# Forecast errors of bivariate EqCMs and dVARs

In this section, we illustrate how the forecast errors of an EqCM and the corresponding dVAR are affected differently by structural breaks. Practical forecasting models are typically open systems, with exogenous variables. Although the model that we study in this section is of the simple kind, its properties will prove helpful in interpreting the forecasts errors of the large systems in Section 11.2.3.

A simple DGP This book has taken as a premise that macroeconomic time-series can be usefully viewed as integrated of order one, I(1), and that they also frequently include deterministic terms allowing for a linear trend. The following simple bivariate system (a first-order VAR) can serve as an example:

Vt = K + AlVt_1 + AXxt_1 + ey, t, (11.3)

xt = p + xt_i + ex, t, (11.4)

where the disturbances ey, t and ex, t have a jointly normal distribution. Their variances are and a“X respectively, and the correlation coefficient is denoted by py, x. The openness of practical forecasting models is captured by xt which is (strongly) exogenous. xt is integrated of order one, denoted I(1), and contains a linear deterministic trend if p = 0. We will assume that (11.3) and (11.4) constitute a small cointegrated system such that Vt is also I(1) but cointegrated with xt. This entails that 0 < Ai < 1 and X = 0. With a change in notation, the DGP can be written as

Ayt = – a[vt_i – pxt_i – Z]+ ey, t, 0 < a < 1, (11.5)

Axt = p + ex, t, (11.6)

where a = (1 — Ai), в = A2/a, and Z = к/a. In equation (11.5), a is the equilibrium-correction coefficient and в is the derivative coefficient of the cointegrating relationship.

The system can be re-written in ‘model form’ as a conditional equilibrium – correcting model for yt and a marginal model for xt.

Ayt = Y + nAxt — a[yt-i — ext-i — Z ]+ £y, t, (11.7)

Axt = p + ex, t, (11.8)

where

7 y

П = Py, x– ,

7x

Y = —Pn?

£y, t = ey, t nex, t

from the properties of the bivariate normal distribution.

We define two parameters, /л and n, such that E[yt — ext] = Ц and E[Ayt] = n. By taking expectations in (11.6) we see that E[Axt] = p. Similarly, by taking expectations in (11.5) and substituting for these definitions, noting that n = вр, we find the following relationship between these parameters:

вР = a(Z — P)-

Solving with respect to p yields

[Зр к — [Зр

P = Z———- =————-

aa

In the case when p = 0, both series contain a deterministic trend which stems from the xt-process and conversely, if p = 0 there is no deterministic growth in either of the variables. In the latter case we see from (11.10) that p = Z.

The case with a linear deterministic trend is relevant for many variables of interest for forecasters. In the empirical part of this chapter, Section 11.2.3, we will show examples of both cases. Typical examples of exogenous variables associated with positive drift are indicators of foreign demand, foreign price indices, and average labour productivity, while the zero drift assumption is the most appealing one for variables like, for example, oil prices and monetary policy instruments, that is, money market interest rates and exchange rates.

EqCM and dVAR models of the DGP The purpose of this section is to trace the impact of parameter changes in the DGP on the forecasts of two models of the DGP. First, the equilibrium correction model, EqCM, which coin­cides with the DGP within sample, that is, there is no initial mis-specification, and second, the dVAR.

The EqCM is made up of equations (11.7) and (11.8). Equation (11.7) is the conditional model of yt (see, for example, Hendry 1995a: ch. 7), which has many counterparts in practical forecasting models, following the impact
of econometric methodology and cointegration theory on applied work. Equa­tion (11.8) is the marginal equation for the explanatory variable xt. The dVAR model of yt and xt (wrongly) imposes one restriction, namely that а = 0, hence the dVAR model consists of

Ayt = Y + nAxt + ty, t, (11.11)

Axt = p + ex, f (11.12)

Note that the error process in the dVAR model, ey, t (=£y, t-a[yt-i-pxt- —(]), will in general be autocorrelated provided there is some autocorrelation in the omitted disequilibrium term (for 0 < а < 1).

We further assume that

• parameters are known;

• in the forecasts, AxT+ = p (j = 1,…,h);

• forecasts for the periods T + 1, T + 2,. ..,T + h, are made in period T.

The first assumption abstracts from small sample biases in the EqCM and inconsistently estimated parameters in the dVAR case. The second assump­tion rules out one source of forecast failure that is probably an important one in practice, namely that non-modelled or exogenous variables are poorly fore­casted. In our framework systematic forecast errors in AxT+j are tantamount to a change in p.

Although all other coefficients may change in the forecast period, the most relevant coefficients in our context are а, в, and Z, that is, the coefficients that are present in the EqCM but not in the dVAR. Among these, we concentrate on a and Z, since в represents partial structure by virtue of being a cointegration parameter; see Doornik and Hendry (19976) and Hendry (1998) for an analysis of the importance and detectability of shifts.

In the following two sections we derive the biases for the forecasts of EqCM and dVAR, when both models are mis-specified in the forecast period. We dis­tinguish between the case where the parameter change occurs after the forecast is made (post-forecast break) and a shift that takes place before the forecast period (pre-forecast break).

Parameter change after the forecast is prepared We first assume that the intercept Z in (11.5) changes from its initial level to a new level, that is, Z ^ Z*, after the forecast is made in period T. Since we maintain a constant a in this section, the shift in Z is funda­mentally the product of a change in к, the intercept in equation (11.3). In equilibrium correction form, the DGP in the forecast period is
therefore

Аут+h = Y + пАхт+h – а[ут+h-1 – вхт+h-1 – С*] + £y, T+h,

Ахт+h = ¥ + ex, T+h,

where h = 1,. ..,H. The 1-period forecast errors for the EqCM and the dVAR models can be written:

Ут+l – yT +l, EqCM = – a[Z – Z*] + ey, т+l, (11.13)

Ут+і – ут+gdVAR = – а[ут – вхт – С*] + еу, т+і. (11.14)

In the following, we focus on the bias of the forecast errors. The 1-step biases are defined by the conditional expectation (on Іт) of the forecast errors and are denoted bias т +1jEqCM and bias т +1jdVAR respectively:

biasт+i. EqCM = – а[С – С*], (11.15)

biasт +i, dVAR = – а[ут – вхт – С*]. (11.16)

Let Xj, denote the steady-state values of the xj-process. The corresponding steady-state values of the yt-process, denoted yj, are then given by

(11.18)

Note that both EqCM and dVAR forecasts are harmed by the parameter shift from С to С*; see Clements and Hendry (1996). Assuming that the initial values’ deviations from steady state are negligible, that is, хт « хт and ут « ут, we can simplify the expression into

bias т +1,dVAR = + bias т +1,EqCM. (11.19)

The two models’ 1-step forecast error biases are identical if ут equals its long-run mean ут. An example of such a case will be ordinary least squares (OLS)-estimated unrestricted dVAR (see Clements and Hendry 1998: ch. 5.4).

For comparison we also write down the biases of the 2-period forecast errors (maintaining the steady-state assumption).

~ feF(a + ^(1)) + bias T+2,EqCM = 2@<p + bias т+2,EqCM,

where S(i) = 1 + (1 — a).

More generally, for h-period forecasts we obtain the following expressions

bias T+h, EqCM = — a^(h—1)[Z — C*]j (11.22)

bias T+ h, dVAR = Р^(аФ(Н-2) — ^(h—1)) — a5(h-1)[(yT — yT)

— в(хт — xT ) + (Z — Z*)] (11.23)

for forecast horizons h = 2, 3,…, where 5h—1 and фh—2 are given by

h—1

<*(h—1) = 1 + X)(1 — a)j, S(°) = 1 (11.24)

j = 1

= 1 + (1 — a)^(h—2), h-2

^h—2) = 1 + 53 д(j), ф(°) = 1, ф( —1) =0 (11.25)

j=1

= (h — F) + (1 — а)ф^—3)

and we have again used (11.17). As the forecast horizon h increases to infinity, J(h—1) ^ 1/a, hence the EqCM-bias approaches asymptotically the size of the shift itself, that is, biasT+h, EqCM ^ Z* — Z.

Assuming that xT « хф and yT « уф, we can simplify the expression and the dVAR forecast errors are seen to contain a bias term that is due to the growth in xt and which is not present in the EqCM forecast bias, cf. the term (Зф(аф(ь—2) + 5(h—1)) in (11.23). We can simplify this expression, since the term in square brackets containing the recursive formulae S(h—1) and ф(^—2) can be rewritten as [аф^—2) + S(h— 1)] = h, and we end up with a simple linear trend in the h-step ahead dVAR forecast error bias in the case when = 0, thus
generalising the 1-step and 2-step results[108]:

biasт+h, dVAR = — aS(h-i)[(yT — yT) – в(хт — XT)] + biast+h, EqCM•

(11.26)

We note furthermore that the two models’ forecast error biases are identical if there is no autonomous growth in xt (<p = 0), and yT and xT equal their steady – state values. In the case with positive deterministic growth in xt (<p > 0), while maintaining the steady-state assumption, the dVAR bias will dominate the EqCM bias in the long run due to the trend term in the dVAR bias.

Change in the equilibrium-correction coefficient a Next, we con­sider the situation where the adjustment coefficient a changes to a new value, a*, after the forecast for T + 1,T+2,…,T+ h have been prepared. Conditional on IT, the 1-step biases for the two models’ forecasts are:

bias T+i, EqCM = —(a* — а)[ут — вхт — Z ], (11.27)

bias T+i, dVAR = —а*[ут — вхт — Z ]• (11.28)

Using the steady-state expression (11.17), we obtain

In general, the EqCM bias is proportional to the size of the shift, while the dVAR bias is proportional to the magnitude of the level of the new equilibrium – correction coefficient itself. Assuming that хт « хТ and yT « yT, we can simplify the expression into

bias T+1,dVAR = вA + bias T+1,EqCM. (11.31)

Hence, the difference between the dVAR and EqCM 1-step forecast error biases is identical to (11.19). For the multi-period forecasts, the EqCM and dVAR

 bias T+h, EqCM — ф*{Н-2) – аф(Ь,-2)) – (a* S*h-1) – aS{h-1))

forecast error biases are

T+h, dVAR — @¥а* ^*h-2)

h-1

S*h-i) = 1 + £(1 – a*)j, S*0) = 1,

j=1

h-2

Ф*h-2) = 1 + 52 S*j), Ф*0) = 1, Ф*-1) = 0.

j=1

To facilitate comparison we again assume that xT « xT and yT « yT, and insert (11.33) in (11.32). Using a similar manipulation as when deriving (11.26), we arrive at the following bias T+h, dVAR-expression:

bias t+h, dVAR = e^h + bias t+h, EqCM.

We see that under the simplifying steady-state assumption, the difference between dVAR and EqCM h-step forecast error biases is identical to (11.26). Hence there will be a linear trend in the difference between the dVAR and EqCM forecast error biases due to the mis-representation of the growth in xt in the dVAR.

Parameter change before the forecast is made This situation is illustrated by considering how the forecasts for T + 2,T + 3,…,T + h +1 are updated conditional on outcomes for period T +1. Remember that the shift Z ^ Z* first affects outcomes in period T + 1. When the forecasts for T + 2,T + 3,… are updated in period T +1, information about parameter non-constancies will therefore be reflected in the starting value yT +1.

Change in the intercept Z Given that Z changes to Z* in period

T +1, the (updated) forecast for yT+2, conditional on yT +1 yields the following

forecast error biases for the EqCM and dVAR models:

bias t+2,EqCM I 1-T+1 = – a[(Z – Z* )L (11.34)

biast+2,dVAR | It+1 = —а[ут +1 – P’xt+1 – Z*]. (11.35)

Equation (11.34) shows that the EqCM forecast error is affected by the para­meter change in exactly the same manner as before, cf. (11.15), despite the fact
that in this case the effect of the shift is incorporated in the initial value yT +i. Manifestly, the EqCM forecasts do not correct to events that have occurred prior to the preparation of the forecast. Indeed, unless the forecasters detect the parameter change and take appropriate action by (manual) intercept correction, the effect of a parameter shift prior to the forecast period will bias the forecasts ‘forever’. The situation is different for the dVAR.

Using the fact that

yT+i — A* + fixT+ii

where

Equation (11.35) can be expressed as

under the steady-state assumption. We see that if there is no deterministic growth in the DGP, that is, — 0, the dVAR will be immune with respect to the parameter change. In this important sense, there is an element of inherent ‘intercept correction’ built into the dVAR forecasts, while the parameter change that occurred before the start of the forecast period will produce a bias in the 1-step EqCM forecast. A non-zero drift in the xt-process will, however, produce a bias in the 1-step dVAR forecast as well, and the relative forecast accuracy between the dVAR model and the EqCM will depend on the size of the drift relative to the size of the shift.

The expression for the h-period forecast biases, conditional on IT +l, takes the form:

bias T+(h+l),EqCM 1 1-T+l = – a^(h-l)[C – z *] (11.38)

bias T+(h+l),dVAR 1 1-T+l = e^h – aS(h-l)[(yT+l – yT +l) – в(хТ+l – xT +l)]

(11.39)

for h — 1, 2,…. This shows that the EqCM forecast remains biased also for long forecast horizons. The forecast does ‘equilibrium correct’, but unfortunately towards the old (and irrelevant) ‘equilibrium’. For really long (infinite) forecast horizons the EqCM bias approaches the size of the shift [(Z* – Z)] just as in the case where the parameter changed before the preparation of the forecast and therefore was undetectable.

For the dVAR forecast there is once again a trend in the bias term that is due to the growth in xt. In the case with no deterministic growth in the DGP, the dVAR forecasts are unbiased for all h.

Change in the equilibrium-correction coefficient a Just as with the long-run mean, the EqCM forecast do not adjust automatically when the change a ^ a* occurs prior to the preparation of the forecasts (in period T +1). The biases for period T + 2, conditional on IT +i, take the form

(ут +i – yT +i) – e(xT+i – xT +i)

О n{ О

– yT+i) – P(xT +i – xT+i)

a (11.41)

where we have used (11.17).

So neither of the two forecasts ‘intercept correct’ automatically to parameter changes occurring prior to the preparation of the forecast. For that reason, the 1-step biases are functionally similar to the formulae for the case where a change to a* after the forecast has been prepared. The generalisation to multi-step forecast error biases is similar to previous derivations.

Estimated parameters In practice both EqCM and the dVAR forecast­ing models use estimated parameters. Since the dVAR is mis-specified relative to the DGP (and the EqCM), estimates of the parameters of (11.11) will in general be inconsistent. Ignoring estimated parameter uncertainty, the dVAR model will be

Ayt = 7* + n*Axt + e*y, t, (11.42)

Axt = p + ex, t, (11.43)

where g* and n* denote the probability limits of the parameter estimates. In the forecast period g* + n* AxT+h = g = 0, hence the dVAR forecast of yT+h will include an additional deterministic trend (due to estimation bias) which does not necessarily correspond to the trend in the DGP (which is inherited from the xt-process).

The parameter bias may be small numerically (e. g. if differenced terms are close to orthogonal to the omitted equilibrium correction term), but can nonetheless accumulate to a dominating linear trend in the dVAR forecast error bias.

One of the dVAR-type models we consider in the empirical section, denoted dRIM, is a counterpart to (11.42). The empirical section shows examples of how dVAR-type models can be successfully robustified against trend-misrepresentation.

Discussion Although we have looked at the simplest of forecasting systems, the results have several traits that one might expect to be able to

recover from the forecast errors of full sized macroeconomic models that we consider in Section 11.2.2.

The analysis above shows that neither the EqCM nor the dVAR protect against post-forecast breaks. In the case we have focused upon, where the dVAR model excludes growth when it is present in the DGP, the dVAR forecast error biases contain a trend component. Even in this case, depending on initial condi­tions, the dVAR may compete favourably with the EqCM over forecast horizons of moderate length.

One suggestion is that the relative merits of EqCMs and dVARs for forecasting depends on

• the ‘mix’ of pre – and post-forecast parameter changes

• the length of the forecast horizon.

In the next section we use this perspective to interpret the forecast outcomes from a large-scale model of the Norwegian economy.