Polynomial Distributed-Lag (PDL) Model

Firstly, download the excel file called " sales_inventory" from this page.

Then create a Workfile,  import the data from the excel data file and specify the variables as m1, m2, irate, cpi and ip as follow box.

Firstly, we postulate the distributed-lag model (Example of Table 17.3 in Gujarati (1995) textbook) as:

Y_t = α + β₀X_t + β₁X_t-1 + β₂X_t-2 + β₃X_t-3 + e_t         

Also, assume the β_i can be approximated by a second-degree of polynomial in i, the length of the lag, as: 

β_i = a₀ + a₁i + a₂i²

It means    β₀ = a₀ 

                     β₁ = a₀ + a₁ + a₂

                β₂ = a₀ + a₁2 + a₂4

                β₃ = a₀ + a₁3 + a₂9

Through substituting into distributed-lag model and transformation, it becomes  

Y_t =α + a₀X_t + (a₀ + a₁ + a₂)X_t-1 + (a₀ + a₁2 + a₂4)X_t-2 + (a₀ + a₁3 + a₂9)X_t-3 + e_t  

Y_t =α + a₀(X_t + X_t-1+ X_t-2+ X_t-3) + a₁(X_t-1+ 2X_t-2 + 3X_t-3) + a₂ (X_t-1+ 4X_t-2+ 9X_t-3)  + e_t  

Y_t =α + a₀Z_0t + a₁Z_1t + a₂Z_2t + e_t  

where Z_i variables are constructed by using the various lagged values of X_i

                Z_0t = X_t + X_t-1+ X_t-2+ X_t-3
            Z_1t = X_t-1+ 2X_t-2+ 3X_t-3
             Z_2t = X_t-1+ 4X_t-2+ 9X_t-3

So to run the regression of  Y_t =α + a₀Z_0t + a₁Z_1t + a₂Z_2t + e_t  

From the estimated coefficients of Z_i, given from the above we can calculate the estimated coefficients β_i 's are:

Estimated β₀ = 0.661248

Estimated β₁ = (0.661248 + 0.902049 - 0.43215) = 1.13114

Estimated β₂ = (0.661248 + 2x0.902049 - 4x0.43215) = 0.7364

Estimated β₃ = (0.661248 + 3x0.902049 - 9x0.43215) = -0.5220

Therefore, the estimated PDL model is:

Estimated Y = -7140.754 + 0.66125X_t + 1.13114X_t-1 + 0.73673X_t-2 - 0.5220X_t-3

Since EVIEWS also provides a command of PDL function to run the polynomial distributed-lag regression equation directly. We can simply specify the regression equation with the command PDL(x, k, d)  (where x is the independent variable, k is the lags in the model, d is the degree of polynomial) on the right hand side as: 

Where the "SALES" is the independent variable, "3" is the number of independent variable's lagged terms and "2" is the second-degree of polynomial. 

The result of PDL regression is:

As blue circled coefficients of the PDL0, PDL02, and PDL03, they indicate the variables Z0, Z1 and Z2, respectively. And the more important of this result is on the bottom part with red circle where indicate the estimated coefficients of β_i which are the original coefficients of the distributed-lag model. Thus, the estimated polynomial distribution model is:

Estimated Y = -7140.754 + 1.13114Z_0t + 0.03773Z_1t - 0.0432Z_2t

Estimated Y = -7140.754 + 0.66125X_t + 1.13114X_t-1 + 0.73673X_t-2 - 0.5220X_t-3

(Note: The calculations of Z_i in EVIEWS are based on the followings:

β_i = a₀ + a₁(i - c) + a₂(i - c )² + a₃(i - c )³ + ... + a_p(i - c )^p

where c is a pre-specified constant given by  c = p/2 if p is even, and c = (p-1)/2 if p is odd, therefore

Z₁ = X_t + X_t-1 + ... + X_t-k

Z₂ = -cX_t + (1-c)X_t-1 + ... + (k-c)X_t-k

_...

Z_p+1 = (-c)^pX_t + (1-c)^pX_t-1 + ... + (k-c)^pX_t-k

Thus the estimation result is different to the result that is obtained from the OLS regression of Y against all Z_i, but the final estimated coefficients β_i are the same.)

The End