5 A re-examination of the exchange rate predictability
Appendix 3: Bootstrap algorithm to estimating the optimal weight
The bootstrap procedures are to obtain robust estimates of the parameters appearing in the optimal shrinking factor. The detailed illustrations are as follows:
1. Estimate △eit+k = βixit+ εit+k and εi,t+k =Pm
l=1ρilεi,t+k−l+ vi,t+k with OLS, in which the lag order m is chosen by the AIC criteria, and obtain ˆβi, ˆρil, ˆεi,t+k and ˆvi,t+k. 2. Now draw from {ˆvi,t+k} with replacements, and generate re-samples using the
data-generating process: △e∗it+k = ˆβixit+ ˆε∗it+k and ˆε∗i,t+k =Pm
l=1ρˆilεˆ∗i,t+k−l+ ˆvi,t+k.
3. Regress △ˆe∗it+k against xit with OLS and obtain the bootstrap estimates of the slope coefficient, denoted by ˆβi∗.
4. Repeat steps 2 and 3 B0 times. Now compute the average bias by
PB0 b=1βˆi− ˆβi,b∗
B0 , and deduct it from ˆβi. This is the bootstrap bias-corrected estimate for the slope coefficient, labeled as ˆˆβi. Then the biased-corrected residuals can be computed accordingly by ˆˆεit+k = △eit+k − ˆˆβixit.
5. Now generate the bootstrap samples based on βˆˆi, ˆˆεi,t+k, xi as if they are true ones by repeating B0 times steps 2, 3, and 4. Thus, a sequence of {ˆˆβi∗} is generated. Repeat the same procedures for different countries other than i, and obtain the bootstrap grand average sequence of { ¯¯β∗} for the N countries.
Based on the two sequences of {ˆˆβi∗} and { ¯¯β∗} obtained from the aforementioned procedures, the bootstrap estimate for the derived optimal shrinkage factor is computed as
ˆ
ωi∗ = P( ¯¯β∗− ˆˆβi)2−P(ˆˆβi∗− ˆˆβi)( ¯¯β∗− ˆˆβi)
P(ˆˆβi∗− ˆˆβi)2+P( ¯¯β∗− ˆˆβi)2− 2P(ˆˆβi∗− ˆˆβi)( ¯¯β∗− ˆˆβi) .
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-0.05 0.00 0.05 0.10 0.15 5
10 15 20 25
30 OLS Stein
Null Alternative
Figure 1: The small-sample distributions of the shrinkage and OLS Estimators under H0
and Ha
22
Table 1: Relative estimation risk (Shrinkage/LS)
Country/Horizons 1 4 8 12 16
H0 b Ha c H0 Ha H0 Ha H0 Ha H0 Ha
MSE:a
Canada 0.829 0.830 0.817 0.822 0.801 0.836 0.817 0.857 0.848 0.865 Germany 0.667 0.669 0.718 0.722 0.672 0.754 0.687 0.774 0.695 0.827 Japan 0.621 0.624 0.701 0.774 0.655 0.809 0.682 0.840 0.706 0.861 Switzerland 0.800 0.840 0.822 0.851 0.806 0.858 0.816 0.862 0.852 0.892 bias2:
Canada 0.875 0.763 0.422 1.209 0.912 1.502 0.733 1.423 1.236 1.201 Germany 0.626 0.891 0.963 1.104 0.802 0.825 0.832 0.861 1.153 1.424 Japan 1.632 1.230 0.870 1.429 0.637 1.410 0.816 1.520 0.822 1.535 Switzerland 1.422 1.229 1.352 1.341 0.403 0.854 0.534 0.732 0.833 0.778 variance:
Canada 0.830 0.830 0.818 0.813 0.802 0.784 0.817 0.805 0.848 0.818 Germany 0.667 0.669 0.718 0.720 0.672 0.768 0.687 0.785 0.695 0.805 Japan 0.621 0.624 0.701 0.707 0.655 0.749 0.683 0.779 0.707 0.802 Switzerland 0.800 0.837 0.822 0.841 0.846 0.864 0.876 0.874 0.893 0.903
a MSE is defined as the sum of the bias squared and the variance of the parameter estimates:
MSE( ˜βi) = Eh
( ˜βi− βi)2i
= E
h
( ˜βi− E( ˜βi)) + (E( ˜βi) − βi)i2
= variance( ˜βi) + bias( ˜βi)2.
bThe column gives the ratios of the MSE, bias squared, and variance of the Shrinkage estimates to by those of the OLS counterparts, under the null of no predictability.
c The column gives the ratios of the MSE, bias squared, and variance of the Shrinkage estimates to by those of the OLS counterparts, under the alternative of predictability.
Table 2: Power performance of the shrinkage estimator (a) 5% significance level
Country/Horizons 1 4 8 12 16
Canada 0.845a 0.850 0.854 0.792 0.723
(1.095)b (1.108) (1.131) (1.145) (1.172)
Germany 0.616 0.680 0.607 0.540 0.507
(1.279) (1.444) (1.317) (1.239) (1.213)
Japan 0.826 0.778 0.863 0.807 0.760
(1.309) (1.319) (1.214) (1.171) (1.193)
Switzerland 0.735 0.715 0.757 0.720 0.628
(1.030) (1.033) (1.080) (1.136) (1.150) (b) 10% significance level
Country/Horizons 1 4 8 12 16
Canada 0.925 0.941 0.925 0.868 0.790
(1.085) (1.079) (1.087) (1.081) (1.078)
Germany 0.802 0.783 0.804 0.770 0.713
(1.303) (1.288) (1.247) (1.207) (1.135)
Japan 0.912 0.896 0.920 0.907 0.847
(1.078) (1.106) (1.108) (1.106) (1.107)
Switzerland 0.896 0.849 0.867 0.818 0.754
(1.083) (1.060) (1.074) (1.068) (1.094)
a The entries represent size-adjusted power of the Shrinkage estimator against the alternative of exchange rate predictability.
bThe entries in parentheses represent the power performance of the Shrinkage estimator relative to that of the OLS estimator (Shrinkage/OLS).
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Table 3: Full-sample estimation results and tests for cointegration
statistics β˜ia p-valueb R2 p-valuec βˆi β¯ ωˆi std( ˆωi)d Canada:
1 0.035 0.000 0.035 0.019 0.029 0.052 0.772 0.311
4 0.131 0.000 0.104 0.010 0.106 0.207 0.734 0.285
8 0.285 0.000 0.218 0.011 0.237 0.428 0.727 0.289
12 0.370 0.011 0.194 0.031 0.230 0.625 0.742 0.314
16 0.379 0.012 0.133 0.070 0.291 0.790 0.813 0.342
Germany:
1 0.051 0.000 0.036 0.032 0.045 0.052 0.125 0.340
4 0.195 0.021 0.119 0.004 0.178 0.207 0.183 0.295
8 0.412 0.037 0.214 0.018 0.385 0.428 -0.035 0.287
12 0.599 0.042 0.340 0.013 0.617 0.625 -0.047 0.320
16 0.767 0.004 0.483 0.009 0.832 0.790 0.083 0.346
Japan:
1 0.051 0.000 0.037 0.027 0.049 0.052 0.100 0.328
4 0.199 0.042 0.121 0.006 0.207 0.207 0.008 0.301
8 0.408 0.032 0.229 0.010 0.454 0.428 -0.067 0.310
12 0.586 0.062 0.311 0.012 0.717 0.625 -0.111 0.324 16 0.734 0.006 0.376 0.019 0.947 0.790 -0.145 0.347 Switzerland:
1 0.072 0.000 0.064 0.000 0.087 0.052 0.578 0.305
4 0.275 0.000 0.221 0.000 0.336 0.207 0.557 0.321
8 0.542 0.000 0.366 0.001 0.634 0.428 0.589 0.301
12 0.787 0.000 0.520 0.000 0.874 0.625 0.684 0.305
16 1.048 0.001 0.722 0.000 1.090 0.790 0.874 0.320
a Shrinkage estimates is defined as ˜βi= ˆωiβˆi+ (1 − ˆωi) ¯β, where ˆβi is the OLS estimate of the slope and the ¯β is the grand average of the OLS estimates of the slopes of the 4 countries; ˆωi is the estimated optimal weight.
b,cP-value under the null of no exchange rate predictability ( ˜βi=0 and R2=0, respectively). Bold-faced numbers refer to p-values less than 10%.
dStandard deviation of the optimal weight estimates (ˆωi).
Table 4: Out-of-Sample forecast evaluations: DM Statistic
Country k DM(A)a p-value p-valueK c DM(20)b p-value p-valueK d
Canada 1 1.568 0.015 0.041 5.719 0.000 0.027
4 1.500 0.027 0.057 1.786 0.030 0.048
8 1.269 0.065 0.015 1.269 0.070 0.016
12 -0.518 0.333 0.064 -0.523 0.319 0.070
16 -1.378 0.655 0.110 -1.311 0.603 0.117
maxe 1.568 0.052 0.060 5.719 0.005 0.056
Germany 1 0.534 0.095 0.151 0.841 0.093 0.141
4 0.438 0.144 0.162 0.529 0.139 0.160
8 0.433 0.169 0.218 0.433 0.180 0.216
12 0.627 0.186 0.249 0.603 0.196 0.250
16 0.796 0.191 0.321 0.777 0.195 0.314
max 0.796 0.288 0.273 0.841 0.300 0.268
Japan 1 0.770 0.080 0.082 0.990 0.094 0.104
4 0.614 0.134 0.151 0.745 0.131 0.144
8 0.855 0.129 0.133 0.890 0.136 0.133
12 0.772 0.161 0.270 0.747 0.180 0.269
16 0.729 0.170 0.451 0.717 0.180 0.433
max 0.855 0.283 0.240 0.990 0.280 0.253
Germany 1 2.658 0.003 0.010 3.482 0.003 0.023
4 2.586 0.004 0.019 2.718 0.009 0.024
8 1.997 0.033 0.045 2.599 0.018 0.039
12 1.744 0.045 0.063 1.981 0.034 0.064
16 1.447 0.063 0.077 1.482 0.080 0.097
max 2.658 0.038 0.091 3.482 0.021 0.089
Notes: The DM statistic is defined as DM = ¯d/
q
2π ˆfd(0)/Nf, where ¯d = Nf−1PT
t=t0+k(u2r,t−u2m,t) with ur,tand um,t
refer to the forecast errors of the random walk model and the monetary model, respectively. Nf is the number of recursive forecasts, and t0 is the first date of forecast. fd(0) is the spectral density of (u2m,t−u2r,t) evaluated at frequency 0. Its consistent estimate ˆfd(0) is obtained using Newey and West (1987).
a DM statistic computed with truncation lags under Bartlett window set to 20.
b DM statistic with truncation lags under Bartlett window set by Andrews’s (1991) algorithm.
c,d The corresponding p-value in Kilian (1999). Bold-faced numbers are those significant at 10% level.
e Joint test statistic proposed by Mark (1995), taking the maximum of a sequence of DM statistics indexed by k.
26
Table 5: Out-of-sample forecast evaluations: Theil’s U Country k Theil’s U p-value p-valueK a
Canada 1 0.971 0.009 0.042
Notes: Theil’s U-statistic is defined as the ratio of the root-mean-square prediction error of the monetary model based on the shrinkage estimator to that of the random walk model. The null hypothesis is that the two models provide forecasts of equal accuracy (U=1). The alternative hypothesis is that the monetary fundamentals is more accurate (U<1).
a The corresponding p-values in Kilian (1999). Bold-faced numbers are those signif-icant at 10% level.
b Joint test statistic according to Mark (1995), which takes the minimun of a se-quence of Theil’s U statistics indexed by k.