Distributed lag model

Distributed lag model

1. PRESENTATION ON DISTRIBUTED LAG MODEL

2. Meaning of Lag Distribution

3. Meaning of Lag Distribution and Models with Lag Distribution In statistics and econometrics, a  distributed lag model  is a model for time series data in which a regression equation is used to predict current values of a dependent variable based on both the current values of an explanatory variable and the lagged (past period) values of this explanatory variable .

4. The starting point for a distributed lag model is an assumed structure of the form

5. where   y t  is the value at time period  t  of the dependent variable  y ,   a  is the intercept term to be estimated, and  w i  is called the lag weight (also to be estimated) placed on the value  i  periods previously of the explanatory variable  x . 

6.  In the first equation, the dependent variable is assumed to be affected by values of the independent variable arbitrarily far in the past, so the number of lag weights is infinite and the model is called an  infinite distributed lag model . In the alternative, second, equation, there are only a finite number of lag weights, indicating an assumption that there is a maximum lag beyond which values of the independent variable do not affect the dependent variable; a model based on this assumption is called a  finite distributed lag model .

7. One type of dynamic model is the Distributed Lag Model. Finite Distributed Lag Model: Y t = α + β X t + β 1 X t-1 + β 2 X t-2 + ε t General Form of Finite Distributed Lag Model: Y t = α + β X t + β 1 X t-1 + β 2 X t-2 +….. + β k X t -k The Standard Multiplier is defined as: β i *= β i / ∑ β i

8. Significance of lags Psychological Imperfect Knowledge Institutional

9. Psychological A change in the independent variable doesnot necessarily lead to an immediate change in the dependent variable. The reaction to each change may be different depending on psychological reasons

10. Imperfect knowledge May lead to lags in a decision making process such as consumption and/or investment. Institutional Contractual obligations may prevent agents from changing their economic behaviour

11. Types of lagged variable There are two types of lagged variable Endogenous lag variable Are those variables whose value is estimated within the model .They are dependent variables.Yt-1 Yt-2 are endogenous lagged variable Exogenous lag variable Are those variables whose value is estimated outside the model.Xt-1,Xt-2..are exogenous lagged variable. Technical reasons Institutional reasons Psychological reasons

12. Distributed lag model

13. The general form of a distributed lag mode with only lagged exogenous variable is written as Xt = Ao + BoXt + B1Xt-1 + B2Xt-2 + ……..+B5Xt-6 + U The number s may be finite or infinite ,but it is supposed to be finite .The coefficient Bo is known as short run or impact or multiplier since Yt and Xt are related in same period. Similarly Σ Bi=Bo + B1 +……+Bs = B is called long run distributed lag multiplier. B1,B2…..are delay or interim multipliers

14. ESTIMATION OF DISTRIBUTED LAG MODEL

15. There are different problems in estimating distributed lag model . If we use OLS methods to estimate the model there arises two problems. If number of lags is large and sample size is small it may not be possible to estimate the parameters. With every additional lagged variable one observation is lost leading to the fall in degree of freedom. Secondly there is strong multicollinearity between explanatory variables.

16. Ad hoc estimation Methods of Estimation

17. : Ad hoc estimation Since E( Xt , ε t ) = 0 , we can conclude that Xt ,Xt-1 , Xt-2 and so on are non stochastic. This implies we can use OLS to estimate the distributed lag model. It is estimated in a sequential procedure:- First step:- estimate Yt = f(Xt) Next step:- estimate Yt = f(Xt ,Xt-1 ) Next step:-estimate Yt = f(Xt ,Xt-1, Xt-2 )

18. : Drawbacks of ad hoc estimation There is no rule as to what might be the maximum lag length Choosing a long lag length means losing degrees of freedom which can lead to mis-specified models if the researchers does not have enough observations If successive lags are correlated then multicollinearity can be a problem Problem of data mining

19. THE KOYCK MODEL The most common type of structured infinite distributed lag model is the  geometric lag , also known as the  Koyck lag . In this lag structure, the weights (magnitudes of influence) of the lagged independent variable values decline exponentially with the length of the lag; while the shape of the lag structure is thus fully imposed by the choice of this technique, the rate of decline as well as the overall magnitude of effect are determined by the data .

20. Specification of the regression equation is very straightforward: one includes as explanators (right-hand side variables in the regression) the one-period-lagged value of the dependent variable and the current value of the independent variable

21. Features of The Koyck Transformation:  We started a distributed-lag model but ended up with an autoregressive model because Y t – 1  appears as one of the explanatory variables. The appearance of Y t – 1  is likely to create some statistical problems.   In the original model, the disturbance term was u t , whereas in the transformed model it is v t  = ( u t  – λu t  – 1). The presence of lagged Y violates one of the assumptions underlying the Durbin-Watson d Test.

22. Problems with Koyck model: If the errors are serially correlated, Koyck estimates are biased and inconsistent. As a general rule Koyck , transformations will induce serial correlation if the original error specification is not autocorrelated . However, the “good” news is that if the original error term is (first-order) serially correlated, the Koyck estimates may alleviate the problem, e.g., if et = ut + let-1, then the transformed error (et - let-1) = ut On the other hand, (et - let-1) and Yt-1 are correlated. The presence of a lagged dependent variable leaves OLS biased and inconsistent.

23. Assumption of KOYCK Model All the β’s are of the same sign The β’s decline geometrically as β k = β λ k , where K = 0, 1, 2, 3…… λ such that 0<λ<1, denotes the rate of decline of the distributed lag and 1-λ is the ‘speed of adjustment’ The expression β k = β λ k implies that each successive β coefficient is numerically less than each preceding β As λ 1, the slower the rate of decline in β k , whereas if λ 0, the more rapid the decline in β k Impact of explanatory variables on Y t in the most distant part is less than what is in more recent periods

24. Estimation of Coefficient of Distributed Lagged Model

25. Estimation of Coefficient of Distributed Lagged Model

26. Cont… Problem of autocorrelation Problem of heteroscedasticity