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15.4 OPTIMALITY TESTS UNDER UNKNOWN LOSS

If the shape of the forecaster’s loss function is not specified up to a few unknown parameters, it can be very difficult to conduct tests of forecast optimality. One strategy is to impose restrictions on the loss function and/or on the underlying DGP and develop testable implications subject to such restrictions.

Patton and Timmermann (2007b) characterize the optimal forecast when both the loss function and the underlying data-generating process can be restricted in this manner. They show that if predictability is constrained to the mean of the outcome, then the optimal forecast takes a very simple form for loss functions that depend only on the forecast error.

Specifically, suppose the loss function depends only on the forecast error,

$L (f, y) = L (e), (15.31)$

while the data-generating process has dynamics only in the conditional mean,

$Y_{t + h} = μ_{t + h ∣ t} + σ_{ε, h} ε_{t + h}, ε_{t + h} ∣ Z_{t} \sim p_{ε, h} (0, 1), (15.32)$

where $p_{ε, h} (0, 1)$ is the density of $ε_{t + h}$ which has mean 0 and variance 1. This density may depend on h, but does not depend on $Z_{t}$ . Patton and Timmermann (2007b) show that, under (15.31) and (15.32),

the optimal forecast takes the form

$f_{t + h ∣ t} = μ_{t + h ∣ t} + α_{h},$

where the constant $α_{h}$ depends only on the density $p_{ε, h}$ and the loss function; 2. forecast errors associated with the optimal forecasts, $e_{t + h ∣ t}$ , are independent of all $z_{t} \in Z_{t};$ in particular, $Cov (e_{t + h ∣ t}, e_{t + h - j ∣ t - j}) = 0$ for all $j \geq h$ and any $h > 0;$ 3. the variance of the forecast error associated with the optimal forecast, $V (e_{t + h ∣ t})$ , is nondecreasing in the forecast horizon, h.

The first result has also been shown by Granger (1969b) and Christoffersen and Diebold (1997), while the result that the optimal bias is time invariant follows by noting that the forecast error associated with the optimal forecast takes the form

$e_{t + h ∣ t} = y_{t + h} - f_{t + h ∣ t} = σ_{ε, h} ε_{t + h} - α_{h},$

where $α_{h}$ is a constant and so is independent of all $z_{t} \in Z_{t}$ . This property can be tested through the constraint $β_{h} = 0$ in regressions such as

$e_{t + h ∣ t} = α_{h} + β_{h}^{'} v_{t} + u_{t + h}, (15.33)$

where $v_{t} = g (z_{t})$ . The regression in (15.33) generalizes the conventional efficiency regression in (15.21) to allow for a nonzero constant. A less formal test could be based on a plot of the forecast against the realized value. The best fitting line should be parallel with the $4 5^{\circ}$ line, although it need not fall on top of it since the optimal bias could be nonzero.

The three properties in (1)–(3) make the strong assumption of no predictive dynamics in second- or higher-order moments of the DGP. This assumption is too strict for many macroeconomic and financial time series which display time-varying heteroskedasticity. On the other hand, the result only imposes weak assumptions on the loss function. If more structure can be imposed on the loss function, the assumptions on the DGP can be relaxed. Patton and Timmermann (2007b) provide forecast optimality results for a more general class of DGPs with time-varying mean and variance.

Specifically, suppose the loss function is homogeneous in the forecast error,

$L (a e) = g (a) L (e), (15.34)$

while the DGP can have conditional mean and variance dynamics,

$y_{t + h} = μ_{t + h ∣ t} + σ_{t + h ∣ t} ε_{t + h}, ε_{t + h} ∣ Z_{t} \sim p_{ε, h} (0, 1), (15.35)$

where the density $P_{ε, h} (0, 1)$ has zero mean and unit variance and can depend on h, but not on elements in $Z_{t}$ . Then the optimal forecast takes the following form:

$f_{t + h ∣ t} = μ_{t + h ∣ t} + σ_{t + h ∣ t} \cdot γ_{h},$

where the constant $γ_{h}$ depends only on the density $p_{ε, h}$ and the loss function. This result relies heavily on homogeneity of the loss function, (15.34), combined with the conditional scale-location properties of the underlying DGP in (15.35).

Let the standardized forecast error associated with the optimal forecast be given by $\tilde{e}_{t + h ∣ t} = e_{t + h ∣ t} / σ_{t + h ∣ t}$ . Then $\tilde{e}_{t + h ∣ t}$ is independent of any element $z_{t} \in Z_{t}$ . In particular, $Cov (\tilde{e}_{t + h ∣ t}^{r}, \tilde{e}_{t + h - j ∣ t - j}^{s}) = 0$ for all $j \geq h$ , any $h > 0;$ , and all $r, s$ for which the covariance exists.

This result follows from noting that $\tilde{e}_{t + h ∣ t} = ε_{t + h} - γ_{h}$ , where $γ_{h}$ is a constant for fixed h and, by assumption, $ε_{t + h}$ is independent of all elements in $Z_{t}$ and has unit variance. Data-generating processes covered by this result include ARCH models like those proposed by Engle (1982).

Even if the assumptions underlying these results are valid, it follows that the forecast error will generally not be unbiased, serially uncorrelated, or homoskedastic. However, the forecast error scaled by the conditional standard deviation, $\tilde{e}_{t + h ∣ t} .$ is independent of any $z_{t} \in Z_{t}$ and $s$ this scaled forecast error, $\tilde{e}_{t + h ∣ t}$ , will be serially uncorrelated and homoskedastic. Complicating matters, tests based on $\tilde{e}_{t + h ∣ t}$ require that an estimate of $σ_{t + h ∣ t}$ is available. For many types of financial data, the conditional variance can be estimated from time series on the outcome variable using GARCH-type models, nonparametric methods, or a realized volatility estimator; see chapter 13.

For situations where a reliable estimate of $σ_{t + h ∣ t}$ is difficult to construct, Patton and Timmermann (2007b) show that under the same conditions leading to the previous results, the optimal forecast can be expressed as a conditional quantile of the outcome variable. Specifically, while the optimal forecast depends on both the forecast horizon and on the shape of the loss function, it is constant over time. Let $P_{t + h ∣ t}$ be the conditional distribution function for $y_{t + h}$ given $Z_{t}$ and suppose that either (i) the loss function depends only on the forecast error $L (e)$ as in (15.31), while the DGP has dynamics only in the conditional mean as in (15.32); or (ii) the loss function is homogeneous in the forecast error as in (15.34), while the DGP can have conditional mean and variance dynamics as in (15.35). Then the optimal forecast, $f_{t + h ∣ t}^{*},$ has the property that, for all $t,$

$P_{t + h ∣ t} (f_{t + h ∣ t}^{*}) = q_{h}^{*}, (15.36)$

where $q_{h}^{*} \in (0, 1)$ depends only on the density $p_{ε, h}$ and the loss function. If $P_{t + h ∣ t}$ is continuous and strictly increasing, then we can alternatively express this as

$f_{t + h ∣ t}^{*} = P_{t + h ∣ t}^{- 1} (q_{h}^{*}) . (15.37)$

To facilitate tests of this property, define the indicator variable $1 (y_{t + h} \leq f_{t + h ∣ t})$ which equals 1 if the forecast is greater than or equal to the outcome, and 0 otherwise. It follows that, for an optimal forecast, $⌊ (y_{t + h} \leq f_{t + h ∣ t})$ is independent of all $z_{t} \in Z_{t}$ and $1 (y_{t + h} \leq f_{t + h ∣ t})$ is a martingale difference sequence with respect to all information in $Z_{t}$

Even though $q_{h}^{*}$ is typically unknown, it is easy to construct a test based on this result by projecting the indicator function $1 (y_{t + h} \leq f_{t + h ∣ t}^{*})$ on elements $v_{t} = g (z_{t}) \in$ $Z_{t},$ , and an intercept and test that $β_{h} = 0$ in the regression

$1 (y_{t + h} \leq f_{t + h ∣ t}^{*}) = α_{h} + β_{h}^{'} v_{t} + u_{t + h} . (15.38)$

The likelihood ratio test of independence proposed by Christoffersen (1998) could also be employed to test for serial dependence in the indicator variable 1 $(y_{t + h} \leq$ $f_{t + h ∣ t}^{*})$ . This test does not require knowledge of the unknown conditional distribution, $P_{t + h ∣ t}$ , nor does it require that the conditional mean or variance of Y is known.

练习题

Given the loss function $L (f, y) = L (e)$ and the DGP $Y_{t + h} = μ_{t + h ∣ t} + σ_{ε, h} ε_{t + h}$ , where $ε_{t + h} ∣ Z_{t} \sim p_{ε, h} (0, 1)$ , what is the form of the optimal forecast $f_{t + h ∣ t}$ ?

f_{t + h ∣ t} = μ_{t + h ∣ t}

f_{t + h ∣ t} = μ_{t + h ∣ t} + α_{h}

f_{t + h ∣ t} = μ_{t + h ∣ t} + σ_{ε, h} ε_{t + h}

f_{t + h ∣ t} = μ_{t + h ∣ t} - α_{h}

Which of the following statements is true about the forecast errors associated with the optimal forecasts under the conditions $L (f, y) = L (e)$ and $Y_{t + h} = μ_{t + h ∣ t} + σ_{ε, h} ε_{t + h}$ ?

A. Forecast errors are correlated with

z_{t} \in Z_{t}

Cov (e_{t + h ∣ t}, e_{t + h - j ∣ t - j}) = 0

for all

j \geq h

and any

h > 0

C. The variance of the forecast error decreases with the forecast horizon

h

D. Forecast errors are dependent on the constant

α_{h}

Which of the following statements are true about the optimal bias under the conditions $L (f, y) = L (e)$ and $Y_{t + h} = μ_{t + h ∣ t} + σ_{ε, h} ε_{t + h}$ ?

A. The optimal bias is time-varying.

B. The optimal bias is time-invariant.

C. The optimal bias depends on

z_{t} \in Z_{t}

D. The optimal bias is a constant

α_{h}

Under the conditions $L (f, y) = L (e)$ and $Y_{t + h} = μ_{t + h ∣ t} + σ_{ε, h} ε_{t + h}$ , the forecast errors associated with the optimal forecasts are independent of the information set $Z_{t}$ .

The variance of the forecast error associated with the optimal forecast is nonincreasing in the forecast horizon $h$ under the conditions $L (f, y) = L (e)$ and $Y_{t + h} = μ_{t + h ∣ t} + σ_{ε, h} ε_{t + h}$ .

The optimal forecast under the loss function $L (f, y) = L (e)$ and the DGP $Y_{t + h} = μ_{t + h ∣ t} + σ_{ε, h} ε_{t + h}$ takes the form $f_{t + h ∣ t} = μ_{t + h ∣ t} +$ ___.

The property that the variance of the forecast error associated with the optimal forecast is ___ in the forecast horizon $h$ is a key result under the conditions $L (f, y) = L (e)$ and $Y_{t + h} = μ_{t + h ∣ t} + σ_{ε, h} ε_{t + h}$ .

Explain how the time invariance of the optimal bias can be tested under the conditions $L (f, y) = L (e)$ and $Y_{t + h} = μ_{t + h ∣ t} + σ_{ε, h} ε_{t + h}$ .

What is the form of the optimal forecast when the loss function is homogeneous in the forecast error, $L (a e) = g (a) L (e)$ , and the DGP can have conditional mean and variance dynamics?

Consider a forecasting model where the loss function depends only on the forecast error $L (f, y) = L (e)$ and the data-generating process has dynamics only in the conditional mean $Y_{t + h} = μ_{t + h ∣ t} + σ_{ε, h} ε_{t + h}$ , with $ε_{t + h} ∣ Z_{t} \sim p_{ε, h} (0, 1)$ . According to Patton and Timmermann (2007b), which of the following statements about the optimal forecast is true?

A. The optimal forecast takes the form

f_{t + h ∣ t} = μ_{t + h ∣ t} - α_{h}

, where

α_{h}

depends on the density

p_{ε, h}

and the loss function.

B. The optimal forecast takes the form

f_{t + h ∣ t} = μ_{t + h ∣ t} + α_{h}

, where

α_{h}

depends on the density

p_{ε, h}

and the loss function.

C. The optimal forecast is always unbiased, meaning

E [e_{t + h ∣ t}] = 0

regardless of the loss function.

D. The optimal forecast errors are serially correlated for all

j \geq h

Which of the following statements are true regarding the properties of optimal forecasts under the assumptions that the loss function depends only on the forecast error and the DGP has dynamics only in the conditional mean?

A. Forecast errors associated with the optimal forecasts are independent of all

z_{t} \in Z_{t}

B. The variance of the forecast error associated with the optimal forecast is nondecreasing in the forecast horizon

h

C. The optimal forecast takes the form

f_{t + h ∣ t} = μ_{t + h ∣ t} - σ_{ε, h} \cdot γ_{h}

D. The forecast errors are serially correlated for all horizons

h

E. The optimal bias is time-invariant.

True or False: If the loss function is homogeneous in the forecast error, $L (a e) = g (a) L (e)$ , and the DGP has conditional mean and variance dynamics, the optimal forecast takes the form $f_{t + h ∣ t} = μ_{t + h ∣ t} + σ_{t + h ∣ t} \cdot γ_{h}$ , where $γ_{h}$ depends only on the density $p_{ε, h}$ and the loss function.

Under the assumption that the loss function depends only on the forecast error and the DGP has dynamics only in the conditional mean, the variance of the forecast error associated with the optimal forecast, $V (e_{t + h ∣ t})$ , is _______ in the forecast horizon $h$ .

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15.4 OPTIMALITY TESTS UNDER UNKNOWN LOSS