IV_C&D_LAGVAR_2011

IV_C&D_LAGVAR_2011 - IV.C 1 James B. McDonald Brigham...

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IV.C 1 James B. McDonald Brigham Young University 10/11/2011 IV. Miscellaneous Topics C. Lagged Variables Individuals frequently respond to a change in independent variables with a time lag. Consequently, economic models describing individual behavior as well as models which attempt to represent the relationships between aggregated variables will often include lagged independent variables or lagged dependent variables. We first consider models which include lagged independent variables (distributed lag models ) and then investigate models containing lagged dependent variables (autoregressive models ). Distributed lag and autoregressive models attempt to model dynamic behavior. 1. Lagged Independent Variables - Distributed Lag Models a. Distributed lag models are of the form: y t = δ + β 0 x t + β 1 x t-1 + . .. + β s x t-s + u t where y t / x t = β 0 denotes the immediate impact of a change in x on y, y t / x t-i = β i denotes the impact of a change in x on y after i periods. Thus, the β i ‟s indicate the distributional (over time) impact of changes in x on y. (1) Distributed lag models can be estimated using least squares if n (sample size) > number of coefficient parameters (s + 2 = # lags +2 (for 0 and )) and yields BLUE if u t are independently and identically distributed as N (0, ζ 2 ). (2) Several possible problems can arise in distributed lag models: (a) how many lags should be used (s=?), (b) the degrees of freedom (n - k) = n - 2s - 2 may be small for large lags (s), and (c) a serious multicollinearity problem can
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IV.C 2 arise if the x's are strongly intercorrelated with the corresponding ˆ i being behaving erraticaly. b. Alternative Estimation Procedures: Alternative estimation procedures have been proposed to "circumvent" the impact of possible multicollinearity by imposing "reasonable" patterns on the β i 's in the estimation process. Ideally, the validity of these hypothesized constraints would be tested. Two of the most commonly encountered patterns for the β i 's are the Koyck scheme and Almon polynomial weights. The Koyck model assumes that the β i 's decline geometrically and the Almon formulation assumes that the patterns in the β i 's can be modeled with a polynomial in "i". We will first discuss the Koyck model, then the Almon procedure, and then consider an application of these procedures to estimating the relationship between sales and advertising expenditure. (1) Koyck Scheme Model: y t = δ + β 0 x t + β 1 x t-1 + . .. + u t Koyck suggested that the β i be approximated by β i = β 0 λ i which can be visualized as β i = β 0 λ i β i
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IV.C 3 The Koyck weights (β i ) decline geometrically for 0 < λ < 1. If we impose the Koyck weights in the original model specification, the corresponding model can be transformed into the following form: y t = δ(1 - λ) + β 0 x t + λy t-1 + v t where v t = u t - λu t-1 . The transformed model can be easily estimated using OLS with the commands reg y x y1 where y1 (generate y1=l.Y) is the lagged value of Y. Estimates of 0 , , and can be obtained from the least squares estimates of the the transformed model.
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This note was uploaded on 02/29/2012 for the course ECON 388 taught by Professor Mcdonald,j during the Winter '08 term at BYU.

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IV_C&D_LAGVAR_2011 - IV.C 1 James B. McDonald Brigham...

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