The Role of Theoretical Restrictions in Price Forecasting with Inverse Demand Models
H. Allen Klaiber Department of Agricultural and Resource Economics North Carolina State University Box 8109, Raleigh, NC 27695-8109, USA. Telephone: 919-513-3864. Fax: 919-515-6268. E-mail:
[email protected].
Matthew T. Holt Department of Agricultural Economics Purdue University 403 W. State Street, West Lafayette, IN 47907–2056, USA. Telephone: 765-494-7709. Fax: 765-494-9176. E-mail:
[email protected].
Selected Paper prepared for presentation at the American Agricultural Economics Association Annual Meeting, Portland, OR, July 29-August 1, 2007
Copyright 2007 by H. Allen Klaiber and Matthew T. Holt. All rights reserved. Readers may make verbatim copies of this document for non-commercial purposes by any means, provided that this copyright notice appears on all such copies.
The Role of Theoretical Restrictions in Price Forecasting with Inverse Demand Models∗
H. Allen Klaiber† Department of Agricultural and Resource Economics North Carolina State University
Matthew T. Holt‡ Department of Agricultural Economics Purdue University
This Draft: May 31, 2007
Abstract In recent years the theoretical restrictions of consumer demand have been examined in post sample forecasting exercises. However, this work has uniformly ignored the concavity restrictions of consumer demand. In this paper we evaluate a series of Normalized Quadratic Inverse Demand System (NQIDS) specifications using rolling windows and generating one– to four–step ahead forecasts. To estimate the models, eleven categories of South Atlantic fish are used from 1980 through 2001. In addition to the NQIDS, we also examine the forecasting performance of a purely time series model. We find that the best predictions are achieved using a composite forecast. Keywords: Composite Forecasts; Concavity; Inverse Demand; Fish Demand; Rank Reduction; Theoretical Restrictions
JEL Classification Codes: C52; C53; Q11; Q22
∗
The research has been partially supported by SEA Grant Project R/BS–16 at North Carolina State University and by the Agricultural Experiment Station at Purdue University. † Corresponding Author: Department of Agricultural and Resource Economics, Box 8109, North Carolina State University, Raleigh, NC 27695-8109, USA. Telephone: 919-513-3864. Fax: 919-515-6268. E-mail:
[email protected]. ‡ Department of Agricultural Economics, Purdue University, 403 W. State Street, West Lafayette, IN 47907– 2056, USA. Telephone: 765-494-7709. Fax: 765-494-9176. E-mail:
[email protected].
Introduction An area of continuing interest in empirical demand analysis is the role of theory in model specification and estimation. More precisely, many authors have specified and estimated aggregate demand equations for various food items and/or commodities, and in so doing have, more often than not, specified models that impose the salient features of demand theory, most notably price and income homogeneity and Slutsky symmetry. While there are many examples of this type of work, several more notable studies along these lines would include Alston et al. (1990), Attfield (1997), Blanciforti and Green (1983), Deaton and Mueelbauer (1980), Eales and Unnevehr (1993), Holt (2003), and Piggott (2003), among many others. Early on, economists were primarily concerned with testing the implications of economic theory, typically by performing (parametric) tests of homogeneity and symmetry restrictions. Several examples along these lines include the studies by Barten (1969), Deaton and Muellbauer (1980), and Green, Hassan, and Johnson (1980). See Keuzenkamp and Barten (1995) for a thorough review. A typical result is that the basic tenants of economic theory are rejected, most notably because homogeneity restrictions are rejected (Laitinen, 1978; Ng, 1995). The failure to find in favor of theory in empirical demand work has, in turn, lead economists in several directions. On one hand, and in may instances, theoretical restrictions are simply imposed in a rote manner without additional testing.1 Alternatively, there has also been increasing attention given to the underlying time-series properties of the data, and the fact that many price series apparently exhibit near unit root behavior. Empirical demand work along the lines of dynamic demand systems, including tests of theoretical restrictions, has been reported on by, among others, Anderson and Blundell (1983), Attfield (1997), Holt and Goodwin (1997), Karagriannis and Mergos (2002), and Karagriannis and Velentzas (1997), among others. In general there is more support for economic theory and in particular, the homogeneity restriction, once proper (and flexible) dynamics are built into the system. One fundamental problem with this approach, however, is there is no reason a priori to expect budget shares to follow a unit root process since by construction they must be bounded between zero and one (Davidson and Ter¨asvirta, 2002). There is, of course, a third way. There is a growing school of thought that model validation 1
As well, several authors have examined the imposition of theoretical restrictions through various applications of Bayesian methods. See, for example, Chalfant, Gray, and White (1991).
1
should involve some form of post-sample validation exercise, and especially when time series data are employed. This case has been put forward in a rather elegant and convincing way by, among others, Granger (1999), who notes that when building models theory–and perhaps more to the point, complexity–is useful only to the extent that it helps the modeler better explain economic decisions and outcomes. And, moreover, the best “test” of this “ability to explain” will often derive from a post–sample forecasting exercise, in part because models, if they truly have any merit, are typically used to address policy questions/issues that lie beyond the range of the data. In this spirit a limited number of studies have sought to examine the forecasting properties of demand models, both with and without the basic structure implied by microeconomic principle imposed. Employing aggregate consumption data from the United Kingdom, Chambers (1990) estimated six models including the Linear Expenditure System (LES), a dynamic version of the LES, and a version of the linearized Almost Ideal Demand System (LAIDS). He compared the performance of these models with a simple random walk model, a vector autoregression (VAR), and a dynamic LAIDS of the sort considered by Anderson and Blundell (1983). He found the more parsimonious LES that included simple habit terms to have superior forecasting performance. Kasterns and Brester (1996) performed a similar analysis using U.S. aggregate annual food consumption and price data. They compared Barten and Theil’s Rotterdam model with a first differenced version of the LAIDS and a first differenced double-log model. The later model does not include the usual homogeneity and symmetry restrictions. Overall, they found the forecasts derived from the double-log model were generally superior to those from either the LAIDS or the Rotterdam models. Wang and Bessler (2001) examined the role of homogeneity restrictions in forecasting U.S. meat demand using VAR–type models. Overall, they find that the homogeneity restriction helps in the post–sample forecasting exercise when it is not rejected in the sample data. But aside from imposing homogeneity, the models estimated by these authors were not otherwise consistent with traditional (static) demand theory. Fisher, Fleissig, and Serletis (2002), using aggregate U.S. consumption data, examined the forecast performance of several locally and globally flexible functional forms, including Generalized Leontief and Laurent Models, the Almost Ideal Model (AIM), and the Fourier Flexible Form (FFF). Overall they found that the more flexible models such as the AIM and the FFF tended to provide better forecast performance overall, although they did not compare the forecast performance of their
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models against an atheoretical benchmark model such as the double-log, as did Kastens and Brester (1996). Finally, Li, Song, and Witt (2006) estimate various types of LAIDS models for UK tourism demand, including a version that allows for time–varying parameters and error correction terms. They find that allowing for time–varying parameters results in dramatic improvements in forecast performance relative to models with fixed parameters. While the aforementioned studies provide useful insights into the role of economic theory in forecasting with demand models, there is scope for further work. To begin, with the exception of Wang and Bessler (2001) we are unaware of any previous study on forecasting with demand models that has actually been conducted in a true forecasting environment; instead, models are typically evaluated in the context of a conditional forecasting environment, where actual, as opposed to predicted, prices and income are employed. As Clements and Hendry (2005) discuss, using conditional forecasts to evaluate econometric models might be especially problematic when multi–step–ahead forecasts are being evaluated in the context of a dynamic model. It would be preferable to model and predict the right–hand–side variables, too, in any forecasting exercise. Secondly, microeconomic demand theory entails more than simply the homogeneity and symmetry restrictions. Curvature conditions as pertains to the underlying expenditure function also come into play (Diewert and Wales, 1988a). In other words, aside from homogeneity and symmetry, theory also dictates that the underlying expenditure function should be concave in prices. And yet, no attention has been given to this issue in prior investigations of the forecasting performance of demand systems. As well, prior studies have been conducted for the case where it is assumed that quantities are endogenous and prices (and income) are exogenous. But for some products, and especially for many perishable food items, it is reasonable to deal with an inverse demand system, that is, one for which prices are endogenous are quantities are exogenous (Holt, 2003). Finally, all prior studies in this vein have been conducted in an either–or context. That is, one model is determined to perform “best,” in which case it is deemed to have the most to say about whether or not theory is informative for forecasting purposes. And yet, the empirical evidence often suggests that combinations of forecasts from several models might be preferred those from any single model. In this paper we report on a forecasting exercise that utilizes an inverse demand system applied to fish landed in the South Atlantic. The data we use are an updated version of the monthly data on fish landings and prices used previously by Beach and Holt (2001). In the fore-
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casting exercise we explicitly examine the role that curvature (i.e., negativity) of the underlying distance function plays in forecasting performance. Specifically, we use the locally flexible but globally concave Normalized Quadratic Inverse Demand System (NQIDS) of Holt and Bishop (2002), as well as their Semiflexible NQIDS (SNQIDS). An important feature of the SNQIDS is that while negativity restrictions are maintained, the rank of the Antonelli substitution matrix is reduced, thereby decreasing the total number of overall estimated parameters. Our working hypothesis is that reducing the rank of the Antonelli substation matrix might actually aid in forecast performance. We also obtain unconditional forecasts from our inverse demand models by treating quantities as weakly endogenous; we model quantities by using simple autoregressive processes. In addition, forecasts from our demand system are compared to a simple model that is consistent with preferences being associated with a Cobb–Douglas utility function. Last of all, we examine the potential for several composite forecasts. The plan of the paper is as follows. In the next section we develop the basic modeling framework. In the following section we discuss the data. In the penultimate section we present the in-sample estimation results and forecast results. The final section concludes.
A Globally Concave Inverse Demand System: The SNQIDS This paper uses Holt and Bishop’s (2002) semiflexible normalized quadratic inverse demand system (SNQIDS) as the basic modeling framework for estimating and evaluating inverse demand systems for fish landed in South Atlantic ports in the United States. In this section we briefly describe the specification and derivation of the SNQIDS–additional details may be found in Holt and Bishop (2002). Let qit denote the quantity landed in time period t of fish species i, qit the corresponding vessel-level price, qt = (q1t , . . . , qnt ) the n-vector of quantities in time period t, pt the corresponding price vector, and mt = pTt qt the group expenditure in time t. Also, let q ∗ denote some (possibly arbitrary) base–period or reference quantity vector. Ignoring for the moment the time subscripts, the SNQIDS is derived from the following normalized quadratic distance function: 1 T −1 T T D(u, q) = c q + b q + (α q) q Aq , 2 T
(1)
where α = (α1 , . . . , αn ) is a pre–determined parameter vector; c = (c1 , . . . , cn ) and b = 4
(b1 , . . . , bn ) are vectors of estimable parameters; A = [aij ] is a n × n parameter matrix; u is an unobservable utility index, and a superscripted T denotes vector/matrix transposition. As Holt and Bishop (2002) discuss, the distance function in (1) must also satisfy the following conditions αT q ∗ = 1,
α ≥ 0n
cT q ∗ = 0, Aq ∗ = 0n ,
(2a)
and
(2b)
AT = A,
(2c)
where 0n is a nx1 null vector. Applying the Shephard–Hanoch lemma to 1 gives a system of compensated inverse demands: the Antonelli demands. Specifically, πia (u, q) =
∂D(u, q) = ci + bi + (αT q)−1 ∂qi
n X j=1
1 aij qj − (αT q)−2 q T Aq u−1 , i = 1, . . . , n, (3) 2
where πi = pi /m denotes the ith normalized price. By construction the Antonelli demands in (3) are homogeneous of degree zero in quantities. Of course Antonelli demands are not immediately useful for empirical work because utility index u is unobserved. Uncompensated inverse demands that are, in fact, a function of observed data may be obtained in the following manner. First, as Deaton (1979) notes, the distance function implicitly defines the consumer’s utility function. Specifically, D(u, q) = 1 at the optimum, which implies that (1) may be solved explicitly for the utility index u as " U (q) =
# [bT q + 21 (αT q)−1 q T Aq] . 1 − cT q
(4)
Utility function (4) may then be used to substitute for u in (3), giving h πi (q) = ci +
bi + (αT q)−1
Pn
1 T −2 T j=1 aij qj − 2 (α q) q Aq bT q + 21 (αT q)−1 q T Aq
i 1 − cT q , i = 1, . . . , n,
(5)
an estimable system of inverse demands. Several additional restrictions on the parameters in (5) are required in estimation. To begin,
5
the system in (5) is homogeneous of degree zero in the parameter vector b and the parameter matrix A. To achieve identification, we simply require that bT q ∗ = 1,
(6)
which would be imposed in addition to the parameter restrictions noted in (2) above. Holt and Bishop (2002) also show that the matrix A must be negative semi–definite for distance function (1) to be (globally) concave in quantities. If this requirement is not automatically satisfied it e denote a (n − 1) × (n − 1) matrix obtained from may be imposed in the following manner. Let A A by deleting the last row and column; these terms may be recovered by using the restrictions e is negative semi–definite, then A will also be negative in (2a). The implication is that if A e as semi–definite. We may then redefine A e = −SS T , A
S = [sij ],
sij = 0 ∀ i > j.
(7)
e To implement the model In other words, S is the (n − 1) × (n − 1) Cholesky decomposition of A. the sij parmaeters are estimated in lieu of the aij parameters (Moschini, 1998). As a practical matter, if negativity must be imposed by using the Cholesky decomposition, it is typically the case that the positive eigenvalues associated with the unrestricted estimates e while now negative, will lie very close to zero. It may therefore be desirable to further of A, e Following Diewert and Wales (1988b), let K < (n − 1) be the rank of A. e reduce the rank of A. e is associated with a K -column Cholesky decomposition. That In the case where K < (n − 1), A is, S is defined according to S = [sij ],
sij = 0 for 1 ≤ i ≤ j ≤ n − 1, and for j = K + 1, . . . , n − 1.
(8)
In other words, S is a lower triangular (n − 1) × (n − 1) matrix with zeros in its final (n − 1) − K columns. The combination of (1)–(2) and (6)–(8) yields the SNQIDS. Finally, and as previously noted, it is not possible to use either the NQIDS or the SNQIDS to examine the implications of homogeneity in quantities, as this condition is imposed on the Antonelli demands in (3) by construction. Even so, the estimable demands in (5) are not homo-
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geneous of degree zero in quantities.2 Alternatively, the symmetry and/or concavity restrictions implied in (7) or (8) may be relaxed, both for in–sample evaluations as well as for post–sample forecasting. By doing so, and given that (5) is not homogenous of degree zero in quantities, we obtain an atheoretical inverse demand model.
A Simple Alternative Because the ultimate goal of this study is to examine the performance of the NQIDS and variants of the SNQIDS in a forecasting exercise, it is useful to have a relatively simple base or reference model. One such possibility in the case of demand analysis is to assume that preferences follow a Cobb–Douglas utility function, where it may be shown that πi (q) = and where
Pn
j=1 βj
βi , qi
i = 1, . . . , n,
(9)
= 1, and where 0 ≤ βi ≤ 1 for all i. Of course the Cobb–Douglas model is
very restrictive in terms of its implications for substitutability. Moreover, it may be converted to a system of share equations by simply multiplying both sides of (9) by qi , the parameters in the system may simply be estimated from the sample means of the budget shares. As restrictive as this model may be from an economic point of view, the forecasting literature is replete with examples of rather more restrictive and parsimonious models outperforming those that are more richly specified (and presumably containing more parameters) in a post–sample setting. See, for example, Clements and Hendry (1998). We therefore use the Cobb–Douglas model in (9) as our reference model for forecasting purposes.
Estimation Framework Inverse Demand Models To render the SNQIDS an estimable system of equations, it is necessary to consider more carefully several model specification issues as well as the stochastic properties of the data. To begin, all quantities are expressed in per capita terms by first dividing by total U.S. population. As 2
The inverse demands that represent the Marshallian counterparts to the Antonelli demands are not required to be homogeneous of degree zero in quantities. See, for example, Deaton (1979) for additional details.
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well, prior to estimation all quantities are normalized to have unit means, that is, all quantities are divided by their respective mean values. As well, both sides of equation (5) are multiplied by qit so that the system may be rewritten as3 h wit = ci qit +
bi qit + (αT qt )−1
Pn
1 T −2 T j=1 aij qit qjt − 2 (α qt ) qt Aqt bT qt + 12 (αT qt )−1 qtT Aqt
i 1 − cT qt + υit ,
(10)
where wit denotes the i th budget share at time t, υit is an additive error term such that υit ∼ iid(0, σi2 ), and i = 1, . . . , n and t = 1, . . . , T . The system in (10) may be used to formulate estimable versions of the SNQIDS. Specifically, let υt = (υ1t , . . . , υnt )T , a (n × 1) vector of normally distributed error terms at time t. The usual stochastic assumptions for υt are then E(υt ) = 0n ∀t, E(υt υtT ) = Ω ∀t, and E(υt υsT ) = 0 ∀ t 6= s, where Ω is the contemporaneous covariance matrix of the disturbance terms. The usual assumption then is that υt ∼ N (0n , Ω), whereby the unknown parameters in the SNQIDS may be estimated by using standard maximum likelihood methods. Of course this typically involves deleting an equation from the model for purposes of estimation due to the adding–up restriction. A remaining issue is how to specify the reference bundle q ∗ . Given that the quantity variables have been normalized to have unit means, we take q ∗ = ιn as the reference bundle, where ιn is a (n × 1) unit vector. Using this reference bundle, the parameter restrictions in (2) reduce to αT ιn = 1,
α ≥ 0n
cT ιn = 0, Aιn = 0n ,
(11a)
and
(11b)
AT = A,
(11c)
We follow prior authors (i.e., Diewert and Wales, 1988a) in simply defining α as α = ι/n, although clearly other normalizations would work as well. Finally, and as a practical matter, it may be useful to allow for systematic trends in the budget shares over time. To accomplish this we consider a specification wherein the parameter ci in (10) is specified as ci = ci1 + ci2 t, where again t = 1, . . . , T . Of course the restriction in (11b) is still employed.4 3 A second subscript, t has now been added to indicate the time series nature of the data. Specifically, t is indexed so that t = 1, . . . , T , where here T denotes sample size. 4 Of course since monthly data are being employed there may be scope for systematic seasonality as well. While we experimented with allowing the ci parameters to also vary with monthly dummies, as explained in more detail in what follows we ultimately settled on a specification that simply accounted for autocorrelation at an annual frequency. We note that a similar specification was employed by Holt and Bishop (2002) in their investigation of
8
Inasmuch as monthly time series data are being employed, the assumption that E(υt υsT ) = 0 ∀ t 6= s is frequently violated, as is the case here. That is, we find in initial estimation runs that the error terms in (10) are autocorrelated, and in particular autocorrelated at lags one and twelve. Although various methods are available for correcting for autocorrelation in a system of demand equations, we follow the framework developed by Holt (1998). Specifically, let a superscripted n be an operator that deletes that last row (row and vector) from a vector (matrix). If we let φ denote the vector of structural parameters to be estimated in the SNQIDS and f (.) the particular functional form, then the estimable version of (10) may be expressed as wtn = f n (qt , φ) + υtn ,
(12)
where, for example, wtn = (w1t , . . . , wn−1,t )T . Of course (12) is not appropriate if, in fact, the errors are autocorrelated. Assuming that a model that accounts for residual autocorrelation at lags one and twelve is appropriate, we may respecify (12) in the following manner n n n wtn = f n (qt , φ) + R1n wt−1 − f n (qt−1 , φ) + R12 wt−12 − f n (qt−12 , φ) + nt .
(13)
n are (n × n) autocovariance matrices for, respectively, lags one and twelve. In (13) R1n and R12 n in singular equation systems, here we use While there are various ways to specify R1n and R12
Holt’s (1998) symmetric and positive semi–definite specifications. The result is that for each autocovariance matrix an additional (n − 1) × (n − 2) estimable parameters are added to the model. See Holt (1998) for additional details.
Auxiliary Models for Quantities Because our ultimate goal is to use the various demand models to obtain unconditional forecasts of budget shares for fish landed in the U.S. South Atlantic region, it is useful to have some consistent way of obtaining forecasts for per capita quantities. For present purposes we assume that quantities are weakly exogenous variables, that is, quantities (i.e., per capita fish landings) may affect current prices but note vice versa. A typical specification for per capita quantities fish landings in the U.S. Great Lakes region.
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might then be written as
ln qit = β0,s +
k X j=1
βj ln qit−j +
p X
λj ln πit−j + ωit ,
i = 1, . . . , n,
(14)
j=1
where, β0,s is an intercept term that varies both with season (quarter) and with time; ln denotes natural logarithm; and πit−j denotes the i th normalized price at lag j. As well, ωit is a mean– zero exogenous shock. In keeping with the notion of weak exogeneity, in estimation we also restrict the covariance terms between ωit in (14) and υit in (10) to zero (or, alternatively, with all elements in nt in (13)). Initial estimations revealed that lagged normalized prices apparently play little if any role in determining current fish landings, that is, all λj are restricted to zero for all species. As well, once seasonality was accounted for, a sufficient lag order k on all quantities was determined to be one. Taken together, these restrictions imply that, at least for present purposes, quantities may be viewed as being strictly exogenous.
Data To implement the estimation framework, monthly data on finfish landed on the South Atlantic coast of the United States is used. This data series was obtained from the National Marine Fisheries Service and contains landings in pounds and total value of landings in dollars for 261 months covering March 1980 through November 2001. In order to obtain per capita landings data, quantities are divided by total U.S. population. Data for all finfish species present in this data was aggregated into 11 groups consisting of billfish, drum, flat fish, grouper, jack, mackerel, other, mullet, sea bass, snapper, and tuna. The other category contains fish that were not consistently landed or were landed at more than an order of magnitude smaller than the other categories resulting in extremely small budget shares. Several notable types of fish within the other category are cobia, grunt, pompano, and porgies. Table 1 displays summary statistics for the budget shares corresponding to the 11 fish groups. From Table 1 we see that flat fish has the largest mean budget share at slightly over 18% and is followed closely by the billfish and other category at approximately 16% and 15%, respectively. The large disparity between the minimum and maximum budget shares for each species is reflective of the very cyclical nature of the data. This large variability in the data will allow us
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to identify the systematic relationships between the 11 fish categories.
Results After estimating the unrestricted (neither concavity nor symmetry is imposed) NQIDS model, we observe that the Antonelli matrix is not negative semi-definite. This is due to positive eigenvalues for both the flat fish and jack fish categories. Imposing curvature forces these eigenvalues to be non-positive, but near zero, while leaving the eigenvalues for the other fish groups relatively unchanged. As a result of this process, all own-Antonelli flexibilities become negative. The near-zero eigenvalues that result after imposing concavity lead us to examine the effects of rank reduction on model fit and forecasting performance. Table 2 shows individual equation R2 measures, system-wide log likelihood (LL) values, as well as the Akaike (AIC), Schwarz (SIC), and Hannan-Quinn (HQC) information criteria values for QNIDS and SNQIDS models with no restrictions imposed, only symmetry imposed, and for all ranks ranging from the full rank (K =10) model to the K =1 rank reduced model, each of course with symmetry and concavity imposed. The individual equation R2 values are fairly consistent across model specifications, but vary across different equations. Almost all R2 values are above 0.8 for every model with many values above 0.9. This indicates that the various model specifications appear to fit the sample data well. To help differentiate among models, we turn to the log likelihood, AIC, SIC, and HQC measures. As expected, imposing symmetry and concavity causes a substantial drop in the log likelihood value due to the added model restrictions. The loss of fit is also evident in the AIC, but not for the SIC and HQC. Specifically, when using the AIC the NQIDS without symmetry and concavity imposed (i.e., the NCNS) is preferred. Alternatively, the SIC chooses the SNQIDS with rank k =2 as the preferred model, while the HQC picks the SNQIDS with rank k =4. This outcome is not surprising in that, as Greene (2002) points out, the AIC tends to prefer more heavily parameterized models while the SIC prefers more parsimonious models, and the HQC represents something of a compromise between the two. Ignoring for the time being the theoretically inconsistent models that do not not maintain concavity and symmetry, there are
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considerable differences in model fit amongst the rank reduced models. Within the class of rank reduced models, it becomes clear that from a model fit perspective some rank reduction improves fit but reducing the rank by too much apparently results in poor fit due to loss of model flexibility. To illustrate, the log likelihood value is highest for the K =6 model followed by the K =5 and then K =4 model. The Akaike criterion chooses the K =4 model followed by the K =6 model and then the K =5 model. The Schwarz criterion prefers the more simplistic K =2 model followed by the K =4 and K =3 model. Lastly, the HQC picks the K =4 model followed by the K =5 and K =6 model. All four of these measures choose rank reduced models around K =4 while none of them have the K =10 model in their top three. To further examine the effects of imposing theoretical restrictions and rank reduction, we perform out of sample forecasts and compare the mean squared forecast errors obtained by using a four–step forecast horizon. To do this, we withheld 60 months worth of data from December 1996 through November 2001. Over this time period, we forecast over a 1 to 4 month horizon for 57 time periods. In addition to the various NQIDS and SNQIDS models, we also estimate and perform out of sample forecasts for a simpler Cobb-Douglas (CD) model that serves as a baseline to gauge the forecasting performance of the more complicated models. Table 3 shows the mean squared forecast errors for all of the estimated models as well as for several composite forecasts.5 Looking first at the non–composite predictions, the Cobb-Douglas model with vector autoregressive errors at lags one and twelve forecasts substantially better than any of the NQIDS or SNQIDS models, with notably larger gains in forecast performance at lags one and four. Of the SNQIDS models, the rank reduced K =4 model forecasts the best followed by the K =10 and K =7 versions, respectively. Contrary to the in–sample model rankings resulting from the log likelihood and information criteria, the theoretically inconsistent models without concavity and symmetry imposed are amongst the worst models in terms of overall forecasting ability. Another difference between the two model selection approaches is that while the forecasting approach agrees in picking a rank reduced model, it tends to favor a lesser degree of rank reduction than the log likelihood and the AIC information criteria would indicate.6 In light of the exceptional performance by the Cobb-Douglas model, we conducted forecast 5
The mean squared forecast errors reported in Table 3 are obtained at each forecast horizon by evaluating the trace of the forecast covariance matrix. This approach has been used to evaluate forecast performance in a multivariate setting by, for example, Clements and Hendry (1995). 6 Indeed, the reduced rank k =4 SNQIDS was picked in–sample only by the HQC.
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encompassing tests to determine if the Cobb-Douglas model provides fundamentally different forecasting information from any of the SQNIDS models. The system forecast encompassing results, reported in Table 4, are obtained following Wang and Bessler (2004). The rows correspond to the left hand side forecasting error for a given fish. The columns correspond to the difference between the row error and the given column error. For example, row 3 column 4 corresponds to the following regression e3t = Γ(e3t − e4t ) + it .
(15)
The values reported in the table are p-values associated with a likelihood ratio test comparing (15) with the following model e3t = it .
(16)
A significant p-value indicates that the models do not encompass each other and forecasts can be improved by combining the models. These results suggest that the Cobb–Douglas model and the SQNIDS models do not encompass each other and therefore composite forecasts should improve forecasting performance. Returning to Table 3, the mean squared forecast errors for several composite forecasts are also reported.7 The forecasting performance of the composite models is greatly enhanced by combining the Cobb–Douglas and a single SQNIDS model. As expected, combining the best SQNIDS model of K =4 with the Cobb-Douglas model results in the best forecasting performance, followed closely by the the combination of the Cobb–Douglas with the NQIDS (i.e., the full–rank model). It is also worth noting that if too many models are combined the improvement from composite forecasting declines.
Conclusions In recent years there has been considerable interest in testing the basic principles of economic theory by using post–sample evaluations. In the context of systems of demand equations estimated with time series data, considerable attention has focused on the role of the homogeneity restriction. Even so, prior studies have not examined the role that curvature conditions and 7
In constructing the composite forecasts we follow the usual convention of simply applying equal weights to each set of forecasts.
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rank reduction might play in demand forecasting. In this paper we examine both the in–sample and post–sample performance of a family of inverse demand models for which curvature can be maintained globally, that is, the NQIDS and the SNQIDS. We compare these models to a simple benchmark: the Cobb–Douglas with autoregressive errors. The in–sample results indicate considerable discrepancy. Specifically, the AIC leads us to choose the NQIDS model without curvature or symmetry conditions imposed as best, while the SIC prefers a more parsimonious specification (i.e., the rank k =2 model) albeit with symmetry and curvature imposed. The post–sample results present an entirely different picture. Averaging across all forecast horizons, the restrictive Cobb–Douglas specification dominates. For those that prefer richer model specifications this would appear to be a less than encouraging conclusion. Forecast encompassing tests reveal, however, that the Cobb–Douglas forecasts do not encompass those from any of the variants of the NQIDS and the SNQIDS models. Several simple composites were examined in which case it was found that, in terms of mean squared forecast error, that combinations of forecasts were preferred. This is especially the case for simple combinations of forecasts from the Cobb-Douglas and the rank k =4 SNQIDS. What can therefore be concluded? Does theory really matter, at least in this instance? The answer would appear to be a resounding maybe, or perhaps more to the point, sort of. As many have found in previous forecasting exercises, relying exclusively on predictions from a single model is not the preferred approach. Therefore, looking for a simple “yes/no” answer to the question of “Does theory matter?” seems to result in a Type III error: the wrong question has been asked. Based on our findings it seems that theory matters, but only in a partial way, and only when forecasts from an appropriately specified demand model are combined with those from a much more restrictive model specification. Indeed, this basic conclusion is similar in spirit to the “Thick Modelling” strategy promoted recently by Granger and Jeon (2004), wherein results from several models are averaged for purposes of policy analysis. We believe that for purposes of modelling fish demands that this general approach holds merit. Of course the results reported here are specific to the data employed and the time–period examined, both of which are rather specific. It is therefore our hope that future research will focus on applying similar procedures in a wider variety of settings.
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Holt, M. T., and R. C. Bishop (2002). “A semiflexible normalized quadratic inverse demand system: an application to the price formation of fish.” Empirical Economics, 27, 23–47. Holt, M.T., and B.K. Goodwin (1997). “Generalized habit formation in an inverse almost ideal demand system: An application to meat expenditures in the united states.” Empirical Economics, 22, 293–320. Karagiannis, G., and G. J. Mergos (2002). “Estimating theoretically consistent demand systems using cointegration techniques with application to greek food data.” Economics Letters, 74, 137–143. Karagiannis, G., and K. Velentzas (1997). “Explaining food consumption patterns in greece.” Journal of Agricultural Economics, 48, 83–92. Kastens, T. L., and G. W. Brester (1996). “Model selection and forecasting ability of theoryconstrained food demand systems.” American Journal of Agricultural Economics, 78, 301–312. Kauzenkamp, H. A., and A. P. Barten (1995). “Rejection without falsification on the history of testing the homogeneity condition in the theory of consumer demand.” Journal of Econometrics, 67, 103–127. Laitinen, K. (1978). “Why is demand homogeneity so often rejected?” Economics Letters, 1, 187–191. Li, G., H. Song, and S. F. Witt (2005). “Time varying parameter and fixed parameter linear aids: An application to tourism demand forecasting.” International Journal of Forecasting, 22, 57–71. Moschini, G. (1998). “the semiflexible almost ideal demand system.” European Economic Review, 42, 349–364. Ng, S. (1995). “Testing for homogeneity in demand systems when the regressors are nonstationary.” Journal of Applied Econometrics, 10, 147–163. Piggott, N. E. (2003). “The nested piglog model: An application to u.s. food demand.” American Journal of Agricultural Economics, 84, 1–15.
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Tables Table 1: Summary Statistics for Budget Shares Category Billfish Drum Flat Fish Grouper Jack Mackerel Other Mullet Sea Bass Snapper Tuna
Mean 0.1604 0.1374 0.1815 0.0815 0.0131 0.1092 0.1486 0.0385 0.0291 0.0641 0.0367
Monthly Deviation 0.0953 0.0763 0.1143 0.0388 0.0102 0.0494 0.0438 0.0371 0.0208 0.0285 0.03
Shares Min 0.0242 0.0162 0.0142 0.0149 0.0005 0.0134 0.0636 0.005 0.0014 0.0178 0.0001
Max 0.4885 0.3663 0.4756 0.1906 0.0449 0.3037 0.3003 0.195 0.1059 0.1761 0.1767
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Monthly Landings (pounds) Mean Deviation Min Max 251669 171181 53784 855390 1602749 1114385 89018 5022073 802668 790178 37409 4880475 204906 94500 48790 581986 133838 109115 10153 654850 671332 725785 45632 5401482 1607479 891324 362980 5365642 461955 448180 96396 3482466 131916 126947 4837 1020646 147795 77184 36810 465440 115089 82608 686 408544
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K =2 0.9670 0.9363 0.9762 0.9774 0.9396 0.8470 0.8135 0.9235 0.9633 0.9795 0.8292 10018.98 -117.26 -114.08 -115.98
K =3 0.9675 0.9385 0.9762 0.9768 0.9425 0.8561 0.8226 0.9247 0.9659 0.9790 0.8275 10033.78 -117.32 -114.03 -115.99
b
NC = no concavity NCNS =noconcavity, no symmetry c CD = Cobb-Douglas
a
Category K =1 Billfish R2 0.9572 Drum R2 0.9334 Flat Fish R2 0.9776 0.9783 Grouper R2 Jack R2 0.9399 Mackerel R2 0.8360 Other R2 0.7836 0.9240 Mullet R2 Sea Bass R2 0.9640 Snapper R2 0.9791 Tuna R2 0.8217 Log Like 9983.67 AIC -117.05 SIC -114.00 HQ -115.82
Table 2: Measures of Fit K =4 0.9674 0.9393 0.9775 0.9793 0.9445 0.8472 0.8290 0.9235 0.9669 0.9787 0.8263 10057.50 -117.45 -114.06 -116.09
K =5 0.9666 0.9388 0.9776 0.9784 0.9451 0.8454 0.8282 0.9240 0.9677 0.9783 0.8256 10061.64 -117.44 -113.96 -116.04
K =6 0.9672 0.9414 0.9775 0.9785 0.9453 0.8466 0.8342 0.9235 0.9693 0.9791 0.8270 10067.44 -117.44 -113.90 -116.01
K =7 0.9677 0.9402 0.9773 0.9785 0.9449 0.8511 0.8306 0.9239 0.9690 0.9791 0.8276 10050.46 -117.27 -113.67 -115.82
K =8 0.9681 0.9378 0.9778 0.9784 0.9380 0.8470 0.8343 0.9240 0.9692 0.9788 0.8274 10041.35 -117.18 -113.53 -115.71
K =9 0.9672 0.9409 0.9777 0.9778 0.9453 0.8504 0.8335 0.9240 0.9690 0.9787 0.8277 10049.72 -117.23 -113.56 -115.75
K =10 0.9666 0.9378 0.9775 0.9785 0.9458 0.8464 0.8248 0.9237 0.9685 0.9792 0.8268 10053.76 -117.25 -113.57 -115.77
NCa 0.9659 0.9396 0.9779 0.9777 0.9436 0.8446 0.8268 0.9231 0.9702 0.9794 0.8321 10082.00 -117.48 -113.79 -116.00
NCNSb 0.9693 0.9401 0.9800 0.9777 0.9475 0.8691 0.8251 0.9255 0.9690 0.9825 0.8437 10161.98 -117.76 -113.44 -116.02
CDc 0.6247 0.6166 0.5468 0.7325 0.7046 0.0870 0.3844 0.6696 0.5463 0.6942 0.5242 8643.06 -106.44 -103.67 -105.33
Table 3: Mean Squared Forecast Error Step Model K =1 K =2 K =3 K =4 K =5 K =6 K =7 K =8 K =9 K =10 NC NCNS CD CD/Fulla CD/K =4a CD/K =3a All Except NCNS/CDa All Except NCNSa All Modelsa a
1 0.024127 0.023920 0.023775 0.023523 0.023654 0.023722 0.023625 0.023642 0.023622 0.023524 0.024406 0.024357 0.023327 0.021502 0.021496 0.021663 0.023746 0.023132 0.023192
2 0.024292 0.023856 0.023850 0.023429 0.023698 0.023650 0.023660 0.023656 0.023655 0.023683 0.024510 0.025385 0.019965 0.019256 0.019169 0.019405 0.023758 0.022675 0.022828
Composite forecasts
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3 0.023667 0.022962 0.022670 0.022320 0.022422 0.022391 0.022372 0.022406 0.022387 0.022377 0.023136 0.024112 0.022158 0.019889 0.019883 0.020112 0.022584 0.021842 0.021951
4 0.022845 0.022011 0.021687 0.021347 0.021350 0.021351 0.021288 0.021343 0.021307 0.021310 0.022124 0.022856 0.019087 0.018162 0.018202 0.018404 0.021569 0.020755 0.020851
Sum 0.094931 0.092749 0.091982 0.090619 0.091124 0.091114 0.090945 0.091047 0.090971 0.090894 0.094176 0.096710 0.084537 0.078809 0.078750 0.079584 0.091657 0.088404 0.088822
Table 4: System Encompassing Tests (p-values for Γ = 0) Model K =1 K =2 K =3 K =4 K =5 K =6 K =7 K =8 K =9 K =10 NC NCNS CD Step 1 K =1 K =2 K =3 K =4 K =5 K =6 K =7 K =8 K =9 K =10 NC NCNS CD
1 0 0 0 0 0 0 0 0 0 0 0 0
0a 1 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0.0001 0.0001 0 1 0 0.0022 0.0001 0 1 0.1632 0.0001 0.0021 0.1731 1 0.0027 0.0002 0.0006 0.0068 0.0019 0.0003 0.0005 0.0276 0.0001 0 0.0001 0.0008 0 0 0 0 0 0 0 0 0 0 0 0
Step 2 K =1 K =2 K =3 K =4 K =5 K =6 K =7 K =8 K =9 K =10 NC NCNS CD
1 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0.001 0.0004 0 0 0
Step 3 K =1 K =2 K =3 K =4 K =5 K =6 K =7 K =8 K =9 K =10 NC NCNS CD
1 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0
Step 4 K =1 K =2 K =3 K =4 K =5 K =6 K =7 K =8 K =9 K =10 NC NCNS CD
1 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0
a
0 0 0 0.0022 0.0003 0.0005 0.0277 0.0023 1 0.0017 0 0 0
0 0 0 0.0001 0 0.0001 0.0008 0 0.0017 1 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.001 0 0 0 0 1 0.0001 0.0079 0.0003 0.0001 1 0 0.0163 0.0077 0 1 0.0023 0.0003 0.0172 0.0025 1 0.0001 0 0.0007 0.2074 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0.0005 0 0.0001 0 0.0007 0.2279 1 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0.0002 0.0002 1 0 0 0.0123 0.1476 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0.0118 0 0.1473 0 0 0 1 0.0002 0.0002 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0.0007 0 0.0012 0 1 0.0065 0.0217 0 0.0008 0.0062 1 0.0004 0.0002 0 0.022 0.0004 1 0.0013 0.0013 0 0.0002 0.0012 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 1
A 0 represents p-values < 0.0001.
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0 0 0 0.0035 0.0003 0.0007 0.0076 1 0.0025 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0
