Forecast construction

As mentioned, we assume the forecaster specifies the forecast model based on the condi- tional expectationEt[Yt+1] of the observed process{Yτ}. This is as a consequence of the

well known fact that the prediction with the smallest MSFE is, in fact, the conditional expectation Et[Yt+1], ([64], p. 72). For example, for a DGP with additive innovations of

the form

DGP : Yt+1=ψ(Xt) +Ut+1, (2.4.1)

where _{Uτ} is the innovation process with Eτ[Uτ] = 0 and V ar(Uτ) = σU2 < ∞ for

all τ, Et[Yt+1] = ψ(Xt). It is common practice in econometrics to assume a specific

form for Et[Yt+1] in estimation and forecasting. In what follows, we use the notation

Ψt,n≡(Et[Yt−n+1], . . . , Et[Yt])> ∈Rn×1 and Ut,n= (Ut−n+1, . . . , Ut)> ∈Rn×1, so that

Yt,n=g(Ψt,n, Ut,n), (2.4.2)

for some functional form g. Not knowing the exact form of the DGP, we assume the forecaster’s prediction ofYt+1 is based on a model forEt[Yt+1] which is linear in Xt, and

an innovation process _{Vτ} such that

Yt+1=β>Xt+Vt+1, (2.4.3)

where β is anm×1 parameter vector, β ∈ B, B compact in Rm_{, andVτ is such that} Eτ[Vτ] = 0.

It is important to note that — while being linear — the forecasting model is not assumed to be correctly specified. In other words, we do not make the assumption that Et(Yt+1) is a linear function of Xt. In fact, a major aim of the work in this thesis is

to investigate the ramifications of the phenomena of misspecification in the context of forecasting. Misspecification of the forecasting model can result from a variety of causes. For example, in her choice of Xt, the forecaster might omit some of theFt–measurable

variables that enterEt(Yt+1); in this case, the forecasting model is dynamically misspec-

ified. Moreover, even ifEt(Yt+1) is a function of Xt alone, its functional form might be

highly nonlinear; in this case the forecasting model is functionally misspecified.

The parameter β is assumed to be estimated by an ordinary least squares (OLS) estimator. The OLS estimator ofβcan be computed by using sample sets of various sizes. When the sample set is a continuous interval in time, we refer to it as an observation window. In the construction of a forecast, one important aspect to determine is the nature of the sample used for the estimation of the forecast model. In the case of an observation window, this corresponds to determining its length. In chapters to follow, we develop quantitative methods for determining the length of an observation window used in the forecasting problem. Two prevalent methods found in the literature are: (1) a rolling window forecasting scheme, or (2) a recursive (also known as expanding window) forecasting scheme. Under the rolling window forecasting scheme, the forecaster re-estimates the parameter β of the linear forecasting model in (2.4.3) at each point t, T ₋R _≤t < T. The estimation sample contains the nmost recent observations—Xt−n

to Xt−1 andYt−n+1 toYt— so the OLS estimator of β has the form

ˆ

βt,n≡Q−t,n1Xt,n>Yt,n. (2.4.4)

(Pt_s−₌1_t−nX2

s)−1 ( Pt−1

s=t−nYs+1Xs). The above OLS estimator ˆβt,n is then used to con-

struct the forecast ˆYt+1,n ofYt+1 as follows

ˆ

Yt+1,n = ˆβt,n> Xt. (2.4.5)

This procedure is repeated R times over the out-of-sample period [T ₋R, T], and the forecaster re-estimates β each time there are new observations available. The value of n — which enters the forecast ˆYt+1,n through the OLS estimator ˆβt,n — is most often

chosen in an ad hoc manner, since there are no systematic methods in the literature to obtain an optimal value.

The recursive (or expanding) window scheme involves using all past observations available, i.e., the observations from date 1 tot. Hence, if the forecaster uses a recursive window forecasting scheme, at any timet,T₋R_≤t < T, she computes ˆβt,t≡Q−t,t1Xt,t>Yt,t,

and constructs ˆYt+1,t = ˆβt,t>Xt. In other words, the recursive scheme corresponds to the

case wheren=tin the OLS expression (2.4.4) above. As previously, the OLS estimator ˆ

βt,tis computedT times, only now the estimate ofβ relies on all the data prior to timet.

Both the rolling window and recursive forecasting schemes have great shortcomings. For instance, neither of these schemes is likely to be optimal if the DGP for the time series

{Y_s}undergoes a structural break. A rolling window of a short fixed size might work well immediately after the break but valuable information will be lost as the distance from the break increases. The recursive scheme will produce significantly biased forecasts after the break, until the post break information out weighs the pre-break information. It is our ultimate goal to develop and evaluate a new optimal forecasting scheme which relies on the nature of the processes {Ys} and {Xs} for the choice of the forecasting window. Before tackling this in the chapters that follow, we examine forecast evaluation based on the decomposition of the MSFE.