Distributed lag model
Download as PPTX, PDF12 likes17,298 views
This document provides an overview of distributed lag models. It defines distributed lag models as models where the current value of a dependent variable is predicted based on current and past values of an explanatory variable. It discusses finite and infinite distributed lag models. Methods for estimating distributed lag models like ad hoc estimation and the Koyck model are described. The Koyck model specifies an exponential decline in lag weights. Problems with estimation like multicollinearity, serial correlation, and heteroscedasticity are also summarized.
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 yt is the value at time period t of the dependent variable y, a is the intercept term to be estimated, and wi 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: Yt = α + β0Xt + β1Xt-1 + β2Xt-2+ εt • General Form of Finite Distributed Lag Model: Yt = α + β0Xt + β1Xt-1 + β2Xt-2 +….. + βk Xt-k • The Standard Multiplier is defined as: βi*= βi / ∑βi
- 8. Significance of lags 1. Psychological 2. Imperfect Knowledge 3. 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 1. 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 1. Exogenous lag variable • Are those variables whose value is estimated outside the model.Xt-1,Xt-2..are exogenous lagged variable. i. Technical reasons ii. Institutional reasons iii. Psychological reasons
- 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 , εt ) = 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 Yt – 1 appears as one of the explanatory variables. • The appearance of Yt – 1 is likely to create some statistical problems. • In the original model, the disturbance term was ut, whereas in the transformed model it is vt = (ut – λut – 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 βk= β0λk, 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 βk = β0λk implies that each successive β coefficient is numerically less than each preceding β • As λ 1, the slower the rate of decline in βk, whereas • if λ 0, the more rapid the decline in βk • Impact of explanatory variables on Yt 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