1) Number of observations: 575 Number of levels: 2
Summary Statistics for GVDelta lag(Inv_10_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 473 -0.00235475 0.000840287 0.0289877 -0.10753 1 102 -0.00790182 0.00134391 0.0366594 -0.14001 ---------------------------------------------------------------------------------------------------------------- Total 575 -0.00333875 0.000931937 0.0305276 -0.14001 lag(Inv_10_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.12503 0.23256 -0.591573 8.99359 1 0.138661 0.278671 -0.303424 6.75281 ---------------------------------------------------------------------------------------------------------------- Total 0.138661 0.278671 -1.09382 12.7949
Means and 95.0 Percent LSD Intervals
lag(Inv_10_yr,1)
GV
Del
ta
0 1-13
-10
-7
-4
-1
2(X 0.001)
Appendix 5: ANOVA on 10-Year Inversion
Page 50 of 58
One-Way ANOVA - Relative Performance of Growth and Value cont. ANOVA Table for GVDelta by lag(Inv_10_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00258179 1 0.00258179 2.78 0.0961 Within groups 0.53235 573 0.000929058 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 0.534932 574 The F-ratio, which in this case equals 2.77894, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean GVDelta from one level of lag(Inv_10_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for GVDelta by lag(Inv_10_yr,1) lag(Inv_10_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 473 293.043 1 102 264.613 --------------------------------------------------------------- Test statistic = 2.45729 P-Value = 0.116977 Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.
Appendix 5: ANOVA on 10-Year Inversion
Page 51 of 58
Box-and-Whisker Plot
BondGDelta
lag(
Inv_
10_y
r,1)
0
1
-0.17 -0.07 0.03 0.13 0.23 0.33
One-Way ANOVA - Relative Performance of LT Bonds and Growth Analysis Summary Dependent variable: BondGDelta Factor: lag(Inv_10_yr,1) Number of observations: 575 Number of levels: 2
Summary Statistics for BondGDelta lag(Inv_10_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 3 -0.00665392 0.00211249 0.0459618 -0.136733 1 2 0.00355696 0.00330558 0.0574941 -0.163053 ---------------------------------------------------------------------------------------------------------------- Total 5 -0.0048426 0.00233398 0.0483113 -0.163053 lag(Inv_10_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.30037 0.437104 7.09525 19.6985 1 0.162041 0.325094 -0.116461 0.927028 ---------------------------------------------------------------------------------------------------------------- Total 0.30037 0.463424 5.91763 15.4029
Means and 95.0 Percent LSD Intervals
lag(Inv_10_yr,1)
Bon
dGD
elta
0 1-10
-6
-2
2
6
10
14(X 0.001)
Appendix 5: ANOVA on 10-Year Inversion
Page 52 of 58
One-Way ANOVA - Relative Performance of LT Bonds and Growth cont. ANOVA Table for BondGDelta by lag(Inv_10_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00874821 1 0.00874821 3.77 0.0528 Within groups 1.33096 573 0.00232279 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.3397 574 The F-ratio, which in this case equals 3.76626, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean BondGDelta from one level of lag(Inv_10_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BondGDelta by lag(Inv_10_yr,1) lag(Inv_10_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 473 281.617 1 102 317.598 ------------------------------------------------------------ Test statistic = 3.93572 P-Value = 0.0472676 Since the P-value is less than 0.05, there is a statistically significant difference amongst the medians at the 95.0% confidence level.
Appendix 5: ANOVA on 10-Year Inversion
Page 53 of 58
Means and 95.0 Percent LSD Intervals
lag(Inv_10_yr,1)
Bon
dVD
elta
0 1-13
-9
-5
-1
3(X 0.001)
Box-and-Whisker Plot
BondVDelta
lag(
Inv_
10_y
r,1)
0
1
-0.21 -0.11 -0.01 0.09 0.19 0.29
One-Way ANOVA - Relative Performance of LT Bonds and Value Analysis Summary Dependent variable: BondVDelta Factor: lag(Inv_10_yr,1) Number of observations: 575 Number of levels: 2
Summary Statistics for BondVDelta lag(Inv_10_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 473 -0.00900867 0.00204726 0.0452466 -0.202883 1 102 -0.00434487 0.00304515 0.0551829 -0.208345 ---------------------------------------------------------------------------------------------------------------- Total 575 -0.00818135 0.00222246 0.0471429 -0.208345 lag(Inv_10_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.258857 0.461741 2.44631 14.9675 1 0.105941 0.314285 -2.50202 3.31854 ---------------------------------------------------------------------------------------------------------------- Total 0.258857 0.467202 0.497432 13.9685
Appendix 5: ANOVA on 10-Year Inversion
Page 54 of 58
One-Way ANOVA - Relative Performance of LT Bonds and Value cont. Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00182505 1 0.00182505 0.82 0.3653 Within groups 1.27387 573 0.00222315 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.27569 574 The F-ratio, which in this case equals 0.820927, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean BondVDelta from one level of lag(Inv_10_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BondVDelta by lag(Inv_10_yr,1) lag(Inv_10_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 473 284.482 1 102 304.314 ------------------------------------------------------------ Test statistic = 1.19565 P-Value = 0.274191 Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.
Appendix 5: ANOVA on 10-Year Inversion
Page 55 of 58
Means and 95.0 Percent LSD Intervals
lag(Inv_10_yr,1)
Bill
GD
elta
0 1-12
-7
-2
3
8
13
18(X 0.001)
Box-and-Whisker Plot
BillGDelta
lag(
Inv_
10_y
r,1)
0
1
-0.21 -0.11 -0.01 0.09 0.19 0.29
One-Way ANOVA - Relative Performance of T-Bills and Growth Analysis Summary Dependent variable: BillGDelta Factor: lag(Inv_10_yr,1) Number of observations: 575 Number of levels: 2
Summary Statistics for BillGDelta lag(Inv_10_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 473 -0.00812609 0.00195698 0.0442378 -0.137027 1 102 0.00669335 0.00325716 0.0570715 -0.205777 ---------------------------------------------------------------------------------------------------------------- Total 575 -0.00549725 0.00221445 0.0470579 -0.205777 lag(Inv_10_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.245906 0.382933 5.13232 10.634 1 0.14565 0.351426 -1.034 2.39953 ---------------------------------------------------------------------------------------------------------------- Total 0.245906 0.451683 3.91312 9.52679
Appendix 5: ANOVA on 10-Year Inversion
Page 56 of 58
One-Way ANOVA - Relative Performance of T-Bills and Growth cont. ANOVA Table for BillGDelta by lag(Inv_10_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.0184271 1 0.0184271 8.43 0.0038 Within groups 1.25267 573 0.00218616 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.27109 574 The F-ratio, which in this case equals 8.429, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is less than 0.05, there is a statistically significant difference between the mean BillGDelta from one level of lag(Inv_10_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BillGDelta by lag(Inv_10_yr,1) lag(Inv_10_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 473 278.638 1 102 331.412 ------------------------------------------------------------ Test statistic = 8.46667 P-Value = 0.00361597 Since the P-value is less than 0.05, there is a statistically significant difference amongst the medians at the 95.0% confidence level.
Appendix 5: ANOVA on 10-Year Inversion
Page 57 of 58
Box-and-Whisker Plot
BillVDelta
lag(
Inv_
10_y
r,1)
0
1
-0.23 -0.13 -0.03 0.07 0.17 0.27
Means and 95.0 Percent LSD Intervals
lag(Inv_10_yr,1)
BillV
Del
ta
0 1-14
-10
-6
-2
2
6(X 0.001)
One-Way ANOVA - Relative Performance of T-Bills and Value Analysis Summary Dependent variable: BillVDelta Factor: lag(Inv_10_yr,1) Number of observations: 575 Number of levels: 2
Summary Statistics for BillVDelta lag(Inv_10_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 473 -0.0104808 0.00182653 0.042738 -0.207198 1 102 -0.00120847 0.00298321 0.0546188 -0.224761 ---------------------------------------------------------------------------------------------------------------- Total 575 -0.00883601 0.00203945 0.0451602 -0.224761 lag(Inv_10_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.204393 0.411591 0.988825 10.2473 1 0.149888 0.374649 -2.18256 5.10111 ---------------------------------------------------------------------------------------------------------------- Total 0.204393 0.429154 -0.329068 11.8175
Appendix 5: ANOVA on 10-Year Inversion
Page 58 of 58
One-Way ANOVA - Relative Performance of T-Bills and Value cont. ANOVA Table for BillVDelta by lag(Inv_10_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00721398 1 0.00721398 3.55 0.0599 Within groups 1.16343 573 0.00203042 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.17064 574 The F-ratio, which in this case equals 3.55296, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean BillVDelta from one level of lag(Inv_10_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BillVDelta by lag(Inv_10_yr,1) lag(Inv_10_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 473 281.693 1 102 317.245 -------------------------------------------------------------- Test statistic = 3.84241 P-Value = 0.0499681 The StatAdvisor --------------- Since the P-value is less than 0.05, there is a statistically significant difference amongst the medians at the 95.0% confidence level.