The Limits of Mathematics and NP Estimation in ... - Chichilnisky

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Using the SUR Model of Tourism Demand for Neighbouring Regions in Sweden and Norway 105Y X e2 2 2 2Y X em m m mand then combined into a larger model written as:Y1 X1 0 0 0 1 e1 Y2 0 X2 0 0 2 e2 ... 0 0 ... 0 ... ... Y m 0 0 0 X m m e mThis model can be rewritten compactly as:(8)YX Be(9)where Y and e are of dimension (TM 1), X is of dimension (TM n), n =dimension (K 1).At this stage, I make the following assumptions:a. X i is fixed with rank n i .M nii1, and B is ofb. P lim1 ('XX i i )T Q ii is nonsingular with finite and fixed elements, i.e. invertible.c. addition, I assume that Plimd.1 ('XX i j ) Q ij is also nonsingular with finite and fixedTelements.,Eee ( i i) ijIT, where ij designates the covariance between the i th and j th equations foreach observation in the sample.The above expression can be written as: 11 12 1MEe () 0and ,21 22 2MEee ( ) IT, where is an M M positiveM1 M2 MMdefinite symmetric matrix and is the Kronecker product. Thus, the errors at eachequation are assumed homoscedastic and not autocorrelated, but there is contemporaneouscorrelation between corresponding errors in different equations.The OLS estimator of B in (9) is:with the variance 1ˆOLS XX XY Var( ˆ ) X X X X( X X)OLS 1 1 The SUR Generalized Least Squares (GLS) estimator of B is given by:
106Advances in Econometrics - Theory and Applicationsand the variance is given by: 1ˆ 1 1GLS ( T ) ( T )X I X X I Y ˆ ( 1) 1GLS TV X I XHowever, the system of the five equations for Sweden and Norway are as follows:Y it= i + S i + XitB i+ Y it-q iq + e ,it i 1, 2, 5 q 1, 2, 12(10)where Y it is a T 1 vector of observations on the dependent variable, e it is a T 1 vector ofrandom errors with E(e t ) = 0 , and S i are monthly dummy variables that take valuesbetween 1 and 11 (the twelfth month is the base). X it is a T nimatrix of observations on n inonstochastic explanatory variables, and B i is an ni 1 dimensional vector of unknownlocation parameters. T is the number of observations per equation, and n i is the number ofrows in the vector B i . iq is a parameter vector associated with the lagged dependentvariable for the respective equation.The dependent variables Y i are the natural logarithms of the number of monthly visitorsfrom Denmark, the UK, Switzerland, Japan, and the US to either Sweden or Norway. Thematrix X i is the natural logarithm of three vectors that contains monthly information aboutthe CPI in Sweden (or Norway), the exchange rate (Ex) in Sweden (or Norway), and relativeprice (Rp) for Sweden (or Norway) with respect to each of the abovementioned countries.Another objective of this study is to test for the existence of any contemporaneouscorrelation between the equations. If such correlation exists and is statistically significant,then least squares applied separately to each equation are not efficient and there is need toemploy another estimation method that is more efficient.The SUR estimators utilize the information present in the cross regression (or equations)error correlation. In this chapter, we estimated the model in Equation (10) by using the OLSmethod for each equation separately to achieve the best specification of each equation. Wethen estimate the whole system using ISUR, see Tables 2 and 3. The ISUR techniqueprovides parameter estimates that converge to unique maximum likelihood parameterestimates and take into account any possible contemporaneous correlation between theequations.To test whether the estimated correlation between these equations is statistically significant,we apply Breusch and Pagan’s (1980) LM statistic. If we denote the covariances between thedifferent equations as 12 , 13 … 45 , the null hypothesis is:H 0 : 12 = 13 . . . = 45 = 0, against the alternative hypothesis,H 1 : at least one covariance is nonzero.In our three equations, the test statistic is: = N(r 2 12 + r 2 13 +…r 2 45), where r 2 ij is the squared correlation,r 2 ij = 2 ij / ii jj .Under H 0 , has an asymptotic 2 distribution with five degrees of freedom. I may reject H 0for a value of greater than the critical value from a 2 (45) distribution (i.e. with 45 degrees offreedom) for a specified significance level. In this study, the calculated 2 value for Sweden
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The Limits of Mathematics and NP Estimation in ... - Chichilnisky

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