make sense in terms of improving performance. It is not clear why all the weights on forecasts should be equal, as is the case in the consensus forecast. However, a forecast portfolio with optimal weights derived analytically, usually fails to outperform the consensus, which was noted in surveys by Clemen (1989), Diebold and Lopez (1996), and Timmermann (2006). This point has not been examined in Japan, as far as we know. Another to be considered is to incorporate covariance matrix of disturbances of variables forecasted into aggregation of RMSEs because these variables do not move around independently, but are connected through some rules, such as identities, Phillips curve and Okun's law.

## 4 Testing Rational Expectations Hypothesis (REH)

Now let’s turn to test the REH, $ E(e_{ijt}^{k}|\Omega_{ij})=0 $ , where $ \Omega_{ij} $ is the information set held by forecaster i at time t-j. Using the ESPF data, Ban (2009) conducted two kinds of tests in quarterly GDP growth rate by the following equation,

$$ y_{t}\quad=\quad\beta_{i0}+\beta_{i1}f_{i j t}+\beta_{i2}Z_{i j}+v_{i j t}, $$

unbiasedness with the null of $ (\beta_{i0},\beta_{i1})=(0,1) $ , given $ \beta_{i2}=0 $ ; and efficiency with the null of $ (\beta_{i0},\beta_{i1},\beta_{i2})=(0,1,0) $ for any variable $ Z_{ij}\in\Omega_{ij} $ . Note these hypotheses should be tested in individual regressions unless micro-homogeneity is satisfied.

Here we define Z as real GDP growth rates in the previous four quarters at the time of forecasting. Table 3 shows two-thirds and about 90 per cent out of 33 individual regressions are rejected in unbiasedness and efficiency tests, respectively. However, interestingly, as the horizon shrinks, the number of rejections decreases: if these hypotheses are tested in each of zero to two months forecast horizons, only a few are rejected. The failure of efficiency hypothesis may imply forecasters use the models different from traditional time series models.

## (Table 3 Tests of Individual Forecasts)

Confirming micro-homogeneity across 33 forecasters, $ \beta_{i0} = \beta_{0} $ and $ \beta_{i1} = \beta_{1} $ , Ban (2009) went on to test the hypotheses in pooled data. Otherwise, consistency could not be ensured, except for a trivia case, $ (1/N)\sum_{i}\beta_{i1} = \beta_{1} $ . The hypotheses are rejected in all the forecast horizons in the pooled data (Table 3). The different results for the short horizons might be affected by inconsistent pooled coefficients due to constant cross-sectional realization (Zarnowitz, 1985). Note Zarnowitz (1985)'s argument holds, even if forecasters are homogeneous, as long as all the variables are stationary (Bonham and Cohen, 2001).

Use of the consensus requires care. Even if micro-homogeneity holds, there is still a risk of private information bias (Bonham and Cohen, 2001). Fingleski and Watchel (1983) showed the unbiasedness test by applying OLS to Eq.(5) with the consensus as the regressor, $ f_{ijt} = f_{cjt} $ , may produce inconsistent estimator because the presence of private information may cause correlation between the regressor, $ f_{cjt} $ , and the disturbance, $ v_{cjt} $ . Hence, Ban (2009) tested unbiasedness of the consensus by IV with real GDP growth rate in the previous eight quarters at the time of forecasting as instrument.