Appendix C: Proofs for Section 6.1 | RDP 2026-05: Real GDP Forecasts by International Organisations

RDP 2026-05: Real GDP Forecasts by International Organisations Appendix C: Proofs for Section 6.1

Martin McCarthy, Umberto Collodel and Prakash Loungani

September 2026

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Proof of Theorem 2. The rational forecast is the sum of the current practice forecast and an extra term.

(C1)

\begin{array}{l} \underset{Rational forecast}{\underset{︸}{E [y_{t + h}]}} & = E [r (x_{t + h})] \\ \begin{array}{l} = \int_{- \infty}^{\infty} r (x) p (x) d x \\ = \int_{- \infty}^{\infty} r (μ) p (x) d x + \int_{- \infty}^{\infty} (r (x) - r (μ)) p (x) d x \\ = r (μ) + \int_{- \infty}^{\infty} (r (x) - r (μ)) p (x) d x \\ = \underset{Current practice forecast}{\underset{︸}{r (μ)}} + \underset{Extra term}{\underset{︸}{\int_{- \infty}^{\infty} (r (x) - r (μ)) p (x) d x}} \end{array} \end{array}

If the future explanatory variable has a degenerate distribution, it is guaranteed to equal $μ$ , so the rational forecast and current practice forecast are equal. Suppose that the future explanatory variable has a non-degenerate distribution. We will consider cases (a) and (b) separately.

Case (a)

In the first case, the response function is $r (x) = a + b x$ . The extra term is zero, so the rational forecast equals the current practice forecast regardless of the distribution of x.

\begin{array}{l} \int_{- \infty}^{\infty} (r (x) - r (μ)) p (x) d x & = \int_{- \infty}^{\infty} (a + b x - a - b μ) p (x) d x \\ \begin{array}{l} = b \int_{- \infty}^{\infty} x p (x) d x - b \int_{- \infty}^{\infty} μ p (x) d x \\ = b μ - b μ = 0 \end{array} \end{array}

Case (b)

In the second case, the extra term is also zero. The first equality is given by performing integration by substitution with $u (x) = x - μ$ , so $u^{'} (x) = 1$ . The third equality is given by performing integration by substitution on the first term with $v (x) = - u (x)$ , so $v^{'} (x) = - 1$ . The fourth equality holds by the symmetry of the conditional pdf around the mean, $p (- v + μ) = p (v + μ)$ . The final equality holds by the rotational symmetry of r around the point $(μ, r (μ))$ .

\begin{array}{l} \int_{- \infty}^{\infty} (r (x) - r (μ)) p (x) d x & = \int_{- \infty}^{\infty} (r (u + μ) - r (μ)) p (u + μ) d u \\ \begin{array}{l} = \int_{0}^{\infty} (r (u + μ) - r (μ)) p (u + μ) d u + \int_{- \infty}^{0} (r (u + μ) - r (μ)) p (u + μ) d u \\ = \int_{0}^{\infty} (r (u + μ) - r (μ)) p (u + μ) d u + \int_{0}^{\infty} (r (- v + μ) - r (μ)) p (- v + μ) d v \\ = \int_{0}^{\infty} (r (u + μ) - r (μ)) p (u + μ) d u + \int_{0}^{\infty} (r (- v + μ) - r (μ)) p (v + μ) d v \\ = \int_{0}^{\infty} (r (u + μ) - r (μ)) p (u + μ) d u + \int_{0}^{\infty} - (r (v + μ) - r (μ)) p (v + μ) d v \\ = 0 \end{array} \end{array}

Proof of Theorem 3. If the response function attains a strict global minimum at $μ$ , the ‘extra term’ from Equation (C1) will be positive, so the rational forecast will exceed the current practice forecast. To see this, note that $r (x) - r (μ) \geq 0$ for all x, with strict inequality whenever $x \neq μ$ . Since the future explanatory variable has a non-degenerate distribution, $P (x \neq μ) > 0$ . Therefore, $E [r (x) - r (μ)] > 0$ , so the extra term is strictly positive.

If the response function attains a strict global maximum at $μ$ , the extra term is negative by the same argument, and hence the rational forecast is lower than the current practice forecast.

Suppose the response function r(x) is strictly convex. Since x has a non-degenerate distribution, strict Jensen's inequality implies:

\begin{array}{l} \int_{- \infty}^{\infty} (r (x) - r (μ)) p (x) d x & = E [r (x)] - r (μ) \\ \begin{array}{l} > r (E [x]) - r (μ) \\ = 0 \end{array} \end{array}

Hence, the extra term is strictly positive, so the rational forecast is strictly greater than the current practice forecast.

If the response function r(x) is strictly concave, we can make the same argument to show that the extra term is strictly negative, and hence that the rational forecast is strictly less than the current practice forecast.