forecasting consumption, income and real interest rates from alternative state space models

ELSEVIER International Journal of Forecasting 11 (1995) 217-231

Forecasting consumption, income and real interest rates from alternative state space models

H . D . V i n o d * , P a r a n t a p B a s u

Economics Department, Fordham University, Bronx, NY 10458, USA

Abstract

The well-known difficulties in forecasting real interest rates are addressed by a comparison between two classes of models: (i) where real interest rates and income are exogenously determined from a vector autoregression (VAR)o as inputs to a state space (SS) model determining consumption, and (ii) where consumption, income and real in(crest rates are all endogenous. The first model resembles Hall (Journal of Political Economy, 1978, 86, 971-986) and Flavin (Journal of Political Economy, 1981, 89,974-1009), while the second model approximates the dynamics of a real business cycle model similar to Christiano (American Economic Review, Papers and Proceedings, 1987, 77, 337-341). We find that the second model yields superior forecasts of real interest rates. Furthermore, greater smoothness of consumption in the second model has implications for some topical issues of real business cycles and policy.

Keywords: Real business cycle model; Smoothness index; Hankel matrix; VAR forecasts; Cointegration

1. Introduction

Consider a consumer forecasting his consumption in two alternative scenarios: (a) in a model where both income and real interest rates are exogenously specified, and (b) in a model where both income and real interest rates are endogenous. Which scenario generates bet ter forecasts for macroeconomic aggregates and greater smoothness of aggregate consumption? This issue is important, because in the theoretical work on aggregate consumption there are basi- cally two strands of literature. The first one assumes that both real income and real interest

* Corresponding author.

rates are exogenous: Hall (1978), Flavin (1981) and Campbell and Mankiw (1990). The second approach, which follows the real business cycle methodology, Christiano (1987), uses a general equilibrium analysis and treats income and real interest rates as endogenous. In the latter framework, consumption, income and real interest rates are jointly determined by the endogenous labor supply and capital accumulation decisions.

This paper evaluates a consumption smoothness index described below (called SI in Subsec- tion 4.1) and the relative forecasting performances of the two approaches detailed above. We use the relatively new state space models pro- posed by Aoki (1987). The state space formulation is useful for this purpose, because it is

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218 H.D. Vinod, P. Basu / International Journal of Forecasting 11 (1995) 217-231

general enough to represent the equilibrium dynamics of the recursive competitive equilibrium. We estimate two state space models with three variables: consumption, real interest rates and income. In the first setting (called Exo-i in Eq. (2.1)), we impose a restriction on the state transition matrix to make income and real interest rates predetermined for consumption. In other words, we do not allow any feedback from consumption to the other two variables. This formulation resembles the theoretical model of permanent income used by many authors who follow a partial equilibrium approach to test the permanent income hypothesis referred to above. In the second setting (called Endo-i and defined in Eq. (2.3)), we estimate a general state space model without imposing any restriction on the state transition matrix. This formulation makes all three variables in the system jointly determined in a state of general equilibrium, which comes close to the real business cycle methodology of Christiano (1987). We then compare the forecasting performances of these classes of state space models. In addition, we investigate whether alternative forecasting rules have any implications for the smoothness property of U.S." aggregate consumption documented in the literature by Campbell and Deaton (1989) and Christ- iano (1987), among others.

The present state space formulation has cer- tain advantages over both the usual tests using Euler equations and the standard calibration procedure of Kydland and Prescott (1982). In the latter approaches, one needs to rely on a specific parametric form of the utility and production functions. The nonlinear cross equation restrictions imposed by a specific utility or production function are very likely to be rejected by the data, and this partly accounts for the inability of many of the equilibrium business cycle models to replicate the data. The state space formulation is a good compromise between theory and the data. The dynamics of the endogenous variables of an equilibrium business cycle model can easily be given a state space representation because of the cointegration among the endogenous variables (see Stock and Watson, 1988). State space formulation can therefore be justified on theoretical grounds. At the same time, we do not need

to impose cross equation restrictions on the structural parameters which are typically present in estimating Euler equations.

The paper is organized as follows. The following section provides a theoretical framework for forecasting the relevant variables. Section 3 reviews Aoki's (1987) state space model in 'innovations form' and Section 4 applies it to the consumption model discussed in Section 2. Sec- tion 4 also discusses the implications of the alternative state space forms for forecastability and consumption smoothness. Section 5 ends with concluding comments.

2. A theoretical framework

Although we expect that better forecasting follows when models satisfy reasonable economic principles, we admit that there may be excep- tions. In order to bridge the gap between atheoretical forecasting and the underlying economic principles, let us consider the following two models drawn from the optimal growth literature and abbreviated for easy reference as follows: (i) Exo-i, where both income and real interest rates are treated as exogenous for consumption, and (ii) Endo-i, where consumption, income and interest rates are all endogenous. The Exo-i model can be stated formally in terms of the following standard dynamic optimization problem:

c¢

max E o ~ fl'U(Ct) (2.1) t = 0

subject to Wt+ I = (1 + rt)(W , + Yt - Ct),

where E 0 is the expectation at date 0, W t is consumer's wealth at date t, Yt is income at date t, C, is consumption at date t, r, is the interest rate at date t and (1 + rt) is called the gross interest rate. The processes driving Yt and r, are specified in terms of the following vector autoregression (VAR):

O(L )(l°gr Yt) = ~t (2.2)

where E, are errors and the matrix polynomial in the lag operator is denoted by D(L). Notice that

H.D. Vinod, P. Basu / International Journal of Forecasting 11 (1995)217-231 219

11, and r, are parts of an exogenously specified bivariate process. In other words, in this setting we do not take into account the effect of the agent's consumption decision on interest rates. Also, since there is no discussion of production, the income process {Y~} is also exogenous. In other words, the system has a recursive nature, because income and interest rates are predetermined for consumption. Because of the additive separability of the utility function and the Mar- kov nature of the forcing process (1I , r ,} , the optimal consumption, C, at date t, will be a function of the current state, (Y, rt) in this specification. This is the partial equilibrium model used by Hall (1978), Flavin (1981), Camp- bell and Mankiw (1990) and others.

Next, we review the formulation of our Endo-i model , where consumption, income and real interest rates are all endogenously determined. The framework of analysis is nothing but a standard real business cycle model, which in- corporates production and capital accumulation as follows:

max E 0 t - -O

subject to

and lit

[3'U(Ct, 1 - n,) (2.3)

c , + I, = v , ,

= 7 t F ( K . n t),

where K, is capital stock at date t, I t is physical investment, 6 is the rate of depreciation of capital E (0, 1), n, is the labor supply at date t, 7/, is an idiosyncratic productivity shock realized at date t and F ( K . nt) is a standard neoclassical product ion function subject to constant returns to scale) The source of uncertainty is the technology shock, 7,. Notice that income, Y,, is now endogenous, which is determined by use of the

~Although we are assuming a Hicks-neutral technical progress, extending the model to incorporate endogenous technical progress as in Romer (1986) or King and Rebelo (1988) has no important implications for the cointegrating relationship between consumption and income in (2.5). The only difference is that, in the case of exogenous technical progress, the common trend in consumption and income is due to the permanent component of the exogenous productivity shock, "0,, while for an endogenous growth model that common trend is attributable to endogenous growth of the capital stock.

production technology 7,F(K,, n,). In equilibrium, the real interest rate, r,, will be equal to the marginal product of capital, 7 f l (K,, n,).

In this economy it can be shown, using standard results of dynamic programming (Stokey and Lucas, 1989, ch.4), that the optimal consumption is a function of the current state summarized by K t and 7t- Since r t =7tFI(K, , nt) , it is also clear that the real interest rate is determined by the same state vector (Kt,7,). Since 11, is related to K, and 7,, the same state vector will also characterize the evolution of output in equilibrium. Finally, due to the Mar- kov nature of the problem, the capital stock will evolve as a first-order Markov process as follows:

K,+, = h(K,, 7,) , (2.4)

where h ( - ) is the optimal policy function for the capital stock.

The above real business cycle model is stated in general terms. Analytical solutions are available only for restrictive configurations. However , in several papers on real business cycle models it has been shown that in equilibrium there exists a long-run relationship between the endogenous variables which include Ct, 11, and r, (see, for example, Neusser, 1991). This suggests the possi- bility of one or more cointegrating relations between these three variables. 2 Defining Xt as a (3 × 1) vector: (log C , log Yt, rt)', their cointegration means that there exists an p × 3 matrix 7 such that 7X, is stationary, where p denotes the number of cointegrating vectors.

It is also possible to give such a cointegrated system a common trend representation as in Stock and Watson (1988), as follows:

X~ = Ar t + tT~ (~7, is stat ionary), (2.5)

where ~-~ is a random walk with drift and A is a vector from the space of three variables. Since

2it is also well known that observed C, and Y, are cointegrated (Hamilton, 1994, pp. 610-612). We also find this to be the case with our long historical data (see Appendix A, part 4). Moreover, the ex post real interest rates also turn out to be stationary for our sample period (see Appendix A, part 1). Since C, and 11, are cointegrated and r, is stationary, it is evident that there are two cointegrating vectors, Indeed, our Appendix A, parts 6 and 7, as well as our state space estimations, confirm this.

220 H.D. Vinod, P. Basu / International Journal o f Forecasting 11 (1995) 217-231

~', = a + r t_ l + vt (random walk with drift) ,

(2.6)

we can reduce (2.5) to the following state space form:

X t = A X t _ ~ + B u t , (2.7)

where A, B and u t are transformations of .4, a and u, appearing above.

The Endo-i model in (2.7) has a three-way feedback among the X, components: log C t, log Y, and r t. Since the Exo-i model of (2.1) makes log Y, and r, exogenous, any feedback from log C t to them is ruled out a priori, making (2.1) a special case of (2.7) at the theoretical level. The following section explains how the two formula- tions are implemented by two separate Aoki- type SS models, with two independent choices of (minimum) unobservable states (as eigenvectors based on two sets of singular values to approximate two Hankel matrices). We shall see that the forecasts of the important out-of-sample case have larger mean squared errors (MSE) for the Exo-i model than for the Endo-i model. For in-sample data, the MSE merely compares the quality of the fit of the SS model (Endo-i) with the VAR (Exo-i) fit.

3. State space model

This section illustrates a relatively new state space model used to reveal the interplay among economic variables and the behavior of economic agents. The terminology of state space modelling and filtering was originally developed by control engineers, and arises from engineering applications. It has been successfully applied in physical sciences in diverse fields. The term 'filtering' refers to 'filtering out' the noise elements and obtaining estimates of the true underlying relationships, and the term 'state' refers to the condition--state of nature-- in which a dynamic system happens to be at a point in time. Aoki (1987) provides a bridge between the 'system-theoretic' and econometric literatures, which is called a bottom-up approach because his state space (SS) model is obtained directly from

the data. The interrelated stochastic data are Yi, for i - - 1 , 2 . . . . , p and t = 0 , 1 , 2 , . . . , T . Deter- ministic components such as a trend are usually removed and differencing of the series is usually performed before writing y , as a non-determinis- tic time series. The observable Y~t is then a sample path of a N(0,,~) stationary Gaussian process with a rational spectral density. It can be proved that such a stochastic process with rational spectral density can always be represented by a linear recursive scheme. The following scheme treats the basic economic variables themselves as states, rather than treating regression coefficients as states. Our notation, similar to Aoki's (1987), follows several features of the engineering tradi- tion. The following state space form is called 'innovations form' by Aoki. 3 See Cerchi and Havenner (1988) and Criddle and Havenner (1989) for illustrative examples:

xt+ 1 = A x t + B u t ,

state (transition) equat ion, (3.1)

y , = C x , + u , , u , ~ N(0, ,Y),

observation equat ion, (3.2)

where A, B, C and ,Y are all p × p matrices and x t , Yt and u t are all p × l vectors. When the number of states p is smaller than the number of variables p, as in our application, the matrix dimensions are adjusted appropriately. The innovations form has the subscript t + 1 and the matrix B in the state equation, while the common error term u t occurs in both (3.1) and (3.2). The lag operator L may be used to write (3.1) as L - i x t = A x , + B u t or ( I L -1 _ A ) x , = B u , where we have inserted the identity matrix needed for the multivariate case here. Substituting the im- plied value of xt in (3.2), we have

Yt = C ( I L - 1 _ A ) - I B u , + u t

= [ I + C ( I L -1 - A ) - I B ] u , (3.3)

in the form of a system representation which generates the output y , from the white noise

3 Aoki's innovations form is less well known than those related to Kalman filters. For a recent list of references to this latter literature, see Hamilton (1994, ch. 13).

H.D. Vinod, P. Basu / International Journal of Forecasting 11 (1995) 217-231 221

input ut, with the 'transfer function matrix' given in the brackets.

When selecting a parameterization, it is clearly desirable to avoid both unnecessary parameters and unnecessary dimensions. Kalman et al. (1969) define a 'minimal' representation, which is used to determine the minimum number of states required for a state space model. Among various linear system triplets (A, C, B) that 'realize' a pattern represented by a sequence of transfer function matrices, the minimal systems are as simple as possible with reference to data accura- cy and dynamics. In traditional time series and Aoki's SS models, one estimates the sample autocovariances to determine the minimal representation. An application to stock market data is reported by Cerchi and Havenner (1988) and to energy data by Vinod (1991). Let y ~ ' =

? (Y~, Y,+1, .-.) denote a vector of future values which we would like to predict from a similar vector Yt-t- ' : (YI-1, Y,-2,' ...) of past data. A key to Aoki's bottom-up estimation technique is an infinite dimensional Hankel matrix H = E(y~ yj__~') of autocovariances Aj = E(yt+jy;). It has a block band counter diagonal structure:

-A A 2

A 2 A3 H A , p ' ~ - " " • .- Ap._; 1 • . . A p , + l .

• " " A2 '-1

(3.4)

The corresponding sample counterpart is denoted by a 'hat' as l?la, p,, where we let p'---~w. From the y data we can easily estimate this sample Hankel matrix from the sample autocovariances. Note that p ' should be sufficiently large to summarize the information contained in the data. An intuitive appeal of the Hankel form is that it works directly (bottom up) with co- variances of the past (Yt-1) and the future (y~+).

Next, we turn to the determination of the dimension p of the state space, which is smaller than p (number of variables in y) when the variables are cointegrated. Aoki approximates the observed /:/A. U' by a representation having dimension p by using the so-called singular value decomposition (SVD). Computer algorithms for SVD are not difficult, and Vinod and Ullah

(1981) discuss the regression model within the SVD framework. Once the singular values are found, the approximation ignores the smallest singular values (which are close to zero) and reconstructs the matrix of rank p. It is well known that formal statistical testing of eigenvalues (squared singular values) is difficult. Aoki and Havenner (1991, pp. 16-18) refer to the rank result called the Kronecker theorem and suggest (analogous to a condition number) that we compute a ratio of the (r + 1)th singular value to the first singular value that is of order (1/~/T). We can supplement Aoki's method by using the formal cointegration tests in Appendix A to determine p the number of states, which equals the rank of the matrix used to approximate the observed Hankel matrix.

Gantmacher (1959, p. 207) notes a connection between infinite Hankel matrices and rational functions. Aoki and Havenner (1991, p. 31) show why the specification (3.1) and (3.2) is likely to be efficient. They also define 'balance' with the help of Grammian matrices and Lyapunov forms. These theoretical arguments suggest that the state space methods are a worthy alternative to the traditional VAR-type models in applied econometrics. The estimation requires further manipulations (Aoki and Haven- ner, 1991, pp. 18-23), including the shifting of the estimated Hankel matrix. Thus one finds estimates of the matrices A, B and C and some statistics for studying their sampling properties.

4. Estimation results and implications

In this section we apply the state space formulation developed in the preceding section to address our forecasting problem. The main ques- tion is whether alternative forecasting rules af- fect the forecastability of the relevant aggregates. To address this issue, our state space formulation includes three variables: (i) y~, = log of real consumption (log RC); (ii) Y2t = log of real income (log R G N P ) ; and (iii) Y3t = R I = (gross) real interest rates• We do not use first differences of consumption and income variables, because the state space models have the

222 H,D. Vinod, P. Basu / International Journal of Forecasting 11 (1995) 217-231

flexibility to deal with the non-stationarity arising from common trends. In an earlier version, we used seasonally adjusted (SA) quarterly data and found the results summarized in the abstract. We do not report 4 the quarterly results, because Maravall (1993) shows that, if SA data are used, the estimates often have non-invertible factors rendering some unit root tests inapplicable and VAR models are 'inappropriate'. Unfortunately, seasonally unadjusted (SUA) quarterly data are simply not available in the United States for the GNP deflator as pointed out by many authors, including McMillan (1991, p. 636). An obvious way to avoid SA data is to use annual data, for which we also have longer time series, but this also involves unavoidable temporal aggregation (Ermini, 1988). On the other hand, a long span of (annual) data is more suitable for cointegration, according to Hakkio and Rush (1991), and p the number of states in Aoki's SS representation also equals the number of cointegrating vectors. Our in-sample data use 1896-1982; our out-of-sample data referring to 1983-1991. RC and RGNP data are from Backus and Kehoe (1992). Their sample from 1888 to 1988 is up- dated, using the quarterly GNP and consumption series from the Federal Reserve Bank of St. Louis data bank to compute the annual growth rates of consumption and GNP. The annual growth rates are used to update the Backus and Kehoe data set. The nominal interest rates are the yield on corporate bonds from Gordon (1993). The rates of inflation were computed using the annual GDP deflator series, also from Gordon (1993). The real interest rate was calcu- lated by subtracting the rate of inflation from the nominal interest rate.

Parts 1-3 of Appendix A report our unit root test results, suggesting support for R I ~ I(0), log R C - I ( 1 ) and log R G N P - I ( 1 ) in the majority of cases with and without trend. Fig. 1 presents RC (solid line) and RGNP (dotted line) against date t, where we note that observed consumption is relatively smoother than in-

4 We are grateful to the referees for this point regarding the seasonal adjustment.

4,500 r I/ 2500.

t l I

1o0o jt

, . . . . , 0 , H H . , 0 n . , . ° , . . . . . . . . . . . . . . . . , . . . . . . . , , , , , . . . . . . . . . . . . . , , . . . . . . . , . . . . . . . . . . . . ',

1900 1910 1920 1930 1940 1950 1960 1970 1983 1991

R c - - . A o., 0Np I

Fig. 1. Actual real consumption and real GNP (in levels).

come--a stylized fact documented by many authors. Also note that they are visually non- stationary and appear to be cointegrated. Figs. 4 and 7 (below) plot actual real interest rates (RI) as dotted lines, which are visually stationary. We formulate two versions, called Exo-i and Endo-i, of the state space model outlined in Section 2 and report them in Tables 1 and 2, respectively. In the Exo-i formulation, the income and real interest rates are treated as a purely exogenous process. An appropriate vector autoregressive (VAR) model is fitted to the real interest rate and income series, chosen with reference to forecasting performance. The detailed results for the bivariate (log RGNP and RI) VAR estimation based on (2.2) are included in Appendix B. The matrix polynomial D(L) of (2.2) is of order 3, since we use lags 1-3. We use the estimated VAR process to obtain the predicted values of RI and log RGNP. Our Exo-i model uses Y2t = VAR-predicted log RGNP and Y3~ =VAR-predicted RI, not the observed data on log RGNP or RI. That is, y ' = (log RC, VAR-predicted log RGNP, VAR-predicted RI) at each time t. The Hankel matrix with p = 3 and 3 lags is 6 x 6, and its singular values (reported in Table 1) are close to zero except for the first. Hence a 'minimal' representation is obtained from an SS model with p = 1, according to Aoki's methods. Conse- quently, we have


x,+ l = A x t + B u t

state (transition) equation, (4.1)

y, = Cx, + u , , u, ~ N(0, ,Y,),

observation equation, (4.2)

with the dimensions A(p x p), B(O x3) and C(3 x P). Table 1 provides the estimation results for this Exo-i model. In our plotting we compute the antilogarithms for comparison with the actual levels. Fig. 2 presents the actuals and forecasts from our Exo-i model for R C (plotted after anti-logs), where we also indicate out-of-sample forecasts. For the Exo-i model where the VAR

000'

2500 In Sample,

[ #

..., i 1500 / / j

1900 1910 1920 1930 1940 1950 1960 1970 1983 1991

I ~ Actual RC - - - SS Forecast RC 1

Fig. 2. Actual and SS forecast RC, Exo-i model.

Table 1 US data Exo-i model: Iog RC, Iog RGNP, RI state space est imation details Lag pa ramete r is 2 and one state is included (p = 1). We use 87 observat ions for (in-sample) fitting and reserve 9 for out-of-sample forecasting. The singular values are 6.75, 0.41, 0.13, 0.03, 0.01 and 0.01, suggesting that one dimension is mos t impor tant and that the choice p = 1 should not be rejected. Since the eigenvalue of A (0.9543) is <1 the system is said to be observable and controllable. The est imated matr ices are reported vectorized by row. For example, along the row marked A are e lements air. Similarly, along row B one reads b ~ , bz2 and ha3; whereas along C are c,~, c2~ and

C31

A = 0.9543 B = - 1.3651 -0 .5590 2.3124 C = -0 .5237 -0 .5673 0.0037 g2 = Covariance between state variables and observations:

-0 .9578 - 1.0331 0.0087 F~, = Uncondi t ional covariance matrix of the data:

0.5242 0.5626 -0 .0032 0.5626 0.6100 -0 .0045 -0 .0032 -0 .0045 0.0011

= Condit ional covariance matrix = covariance of the errors with errors:

0.0415 O. 0396 O. 0003 0.0396 0.0434 -0 .0008 0.0003 -0 .0008 0.0010

=Var iance of the state: 1.7602

Asymptot ic statistic called Dor fman ' s M = 17361 exceeds X2(18) = 28.8693 at the 5% level and Havenner ' s A = 9.7534 exceeds X-'(1) = 3.8415. They both imply that p = 1 chosen here is not too large (Dor fman and Havenner , 1989). The Criddle ratio = det(q~)/det(F0) = 0.07088,~ 1 (see Criddle and Havenner , 1989), which would have suggested rejection of the model if the ratio had exceeded unity. Smallness of the Criddle ratio supports our overall model specification.

forecasts of Appendix B are shown as solid lines, Figs. 3 and 4 are similar to Fig. 2 for income and interest rates, respectively.

The same forecasting experiment is also car- ried out for an alternative SS formulation called Endo-i, where the real interest rate ( R I ) series is treated as endogenous. In other words, the feedbacks among consumption, income and R I

are explicitly taken into account in this formulation. Since RI is now endogenous, this Endo-i model has y ' = (log RC, log RGNP, RI) at each time t. The dimensions of the matrices A, B and C are the similar to those in (4.1) and (4.2), except that p = 2 here. Note that in Appendix A,

as00.

~000.

2000

150(

1O111

Out of 8amp le - - - . - - - - -

I - - Actual RGNP - - - Forecast RGNP ]

Fig. 3, Actual and VAR forecast RGNP, Exo-i model.

224 H.D. Vinod, P. Basu / International Journal o f Forecasting I1 (1995) 217-231

1.2

1.16'

1.1'

1.05

t '

O ~

0.9

O.85

0.8'

03S Out of ~am~1~----------

1900 1910 1920 1930 1940 1950 1960 1970 1983 1991

l - - Actual I~l . . . . VN~I Forecast RI I

Fig. 4. Actual and VAR forecast RI, Exo-i model.

parts 6 and 7, Johansen and Juselius' (1990) 'maximum eigenvalue' and 'trace' tests reject p = 0 against p = 1 at the 5% level and reject p ~< 1 in favor of p = 2 at the 10% level. Since the 5% critical value is very close to the observed statistic, the choice p = 1 versus p = 2 is not definitive. Because of this cointegration, the dimensionality is reduced (from p = 3 to p = 2) and two states at most are assumed to be adequate to represent the dynamics of the Endo- i model.

Table 2 summarizes the estimation results for the Endo-i model, while Figs. 5 -7 present its in-sample and out-of-sample forecasts of consumption, income and interest rates. Table 3 reports the usual numerical measures for com- paring forecasting performance.

The out-of-sample forecasting performance of our third variable, real interest rates (RI), seems to be of interest. It can be seen from the lower half of Table 3 for out-of-sample forecast errors that the Endo-i model is superior in reducing the mean squared error (MSE) and root MSE for RI. The in-sample mean absolute deviations (MAD) decreases from 0.04 to about 0.02. The squared correlation between the estimate and the forecast (RSQ) is a linear measure and is slightly larger for the RI from Exo-i. Appendix C and the footnote to Table 3 mention the inconclusive results of testing for the correct 'direction' of forecasts. Even minor superiority

Table 2 SS estimates, Endo-i model: log RC and VAR estimates of Iog RGNP and RI: Lag parameter is 2 and 2 states are included. We use 87 observations for in-sample fitting and reserve 9 for out-of-sample forecasting. The singular values of the Hankel matrix are: 6.72, 0.59, 0.09, 0.03, 0.02 and 0.005. Since all are small except for the first two, p = 2 seems reasonable. This choice is not rejected by cointegration tests. From the eigenvalues of A (0.9544 and 0.3324), and since neither moduli is ->1, the system appears to be observable and controllable. The same conclusion is supported by the eigenvalues of a covariance matrix. The estimated matrices are reported vectorized by row. For example, along the row marked A are elements all , a12, a2] and a22 , respectively. Similarly, along row B read: b11, b12, b13, bzl, b:2 and b23; whereas along C read: cH, ci2, cz~, czz, c3~ and c32

A = 0.9552 0.0132 -0.0385 0.3315 B = -0.6066 -1.2732 1.0167 5.5217 -5.5173 15.1864 C = -0.5235 0.0672 -0.5681 0.0240 0.0041 0.0405 /2 = Covariance between state variables and observations:

-0.9569 -1.0426 0.0116 0.0342 -0.0011 0.0415

I o = Unconditional covariance matrix of the data: 0.5242 0,5657 -0.0028 0.5657 0.6162 -0.0048

-0.0028 -0.0048 0.0032 --- Conditional covariance matrix = covariance of the errors

with errors: 0.0392 0.0406 -0.0001 0.0406 0.0461 - 0.0004

-0.0001 -0 .0004 0.0020 _~ = Covariance of the states with states:

1.7698 0.0381 0.0381 0.7202

Similar to Table 1, Dorfman's M = 1 3 3 . 1 5 > x Z ( d f = 2 7 ) = 40.1133 at 5% and Havenner 's A = 7 . 0 6 > X z ( 3 ) = 6.2514 at 10% (but A < 7.8197 at 5%). At the 5% level, Havenner ' s M and Dorfman's A disagree, but at 10% both imply that p = 2 chosen here is not too large. The Criddle ratio, which would suggest rejection of the model if it exceeded unity, equals det (~) /det (F0) = 0.03763. As in Table 1, its smallness supports the overall model specification.

in forecasting RI out-of-sample deserves further attention, due to well-known difficulties in forecasting interest rates in light of the apparent volatility since the 1970s. However , this point is unimportant from a behavioral viewpoint. The Exo-i model agents are not actually using the state space forecasts of the two exogenous variables related to income and interest rates. They are assumed to be relying on best fitting in- sample VAR forecasts. However , in the context


15oo.

'.L In Sample /~ . j/~/~

Out o~ ~ i ~ 1900 1910 lg20 1930 1940 1950 1960 1970 1983 1991

I ~ Actual RC - - - SS Forecast RC I

Fig. 5. Actual and forecast RC, Endo-i model.

1.2

1.15 1.1

1.052 1"

0.1~"

0.9'

0.1~'

0.8

0.75

In Sampla

~ 9 9 1

1- - A=uat•, - - SSFor~,~, 1

Fig. 7. Actual and forecast RI, Endo-i model.

of model comparisons, Table 3 includes these and in-sample results for completeness.

4. I. Implications about consumption smoothness

Although both Endo-i and Exo-i models have similar forecasting performance for consumption and income, we wish to know if they imply different things about the relative smoothness of consumption, with respect to innovations to income. Christiano (1987) demonstrates that, once real interest rates and income are endogen- ized, the simulated ratio trc/try of standard deviations of consumption to income is 0.44, whereas in his quarterly 1956:11-1984:1 data it is 0.49, i.e.

4500,

4000 In Sample I ~,s

I / /

o ~ ~--- O~ o¢ S a m ~ e - ~ - ~ 1900 1910 19"20 1930 1940 1950 1960 1970 1983 1991

~ - - Actual RGNP - -- SS Forecut RGNP [

Fig. 6. Actual and forecast RGNP, Endo-i model.

slightly larger. In our annual 1896-to 1982 in- sample data, the standard deviations of ~x log RC and A log RGNP are 0.0427 and 0.0622, respectively, correct to four decimal places. Their ratio after rounding, 0.6872, is less than unity, suggesting that real consumption has greater smoothness (lesser variability) than real income in the annual data. By analogy with Christiano, we compute the following summary statistic (smoothness index, SI) for the relative smoothness of consumption using alternative forecasts from our Exo-i and Endo-i models:

SI = (standard deviation of A log/~C)

/(standard deviation of Iog[RGNP/I~GNP]),

(4.3)

where /~C denotes the forecasts of RC and I~GNP denotes the forecasts of RGNP. Note that Iog[RGNP/I~GNP] measures proportional income innovations. Our smoothness measure, SI, is close to Deaton's (1987) measure of consumption smoothness. The denominator of S! repre- sents the standard deviation of the proportional income innovation. When SI = 0, the predicted consumption displays maximum smoothness, in the sense that A log /~C is fixed with zero variance. When SI = 1, both consumption (A log /~C) and proportional income innovations have the same variability (variance). For the in-sample Endo-i model, SI is 0.6589, which is also smaller


Table 3 Forecasting performance of SS model

Series Pes-Tim UVAR CVAR MSE RMSE RSQ AVGERR MAD

Exo-i, in sample: log RC 1.0815 0.5242 0.0370 0.0086 0.0925 0.9857 -0.0252 0.0718 log RGNP NA 0.6162 NA 0.0034 0.0583 0.9945 0.0009 0.0437 R1 NA 0.0032 NA 0.0021 0.0462 0.3375 -0.0001 0.0265

Endo-i, in sample: log RC 1.3710 0.5242 0.0386 0.0050 0.0710 0.9928 0.0252 0.0575 log RGNP 0.7818 0.6162 0.0457 0.0067 0.0816 0.9914 0.0266 0.0672 RI 0.0000 0.0032 0.0020 0.0021 0.0458 0,3485 0.0005 0.0275

Exo-i, out of sample: logRC NA 0.5242 0.0370 0.0318 0.1784 0.9839 -0.1769 0.1769 IogRGNP NA 0.6162 NA 0.0114 0.1068 0.9472 -0.1016 0.1016 R1 NA 0.0032 NA 0.0018 0.0420 0.4847 -0.0407 0.0407

Endo-i, out of sample: log RC NA 0.5242 0.0386 0.0163 0.1275 0.9714 0.1267 0.1267 IogRGNP NA 0.6162 0.0457 0.0085 0.0921 0.9686 0.0888 0.0888 RI NA 0.0032 0.0020 0.0004 0.0207 0.4263 0.0175 0.0175

Note: NA = not applicable/available. UVAR is unconditional covariance between actuals and forecasts (see F o in Table 1), CVAR is the corresponding conditional covariance (see ~ in Table 1). MSE is mean squared error, RMSE is root MSE. RSQ is simple correlation squared between observed and forecast series. AVGERR is average error and MAD is mean of absolute errors. The forecasting performance of SS methods is reasonable for log RC, log RGNP and RI. Column entitled Pes-Tim, row log RC reports Pesaran-Timmermann's (1992) statistic for predicting the direction of forecasts, explained in Appendix 3. For log RC (in-sample) value of 1.08 for the Exo-i increases to 1.37 for the Endo-i. However, despite the superiority of the Endo-i model, Pes-Tim = 1.37 is not significant at the conventional significance levels. Henriksson and Merton's (1981) confidence level for in-sample Endo-i is 0.935 and for in-sample Exo-i models it is 0.847; similar out-of-sample values are both 0.999. Exo-i model uses VAR forecasts of log RGNP and RI, hence their CVAR's are obviously not applicable (NA).

t h a n the observed smoothness , 0.6872. By con- trast , the Exo-i value of SI is 1.3663, not 'close e n o u g h ' to the observed 0.6872. This seems to help confirm our conclus ion that the Endo- i

mode l is superior . 5 Fo rma l test ing of differences be tween SI esti-

ma tes is a n o n - s t a n d a r d p rob lem, even assuming no rma l i t y , for m a n y reasons inc luding the pres- ence of two different Sl ' s and n o n - i n d e p e n d e n c e of their n u m e r a t o r s and denomina to r s . It would be a useful ex tens ion 6 to cons ider Ef ron ' s (1979)

5 When the nine years of the out-of-sample data are also included in a combined dataset, the SI for Endo-i is 0.6263, again smaller than 0.6872 (mentioned above). The corresponding SI for the Exo-i is 0.7653, which remains larger, but is closer than the in-sample value noted above. Note that the SI is a long-term concept and may not be reliable if computed from just the out-of-sample years.

6.Another extension suggested by a referee is to modify the Hausman-type specification testing to compare VAR estimation of the Exo-i model and SS estimation of the Endo-i model.

boots t rap , surveyed in Vinod (1993), in o rder to

study sampl ing d is t r ibut ion of this n o n - s t a n d a r d p rob lem. However , in this example , we have no reason to expect that formal test ing with the boots t rap would yield any different conclusions.

5. Concluding comments

This paper is conce rned with endogene i ty of income and interes t rates in models of consump-

t ion and income in the context of forecasting. O ur Endo- i model uses a three-var iab le state

space model with logs of consumpt ion , logs of income and in teres t rates, whereas our Exo-i model uses V A R forecasts of logs of income and interest rates. We apply Aok i ' s (1987) state space formula t ion , which has novel features and desirable propert ies . For example , it pe r fo rms a direct es t imat ion f rom the data and exhibi ts pa rs imony, due to a 'm i n i ma l ' r ep resen ta t ion which incorpora tes the co in tegra t ion be t w e e n

H.D. Vinod, P. Basu / International Journal of Forecasting 1i (I995) 217-231 227

income and consumption. The forecasting performance of SS models for interest rates is impressive, in view of the notorious fluctuations in the observed real interest rates. Our result may be pleasing to an applied econometrician facing the curse of dimensionality.

In addition to addressing this particular forecasting problem, the paper attempts a marriage between real business cycle literature and relatively new Aoki-type SS estimation. Our empiri- cal results shed some light on a topical issue in the macro consumption literature. The issue is whether macro forecasts and consumption smoothness properties differ if agents recognize that their savings and labor supply decisions have feedbacks on real interest rates and income. In our Exo-i specification, economic agents do not recognize any such feedback and treat income and real interest rates as strictly exogenous processes. In our Endo-i specification, the agent takes into account these feedbacks while forecasting consumption. Our state space estimation

results suggest that the forecasting abilities of these two models are similar, although the Endo- i model has some edge over its Exo-i counterpart as far as forecasting the real interest rate is concerned. On the other hand, the smoothness property of consumption is remarkably different in these two scenarios. The Endo-i model displays less consumption volatility, because endogenous movements of real interest rates smooth the consumption stream. The policy implications of these two models would, therefore, be different. For example, Basu (1995) demonstrates that a tax policy uncertainty might result in a greater degree of consumption smoothness in a real business cycle model with endogenous changes in interest rates.

Acknowledgements

We are grateful to three referees for detailed and very helpful comments.

Appendix A: Discussion of unit root and cointegration tests

For computations in this appendix we use Microfit software. The first three parts report our unit root tests of the three variables. Parts 4 and 6 report cointegration tests using the maximum eigenvalue statistic and parts 5 and 7 report similar tests based on a trace statistic.

Unit root tests: 95% critical values are in parentheses; DF = Dickey-Fuller and ADF = augmented DF statistics.

Part 1: For gross real interest variable RI (no log): we reject the unit root null.

Statistic Sample T Without trend With trend

DF 1891-1988 98 -5.0997 (-2.8909) ADF(1) 1892-1988 97 -5.1042 (-2.8912) ADF(2) 1893-1988 96 -4.1599 (-2.8915) ADF(3) 1894-1988 95 -3.9486 (-2.8918) ADF(4) 1895-1988 94 -3.3385 (-2.8922)

-5.0626 (-3.4557) -5.0707 (-3.4561) -4.1228 (-3.4566) -3.9082 (-3.4571) -3.2664 (-3.4576)


Part 2: Unit root tests for variable log RC: we cannot reject the unit root null.

Statistic Sample t Without trend With trend

DF 1891-1988 98 -0.56823 (-2.8909) -2.8079 (-3.4557) ADF(1) 1892-1988 97 -0.38515 (-2.8912) -2.7168 (-3.4561) ADF(2) 1893-1988 96 -0.30099 (-2.8915) -2.9976 (-3.4566) ADF(3) 1894-1988 95 -0.43811 (-2.8918) -3.0083 (-3.4571) ADF(4) 1895-1988 94 -0.77187 (-2.8922) -3.3664 (-3.4576)

Part 3: Unit root tests for variable log RGNP: we cannot reject the unit root null.

Statistic Sample t Without trend With trend

DF 1891-1988 98 -0.52130 (-2.8909) -2.6812 (-3.4557) ADF(1) 1892-1988 97 -0.58307 (-2.8912) -3.8689 (-3.4561) ADF(2) 1893-1988 96 -0.53329 (-2.8915) -4.0121 (-3.4566) ADF(3) 1894-1988 95 -0.60538 (-2.8918) -3.4758 (-3.4571) ADF(4) 1895-1988 94 -0.77028 (-2.8922) -3.2271 (-3.4576)

Cointegration tests: Johansen Maximum Likelihood Procedure (trended case, with trend in DGP). Cointegration LR test, 96 observations from 1893 to 1988. Maximum lag in VAR is 3.

Part 4: Maximal eigenvalue of the stochastic matrix, two variables: log RC and log RGNP. List of eigenvalues in descending order: 0.16370, 0.0010321.

Null Alternative Statistic 95% critical value 90% critical value

p = 0 p = 1 17.1616 14.0690 12.0710 p -< l p = 2 0.099133 3.7620 2.6870 Reject p = 0 in favor of p = 1, suggesting one cointegrating vector between log RC and log RGNP.

Part 5: Trace of the stochastic matrix, two variables: log RC and log RGNP. List of eigenvalues in descending order: 0.16370, 0.0010321.


p = 0 p --> 1 17.2607 15.4100 13.3250 p --< 1 p = 2 0.099133 3.7620 2.6870 Reject p = 0 in favor of p -> 1, suggesting one cointegration vector between log RC and log RGNP.

Part 6: Maximal eigenvalue of the stochastic matrix, three variables: log RC, log RGNP and RI. List of eigenvalues in descending order: 0.24092, 0.13543, 0.0011514.


p = 0 p = 1 26.4621 20.9670 18.5980 p -< 1 p = 2 13.9699 14.0690 12.0710 p ~ 2 p = 3 0.11060 3.7620 2.6870 Reject p = 0 against O = 1 (one cointegrating vector) at 5%; reject t9-< 1 against p = 2 (two cointegrating vectors among the three variables: log RC, log RGNP and RI) at 10% level of significance.

H.D. Vinod, P. Basu / International Journal of Forecasting 11 (1995) 217-231

Part 7: Trace of the stochastic matrix, three variables: log RC, log R G N P and RI. List of eigenvalues in descending order: 0.24092, 0.13543, 0.0011514.

229


p = 0 p ~ l 40.5425 29.6800 26.7850 p ~ 1 p ~ 2 14.0804 15.4100 13.3250 p ~ 2 p = 3 0.11060 3.7620 2.6870 Reject p = 0 against p = 1 (one cointegrating vector) at 5%; reject p ~ 1 against p = 2 (two cointegrating vectors among the three variables: log RC, log R G N P and RI) at 10% level of significance.

Appendix B: Discussion of vector autoregressive (VAR) model results using annual data

(1) Dependent variable = log RGNP, 1895-1982 annual data with t = 88, degrees of f reedom (d f )= 81. R 2= 0.9946, adjusted R 2= 0.9942, sum of squared residuals ( S S R ) = 0.3027, standard error of estimates (SEE) = 0.0611, Durbin-Watson (DW) statistic = 1.9483, Q statistic = 28.0378 with a significance level of 0.4090.

No. Label Lag Coefficient Stand. error t-statistic

1 R G N P 1 1.297841 0.1106958 11.72440 2 R G N P 2 -0.3191154 0.1787479 - 1.785282 3 R G N P 3 0.1398875E-01 0.1118172 0.1251037 4 R I 1 0.6302715E-03 0.1416011E-02 0.4451034 5 RI 2 - 0.6739383E-03 0.1638196E-02 - 0.4113905 6 RI 3 0.7554740E-03 0.1404874E-02 0.5377523 7 Constant 0 0.7007177E-01 0.5805339E-01 1.207023

(2) Dependent variable = RI, 1895-1982 annual data with t = 88, degrees of f reedom ( d f ) = 81. R 2-- 0.3416, adjusted R 2= 0.2928, sum of squared residuals ( S S R ) = 1855.3893, standard error of estimates (SEE) = 4.7860, Durbin-Watson (DW) statistic = 1.9721, Q statistic = 20.7129 with a significance level of 0.7995.

No. Label Lag Coefficient Stand. error t-statistic

1 R G N P 1 -12.69296 8.665868 -1.464707 2 R G N P 2 14.46181 13.99335 1.033477 3 R G N P 3 -2.010481 8.753657 -0.2296732 4 R 1 0.6311703 0.1108531 5.693757 5 R 2 -0.1828202 0.1282469 -1.425533 6 R 3 0.5030691E-01 0.1099811 0.4574141 7 Constant 0 3.069069 4.544734 0.6753022

Note: These results are produced by using RATS software.

Appendix C: Pesaran and Timmermann (1992) predictive performance test

This test is suggested as a superior alternative to Henriksson and Merton 's (1981) test for direction of forecasts. Let lower case x t denote the forecast series and yt denote the actuals. The upper case X, and


Y, are unity for positive values and zero otherwise. Z, is unity if both have the same sign (yrr, > 0). /5 denotes the proportion of times the sign is predicted correctly and equals the average of Z values or 2. Under the null hypothesis that x, has no power in predicting y,, the quantity nP has a Binomial distribution. In general, a Hausman-type statistic compares P with P* based on the null. If all actual directions are positive, the statistic is not defined and this is what makes our out-of-sample forecasts not available (NA). The Pesaran-Timmermann statistic has 15_ p . in the numerator and a complicated variance expression in the denominator. The statistic is distributed as a unit normal N(0, 1) variable, so the numbers above 1.96 suggest significant predictive power concerning the directions of the forecasts.

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Biographies: Hrishikesh D. VINOD (alias Rick Vinod) is Professor of Economics at Fordham University. He completed his Ph,D, at Harvard University in 1966. His main areas of concentration are Econometrics and Industrial Organization. He was an expert witness in the Bell System divestiture, a past Vice President of the Indian Econometric Society and a past President of a charity called the

Maharashtra Foundation. He has published extensively (about 80 research papers) in refereed journals, including almost all the top journals in Econometrics and Statistics. With a fourth paper in the Journal of Econometrics, he is scheduled to become a fellow of the journal in 1995. His second book, Handbook of Statistics: Econometrics (North- Holland, 1993), was co-edited with C . R Rao and G.S~ Maddala. Parantap BASU is an Associate Professor of Economics at Fordham University. He completed his Ph.D. at the Uni- versity of California at Santa Barbara in 1985. His main area of concentration is Macro and Monetary Theory. He has published in several journals including International Econ- omic Review, Journal of Macroeconomics, Journal of De- velopment Economics, Southern Economic Journal, Scan- dinavian Journal of Economics, Kyklos and others.

forecasting consumption, income and real interest rates from alternative state space models

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