Bootstrap results for regressions
Compare 95% confidence intervals and P-values for the slope
  for least-squares and Theil`s regression with bootstrap
  P-values and bootstrap percentile confidence intervals for
     (i) bootraps of residuals
    (ii) bootraps of observations
 
Initializing the uniform random-number generator at
    12345678
Initializing the normal random-number generator at
    12345678
(In MATLAB, rand() and randn() must both be seeded.)
 
Five U(0,1)s:  0.0478904   0.996547   0.501069   0.617014   0.556112
Five N(0,1)s:  -1.57019   -0.362043   0.47415   -0.245975   -1.20185
 
Generating (X,Y) for a normal regression
  with  Y = 5 X + 10 + 100*N(0,1), n=30 pairs

Data (n=30)  (X then Y):
     1.   2.96     20.64        16.  11.67     29.85
     2.  18.86     37.82        17.  18.02     53.74
     3.  19.14    231.35        18.   2.52    -16.71
     4.  16.43     71.36        19.  15.57    226.43
     5.  19.50    188.43        20.   2.08     82.18
     6.   2.11     45.31        21.  16.59    238.19
     7.   7.13    -18.91        22.  17.61      3.93
     8.   5.63     67.92        23.  10.41   -121.86
     9.   0.33     94.79        24.   6.68    181.14
    10.   4.38    -33.16        25.   8.79     85.91
    11.  18.16     56.48        26.   9.65   -109.22
    12.   4.72    -25.45        27.  17.85     54.74
    13.   9.44     98.89        28.  10.41    112.11
    14.   1.21   -137.05        29.  16.95    115.14
    15.  12.08    190.32        30.   4.58    100.24

Within-sample tie groups:  X: 1   Y: 0
  X(23)=X(28)=10.41 changed to  10.406  10.414
  1 X tie(s) corrected.

LEAST-SQUARES REGRESSION:  Y = 6.21565 X + -0.379163
 
Normal-theory test of H_0:beta=0  (Student-t test)
  beta_hat=6.2156   T=2.3429   P=0.0265  (two-sided, df=28)
  95% Conf.Int. for beta:  (0.7812   6.2156   11.6501)

For nb=10000 bootstrap replications:
(LEAST SQUARES) BOOTSTRAP ON RESIDUALS (beta=6.21565)
  95% Cred.Interval  (1.2399   6.1236   11.0562)
  Bootstrap bias: 5148/10000 = 0.5148
  Bootstrap P-value for H_0:beta=0:
    One-sided P:  P=79/10000=0.00790
    Two-sided P:  P=0.01580   95% CI: (0.01233, 0.01927)
(LEAST SQUARES) BOOTSTRAP ON OBSERVATIONS (beta=6.21565)
  95% Cred.Interval  (1.6223   6.1716   11.2460)
  Bootstrap bias: 5073/10000 = 0.5073
  Bootstrap P-value for H_0:beta=0:
    One-sided P:  P=26/10000=0.00260
    Two-sided P:  P=0.00520   95% CI: (0.00320, 0.00720)


THEIL REGRESSION:  Y = 5.6129 X + 6.38032
  Theil`s 95% Confidence interval (with median of S-ratios):
    (-0.4472    5.6129    11.4823)
  with offsets  163  217.5  273  in sratio array  (n=435)
 
Theil`s test of H_0:beta=0  (same as Kendall`s test)
(NOT assuming that X(i) are increasing)
  C=105   Z=1.8733   P=0.0610 (Assuming no ties)

For nb=10000 bootstrap replications:
(THEIL) BOOTSTRAP ON RESIDUALS (beta=5.6129)
  95% Cred.Interval  (0.4113   5.6129   10.8199)
  Bootstrap bias: 5536/10000 = 0.5536
  Bootstrap P-value for H_0:beta=0:
    One-sided P:  P=181/10000=0.01810
    Two-sided P:  P=0.03620   95% CI: (0.03097, 0.04143)
(THEIL) BOOTSTRAP ON OBSERVATIONS (beta=5.6129)
  95% Cred.Interval  (0.4501   5.6129   11.4823)
  Bootstrap bias: 5034/10000 = 0.5034
  Bootstrap P-value for H_0:beta=0:
    One-sided P:  P=207/10000=0.02070
    Two-sided P:  P=0.04140   95% CI: (0.03582, 0.04698)