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19.6.3 Forecast Combination and Model Instability

A number of authors, including Diebold and Pauly (1987), Min and Zellner (1993), Hendry and Clements (2004), and Aiolfi, Capistrán, and Timmermann (2011) suggest that parameter instability could be among the reasons that forecast combinations are found to perform well in empirical studies. As an illustration, Aiolfi, Capistrán, and Timmermann (2011) consider the simple mixture model with two Gaussian predictors:

$y_{t + 1} = (1 - s_{t + 1}) x_{0 t + 1} + s_{t + 1} x_{1 t + 1} + ε_{t + 1}, f_{0 t + 1} = x_{0 t + 1}, f_{1 t + 1} = x_{1 t + 1} .$

Suppose that $x_{i t + 1} \sim N (0, σ_{i}^{2}), i = 0, 1, ε_{t + 1} \sim N (0, σ_{ε}^{2}), E [ε_{t + 1} x_{i t + 1}] = 0$ , and $E [x_{0 t + 1} x_{1 t + 1}] = 0$ . Finally, let $s_{t} = 0$ with probability $p,$ , while $s_{t} = 1$ with probability $(1 - p)$ . The best forecast in this model is determined by the outcome of the state variable, $s_{t + 1} . If s_{t + 1} = 0, f_{0 t + 1}$ is best, while $f_{1 t + 1}$ is best if $s_{t + 1} = 1$ . The possibility of switching between states captures the notion of instability in the underlying data-generating process. Aiolfi, Capistrán and Timmermann show that the population MSE of the equal-weighted forecast combination $(f_{0 t + 1} + f_{1 t + 1}) /2$ is lower than the population MSFE of the best individual forecasting model, $f_{0 t + 1}$ or $f_{1 t + 1}$ provided that

$\frac{1}{3} (\frac{p}{1 - p})^{2} < \frac{σ _{1}^{2}}{σ _{0}^{2}} < 3 (\frac{p}{1 - p})^{2} . (19.29)$

These inequalities ensure that neither of the forecasts dominates the other forecast “too much in population,” in which case it would clearly be optimal to simply use the dominant forecast and not combine. In the simple case where $σ_{1}^{2} = σ_{0}^{2}$ , the equal-√ weighted forecast combination is optimal as long as $(p / (1 - p)) \overset{ˉ}{\in} (1/3, 3), i.e.,$ as long as neither of the states (forecasting models) occurs far more often than the other one. Note that the result in equation (19.29) ignores estimation error, another reason often used to explain the good performance of equal-weighted combinations.

In an empirical analysis Clark and McCracken (2010) combine forecasts of output, inflation, and short-term interest rates across VAR specifications that allow for model instability in different ways, including the use of recursive lag selection, rolling estimation windows, and intercept corrections. They find that simple equal-weighted averaging yields consistently good results. In contrast, least squares estimation of the combination weights tends to yield poor forecasting results. These findings are consistent with Rossi (2013a) who finds that equal-weighted forecast combinations produce relatively good out-of-sample forecasts.

练习题

According to Aiolfi, Capistrán, and Timmermann (2011), which of the following is a reason for the good performance of forecast combinations in empirical studies?

A. The use of advanced computational algorithms

B. Parameter instability in the underlying data-generating process

C. The simplicity of the forecasting models used

D. The use of a large number of predictors

In the simple mixture model with two Gaussian predictors, what determines the best forecast?

A. The variance of the predictors

x_{0 t + 1}

and

x_{1 t + 1}

B. The outcome of the state variable

s_{t + 1}

C. The value of

ε_{t + 1}

D. The correlation between

x_{0 t + 1}

and

x_{1 t + 1}

Which of the following conditions ensure that the population MSE of the equal-weighted forecast combination $(f_{0 t + 1} + f_{1 t + 1}) /2$ is lower than the population MSFE of the best individual forecasting model?

\frac{1}{3} (\frac{p}{1 - p})^{2} < \frac{σ _{1}^{2}}{σ _{0}^{2}}

\frac{σ _{1}^{2}}{σ _{0}^{2}} < 3 (\frac{p}{1 - p})^{2}

\frac{σ _{1}^{2}}{σ _{0}^{2}} = \frac{p}{1 - p}

\frac{1}{3} (\frac{p}{1 - p})^{2} > \frac{σ _{1}^{2}}{σ _{0}^{2}}

In the simple case where $σ_{1}^{2} = σ_{0}^{2}$ , the equal-weighted forecast combination is optimal as long as $(p / (1 - p)) \in / (1/3, 3)$ .

Clark and McCracken (2010) find that simple equal-weighted averaging yields consistently good results, while least squares estimation of the combination weights tends to yield poor forecasting results. This finding is consistent with the work of ___.

Explain why parameter instability might lead to the good performance of forecast combinations.

Which of the following is a key assumption in the simple mixture model with two Gaussian predictors?

E [ε_{t + 1} x_{i t + 1}] \neq = 0

for

i = 0, 1

E [x_{0 t + 1} x_{1 t + 1}] \neq = 0

E [ε_{t + 1} x_{i t + 1}] = 0

for

i = 0, 1

x_{i t + 1} \sim N (1, σ_{i}^{2})

for

i = 0, 1

The optimality of the equal-weighted forecast combination when $σ_{1}^{2} = σ_{0}^{2}$ depends on the relative frequency of the states $s_{t + 1} = 0$ and $s_{t + 1} = 1$ .

What is the main advantage of using forecast combinations in the presence of model instability?

Which of the following methods did Clark and McCracken (2010) use to allow for model instability in their empirical analysis?

A. Recursive lag selection

B. Fixed proportion estimation window

C. Rolling estimation windows

D. Intercept corrections

In the simple mixture model with two Gaussian predictors, what determines the best forecast for $y_{t + 1}$ ?

A. The variance of

ε_{t + 1}

B. The outcome of the state variable

s_{t + 1}

C. The correlation between

x_{0 t + 1}

and

x_{1 t + 1}

D. The mean of

x_{0 t + 1}

and

x_{1 t + 1}

Which of the following are methods used by Clark and McCracken (2010) to allow for model instability in their empirical analysis of forecast combinations?

A. Recursive lag selection

B. Rolling estimation windows

C. Discounted least squares estimation

D. Intercept corrections

In the simple mixture model, when $σ_{1}^{2} = σ_{0}^{2}$ , the equal-weighted forecast combination is optimal regardless of the value of $p$ .

The population MSE of the equal-weighted forecast combination $(f_{0 t + 1} + f_{1 t + 1}) /2$ is lower than the population MSFE of the best individual forecasting model, $f_{0 t + 1}$ or $f_{1 t + 1}$ , provided that $\frac{1}{3} (\frac{p}{1 - p})^{2} < \frac{σ _{1}^{2}}{σ _{0}^{2}} < 3 (\frac{p}{1 - p})^{2}$ . This condition ensures that neither forecast dominates the other ___.

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19.6.3 Forecast Combination and Model Instability