An Improved Specification Test for AR(1) versus MA(1) Disturbances in Linear Regression Models

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1 An Improved Specification Test for AR(1) versus MA(1) Disturbances in Linear Regression Models Pierre Nguimkeu Georgia State University Abstract This paper proposes an improved likelihood-based method to test the hypothesis that the disturbances of a linear regression model are generated by a first-order autoregressive process against the alternative that they follow a first-order moving average scheme. Compared with existing tests which usually rely on the asymptotic properties of the estimators, the proposed method has remarkable accuracy, particularly in small samples. Simulations studies are provided to show the superior accuracy of the method compared to the traditional tests. An empirical example using Canada real interest rate data is also provided to illustrate the implementation of the proposed method in practice. Keywords: Autoregressive errors; Moving average errors; Likelihood Analysis; p-value JEL Classification: C22; C52 1 Introduction Consider the linear regression model y t = x tβ + u t, t = 1, 2,..., n (1) where y t is the dependent variable, x t is a k dimensional vector of linearly independent regressors, β is an unknown k-dimensional parameter vector, and u t is the error term. A common practice among applied researchers is to test for serial correlation in the disturbance term u t in the above model. However, it is well known that in practice the same Lagrange Multiplier test statistic is used for testing the null hypothesis of zero first-order serial correlation whether the alternative is AR( 1) or MA(1) (see Breusch and Godfrey 1981). It is therefore important for the practitioner to be able to distinguish which process the disturbances of the model actually follow. Suppose we are interested in testing the null hypothesis that the components of u are generated by the stationary AR (1) process H 0 : u t = ρu t 1 + ɛ t, ρ < 1, Department of Economics, Georgia State University, PO Box 3992, Atlanta GA , USA; nnguimkeu@gsu.edu, Phone: 1(404) ; Fax: 1(404)

2 against the alternative hypothesis that the components of u are generated by the MA (1) process H 1 : u t = ɛ t + γɛ t 1, γ < 1, where it is assumed that ɛ t NID(0, σ 2 ) in both cases. The two hypothesis H 0 and H 1 are clearly non-nested in the sense of Cox (1961, 1962). Denote by ( ˆβ, ˆρ, ˆσ 2 ) the maximum likelihood estimates of Model (1) under H 0 with the associated residuals û t = y t x ˆβ, t the associated predictions ŷ t = x ˆβ t + ˆρû t 1 and prediction errors ˆɛ t = y t ŷ t, and by ( β, γ, σ 2 ) the maximum likelihood estimates of Model (1) under H 1 with the associated residuals ũ t = y t x β, t the associated predictions ỹ t = x β t + γ ɛ t 1 and prediction errors ɛ t = y t ỹ t. The problem of testing AR(1) against MA(1) in the present context has been considered in the literature; see King (1983), King and McAleer (1987), Burke, Godfrey, and Termayne (1990), Baltagi and Li (1995), McKenzie, MacLeer and Gill (1999). However, these methods have the common feature that they are only asymptotically valid and often not suitable for models with small samples. They can therefore be misleading in instances where only few observations are available, with adverse consequences on forecasting. This paper proposes an improved procedure for testing AR( 1) against MA(1) in linear regression using small-sample likelihood asymptotic inference methods. These methods were developed by Fraser and Reid (1995) and have been proven to possess high accuracy compared to the traditional asymptotic methods, and are therefore preferable in small samples. Theoretically, the proposed method is a third-order inference procedure which means that the rate of convergence is O(n 3/2 ), whereas the commonly used methods converge at rate O(n 1/2 ). Section 2 presents the third-order method for obtaining p-value functions for any scalar parameter of interest from a general model estimated by maximum likelihood. In Section 3, we show how the method can be applied to test for AR( 1) against MA(1) disturbances in the linear regression model. Numerical studies including Monte Carlo simulations and an empirical example are provided in Section 4. Some concluding remarks are given in Section 5. 2 Overview of the the third-order likelihood based approach Suppose we have a parametric model with log-likelihood function l(θ) where the parameter vector θ can be written as θ = (ψ, λ). Let ψ = ψ(θ) be the scalar parameter of interest and λ = λ(θ) the vector of nuisance parameters. To test for a particular fixed value of ψ = ψ(θ) one can use the log-likelihood function to derive the signed log-likelihood ratio test statistic (r) as follows 2

3 r = r(ψ) = sgn( ˆψ ψ)[2{l(ˆθ) l(ˆθ ψ )}] 1/2. (2) where ˆθ is the overall maximum likelihood estimate of θ, ˆψ = ψ(ˆθ) and ˆθ ψ is the constrained maximum likelihood estimate of θ at a given ψ. Denote by ĵ θθ (ˆθ) and ĵ λλ (ˆθ ψ ) the observed information matrix evaluated at ˆθ and observed nuisance information matrix evaluated at ˆθ ψ, respectively. The statistic given in (2) has the standard normal limiting distribution with an O(n 1/2 ) rate of convergence. This approximation is thus accordingly known as first-order approximation. Tail probabilities for testing a particular value of ψ can be approximated using this statistic with the cumulative standard normal distribution function Φ( ), i.e. Φ(r). The accuracy of this test and the other existing methods mentioned above suffer from the typical drawbacks of requiring a large sample size and an original distribution that is close to normal. However, third-order tail probability approximations for testing a particular value of ψ can be derived using the Barndorff-Nielsen (1991) saddlepoint approach defined by ( p(ψ) = Φ(r (ψ)) = Φ r 1 ( )) r r log, (3) q where the statistic r = r(ψ) is the signed log-likelihood ratio test statistic given in (2), and q = q(ψ) is a modified maximum likelihood departure given in Equation (5) below, derived from the following steps: 1. Define a vector of ancillary directions v: v = ( ) 1 ( ) z(x; θ) z(x; θ) x θ, ˆθ where the variable z(x; θ) represents a pivotal quantity of the model whose distribution is independent of θ. 2. Use the ancillary directions to calculate a locally defined canonical parameter, ϕ: [ ] l(θ) ϕ (θ) = v. x 3. Given this canonical reparameterization, define a new parameter χ: where ψ ϕ (θ) = ψ(θ)/ ϕ = ( ψ(θ)/ θ )( ϕ(θ)/ θ ) 1. χ(θ) = ψ ϕ (ˆθ ψ ) ϕ(θ), (4) ψ ϕ (ˆθ ψ ) The modified maximum likelihood departure is then constructed in the ϕ space. The expression for q 3

4 is given by { } 1/2 q = q(ψ) = sgn( ˆψ ĵ ϕϕ (ˆθ) ψ) χ(ˆθ) χ(ˆθ ψ ), (5) ĵ (λλ )(ˆθ ψ ) where ĵ ϕϕ and ĵ (λλ ) are the observed information matrix evaluated at ˆθ and observed nuisance information matrix evaluated at ˆθ ψ, respectively, calculated in terms of the new ϕ(θ) reparameterization. In fact, the determinants in Equation (5) can be computed as follows: ĵ ϕϕ (ˆθ) = ĵ θθ (ˆθ) ϕ θ (ˆθ) 2 and ĵ (λλ )(ˆθ ψ ) = ĵ λλ (ˆθ ψ ) ϕ λ(ˆθ ψ )ϕ λ (ˆθ ψ ) 1. The statistic r (ψ) = r(ψ) 1 ( ) r(ψ) r(ψ) log defined in Equation (3) is due to Barndorff-Nielsen (1991) q(ψ) and is known as the modified signed log-likelihood ratio statistic. The statistic r is the signed log-likelihood ratio statistic defined in (2) and the statistic q is a standardized maximum likelihood departure whose expression depends on the type of information available. Fraser and Reid (1995) showed that the approximation given in (3) has an O(n 3/2 ) rate of convergence and is thus referred to as a third-order approximation. It thus provides a more accurate way to perform inference in the presence of small samples. The following section shows how this procedure can be used to obtained an improved test of AR(1) against MA(1) in the linear regression model disturbances. 3 The test procedure The method proposed uses an approach similar to Davidson and MacKinnon (1981). From the linear model (1) and hypotheses H 0 and H 1, a comprehensive artificial auxiliary model can be specified as follows y t = x t β + (ρ ˆρ)û t 1 + ψ(ŷ t ỹ t ) + ɛ t, (6) where y t = y t ˆρy t 1, x t = x t ˆρx t 1 and ψ is a scalar parameter. When ψ = 0, the above equation corresponds to Model (1) with AR(1) errors and when ψ = 1 it corresponds to Model (1) with MA(1) errors. The proposed procedure therefore consists in testing the significance of the maximum likelihood estimate of the scalar parameter ψ in the auxiliary regression (6) using the third-order approach described in Section 2. Applying the usual t-test statistic to test the significance of ψ in this regression yields the P -statistic proposed by Davidson and MacKinnon (1981), which is asymptotically distributed as N(0, 1) with a O(n 1/2 ) distributional accuracy. The P -statistic for testing ψ = 0 is the same as the difference of prediction error test (DOP1) of MacKenzie et al.(1999) and is also asymptotically equivalent to the Hatanaka s (1974) two steps estimator for Model (1) under H 0. These test statistics rely on asymptotic properties of the estimators and are therefore less accurate in the presence of small samples. In contrast, the proposed procedure has a O(n 3/2 ) distributional accuracy and is thus more appropriate to statistically distinguish between the AR(1) 4

5 and the MA(1) error structures in small samples regression models. Let θ = (ρ, β, σ 2, ψ) be the parameter vector in the auxiliary regression (6). Our parameter of interest in this auxiliary regression is ψ = ψ(θ), whereas λ = λ(θ) = (ρ, β, σ 2 ) is the vector of nuisance parameters. Following the above authors, we can claim that the MA(1) structure of the disturbances in the linear regression (1) is rejected against the AR(1) structure if we fail to reject the hypothesis H 0 : ψ = 0. For the clarity of the exposition let adopt the following notations: y = (y 1,..., y n), X = (x 1,..., x n), û 1 = (û 0,..., û n 1 ), ˇy = (ŷ 1 ỹ 1,..., ŷ n ỹ n ) and x = (y ; X ; û 1 ; ˇy). Equation (6) then takes the matrix form y = X β + (ρ ˆρ)û 1 + ψˇy + ɛ (7) The log-likelihood function of the auxiliary model (6) or (7) is then defined by l(θ; x) = n 2 log 2π n 2 log σ2 1 2σ 2 ( y X β (ρ ˆρ)û 1 ψˇy ) ( y X β (ρ ˆρ)û 1 ψˇy ) Taking the first-order derivatives of the log-likelihood function with respect to the parameter vector gives: l ρ = 1 ( σ 2 û 1 y X β (ρ ˆρ)û 1 ψˇy ) l β = 1 σ ˆX ( 2 y X β (ρ ˆρ)û 1 ψˇy ) l σ 2 = n 2σ σ 4 ( y X β (ρ ˆρ)û 1 ψˇy ) ( y X β (ρ ˆρ)û 1 ψˇy ) l ψ = 1 σ 2 ˇy ( y X β (ρ ˆρ)û 1 ψˇy ) (8) The overall maximum likelihood estimates (MLE) ˆθ ML of θ is derived by solving the first-order conditions obtained by equating these first-order derivatives with zero. The overall MLE, ˆθ ML = (ˆρ ML, ˆβ ML, ˆσ 2 ML, ˆψ ML ), is given by ˆρ ML = ˆρ + û 1M ˇy ˇy My û 1My ˇy M ˇy (û 1 M ˇy)2 û 1 Mû 1 ˇy M ˇy ˆψ ML = û 1My û 1M ˇy û 1Mû 1 ˇy My (û 1 M ˇy)2 û 1 Mû 1 ˇy M ˇy ˆβ ML = (X X ) 1 X ( y (ˆρ ML ˆρ)û 1 ˆψ MLˇy ) (9) ˆσ 2 ML = 1 n( y (ˆρ ML ˆρ)û 1 ˆψ MLˇy ) M ( y (ˆρ ML ˆρ)û 1 ˆψ MLˇy ), where M is the n n projection matrix defined by M = I X (X X ) 1 X. The constrained maximum likelihood estimator (cml), ˆθ cml = (ˆρ cml, ˆβ cml, ˆσ 2 cml) obtained by solving the first-order conditions at a fixed ψ(θ) = ψ is given by 5

6 ˆρ cml = ˆρ + û 1M ( y ψˇy ) û 1 Mû 1 ˆβ cml = (X X ) 1 X ( y (ˆρ cml ˆρ)û 1 ψˇy ) (10) ˆσ cml 2 = 1 ( y (ˆρ cml ˆρ)û 1 ψˇy ) ( M y (ˆρ cml ˆρ)û 1 ψˇy ) n As defined in Section 2 the statistic r(ψ) = r(ψ) can be obtained from (2) as r = r(ψ) = sgn( ˆψ ML ψ)[2{l(ˆθ ML ) l(ˆθ cml )}] 1/2. To construct the statistic q, we define the following pivotal quantity z(θ, y) = y X β (ρ ˆρ)û 1 ψˇy σ The canonical parameters are then given by ϕ(θ) = ( ϕ 1 (θ), ϕ 2 (θ), ϕ 3 (θ), ϕ 4 (θ) ) with ϕ 1 (θ) = 1 σ 2 ( y X β (ρ ˆρ)û 1 ψˇy) û 1 ϕ 2 (θ) = 1 ( y σ 2 X β (ρ ˆρ)û 1 ψˇy ) X 1 ( ϕ 3 (θ) = 2σ 2ˆσ y ML 2 X β (ρ ˆρ)û 1 ψˇy ) ( y X ˆβML (ˆρ ML ˆρ)û 1 ˆψ ) MLˇy (11) ϕ 4 (θ) = 1 σ 2 ( y X β (ρ ˆρ)û 1 ψˇy ) ˇy The first-order derivative of the canonical parameter with respect to θ can now be obtained by ϕ 1 (θ)/ ρ ϕ 1 (θ)/ β ϕ 1 (θ)/ σ 2 ϕ 1 (θ)/ ψ ϕ ϕ θ (θ) = 2 (θ)/ ρ ϕ 2 (θ)/ β ϕ 2 (θ)/ σ 2 ϕ 2 (θ)/ ψ ϕ 3 (θ)/ ρ ϕ 3 (θ)/ β ϕ 3 (θ)/ σ 2 ϕ 3 (θ)/ ψ ϕ 4 (θ)/ ρ ϕ 4 (θ)/ β ϕ 4 (θ)/ σ 2 ϕ 4 (θ)/ ψ (12) The derivative of the canonical parameter with respect to the nuisance parameter vector, denoted ϕ λ (θ), is obtained by deleting the last column of the matrix defined in (12). Moreover, the vector ψ ϕ (θ) featuring in Equation (4) is obtained by taking the last row of the matrix ϕ 1 θ (θ). Finally, the information matrix j θθ and the nuisance information matrix j λλ are defined by j θθ (θ) = l ρρ l ρβ l ρσ 2 l ρψ l βρ l ββ l βσ 2 l βψ l σ 2 ρ l σ 2 β l σ 2 σ 2 l σ 2 ψ l ψρ l ψβ l ψσ 2 l ψψ and j λλ (θ) = l ρρ l ρβ l ρσ 2 l βρ l ββ l βσ 2 l σ2 ρ l σ2 β l σ2 σ 2, where the elements of these matrices are given by the following equations 6

7 l ρρ = 1 σ 2 û 1û 1 ; l ρβ = 1 σ 2 X û 1 ; l ρσ 2 = 1 σ 4 û 1( y X β (ρ ˆρ)û 1 ψˇy ) ; l ρψ = 1 σ 2 ˇy û 1 ; l ββ = 1 σ 2 X X ; l βσ 2 = 1 σ 4 ˆX ( y X β (ρ ˆρ)û 1 ψˇy ) l βψ = 1 σ 2 ˇy X ; l σ2 σ 2 = n 2σ 4 1 σ 6 ( y X β (ρ ˆρ)û 1 ψˇy ) ( y X β (ρ ˆρ)û 1 ψˇy ) ; l σ2 ψ = 1 σ 4 ( y X β (ρ ˆρ)û 1 ψˇy ) ˇy; lψψ = 1 σ 2 ˇy ˇy. It can be seen from the first-order conditions (8) that the mean and the variance parameters are observedorthogonal so that the observed constrained information matrix j λλ (θ) can be written in a simple blockdiagonal form. This is computationally more convenient than using a full matrix expression. This also yields a simple and easy-to-compute expression for the observed information matrix j θθ (θ). Hence the statistic q(ψ) can be obtained from (5) and the test for ψ can be approximated with third-order accuracy using the p value function of the modified signed likelihood ratio test statistic r (ψ) given in (3). (13) 4 Numerical Studies In this section, we provide both a Monte Carlo simulation study to gain a practical understanding of the performance of our testing procedure and results of an empirical example using real data. The focus of the simulation is to compare the results from the proposed method based on the Barndorff-Nielsen thirdorder approximation (BN) with existing tests. For comparison, we consider the Lagrange Multiplier statistic (LM) for testing AR(1) (or MA(1) ) against the ARMA(1,1) alternative (see King and MacLeer 1987), the difference of prediction error test (DOP) of MacKenzie et al.(1999), the τ test of Burke and al. (1990), and the point optimal invariant test (POI) of King and MacLeer (1987). 1 The accuracy of the different methods are evaluated by computing their empirical sizes and powers estimated by the rejection frequencies obtained under H 0 and H 1, respectively. 4.1 Monte Carlo simulation results The setup of the Monte Carlo simulation is similar to the one considered by King and MacLeer (1987), Godfrey and Tremayne (1988), Burke et al.(1990), and McKenzie et al.(1999). The data generating process is given by y t = β 0 + β 1 x 1t + β 2 x 2t + u t, t = 1,..., n (14) u t = ρu t 1 + ɛ t + γɛ t 1 ɛ t NID(0, σ 2 ) (15) The design matrix of regressors is given in Table 1. The explanatory variables x 1t and x 2t are the log of a real income measure and the log of a relative price index that refer to the UK for the period (see 1 The DOP test is equivalent to the P test of Davidson and MacKinnon (1981). 7

8 Table 1: Design Matrix for Simulation study t x 1 x 2 t x 1 x 2 t x 1 x Durbin and Watson 1951). The true values of the coefficients at set to β 0 = β 1 = β 2 = 0, and σ 2 = 1 when generating data from the linear regression model (14) under H 0 and H 1. We consider various values for the autocorrelation coefficient ρ {0.9; 0.6; 0.3; 0.0} and for the moving average coefficient γ {0.0; 0.2; 0.5; 0.7}. The nominal size is set to 5% and rejection frequencies are calculated using 10, 000 replications and sample sizes are set at n = 15, n = 30 and n = 60. Both the AR(1) and MA(1) models are estimated by Maximum likelihood estimation. Rejection frequencies for the AR(1) model as the null are obtained for the combination of sample sizes and coefficient values, and presented in Table 2. Simulations from different values of ρ with γ fixed at γ = 0 provide estimates of the empirical sizes of the tests, whereas simulations from different values of γ with ρ fixed at ρ = 0 provide estimates of the empirical powers of the tests under false models. Figure 1 gives a graphical illustration of the behavior of the sizes of the tests with respect to the sample size and the different values of ρ for the AR(1) model as the null. An examination of this figure indicates that the proposed test, BN, has very accurate sizes for each value of the autocorrelation coefficient ρ at each sample size including very small sample sizes like n = 15. On the other hand, the other tests tend to under reject. In particular, while the τ test of Burke et al. (1990) performs the worst when the sample gets smaller, the DOP test performs the worst for larger samples. Moreover, it can be seen from Figure 1 that the proposed method, BN, gives results that are stable around the nominal size while the other methods 8

9 Table 2: Results for Simulation Study Estimated size and power functions for testing H 0 : u t = ρu t 1 + ɛ t against H 1 : u t = ɛ t + γɛ t 1, at 5% significance Sample size Parameters DOP LM P OI τ BN n =15 ρ = γ = n = 30 ρ = γ = n = 60 ρ = γ = are unstable and less satisfactory especially as the values of ρ get closer to 1. Figure 2 gives a graphical illustration of the behavior of the power of the tests with respect to the sample size and the different values of γ when the AR(1) model is taken as the null. Increasing values of γ imply stronger misspecification. Thus when γ takes values 0.2, 0.5, 0.7 and 0.9, the rejection frequencies are expected to be increasingly higher. The results show that while the proposed method, BN, has reasonable power for larger samples, it outperforms all the other tests as the sample gets smaller. The DOP test performs the worst: although it has reasonable power for larger samples, it has no power for smaller samples. In general, the power of all the tests considered tend to be equivalently higher for larger samples. The difference in their performance and thus the superior accuracy of the proposed test is, as expected, noticeable only when the sample is relatively small. 9

10 The simulations results clearly confirm that for small samples, the proposed method, BN, performs the best and should be preferred to the existing methods. Figure 1: Empirical sizes of the tests Estimated size functions for testing H 0 : u t = ρu t 1 + ɛ t against H 1 : u t = ɛ t + γɛ t 1, at 5% significance level n = 15 n = 30 n = DOP LM POI TAU BN ! DOP LM POI TAU BN ! DOP LM POI TAU BN ! Figure 2: Empirical powers of the tests Estimated power functions for testing H 0 : u t = ρu t 1 + ɛ t against H 1 : u t = ɛ t + γɛ t 1, at 5% significance level DOP LM POI TAU BN n= DOP LM POI TAU BN n= Student Version of MATLAB DOP LM POI TAU BN n= ! ! ! 4.2 Empirical Example: A model for Canadian interest rate In this section, an application of the third-order method is illustrated with a simple linear model for Canadian real interest rates. Consider the quarterly regression model r t = β 0 + β 1 S 1t + β 2 S 2t + β 3 S 3t + u t, (16) 10 Student Version of MATLAB

11 where denotes the first difference, and r t is the ex-post real interest rate, defined by r t = R t Π t, the difference between the nominal interest rate R t measured by the 90-day bank accepted treasury bill rate and the annual inflation rate Π t, measured by the annual percentage change in the consumer price index. The variables S 1t, S 2t and S 3t are quarterly seasonal dummy variables. This is a random walk model with seasonal drift as postulated by Kinal and Lahiri (1988) to estimate expected real interest rates and forecast inflation rate in the US over the period Estimation of this model by ordinary least squares regression using quarterly Canadian data for the period yields the following results: ˆr t = S 1t S 2t S 3t (0.369) (0.362) (0.361) (0.370) R 2 = d = The DW statistic, d = 2.358, suggest the presence of negative serial correlation among the residuals. Although Kinal and Lahiri (1988) provide an economic argument that the error terms should follow a MA(1) process, one may assume as is common in practice that it could also be an AR(1) process. Hence our procedure consists in testing AR(1) against a MA(1) alternative. Table 3: Results for the Maximum Likelihood estimation of model (16) with serially correlated errors Under H 0 : u t = ρu t 1 + ɛ t Under H 1 : u t = ɛ t + γɛ t 1 Parameters Estimates Standard Error Estimates Standard Error β β β β ρ n.a. n.a. γ n.a. n.a σ Our testing procedure requires to compute the Maximum Likelihood Estimators of the model assuming both AR(1) errors and MA(1) errors separately. The results of the estimation are given in Table 3. The p-value of the proposed test statistic is then calculated at which is lower than the significance level of 5%, so that we reject H 0 in favour of H 1. Except for the DOP, all the other tests also reject H 0 in favour of H 1. Our test results therefore suggest that the disturbances terms in the Canadian interest rate model specified in (16) are closer to an MA(1) process with a moving average coefficient estimated at

12 5 Concluding remarks This paper proposes a procedure to select between AR(1) and MA(1) error structure in linear regression models. The proposed procedure is based on recent likelihood-based inference theory that is known to deliver third-order accuracy, and is therefore appropriate for small sample models. A Monte Carlo experiment is carried out to compare the performance of the proposed third-order method with several existing ones. The results show that while the proposed method produces competitive power performance compared to the other methods, its size clearly outperforms existing methods in smaller samples. An illustration is provided by testing an empirical model of the Canadian interest rate using quarterly data over the period References [1] Baltagi, B.H. and Li, Q., 1995, Testing AR(1) against MA(1) disturbances in an error component model. Journal of Econometrics, 68, [2] Barndorff-Nielsen, O., 1991, Modified Signed Log-Likelihood Ratio, Biometrika 78, [3] Breusch, T.S. and L.G. Godfrey, 1981, A review of recent work on testing for autocorrelation in dynamic simultaneous models, in: D.A. Currie, R. Nobay, and D. Peel, eds., Macroeconomic analysis: Essays in macroeconomics and economics (Croom Helm, London). [4] Burke, S.P.. L.G. Godfrey, and A.R. Termayne. 1990, Testing AR(l) against MA( 1) disturbances in the linear regression model: An alternative procedure, Review of Economic Studies 57, [5] Davidson, R. and J. MacKinnon, 1981, Several tests for model specification in the presence of alternative hypotheses, Econometrica, 49(3), [6] Fraser, D., Reid, N., 1995, Ancillaries and Third Order Significance, Utilitas Mathematica 47, [7] Hatanaka, M., 1974, A dynamic two step estimator for the dynamic adjustment model with autoregressive errors. Journal of Econometrics, 2(3), [8] Kinal, T., and Lahiri, K., 1988, A Model for Ex Ante Real Interest Rates and Derived Inflation Forecasts, Journal of the American Statistical Association, 83, [9] King, M.L., 1983, Testing for autoregressive against moving average errors in the linear regression model, Journal of Econometrics 21, [10] King, M.L. and M. McAleer, 1987, Further results on testing AR(l) against MA(l) disturbances in the linear regression model, Review of Economic Studies 54, [11] Mckenzie, C.R., McAleer, M. and Gill, L., 1999, Simple procedures for testing autorregressive versus moving average errors in regression models. Japanese Economic Review, 50(3),