Econometric analysis of the general linear model with Gretl

CLet’s consider the example on page 242 of Econometrics by Alfonso Novales in which he studies a total consumption regression model of total consumption (public and private) as a function of GDP.

Firstly, we enter the data in ASCII format to be imported into Gretl later. We consider two columns of data as the first for GDP and the second for consumption, whose separation is determined by the tabulator.

Once the data is recovered with the Gretl econometric software, firstly, we will represent a scatter graph of consumption against GDP so that we see that adjusting the model using OLS is an ideal choice. So we proceed to estimate the model using this method. The results are:

Consumption(t) = 76'53 + 0'7689 * GDP(t),     t = 1954,...,1988,

with R-squared of 0'9973 and the gradient coefficient being significant.

To study the possible existence of autocorrelation in the model, we use graphical methods and represent the temporal graph of the residuals and the scatter graph plot of the same against its delay. In the first, we can see points above and below average (zero) and in the second, a clear upward trend, then we consider the option of the present model having positive autocorrelation.

To make a decision on the existence or not of autocorrelation in the model, we used the Durbin-Watson contrast, so that on the screen, where you got the estimate from the model, we estimate the value of the Durbin-Watson statistic (0'3388). Given the number of observations (35) and the number of regressors excluding the constant (1), the limits obtained from the Durbin-Watson table are: du=1'5191 and dl=1'4019. Then the model clearly shows positive autocorrelation.

To correct the autocorrelation, we need to transform the model:

Y(t) = Consumption(t) - ro * Consumption(t-1),       X = GDP(t) - ro * GDP(t-1),

then we have to determine the value of ro. To this end we estimate the model u(t) = ro * u(t-1) + e(t), obtaining that ro = 0'824911.

With this information we transform the data and estimate the new model obtained using OLS:

Y(t) = 3'51579 + 0'772551 * X(t),     t = 1955,...,1988,

with an R square of 0'997151 and the coefficient gradient being significant. We also know that the estimate of the original model’s constant is comparable to that of the second model. For the first, we have to divide the second between 1 - ro.

On the other hand, from the time graph for the residuals and the scatter graph for the same, , we can see that they are distributed randomly, which makes us think that the residuals are uncorrelated. This hypothesis is confirmed by taking the value of the Durbin-Watson statistic (1'808495) and limits du=1'5136 and dl=1'3929 into account.

Finally, they show the Cochrane-Orcutt and Prais-Winsten iterative processes to estimate the presence of autocorrelation in models. Notice that you must enter the original variables and the difference between the two is how to transform the data. While the former ignores the first observation, the second considers it as something we already know differently from the others. We see that both methods converge quickly and provide similar values ​​for the gradient of the regression and the Durbin-Watson statistic (obviously correcting autocorrelation).