正在学习

17.1 FORECAST ENCOMPASSING TESTS

The general idea of encompassing relates to the notion that one model not only fully explains what another model can explain but provides additional explanation of some phenomenon. In a predictive context, the idea of forecast encompassing is similar to the notion of orthogonality in the efficiency regressions discussed in chapter 15, the only difference being that the additional information now consists of forecasts or forecast errors from other forecast methods. If an additional forecast contains information relevant for predicting the outcome and not already contained in the original forecast, then such a forecast will enter the orthogonality regression with a nonzero weight. In this situation, the original forecast did not include all relevant information. Conversely, if orthogonality holds, then the original forecast is said to “encompass” the other forecast since it incorporates all relevant information that the other forecast embodies.

With two competing forecasts, the condition that $f_{1 t + 1∣ t}$ encompasses $f_{2 t + 1∣ t}$ can, under general loss, be stated as

$E [L (f_{1 t + 1∣ t}, y_{t + 1})] \leq g (\cdot) min E [L (g (f_{1 t + 1∣ t}, f_{2 t + 1∣ t}), y_{t + 1})] . (17.3)$

If (17.3) holds, then the first forecast is “sufficient” given the pair of forecasts $(f_{1 t + 1∣ t}, f_{2 t + 1∣ t})$ in the sense that there is no information in the second forecast that is useful once we have access to the first forecast.

Note that forecast encompassing depends on the loss function, L. Hence, we might well find that forecast $f_{1 t + 1∣ t}$ encompasses $f_{2 t + 1∣ t}$ for some loss function, $L_{1}$ but not for another loss function $L_{2} \neq = L_{1}$ ·

Chong and Hendry (1986) consider forecast encompassing under MSE loss. For this case (17.3) reduces to

$E [(y_{t + 1∣ t} - f_{1 t + 1∣ t})^{2}] = g (\cdot) min E [(y_{t + 1∣ t} - g (f_{1 t + 1∣ t}, f_{2 t + 1∣ t}))^{2}] . (17.4)$

Under the null of forecast encompassing, it follows that the expected loss from the forecast that is encompassed exceeds the expected loss associated with the forecast that encompasses it. This means that if $f_{1 t + 1∣ t}$ encompasses $f_{2 t + 1∣ t} :$ , then the MSE from $f_{1 t + 1∣ t}$ is less than that for $f_{2 t + 1∣ t}$ . If the MSE for $f_{2 t + 1∣ t}$ were less than that for $f_{1 t + 1∣ t},$ clearly $E (y_{t + 1} - f_{1 t + 1 ∣ t})^{2} > E (y_{t + 1} - f_{2 t + 1 ∣ t})^{2} \geq min_{g (\cdot)} E (y_{t + 1} -$ $g (f_{1 t + 1∣ t}, f_{2 t + 1∣ t}))^{2}$ and so $f_{1 t + 1∣ t}$ could not encompass $f_{2 t + 1∣ t}$ . Under the alternative, both forecasts are useful and hence it would be better to combine the forecasts rather than use a single forecast.

练习题

What does the general idea of encompassing refer to?

A. One model explains less than another model

B. One model fully explains what another model can explain and provides additional explanation

C. Two models explain the same phenomenon without any additional information

D. One model is completely unrelated to another model

In a predictive context, what does forecast encompassing relate to when considering additional forecasts?

A. The sum of the forecasts

B. The difference between the forecasts

C. The notion of orthogonality in efficiency regressions, where additional information consists of forecasts or forecast errors from other methods

D. The average of the forecasts

What is the condition for $f_{1 t + 1∣ t}$ to encompass $f_{2 t + 1∣ t}$ under general loss?

E [L (f_{1 t + 1∣ t}, y_{t + 1})] \geq min_{g (\cdot)} E [L (g (f_{1 t + 1∣ t}, f_{2 t + 1∣ t}), y_{t + 1})]

E [L (f_{1 t + 1∣ t}, y_{t + 1})] \leq min_{g (\cdot)} E [L (g (f_{1 t + 1∣ t}, f_{2 t + 1∣ t}), y_{t + 1})]

E [L (f_{1 t + 1∣ t}, y_{t + 1})] = min_{g (\cdot)} E [L (g (f_{1 t + 1∣ t}, f_{2 t + 1∣ t}), y_{t + 1})]

E [L (f_{1 t + 1∣ t}, y_{t + 1})] \neq = min_{g (\cdot)} E [L (g (f_{1 t + 1∣ t}, f_{2 t + 1∣ t}), y_{t + 1})]

If (17.3) holds, then the first forecast is sufficient given the pair of forecasts $(f_{1 t + 1∣ t}, f_{2 t + 1∣ t})$ in the sense that there is no information in the second forecast that is useful once we have access to the first forecast.

Forecast encompassing is independent of the loss function $L$ .

What are the implications under the null of forecast encompassing? (Select all that apply)

A. The expected loss from the encompassed forecast exceeds the expected loss from the encompassing forecast

B. The MSE from the encompassing forecast is less than that for the encompassed forecast

C. The MSE from the encompassing forecast is greater than that for the encompassed forecast

D. The expected loss from the encompassing forecast exceeds the expected loss from the encompassed forecast

Under MSE loss, the condition for forecast encompassing (17.3) reduces to $E [(y_{t + 1∣ t} - f_{1 t + 1∣ t})^{2}] = min_{g (\cdot)} E [(y_{t + 1∣ t} - g (f_{1 t + 1∣ t}, f_{2 t + 1∣ t}))^{2}]$ . The function $g (\cdot)$ represents a ___.

Explain what happens if the MSE for $f_{2 t + 1∣ t}$ is less than that for $f_{1 t + 1∣ t}$ .

What is the recommended action under the alternative situation where both forecasts are useful?

Which of the following statements are correct regarding forecast encompassing and loss functions? (Select all that apply)

A. Forecast encompassing is invariant to the choice of loss function

B. A forecast may encompass another for one loss function but not for a different loss function

C. The MSE loss function is commonly used in forecast encompassing tests

D. The concept of forecast encompassing is only applicable under the MSE loss function

In the context of forecast encompassing, if $f_{1 t + 1∣ t}$ encompasses $f_{2 t + 1∣ t}$ under MSE loss, which of the following statements is true?

E [(y_{t + 1} - f_{1 t + 1∣ t})^{2}] > E [(y_{t + 1} - f_{2 t + 1∣ t})^{2}]

E [(y_{t + 1} - f_{1 t + 1∣ t})^{2}] = E [(y_{t + 1} - f_{2 t + 1∣ t})^{2}]

E [(y_{t + 1} - f_{1 t + 1∣ t})^{2}] < E [(y_{t + 1} - f_{2 t + 1∣ t})^{2}]

E [(y_{t + 1} - f_{1 t + 1∣ t})^{2}]

and

E [(y_{t + 1} - f_{2 t + 1∣ t})^{2}]

are unrelated

Which of the following statements are true regarding forecast encompassing?

A. If

f_{1 t + 1∣ t}

encompasses

f_{2 t + 1∣ t}

, then

f_{1 t + 1∣ t}

is sufficient given the pair of forecasts

(f_{1 t + 1∣ t}, f_{2 t + 1∣ t})

B. Forecast encompassing depends on the loss function used.

C. If the MSE for

f_{2 t + 1∣ t}

is less than that for

f_{1 t + 1∣ t}

, then

f_{1 t + 1∣ t}

encompasses

f_{2 t + 1∣ t}

D. Under the alternative hypothesis, both forecasts are useful and should be combined.

E. Forecast encompassing is similar to the notion of orthogonality in efficiency regressions.

登录后解锁笔记、知识点解析、AI 问答

立即登录

17.1 FORECAST ENCOMPASSING TESTS