Abstract
Motivated by the risk inherent in intraday investing, we propose several ways of quantifying extremal behavior of a time series of curves. A curve can be extreme if it has shape and/or magnitude much different than the bulk of observed curves. Our approach is at the nexus of functional data analysis and extreme value theory. The risk measures we propose allow us to assess probabilities of observing extreme curves not seen in a historical record. These measures complement risk measures based on point-to-point returns, but have different interpretation and information content. Using our approach, we study how the financial crisis of 2008 impacted the extreme behavior of intraday cumulative return curves. We discover different impacts on shares in important sectors of the US economy. The information our analysis provides is in some cases different from the conclusions based on the extreme value analysis of daily closing price returns.
Appendices
A Variances of V a R ˆ α and E S ˆ α
A Denote
and
large sample standard errors or confidence intervals for
Hence, by the delta method,
and
where
and
evaluated at
B Estimation of the angular density
This section explains how the density of
where
C Elaboration on the measures χ and χ ‾
This section provides details of inference based on the extreme dependence measures χ and
We begin by providing two examples where χ > 0 (asymptotic dependence) and χ = 0 (asymptotic independence).
Example C.1.
In this example, X and Y have a common risk factor. Let Z,
for u > 0. Consider the simple factor model
where
and
Thus, the larger the ratio
Example C.2.
Suppose now that (X,Y) are jointly Gaussian with correlation coefficient
Lemma C.1.
Let
where
where
Consequently, the tail-dependence coefficient between X and Y equals
where
Proof.
We shall first obtain Relation eq. (16) as a consequence of eq. (15). For simplicity, let
and similarly
Thus,
Now, by applying eq. (15) with
Relation eq. (15) is well-known but we give a proof for completeness. This result can be understood in terms of the so-called one big jump heuristic. Namely, since the
as
We will make the above heuristic precise using the notion of multivariate regular variation. We start with some terminology. A set
for all measurable bounded away from
In the simple case above, it is easy to see that the vector
Thus eq. (17) holds with
where
With this general tool, we can establish the tail behavior of various functionals of
where
as
This, in view of eq. (18), yields the formula eq. (15) and completes the proof.
Equivalent definition of χ
Before we discuss any estimation methods of χ and
where
Estimation of χ and χ ‾
Ledford and Tawn (1996 and 1998) established that under weak conditions
where 0 < η ≤ 1 is a constant and
and if
Before we estimate χ, it is important to decide if there exists an asymptotic dependence. We thus first test the null hypothesis
Testing χ ‾ = 1
There are basically two major different ways to test if
where
as the sample size in the study is big enough. Thus, the hypothesis
Estimation of χ
Once the above test shows no significant evidence to reject
where n is the sample size and
D Large tables and displays
ETF | Prd | u | Parameter | |||||
---|---|---|---|---|---|---|---|---|
Bf | 0.8 | 54 | 0.12 (0.16) | 2.46 (0.16) | 3.08 (0.13) | 3.43 (0.94) | 4.18 (0.34) | |
XLF | Dr | 2.7 | 71 | 7.83 (0.20) | 8.40 (0.70) | 8.78 (0.43) | 9.15 (0.97) | |
(Financials) | A1 | 2.5 | 33 | 6.81 (0.37) | 7.54 (1.57) | 8.02 (1.23) | 8.52 (2.56) | |
A2 | 1.3 | 51 | 0.02 (0.16) | 3.11 (0.13) | 3.65 (0.19) | 3.98 (0.69) | 4.54 (0.35) | |
Bf | 0.8 | 67 | 2.13 (0.04) | 2.39 (0.09) | 2.56 (0.13) | 2.79 (0.17) | ||
XLK | Dr | 1.6 | 69 | 4.96 (0.23) | 5.64 (0.43) | 6.06 (0.79) | 6.66 (0.74) | |
(Technology) | A1 | 1.1 | 73 | 0.16 (0.12) | 3.08 (0.19) | 3.90 (0.19) | 4.33 (1.09) | 5.39 (0.65) |
A2 | 0.8 | 68 | 1.99 (0.03) | 2.23 (0.11) | 2.37 (0.14) | 2.57 (0.21) | ||
Bf | 0.6 | 78 | 0.09 (0.11) | 1.91 (0.06) | 2.33 (0.05) | 2.58 (0.31) | 3.06 (0.12) | |
XLY | Dr | 2.2 | 58 | 7.06 (0.69) | 8.29 (1.11) | 9.04 (2.92) | 10.22 (2.03) | |
(Cons. Dis.) | A1 | 1.4 | 63 | 0.10 (0.17) | 4.20 (0.41) | 5.18 (0.33) | 5.74 (2.49) | 6.88 (0.64) |
A2 | 0.9 | 69 | 2.56 (0.08) | 2.92 (0.16) | 3.14 (0.35) | 3.47 (0.31) | ||
Bf | 0.6 | 50 | 1.34 (0.01) | 1.47 (0.08) | 1.55 (0.04) | 1.65 (0.15) | ||
XLP | Dr | 0.6 | 185 | 0.20 (0.08) | 5.38 (0.82) | 7.27 (0.43) | 8.25 (4.41) | 10.84 (1.37) |
(Cons. Stap.) | A1 | 0.8 | 72 | 1.92 (0.02) | 2.13 (0.07) | 2.27 (0.09) | 2.45 (0.14) | |
A2 | 0.6 | 66 | 0.12 (0.14) | 1.51 (0.04) | 1.84 (0.06) | 2.03 (0.21) | 2.43 (0.21) | |
Bf | 1.3 | 50 | 2.46 (0.05) | 2.71 (0.16) | 2.87 (0.23) | 3.09 (0.32) | ||
XLE | Dr | 2.5 | 51 | 5.79 (0.16) | 6.27 (0.60) | 6.58 (0.49) | 6.91 (0.94) | |
(Energy) | A1 | 1.7 | 51 | 3.45 (0.05) | 3.71 (0.23) | 3.88 (0.14) | 4.06 (0.39) | |
A2 | 1.3 | 57 | 2.93 (0.05) | 3.24 (0.16) | 3.44 (0.19) | 3.70 (0.30) | ||
Bf | 0.5 | 66 | 0.11 (0.14) | 1.74 (0.09) | 2.24 (0.18) | 2.51 (0.54) | 3.16 (0.77) | |
XLV | Dr | 1.5 | 53 | 4.46 (0.29) | 5.15 (0.58) | 5.59 (1.31) | 6.22 (1.08) | |
(Health) | A1 | 0.8 | 95 | 0.12 (0.11) | 3.13 (0.20) | 3.93 (0.13) | 4.38 (1.01) | 5.35 (0.30) |
A2 | 0.7 | 71 | 0.11 (0.11) | 2.20 (0.08) | 2.73 (0.08) | 3.03 (0.41) | 3.66 (0.19) | |
Bf | 0.8 | 48 | 1.69 (0.02) | 1.87 (0.10) | 1.98 (0.08) | 2.12 (0.20) | ||
XLI | Dr | 1.5 | 88 | 0.14 (0.13) | 5.28 (0.68) | 6.67 (0.41) | 7.44 (3.92) | 9.17 (0.88) |
(Industrials) | A1 | 1.5 | 65 | 3.83 (0.10) | 4.28 (0.22) | 4.57 (0.31) | 4.95 (0.39) | |
A2 | 1.2 | 48 | 2.42 (0.04) | 2.69 (0.14) | 2.85 (0.17) | 3.08 (0.28) | ||
Bf | 0.8 | 85 | 0.09 (0.13) | 2.75 (0.16) | 3.35 (0.09) | 3.70 (0.81) | 4.39 (0.18) | |
XLB | Dr | 2.0 | 74 | 0.02 (0.14) | 5.83 (0.51) | 6.87 (0.57) | 7.49 (2.40) | 8.57 (1.07) |
(Materials) | A1 | 1.5 | 70 | 3.99 (0.17) | 4.54 (0.30) | 4.88 (0.71) | 5.38 (0.54) | |
A2 | 1.2 | 64 | 2.84 (0.03) | 3.06 (0.13) | 3.21 (0.07) | 3.36 (0.21) | ||
Bf | 0.8 | 57 | 2.17 (0.07) | 2.52 (0.10) | 2.74 (0.32) | 3.08 (0.20) | ||
XLU | Dr | 0.9 | 163 | 0.13 (0.09) | 5.78 (0.75) | 7.35 (0.34) | 8.22 (3.53) | 10.13 (0.67) |
(Utilities) | A1 | 1.1 | 56 | 2.83 (0.08) | 3.24 (0.17) | 3.49 (0.32) | 3.86 (0.32) | |
A2 | 0.7 | 74 | 0.16 (0.14) | 2.17 (0.12) | 2.77 (0.17) | 3.10 (0.74) | 3.88 (0.73) |
ETF | Prd | u | Parameter | |||||
---|---|---|---|---|---|---|---|---|
Bf | 0.7 | 60 | 0.10 (0.14) | 2.30 (0.12) | 2.86 (0.10) | 3.17 (0.64) | 3.82 (0.21) | |
XLF | Dr | 2.5 | 69 | 7.57 (0.18) | 8.11 (0.69) | 8.46 (0.38) | 8.79 (0.95) | |
(Financials) | A1 | 2.3 | 34 | 6.72 (0.40) | 7.49 (1.58) | 8.00 (1.35) | 8.54 (2.60) | |
A2 | 1.2 | 57 | 0.02 (0.15) | 2.98 (0.12) | 3.48 (0.16) | 3.78 (0.58) | 4.30 (0.30) | |
Bf | 0.9 | 49 | 2.07 (0.04) | 2.34 (0.10) | 2.50 (0.16) | 2.74 (0.20) | ||
XLK | Dr | 1.6 | 60 | 4.91 (0.29) | 5.66 (0.52) | 6.13 (1.14) | 6.82 (0.94) | |
(Technology) | A1 | 1.0 | 76 | 0.14 (0.12) | 3.05 (0.18) | 3.84 (0.16) | 4.27 (0.99) | 5.25 (0.43) |
A2 | 0.9 | 45 | 1.97 (0.06) | 2.26 (0.13) | 2.44 (0.33) | 2.72 (0.27) | ||
Bf | 0.6 | 68 | 0.12 (0.13) | 1.87 (0.08) | 2.32 (0.07) | 2.58 (0.40) | 3.12 (0.22) | |
XLY | Dr | 2.2 | 51 | 6.97 (0.64) | 8.14 (1.17) | 8.86 (2.56) | 9.95 (2.13) | |
(Cons. Dis.) | A1 | 1.0 | 108 | 0.06 (0.11) | 4.04 (0.28) | 4.88 (0.21) | 5.37 (1.23) | 6.30 (0.37) |
A2 | 1.0 | 48 | 2.45 (0.05) | 2.74 (0.20) | 2.92 (0.21) | 3.14 (0.39) | ||
Bf | 0.6 | 44 | 1.31 (0.01) | 1.43 (0.07) | 1.50 (0.03) | 1.59 (0.15) | ||
XLP | Dr | 0.5 | 188 | 0.19 (0.08) | 5.22 (0.77) | 7.06 (0.40) | 8.03 (4.06) | 10.54 (1.24) |
(Cons. Stap.) | A1 | 0.9 | 48 | 1.89 (0.03) | 2.13 (0.08) | 2.28 (0.12) | 2.49 (0.17) | |
A2 | 0.5 | 90 | 0.13 (0.12) | 1.47 (0.04) | 1.81 (0.07) | 2.01 (0.21) | 2.43 (0.29) | |
Bf | 1.2 | 52 | 2.39 (0.02) | 2.57 (0.15) | 2.68 (0.08) | 2.80 (0.26) | ||
XLE | Dr | 2.5 | 45 | 5.62 (0.24) | 6.24 (0.71) | 6.64 (0.96) | 7.14 (1.26) | |
(Energy) | A1 | 1.7 | 46 | 3.40 (0.05) | 3.66 (0.24) | 3.83 (0.14) | 4.01 (0.42) | |
A2 | 1.2 | 57 | 2.85 (0.04) | 3.13 (0.15) | 3.31 (0.13) | 3.52 (0.27) | ||
Bf | 0.5 | 58 | 0.15 (0.16) | 1.69 (0.09) | 2.17 (0.16) | 2.43 (0.53) | 3.05 (0.66) | |
XLV | Dr | 1.7 | 35 | 4.31 (0.18) | 4.81 (0.80) | 5.14 (0.79) | 5.51 (1.45) | |
(Health) | A1 | 0.7 | 104 | 0.10 (0.10) | 3.07 (0.18) | 3.83 (0.12) | 4.26 (0.84) | 5.16 (0.24) |
A2 | 0.7 | 59 | 0.08 (0.11) | 2.16 (0.08) | 2.66 (0.08) | 2.94 (0.35) | 3.51 (0.16) | |
Bf | 0.8 | 39 | 1.64 (0.01) | 1.76 (0.10) | 1.84 (0.03) | 1.93 (0.19) | ||
XLI | Dr | 1.3 | 88 | 0.11 (0.12) | 5.14 (0.60) | 6.43 (0.42) | 7.16 (3.15) | 8.70 (0.82) |
(Industrials) | A1 | 1.6 | 53 | 3.80 (0.12) | 4.30 (0.26) | 4.62 (0.45) | 5.07 (0.48) | |
A2 | 1.2 | 41 | 2.37 (0.05) | 2.65 (0.18) | 2.82 (0.24) | 3.06 (0.38) | ||
Bf | 0.8 | 76 | 0.13 (0.15) | 2.74 (0.21) | 3.44 (0.13) | 3.83 (1.20) | 4.68 (0.38) | |
XLB | Dr | 1.5 | 94 | 5.65 (0.39) | 6.52 (0.52) | 7.06 (1.51) | 7.86 (0.91) | |
(Materials) | A1 | 1.6 | 48 | 3.94 (0.07) | 4.31 (0.27) | 4.55 (0.17) | 4.81 (0.43) | |
A2 | 1.3 | 50 | 2.79 (0.03) | 3.03 (0.15) | 3.19 (0.09) | 3.36 (0.26) | ||
Bf | 0.8 | 50 | 0.06 (0.19) | 2.16 (0.11) | 2.60 (0.09) | 2.86 (0.62) | 3.34 (0.16) | |
XLU | Dr | 0.9 | 136 | 0.06 (0.08) | 5.35 (0.48) | 6.56 (0.34) | 7.26 (1.97) | 8.60 (0.61) |
(Utilities) | A1 | 1.1 | 50 | 2.80 (0.10) | 3.27 (0.18) | 3.55 (0.48) | 4.00 (0.35) | |
A2 | 0.6 | 93 | 0.11 (0.12) | 2.04 (0.08) | 2.53 (0.06) | 2.81 (0.41) | 3.39 (0.17) |
ETF | Prd | u | Parameter | |||||
---|---|---|---|---|---|---|---|---|
Bf | 1.1 | 59 | 0.06 (0.18) | 4.58 (0.65) | 5.72 (0.47) | 6.28 (3.61) | 7.53 (0.97) | |
XLF | Dr | 5.3 | 53 | 15.53 (1.37) | 16.00 (2.77) | 17.95 (3.82) | 17.96 (4.27) | |
(Financials) | A1 | 4.9 | 26 | 13.52 (2.03) | 14.46 (4.77) | 16.47 (7.46) | 16.89 (8.80) | |
A2 | 2.4 | 53 | 7.77 (0.90) | 9.15 (1.08) | 10.06 (3.94) | 11.40 (2.16) | ||
Bf | 1.4 | 40 | 2.83 (0.07) | 3.16 (0.08) | 3.44 (0.25) | 3.76 (0.17) | ||
XLK | Dr | 2.9 | 58 | 0.10 (0.14) | 9.20 (1.77) | 11.75 (1.59) | 12.72 (9.59) | 15.66 (3.46) |
(Technology) | A1 | 2.1 | 46 | 5.05 (0.19) | 5.51 (0.30) | 6.11 (0.61) | 6.48(0.57) | |
A2 | 1.8 | 48 | 0.02 (0.18) | 4.89 (0.41) | 5.84 (0.40) | 6.37 (2.17) | 7.34 (0.85) | |
Bf | 1.1 | 57 | 3.26 (0.15) | 3.71 (0.17) | 4.06 (0.61) | 4.48 (0.35) | ||
XLY | Dr | 3.4 | 50 | 9.50 (1.08) | 10.90 (1.33) | 11.98 (4.38) | 13.29 (2.61) | |
(Cons. Dis.) | A1 | 2.5 | 43 | 5.95 (0.23) | 6.31 (0.42) | 7.07 (0.69) | 7.28 (0.76) | |
A2 | 2.0 | 42 | 5.35 (0.35) | 6.11 (0.50) | 6.75 (1.49) | 7.43 (1.02) | ||
Bf | 0.8 | 51 | 0.04 (0.14) | 2.27 (0.09) | 2.76 (0.08) | 2.99 (0.41) | 3.50 (0.16) | |
XLP | Dr | 1.8 | 56 | 0.11 (0.15) | 6.65 (1.11) | 8.64 (1.01) | 9.43 (6.23) | 11.75 (2.19) |
(Cons. Stap.) | A1 | 1.2 | 60 | 3.24 (0.08) | 3.54 (0.13) | 3.91 (0.24) | 4.16 (0.24) | |
A2 | 1.2 | 42 | 0.00 (0.16) | 3.25 (0.16) | 3.87 (0.19) | 4.24 (0.78) | 4.86 (0.41) | |
Bf | 1.7 | 67 | 3.82 (0.11) | 3.89 (0.17) | 4.34 (0.41) | 4.33 (0.35) | ||
XLE | Dr | 3.5 | 64 | 0.04 (0.14) | 15.68 (5.60) | 19.38 (5.57) | 21.36 (27.57) | 25.29 (11.57) |
(Energy) | A1 | 2.7 | 51 | 6.16 (0.21) | 6.57 (0.34) | 7.30 (0.58) | 7.59 (0.60) | |
A2 | 2.3 | 57 | 6.90 (0.57) | 8.00 (0.68) | 8.77 (2.24) | 9.82 (1.32) | ||
Bf | 0.7 | 64 | 2.53 (0.08) | 2.78 (0.17) | 3.06 (0.31) | 3.25 (0.32) | ||
XLV | Dr | 2.2 | 47 | 0.06 (0.16) | 7.84 (1.47) | 9.91 (1.56) | 10.88 (7.99) | 13.16 (3.41) |
(Health) | A1 | 1.6 | 38 | 3.98 (0.12) | 4.25 (0.26) | 4.77 (0.37) | 4.93 (0.50) | |
A2 | 1.4 | 57 | 4.44 (0.29) | 5.17 (0.33) | 5.68 (1.26) | 6.38 (0.67) | ||
Bf | 1.2 | 46 | 3.04 (0.11) | 3.33 (0.17) | 3.68 (0.45) | 3.92 (0.36) | ||
XLI | Dr | 2.9 | 55 | 8.21 (0.45) | 8.67 (0.81) | 9.67 (1.35) | 9.92 (1.36) | |
(Industrials) | A1 | 2.3 | 58 | 6.42 (0.35) | 7.03 (0.51) | 7.77 (1.25) | 8.25 (0.94) | |
A2 | 2.2 | 45 | 5.87 (0.31) | 6.41 (0.52) | 7.13 (1.09) | 7.55 (0.98) | ||
Bf | 1.3 | 59 | 3.72 (0.09) | 3.87 (0.21) | 4.31 (0.28) | 4.36 (0.37) | ||
XLB | Dr | 4.1 | 40 | 10.59 (0.70) | 10.93 (1.48) | 12.37 (1.98) | 12.39 (2.46) | |
(Materials) | A1 | 2.6 | 58 | 5.51 (0.07) | 5.19 (0.20) | 6.03 (0.16) | 5.58 (0.32) | |
A2 | 2.6 | 40 | 6.66 (0.53) | 7.40 (0.77) | 8.22 (2.40) | 8.83 (1.60) | ||
Bf | 1.3 | 43 | 3.80 (0.21) | 4.15 (0.39) | 4.62 (0.95) | 4.88 (0.82) | ||
XLU | Dr | 2.4 | 51 | 0.11 (0.17) | 9.60 (2.84) | 12.66 (2.66) | 13.91 (17.54) | 17.51 (6.12) |
(Utilities) | A1 | 1.7 | 51 | 3.59 (0.05) | 3.52 (0.14) | 4.03 (0.16) | 3.86 (0.27) | |
A2 | 1.1 | 70 | 0.19 (0.14) | 3.90 (0.44) | 5.40 (0.58) | 5.81 (2.87) | 7.75 (1.78) |
ETF | Prd | u | (+, +) | (+, | ( | ( | P value | |
---|---|---|---|---|---|---|---|---|
Bf | 0.8 | 52 | 15 | 8 | 15 | 14 | 0.27 | |
XLF | Dr | 2.7 | 68 | 13 | 21 | 18 | 16 | 0.62 |
(Financials) | A1 | 2.5 | 29 | 6 | 7 | 6 | 10 | 0.47 |
A2 | 1.3 | 50 | 14 | 10 | 12 | 14 | 0.78 | |
Bf | 0.8 | 61 | 14 | 12 | 15 | 20 | 1.00 | |
XLK | Dr | 1.6 | 64 | 18 | 15 | 15 | 16 | 1.00 |
(Technology) | A1 | 1.1 | 69 | 16 | 14 | 22 | 17 | 0.47 |
A2 | 0.8 | 66 | 16 | 13 | 20 | 17 | 0.62 | |
Bf | 0.6 | 74 | 19 | 16 | 18 | 21 | 1.00 | |
XLY | Dr | 2.2 | 53 | 12 | 16 | 12 | 13 | 0.59 |
(Cons. Dis.) | A1 | 1.4 | 61 | 18 | 12 | 16 | 15 | 0.44 |
A2 | 0.9 | 63 | 18 | 15 | 10 | 20 | 0.44 | |
Bf | 0.6 | 47 | 13 | 8 | 13 | 13 | 0.56 | |
XLP | Dr | 0.6 | 171 | 45 | 48 | 37 | 41 | 0.65 |
(Cons. Stap.) | A1 | 0.8 | 67 | 14 | 17 | 16 | 20 | 0.47 |
A2 | 0.6 | 60 | 15 | 12 | 14 | 19 | 1.00 | |
Bf | 1.3 | 46 | 13 | 12 | 8 | 13 | 0.56 | |
XLE | Dr | 2.5 | 48 | 12 | 11 | 8 | 17 | 0.38 |
(Energy) | A1 | 1.7 | 50 | 12 | 15 | 13 | 10 | 1.00 |
A2 | 1.3 | 54 | 16 | 8 | 15 | 15 | 0.27 | |
Bf | 0.5 | 63 | 15 | 17 | 16 | 15 | 1.00 | |
XLV | Dr | 1.5 | 51 | 16 | 8 | 13 | 14 | 0.39 |
(Health) | A1 | 0.8 | 92 | 25 | 19 | 29 | 19 | 0.14 |
A2 | 0.7 | 65 | 17 | 10 | 19 | 19 | 0.32 | |
Bf | 0.8 | 44 | 10 | 11 | 12 | 11 | 1.00 | |
XLI | Dr | 1.5 | 77 | 21 | 18 | 17 | 21 | 1.00 |
(Industrials) | A1 | 1.5 | 63 | 16 | 15 | 17 | 15 | 0.80 |
A2 | 1.2 | 46 | 10 | 11 | 14 | 11 | 1.00 | |
Bf | 0.8 | 81 | 21 | 19 | 20 | 21 | 1.00 | |
XLB | Dr | 2.0 | 67 | 19 | 12 | 11 | 25 | 0.61 |
(Materials) | A1 | 1.5 | 60 | 13 | 15 | 17 | 15 | 1.00 |
A2 | 1.2 | 61 | 20 | 11 | 13 | 17 | 0.61 | |
Bf | 0.8 | 55 | 15 | 12 | 15 | 13 | 0.59 | |
XLU | Dr | 0.9 | 147 | 39 | 38 | 34 | 36 | 1.00 |
(Utilities) | A1 | 1.1 | 53 | 11 | 13 | 14 | 15 | 0.78 |
A2 | 0.7 | 69 | 20 | 18 | 13 | 18 | 0.81 |
ETF | Prd | u | P(+, +) | P(+, | P( | P( | P value | |
---|---|---|---|---|---|---|---|---|
Bf | 0.8 | 52 | 0.29 | 0.15 | 0.29 | 0.27 | 0.27 | |
XLF | Dr | 2.7 | 68 | 0.19 | 0.31 | 0.26 | 0.24 | 0.62 |
(Financials) | A1 | 2.5 | 29 | 0.21 | 0.24 | 0.21 | 0.34 | 0.47 |
A2 | 1.3 | 50 | 0.28 | 0.20 | 0.24 | 0.28 | 0.78 | |
Bf | 0.8 | 61 | 0.23 | 0.20 | 0.25 | 0.33 | 1.00 | |
XLK | Dr | 1.6 | 64 | 0.28 | 0.23 | 0.23 | 0.25 | 1.00 |
(Technology) | A1 | 1.1 | 69 | 0.23 | 0.20 | 0.32 | 0.25 | 0.47 |
A2 | 0.8 | 66 | 0.24 | 0.20 | 0.30 | 0.26 | 0.62 | |
Bf | 0.6 | 74 | 0.26 | 0.22 | 0.24 | 0.28 | 1.00 | |
XLY | Dr | 2.2 | 53 | 0.23 | 0.30 | 0.23 | 0.25 | 0.59 |
(Cons. Dis.) | A1 | 1.4 | 61 | 0.30 | 0.20 | 0.26 | 0.25 | 0.44 |
A2 | 0.9 | 63 | 0.29 | 0.24 | 0.16 | 0.32 | 0.44 | |
Bf | 0.6 | 47 | 0.28 | 0.17 | 0.28 | 0.28 | 0.56 | |
XLP | Dr | 0.6 | 171 | 0.26 | 0.28 | 0.22 | 0.24 | 0.65 |
(Cons. Stap.) | A1 | 0.8 | 67 | 0.21 | 0.25 | 0.24 | 0.30 | 0.47 |
A2 | 0.6 | 60 | 0.25 | 0.20 | 0.23 | 0.32 | 1.00 | |
Bf | 1.3 | 46 | 0.28 | 0.26 | 0.17 | 0.28 | 0.56 | |
XLE | Dr | 2.5 | 48 | 0.25 | 0.23 | 0.17 | 0.35 | 0.38 |
(Energy) | A1 | 1.7 | 50 | 0.24 | 0.30 | 0.26 | 0.20 | 1.00 |
A2 | 1.3 | 54 | 0.30 | 0.15 | 0.28 | 0.28 | 0.27 | |
Bf | 0.5 | 63 | 0.24 | 0.27 | 0.25 | 0.24 | 1.00 | |
XLV | Dr | 1.5 | 51 | 0.31 | 0.16 | 0.25 | 0.27 | 0.39 |
(Health) | A1 | 0.8 | 92 | 0.27 | 0.21 | 0.32 | 0.21 | 0.14 |
A2 | 0.7 | 65 | 0.26 | 0.15 | 0.29 | 0.29 | 0.32 | |
Bf | 0.8 | 44 | 0.23 | 0.25 | 0.27 | 0.25 | 1.00 | |
XLI | Dr | 1.5 | 77 | 0.27 | 0.23 | 0.22 | 0.27 | 1.00 |
(Industrials) | A1 | 1.5 | 63 | 0.25 | 0.24 | 0.27 | 0.24 | 0.80 |
A2 | 1.2 | 46 | 0.22 | 0.24 | 0.30 | 0.24 | 1.00 | |
Bf | 0.8 | 81 | 0.26 | 0.23 | 0.25 | 0.26 | 1.00 | |
XLB | Dr | 2.0 | 67 | 0.28 | 0.18 | 0.16 | 0.37 | 0.61 |
(Materials) | A1 | 1.5 | 60 | 0.22 | 0.25 | 0.28 | 0.25 | 1.00 |
A2 | 1.2 | 61 | 0.33 | 0.18 | 0.21 | 0.28 | 0.61 | |
Bf | 0.8 | 55 | 0.27 | 0.22 | 0.27 | 0.24 | 0.59 | |
XLU | Dr | 0.9 | 147 | 0.27 | 0.26 | 0.23 | 0.24 | 1.00 |
(Utilities) | A1 | 1.1 | 53 | 0.21 | 0.25 | 0.26 | 0.28 | 0.78 |
A2 | 0.7 | 69 | 0.29 | 0.26 | 0.19 | 0.26 | 0.81 |
ETF | Prd | q | P value | |||
---|---|---|---|---|---|---|
Bf | 0.80 | 63 | 0.59 (0.43) | 0.21* | 0.40 (0.04) | |
XLF | Dr | 0.70 | 106 | 0.25 (0.32) | 0.00 | 0 |
(Financials) | A1 | 0.80 | 71 | 0.53 (0.42) | 0.15* | 0.43 (0.05) |
A2 | 0.80 | 72 | 0.73 (0.41) | 0.39* | 0.35 (0.04) | |
Bf | 0.80 | 63 | 0.67 (0.43) | 0.32* | 0.38 (0.04) | |
XLK | Dr | 0.70 | 105 | 0.01 (0.33) | 0.00 | 0 |
(Technology) | A1 | 0.80 | 71 | 0.00 | 0 | |
A2 | 0.75 | 89 | 0.21 (0.33) | 0.00 | 0 | |
Bf | 0.70 | 94 | 0.62 (0.34) | 0.15* | 0.42 (0.04) | |
XLY | Dr | 0.75 | 87 | 0.28 (0.39) | 0.02 | 0 |
(Cons. Dis.) | A1 | 0.65 | 125 | 0.31 (0.28) | 0.00 | 0 |
A2 | 0.75 | 90 | 0.37 (0.37) | 0.04 | 0 | |
Bf | 0.80 | 63 | 0.29 (0.41) | 0.04 | 0 | |
XLP | Dr | 0.60 | 140 | 0.62 (0.33) | 0.13* | 0.42 (0.03) |
(Cons. Stap.) | A1 | 0.85 | 54 | 0.01 | 0 | |
A2 | 0.75 | 90 | 0.42 (0.38) | 0.05 | 0 | |
Bf | 0.80 | 63 | 0.00 | 0 | ||
XLE | Dr | 0.80 | 70 | 0.15 (0.40) | 0.01 | 0 |
(Energy) | A1 | 0.80 | 71 | 0.21 (0.37) | 0.01 | 0 |
A2 | 0.70 | 108 | 0.61 (0.35) | 0.14* | 0.39 (0.03) | |
Bf | 0.80 | 63 | 0.97 (0.50) | 0.93* | 0.38 (0.04) | |
XLV | Dr | 0.80 | 70 | 0.00 | 0 | |
(Health) | A1 | 0.70 | 106 | 0.25 (0.30) | 0.00 | 0 |
A2 | 0.85 | 54 | 0.19 (0.42) | 0.02 | 0 | |
Bf | 0.85 | 47 | 0.54 (0.47) | 0.22* | 0.37 (0.05) | |
XLI | Dr | 0.80 | 70 | 0.00 | 0 | |
(Industrials) | A1 | 0.80 | 70 | 0.00 | 0 | |
A2 | 0.80 | 72 | 0.23 (0.39) | 0.02 | 0 | |
Bf | 0.85 | 47 | 0.44 (0.50) | 0.16* | 0.35 (0.05) | |
XLB | Dr | 0.80 | 70 | 0.00 | 0 | |
(Materials) | A1 | 0.80 | 71 | 0.00 | 0 | |
A2 | 0.80 | 72 | 0.13 (0.36) | 0.01 | 0 | |
Bf | 0.80 | 62 | 0.07 (0.37) | 0.00 | 0 | |
XLU | Dr | 0.80 | 70 | 0.09 (0.38) | 0.01 | 0 |
(Utilities) | A1 | 0.80 | 71 | 0.31 (0.40) | 0.04 | 0 |
A2 | 0.80 | 72 | 0.87 (0.45) | 0.70* | 0.36 (0.04) |
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