Rationality and Heterogeneity of Survey Forecasts of the Yen-Dollar Exchange Rate: A Reexamination

Abstract

This chapter examines the rationality and diversity of industry-level forecasts of the yen-dollar exchange rate collected by the Japan Center for International Finance. In several ways we update and extend the seminal work by Ito (1990, American Economic Review 80, 434–449). We compare three specifications for testing rationality: the “conventional” bivariate regression, the univariate regression of a forecast error on a constant and other information set variables, and an error correction model (ECM). We find that the bivariate specification, while producing consistent estimates, suffers from two defects: first, the conventional restrictions are sufficient but not necessary for unbiasedness; second, the test has low power. However, before we can apply the univariate specification, we must conduct pretests for the stationarity of the forecast error. We find a unit root in the 6-month horizon forecast error for all groups, thereby rejecting unbiasedness and weak efficiency at the pretest stage. For the other two horizons, we find much evidence in favor of unbiasedness but not weak efficiency. Our ECM rejects unbiasedness for all forecasters at all horizons. We conjecture that these results, too, occur because the restrictions test sufficiency, not necessity.

We extend the analysis of industry-level forecasts to a SUR-type structure using an innovative GMM technique (Bonham and Cohen 2001, Journal of Business & Economic Statistics, 19, 278–291) that allows for forecaster cross-correlation due to the existence of common shocks and/or herd effects. Our GMM tests of micro-homogeneity uniformly reject the hypothesis that forecasters exhibit similar rationality characteristics.

Appendix 1: Testing Micro-homogeneity with Survey Forecasts

The null hypothesis of micro-homogeneity is that the slope and intercept coefficients in the equation of interest are equal across individuals. This chapter considers the case of individual unbiasedness regressions such as Eq. 43.2 in the text, repeated here for convenience,

$$ {s}_{t+h}-{s}_t={\alpha}_{i,h}+{\beta}_{i,h}\left({s}_{i,t,h}^e-{s}_t\right)+{\varepsilon}_{i,t,h} $$

(43.17)

and tests H ₀ : α ₁ = α ₂ = … = α _N and β ₁ = β ₂ = … = β _N .

Stack all N individual regressions into the Seemingly Unrelated Regression system

$$ S=\mathbf{F}\theta +\varepsilon $$

(43.18)

where S is the NT × 1 stacked vector of realizations, s _t+h , and F is an NT × 2N block diagonal data matrix:

$$ \mathbf{F}=\left[\begin{array}{ccc}\hfill {\mathbf{F}}_1\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \ddots \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill {\mathbf{F}}_N\hfill \end{array}\right]. $$

(43.19)

Each F _i = [ι s ^e _i,t,h ] is a T × 2 matrix of ones and individual i’s forecasts, θ = [α ₁ β ₁ … α _N β _N ]′, and ε is an NT × 1 vector of stacked residuals. The vector of restrictions, Rθ = r, corresponding to the null hypothesis of micro-homogeneity is normally distributed, with Rθ − r ∼ N[0, R(F′F)⁻¹ F′Ω F(F′F)⁻¹ R′], where R is the 2(N − 1) × 2N matrix

$$ R=\left[\begin{array}{cccccc}\hfill 1\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill \vdots \hfill \\ {}\hfill \vdots \hfill & \hfill 0\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill -1\hfill \end{array}\right], $$

(43.20)

and r is a 2(N − 1) × 1 vector of zeros. The corresponding Wald test statistic, ${\left(R\widehat{\theta}-r\right)}^{\prime}\left[R{\left(\mathbf{F}\mathit{\hbox{'}}\mathbf{F}\right)}^{-1}{\mathbf{F}}^{\mathbf{\prime}}\widehat{\varOmega}\mathbf{F}{\left({\mathbf{F}}^{\mathbf{\prime}}\mathbf{F}\right)}^{-1}{R}^{\prime}\right]\left(R\widehat{\theta}-r\right)$, is asymptotically distributed as a chi-square random variable with degrees of freedom equal to the number of restrictions, 2(N − 1).

For most surveys, there are a large number of missing observations. Keane and Runkle (1990), Davies and Lahiri (1995), Bonham and Cohen (1995, 2001), and to the best of our knowledge all other papers which make use of pooled regressions in tests of the REH have dealt with the missing observations using the same approach. The pooled or individual regression is estimated by eliminating the missing data points in both the forecasts and the realization. The regression residuals are then padded with zeros in place of missing observations to allow for the calculation of own and cross-covariances. As a result, many individual variances and cross-covariances are calculated with relatively few pairs of residuals. These individual cross-covariances are then averaged. In Keane and Runkle (1990) and Bonham and Cohen (1995, 2001) the assumption of 2(k + 1) second moments, which are common to all forecasters, is made for analytical tractability and for increased reliability. In contrast to the forecasts from the Survey of Professional Forecasters used in Keane and Runkle (1990) and Bonham and Cohen (1995, 2001), the JCIF data set contains virtually no missing observations. As a result, it is possible to estimate each individual’s variance-covariance matrix (and cross-covariance matrix) rather than average over all individual variances and cross-covariance pairs as in the aforementioned papers.

We assume that for each forecast group i,

$$ \begin{array}{c}E\left[{\varepsilon}_{i,t,h}{\varepsilon}_{i,t,h}\right]={\sigma}_{i,0}^2\ \mathrm{for}\ \mathrm{all}\ i,t,\\ {}E\left[{\varepsilon}_{i,t,h}{\varepsilon}_{i,t+k}\right]={\sigma}_{i,k}^2\ \mathrm{for}\ \mathrm{all}\ i,t,k\kern0.5em \mathrm{such}\ \mathrm{that}\ 0<k\le h,\\ {}E\left[{\varepsilon}_{i,t,h}{\varepsilon}_{i,t+k}\right]=0\kern0.5em \mathrm{for}\ \mathrm{all}\ i,t,k\kern0.5em \mathrm{such}\ \mathrm{that}\ k>h,\end{array} $$

(43.21)

Similarly, for each pair of forecasters i and j, we assume

$$ \begin{array}{c}E\left[{\varepsilon}_{i,t,h}{\varepsilon}_{j,t}\right]={\delta}_{i,j}(0)\kern1em \forall i,j,t,\\ {}E\left[{\varepsilon}_{i,t,h}{\varepsilon}_{j,t+k}\right]={\delta}_{i,j}(k)\kern0.5em \forall i,j,t,k\kern0.5em \mathrm{such}\ \mathrm{that}\ k\ne 0,\mathrm{and}-h\le k\le h.\\ {}E\left[{\varepsilon}_{i,t,h}{\varepsilon}_{j,t+k}\right]=0\kern2em \forall i,j,t,k\kern0.5em \mathrm{such}\ \mathrm{that}\ k>\left|h\right|.\end{array} $$

(43.22)

Thus, each pair of forecasters has a different T × T cross-covariance matrix:

$$ {\mathbf{P}}_{i,j}=\left[\begin{array}{ccccc}\hfill {\delta}_{i,j}(0)\hfill & \hfill {\delta}_{i,j}\left(-1\right)\hfill & \hfill \dots \hfill & \hfill {\delta}_{i,j}\left(-h\right)\hfill & \hfill 0\hfill \\ {}\hfill {\delta}_{i,j}(1)\hfill & \hfill {\delta}_{i,j}(0)\hfill & \hfill {\delta}_{i,j}\left(-1\right)\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ {}\hfill \hfill & \hfill \dots \hfill & \hfill {\delta}_{i,j}(1)\hfill & \hfill {\delta}_{i,j}(0)\hfill & \hfill {\delta}_{i,j}\left(-1\right)\hfill \\ {}\hfill 0\hfill & \hfill {\delta}_{i,j}(h)\hfill & \hfill \dots \hfill & \hfill {\delta}_{i,j}(1)\hfill & \hfill {\delta}_{i,j}(0)\hfill \end{array}\right], $$

(43.23)

Finally, note that P _i,j ≠ P _j,i , rather P ^' _i,j = P _j,i . The complete variance-covariance matrix, denoted Ω, has dimension NT × NT, with matrices Q _i on the main diagonal and P _i,j off the diagonal.

The individual Q _i , variance-covariances matrices are calculated using the Newey and West (1987) heteroscedasticity-consistent, MA(j)-corrected form. The P _i,j matrices are estimated in an analogous manner.