An Illustration of the Koyck model

Although very useful in Economics, a distributed lag-model with a not defined length of the lag (infinite lag-model) or rather how far back we want go  poses some serious problem of estimation. Ad Hoc estimation implies a sequential procedure. Therefore, first regress Y_t on X_t then regress Y_t on X_t-1 and so on.  

A test significance of β_i (the coefficient of the lagged variable) is run each time until the coefficient start becoming statistically insignificant. This approach, however suffers from many drawbacks such as there is no prior guide of what is the maximum length of the lag; as one estimate successive lags, there are fewer degrees of freedom left; successive values (lags) tend to be highly correlated and the sequential search for the lag length opens the researcher to the charge of data mining.  

The Koyck model proposed a method to estimate distributed-lag model. It assumes an infinite lag distributed-lag model.

Y_t = α + β₀ X_t+ β₁X_t-1+ β₁X_t-2+ …. + μ_t

Hence assuming that the β’s are all of the same sign, Koyck assumes that they decline geometrically. 

β_k= β₀λ^k    where k=0,1,… and  0<λ<1  where λ : rate of decline and (1-λ): speed of decline 

This equation postulates that each successive coefficient (β) is numerically less than each preceding β since λ<1, implying that as one goes back into distant past, the effect of that lag on Y became smaller . Therefore the closer λ is to 1, the slower the rate of decline. 

As a result, the infinite lag model can be written as 

Y_t = α + β₀ X_t+ β₀ λX_t-1+ β₀ λ²X_t-2+ …. + μ_t (1)

The model cannot be easily estimated in this form therefore, Koyck transformation proceeds with these steps:

 It lags one period to obtain 

 Y_t-1 = α + β₀ X_t-1+ β₀ λX_t-2+ β₀ λ²X_t-3+ ….+ μ_t-1 (2)    Then he multiplies by λ to obtain

 λ Y_t-1 = λ α + λ β₀ X_t-1+ λ β₀ λX_t-2+ λ β₀ λ²X_t-3+ ….+ λ μ_t-1              (3)   Subtracting (3) from (1) 

he obtains

 Y_t- λ Y_t-1 = α (1- λ) + β₀ X_t+ (μ_t - λ μ_t-1)

Thus, rearranging 

Y_t= α (1- λ) + β₀ X_t+ λ Y_t-1+ v_t                                                    where v_t= μ_t - λ μ_t-1 

Therefore we have converted a distributed model into an auto regressive model. This transformation implies that we know have only three unknowns (α, β₀, λ); there is no reason to expect multi-collinearity, although we may have a problem of serial correlation because of v_{t .}