confidence intervals for linear combinations of forecasts from dynamic econometric models

ntervals for ear

conometric

Julia Campos, Banco Central de Venezuela

This paper derives the approximate distribution of a vector of forecast errors from a dynamic simultaneous equations econometric model. The system may include exogenous variables with known and/or unknown future values. For the latter set of exogenous variables, a stationary and invertible ARMAX process is assumed. Con- fidence regions are derived for vectors of linear combinations of forecasts. These confidence regions are particularly useful for designing tests of super exogeneity via tests of predictive failure. To illustrate the use of confidence intervals, forecasts are generated for an oil price and for a Venezuelan consumer price index.

1. INTRODUCTION

Knowing the potential future evolution of economic variables allows policy makers to anticipate undesirable economic situations and thereby to implement policies that would change the path of economic variables. However, knowledge in economics is not exact, so policies must be based on variables’ forecasts, which are subject to uncertainty. Forecasts are not exact, and all we know is that a particular future observation ought to lie between two values with some probability. Those two values, together with that associated probability, define a confidence interval, which can be derived using statistical techniques.

Forecast confidence intervals themselves, as well as the ex ante forecasts, may be informative to policy makers. To illustrate, consider a country like Venezuela, which earns its international reserves almost entirely from exports of oil. Suppose that the country requires $lO,ooO million to pay for imports, that a crude oil price of $30 a barrel provides

Address correspondence to Julia Campos. Apartado 5162. Carmelitas. Caracas 1010, Venezuela.

The views in this paper do not necessarily coincide with those of the Banco Central de Venezuela. I am grateful to R. Campos, N .R. Ericsson, M. Nerlove, and an anonymous referee

for valuable comments, and to L. Barrow and R. Zerpa for help in preparing the graphs. Hoqvever,

I am solely responsible for any errors. Received May 1991; final draft accepted September 199 1.

Journal of Policy Modeling 14(4):535-560 ( 1992) 535

0 Society for Policy Modeling, 1992 0161~8938/92/$5.00

536 9. Camps

$12,000 million, and that the short-run price elasticity of oil is low. Forecasting that oil prices will be $29-34 a barrel with a 95percent probability in the near future will very likely lead decision makers to keep related policies unchanged. Qn the other hand, policy makers may be inclined to implement policies to cover excess imports if oil prices were forecast to lie between $15-20 a barrel wi.th a 95-percent probability. Finally, a 95percent confidence interval of $10-60 a barrel would indicate great uncertainty, and could induce yet different policies, depending upon the policy-makers’ attitude towards risk.

Lucas (1976) questions the use of econometric models for policy analysis when the processes of the forcing variables change. However, we do not know for sure whether those processes have actually changed and, even if they have, whether those changes have affected the parameters of our econometric model. Forecasts provide us with indirect information on whether the Lucas critique applies by allowing us to test for parameter constancy (see Hendry, 1988). The distribution of forecasts can be used to define suitable statistics for testing parameter constancy. These statistics can be applied to both the estimated process of the forcing (or control) variables and to the econometric model, in which the latter has the forcing variables as determinants. If the forcing- variable process appears empirically nonconstant but the econometric model is constant, then this is evidence that the econometric model can be used validly for policy simulation, i.e., that the forcing variables are ‘ ‘super exogenous’ ’ for the interventions that occurred in sample (Engle, Hendry, and Richard, 1983; and Hendry, 1988).

Even for a linear model, derivation of the forecast-based test statistics requires assumptions on the distribution of the model’s innovations. Further, obtaining exact results is not straightforward, so authors have resorted to using large sample theory, even though the latter renders only approximate results [usually to o&T- “2)]. Schmidt (1974) rinds the asymptotic distribution of a vector of forecast errors for a system of dynamic equations with exogenous variab!es (see also Chong and Hendry, 1986). Yamamoto (1976) finds the asymptotic mean square error of the h-step ahead predictor of a variable from a pure autoregressive (AR) model. Baillie ( 1979a) derives it for variables from models with AR disturbances and fixed, stochastic, or both kinds of regressors. That leads to an autoregressive distributed lag (AD) model, generalizing Yamamoto’s results. Baillie ( 1979b) extends the former derivation to a multivariate AR process (see also Reins& 1980). Ya- mamoto (1980) obtains the asymptotic distribution of the multi-step ahead prediction error for a simultaneous dynamic model with either AR or moving average (MA) disturbances, assuming known future

CONFIDENCE INTERVALS 537

values of the exogenous variables. Yamamoto ( 198 1) derives the asymptotic mean square error of the h-step forecast for a multivariate autoregressive moving average of order (p,q) [(ARMA@,q)] model.

This paper extends Yamamoto’s ( 1980, 198 1) results in three com- plementary ways. First, we extend Yamamoto ( 198 1) by including variables with known future values, thereby allowing for constants and seasonals, which are common in econometric models. Second, we extend Yamamoto (1980) by allowing for exogenous variables with unknown future values. Under the assumptions made, that leads to a multivariate ARMA process with exogenous variables (ARMAX) of the kind addressed by Yamamoto (198 I). However, the matrices in- volved in this ARMAX process are special. They contain a number of zeros, and their other elements are implicit functions of the model’s reduced form parameters. Hence, we can find simple expressions for those matrices, writing them as linear functions of those parameters. By doing this, we can obtain their covariance matrix directly by estimating separately the two original systems of equations: that of the variables of interest and that of the strongly exogenous variables with unknown future values. Joint estimation would require a large number of observations, which generally is not available. Third, available econometric models may yield forecasts at a periodicity or in a func- tional form different from that of interest to policy makers. Thus, the asymptotic distribution of linear combinations of forecasts is derived, from which we can obtain corresponding confidence intervals.

Section 2 describes the model, and presents three lemmas and a theorem on the properties of that model’s forecasts. Section 3 illustrates the resulting formulas with models of an oil pnce and a Venezuelan CPL Appendices &4-E prove the lemmas and theorem, Appendix F describes the data, and Appendix G summarizes the notation in Section 2 and the appendices. Sections 2 and 3 are reasonably modular, so a reader primarily interested in applying the results can skip directly to Section 3, referring to Section 2 as necessary.

2. THE MODEL, THE PREDICTOR, AND THE PREDICTOR’S DISTRIBUTION

We ‘wish to consider forecasting future values of endogenous variables from an ARMAX model. Each strongly exogenous variable in that model may have either known or unknown future values. We assume that the strongly exogenous variables with unknown future values are generated by a separate ARMAX process. Because forecasts are computed from reduced form equations, we express the system as

538 J. Campos

a moving average with exogenous variables (MAX), and specify the optimal linear predictor in terms of observed data. Thus, the forecast error is written in terms of observed variables, and estimators whose asymptotic distributions are known. From these results, we can straightforwardly derive the asymptotic distribution of a linear combination of a vector of forecast errors and its corresponding confidence intervals. This section describes the model, the predictor, and the predictor’s distribution, relegating proofs to appendices.

Let us assume that the econometric model in its structural form is ARMAX:

B,,z, = $ B,z,_, + 2 C,+x,+-, + C,,x,, + cp,, (1) 1-I I - 0

where i,, x,“. and x,, are respectively tr, endogenous variables, n2 strongly exogenous variables whose future values are unknown, and #, strongly exogenous variables whose values over the forecast period are known exactly, all at time t. The Bi’s, CT’s and Co are matrices of parameters of dimensions n, X n,, n, X n2 and n, X K,, respectively. B, is nonsingular, and cp, is an n, column-vector of disturbances normally and independently distributed with zero mean and variance covariance matrix I&.

The system generating the exogenous variables x,+ is also assumed to be ARMAX:

P3

c @,x,‘, = Y&t + i @,P, I’ (2) , - 0 ,-0

where x2, is a vector of the tth observation on K-, strongly exogenous variables with known future values. The @i’s and 8,‘s are (n, X n,) matrices with a, = 8, = I,,, (without loss of generality), Y is an n2 X K2 matrix, F, is an n2 column-vector of innovations normally and independently distributed with zero mean and variance covariance matrix C,, and E(cp,&) = 0 for all t, S.

In what follows, (1) and (2) are written as a single system in MAX representation, thereby relating the full set of modeled variables y, [ = (z’,:x,+ ‘)‘I to the exogenous variables with known future values x, I= (x~,:x’~,)‘] and a disturbance e,. From that relationship, we derive optimal predictors and their asymptotic distribution, and define confidence intervals. Lemma 1 provides the MAX representation.

al

Assuming stationarity. the _ systems in ( ! ) and (2) have the following MAX representation:


where T,(L) lag operator

.v, = IX%, + I&%,. (3)

.(L) are matrices L. (See Appendix A.)

of one-sided polynomials in the

The MAX representation in (3) expresses y, as a function of present and past reduced form shocks (e,-k), and present and past known values of the set of exogenous variables with known future values (x,-k). From (3), we can straightforwardly derive the optimal linear predictor.

Let us assume we wish to forecast observation y,,,,,, (N 2 T, h > 0), where T is the sample size used for estimating the parameters in (1) and (2), and y, and x, are known through time N and N + h respectively. From (3), the (N + h)th observation is

so we can state the following lemma.

Lemma 2

Conditional upon past observations and all known values of x, the optimal linear predictor of the (N + h)th observation y,,,,, is

4;v.h = r? *tLkN + r2(L)xN+~~~ (5)

where T?*(L) is a matrix of polynomials in L. (See Appendix B.)

The forecasts in (5) depend upon the unobserved disturbances, so it is useful to re-express the latter in terms of observed data. Solving for the disturbances in (3), we obtain

where

'P,(L) = I,, + c H'F"-'(F - A)HL" h-1

and

'k(L) = -c H'F“HCL" (8) 1, -0

for suitable matrices H, F, A, and C, defined in Appendix A. Thus, from (6), the optimal predictor in (5) can be written as

YN 11 = [rI(L)-~N+/, + r~*(~m(L)_~N~ + ry(L)*,(~)~,~ (9)

= w4x, + q(L)yv.

540 J. Campos

6(L) = 2 H’Ah +kHCLh + c HIAh - ‘Fh t ‘HCL”

h- -h I - 0

(see Appendix C), and similarly,

(10)

q(L) = i H’Ah - ‘F”(A - F)HL”, b > 1. . - (11) h-0

The optimal predictor in (9’) is a function of unknown parameters. Hence, we define the feasible predictor as

%v.h = is(L).& + ij(L)>lN. (12)

The polynomials 6(L) and q(L) are 8(L) and q(L) with their unknown parameters replaced by suitable estimates.

Equations (3), (5), and (12) express the unknown future value y,,,+,,, its optimal predictor yN,,#, and its feasible predictor j$,,h in terms of observables and disturbances. From these equations, we derive the forecast error in terms of disturbances over the forecast period, observables, and estimates of the model parameters.

The error in forecasting observation yN+h is

where Q is a vector of disturbances, W(,,, depends on observables, Q 01) and M are functions of estimated parameters, S is a selection matrix, and p is the vector of reduced form parameters. (See Appendix D.)

Using lemma 3, the vector of m prediction errors is

Y -G- - + P(fi - p,, where

01 - .9>’ = I&v+, - _i)N.,,’ - * . (vN*,,, - jN,,,)‘l.

E’ = [E’~,, . . . &,,,I, and

P’ = MWQ:d':,, . - . Q:,JCJ.

(14)

Lemma 3 leads to the following theorem, which describes the dis- of the forecast errors and the confidence region for linear

combinations of future observations.


Theorem

Let

i) The us assume we wish to forecast m observations &_ ,

vector of forecast errors is approximately normal

w - s, - N(0, A + T- ’ plimPfW ‘), aPP

where A = E&E’) is the variance covariance mat is the asymptotic covariance matrix of g .

ii) A confidence region for the vector of li ear combinations of future observations is given by

where in the

iii) In particular, the confidence interval future observations d,!y is given by

D xf

is an r x nm matrix of fixed ele ents, xz is the crit distribution corresponding to the significance level

s -f = D( 1 + T- ’ plimPCP’)D’.

f0r a 1 inear combination of

:ical point a, and

where d,f is the i-th row of D, a is the confidence level, & is the 01/2 quantile of the normal distribution, and v is the (i,i) X f’

Part (ii) of the theorem inter alia implies that the mean square forecast error (MSFE) is not invariant to linear transformations of the variables being forecast, noting that the diagonal elements of & in general depend on D. Further, the ranking of empirical MSFEs across models is not invariant to the choke of transformation. For instance, one model may have the smallest; MSFE for the changes of a given variable, when compared with several other models. Yet, that same model could have the largest MSFE for the levels of the same variable, even over the same sample period, against the same models, and based on the same underlying forecasts. The reason is that the MS the off-diagonal elements of the empirical equivalent to Z&. T is to look at &- ?rz its entirety, rather than certain pieces L: it. Q-Q Li** Hendry (1991) for further details.

3. TWO SIMPLE EXAMPLJfiS

In the following examples, we entertain A ( 1) models for he price

of Texas Intermediate crude oil and Venezuelan inflation. Both models are misspecified, thereby allowing us to illustrate the value of con-

542 J. Campos

strutting forecast confidence intervals (a) in testing against specifica- tion errors and (b) in evaluating the usefulness of such a model for policy. In Section 3A, we forecast changes in the logarithm of monthly oil prices, quarterly averages of those changes, the logarithm, and the price itself. Section 3B computes monthly and quarterly forecasts for Venezuelan inflation and for the underlying monthly CPI. (See Ap- pendix F for data description.) Both models are AR( 1) in the change in the logarithm of the relevant price, so we begin by specifying that model, giving the expressions for the forecasts and for all matrices relevant to computing the covariance matrix of the forecasts errors.

Denoting monthly changes in logarithms of prices by Y,, the AR( 1) model with drift is

(1 - cw,L):9, = (Y, + e,, e, - lN(O,o~), (18)

which is (3) [and also (l)] with A = CY,, F = 0, C = ao, H = 1, Yr = z x, = Xl, cw, x;:T,

= 1 (W) and Zp = g. Because F = 0, S(L) = (~L:+~L~, and q(L) = a:, it follows that the h-step forecast

is h- I

Li A

YN,h = % c kL(I + &:I?)N, (19) , =o

which tends to the estimated unconditional mean &, /( 1 - &,) as the forecast horizon h + 00.

Defining p - p = (&, - (x,: 8, - olO)’ and taking J = 1, the (h + 4) row-vector Wfh, is (1 1 . . . 1 yNyN_ ,) and the (h + 4) x 3 matrix & is

Q (h) =

i=O

. . .

h-2

0 b- I

c h-l-i-r 011 at

i=O

0

0 1

0 011

. . . . . .

0 I1 -

aI

h- I

&OQLl 0

0 0

-ay 0

h- I

4% 0

-

I

I -

(20)


for h = 2, 3,. . . , m. The matrix Q (1) is Qfh, but missing rows 2 through h. In addition, A is a symmetric matrix with elements A,, = o~(o$’ xi=, af’(-“) for j zz i. The matrix S-M is a 3 x 2 matrix of

zeros except for its elements (I, 1) and (3,2), which are unity. From the estimates of (cxo:(x,)‘, their covariance matrix 0, the start-

ing value yN, and the expressions for Q ,,,,, W(,,,, S, M, and A, we can compute forecasts of monthly changes in log prices and the covariance matrix of the forecast errors. As h + x, the contribution of parameter uncertainty to the variance of the h-th forecast error tends to

T- ‘{[iiu/( 1 - &)‘I’ V(k) + 21&N - &,,‘I COV(&, G,) + [I/( 1 - &,)]’ V(&,,)}.

3A. Confidence Intervals for the Price of Texas Intermediate Crude Oil

Let us denote changes in the logarithms of the oil price by Aop. Its h-step ahead forecast is (19), with y replaced by Aop. The covariance matrix of the m corresponding forecast errors can be computed from A, Wf,l,, Q, ,,,, S, and M, defined above.

The model in (18) is estimated by OES from 1984:4 through 1989: 12, yielding

.A Aop, = 0.26 Aop, , - 0.004

(0.21) (0.01)

T = 1984:4- 1989: 12 R' = 0.07 & = 8.69 percent COV(&,,,&, ) = 0.000085

q,(5, 62) = 0.58 qz(2, 64) = 3.75 e,(2) = 55.32 q,(15, 67) = 2.08

&( 15)/15 = 2.18.

The numbers below the equation’s coefficients are White’s standard errors, and COV(&,, &,) is the covariance between 6~~ and &, . ii is the standard error of the regression. The q and 5 statistics are distributed as F and xX distributions under the null hypothesis of correct specifi- cation, with the degrees of freedom indicated in parentheses. Except for residual correlation (q,), the model fails to satisfy all the test criteria, which GXZ against: heteroscedasticm *ty (q,); non-normality (s,); and parameter nonconstancy (the Chow statistic q3 and the x2 forecast statistic c2). This is despite a very small R’.

From the change in oil prices for 1988: 12, and the estimates of (x0, cxl, and their covariance matrix a, we compute h-step ahead forecasts of changes in the logarithms of oil prices and the covariance matrix of the corresponding forecast errors, for January 1990 through NIarch 1991. Figure 1 plots actual and h-step ahead forecasts, and rhe con-

J. Campos

-20 Jan Feb Mar Apr May Jun Jul Aug Sep Ott Nov Oec Jon Feb Mar

I 1990 I 1991 I

-.- Forecast - Actual - Lower limit - Upper limit

Figure 1. Month11 changes in the logarithm of oil prices ( x 100).

fidence bands for actual monthly changes in oil prices, scaled by 100 to obtain percentages. Forecasts and their standard errors reach their limit values rapidly due to the small value of &,.

The confidence bands show that the AR( 1) model substantially underpredicts August 1990 and, to a lesser extent, September 1990. This is not surprising because there is nothing in the model to pick up the effect of the Gulf War, which started in August 1990. The model also overpredicts February 1991. Both episodes suggest misspecification. However, h-step ahead forecast errors must be interpreted with care because they are not independent, as they are linear combinations of l-step ahead forecast errors. Chong and Hendry (1986) derive the joint test statistic for h-step ahead forecasts, which is

&&zm) = (y - j)‘(A + T-’ plimPRP’)-‘(y - 9).

If the parameters were constant, &, would be approximately distributed as a x2 with nm degrees of freedom. Here, $( 15) = 33.68, suggesting that the parameters are nonconstant.

In policy making, future observations are unknown, in which case confidence intervals provide an ex ante measure of the forecasts’ uncertainty. The results in Figure 1 show that, with a probability of .95, actual monthly changes in oil prices from January 1990 to March 199 1 could lie between - 18.0 and + 18.0 percent. This confidence interval is large, partly because of misspecification. However, even correctly specified models may render wide confidence intervals for future observations.

We may also be interested in forecasts of the average change in the oil price over the quarter. To calculate those forecasts and their con-


-20 L I 1 1 I

9’ 02 03 04 . . 01 1990 I 19&l

- * Forecast -- Actual - Lower limit - Upper limit

Figure 2. Quarterly changes in the logarithm of oil prices ( x 100, at monthly rates).

fidence intervals, D is defined as a 5 x 15 selection matrix of zeros and 1/3’s, where the latter elements average the required elements of the vector of monthly forecasts. The results are plotted in Figure 2. With 95percent probability, quarterly oil price inflation should lie between - 12.0 and + 12.0 percent (at monthly rates). Again, the model underpredicts in the third quarter of 1990.

Instead of changes in logarithms of oil prices, let us now consider computing paths for the logarithms of monthly prices. The linear combination is defined by a lower triangular D matrix with all its nonzero elements being unity. Thus, the vector of price forecasts is

e = D l (Ao,p,., . . - ~N.,5)’ + op.&l - L,

where N is 1989: 12 and L’ = ( 1 . . . 1). The confidence interval for h- periods ahead is

(op,,, + x Ao~~.,) -c 1.96u:“,

where vh is the variance of the h-step ahead forecast error of the logarithms of oil prices, which is the sum of variances and covariances of forecast errors of changes in log oil prices up to time N + h, inclusive. Hence, uh increases with h unless negative ccvariances offset the values of the newly added variances. So, confidence intervals for prices are likely to become wider as the forecast horizon increases. Figure 3 shows that they do for this example.

Confidence intervals for monthly prices themselves can be computed from forecasts of their logarithms although price forecasts so obtained may be biased. Those intervals al-e

546 J. Campos

Jan feb Mar Apr May Jun Jul Aug Sep Ott Nov Dee Jan feb Mar

1990 1991

. forecast _ Actuc’ - lower Limit - Upper Limit

Figure 3. The logarithm of oil prices.

OPN + c A0pN.t + VL’2gl ; exp I i “A OPN + c hop,, + v:/'g2 ,

1=1 r-l II where g, and g, are two values such that

and uF2 + (1/2)(& + g,) = 0

I 5

(27C)-l'2e-'.2'2dv = y,

*I

and where y is the probability associated with the interval, e.g., y =

35. For these confidence intervals to be optimal, g, must be different from g, in absolute value. Nonetheless, for this example, we chose g2 = -3 = 1.96 for computational ease. Figure 4 plots actual and h-step ahead forecasts of oil prices with these confidence intervals.

0 ’ 1 1 / i I I J

Jan Feb Mar Apr May Jun Jul Aug Sep Ott Nov Dee Jan Feb Mar

1990 1991 '

. forecast - Lower Limit - Upper Limit

Oil prices (EWbarrel).

CONFIDENCE INTERVAI S 547

- Forecost . Actual - Lower t .ry, ? - Upper tlmk(

Figure 5. Monthly inflation (percent).

3B. Confidence Intervals for Venezuelan Inflation

Let us denote Venezuelan inflation by Ap, defined as the change in the logarithm of the monthly consumer price index for Caracas. We construct forecasts and confidence intervals for monthly and quarterly inflation, for the logarithms of prices, and for prices.

Estimating (18) by OLS, we obtain

2 = 0.40 Ap,_, + 0.004

(0.09) (0.001)

T = 1968:4-1983:12 R’ = 0.16 6 = 0.6 percent COV(&,,&,) = -0.00003

q,(7, 180) = 2.66 ~~(2, 184) = 6.02 e,(2) = 223.21 ~~(24, 187) = 2.16

(,(24)/24 = 2.17.

These results suggest that there is residual correlation, heteroscedas- ticity , nonnormality, and parameter nonconstancy.

Figure 5 shows actual and h-step ahead inflation forecasts and the confidence bands for actual inflation from January 1984 to December 1985, scaled by 100 to obtain percentages. The model substantially underpredicts inflation for April and September 1984. Large price increases occurred in the second half of 1984 due to exchange rate changes and the removal of subsidies on some goods. Our simple AR( 1) model does not pick up those effects. Chong and Hendry’s (1986) statistic b(24) is 52.;. -iom Figure 5. actual monthly inflation should lie between a small negative value a;ld 2 percent per month, with 95percent probability. Wenee, assuming our model were correctiy specified and the decision maker were risk-averse, he might decide to implement policies to control inflation. Figure 6 shows that the model under-predicts average quarterly inflation for the second and third quar-

548

5

J. Campos

-1 01 02 03 04 01 02 03 G4

,984 ’ 985

Figure 6. Quarterly inflation (percent, at monthly rates).

ters of 1984. With 95-percent probability, quarterly inflation at monthly rates should lie between - 0.3 and + 1.6 percent. Figure 7 plots the results for the logarithms of the consumer price index: Confidence intervals increase with the forecast horizon. Figure 8 shows comparable results for the consumer price index itself.

4. CONCLUSION

Confidence intervals for future observations of economic variables are defined, based on their forecasts and the distribution of forecast errors. Those forecasts are obtained from a dynamic simultaneous equations econometric model (potentially) including two sets of exogenous variables: those with known future values and those whose forecasts must be computed. A stationary and invertible ARMAX process is assumed to generate the latter set of variables.

61

57 ’ 1

JFMAMJJASONDJFMaMJJASOND 1984 I 1985

. Forecast - AcIlia, - I OWPT I imll - Upbsr hmit

igure 7. The logarithm of the consumer price index.


320 1 I 1 1

JFMAMJJASONDJF~ +MJJASOND 1984 I i 985

. Forewst . Actual -- Lower Limit - Upper limit

Figure 8. The consumer price index ( 1968 = 100).

Confidence intervals show decision makers the most likely future paths of economic variables, thereby helping them implement policies suitable for attaining economic goals. However, the closeness of the calculated confidence intervals to the unknowrt exact confidence intervals will ckzpend upon the model, the sample size, the estimator employed, and the region of the parameter space. Furthermore, the independence and normality of the disturbances, the consistency and asymptotic n(Jrmality of the estimators, together with the exogeneity of x,+ all underpin the validity of the confidence intervals for sample sizes available in econometric studies of time series. Conversely, confidence intervals themselves may provide us with evidence on whether c;r not those assumptions are satisfied.

APPENDIX A. PROOF OF LEMMA 1

Ex ante forecasts and their standard errors are based upon the reduced Term of (1) and (2), which we write in a compact form as the multi- Jariate ARMAX(p,q) process:

A$y, = ATy,_, + . . . + A;y,-, + C$x,+o, 9

(AU

0, = +, + Fli’ll,r,_, + . . . + F;$,.q

where p = max(p,,p,,p,), Bi = 0 if i > pl, Cr z= @,- = 0 if i > p3; and y, = (z:xI+')',x, = (x;,x;,)~, an

550 J. Campos

For n =n, + n2andK= K, + Kz, A”, C$ and F” are n x n, n x K and n X n matrices as follows: F” = diag(O,@) (i = 1, . . . , 4)

and

A$= : - G [ _I I , ,!,

C’ A:= iI-6 ,

[ 1 (i=l,.... p), ,

and C$ = diag (Co Y) for n, 2 1. A” = Bi and C$ = Co for nz = 0.

Hence, the reduced form of (Al) is the ARMAX(p,q) process:

V, = A,!,-, + . . . + A,,!, ,, + Cx, + 14, .

(AD

M, = e, + F,e,_, + . . . + Fqe,-v

where Ai = (A$)- 'A* (i = 1, . . . , p), C = (AZ) - ‘C$, U, =

A$ -b,, F, = (A$)-‘FFA,* (i = 1,. . . , q) and e, = (A$)-‘+, . P 4

Defining A(L) = C A,L’ and F(L) = C FjL’ with A0 = F. = I,,, j=O j=O

then the model in (A2) can be written as follows:

A@.& = F( L)e, + C-x,. (A3)

Assuming that all roots of A(L) lie outside the unit circle, the MAX representation of (A3) is

,v, = LWe, + r,(L)x,, 644)

where

r,(L) = I,, + i H’A”-‘(A - F)HL” (A5) 1-I

[see Yamamoto (1981)]. The polynomial T,(L) can be derived by following the same lines of the proof for T,(L), yielding

T,(L) = c H’AhHCLh. uw

h 0

Also, H’ = (Zn 0) is an n x ns matrix; and


A, I,,0 . ..O A2 0 I,, . . . 0

A = i ii :

A,_, 0 0 . . . ;,, A, OO...O

and

F= i ii i

[

-F, I,,0 . ..o -F, 0 I,,...0

-F,_, 0 0 . . . I,, -F, 0 0 . ..O

are ns X ns matrices with s = max@,q).

APPENDIX B. TIIE OPTIMAL PREDICTOR

The predictor that minimizes a positive definite quadratic form in the forecast error is shown to be the conditional expectation of the random variable we wish to forecast, given past information. Thus,

1’N.h = E(_-v~+,~~x~+~, x,~+~,. ,. . . . ; qv, qv. ,. . . . 1. W)

Noting that h- I

L(L) = c I-,,Lh + i: IyhrhLh+h, 1. -0 I. -0

we can write the (N + h)th observation in (4) as follows:

where r?(L) = c lY,,L” and k=O

r-x4 = C r,.,+,x. I. - 0

Hence (Bl) and (B2) imply that the optimal linear predictor of Ye++ is

552 J. Campos

Because of the assumptions made on the systems in ( 1) and (2), the e’s are serially independent and are independent of the X’S, so that

E(T~(L)e,+,I.r,+,, . . . ; eN, eN_,, . . . ) = E(T~(L)e,+,,) = 0

and

E(r~*(L)e,Ix,, ,,, .r,+ ,, _ , , . . . ; e,v, 4, , . . . . ) = I‘f-*Ok..

Thus, the optimal linear predictor in (B4) is

VN.h = 17F*(L)ev + r2(L)-rN+ ,,’

See Harvey (198 1, pp? 159- 160), who finds the optimal predictor for a pure ARMA model.

APPENDIX C. DEItIVATION OF THE LAG I’OLYNOMIAL OF xN

Let us first find an expression for S,(L) = ry*(L)%(L). From (8) and (B3),

= c &A Lh, 1. 0

where h

6 3 = c r I.h+,*_.h -,

, 0

= - H’A” - ‘(Ah + ’ - Fh + ‘)HC, (Cl)

using the expressions of the T,‘s and V2’s in (A5) and (8) and noting that (A - F)HH’ = (A - F)

Finally, the whole polynomial of XN in (9) can be written as follows:

ifi(L) = hi rzhLh h + c (r2.h+L + tizh)Lh. h 0 h -0

From the definition of the rzk’s and SZk’s in (A6) and (Cl), we have

r -'.hch + hh = H’Ah - ‘Fk + ‘HC, (W

so that (A6) and (C2) yield


,

6(L) = i H’A”“HCL” -I- x H’Ah ‘F’- ‘HCL’. A h k - 0

APPENDIX D. PROOF OF LEMMA 3

From (4), (S), (9) and (12), the forecast error is *

_.‘N + h - yN.h = (v,v + h - !A .I,) + @.v It - L)

= IT,(LlL h - I‘f*‘(L)]e, - [ii(L) - 6(L)]&

- Iti - q(L)l_b. (DI)

We take the polynomials in the lag operator L in (Dl ) one by one. First, from (B3),

T,(L)L lJ - T?“(L) = hi T,,L” h. (W 1-o

Second, from ( lo),

= c (x;. h@I,,)~ec(H’/ih + ‘He - H’A” + “HC) h- h

- H’Ah - ‘Fh + ‘HC), (03;

where vet stands for a column vectoring operation. Third, and similarly, from (1 I),

vec[ij(L) - I]?:, z x (_d’- h@k,) 1. - 0

vec(H’a” ‘&a - k)H - H’A” ‘F”(A - F)HJ. (D4)

Hence, from (D2), (D3), and (D4). the prediction error in (Dl) is

yN+h - 9N.h = $h) + W&&h, - Q,,,,), (D5)

where h-l

E (h) = & . . v rl&+h-kr

k-0

W)

554 J. Campos

and (4, - a,,,,) is a (h + 2(5 + 1)) column vector with vec(H’kH~ - H’A$=) (i = 0,. . . , h - 1) being its first h elements, vet (H’Ah-‘pHc - H’Ah-‘F’HC) (i = 1, . . . , J + 1) being the next (J + 1) elements, and vec(H’/?‘p(A - lf)H - H’A’-‘p(A - F)H) ( i = O,... ,J) being the last (J + 1) elements.

For the first h rows of vets in (Ci’h) - a(h)), we have

H?i'Ht - H'A'HC = HQi' - A')Hd + H’A’H(t - C)

= H'A'H(i‘ - C) + H' '~'k(ii - A)A'-'-'He, (D8) , - 0

(e = O,...,h - I)

becaue 4-I

(A’ - A’) = 2 &A - A)&‘-‘, t’=, 1, (D9)

where only the first term in (D8) should appear for 4T’ = 0. For the following (J + 1) rows of vets in (&) - qh)), we have

H’ah - ‘&fc _ H’Ah- ‘FCfJc = H’(Ah- ’ - Ah- ‘)t?6Hc

+ ,'A'-'@ - F')Ht + H'Ah-'FtH(t - C)

= H'Ah-'F'H(t - C) h-2

+ H' c k(/i - A)Ah-2-Jj?Hc

1’0

Y- I

+ HIAh-' c p(fi - F)FY-'-JHc, (DlO) J’o

(e = 1,. . .,J + 1)

where the first summation does not appear for h = 1. Finally, for the last (J + 1) rows:

WA”- ‘P(A - F)H - H'Ah-'ti(A - F)H = H'@-' - A"-')%$ - k)H

+ H'Ah-'(p - F')(/i - $)H

+ H'A"-'ti[(ti - h - (A - F)]H

= H'Ah-'F'[(ti - I+) - (A - F)]H

+ H' hi' ,,$(A _ A)Ah-*-Jkf(a _ jt)H I=0

(-I + H'Ah-' c $'(p - @-'-'(A - @)H, (Dll)

J=c

where the second term would not appear for h = 1 and the third term should be deleted for e = 0. Using (D8), (DlO), and (Dll), (&I, - a& is given by


(48, - a(,,,) = Qth, _ W2)

The matrix Q(,,, is a 3-column partitioned matrix of size [(.I + h + 1)Kn + (J + l)n’] x (2n’s2 + Kn) with its submatrices defined as

Q,h,(W = 0, 0)

Q,,<e + 2,l) = 2 [(A’-Wt)‘@H’ii’] (4) = 0, . . . , h - 2; h > I), (ii) J=t,

h-2

Qch,(h + e.1) = 2 [(AhbJw2pH&@H’,$] (e = 1, . . . , J + I; h > I), (iii) J=o

Q,,,(h + J + 4 + 1,l) = (H’@H’Ah-IF+‘) h-2

+ c [(Ah-‘-2j+‘(h

J=o

(t = l,...,J + I),

Qt&,Zj = 0 (i = 1, . h) *., , t- I

Qdh + C2) = c ((F’ -‘- ‘H@@H’A- ‘p] J =- 0

Q,h,(h + J + 2,2) = - (H’OH’A” ‘)

Q,h,(h + J + Q + 2,2) = -(H’@H’A”- ‘t’) t I

- &f)‘@H’i’)

(t = I,..., J + I),

+ c [(F’ ’ -‘(ii - &f)‘@H’A” ‘F’] ‘-0

(4 = l,..., J),

Q ,,,, (t + 1,3) = (I,@H’A’H) (t = 0, . . . 9 h - I),

Q,,,,(h + Y.3) = (I,@H’Ah-‘F’H) (Q = l,...,J + l),

Q,,,(i + h + J + 1,3) = 0 (i = l,...,J + 1).

(iv)

w

(iv)

(vii)

(viii)

(ix)

6)

(xi)

Expressions (i), (ii), (iii), (v), and (vi) are (Kn x n’s*) matrices; (iv), (vii), and (viii) are (n2 x n2s2) matrices; (ix) and (x) are (Kn x Kn); and (xi) is an (n2 X Kn) matrix. In addition, for h = 1, (iii) is a matrix of zeros, and (ii) and the summation in (iv) do not appear.

The matrices A, F, and C contain a good number of zeros, and their other elements are nonlinear functions of the reduced form parameters of systems (1) and (2). We wish to write those functions explicitly in terms of those parameters so we can derive the variance covariance matrix. In doing so, we write

556 J. Campos

where p is the vector of parameters of the reduced form of system (1) and those of system (2). That is,

p = [(vec(B; ‘B,))’ . . . (vec(B, ‘B,))’ (vec(& ‘C,’ ))’ . . .

(vec(B; ‘C,‘))’ (vec(B; ‘CJ) (vet a,)’ . . .

(vet Cp,)’ (vet 9’)’ . . . (vet 8,)’ (vet Y)‘]‘.

The matrix S is diag(S, SF SC), which is [2n2s2 + nK] X [s(nf + nn,) + sn2n + nlKI + nK,]. &, SF, and S, are selection matrices of zeros and ones picking up the nonzero elements in A, F, and C. Those matrices and the matrix M are defined as follows.

SA is a 3 x 2 partitioned matrix of size n*s* x s(nf + n2n)_ Submatrices (1,2), (2, l), (3, I), and (3,2) are zero. Its (1,l)st submatrix consists of n,s submatrices of nf columns each. The (i,j)th of those submatrices is defined as the Kronecker product of an s x n, matrix of zeros with a one in its (j,i)th position, and the n X n, matrix (In, 0)‘. Its (2,2)nd submatrix is an nn2s X nn2s partitioned matrix of 2sn2 submatrices, defined as follows. The (i,(2j + 1))th (i = 1, . . . ,

. n2;j = 0 ,..., (s - 1)) submatrix is the Kronecker product of an s X n2 matrix of zeros with a one in its (j + 1,i)th position, and the n x n, matrix (I,, 0)‘. The (i,2j)th (i = 1, . . . , n,; j = 1, . . . , s)

submatrix is the Kronecker product of an s x n2 matrix ot’ zeros with a one in its (j,i)th position, and the n x n2 matrix (0 I,&

SF is a 3 x 1 partitioned matrix of size n’s’ X sn2n with all its submatrices being zero except for the (3,2)th one, which is of size nn2s x nn2s and consists of 2s~: submatrices defined in the same way as the (2,2)nd element of &. SF is not in S if n2 = 0.

SC is a 2 x 3 partitioned matrix of size nK X nJK, + nK2 with its (1,l )st element being an nK, X n,K, matrix defined as IK, @ (In, 0)‘. Its (2,2)nd element is an nK2 x n,K2 matrix defined as &, @ (In, 0)‘, and the (2,3)th is of size nKz x n,K, and defined as _&, @ (6 1,J.

M is an (nfs + 2nn2s + n,K, + nK,) X (t2T.s + &z2(s + lj + n,K, + 2nfs + n2K2) matrix consisting of 24 submatrices, all of which are zero except for the f.Alowing. The (1,l)st submatrix is of size nfs X nfs defined by I.S@I,+ The (2,2)nd submatrix is of size nn2s

X n,& + I), defined by two further submatrices: the (1 ,l)st is of size nn2s X n,nz having all elements equal to zero except the ((2i + l),l) (i = 0, l,..., s - 13, which are defined by - (a,!+ @,J; and the (1,2)th submatrix is of size IZH~S x n,nzs defined

as LO(L,“Z 0)’ *

“NCE INTERVALS 557

The (L ,4)th and (3,5)th (rtn,~ x nis) submatrices of A4 are defined as I,@[ - (I&&B, ‘C,‘)) -I~;]‘. The (3,Z)th nn?~ x n&s + 1) submatrk is zero everymwhcre-except at positions (2i + 1 ,I) (i = 0 s- ?“‘) I), which are - (&+ @,J. The (4,2)th (n,K, + nK,) x (s + 1 IA ,n, submatrix has zeros everywhere except the (2,1)st

element, which is (Y’@I,,,). The (4,3)th (n,K, + nK,) X n,K, submatrix of 1M is zero except for its ( 1,l)st element, which is lnIK,. Finally, the (4,6)th (n, + nK,) x n,K, submatrix has its (1 ,l)st element equal to zero. ,nd the (2,1) and (3,1) elements equal to (I&(B; ‘Cl )) and In,K,, respectively.

Having defined all the-matrices in (D 13), we can write the prediction error in (D5) as

yN+h - jN.h = $h, + W,,Q,mWb - P,. (D14)

APPENDIX E. PROOF OF THE THEOREM

The vector 6 is the vector of the estimates of the parameters of the reduced form model and of those of the process for the exogenous variables xl“. We assume g is asymptotically normal, i.e.,

UT@ - p> - Wm. (El) a

The matrix P has dimension (nm x [nf s + n,n,(s + I) + n,K, + 2nfs + n&,1), which does not change with the sample size T. Thus, we can apply Cramer’s (1946, p. 254) linear transformation theorem to obtain

P dF(S - p) ; lV(O,plimP W’). (E2)

Because forecasts are made outside the estimation period, E and P(s - p) in (14) are mutually uncorrelated. Thus, (14) and (E2) lead to

@ - j) a--p N(c), A + T- ’ plimPRP’), (E3)

where A = E(E~‘) is the variance covariance matrix of E with its elements given by

m,dJ = ‘c: r,, 15,,Iyh h - 0

and

E(E,,,E,;,) = C !LX.K., , . h i < .i.

I 0

(E4)

558 J. Campos

This follows because e, - IA&O, 2,) where x,, = (A$) ‘&,(AX’) ’ and & is the variance covariance matrix of +. The elements below the diagonal (i.e., i > j) can be obtained by symmetry.

Let us now assume that D is an Y x nm matrix of fixed elements. Then

0-r - _t> - N(O, 2,). aPP

(E5)

where

2, = D(A + T ‘plimPflP’)D’.

Furthermore, because X,- ’ is positive definite, it can be decomposed as C;’ = R’R where R is an r x r nonsingular matrix. Hence,

RD(_v - .i,, - N(0, I,) aPP

so that

(y - j)‘D’c, ’ D(J - 0, - xf. aPP

(E6)

From (E5) and (E6) we can write down confidence intervals and confidence regions for di ‘y and Qv, respectively, where d,! is the ith row of D.

IX F. DATA DESCRIPTION

OP West Texas Intermediate crude oil price, US$/barrel. Monthly figures from January I984 to April 199 1. Source: International Financial Sta- tistics, IMF, M 11 I M 1776AAZ.

P Consumer price index for Caracas, 1968 = 100. Monthly figures from January 1968 to December 1985. Source: Boletin Mensual, Banco Cen- tral de Venezuela, several issues. Prices on consumers goods have been subject to governmental regulation for most of the sample period, with the share of regulated goods in total expenditure varying over time. The share was 39.2 percent in 1981, and decreased over the next 2 years. Then, the prices of all goods and services, excluding beverages, non- monetary gold, and goods imported through Margarita Island, were put under governmental regulation.

Lower-case letters (op and p) mean logarithms of capitals.

CONFIDENCEINTERVALS 559

APPENDIX G. N

A subscript t denotes “in time period t.” “Exogenous” means ‘ ‘strongly exogenous. ’ ’

A

B, c co C,’ D

P2

P3

9

Q (h)

S

wfh,

&

x;+

XI,

x2r

Yt

YN+h

YN.h

9N.h

Y

P

=t

P e

matrix of functions of the reduced form parameters for the y’s n, X n, matrix of coefficients of z, _ , matrix of the functions of the reduced form para‘meters for the X’S n, X K, matrix of coefficients of x,, n, X n, matrix of coefficients of ~~1 I in the econometric model

matrix of constants. D applied to (e.g.) the vector of forecasts gives us its line : combinations. reduced form innovation in the whole system matrix of functions of the reduced form coefficients of the e’s selection matrix of zeros and ones matrix of functions of parameters in the whole system number of lags in the endogenous variables z, in the econometric model number of lags in the exoger aus variables x+, in the econometric model number of lags in x+ , in the exogenous variables’ process order of the MA process of the disturbances in the exogenous variables’ process matrix of functions of parameters in the whole system selection matrix of zeros and ones matrix of observations on all variables (x~,:x;,)‘~ the vector of variables whose future values are known

n2 stochastic exogenous variables

K, exogenous variables with known future values (in the econometric model) K, exogenous variables with known future values (in the exogenous variables’ process) (z;:x,+ ‘)r, the vector of variables whose future values are not known in advance the actual value of y in period N + h optimal prediction of the (N + h)th observation on y feasible forecast of the (N + h)th observation on y vector of all observations on y in the forecast period. vector of feasible forecasts on y n, endogenous variables in the econometric model vector of the reduced form of parameters vector of qh)9s

J. Camps

vector of linear combinations of h future reduced form innovations

W sz, x n, matrix of coefficients of the MA disturbances in the exogenous variables’ process covariance matrix of l n, vector of disturbances in the exogenous variables’ process (i, i)th element in xf variance matrix of the linear combinations of the forecast errors n2 x K, matrix of coefficients of x,, n, vector of disturbances in the econometric model n2 X n, matrix of coefficients of x,+ in the exogenous variables’ process asymptotic covariance matrix of @, the estimate of p

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confidence intervals for linear combinations of forecasts from dynamic econometric models

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