ifdp · March 31, 1989

Exact and Approximate Multi-Period Mean-Square Forecast Errors For Dynamic Econometric Models

Abstract

Both future disturbances and estimated coefficients contribute to the uncertainty in model-based ex ante forecasts, but only the first source is usually taken into account when calculating confidence intervals for practical applications. Schmidt (1974) and Baillie (1979) provide an easily computable second-order approximation to the mean-square forecast error (MSFE) for linear dynamic systems which recognizes both sources of uncertainty. To assess the accuracy of their approximation, and thus its usefulness, we compare it with three sets of estimates of the exact MSFE for the univariate AR(l) process: Monte Carlo estimates for OLS, analytically based values for OLS, and Monte Carlo estimates for maximum likelihood. We find that the Schmidt-Baillie formula is a good approximation to the exact MSFE, and that it helps explain why the exact MSFE can decrease as the forecast horizon increases. In fact, for dynamics typical to econometric models, the MSFE often has a maximum at a forecast horizon of one to twelve periods, i.e., at horizons that are of principal concern to forecasters and policy makers.

International Finance Discussion Papers Number 348 April 1989

EXACT AND APPROXIMATE MULTI-PERIOD MEAN-SQUARE FORECAST ERRORS FOR DYNAMIC ECONOMETRIC MODELS

Neil R. Ericsson and Jaime R. Marquez

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment by a writer that he has had access to unpublished material) should be cleared with the author or authors.

ABSTRACT

Both future disturbances and estimated coefficients contribute to the uncertainty in model-based ez ante forecasts, but only the first source is usually taken into account when calculating confidence intervals for practical applications. Schmidt (1974) and Baillie (1979) provide an easily computable second-order approximation to the mean-square forecast error (MSFE) for linear dynamic systems which recognizes both sources of uncertainty. To assess the accuracy of their approximation, and thus its usefulness, we compare it with three sets of estimates of the exact MSFE for the univariate AR(1) process: Monte Carlo estimates for OLS, analytically based values for OLS, and Monte Carlo estimates for maximum likelihood. We find that the Schmidt-Baillie formula is a good approximation to the exact MSFE, and that it helps explain why the exact MSFE can decrease as the forecast horizon increases. In fact, for dynamics typical to econometric models, the MSFE often has a mazimum at a forecast horizon of one to twelve periods, i.e.,

at horizons that are of principal concern to forecasters and policy makers.

Key words and phrases: approximations, autoregressive models, confidence intervals, dynamics, forecasts, maximum likelihood, mean-square forecast error, Monte Carlo, statistical inference, time series.

Exact and Approximate Multi-period Mean-square Forecast Errors for Dynamic Econometric Models

Neil R. Ericsson and Jaime R. Marquez!

1. Introduction

In practice, numerous factors contribute to the uncertainty associated with modelbased forecasts, including the inherently stochastic nature of the process generating the data and the imprecision of coefficient estimates.2 Confidence bands for forecasts, if computed, typically take account of the first source of uncertainty, but not the second. In an extensive Monte Carlo study of the univariate AR(1) process, Orcutt and Winokur (1969) obtain unbiased estimates of the exact least-squares based mean-square forecast error (MSFE) which accounts for both these sources of uncertainty. Hoque, Magnus, and Pesaran (1988) (hereafter, HMP) derive an analytical expression for the exact MSFE for the AR(1) srocess. These two papers show that coefficient uncertainty can substantially increase the MSFE over and above the contribution from the inherent uncertainty.

Although both analyses are significant steps in properly interpreting forecasts, they have important limitations. First, as HMP note, numerical evaluation of their exact formula is computationally burdensome for any forecast horizons but very short ones (e.g., for more than four periods ahead). Given the availability of high-frequency data, longer horizons often are of interest in economic, business, and policy applications. Second, their formula is restricted to the univariate first-order process, and exact generalizations to

multivariate multiple-lag econometric systems seem unlikely. Third, although Monte Carlo

!The authors are staff economists in the Division of International Finance. The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as reflecting those of the Board of Governors of the Federal Reserve System or other members of its staff. Helpful discussions with and comments from Julia Campos, Dale Henderson, David Hendry, David Howard, Jan Magnus, Doug McManus, and Ted Truman are gratefully acknowledged. We are indebted to Ned Prescott for assistance in preparing the figures.

2Additional sources of uncertainty include the choice of model specification and errors in data measuzement. As important as they are, those sources are beyond the scope of our paper, so we ignore them.

methods permit estimating the exact MSFE for more general models, such estimates are subject to the imprecision inherent in Monte Carlo simulation and the specificity of choosing a given model and set of coefficients rather than some other. A need exists for a formula for the MSFE which can be implemented and computed with ease for any linear dynamic system at any forecast horizon and which accounts for both inherent and coefficient uncertainty.

Schmidt (1974) and Baillie (1979) provide a solution via a simple approximation to the exact MSFE. Our paper ascertains the accuracy of their approximation by comparing it with the results from Orcutt and Winokur and HMP, and with a further study conducted herein on the MSFE of the maximum likelihood estimator for the stationary AR(1) model. We find that their approximation is remarkably accurate over a wide range of sample sizes, parameter values, and forecast horizons, giving support for its use in empirical practice.

Section 2 briefly reviews the derivation of the Schmidt-Baillie approximation and discusses its analytical properties. In particular, the approximation’s formulation provides an intuitive explanation of why the MSFE can decrease as the forecast horizon increases, behavior which Hoque, Magnus, and Pesaran find surprising. Section 3 shows that the deviations between the approximation and the exact MSFE for the AR(1) model are numerically small for most practical purposes, except in two cases: small samples with extreme values of the autoregressive coefficient, and forecast horizons approaching the distance at which the exact MSFE is infinite. To. assess the sensitivity of these findings, Section 4 compares the Schmidt-Baillie approximation with Monte Carlo estimates of the exact MSFE using an alternative asymptotically equivalent estimator, maximum. likelihood: the approximation is quite accurate, both for short forecast horizons and for longer horizons at which the MSFE for OLS is infinite. As justification for evaluating the exact MSFE for maximum likelihood at such horizons, we show that any truncated estimator has a finite exact MSFE at all forecast horizons, and that, for some truncated estimators such as maximum likelihood, the exact MSFE is bounded, regardless of the forecast horizon. That

identifies how sensitive the condition for the existence of the MSFE for OLS is to minor

changes in distributional assumptions because (e.g.) a truncated OLS estimator might be truncated at only very large values (which occur very infrequently) and yet would have a

finite MSFE at all forecast horizons.

2. An Approximation to the Multi-period Mean-Square Forecast Error

Schmidt (1974) and Baillie (1979) provide an approximation to the MSFE, albeit in two distinct contexts. Schmidt approximates the MSFE for the linear dynamic simultaneous equations model where a subset of the variables are strongly exogenous and are known for the forecast period. Baillie’s framework allows for such strongly exogenous variables but, if they are present, requires that they be forecast as well (i.e., true ez ante forecasting). Our exposition follows Chong and Hendry (1986) because of the the latter’s accessibility. To help in understanding the properties of the ezact MSFE, this section sketches ihe derivation of the Schmidt-Baillie approximation and discusses its analytical properties. For convenience, we denote the exact and approximate MSFE as ExMSFE and AppMSFE respectively.

Derivation. Let y; be an mxl vector of variables generated by a first-order autoregressive process:

(1) y = Ay + y u, ~ IN(O,Q) t=2,....n+s

t with y, given, and where the first n observations are available for estimation and the s—period-ahead forecast is of interest. Bold characters denote vectors (if lower case) and matrices (if upper case). Although (1) appears limited to first-order processes, it is not. If the underlying process is of a higher order, it always can be "stacked" to give a first-order process. Because of that stacking, or for other reasons, the variables of interest for

forecastirg may be a subset (or some linear combination) of y so we introduce a

n+s’ selection matrix S such that S’Ynas is the vector of interest. Further, the matrix A may be restricted (e.g., have zeros), so it is useful to recognize explicitly how the matrix A is a function of its unconstrained elements 8:

(2) a = AY = RO4+r,

where (-)” denotes the column vectorizing operator, and all the elements of R and r are known.

Next, assume that @is estimated by @ which is asymptotically distributed as: (3) ia-(9- 6) 3 N(0,¥) and so (4) Va-(a-a) 2 N(0,P) where lr = RWR’. In finite samples, the approximate distribution of 0 is: (5) 6 ~ N(O, W/n) . For the remainder of the derivation, (5) is treated as if it were the exact distribution of 6, i.e., terms smaller than 0,(n/?) in the distribution of @ are ignored.

Using the data [y1--Y, to forecast Ynis gives: (6) Ings = AY > the ez ante s—step-ahead forecast. By repeated substitution of (1) into itself at successive lags, the actual outcome nis is:

Sal

_ S i (7) Yn+s ~ A’y, + ey An 4ss

where A%I if A=0. Thus, the discrepancy between actual and forecast Yass is:

~ — {Set qi Ss 4S (8) Ones Ines) = (BA as] + (AMY,

Selecting the variable of interest gives the corresponding forecast error Bais’

+ S’(AS—A®)y

nn’

= s/ 3, Aus sl The two terms on the RHS of (9) correspond directly to the two sources of uncertainty being investigated. The first, S’ 3 Mun ysid is the cumulation of the shocks to which y, is subject over the interval [n+1,n+s], where each shock is weighted by the degree to which it influences S’Y¥n ag reflects the uncertainty present from using an estimated value of A rather than its true

the variable being predicted. The second, S’(A°—A*)y,

value in forecasting S’¥nas: We denote these two terms a 5°

5 and Le

Straightforwardly, the variance of the first term is: (10) Var(a. ly.) = $’ 2 Aa(A"))s = AsyMSFE , n,s n 1=0 which is the "asymptotic" (i.e., large n) MSFE. The approximate variance of the second term is: (11) Var(by l¥,) = nt (ley, )[D(s)’TD(s)](Iey,,) where D(s)’ = €(S’A°)”/ da’ = (S’eI) ee Ale(as 7’) . Its derivation is more 1=

complicated and is given in Appendix A.3 Because A® and the u ; are independent (by

n+s— assumption», ans and b, , are as well, so we can add their variances together to obtain the approximate MSFE (AppMSFE):

(12) AppMSFE(&, | .ly,) = 9’ BE Ag(a')|s + nt (Tey/)[D(s)’ FD(s)](Iey,)

Var(a, l¥,,) increases monotonically as s increases, but Var(b, 1¥,) may increase before decreasing “o zero. Hence, AppMSFE(&, , .|Y,) may decrease as well as increase, as s increases. Equation (12) is relatively easy to implement in a computer program because it involves only sums of products of matrices. S and y, are known, and the unknown elements of A, 2, and [ may be replaced by consistent estimates of them.

The univariate AR(1) provides insight into the approximation, i.e., (1) is:

(1°) y = M1 + Y& u, ~ IN(0,o?) t=2,...,n+s with some initial condition for y,, such as (13) y, »~ IN(0,60?) for arbitrary 6. The OLS estimator of Gis asymptotically distributed as: (4’) va-(2—8) 2 N(O0, (1-83) . Thus, (14) A = a= 6 = £ Q = @ R = 1

3Higher-order approximations could be obtained by employing a higher-order Tayler-series expansion in (A.5) and using distributional results in Shenton and Johnson (1965).

I = 0

v = [ = (l1-() = 1

Dis) = sis! .

In that case, the forecast error (9) simplifies to:

sol : *

and the approximate MSFE in (12) is:

(12’) AppMSFE(8,1 .1¥,)

_ G2 s “— + (n-t-y2)-(s@S-1)2. (1-6?) .

1-92

Note that (12’) immediately identifies the separate contributions of the different sources of

uncertainty: the first term on the RHS is the asymptotic term, the second is the part arising from coefficient uncertainty. 4

Analytical Properties. The AppMSFE in (12’) is a function of (02, vw 6, s,n). As forecasters, we are particularly interested in knowing how and why (12’) varies as these determinants vary and in knowing how well (12’) approximates the ExMSFE. Thus, the remainder of this section considers analytical properties of the components of (12’), and the following section compares the AsyMSFE, AppMSFE, and ExMSFE numerically.

The properties of the first term on the RHS of (12’) (i.e., AsyMSFE) are relatively simple and well-known. Starting at the conditional variance of y, (0?) for s=1, the AsyMSFE increases monotonically in s, tending to the unconditional variance of Yt (c2/(1—6?)).5 Because the two sources of uncertainty are additive and independent, both the AppMSFE and the ExMSFE are always larger than the AsyMSFE (with possible equality for the AppMSFFE).

4Although the original derivation of (12’) is difficult to ascertain, it appears as early as 1970 in Box and Jenkins (1970, p. 269).

5For the one-step-ahead forecast, (12’) simplifies to the more familiar formula (cf. Chow A n (1960)): AppMSFE(g, . sly,) = 9? + y2(1-62)/n & ol + yal yo yea).

The structure and properties of the second term on the RHS of (12’) require some examination. Its functional form can be easily explained and interpreted via its derivation. The term fs in (9’), viewed as a function of B, is approximated by a first-order Taylorseries expansion about f to give f() = fs = 6s + D(s)(#-6) + O,(n-t) = 6s + sfs-1(6—B) + O,(n-). Substitution into (GBs) -y,, gives ss-(B-f)-y + O,(n1), from which the second RHS term in (12’) follows immediately, using (4’).6 That term is always nonnegative (and generally positive) for finite n and s, and vanishes as either s or n becomes large. However, for a given sample size n, final observed value Ya and 6#0, it can either decrease monotonically as the forecast horizon s increases, or increase first and then decrease. Its path depends upon the behavior of the sequence {(s/s-!); s=0,1,2,...}, and so upon the particular value of 6. The contribution of coefficient uncertainty to the AppMSFE can be large or small relative to the latter, so the functional relationship between the AppMSFE and the forecast horizon s itself depends upon @ and Yn:

To examine the behavior of the MSFE as a function of s and G, we have evaluated (12’) numerically for a range of values of (vy 6, s,n).7 Figures 1-6 plot the AsyMSFE, AppMSFE, and ExMSFE (stationary case) for 6=(0.2, 0.7, 0.9) in combination with n=(10, 20). Both here and in following sections, the term y2 in (12’) is chosen to be a "representative" value, o2/(1—62), i.e., equal to its unconditional expectation.8 The values

of 6 imply Mr ranging from being nearly white-noise to highly autoregressive.

6The effect of s on the distribution of @s also can be seen through the following analogy with a standardized normal variate x. Var(xs) = (2s)!/(2s-s!), e.g., Var(xs) = 1, 3, 15, 105 for s = 1, 2, 3, 4. Clearly, taking a power of @ can increase its variance dramatically.

"Because o” is a scale factor in (1’) (and hence (12’)), we can set o2=1 without loss of generality: in that case, y, is measured in standard deviations of ut.

8Another justification for this choice is that the unconditional expectation E[((s—(s)y2] is approximately E[(@—(s)?]- Ely2] because fand yy are approximately independent. See Phillips (1979) for an extensive discussion on the conditional and unconditional finite sample distributions of the forecast error.

The figures reveal three distinct patterns which depend upon the values of 6 and n: steadily decreasing AppMSFE, steadily increasing AppMSFE, and an AppMSFE which increases and then decreases. For small 6, the AppMSFE is declining almost uniformly as the horizon increases, with the initial (one-step-ahead) AppMSFE being the largest. That arises because the uncertainty from estimating @ is large (from (4’)), but that uncertainty is unimportant in forecasting Yn4s except for s=1: mathematically, s@s-t in (12’) is approximately zero except for s=1, when it is approximately unity. Because the AsyMSFE changes little as s increases, the AppMSFE is declining from the start (Figures 1 and 2).

For larger values of §, the variance from coefficient uncertainty increases first and then falls because the multiplicative coefficient s in sfs-! dominates for small s but the exponential term (Gs-!) dominates for large s. As the sum of two components, one monotonically increasing and the other increasing and then tending to zero, the AppMSFE can either increase first and then fall towards the unconditional variance of yy (Figures 3-6) or increase monotonically, approaching that asymptotic variance (also increasing in s) from above. The latter would be the case for all the figures if the sample size n were large enough. Thus, the potentially large and varying contribution of coefficient uncertainty to the MSFE explains the puzzling phenomena that HMP (pp. 333-335) note on the behavior of the exact MSFE as s increases; cf. Chong and Hendry (1986, p. 685). The second component of the AppMSFE provides a simple analytical explanation of this behavior, to the extent that the AppMSFE offers a good approximation to the ExMSFE. Although there are some notable discrepancies between the AppMSFE and the ExMSFE in the figures (primarily for n=10 with s=4), the AppMSFE does remarkably well in approximating the ExMSFE, so well that it is difficult to distinguish them at n=20. The accuracy of approximation is the focus of Section 3, which compares the AsyMSFE, AppMSFE, and ExMSFE numerically for a range of values of (02, Vw 6, s, n). Before doing so, we note some empirical implications of these results.

Estimated autoregressive coefficients in dynamic econometric models range from the

very small (e.g., for equations in first differences) to those close to unity (e.g., for equations

8a MEAN-SQUARE FORECAST ERROR

Figure 1. #=.2 n=10 MSFE Figure 2. 6=.2 n=20 MSFE 122 122 ~ Asymptotic MSFE — — — - Approximate MSFE 18 118 o---—~- Exact MSFE for OLS - - Estimated Exact MSFE for MLE 114 114 ul 4] 1060 —— 1060 1020 1020 . 0.98 098

10 20

Forecast horizon (s) Forecast horizon (s)

in levels). Even for the corresponding (and wide) range of values for @, the ‘naximum of the AppMSFE is often between one and twelve periods, precisely the range over which we are interested in forecasting most accurately. That is also the range for which the AsyMSFE appears the poorest approximation to the approximate and exact MSFE. In fact, the AsyMSFE generally underestimates the AppMSFE (and ExMSFE) for finite

horizons, and the former need not even be the main component of the latter.

3. Numerical Properties of the Asymptotic, Approximate, and Exact MSF

This section contrasts the AsyMSFE and AppMSFE with the ExMSFI derived by HMP and with Monte Carlo estimates of the ExMSFE calculated by Orcutt and Winokur (1969). In both of these studies, the AsyMSFE captures much of the variation across experiments, and the AppMSFE does even better in doing so. The inaccuracy of approximation appears related to how close conditions for the existence of the ExMSFE are to being violated, so we discuss the existence of the ExMSFE in the context of both papers.

HMP. Taking advantage of the explicit relationship between the OLS estimator and the disturbances {u,} HMP derive an analytical expression for the exact MSFE for the univariate AR(1) process given by (1’)+(13) for OLS and tabulate it for two cases: (a) §=(1—6?)1 (the "stationary case") and (b) 6=1 (the "non-stationary case"). Tables la and 2a give the percent discrepancies between the AsyMSFE in (10) and the associated ExMSFE from HMP (Tables 1 and 2) under each of those assumptions; Tables 1b and 2b likewise give percent discrepancies between the approximate and exact MSFE.9 Because (10) holds exactly, discrepancies between the approximate and exact MSFE arise because (a) the asymptotic and finite sample variances of GB differ (and hence so do the respective variances of 3 in (9’)), (b) Bis biased in finite samples but not asymptotically (likewise for ps), (c) the first-order Taylor-series approximation of Gs about (8 ignores important terms, and (d) the approximate MSFE is conditional upon y, whereas the exact MSFE treats y,

observations are actually used in estimating (, the calculation of (12’) for the tables uses (n—1) rather than n.

Appendix B describes the calculations here and with respect to Orcutt and Winokur’s experiments, and lists the corresponding values of AsyMSFE and AppMSFE.

8Because only mr

0.60 0.70 0.80 0.90 0.95 0.99

0.50

9a Stationary Case

Table 1a: 100-{1 — (AsyMSFE/ExMSFE)}. 0.00 0.10 0.20 0.30 0.40

Percent Deviations of the Asymptotic and Approximate MSFE from the Exact MSFE: S

ONO CO SH OD 109 mreN

BX 19 00 4 CO HM 19 mon

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Sn COM De) st rd

AoW a oe

Table 1b: 100-[1 — (AppMSFE/ExMSFE)]. 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99

0.00

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9b 0.50 0.60 0.70 0.80 0.90 0.95 0.99

Non—Stationary Case 0.40

Table 2a: 100-[1 — (AsyMSFE/ExMSFE)]. 0.20 0.30

0.10

Percent Deviations of the Asymptotic and Approximate MSFE from the Exact MSFE: 0.00

ee on RMN oo

BR O19 <

2 OO Rt oN

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Table 2b: 100-[1 — (AppMSFE/ExMSFE)). 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99

0.00

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as stochastic.!0 Discrepancies between the asymptotic and exact MSFE arise because Bis not identiceélly @.

The asymptotic formula captures the behavior of the ExMSFE well for small to medium values of 6 and for large n, with deviations of the order of 5-15%, but it does poorly otherwise. The AppMSFE fares better: typical departures are 2-3% or less. As with the AsyMSFE, more sizable discrepancies appear for large s paired with small n and/or large @: the concept of "effective sample size" predicts the possibility of such departures under those conditions (cf. Sims (1974) and Hendry (1984)). Almost invariably, the AppMSFE is smaller than the exact MSFE.

The accuracy and the generality of the approximate MSFE do not in any way belittle HMP’s exact results for the AR(1) process. To the contrary, exact results are highly desirable because they involve no approximation error; and they are essential for assessing the accuracy of approximations such as (12’).

Orcutt and Winokur. In a Monte Carlo study evaluating numerous facets to least- Squares estimation of the univariate AR(1) process, Orcutt and Winokur (1969) estimate the ExMSFE for all combinations of B=(0.0, 0.3, 0.6, 0.9, 1.0), n=(10, 20, 40), and s=(1, 2, 3, 4). Unlike HMP, they include a constant term in the estimation of the AR(1) process: that is easily incorporated into the Schmidt-Baillie approximation by including a non-stochastic variable in the vector Y; which is equal to its own lag and is initialized at unity. Because of the additional uncertainty introduced by estimating a constant, the resulting AppMSFE is always larger than that for equations with a known constant, even if (as in Orcutt and Winokur’s experiments) the constant is zero. Equation (3) is not valid for Orcutt and Winokur’s experiment with G=1, so we use G=0.9999 instead. That should (and does) offer a good approximation, given the difficulty in finite samples in

distinguishing between a unit root and a root close to (but less than) unity.

In actual forecasting, yn is given, in which case the conditional MSFE seems more appropriate than the unconditional MSFE. At another level, Yn is often subject to data revisions, so it may be invalid to treat its (latent) value as known. This exemplifies another source of uncertair*v and it is outside our analysis.

Tables 3a and 3b respectively list the percent discrepancies of the AsyMSFE and AppMSFE from the estimated exact MSFE (ExMSFE) of Orcutt and Winokur (Table VII). The overall pattern parallels that in Tables 1 and 2: the AppMSFE generally fares better than the AsyMSFE, with the latter almost invariably underestimating the estimated exact MSFE. Both the asymptotic and approximate MSFE fare better for larger n and smaller £. Orcutt and Winokur’s estimates of the exact MSFE are subject to sampling errors from the Monte Carlo simulation. As Appendix C shows, the standard error for their estimated ExMSFE is about 4.5%, so the discrepancies between their estimates and the AppMSFE appear to be due almost exclusively to simulation uncertainty. 1

Existence of the MSFE. HMP show that the ExMSFE exists for the AR(1) model if and only if the forecast horizon s is not greater than (n—2)/2 [not greater than (n—3)/2 if a constant term is included in the regression, cf. Magnus and Pesaran (1989)]. For instance, for n=10 in Tables 1 and 2, the ExMSFE exists for s<4 only. The worsening of the approximation error as s increases may be due to the declining number of moments of the forecast error. The effects of the existence of moments are also suggested by Tables 1-2 and Figures 1-6, where the convergence of the exact and approximate MSFE appears faster. than O(n-2) for large s.

Interestingly, the ExMSFE does not exist for Orcutt and Winokur’s experiments with (n=10, s=4), yet the AppMSFE still does quite well at approximating their estimates. We interpret this surprising result as follows. For values of s for which the ExMSFE does not exist, the AppMSFE ‘still can be calculated and may provide accurate confidence intervals for the forecasts. However, because there is a significant probability of 8 being greater than unity and thus causing the forecast error to explode for large s, the tails of the exact density of the forecast error are too thick for its variance to exist. Sargan (1982)

examines a similar situation in which an estimator is well-behaved asymptotically but has

"The accuracy of AppMSFE for the univariate AR(1) processes in HMP and Crcutt and Winokur (1969) adds to Chong and Hendry’s (1986) Monte Carlo evidence on the accuracy of the Baillie-Schmidt approximation for a two-equation model.

lla

Percent Deviations of the Asymptotic and Approximate MSFE from Orcutt and Winokur’s (1969) Monte Carlo Estimates of the Exact MSFE

Table 3a: 100-[1 — (AsyMSFE/ExMSFE)}.

B n s 0.00 0.30 0.60 0.90 1.00 10 1 25.4 23.5 23.8 22.9 22.7 10 2 96 25.7 268 296 27.4 10 3 20.7 24.2 34.0 35.2 35.4 10 4 184 24.0 42.0 41.5 43.4 20 1 #47 +158 9.6 16.0 13.0 20 2 103 86 111 23.8 22.3 0 3 33 23 14.2 23.5 23.3 2 4 #2. 90 188 240 24.7 40 1 11.3 0 56 82. 9.5 40 2 7.0 21 54 103 8.0 40 3 92 100 40 111 106 40 4 22 48 86 149 162

Table 3b: 100-[1 — (AppMSFE/ExMSFE)].

p 0.60 0.90 1.00

jo) co) Oo can) w oe

10 1 8.9 6.5 6.9 5.8 5.5 10 2 —65 10.2 2.9 0 -4.9 10 3 11.9 8.8 9.3 -3.5 —7.7 10 4 9.4 8.4 19.0 -26 -6.9 20 1 —d.4 6.9 1 7.1 3.9 20 2 56 -4 -2.6 8.6 5.9 20 3 -18 -7.1 —9 1.9 —1.0 20 4 5.2 2 3.6 -—3.1 —7.0 40 1 6.7 -8.1 T 3.5 4.9 40 2 46 -2.7 -1.8 1.6 —1.4 40 3 6.8 5.8 -4.38 -1.2 —3.2 40 4 +49 3 2 1 -—1.0

no moments in finite samples. In light of his paper, the approximation in (12) may be interpreted as analogous to the Nagar approximation for the moments of an estimator. Conversely, the lack of existence of the ExMSFE (when that occurs) must be due to terms smaller than Op(n) in the squared forecast error (i.e., op(n-!) and probably Op(n-- 5)) because the AppMSFE accounts for all terms Op(n-!) and larger, and it exists for all s. The existence or otherwise of the ExMSFE at relatively large s is an issue recurring in the

following section.

4, The MSFE for the Maximum Likelihood Estimator

Because the Schmidt-Baillie approximation relies on only the asymptotic distribution of the estimator used, the formulae in Section 2 for OLS are equally valid for all estimators asymptotically equivalent to OLS (cf. (4) and (4’) above). Exact maximum likelihood (ML) is such an estimator, and one which has several desirable features in the present context.!2 In particular, its ExMSFE exists at all forecast horizons s, indeper.dent of the size of the estimation period n. We show this by examining the properties of the ExMSFE for truncated estimators: truncation at an arbitrary value implies the existence of the ExMSFE at all finite forecast horizons, and truncation at the unit circle (as with ML) implies a bound on the ExMSFE, independent of the forecast horizon. The first of these results allows us to assess the generality of the Schmidt-Baillie approximation at forecast horizons longer than those feasible for OLS. However, because the analytical formula for the ExMSFE with ML is unknown, we compare the AppMSFE with Monte Carlo estimates of the exact MSFE. Again, the AppMSFE is a remarkably good approximation, even for short estimation periods and long forecast horizons.

Truncated estimators of B and existence of the exact MSFE. Consider a truncated estimator § such that |8|<y for some positive bound y. The corresponding forecast error is:

(9) Ents = 2, (uy + (B)-y, |

Cf. Maekawa (1987) who shows an equivalence to O(n~!) between the distributions of the forecast error for OLS and approximate ML.

A bound can be placed on the MSFE for & by application of the triangle and Schwartz

n+s inequalities and by noting that E(y?)=02/(1—6?). _ 1-(B?)s . (15) ExMSFE(8,1 .lY,) o? LR + E[(@s—fs)2-y2] 1-(8?)s _ o———_] + Bl(6-Ps)}]-B(y2) 1-62 ,|1(82)§ E/(| Gs8|+ | Bs| )2]-E(y2 ot] | + BILAL) 1 — f2s + (1495)? o2 ne ; | 1-6?

If the bound is the unit circle (y=1), then a slightly looser bound exists which is _ independent of s: i.e., 502/( 1—6?), five times the unconditional variance of the process. The » existence o: the ExMSFE does not’ require that the estimator be consistent for any value | whatsoever, Conversely, because neither bound makes use of the asymptotic properties of G, neither converges to the AsyMSFE as n-w. Even so, the existence of a bound (and so of the ExMSFE) indicates how sensitive the existence conditions are to minor changes in the distributional assumptions of the estimator being used. 13 The MSFE for Mazimum Likelihood. In order to assess the accuracy of the AppMSFE in approximating the ExMSFE for ML without the advantage of exact analytical formulae, we have estimated the exact MSFE by Monte Carlo for a wide range of 6, n, and s, and compared those estimates with the asymptotic and approximate MSFE. Specifically, we chose 6 = (0.0, 0.1, 0.2, ..., 0.8, 0.9, 0.95, 0.99), n =.(10, 15, 20, 25, 40), and s = (1, 2, ..., 30), with 6=o2/(1—@2) to ensure stationarity. This design embeds the range of values evaluated by HMP for OLS with a stationary AR(1) process. However,

because the ExMSFE for ML exists for all forecast horizons, we can compare its values with

BE.g., the truncated estimator based on OLS and with y=103!° (the range permitted by double-precision calculations on a computer) implies existence of the ExMSFE for all s, yet that truncated estimator will look like OLS for virtually all practical purposes.

the AppMSFE at much longer forecast horizons than available to HMP. To obtain reasonably accurate Monte Carlo estimates, we used 10,000 replications per experiment and in addition implemented a control variate; Appendix D provides details.

Tables 4a and 4b respectively list the percent deviations of the AsyMSFE and the AppMSFE from the control variate "pooled" estimates of the exact MSFE for ML (PoMSFE) with n=10. Tables 5a—b, 6a—b, 7a—b, and 8a—b likewise list percent deviations for n = 15, 20, 25, 40. To condense presentation, values for s>10 appear for s a; multiples of five: at long horizons, the exact, asymptotic, and approximate MSFE all change slowly as a function of s in any case.!4 As with OLS, the AsyMSFE does reasonably well for small to medium values of @ and for large n, with the AppMSFE doing better, and over a wider range of G and n. As n increases, both deviations generally decline, as would ke expected because the ExMSFE is tending to the AsyMSFE. Unlike with OLS, the AppMSFE often over-estimates the ExMSFE for @ in the range of [0.80, 0.95]. However, for @ very close to the unit circle (G=0.99), the AppMSFE again under-estimates the ExMSFE. The boundedness of the ML estimator is affecting the ExMSFE for large 6, but little more can be said without considering terms smaller than O(n-!) in the MSFE. Evan so, the AppMSFE offers a remarkably simple and accurate summary of the behavior of the ezact MSFE for ML.

Figures 1-6 graph the estimated values of the ExMSFE for ML as well as the ExMSFE for OLS, the AsyMSFE, and the AppMSFE. For $=0.2 with n=10, the AppMSFE approximates the ExMSFE for ML better than it does the ExMSFI5 for OLS. At medium and large values of § (0.7 and 0.9) with n=10, the AppMSFE still does well, but it over-estimates the ExMSFE at short to medium horizons and under-estimates it at very long horizons. Relatedly, the "hump" so evident for the AppMSFE is less pronounced (but still present) for the ExMSFE for ML. For n=20, the deviations betweer. the exact

and approximate MSFE are much smaller than for n=10, as expected.

14A]l values of the respective MSFEs for s<30 are tabulated in Appendix D.

l4a

Percent Deviations of the Asymptotic and Approximate MSFE from Pooled Monte Carlo Estimates of the Exact MSFE for Maximum Likelihood (n

10)

Table 4a: 100-[1 — (AsyMSFE/PoMSFE)].

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99

0.00

SEN OQ O DH oor a

COMN MH rt

DORAN DOM

SH OD ah tH

MN & SH 19

2 DON 6 191916 tH

DON OD

OMAN

ONIN MONAT

CON EH No

Own Sad sO

Or ODO ri

RIGS

ator

mOOS

seco

ocos

oeas

Table 4b: 100-[1 — (AppMSFE/PoMSFE)].

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99

0.00

HTOIONN loom

oO 4

maN H19

NAOT

Owed

(O Ph CO MD © re

OAL O10

ACM OANA

RWIS toed OOO meco

oocf

ecco

14b

Percent Deviations of the Asymptotic and Approximate MSFE from Pooled Monte Carlo Estimates of the Exact MSFE for Maximum Likelihood (n=15)

Table 5a: 100-[1 — (AsyMSFE/PoMSFE)].

0.10 0.20 0.380 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99

0.00

13.2 23.4 31.3 38.2 43.2

4.0 6.5 8.7 10.7 12.6

DR QP st CO CO I ©€

20 Oh 1d LO CO CO CO

- OOD Or © 21 1 1D SH

fS- Ono OO1IdM

om) O19 CO ( OD OD CON

MAROON OMNIA

202 1D OOON

reonwwn Kr

Qrmown ro

rw? DS cO4

maN OY HD

toh ink

as I oe Bl eo

eeoeoo

oonsc}e

mecesce

OF 0D © re

Nano

400°

ooo o

oo°oo

eeoo

[eee

15 20 25 30

Table 5b: 100-[1 — (AppMSFE/PoMSFE)}.

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99

0.00

Mn wWmoNr mor

moO O1919 ae ae ee!

reo

OH OWN

MN W219

oenes

AOC COooO

CO B- CO MD © re

Nano

meee

cose

Sees

oooo

14c

Percent Deviations of the Asymptotic and Approximate MSFE from Pooled Monte Carlo Estimates of the Exact MSFE for Maximum Likelihood (n

20)

Table 6a: 100-[1 — (AsyMSFE/PoMSFE)}.

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99

0.00

TeAawn 4

Am aa 1d rH

mk OHO

Vea

MN OO SH 1D

moooo

4ecoo

oo°ocoo

CO r= ON © ri

com Ost

COmnriri

NOOO

4eco

coco

ecoco

oees

oooeo

Table 6b: 100-[1 — (AppMSFE/PoMSFE)].

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99

0.00

NM2A4

Teoets

MmN OHO

aecooeo

oceco

or om © ri

CO SH

AOC O

anon an)

oC Co°O

ecco

14d

Percent Deviations of the Asymptotic and Approximate MSFE from Pooled Monte Carlo Estimates of the Exact MSFE for Maximum Likelihood (n

25)

Table 7a: 100-[1 — (AsyMSFE/PoMSFE)].

0.10 0.20 0.30 0.40 0.50 060 0.70 0.80 0.90 0.95 0.99

0.00

Ny)

DORM H

ONON ON

SHeyreT

sto

OO OQNOS

K-OnnToO

NO HO

2AINSOS

an HOO

ocoee

ocece

Or CO MD © ri

ANSS

eoee

cose

oeoce

Table 7b: 100-[1 — (AppMSFE/PoMSFE)].

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99

0.00

mode

aN OO HO

Sop eD

2NSOOS

Son on oe

ooece

cooeo

ooece

1D 4 10 0 —H LO 19 19

WOANwO

dwt oO myo

ANOS

44CO

oooe

ococeo

ooce

coco

l4e

Percent Deviations of the Asymptotic and Approximate MSFE from Pooled Monte Carlo Estimates of the Exact MSFE for Maximum Likelihood (n

40)

Table 8a: 100-[1 — (AsyMSFE/PoMSFE)].

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99

0.00

Qn ton DONLH-N OHINN

No Ht tn

AOwWOM ra OD SH LD uD

aonA~ NOD OO OD

Nato NNANNANN

RP OINN MNT

IQ NAN RMI NN

HON

Aamate

onoes

MANY Ho

AMMO 1D «OB 00 00

Owowowowo

TANTS OOANN

OWwIAN

EQN QS

aAOnoOo

moeoco

ocece

ooece

co fk OD © re

4oc0

eoos

ooce

Table 8b: 100-[1 — (AppMSFE/PoMSFE)].

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99

0.00

HoH M

HoH Onr

LROSLS

WRITS S

aN OO HO

INOS

mWOEWS2

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ooees

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Before concluding, three issues are worth brief mention: sample size, exogenous variables, and model linearity. The smallest sample size for our numerical results (n=10) is very small in the context of empirical work. However, it may be a reasonable number to use for comparison with empirical work, given that only one coefficient is being estimated. The sample size relative to the number of coefficients is a plausible measure in this context (cf. Sargan (1975)); and since much empirical work involves fewer than ten observations per coefficient estimated, n=10 may be large rather than small for practical purposes.

Forecasts of endogenous variables often are based on forecasts (rather than known values) of exogenous variables, adding another source of uncertainty. The algebra of Section 2 readily addresses this because it analyzes a complete system: exogenous variables can be included in the system in the same manner as the endogenous variadles, but the former are not simultaneously determined with the endogenous variables nor are they Granger-caused by the endogenous variables. However, for even relatively srnall systems, the AppMSFE in (12) can become awkward to compute because of the large matrices arising from vectorizing and from Kronecker products. Calzolari (1987) provides an ingenious analytical technique which dramatically reduces the computational burden, making the calculation of the AppMSFE feasible for medium- to large-scale mcdels.

The system in Section 2 is linear. Analytic approximations to confidence intervals could be constructed for nonlinear equations (or systems) as well, but Mariano and Brown (1983) show that simulation may be preferable, not only for the MSFE but for the forecast itself. In the context of (9), both the Waa si and the A would be replicated a number of times by Monte Carlo simulation according to their estimated distributions, and Monte Carlo estimates of the forecast mean and the MSFE would be constructed from the resulting "pseudo-forecasts". Marquez (1988) applies this simulation approach to estimate confidence intervals for the response of the US trade account to alternative exchange rate realizations. That analysis examines the sensitivity of the confidence intervals to the two

types of uncertainty addressed here. His application also demonstrates that the uncertainty

of coefficient estimates can have implications for economic questions other than just those

dealing with forecasts, e.g., paths of dynamic multipliers.

5. Conclusions

The Schmidt-Baillie formula provides a simple, accurate analytical approximation to the exact MSFE for the AR(1) process and conveniently summarizes a wealth of computationally intensive calculations given by Orcutt and Winokur (1969) and Hoque, Magnus, and Pesaran (1988) for OLS with and without a constant term, and given herein for maximum likelihood without a constant term. Further, the approximate MSFE can be used to provide confidence intervals for forecasts in instances where formulae for the exact MSFE are not known (e.g., multi-equation, multiple lag, dynamic simultaneous equations systems), and when the exact MSFE may not even exist. In contrast to the asymptotic MSFE, which increases monotonically with the forecast horizon, the exact MSFE can decrease as well as increase as the forecast horizon increases, and the approximate MSFE simply and accurately captures why that occurs. That non-monotonicity can be present even when the economic process has little dynamics. What is important is that the extent of the dynamics is unknown and so must be estimated.15 For dynamics common to econometric models, the approximate MSFE often has a mazimum at a forecast horizon of one to twelve periods, i.e., at horizons that are of principal concern to forecasters and policy makers. Although exact results are seldom available for realistic econometric models, tine Schmidt-Baillie approximation is easily. calculated and appears more accurate

than the standard asymptotic formula.

15In our framework, autoregressive errors constitute dynamics, even if associated with static models.

17 Appendiz A. The Derivation of V ar(b,, 5 | Yn:

First, note that (ABC)” = (AeC’)B” for conformable matrices A, B, and C, where the Kronecker product ® is defined as (A®B) = (bj ;A). Thus,

(A.1) (S’A*y,) (IS’ A*y,,)”

(Iey’)(S”.A®)”

where I is the identity matrix. Hence the second term on the RHS of (9) is:

(A.2) S’‘(A°-A®)y, = (Iey’)[(S’A‘)”-(S’ AS)”

(Tey) )[f,(8@) - £,(9)]

where f,(@) = (S’A®)” = (S’eI)(A®)”. The derivative Af,(9)/8’ is D(s)’R where: (A.3) D(s)’ a(S’ A®)”/ da’

(S’@1){A(A*)”/da’}

(s’e1)[*5) Alias ty’)

by application of the matrix form of the chain rule, noting that:

(A.4) (AS)” = [ alacas 1] _ [ alecas hy] cay’

for i=0,...,s—l. Expanding £,(8) in a Taylor series about £,(8) gives:

(A.5) va-[f,(8) —£,(8)] = nts)" ge le (#8)

where 6* lies between @ and 8. Because D(s)’ is everywhere continuous in @, then:

(A.6) plim ID(s) "T] 9: = M(s)"I] i

By application of Cramér’s (1946, p. 299) Linear Transformation Theorem and Mann and Wald’s (1943) Corollary 2,

(A.7) va-[f,(4) -£,(9)] 2 N(0, D(s)FD(s)) ,

and so at last we have:

(A.8) Var(b, sly,) = nt-(ley{)[D(s)/TD(s)](Iey,,) ,

ignoring terms of o,(n4).

See Schmidt (1974), Baillie (1979), and Chong and Hendry (1986) for details.

Appendiz B. Asymptotic and Approtimate MSFE for Hoque, Magnus, and Pesaran (1988) and Orcutt and Winokur ( 1969)

This appendix lists the asymptotic and approximate MSFE corresponding to the values of ((,n,s) for which the exact MSFE is numerically evaluated in Hoque, Magnus, and Pesaran (1938, Tables 1 and 2) and for which Orcutt and Winokur (1969, Table VII) conducted Monte Carlo experiments.

For HMP, (12’) in the text above is the basis for the calculations of the AsyMSFE and AppMSFE, reported in Table B.1, noting that the AsyMSFE is the first term on the RHS of that equation. The same asymptotic and approximate MSFE are used for both the stationary and non-stationary cases in Tables 1 and 2 in the text.

For Orcutt and Winokur (1969), the AsyMSFE and AppMSFE are derived below, directly from: (12), and are reported in Table B.2. However, (3) is not valid for their experiment with 6=1.0, so we use §=0.9999 instead. That should offer a good approximation, given the difficulty in finite samples in distinguishing between a unit root and a root close to (but less than) unity.

The derivation of the AsyMSFE and AppMSFE for the AR(1) process with an unknown constant proceeds as follows. Equation (1) is a two-equation system, with the

first equation being the AR(1) process and the second equation defining the constant term:

Vit Ba Unt (B.1) YY = = Ay, + ui= Yy_y + , Yor 0 1 0 where y,, = 1. The vectors and matrices necessary for solving (12) are as follows. (B.2) a = AY = (f0a1)! 6 = (af)’ a2 0 Q = 0 0 S = (10)

o?+a?(1+8)/(1-6) -a(1+f) -a( 1+) (1-62)

01 R - |09 0

1 0

00 r = (0001)

In Orcutt and Winokur’s experiments, a=0, simplifying ¥ and hence I and D(s).

a2 0 B.3 v= Ba LS com)

T= RWR’ = 0 0 o 0 0 0 0 0 Dest 0 0 0 Dis)’ = . 0 0 se oO

The summations in D(s)’ are over j=0,...,s-1. Substituting (B.2)-(B.3) into (12) and

simplifying, the AppMSFE for a univariate AR(1) process with an unknown constant is: 1-(82)s 1-#5)*}

-p? = y

where we have set y?,=02/(1—G2), its unconditional expectation. The first term on the

(B.4) AppMSFE = “|

+ (n*t- n),| (sGs-1)2 +

RHS of (B.4) is the AsyMSFE and is the same as when the constant is known. The first term in the braces is the effect from estimating 6 (as in (12’)); the second term is from estimating and is additional to what appears in (12’). These three sources of uncertainty

can be seen clearly from the generalization of (9’), the equation for the forecast error.

(B5) Onis Tues) = 2 yey + (BB) -y, + oS (GH) + (ayD

The first two terms on the RHS are the same as in (9’), the third is zero for a=0, and the fourth is the contribution from estimating rather than knowing a. Fuller and Hasza (1980)

derived (B.4) directly from (B.5).

10 10 10 10

15 15 15 15

20 20 20 20

25 25

8s88se8 8

Rone | Ppwonme | pone | Pome |] w

Rm WHR

Table B.1: Values of AppMSFE and AsyMSFE for HMP (1988, Tables 1-2)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.111

1.000 1.000 1.000

1.071 1.000 1.000 1.000

1.053 1.000 1.000 1.000

1.042 1.000 1.000 1.000

1.111 1.014 1.010 1.010

1.071 1.013 1.010 1.010

1.053 1.012 1.010 1.010

1.042 1.012 1.010 1.010

1.111 1.058 1.043 1.042

1.071 1.051 1.043 1.042

1.053 1.048 1.042 1.042

1.042 1.047 1.042 1.042

1.111 1.130 1.106 1.100

1.071 1.116 1.103 1.100

1.053 1.109 1.102 1.099

1.042 1.105 1.101 1.099

1.111 1.231 1.211 1.197

1.071 1.206 1.202 1.194

1.053 1.194 1.198 1.193

1.042 1.187 1.195 1.192

1.111 1.361 1.375 1.356

1.071 1.321 1.353 1.346

1.053 1.303 1.342 1.341

1.042 1.292 1.336 1.339

1.111 1.520 1.619 1.619

1.071 1.463 1.573 1.590

1.053 1.436 1.551 1.576

1.042 1.420 1.538 1.567

1.111 1.708 1.970 2.057

1.071 1.630 1.884 1.982

1.053 1.593 1.844 1.947

1.042 1.572 1.820 1.926

1.111 1.924 2.459 2.778

1.071 1.823 2.313 2.611

1.053 1.775 2.244 2.532

1.042 1.747 2.203 2.487

1.111 2.170 3.122 3.942

1.071 2.041 2.888 3.605

1.053 1.981 2.777 3.445

1.042 1.945 2.712 3.352

1.111 2.416 3.901 5.556

RPwhr Oot © crore A ocOoOr

DMWr © “1 CO OT Ord & w

1.042 2.143 3.301 4.510

1.000

1.000 1.000 1.000

1.000 1.010 1.010 1.010

1.000 1.040 1.042 1.042

1.000 1.090 1.098 1.099

1.000 1.160 1.186 1.190

1.000 1.250 1.313 1.328

1.000 1.360 1.490 1.536

1.000 1.490 1.730 1.848

1.000 1.640 2.050 2.312

1.000 1.810 2.466 2.998

1.000 1.980 2.941

3.882

Table B.2: Values of AppMSFE and AsyMSFE for Orcutt and Winokur (1969, Table VII)

n § 0.00 0.30 0.60 0.90 1.00

10 1 1.222 1.222 1.222 1.222 1.222 10 2 1.111 1.818 1.804 2.571 2.889 10 3 1.111 1.821 2.046 3.938 4.999 10 4 1.111 1.823 2.145 5.256 7.553 20 1 1.105 1.105 1.105 1.105 1.105 20 2 1.053 1.198 1.571 2.171 2.421 20 3 1.053 1.204 1.753 3.163 3.946 20 4 1.053 1.205 1.825 4.068 5.682 40 1 1.051 1.051 1.051 1.051 1.051 40 2 1.026 1.143 1.463 1.986 2.205 40 3 1.026 1.150 1.618 2.806 3.461 40 4 1.026 1.151 1.677 3.519 4.819 o 1 1.000 1.000 1.000 1.000 1.000 o 2 1.000 1.090 1.360 1.810 2.000 o 3 1.000 1.098 1.490 2.466 3.000 o 4 1.000 1.099 1.5386 2.998 4.000

Appendiz C. The Imprecision of Monte Carlo Estimates of the MSFE

This appendix provides an approximate lower bound on the uncertainty of Monte Carlo estimates of the exact MSFE. This bound is valid for the AR(1) model both with and without estimating a constant term, and it generalizes straightforwardly to the MSFE from a general linear dynamic system by using a Wishart rather than a x? distribution.

In (9), and so in (9’) and (B.5), the source of "inherent" uncertainty is independent from that of coefficient uncertainty. In a Monte Carlo analysis such as Orcutt and

Winokur’s, all the u,’s are simulated; thus, both the "future" errors u and the @ are

n+s—i simulated. Throughout this appendix, we ignore the latter effect because the first

dominates, at least for large n. From independence, that results in a lower bound on the

associated variability from Monte Carlo simulation. The forecast errors (ignoring

n4s-i? where ¢

coefficient uncertainty) are a linear combination of the future shocks u

denotes the /th of L replications.

t af wv Sst ray. The wt 4.g—; are jointly normal, so the linear combination of them on the RHS is normal. Sol, p. (C.2) S, (6i)-uh,. ~ N(0, AsyMSFE)

The Monte Carlo estimator of the MSFE is the average of the squared forecast errors:

L wt

_ y (C.3) McMSFE = Jy (Yass Sats

}3/L~ AsyMSFE-x%(L)/L.

Because the first two moments of a x2(L) are L and 2L, and because L typically is quite large, we have the following approximation:

(C.4) McMSFE/ExMSFE % McMSFE/AsyMSFE wy OYE» NG, (2/0). That is, the estimated MSFE is unbiased for the ExMSFE (which is an exact result, following directly from the estimated MSFE being a sample mean of the ExMSFE), with a percent standard deviation approximately equal to 100-y(2/L). Monte Carlo simulation indicates that the approximation errors in (C.4) are small for the values of (G,n,s) in (e.g.)

HMP and with L>100. Orcutt and Winokur (1969) use L=1000, so the 95% confidence

interval on a typical estimate in their study is approximately +9.0%.

Appendiz D. Details of the Monte Carlo Simulation of the MSFE Based on the Mazimum Likelihood Estimator

This appendix describes methodological and computational aspects of the Monte Carlo simulation of the MSFE for ML, and tabulates the resulting asymptotic, approximate, and Monte Carlo estimates of the MSFE.

Experimental Design. In the notation and terminology of Hendry (1984), the data generation process is (1’) plus (13), the relationship of interest is (1’) (but would not be if, e.g., the econometric model were mis-specified), the objective of the Monte Carlo study is to estimate the (exact) MSFE over a wide range of the parameter space O x J, so that:

(D.1) 6 (6,07) € ©O (n, s) € fF

(9 | [8|<1; 0?>0)

T (r | n€[na,np], s€[Sa,sv]) ,

where na, Np, Sa, and sp are pre-assigned. We chose o2=1, and without loss of generality. For 6, n, and s, we chose the same values as in Hoque, Magnus, and Pesaran (1988), but included n=40 and s=5,6,...,30 as well, given the interest in the performance of the analytical formulae at medium to long forecast horizons. A full factorial design was adopted for (6,n) given by:

(D.2) B = (0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99)

(10, 15, 20, 25, 40) , with 60 experiments in all. For each experiment, the MSFE was estimated at horizons: (D.3) s = (1,2,3,4,5,6,..., 29, 30) .

Simulation. For each experiment, L replications of n+s normal pseudo-random numbers {(uf, t=1,....n+s), 4=1,...,L} were generated from pairs of uniform pseudo-random numbers using Box and Muller’s (1958) transformation.! For each replication, a set (yé, t=1,....n+s) was created from (1’) and (13) with 62=(1—6?)-1 (i.e., stationary y,’3)s and the

ML estimate was found by solving the cubic in 6 from setting the score of the likelihood

‘The uniform random number generators are Carrier, Atkins, and Taylor’s (1969) mixedcongruential generator RNDM (but converted from COMPASS to FORTRAN) and NAg’s (1984) multiplicative-congruential generator GO5CAF. Different random number generators were used for each number in the pair of uniform pseudo-random numbers in order to avoid potential difficulties with Box and Muller’s transformation: see Neave (1978).

equal to zero; see Koopmans (1942), Anderson (1971, p. 354), and Beach and MacKinnon (1978). Given the ML estimate, the forecast error was calculated for each value of s. Explicitly, let 6=B(Gn,nj,£) denote the ML estimate for the éth replication of the experiment with 6=6, and n=nj, and let (62) (Bu) 8«=1,2,...,30} be the corresponding set of observed forecast errors.

-l =(£) - (4) (2) _ Spiny? Sk_7k).yé The Monte Carlo estimator of the MSFE is: ; L _(£)

which, in Hendry’s (1984) terminology, is the naive Monte Carlo estimator. When normalized by the ExMSFE, it is approximately distributed as N(1, [2/L]). In our design, L is 104, so the standard deviation of McMSFE/ExMSFE is y(2/104) or about 1.4%. Increasing L tenfold would reduce the standard deviation to only about 0.5%, an indication of the difficulties in obtaining precise estimates by such Monte Carlo techniques. Cf. Ansley and Newbold (1980) who compute the McMSFE for several estimators (including ML) of various ARMA processes, but use 1000 or fewer replications per experiment.

Control variates provide a powerful method for variance reduction of naive Monte Carlo estimators; cf. Hammersley and Handscomb (1964) and Hendry (1984) for details. To be useful, a control variate (CV) should be highly correlated with the naive estimator and should have a known distribution.2 Because the purpose of the Monte Carlo study is to estimate a moment which is unknown, those two properties might seem to conflict. However, often it is possible to partition a statistic into an "asymptotic" component and a finite sample one, with the former having an exact distribution; cf. Hendry (1984) on doing so for econometric estimators. The McMSFE has a natural control variate because the first term on the RHS of the equality in (D.4) (and more generally, of (9)) is exactly normal, and

independent of the second term. The implied control variate is:

2Often, knowing the first two moments of the CV suffices.

L s,-1 t (D.6) CyvMSFE = Pa [2 (Bin sil! ; which, as is shown in Appendix C, is exactly distributed as AsyMSFE- y2(L)/L and has a mean of AsyMSFE. The CV is used to reduce the simulation uncertainty by subtracting it from the naive estimator (with which it is positively correlated) and adding back the known mean of

the CV. The resulting Monte Carlo estimator is called a pooled estimator, anc. here is:

(D.7) PoMSFE = McMSFE — CvMSFE + E(CvMSFE) L Sk 7k, _£ = AsyMSFE + EG (f° )-¥q 7/1 ,

By construction, the pooled estimator has the same expectation as the naive estimator. Its variance is smaller than that of the naive estimator by the extent to which the CV is correlated with the naive estimator. In the present case, the reduction in variance is obvious because the CV has eliminated the term in the naive estimator which simulates the AsyMSFE. The efficiency of the CV will vary across experiments, but from a cursory comparison of the fluctuations in the naive and pooled estimates, it is readily apparent that they are considerable for the MSFE.

The entire Monte Carlo study for the MSFE with ML took 12 hours 45 minutes on an IBM PS/2 Model 70 (80386 PC running at 20 MHz with an 80387 math chip).

Tables. Tables D.1—D.5 list the values of the PoMSFE for n=10, 15, 20, 25, and 40 respectively. Tables D.6—D.10 list the corresponding values of AppMSFE. Table D.11 lists

the values of AsyMSFE (applicable to all values of n).

COWONMDORPWNEF

Table D.1: The Pooled Monte Carlo Estimates

of the Exact MSFE for Maximum Likelihood (PoMSFE) for n=10.

1.106 1.049 1.030 1.022 1.012 1.016 1.014 1.015 1.011 1.012

1.011 1.011 1.010 1.010 1.010 1.010 1.011 1.010 1.011 1.010

1.010 1.010 1.010 1.010 1.010 1.010 1.010 1.010 1.010 1.010

1.116 1.081 1.064 1.054 1.050 1.046 1.046 1.043 1.042 1.042

1.042 1.042 1.043 1.042 1.042 1.042 1.043 1.041 1.042 1.042

1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042

0.30

1.104 1.138 1.135 1.118 1.110 1.104 1.106 1.102 1.102 1.102

1.100 1.100 1.099 1.100 1.100 1.099 1.099 1.099 1.099 1.099

1.099 1.099 1.099 1.099 1.099 1.099 1.099 1.099 1.099 1.099

0.40

1.106 1.232 1.237 1.227 1.221 1.214 1.209 1.199 1.200 1.201

1.195 1.193 1.191 1.192 1.194 1.193 1.193 1.192 1.191 1.191

1.191 1.191 1.190 1.191 1.191 1.191 1.191 1.191 1.191 1.191

1.111 1.331 1.392 1.380 1.378 1.378 1.372 1.364 1.351 1.350

1.349 1.346 1.344 1.342 1.339 1.340 1.337 1.337 1.336 1.335

1.335 1.334 1.333 1.334 1.334 1.334 1.335 1.334 1.334 1.334

0.60

1.122 1.472 1.606 1.638 1.634 1.622 1.614 1.614 1.599 1.592

1.587 1.581 1.577 1.576 1.576 1.574 1.571 1.568 1.567 1.568

1.567 1.567 1.567 1.565 1.565 1.564 1.564 1.563 1.563 1.563

0.70

1.091 1.617 1.877 1.990 2.044 2.063 2.071 2.070 2.065 2.048

2.038 2.039 2.026 2.009 2.007 1.994 1.987 1.988 1.985 1.981

1.978 1.979 1.973 1.973 1.974 1.973 1.970 1.971 1.972 1.971

0.80

1.093 1.782 2.227 2.504 2.693 2.801 2.889 2.921 2.957 2.968

2.981 2.951 2.953 2.952 2.925 2.915 2.905 2.903 2.885 2.870

2.863 2.858 2.852 2.844 2.837 2.829 2.823 2.828 2.833 2.827

0.90 0.95

1.071 1.958 2.694 3.312 3.809 4.245 4.597 4.867 5.092 5.292

1.067 2.093 3.077 4.019 4.906 5.770 6.602 7.416 8.165 8.833

5.449 9.491 5.593 10.189 5.709 10.819 5.834 11.413 5.858 11.911 5.942 12.413 5.974 12.905 6.032 13.393 6.046 13.799 6.060 14.197

6.093 14.475 6.090 14.783 6.066 15.058 6.054 15.323 6.032 15.574 6.028 15.829 6.024 16.083 5.985 16.388 5.946 16.678 5.925 16.910

0.99

1.168 2.631 4.403 6.441 8.788 11.221 14.008 17.080 20.418 23.980

27.659 31.724 35.812 40.078 44.643 49.584 54.764 59.840 65.245 70.630

75.965 81.850 88.023 94.019 100.169 106.487 112.841 119.275 125.746 131.839

Table D.2: The Pooled Monte Carlo Estimates of the Exact MSFE for Maximum Likelihood (PoMSFE) for n=15.

p s 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99 1 1.065 1.078 1.083 1.070 1.072 1.067 1.072 1.072 1.062 1.051 1.041 1.152 2 1.014 1.027 1.059 1.127 1.207 1.297 1.447 1.583 1.756 1.941 2.034 2.586 3 1.003 1.016 1.051 1.124 1.220 1.360 1.569 1.839 2.198 2.648 2.977 4.283 4 1.003 1.014 1.046 1.105 1.209 1.369 1.597 1.962 2.477 3.256 3.867 6.280 5 1.000 1.012 1.045 1.104 1.204 1.363 1.607 2.000 2.661 3.756 4.710 8.453 6 1.001 1.010 1.042 1.103 1.198 1.352 1.611 2.032 2.763 4.147 5.491 10.973 7 1.000 1.011 1.042 1.100 1.195 1.348 1.602 2.030 2.820 4.482 6.255 13.634 8 1.000 1.011 1.041 1.100 1.192 1.343 1.590 2.037 2.869 4.737 6.955 16.574 9 1.000 1.010 1.041 1.100 1.193 1.340 1.586 2.027 2.906 4.949 7.618 19.685 10 1.000 1.010 1.042 1.100 1.193 1.342 1.582 2.024 2.917 5.117 8.194 23.056 11 1.000 1.010 1.042 1.100 1.193 1.339 1.580 2.016 2.913 5.267 8.791 26.588 12 1.000 1.010 1.042 1.099 1.192 1.338 1.572 2.003 2.914 5.375 9.358 30.521 13 1.000 1.010 1.042 1.099 1.191 1.335 1.572 1.992 2.905 5.452 9.883 34.334 14 1.000 1.010 1.042 1.099 1.191 1.334 1.569 1.991 2.887 5.515 10.339 38.636 15 1.000 1.010 1.042 1.099 1.191 1.337 1.571 1.985 2.869 5.558 10.714 42.911 16 1.000 1.010 1.042 1.099 1.191 1.335 1.566 1.979 2.862 5.599 11.168 47.296 17 1.000 1.010 1.042 1.099 1.191 1.334 1.567 1.973 2.853 5.641 11.520 51.791 18 1.000 1.010 1.042 1.099 1.190 1.334 1.565 1.972 2.852 5.640 11.864 56.558 19 1.000 1.010 1.042 1.099 1.191 1.334 1.564 1.970 2.844 5.646 12.192 61.286 20 1.000 1.010 1.042 1.099 1.191 1.334 1.564 1.969 2.839 5.641 12.434 66.431 21 1.000 1.010 1.042 1.099 1.191 1.334 1.562 1.969 2.837 5.673 12.763 71.834 22 1.000 1.010 1.042 1.099 1.190 1.335 1.563 1.970 2.831 5.689 13.020 77.132 23 1.000 1.010 1.042 1.099 1.190 1.335 1.563 1.966 2.820 5.688 13.277 82.578 24 1.000 1.010 1.042 1.099 1.191 1.334 1.564 1.962 2.818 5.712 13.518 88.121 25 1.000 1.010 1.042 1.099 1.191 1.334 1.564 1.963 2.811 5.691 13.790 94.025 26 1.000 1.010 1.042 1.099 1.190 1.334 1.564 1.967 2.807 5.712 13.950 99.598 27 =:1.000 1.010 1.042 1.099 1.191 1.334 1.564 1.964 2.804 5.708 14.167 105.510 28 1.000 1.010 1.042 1.099 1.191 1.333 1.563 1.963 2.799 5.694 14.351 111.464 29 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.964 2.797 5.690 14.520 117.633 30 1.000 1.010 1.042 1.099 1.191 1.333 1.563 1.963 2.791 5.677 14.631 123.824

m~m

— CWONDOPWNFE

ea oR WN

Table D.3: The Pooled Monte Carlo Estimates

of the Exact MSFE for Maximum Likelihood (PoMSFE) for n=20.

1.028 1.039 1.896 2.023 2.609 2.937 3.179 3.811 3.653 4.615 4.044 5.336 4.372 6.062 4.640 6.687 4.846 7.265 5.028 7.842

5.170 8.378 5.281 8.841 5.380 9.315 5.457 9.748 5.906 10.133 5.962 10.497 5.607 10.868 5.605 11.204 5.638 11.524 5.672 11.781

5.657 12.027 5.642 12.282 5.651 12.491 5.648 12.713 5.621 12.866 5.611 13.022 5.602 13.183 5.096 13.274 5.990 13.428 5.088 13.540

1.123 2.509 4.146 6.024 8.061 10.375 12.865 15.592 18.504 21.550

24.881 28.284 31.905 35.593 39.527 43.559 47.818 51.968 56.220 60.862

65.446 70.448 75.559 80.547 85.965 90.991 96.572 102.169 107.923 113.626

Table D.4: The Pooled Monte Carlo Estimates

of the Exact MSFE for Maximum Likelihood (PoMSFE) for n=25.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90 0.95

CTOWONMDOLPWNH

NORE RB eRe RRR Ee ee DOoOnnorhwohdre

NONMWNNN Wb AOorPwondre

WN bd bdo owoon

1.038 1.006 1.001 1.001 1.000 1.000 1.000 1.000 1.000 1.000

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

1.045 1.018 1.011 1.010 1.011 1.011 1.010 1.010 1.010 1.010

1.010 1.010 1.010 1.010 1.010 1.010 1.010 1.010 1.010 1.010

1.037 1.049 1.045 1.044 1.042 1.041 1.042 1.042 1.042 1.042

1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042

1.041 1.109 1.103 1.102 1.100 1.100 1.100 1.100 1.099 1.099

1.099 1.099 1.099 1.099 1.099 1.099 1.099 1.099 1.099 1.099

1.038 1.186 1.205 1.198 1.194 1.195 1.192 1.191 1.191 1.191

1.191 1.190 1.190 1.190 1.190 1.190 1.190 1.190 1.190 1.190

1.191 1.191 1.191 1.190 1.190 1.190 1.190 1.190 1.190 1.190

1.040 1.289 1.337 1.346 1.344 1.340 1.338 1.338 1.334 1.335

1.333 1.334 1.334 1.334 1.333 1.333 1.333 1.333 1.333 1.333

1.333 1.333 1.333 1.333 1.333 1.334 1.334 1.334 1.333 1.333

1.038 1.406 1.532 1.574 1.579 1.576 1.579 1.576 1.572 1.570

1.569 1.567 1.565 1.566 1.564 1.563 1.563 1.563 1.563 1.564

1.563 1.563 1.562 1.563 1.563 1.563 1.563 1.563 1.563 1.563

1.042 1.558 1.801 1.920 1.961 1.979 1.994 1.988 1.985 1.986

1.982 1.969 1.964 1.963 1.965 1.969 1.968 1.968 1.967 1.965

1.964 1.963 1.964 1.962 1.960 1.961 1.962 1.961 1.962 1.961

1.040 1.721 2.170 2.451 2.608 2.720 2.801 2.827 2.843 2.852

2.863 2.862 2.861 2.848 2.835 2.840 2.831 2.826 2.821 2.820

2.815 2.817 2.808 2.808 2.802 2.802 2.800 2.795 2.792 2.793

1.034 1.030 1.903 2.008 2.618 2.922 3.199 3.750 3.669 4.529 4.068 5.259 4.378 5.922 4.643 6.532 4.831 7.092 4.988 7.603

5.138 8.107 5.250 8.535 5.357 8.950 5.423 9.356 5.471 9.687 5.489 10.001 5.538 10.296 5.543 10.654 5.559 10.871 5.569 11.101

5.065 11.277 5.583 11.479 5.593 11.653 5.592 11.853 5.573 12.025 5.587 12.194 5.568 12.336 5.576 12.412 5.554 12.469 5.545 12.558

0.99

1.122 2.463 4.033 5.806 7.713 9.822 12.175 14.692 17.384 20.285

23.305 26.582 29.712 33.256 36.877 40.741 44.612 48.773 52.859 57.374

61.868 66.301 71.171 75.694 80.584 85.633 90.631 95.607 100.474 105.541

COWMOONMDMOHPWNH

—e

ry —

NR RRR ep CWO ONDER WN

NNINMWMNMNN ND NOD OTR NF

Wh DD ooo

Table D.5: The Pooled Monte Carlo Estimates

of the Exact MSFE for Maximum Likelihood (PoMSFE) for n=40.

1.000 1.010 1.042 1.099 1.190 1.000 1.010 1.042 1.099 1.191 1.000 1.010 1.042 1.099 1.190 1.000 1.010 1.042 1.099 1.190 1.000 1.010 1.042 1.099 1.190 1.000 1.010 1.042 1.099 1.190 1.000 1.010 1.042 1.099 1.190 1.000 1.010 1.042 1.099 1.190 1.000 1.010 1.042 1.099 1.190 1.000 1.010 1.042 1.099 1.190 1.000 1.010 1.042 1.099 1.190 1.000 1.010 1.042 1.099 1.190 1.000 1.010 1.042 1.099 1.190 1.000 1.010 1.042 1.099 1.190 1.000 1.010 1.042 1.099 1.190

1.000 1.010 1.042 1.099 1.190 1.000 1.010 1.042 1.099 1.190 1.000 1.010 1.042 1.099 1.190 1.900 1.010 1.042 1.099 1.190 1.900 1.010 1.042 1.099 1.190 1.900 1.010 1.042 1.099 1.190

1.026 1.278 1.325 1.338 1.339 1.338 1.337 1.335 1.334 1.334

1.333 1.333 1.333 1.333 1.333 1.333 1.333 1.333 1.333 1.333

1.019 1.392 1.513 1.555 1.566 1.568 1.568 1.568 1.565 1.565

1.565 1.564 1.564 1.564 1.563 1.563 1.563 1.563 1.563 1.562

1.562 1.562 1.562 1.562 1.563 1.563 1.563 1.563 1.563 1.562

1.026 1.532 1.781 1.894 1.945 1.971 1.975 1.981 1.979 1.975

1.970 1.966 1.969 1.964 1.963 1.963 1.964 1.965 1.963 1.963

1.962 1.961 1.962 1.961 1.962 1.961 1.961 1.961 1.961 1.961

1.022 1.691 2.115 2.388 2.566 2.679 2.743 2.776 2.794 2.801

2.809 2.812 2.809 2.808 2.810 2.808 2.806 2.804 2.798 2.794

2.792 2.794 2.794 2.790 2.791 2.789 2.786 2.785 2.784 2.783

1.020 1.021 1.866 1.971 2.583 2.836 3.155 3.626 3.619 4.356 4.018 5.009 4.329 5.635 4.575 6.197 4.791 6.721 4.942 7.185

5.051 7.630 5.142 8.039 5.240 8.417 5.303 8.742 5.340 9.045 5.368 9.335 5.382 9.601 5.407 9.821 5.418 10.034 5.435 10.252

5.457 10.415 5.457 10.596 5.432 10.730 5.436 10.869 5.427 11.002 5.426 11.126 5.416 11.210 5.436 11.307 5.445 11.298 5.440 11.420

1.099 2.371 3.790 9.359 7.088 8.993 11.023 13.225 15.557 17.967

20.450 23.063 25.659 28.448 31.282 34.305 37.339 40.631 43.852 47.145

50.553 54.235 57.833 61.580 65.137 68.910 72.868 76.749 80.818 84.840

eee

Table D.6: The Approximate MSFE (AppMSFE) for n=10.

s 0.00 0.10 0.20 0.30

0.40

6 0.50

0.60

0.70

0.80

0.90 0.95

1.111 1.111 1.111 1.014 1.058 1.130 1.000 1.010 1.043 1.106 1.010 1.042 1.100 1.010 1.042 1.099 . 1.042 1.099 1.000 1.010 1.042 1.099 1.000 1.010 1.042 1.099 1.000 1.010 1.042 1.099 1.000 1.010 1.042 1.099

COON MOFR WHE a oO co) lon) —" jon) He con)

11 1.000 1.010 1.042 1.099 12 1.000 1.010 1.042 1.099 13 1.000 1.010 1.042 1.099 14 1.000 1.010 1.042 1.099 15 1.000 1.010 1.042 1.099 16 =1.000. 1.010 1.042 1.099 17 1.000 1.010 1.042 1.099 18 1.000 1.010 1.042 1.099 19 1.000 1.010 1.042 1.099 20 1.000 1.010 1.042 1.099

21 1.000 1.010 1.042 1.099 22 1.000 1.010 1.042 1.099 23 1.000 1.010 1.042 1.099 24 1.000 1.010 1.042 1.099 25 1.000 1.010 1.042 1.099 26 1.000 1.010 1.042 1.099 27 1.000 1.010 1.042 1.099 28 1.000 1.010 1.042 1.099 29 1.000 1.010 1.042 1.099 30 1.000 1.010 1.042 1.099

1.111 1.231 1.211 1.197 1.192 1.191 1.191 1.190 1.190 1.190

1.190 1.190 1.190 1.190 1.190 1.190 1.190 1.190 1.190 1.190

1.111 1.361 1.375 1.356 1.343 1.337 1.335 1.334 1.333 1.333

1.333 1.333 1.333 1.333 1.333 1.333 1.333 1.333 1.333 1.333

1.111 1.520 1.619 1.619 1.600 1.583 1.573 1.568 1.565 1.564

1.563 1.563 1.563 1.563 1.563 1.563 1.563 1.563 1.563 1.563

1.563 1:563 1.563 1.563 1.563 1.563 1.563 1.563 1.563 1.563

1.111 1.708 1.970 2.057 2.066 2.047 2.023 2.002 1.988 1.977

1.971 1.967 1.964 1.963 1.962 1.961 1.961 1.961 1.961 1.961

1.961 1.961 1.961 1.961 1.961 1.961 1.961 1.961 1.961 1.961

1.111 1.924 2.459 2.778 2.946 3.016 3.030 3.012 2.981 2.946

2.912 2.883 2.858 2.838 2.823 2.811 2.802 2.795 2.790 2.787

2.784 2.782 2.781 2.780 2.779 2.779 2.779 2.778 2.778 2.778

1.111 1.111 2.170 2.304 3.122 3.532 3.942 4.759 4.624 5.958 5.171 7.109 5.597 8.197 5.915 9.210 6.141 10.144 6.291 10.993

6.379 11.758 6.419 12.438 6.421 13.037 6.395 13.556 6.348 14.001 6.288 14.375 6.219 14.584 6.146 14.932 6.071 15.125 5.996 15.267

5.924 15.364 5.856 15.420 5.792 15.440 5.732 15.428 5.678 15.388 5.628 15.324 5.983 15.238 5.043 15.136 5.007 15.018 5.476 14.889

eee

0.99

1.111 2.416 3.901 5.556 7.368 9.327 11.422 13.642 15.979 18.423

20.965 23.596 26.309 29.095 31.948 34.860 37.825 40.836 43.886 46.970

50.083 53.219 56.373 59.540 62.716 65.897 69.077 72.254 75.424 78.583

COMONIMDoOPWNH

Table D.7: The Approximate MSFE (AppMSFE) for n=15.

1.071 1.071 1.071 1.071 1.071 1.071 1.071 1.071 1.000 1.013 1.051 1.116 1.206 1.321 1.463 1.630 1.000 1.010 1.043 1.103 1.202 1.353 1.573 1.884 1.000 1.010 1.042 1.100 1.194 1.346 1.590 1.982 1.000 1.010 1.042 1.099 1.192 1.339 1.583 2.008 1.000 1.010 1.042 1.099 1.191 1.336 1.575 2.006 1.000 1.010 1.042 1.099 1.191 1.334 1.569 1.996 1.000 1.010 1.042 1.099 1.190 1.334 1.566 1.985 1,000 1.010 1.042 1.099 1.190 1.333 1.564 1.977 1,000 1.010 1.042 1.099 1.190 1.333 1.563 1.971

1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.967 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.964 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.963 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.962 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961

1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961

1.071 1.823 2.313 2.611 2.779 2.863 2.896 2.901 2.891 2.874

2.857 2.841 2.826 2.815 2.805 2.798 2.793 2.789 2.786 2.783

2.782 2.781 2.780 2.779 2.779 2.778 2.778 2.778 2.778 2.778

1.071 1.071 2.041 2.160 2.888 3.241 3.605 4.292 4.197 5.300 4.673 6.254 5.048 7.146 5.334 7.972 5.545 8.729 5.695 9.417

5.796 10.036 5.856 10.589 5.886 11.078 5.892 11.506 5.881 11.877 5.858 12.194 5.826 12.462 5.788 12.684 5.748 12.864 5.707 13.007

5.666 13.115 5.626 13.192 5.588 13.243 5.553 13.269 5.920 13.273 5.490 13.260 5.463 13.230 5.438 13.186 5.416 13.131 5.396 13.065

1.071 2.260 3.558 4.958 6.453 8.035 9.698 11.436 13.242 15.111

17.037 19.015 21.040 23.106 25.210 27.346 29.511 31.700 33.910 36.136

38.376 40.626 42.883 45.144 47.407 49.667 51.924 54.173 56.414 58.645

eee

Table D.8: The Approximate MSFE (AppMSFE) for n=20. 6B

eee

s 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99 ena 1 1.053 1.053 1.053 1.053 1.053 1.053 1.053 1.053 1.053 1.053 1.053 1.053 2 1.000 1.012 1.048 1.109 1.194 1.303 1.436 1.593 1.775 1.981 2.092 2.186 3 1.000 1.010 1.042 1.102 1.198 1.342 1.551 1.844 2.244 2.777 3.103 3.396 4 1.000 1.010 1.042 1.099 1.193 1.341 1.576 1.947 2.532 3.445 4.071 4.675 5 1.000 1.010 1.042 1.099 1.191 1.337 1.575 1.981 2.700 3.994 4.988 6.019 6 1.000 1.010 1.042 1.099 1.191 1.335 1.571 1.987 2.790 4.437 5.849 7.423 7 1.000 1.010 1.042 1.099 1.191 1.334 1.567 1.983 2.833 4.787 6.648 8.882 8 1.000 1.010 1.042 1.099 1.190 1.334 1.565 1.977 2.848 5.058 7.385 10.391 9 1.000 1.010 1.042 1.099 1.190 1.333 1.564 1.972 2.848 5.263 8.059 11.946 10 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.968 2.841 5.413 8.670 13.543 11 =1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.965 2.831 5.519 9.221 15.177 12 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.963 2.821 5.590 9.714 16.845 13 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.962 2.811 5.633 10.151 18.544 14 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.962 2.804 5.654 10.536 20.269 15 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.797 5.660 10.871 22.018 16 =1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.792 5.654 11.162 23.787 17 =1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.788 5.639 11.410 25.572 18 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.786 5.619 11.619 27.372 19 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.783 5.595 11.794 29.184 20 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.782 5.570 11.936 31.004 21 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.781 5.543 12.050 32.831 22 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.780 5.517 12.137 34.661 23 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.779 5.492 12.202 36.493 24 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.779 5.468 12.246 38.325 25 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.778 5.445 12.272 40.155 26 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.778 5.425 12.282 41.979 27 ~=1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.778 5.406 12.278 43.798 28 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.778 5.388 12.262 45.609 29 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.778 5.373 12.236 47.410 30 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.778 5.359 12.202 49.200

COMOONDOPWNEH

0.00

1.042 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

Table D.9: The Approximate MSFE (AppMSFE) for n=25.

ree

eee

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90 0.95

0.99

= TT eS

1.042 1.012 1.010 1.010 1.010 1.010 1.010 1.010 1.010 1.010

1.010 1.010 1.010 1.010 1.010 1.010 1.010 1.010 1.010 1.010

1.042 1.047 1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042

1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042

1.042 1.105 1.101 1.099 1.099 1.099 1.099 1.099 1.099 1.099

1.099 1.099 1.099 1.099 1.099 1.099 1.099 1.099 1.099 1.099

1.042 1.187 1.195 1.192 1.191 1.191 1.191 1.190 1.190 1.190

1.190 1.190 1.190 1.190 1.190 1.190 1.190 1.190 1.190 1.190

1.042 1.292 1.336 1.339 1.336 1.334 1.334 1.333 1.333 1.333

1.333 1.333 1.333 1.333 1.333 1.333 1.333 1.333 1.333 1.333

1.042 1.420 1.538 1.567 1.571 1.568 1.566 1.564 1.563 1.563

1.563 1.563 1.563 1.563 1.563 1.563 1.563 1.563 1.563 1.563

1.042 1.572 1.820 1.926 1.965 1.976 1.976 1.972 1.969 1.966

1.964 1.963 1.962 1.961 1.961 1.961 1.961 1.961 1.961 1.961

1.961 1.961 1.961 1.961 1.961 1.961 1.961 1.961 1.961 1.961

1.042 1.747 2.203 2.487 2.654 2.748 2.796 2.817 2.823 2.821

2.815 2.809 2.803 2.797 2.792 2.789 2.786 2.784 2.782 2.781

2.780 2.779 2.779 2.779 2.778 2.778 2.778 2.778 2.778 2.778

1.042 1.042 1.945 2.053 2.712 3.022 3.352 3.942 3.876 4.807 4.300 5.612 4.636 6.358 4.898 7.043 5.099 7.668 5.249 8.235

5.358 8.745 5.434 9.203 5.485 9.610 5.515 9.969 5.531 10.285 5.0935 10.559 5.930 10.796 5.920 10.998 5.506 11.169 5.489 11.312

5.472 11.428 5.454 11.522 5.436 11.595 5.418 11.649 5.402 11.687 5.386 11.711 5.372 11.723 5.359 11.724 5.347 11.715 5.337 11.698

1.042 2.143 3.301 4.510 5.766 7.066 8.405 9.781 11.190 12.628

14.092 15.580 17.088 18.614 20.156 21.710 23.275 24.848 26.427 28.011

29.596 31.182 32.766 34.347 35.924 37.495 39.058 40.613 42.157 43.691

TT eee

Table D.10: The Approximate MSFE (AppMSFE) for n=40. B

s 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 1 1.026 1.026 1.026 1.026 1.026 1.026 1.026 1.026 1.026 1.026 1.026 2 1.000 1.011 1.044 1.099 1.176 1.276 1.397 1.540 1.706 1.893 1.995 3 1.000 1.010 1.042 1.100 1.192 1.327 1.520 1.786 2.144 2.618 2.905 4 1.000 1.010 1.042 1.099 1.191 1.335 1.555 1.896 2.419 3.216 3.754 5 1.000 1.010 1.042 1.099 1.191 1.335 1.564 1.942 2.587 3.704 4.541 6 1.000 1.010 1.042 1.099 1.191 1.334 1.565 1.960 2.686 4.099 5.267 7 1.000 1.010 1.042 1.099 1.190 1.334 1.564 1.965 2.742 4.414 5.934 8 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.965 2.772 4.663 6.543 9 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.964 2.786 4.858 7.097 10 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.963 2.792 5.008 7.598 11 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.962 2.793 5.122 8.050 12 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.962 2.792 5.207 8.456 13 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.790 5.269 8.819 14 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.788 5.312 9.142 15 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.785 5.342 9.427 16 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.784 5.361 9.679 17 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.782 5.371 9.899 18 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.781 5.376 10.091 19 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.780 5.376 10.256 20 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.780 5.373 10.399 21 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.779 5.367 10.520 22 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.779 5.361 10.622 23 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.778 5.353 10.707 24 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.778 5.346 10.777 25 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.778 5.338 10.834 26 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.778 5.331 10.878 27 =1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.778 5.323 10.912 28 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.778 5.317 10.936 29 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.778 5.311 10.953 30 1.000 1.010 1.042 1.099 1.190 1.333 1.563 1.961 2.778 5.305 10.962

0.99

1.026 2.081 3.162 4.268 5.396 6.544 7.709 8.890 10.084 11.290

12.506 13.730 14.960 16.196 17.435 18.676 19.917 21.159 22.398 23.635

24.868 26.096 27.318 28.534 29.741 30.941 32.131 33.311 34.481 35.639

OCWoOoOnNDoOPWNr

NORE eee eee i COON DOHPWNEFH

bo bo Nr

ww) oe Ww

WhO bd Hd bo moO OTE

0.00

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

.000 .000 .000 000 .000 .000 ..000 ]..000 1..000 1.000

1..000 1.000 1.000 1..000 1..000 1.000 1..000 1..000 -..000 -..000

0.10

1.000 1.010 1.010 1.010 1.010 1.010 1.010 1.010 1.010 1.010

1.010 1.010 1.010 1.010 1.010 1.010 1.010 1.010 1.010 1.010

Table D.11:

0.20

1.000 1.040 1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042

1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042

0.30

1.000 1.090 1.098 1.099 1.099 1.099 1.099 1.099 1.099 1.099

1.099 1.099 1.099 1.099 1.099 1.099 1.099 1.099 1.099 1.099

The Asymptotic MSFE (AsyMSFE).

0.40

1.000 1.160 1.186 1.190 1.190 1.190 1.190 1.190 1.190 1.190

1.190 1.190 1.190 1.190 1.190 1.190 1.190 1.190 1.190 1.190

0.50

1.000 1.250 1.313 1.328 1.332 1.333 1.333 1.333 1.333 1.333

1.333 1.333 1.333 1.333 1.333 1.333 1.333 1.333 1.333 1.333

0.60

1.000 1.360 1.490 1.536 1.553 1.559 1.561 1.562 1.562 1.562

1.562 1.562 1.562 1.562 1.562 1.562 1.563 1.563 1.563 1.563

1.563 1.563 1.563 1.563 1.563 1.563 1.563 1.563 1.563 1.563

0.70

1.000 1.490 1.730 1.848 1.905 1.934 1.947 1.954 1.958 1.959

1.960 1.960 1.961 1.961 1.961 1.961 1.961 1.961 1.961 1.961

1.961 1.961 1.961 1.961 1.961 1.961 1.961 1.961 1.961 1.961

0.80

1.000 1.640 2.050 2.312 2.480 2.587 2.656 2.700 2.728 2.746

2.757 2.765 2.769 2.772 2.774 2.776 2.776 2.777 2.777 2.777

2.778 2.778 2.778 2.778 2.778 2.778 2.778 2.778 2.778 2.778

0.90

1.000 1.810 2.466 2.998 3.428 3.777 4.059 4.288 4.473 4.623

4.745 4.843 4.923 4.988 5.040 5.082 5.117 5.145 5.167 5.185

5.200 5.212 5.222 5.230 5.236 5.241 5.245 5.249 5.251 5.254

0.95

1.000 1.902 2.717 3.452 4.116 4.714 5.255 5.742 6.182 6.580

6.938 7.262 7.554 7.817 8.055 8.270 8.463 8.638 8.796 8.938

9.067 9.183 9.288 9.382 9.467 9.544 9.614 9.676 9.733 9.784

0.99

1.000 1.980 2.941 3.882 4.805 5.709 6.596 7.464 8.316 9.150

9.968 10.770 11.556 12.326 13.080 13.820 14.545 15.256 15.952 16.635

17.304 17.959 18.602 19.232 19.849 20.454 21.047 21.628 22.198 22.756

37 References

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Ansley, C.F. and P. Newbold (1980) "Finite Sample Properties of Estimators for Autoregressive Moving Average Models", Journal of Econometrics, 13, 2, 159-183.

Baillie, R.T. (1979) "Asymptotic Prediction Mean Squared Error for Vector Autoregressive Models", Biometrika, 66, 3, 675-678.

Beach, C.M. and J.G. MacKinnon (1978) "A Maximum Likelihood Procedure for Regression with Autocorrelated Errors", Econometrica, 46, 1, 51—58.

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Box, G.E.P. and M.E. Muller (1958) "A Note on the Generation of Random Normal Deviates", Annals of Mathematical Statistics, 29, 2, 610-611.

Calzolari, G. (1987) "Forecast Variance in Dynamic Simulation of Simultaneous Equation Models", Econometrica, 55, 6, 1473-1476.

Carrier, N., E. Atkins, and C. Taylor (1969) "Report on the London Atlas Random Number Generator", mimeo, London, University of London Computing Centre.

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Fuller, W.A. and D.P. Hasza (1980) "Predictors for the First-order Autoregressive Process", Journal of Econometrics, 13, 2, 139-157.

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Hoque, A., J.R. Magnus, and B. Pesaran (1988) "The Exact Multi-period Mean-square Forecast Error for the First-order Autoregressive Model", Journal of Econometrics, 39, 3, 327-346.

Koopmans, T.C. (1942) "Serial Correlation and Quadratic Forms in Normal Variables", Annals of Mathematical Statistics, 13, 1, 14-34.

Maekawa, K. (1987) "Finite Sample Properties of Several Predictors from an Autoregressive Model", Econometric Theory, 3, 3, 359-370.

Magnus, J.R. and B. Pesaran (1989) "The Exact Multi-period Mean-square Forecast Error for the First-order Autoregressive Model with an Intercept", Journal of Econometrics, forthcoming.

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Schmid:, P. (1974) "The Asymptotic Distribution of Forecasts in the Dynamic Simulation of an Econometric Model", Econometrica, 42, 2, 303-309.

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IFDP NUMBER

348

345

344

343

342

341

340

339

338

337

- 39.

International Finance Discussion Papers

TITLES 1989

Exact and Approximate Multi-Period Mean-Square Forecast Errors for Dynamic Econometric Models

Macroeconomic Policies, Competitiveness, and U.S. External Adjustment

Exchange Rates and U.S. External Adjustment in the Short Run and the Long Run

U.S. External Adjustment:

Progress and Prospects

Domestic and Cross-Border Consequences of U.S. Macroeconomic Policies

The Profitability of U.S. Intervention Approaches to Managing External Equilibria: Where We Are, Where We Might Be Headed, and How We Might

Get There

A Note on "Transfers" A New Interpretation of the Coordination Problem and its Empirical Significance

A Long-Run View of the European Monetary System

1988

The Forward Exchange Rate Bias: Explanation

A New

Adequacy of International Transactions and Position Data for Policy Coordination

Nominal Interest Rate Pegging Under Alternative Expectations Hypotheses

The Dynamics of Uncertainty or The Uncertainty of Dynamics: Stochastic J-Curves

AUTHOR (s)

Neil R. Ericsson Jaime R. Marquez

Peter Hooper

William L. Helkie Peter Hooper

Ralph C. Bryant John Helliwell Peter Hooper

Michael P. Leahy

Edwin M. T:cuman

David B. Gordon Ross Levine

Matthew B. Canzoneri Hali J. Edison

Hali J. Edison Eric Fisher

Ross Levine Lois Stekler Joseph E. Gagnon

Dale W. Henderson

Jaime Marquez

Please address. requests for copies to International Finance Discussion Papers, Division of International Finance, Stop 24, Board of Governors of the

Federal Reserve System, Washington, D.C.

20551.

Cite this document

APA

Neil R. Ericsson and Jaime R. Marquez (1989). Exact and Approximate Multi-Period Mean-Square Forecast Errors For Dynamic Econometric Models (IFDP 1989-348). Board of Governors of the Federal Reserve System, International Finance Discussion Papers. https://whenthefedspeaks.com/doc/ifdp_1989-348

BibTeX

@techreport{wtfs_ifdp_1989_348,
  author = {Neil R. Ericsson and Jaime R. Marquez},
  title = {Exact and Approximate Multi-Period Mean-Square Forecast Errors For Dynamic Econometric Models},
  type = {International Finance Discussion Papers},
  number = {1989-348},
  institution = {Board of Governors of the Federal Reserve System},
  year = {1989},
  url = {https://whenthefedspeaks.com/doc/ifdp_1989-348},
  abstract = {Both future disturbances and estimated coefficients contribute to the uncertainty in model-based ex ante forecasts, but only the first source is usually taken into account when calculating confidence intervals for practical applications. Schmidt (1974) and Baillie (1979) provide an easily computable second-order approximation to the mean-square forecast error (MSFE) for linear dynamic systems which recognizes both sources of uncertainty. To assess the accuracy of their approximation, and thus its usefulness, we compare it with three sets of estimates of the exact MSFE for the univariate AR(l) process: Monte Carlo estimates for OLS, analytically based values for OLS, and Monte Carlo estimates for maximum likelihood. We find that the Schmidt-Baillie formula is a good approximation to the exact MSFE, and that it helps explain why the exact MSFE can decrease as the forecast horizon increases. In fact, for dynamics typical to econometric models, the MSFE often has a maximum at a forecast horizon of one to twelve periods, i.e., at horizons that are of principal concern to forecasters and policy makers.},
}