ON EFFICIENT FORECASTING IN LINEAR REGRESSION MODELS

Transcription

1 Journal of Quantitative Economics, Vol. 13, No.2 (July ON EFFICIENT FORECASTING IN LINEAR REGRESSION MODELS SHALABH Department of Statistics, University of Jammu, Jammu , India In this article, we present efficient forecasts for some future values of study variable in a linear regression model and analyze its performance with respect to forecasts derived from least squares and Stain-rule procedures. 1. INTRODUCTION Forecasting the future values of study variable corresponding to a given set of values of explanatory variables is an important aspect of regression analysis. Traditionally these forecasts are constructed from the least squares estimation of parameters employing the available observations. Such forecasts are linear and unbiased for both the actual and average values of the study variable. However, they may often be far less efficient with respect to the criterion of variability in comparison to some non-linear and biased forecasts such as those based on Stain-rule estimation of parameters. This articles presents a biased and nonlinear family of forecasts and analyses its performance properties. The organisation of our presentation it as follows. In Section 2, we specify the model and present the forecasts arising form least squares and Stain-rule estimation of regression coefficients. We also propose an alternative forecasting procedure. This provides a kind of extension of forecasts based on Stain-rule method. Taking the objective as forecasting of a weighted combination of the actual and average values of study variable, we analyze the performance properties of three forecasts in Section 3. All the three forecasts are found to share the same asymptotic properties. We have therefore considered higher order approximations employing the small disturbances asymptotic theory. First the bias vectors are examined and then the average forecasts risks are studied. Sufficient conditions for the superiority of one forecast over the other are derived in each case. The conditions are easy to verify in practice and may possibly help in making a suitable choice of forecasting procedures. 2. MODEL SPECIFICATIONAND FORECASTS

2 134 JOURNAL OF QUANTITATIVE ECONOMICS Let us consider the following linear regression model (2.1 Y = x(3 + (1 U where Y is a n x 1 vector of n observations on the study variable, X is a n x p full column rank matrix of n observations on p explanatory variables, {3 is a p x 1 vector of unknown regression coefficients,(1 is an unknown positive scalar and U is n x 1 vector of disturbances assumed to be independently, identically and normally distributed with mean zero and variance one. In addition to the n observation, we have another column vector Yf of nf unobserved of future values of study variable and a n f x p matrix xf p explanatory variables. Thus we can write (2.2 Yf = Xf{3 + (1 Uf of nf prespecified values of where Ut is a nf x 1 vector of disturbances possessing the same distributional properties as the elements of u. The least squares estimator of (3 in (2.1 is given by (2.3 b = (X' X-1 X'Y which is linear and unbiased. The Stain-rule estimator of {3 characterized by a positive and nonstochastic scalar k is defined by (2.4 b = [ 1-2k Y' (1- H Y ] b' H = X ( X' X -1 X' S (n - p + 2 Y' HY, which is neither linear nor unbiased: see, s.g., Judge and Bock (1978. Based on (2.3 and (2.4, the forecast vectors for the nf values of study variable are formulated as follows: (2.5 2k Y' (I - H Y (2.6 Fs = Xf bs = [1 - (n - p + 2 Y' H Y ] X f' b The vectors F and Fs can be used for forecasting the vector of actual values Yf as well as the average values E ( Yf = Xf p; see, e.g., Srivastavaand Shalabh (1996 and Trenkler and Toutenburg (1992. In order to provide a unified treatment, Shalabh (1995 has considered the following function: (2.7 with A as a nonstochastic scalar between a and 1, and has analyzed the performance

3 ON EFFICIENT FORECASTING IN LINEAR REGRESSION MODELS 135 properties of F and FS as forecasts for T. Notice that T reduces to the vector of average values of study variable when A. = 0 while it becomes the vector of actual values when A. = 1. If we consider gf as the forecast vector for T and choose the scalar such that the quantity (2.8 E (ef - T' (ef - T - e2 tr ( X'X -1 X' X + ( 1-e 2 f3 ' X' X f3+a.2n - ff a ff f is minimum, this yields the optimum value of g as ~ tr (X' X :-1X' X (2.9 = eo (say. It is easy to see that e=1- f f f3' X/ Xf f3 + a 2 tr (X' X -1 X I f X f (2.10 E [Y' (J - H Y ] = (n - p a 2 (2.11 E [y, X (X I X -1 X 'f Xf ( X' X -1 X' Y ] = f3 ' X 'f X f f3+ a 2 tr (X' X -1 X' f X f so that eo can be consistently estimated by '" Y' (1- H Y tr (X' X -1 X' X (2.12 e = 1 - f f o (n - p Y' X (X' X -1 X'f X f (X I X -1 X' Y We thus find eo F as a forecast vector for T. Stemming from it in the spirit of Stain-rule estimation we can define the following family of forecasts: (2.13 F*= [ where 1-2gY'(I-HY Xb (n-p+2 Y'H* yj f (2.14 H * = X (X' X -1 X'f Xf (X' X -1 X' and 9 is any positive nonstochastic scalar characterizing the forecasts. 3. EFFICIENCY PROPERTIES It is easy to see that the forecast vector F is, while Fsand unbiasedin the sensethat E (F - T = 0 F * are not, weakly

4 136 JOURNAL OF QUANTITATIVE ECONOMICS (3.1 E(FS-T=Jt:O E(F*-T=Jt:O. A Next, we consider the average forecast risk associated with any forecast vector T: A A A R(T=E(T-T' (T-T which means that all the nf values are errors are assigned identical weightage. It can be easily seen that forecasted collectively and all the forecasting (3.2 Similar results for Fsand F * can be derived utilizing the normality of disturbances but the expression -in case of F * may be sufficiently intricate and may not permit us to deduce any clear inference. We therefore propose to employ the small disturbance asymptotic theory. Accordingly, it can be easily seen that plim (F-T = plim (Fs - r = plim(f*-t = 0 implying the weak consistency of F, Fs and F* for T. Further, their asymptotic variance covariance matrices are identical and equal to a 2 V with V = fa.2 I p + Xf (X' X -1 X t' ]. Thus all the three forecasts are asymptotically equivalent in the sense that they have same asymptotic distribution. We therefore need to consider higher order approximations so as to discriminate their performance. These approximations are obtained in Appendix and are presented below. Let us first introduce the following notation: m = tr (X' X -1 X f' X f (3.3 q = (3'Xf' Xf(X' X-1 Xt' Xf (3 (3' Xt' Xf(3 Theorem: The small disturbance asymptotic approxiamtions for the bias for forecast

5 ON EFFICIENT FORECASTING IN LINEAR REGRESSION MODELS 137 vectors Fsand F * to order B ( a 2 are given by (3.4 8(FS = E(Fs - T 2 n-p k = - 2 a n B ' X I X~ X f {3 ( ( (3.5 8(F* = E(F*- T ( n-p n-p+2 while their average forecast risks to order 0 ( a 4 are given by (3.6 (3.7 R(Fs=R(F-4a 4 ( n-p n-p+2 4 n-p R(F* = R(F - 4a (n-p+2 ( {3, Xk,X{3 [m - ( k + 2 (BIX:'X,B [m-(g+2q». It is interesting to observe that the relative performance properties of F and Fs remain unaltered whether F, Fsand F * are used for forecasting the actual values or average values or a weighted combination of the two values. The expression (3.4 and (3.5 provide us an idea about the nature of bias in forecasting T by Fsand F *. If we take fj..= k, it is seen that Fs is superior to F * with respect to the criterion of magnitude of bias to the order of approximation so long as the largest eigen values of (X t' X f in the metric of (X I X is less than one. The reverse is true, Le., F *is superior to Fs when the smallest eigen value of (X f' Xf the metric of (X I X exceeds one. Looking at the expression (3.6, we observe that Fs is superior to F according to the criterion of average forecast risk to the order of our approximation when in (3.8 k«m-2;m>2. The converse is true, Le., F is superior to Fs when k exceeds (m - 2. Similarly, it follows from (3.7 that F * has 'smaller risk than F, to the order of our approximation, when (3.9 g < (m - 2q ; m > 2q. As q involves unknown {3, this condition is hard to check in practice. A sufficient

6 138 JOURNAL OF QUANTITATIVE ECONOMICS version of it can, however, be obtained as follows. If C1 s C2 S... s Cpdenote the eigen values of (X f' Xf in the metric of ( X ' X, we observe that (3.10 {3' X ' X ( X' X -1 X ' X {3 < - ff f f < C1 - q - {3' X / X f {3 - Cp' Thus the condition (3,9 is satisfied at least as long as p-1 (3.11 g < (m - 2 C - ( ~ C. - C - 2 P i=1 I P provided that m exceeds 2 Cp' On the other hand, F is better than F * when g exceeds (m - 2q so long as (3.12 p g > (m - 2 C1 = ( ~ i=2 Ci - C1-2. which is satisfied For compari~g Fsand F *, lest us assume that g and k are equal so that from (3.6 and (3.7 we have * 4 (3.13 ( n-p g R(Fs-R(F =4(1 n-p+2 ({3' Xf' Xf {3 {3' Xf' Xf {3 {3' Xf' Xf{3 [(1- B'X'XB (m-g+2(q- /3'X'X/3 J. This expression is positive implying the superiority of F * over Fs when (3.14 {3'X/Xf{3 {3'X/Xf{3 {3'X/Xf{3 (1- B'X'XB g< (1- B'X'XBm+2(q- B'X'XB' This inequality will be satisfied so long as or (3.15 (3.16 g < (1 ~ c g>(c~1 1

7 ON EFFICIENT FORECASTING IN LINEAR REGRESSION MODELS 139 The opposite is true, i.e., FS is superior to F * when the inequality (3.14 holds with a reversed sign. Such condition is satisfied as long as (3.17 (3.18 g>(1~c g«c~1 p p It may be remarked that the conditions such as (3.8, (3.11, (3.12, (3.15, (3.16, (3.17 and (3.18 for the superiority of one forecast over the other with respect to criteria like bias and average forecast risk, to the order of our approximation, are easy to check in- any given application as all that we need is certain eigen values. These conditions may also help us in finding forecasts with improved performance. APPENDIX In order to derive small disturbance asymptotic approximations for functions, we first observe from (2.1 that Y' (I-H Y - a2u' (/-Hu Y'H*Y - f3ix'xh*xf3 + 2aBIX'H*u + a2u'h*u 2 U' (1- H u [ 1 2 f3 I X' H * U 2 U I H * u =a + a +a f3'x'h*xf3 f3'x' H*Xf3 f3'x'h*xf3 bias and risk ] -1 (F- T 2 u'(i-hu 3 2 u'(i-hu, f3ix'h*u 4 =a - a + 0 ( a f3ix'h*xf3 (f3'x'h* X(32 P = Xf(b-f3 - aluf so that we can express = a [Xf (X I X -1 X' U- l u f ] (A.1 2 gu'(l-hu X f3 (F*-T=a[Xf(X'X-1 X'u-lUf] -2a (n-p+2 f3' X' H*Xf3 f Thus we have E(F*-T = -2a2 ( n-p+2 n-p which provides the result (3.5 of Theorem. ( 9 f3' X'H* X(3 Xff3 + 0(a3 + Op(a3

8 140 JOURNAL OF QUANTITATIVEECONOMICS Next, we observe that (A.2 0= (F-T' (F-T - (F*-T' (F*-T 4gy ' (/-H y Y'(I-HYb'X f 'X f b = b'x' (F-T -g [ (n-p+2 Y'H*Y f (n-p+2 Y'H*Y 4gU'(I-Hu [ = a a +a a + 0 a ] (n-p+2[3ix'h*x'f3 3 4 p( where J a3 = [31 X f' [Xf (X' X -1 X' U - AUf [X(X' X' -1-2 H* [3f3'J[X' X (X' X-1 f3'x' H*Xf3 f f X'U-AX'U f f] J g u'(i-hu. f3'xf' Xf[3 (n-p+2 [3'X'H* X[3. Utilizing the stochastic independence of U and U f along with their normality, it is easy to see that E [ U 1 (I - H U a3 = 0 E [u 1 (1- H U a 4 = (n - p tr (X' X -1 X f' X f - (n-p [2f3'X'X X'X-1X'H*X[3 f3'x'h*x[3 f f + g[3' Xt' Xff3J. Substituting these results in '(A.2, we find the result (3.7 stated in Theorem. The other two resultsmentioned in the Theorem can be derived in a similarmanner. REFERENCES JUDGE,G.G.AND M.E.BOCK (1978 : The Statistical Implications of Pre-Test and Stain-Rule Estimator in Econometrics, North-Holland Publishing Company, New York. SHALABH (1995 : 'Performance of Stain-Rule Procedure for Simultaneous Prediction of Actual and Average Values of Study Variable in Unear Regression Model' Proceedings of the Fiftieth Session of the International Statistical Institute, pp SRIVASTAVE,A.K. AND SHALABH (1996 : 'Efficiency Properties of Least Squares and Stein-Rule Predictions in Linear Regression Models' Journal of the Applied Statistical Science, 4, TRENKLAR, C. AND H. TOUTENBURG (1992 : 'Pre-test Procedures and Forecasting in the Regression Model under Restrictions' Journal of Statistical Planning and Inference, 30,

9 ~ c0u. -t'&p ~ d '"t\a'.1:,~ cj ~ ~ 1M ~ p~. YtW 'W\""Y ~ "IN- -to ~- f(1 -t1a <9 ~ ojh ~ C{ ~ D.. I \ t k., t:x.(. I "\.1 ' A,t kcj..~b~ 1 Q ~C{ ~ 00. Uf\M / De.-pa.h.:!:~&-6 tv\.~u, 2< ~-\--a;i-v:>-t1~ 'IA.cJ {0\", IM\itv-:te 01 T~V\{ 1 '0'1- K<A"'fcUL - 2-0'0 Ol~ t ~oti 0.