Reducing Approximation Error in the Fourier. Flexible Form

Transcription

1 Reducing Approximation Error in the Fourier Flexible Form Tristan D. Skolrud September 27, 2013 Abstract The Fourier Flexible form introduced by Gallant (1981; 1982) provides a global approximation to an unknown data generating process. In terms of limiting function specification error, this form is preferable to functional forms based on second-order Taylor series expansions, such as the translog or normalized quadratic. The Fourier Flexible form is most commonly represented as a truncated Fourier series expansion appended to a second-order expansion in logarithms. By replacing the logarithmic expansion with a Box-Cox transformation, we show that the Fourier Flexible form can be made more robust to approximation error and allow for nested testing between some of the most commonly implemented functional forms in empirical work. JEL Classification: C51, D24, Q12 Keywords: Fourier Series, Box-Cox, Functional Form Selection, Specification Bias Graduate Research Assistant, School of Economic Sciences, Washington State University, tskolrud@wsu.edu 1

2 1 Introduction Functional form selection can be difficult in applied work, especially in cases where economic theory is not a useful guide. The Fourier flexible form of Gallant (1981; 1982) has been an adequate solution in many cases, as it allows for accurate, global approximations to the underlying theoretical counterpart. However, there are a few notable issues with this functional form. The first is the lack of testing procedures to investigate its fit and performance relative to competing functional forms. The second, and arguably more significant issue, is its potential for approximation error resulting from boundary and edge effects when approximating non-periodic data, as is often the case in economic applications. The purpose of this paper is to develop a new functional form based on the existing Fourier flexible form that will address these issues while maintaining the desirable characteristics of the original. From the early 1970 s, Diewert-flexible forms such as the generalized Leontief, quadratic (normalized, square-rooted, symmetric) and translog have dominated applied parametric analysis. These functional forms have many desirable properties, placing no restrictions on derived measures that are functions of their first and second derivatives (Creel, 1997). However, these forms are based on second-order Taylor series approximations, which are local in nature. As pointed out by White (1980), unless the true function subject to approximation happens to be in the same family as the approximating function, least squares will not consistently estimate the true value of the function in a global sense. This limitation was addressed by Gallant in a series of papers (Gallant, 1981, 1982) in which the Fourier flexible functional form was introduced. Based on the composition of a truncated Fourier series expansion of orthogonal polynomials and a second-order Taylor-series expansion in logarithms, the Fourier flexible form can globally approximate an unknown function as closely as desired in Sobolev norm (Gallant, 1982). The form has seen applications in many areas of economics, including 2

3 agriculture and production; (Chalfant, 1984), (Skolrud and Shumway, 2013), (Ivaldi et al., 1996), banking and stock price variation; (Mitchell and Onvural, 1996), (Huang and Wang, 2004a,b), (Park, 2010), and the measurement of welfare (Creel, 1997). A notable feature of the Fourier flexible form is that it nests the immensely popular translog functional form popularized by Christensen, Jorgenson, and Lau (1975). This has facilitated a large number of comparisons between the two forms due to the relative ease of nested hypothesis testing. Without exception, studies that have made this comparison find overwhelming support for the Fourier flexible form. The resulting temptation is to think of this form as the ultimate panacea for the functional form selection problem. However, as astutely noted by Mitchell and Onvural (1996), use of the Fourier flexible form trades the problem of functional misspecification for the problem of approximation error, implying that when functional misspecification error is small relative to approximation error, the Fourier flexible form may not be the best choice. To date, only one study has attempted to compare the Fourier flexible form to functional forms other than the translog through statistical hypothesis testing (Skolrud and Shumway, 2013). Their approach was based on the Likelihood Dominance Criterion (LDC) of Pollak and Wales (1991); a likelihood ratio test was used to conclude that the Fourier form was preferable to the translog, and then the LDC was used to determine that the translog was preferable to the normalized quadratic. This argument relies heavily on transitivity, and is not a suitable way to proceed in further research. One contribution of this paper will be to overcome this problem, which will allow for more rigorous tests of functional form in the future. The remaining issue with the Fourier flexible form is its susceptibility to approximation error, especially at the boundaries of the data. The approximation error results from two sources: using trigonometric regression terms to estimate a non-periodic func- 3

4 tion; and Gibbs phenomenon, an issue resulting from a Fourier approximation with a finite order of approximation. Due to finite sample sizes, virtually all applications in economics will require a truncated Fourier series, i.e. a finite order of approximation, and most applications employ non-periodic relationships, so these two sources of approximation error will be present most of the time. To solve this problem, thereby reducing the approximation error inherent in the Fourier flexible form, the second-order logarithmic Taylor-series expansion is replaced with a second-order expansion in the Box-Cox polynomial. As evidenced by Eubank and Speckman (1990), including a low-order polynomial in a trigonometric regression (such as a Fourier series approximation) can dramatically improve approximation error. The exact low-order polynomial that minimizes the approximation error is dependent on the data generating process, and simulation results demonstrate that the precise order can vary widely. By using the Box-Cox function in place of the logarithmic, the ideal low-order polynomial that minimizes the approximation error will be revealed by the true data generating process. As the logarithmic function is a special case of the Box-Cox, the original Fourier flexible form will still be nested in the modified version, and given the data generating process, may still prove to be the functional form of choice for a specific application. Simulation evidence demonstrates that the new functional form, referred to as the Box-Cox Fourier functional form, has reduced approximation error relative to the original Fourier flexible form. This results holds for varying degrees of approximation, sample sizes, and data generating processes. The paper is organized as follows: Section 2 introduces the new functional form and provides the background necessary for the estimation procedure, which has appeared more cumbersome than necessary in previous work. Section 3 describes the simulation, Section 4 gives preliminary results, and Section 5 provides concluding remarks. 4

5 2 The Fourier Flexible Cost Function While the Fourier flexible form has been used in many areas of economics, the most common usage is in the estimation of cost functions, a trend we continue in this paper. Let c = c(w, q) be the cost function resulting from the typical cost minimization problem, where w is a vector of p input prices and q is a vector of m output quantities. The Fourier flexible form of the cost function is given by (Gallant, 1982) g K (z θ) = a 0 + b z + 1 A 2 z Cz + (u 0α + [u α cos(k αz + ν α sin(k αz)]) + ε (1) α=1 where z = (w, q ) is an p+m dimensional vector of scaled logarithmic input prices and output quantities, θ = [ a 0, b, vec(c), (u 0α, u α, ν α ) α=1] A is a vector of parameters (with dimensionality to be determined), K is the order of approximation for the truncated Fourier series, k α is an integer-valued p + m dimensional vector, A is the number of k α vectors in the function g K (z θ), and ε is a stochastic disturbance term with mean 0 and variance σ 2. Note that the data vector z must be scaled to fit in the interval [0, 2π] in order to be expanded in the Fourier series approximation. The order of approximation K, the number of k α vectors A, and the composition of the k α vectors (multi-indices) are central to the statistical identification and performance of the Fourier flexible form. 2.1 Choosing the Order of Approximation in the Fourier Flexible Form Choosing the order of approximation K and the resulting parameters A and k α is a non-trivial matter. Due to the semi non-parametric estimation technique, which ties the number of parameters in the model to the sample size, it is usually best to make the determination of K based on the sample size, n. Eastwood and Gallant (1991) develop 5

6 a simple rule that results in asymptotically normal parameter estimates; setting the number of parameters in the model equal to n 2/3. Starting from this point, we need to determine how many of these parameters can be reserved for the Fourier series approximation, and how many are required for the second-order expansion. In the original Fourier flexible form, as we have noted previously, the second-order expansion is the translog function. In order to have a fully-flexible specification, we must reserve (p + m)(p + m + 1)/2 parameters for the translog portion, leaving 2A = n 2/3 (p + m)(p + m + 1)/2 free for the truncated Fourier series. At this juncture, it is necessary to reveal the relationship between the components of the k α vector and the order of approximation, K. Let k αi be the ith element of the k α vector. To be conformable with z, it must be the case that {i = 1,..., p + m}, so each k α vector is of length p + m. For a Kth order approximation, it must be the case that K n+m i=1 for each α = 1,..., A. Each k α vector must follow an additional set of rules for use in Fourier series expansion. First, k α cannot be the zero vector. Second, in order for the cost function to retain homogeneity of degree one in input prices, n i=1 k αi = 0. Third, no k α vector can contain a common integer divisor. So the vector k α1 = [0, 0, 3, 6] would not be a valid choice. Finally, the first non-zero element of each k α must be positive. Only when all of these conditions are satisfied will the Fourier series expansion be a true approximation of order K. The number of vectors that satisfy these conditions (i.e. A) will depend on the order of approximation K, and the sum of input prices and output quantities p + m 1 Note that A changes rapidly as a function of K, n, and m. Consider first the case of one 1 A MATLAB program that produces all possible k vectors for each choice of K, p, and m is available from the author on request. k αi 6

7 input price, one output quantity, and a second order approximation. Only two vectors satisfy the four conditions for k α, i.e. k 1 = 0 1 and k 2 = 0 2 where the first element of each k α vector corresponds to the input price, and the second element corresponds to the output quantity. For a second-order approximation with p = 3 and m = 3, the case considered in the simulation, there are fifteen multiindices, which are listed in full in the Appendix. This implies that A = 15, thus 2A = 30 parameters are required for the truncated Fourier series (15 u α s and 15 ν α s). Notice that there are two competing sources influencing the selection of A. To maintain statistical identification, flexibility in parameter estimates, and asymptotic normality, A should be chosen such that 2A = n 2/3 (p + m)(p + m + 1)/2. However, the choice of A is also driven by the length of the k α vector and the desired degree of approximation. In practice, the decision is often made through pretesting. To ensure that our simulation results are not influenced by an incorrect choice of A, we first pick the order of approximation and the length of k α, and then we set the sample size so that it complies with the two-thirds rule. 2.2 The Box-Cox Fourier Functional Form The fact that z in equation (1) is expressed in logarithms is rooted in convenience. In Gallant s (1981) paper proving that the Fourier flexible form can approximate a function and its first and second derivatives as closely as desired in Sobolev norm, the leading low-order polynomial was immaterial. As Gallant points out, the low-order polynomial is included to alleviate the approximation effects resulting from Gibbs phenomenon, an observation corroborated by Eubank and Speckman (1990). By choosing a logarithmic 7

8 expansion, the Fourier flexible form nests the popular translog form. 2 While this is convenient for easy comparison to the translog, it can be replaced by an alternative low-order polynomial and still maintain the desirable qualities of the original form. Thus, we can write (1) as c (δ) = a 0 + b x (λ) + 1 A 2 x(λ) Cx (λ) + (u 0α + [u α cos(k αx + ν α sin(k αx)]) + ε (2) α=1 where x = (w, q) is a p + m vector of scaled input prices and output quantities, and x (λ) and c (δ) are Box-Cox transformations, defined as x (λ) = [ ] x λ 1 1,..., xλ p+m 1 λ λ and c (δ) = cδ 1 δ and all other variables are as previously defined. Note that as λ, δ 0, x (λ) and c (δ) become log(x) and log(c) respectively. To satisfy the theoretical properties of a cost function, (2) must be linearly homogeneous, convex, and monotonically increasing in input prices, w. At this juncture, convexity and monotonicity are not imposed a priori, but these properties are checked and confirmed after estimation. Linear homogeneity is imposed in two ways; by input price normalization and the multi-index homogeneity restriction, discussed previously. To differentiate between the two functional forms, we will refer to the original form in equation (1) as the Translog Fourier (TLF), and the new form in equation (2) will be referred to as the Box-Cox Fourier (BCF). 2.3 Empirical Implementation One of the advantages of the TLF is that despite its complexity, it is still linear in parameters, so ordinary least squares can be used with no complications. The BCF 2 To see this, set (u α, ν α ) A α=1 = 0 in equation (1). The constant term in the translog becomes a 0 + A α=1 u 0a. 8

9 has two nonlinear parameters, λ and δ. Using maximum likelihood estimation, these parameters (and the remaining parameters) in the model can be recovered with relative ease. First, we collect the right-hand side variables into the matrix X, such that X includes the second-order expansion in x (λ) and the 2A trigonometric terms. Then we can write the BCF more simply as c (δ) = Xθ + ε. If we assume that the error ε is normally distributed, we can write the likelihood function as L(θ, σ 2, λ, δ; c, X) = n ( 1 exp 1 ( c (δ) θx ) ( c (δ) θx ) ) c (δ) 2πσ 2 2σ 2 δ (3) i=1 Thus, the simplified version of the log-likelihood can be written in the following way log L(θ, σ 2, λ, δ; c, X) = n 2 ( log(2π) + log(σ 2 )) 1 ( c (δ) θx ) ( c (δ) θx ) ) 2σ 2 + (δ 1) n log(c i ) (4) Instead of choosing all parameters in the vector θ, λ, and δ simultaneously, we can concentrate out most of them, simplifying the optimization process greatly. Concentrating out a parameter from an objective function involves replacing the parameter with its optimal solution from solving the system of first-order conditions. This process embeds the optimal choice of the parameter in the objective function itself. The first-order conditions from choosing θ and σ 2 to maximize (4) can be solved to yield the usual estimators; ˆθ(λ, δ) = (X X) 1 X c (δ) i=1 and ŝ 2 (λ, δ) = [ ( c (δ) ˆθ(λ, δ)x) ( c (δ) ˆθ(λ, δ)x) ] /n 9

10 Before both θ and σ 2 are concentrated out, we need to concentrate out just σ 2, so that the information matrix can be derived, yielding the standard errors for each of the estimated parameters. Denote by L σ the log-likelihood function with σ 2 concentrated out: L σ = n n 2 (log(2π) log(n) + 1) + (δ 1) log(c i ) i=1 n 2 log ( c (δ) θx ) ( c (δ) θx ) (5) The first-order conditions from the maximization of (5) are as follows: ŝ 2 X (λ) ε = 0 (6) ŝ 2 ε ε λ = 0 (7) ŝ 2 ε ε δ + ι log(c) = 0 (8) where subscripts indicate partial derivatives with respect to the subscripted argument, and ι is a vector of ones. The Hessian matrix, derived from the system of first-order conditions, is given by H = ŝ 2 X X (X ε λ + X λ ε) X ε δ (ε λ X + ε X λ ) ε ε λλ + ε λ ε λ ε ε λδ + ε δ ε λ (X ε δ ) (ε ε λδ + ε δ ε λ) ε ε δδ + ε δ ε δ 2n 1 ŝ 2 (ι log(c)) 2 (9) The inverse of the negative of the Hessian matrix in (9), evaluated at the parameter values that maximize (5), is the estimated covariance matrix of the parameter estimates. As noted in Spitzer (1982), use of ŝ 2 (X X) 1 as the covariance matrix for ˆθ leads to an underestimation of the standard errors, due to the neglected variance from λ and δ. 10

11 The estimation problem can be further simplified by concentrating out θ from the log-likelihood function. Denote this concentrated likelihood function by L θ,σ : L θ,σ = n 2 (log(2π) log(n) + 1) + (δ 1)ι log(c) n ( ) ε 2 log ε n (10) Now the problem has been reduced to a choice of just two parameters, not the entire set of 2A + (p + m)(p + m + 1)/2 parameters. Denote the values of λ and δ that maximize (10) as ˆλ and ˆδ, respectively. Then, the optimal θ is given by ˆθ(ˆλ, ˆδ). Note that this method is equivalent to solving a series of least squares problems for varying values of λ and δ. If the distribution assumption necessary for maximum likelihood is deemed too restrictive for a given application, alternative estimation strategies can be implemented, such as nonlinear least squares. 3 Simulation The simulation is designed to reveal the approximation error inherent in the TLF functional form at the boundaries of the data. The majority of Fourier studies relying on Monte Carlo simulations estimate derived measures like substitution elasticities at the mean of the simulated data, which does not reveal the main source of approximation error. Starting from a known data generating process, both the TLF and the BCF models are used to estimate the true population model. As the coefficient estimates between the two models are not expected to be directly comparable, cost elasticities are estimated for each model, and are compared to the true cost elasticity. Multiple data generating processes are used to ensure robustness. The first data generating process is based on the generalized Box-Cox model: c (δ) = a 0 + b x (λ) x(λ) Cx (λ) + ε (11) 11

12 which is nested in the BCF. The disturbance term in (11) is distributed normally with mean zero and variance one. Multiple models from this family are used as data generating processes, such as the generalized Leontief (δ = 1, λ = 1/2), the normalized quadratic (δ = λ = 1), and the translog (δ, λ 0). An alternative data generating process was chosen that is not nested in either the TLF or the BCF, specifically c (δ) = a 0 + b y y Cy + ε (12) where y is the vector: y = sin(w 1 ) e w 2 w3 3 cos(q 1 ) e q 2 q 4 4 and the disturbance term is identical to the one in (11). In each repetition of the experiment, the cost elasticity with respect to the first output, q 1, defined as ɛ c,q1 c q 1 q 1 c is estimated at mean prices, sorted on the first output, q 1. Second-order Fourier series approximations are used in both the TLF and BCF models, resulting in 30 distinct Fourier parameters (the u α s and ν α s in equations (1) and (2), and 28 parameters from the second-order expansion to be estimated. The BCF model has an additional two parameters, λ and δ, for a total of 58 parameters in the TLF and 60 parameters in the BCF. Using the two-thirds rule from Eastwood 12

13 and Gallant (1991), the sample size required to yield asymptotically normal parameter estimates and still allow for a full second-order approximation is Results For each data generating process, both models fit remarkably well by conventional standards, with a high average R 2 and a high proportion of significant parameters. When measured at the mean, cost elasticity estimates from each form have negligible bias. However, when measured at each observation, the approximation error in the TLF relative to the BCF becomes evident. Figure 1 shows the absolute bias from the true cost elasticity for both the TLF and the BCF when the true data generating process is generalized Leontief. The y-axis is measured in units of elasticity, and the sorted observations are displayed on the y-axis. Notice that the amount of bias in the center of the data is negligible for each form. However, for the smallest and largest observations, the amount of bias in the TLF form is substantial, up to 1.05, which is approximately twenty percent of the true elasticity at that point. For other data generating processes within the generalized Box-Cox family, results are similar to the generalized Leontief case. On average, the bias at the boundaries of the data ranges from 15 to 20 percent, depending on the true functional form. In all cases, the BCF outperforms the TLF. When the alternative data generating process is used, the results are similar. However, as demonstrated in Figure 2, the approximation error is more persistent, remaining larger for a wider range of the data. At the data means, it remains the case that the bias between the TLF and BCF forms is negligible. 13

14 Distance from True Cost Elasticity BCF TLF Figure 1: DGP: Generalized Leontief Distance from True Cost Elasticity BCF TLF Figure 2: DGP: Alternative 14

15 4.1 Hypothesis Testing One of the benefits to using the BCF functional form is the ability to conduct a wide range of nested hypothesis tests. With the TLF, the only flexible functional form that can be tested as an alternative in a nested hypothesis test is the translog. While other, non-nested specification testing procedures have been used (Skolrud and Shumway, 2013), they are not nearly as robust as nested tests. This lack of flexibility in the TLF may be a large contributor to the rejection of the translog specification in favor of the TLF in most empirical studies. In cases where the true data generating process is simple enough that a flexible, second-order approximation leads to a good fit and robust results, the BCF is more likely to identify the true model through hypothesis testing than the TLF. To illustrate this idea, we conduct a series of hypothesis tests when the true data generating process is either generalized Leontief or the previously discussed alternative data generating process. We find that when the true data generating process is generalized Leontief (i.e. generalized Box-Cox with δ = 1 and λ = 0.5), a likelihood ratio test rejects the BCF in favor of the simpler generalized Box-Cox, with a test statistic of 16.75, which is significant at the one percent level. This is of course the expected result due to the construction of the BCF. However, when the TLF model is used, a likelihood ratio test fails to reject the TLF in favor of the simpler translog specification. Thus, when using the TLF, it is entirely possible to reject the translog in favor of the TLF even when a much simpler model will suffice. The possibility of this type of situation occurring with the BCF is much more limited by comparison due to its improved flexibility. When the alternative data generating process is used, likelihood ratio testing indicates a rejection of the TLF in favor of the BCF at the one percent level of significance. This demonstrates an example of a situation where the true data generating process 15

16 is too complicated to be sufficiently modeled by the generalized Box-Cox, but is more accurately represented by the BCF than the TLF due to its superior performance at the edges of the data. 5 Conclusion With simulation evidence, it has been demonstrated that the TLF functional form of Gallant (1981; 1982) suffers from approximation bias at the boundaries of the data. A new functional form has been proposed, the BCF, which modifies the leading secondorder polynomial of the original function proposed by Gallant, which improves approximation error and allows for nested testing of alternative functional forms. These nested testing procedures will allow for a more accurate assessment of the suitability of a larger class of second-order models. The new functional form adds an additional layer of complexity by requiring the estimation of two nonlinear parameters, but a strategy is presented that minimizes the computational burden. The BCF will be especially useful in situations where derived measures at the minimum and maximum levels of the data are of significant importance. For example, consider the case when returns to scale (a function of cost elasticities) are used to predict growth of the largest firms in an industry. In this case, estimates from the TLF could be biased by as much as twenty to thirty percent for the largest firms, resulting in an inaccurate prediction for industry growth. It is not difficult to imagine a case where returns to scale for the largest firms are slightly less than one, indicating decreasing returns to scale, but the TLF incorrectly estimates a number up to thirty percent higher, indicating increasing returns to scale. Using the BCF will be the best way to proceed in such situations. 16

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18 Marc Ivaldi, Norbert Ladoux, Hervé Ossard, and Michel Simioni. Comparing fourier and translog specifications of multiproduct technology: Evidence from an incomplete panel of french farmers. Journal of Applied Econometrics, 11(6): , Karlyn Mitchell and Nur M Onvural. Economies of scale and scope at large commercial banks: Evidence from the fourier flexible functional form. Journal of Money, Credit and Banking, 28(2): , Cheolbeom Park. How does changing age distribution impact stock prices? a nonparametric approach. Journal of Applied Econometrics, 25(7): , Robert A Pollak and Terence J Wales. The likelihood dominance criterion: a new approach to model selection. Journal of Econometrics, 47(2): , Tristan D Skolrud and C Richard Shumway. A fourier analysis of the us dairy industry. Applied Economics, 45(14): , doi: / John J Spitzer. A primer on box-cox estimation. The Review of Economics and Statistics, 64(2): , Halbert White. Using least squares to approximate unknown regression functions. International Economic Review, 21(1): ,

19 6 Appendix 6.1 Multi-Indices For a second-order Fourier series approximation with three input prices and three output quantities, the set of permissible multi-indices are