**3. Forecasting**

In this section we make some considerations about forecasting and validation of the proposed model. Usually two data bases are used for testing tha forecasting ability of a model: one (in-sample), used for estimation, and the other (out-of-sample) used for comparing forecasts with true values. There is an extra complication in the case of volatility models: there is no unique definition of volatility. Andersen and Bollerslev (1998) show that if wrong estimates of volatility are used, evaluation of forecasting accuracy is compromised. We could use the realized volatility as a basis for comparison, or use some trading system.

We could, for example, have a model for hourly returns and use the realized volatility computed from 15 min returns for comparisons. In general, we can compute *vh*,*<sup>t</sup>* = ∑*ahi*=<sup>1</sup> *<sup>r</sup>*2*t*−*i*, where *ah* is the aggregation factor (4, in the case of 15 min returns). Then use some measure based on *sh* = *<sup>v</sup>*˜*h*,*<sup>t</sup>* − *vh*,*t*, for example, mean squared error, where *<sup>v</sup>*˜*h*,*<sup>t</sup>* is the volatility predicted by the proposed model. See Taylor and Xu (1997), for example.

Now consider Model (3). The forecast of volatility at origin *t* and horizon is given by

$$\begin{aligned} \hat{\sigma}\_t^2(l) &= \mathbb{E}(\sigma\_{t+l}^2 | X\_t) \\ &= \mathbb{E}\left(\mathbb{C}\_0 + \mathbb{C}\_1 \left(r\_{t+l-1} + \dots + r\_{t+l-a\_1}\right)^2 + \dots + \\ &+ \mathbb{C}\_m \left(r\_{t+l-1} + \dots + r\_{t+l-a\_m}\right)^2 + b\_1 \sigma\_{t+l-1}^2 + \dots + b\_p \sigma\_{t+l-p}^2 | X\_t), \end{aligned}$$

where *Xt* = (*rt*, *σt*,*rt*−1, *σt*−1, ...), for *l* = 1, 2, . . .

Since *a*0 = 1 < *a*1 < *a*2 < ... < *am* < <sup>∞</sup>, then we have three cases:

$$\text{(i)}\qquad\text{If }l=1,\dots$$

$$\begin{array}{ll} \hat{\sigma}\_{t}^{2}(l) &=& \mathbb{E}(\mathbb{C}\_{0} + \mathbb{C}\_{1} \left(r\_{t+l-1} + \ldots + r\_{t+l-a\_{1}}\right)^{2} + \ldots, + \\ & & + \mathbb{C}\_{s} \left(r\_{t+l-1} + \ldots + r\_{t+l-a\_{s}}\right)^{2} + \ldots, + \\ & & + \mathbb{C}\_{m} \left(r\_{t+l-1} + \ldots + r\_{t+l-a\_{m}}\right)^{2} + \\ & & + b\_{1} \sigma\_{t+l-1}^{2} + \ldots + b\_{p} \sigma\_{t+l-p}^{2} \left(X\_{t}\right) \\ &=& \mathbb{C}\_{0} + \mathbb{C}\_{1} \left(r\_{t} + \ldots + r\_{t+1-a\_{1}}\right)^{2} + \ldots, + \\ & & + \mathbb{C}\_{s} \left(r\_{t} + \ldots + r\_{t+1-a\_{s}}\right)^{2} + \ldots, + \\ & & + \mathbb{C}\_{m} \left(r\_{t} + \ldots + r\_{t+1-a\_{m}}\right)^{2} + b\_{1} \sigma\_{t}^{2} + \ldots + b\_{p} \sigma\_{t+1-p}^{2}. \end{array}$$

(ii) If *l* is such that *as*−1 < *l* < *as*, *s* = 1, 2, . . . , *m*, then we have,

*σ*ˆ2 *t* (*l*) = *<sup>E</sup>*(*<sup>C</sup>*0 + *C*1 *rt*+*l*−<sup>1</sup> + ... + *rt*+*l*−*a*<sup>1</sup> 2 + ... + + *Cs* (*rt*+*l*−<sup>1</sup> + ... + *rt*+*l*−*as* ) 2 + ... + + *Cm* (*rt*+*l*−<sup>1</sup> + ... + *rt*+*l*−*am* ) 2 + <sup>+</sup>*b*1*σ*<sup>2</sup> *t*+*l*−1 + ... + *bpσ*<sup>2</sup> *<sup>t</sup>*+*l*−*<sup>p</sup>*/*Xt*) = *<sup>E</sup>*(*<sup>C</sup>*0 + ∑*<sup>s</sup>*−<sup>1</sup> *i*=1 *Ci* ∑*ai j*=1 *rt*+*l*−*j* 2 + + ∑*m i*=*<sup>s</sup> Ci* ∑*<sup>l</sup>*−<sup>1</sup> *j*=1 *rt*+*l*−*j* 2 + ∑*m i*=*<sup>s</sup> Ci* ∑*ai j*=*l rt*+*l*−*j* 2 + + ∑*m i*=*<sup>s</sup> Ci* ∑*<sup>l</sup>*−<sup>1</sup> *j*=1 *rt*+*l*−*<sup>j</sup>* ∑*ai j*=*l rt*+*l*−*j* + <sup>+</sup>*b*1*σ*<sup>2</sup> *t*+*l*−1 + ... + *bpσ*<sup>2</sup> *<sup>t</sup>*+*l*−*<sup>p</sup>*/*Xt*) = *<sup>E</sup>*(*<sup>C</sup>*0 + ∑*<sup>s</sup>*−<sup>1</sup> *i*=1 *Ci* ∑*ai j*=1 *<sup>σ</sup>t*+*l*−*j<sup>ε</sup>t*+*l*−*j* 2 + + ∑*m i*=*<sup>s</sup> Ci* ∑*<sup>l</sup>*−<sup>1</sup> *j*=1 *<sup>σ</sup>t*+*l*−*j<sup>ε</sup>t*+*l*−*j* 2 + + ∑*m i*=*<sup>s</sup> Ci* ∑*ai j*=*l rt*+*l*−*j* 2 + + ∑*m i*=*<sup>s</sup> Ci* ∑*<sup>l</sup>*−<sup>1</sup> *j*=1 *<sup>σ</sup>t*+*l*−*j<sup>ε</sup>t*+*l*−*<sup>j</sup>* ∑*ai j*=*l rt*+*l*−*j* + <sup>+</sup>*b*1*σ*<sup>2</sup> *t*+*l*−1 + ... + *bpσ*<sup>2</sup> *<sup>t</sup>*+*l*−*<sup>p</sup>*/*Xt*),

and given the independence of *εt* and *<sup>E</sup>*(*<sup>ε</sup>t*) = 0, we have *E rt*−*irt*−*j* = 0, ∀*i* = *j*; hence,

$$\begin{array}{rcl} \partial\_t^2(l) &=& E(\mathbb{C}\_0 + \sum\_{i=1}^{s-1} \mathbb{C}\_i \sum\_{j=1}^{d\_i} \sigma\_{t+l-j}^2 + \sum\_{i=s}^m \mathbb{C}\_i \sum\_{j=1}^{l-1} \sigma\_{t+l-j}^2 + \\ &+ \sum\_{i=s}^m \mathbb{C}\_i \left(\sum\_{j=l}^{a\_i} r\_{t+l-j}\right)^2 + b\_1 \sigma\_{t+l-1}^2 + \dots + b\_p \sigma\_{t+l-p}^2/\mathbf{X}\_l \\ &=& \mathbb{C}\_0 + \sum\_{i=1}^{s-1} \mathbb{C}\_i \sum\_{j=1}^{a\_i} \sigma\_{t+l-j}^2 + \sum\_{i=s}^m \mathbb{C}\_i \sum\_{j=1}^{l-1} \sigma\_{t+l-j}^2 + \\ &+ \sum\_{i=s}^m \mathbb{C}\_i \left(\sum\_{j=l}^{a\_i} r\_{t+l-j}\right)^2 + b\_1 \sigma\_{t+l-1}^2 + \dots + b\_p \sigma\_{t+l-p}^2. \end{array}$$

*l*

*l*

where, for *i* = 1, . . . , *p*, we have that *σ*˜ 2 *t*+*l*−*i* = *σ*<sup>2</sup> *t*+*l*−*i* , *i* ≥ *σ*ˆ 2 *t*+*l*−*i* , *i* <

(iii) If *l* is such that *l* > *am*, *s* = 1, 2, . . . , *m*, then it follows

$$\begin{aligned} \boldsymbol{\sigma}\_{t}^{2}(\boldsymbol{l}) &= \boldsymbol{E} \left( \mathbf{C}\_{0} + \sum\_{i=1}^{m} \mathbf{C}\_{i} \sum\_{j=1}^{a\_{i}} \boldsymbol{\sigma}\_{t+l-j}^{2} + b\_{1} \boldsymbol{\sigma}\_{t+l-1}^{2} + \dots + b\_{p} \boldsymbol{\sigma}\_{t+l-p}^{2} / \mathbf{X}\_{t} \right) \\ &= \mathbf{C}\_{0} + \sum\_{i=1}^{m} \mathbf{C}\_{i} \sum\_{j=1}^{a\_{i}} \boldsymbol{\sigma}\_{t+l-j}^{2} + b\_{1} \boldsymbol{\sigma}\_{t+l-1}^{2} + \dots + b\_{p} \boldsymbol{\sigma}\_{t+l-p}^{2} \end{aligned}$$

where for *i* = 1, . . . , *p*, we have *σ*˜ 2 *t*+*l*−*i* = *σ*<sup>2</sup> *t*+*l*−*i* , *i* ≥ *l σ*ˆ 2 *t*+*l*−*i* , *i* < *l*
