Question

7. Give an example in which a static time-series model might be appropriate. Briefly explain why it is appropriate.

8. Give an example in which a static time-series model would not be appropriate.

Write down a dynamic model that would deal with the shortcomings of the failed static model.

9. Why are dynamic models with lagged dependent variables biased with OLS? Which of our assumptions do they violate? To answer this question, write out an ADL(1,0) model for time t and time .t+1

Answer 1

7.

A time series model is a static model, if it captures the contemporaneous effect of independent variable (X) on dependent variable (Y). When X at time t changes by 1 unit , it only has effect on Y at same time t

Example : Phillip's Curve - There is contemporaneous relationship between the independent variable (unemployment-rate) and dependent variable (inflation-rate).

inflation-rate(t)= b0 +b1*(unemployment-rate(t)) + u(t)

This is appropriate because, the effect that the unemployment-rate has on inflation rate does not happen with a lag. It shows that time-variant inflation-rate moves up or down along with time-variant unemployment rate in the same period itself.

8.

The example in which static time series model would not be appropriate would be the case when the dependent variable (Y) respond to independent variable (X) with a lapse of time. Such time series model are called distributed -lag model with finite lag of k time periods. For example - When salary of person is increased annually by say $20000, the person would not rush to spend this increase in salary immediately. That means, the consumption expenditure would increase in first year by say $800, in the 2nd year by another $600 and by another $ 400 in the following year, such that by end of third year , the person's annual consumption expenditure in increased by $1800

Thus, in this case, static time series model of effect of independent variable (Income) on the consumption expenditure (Y) would not be appropriate.

Thus, the dynamic model form would be used.

Y(t) = b0 + b1*X(t) + b1X(t-1) + b2X(t-2) + u(t)

Such a form of dynamic model in which Dependent variable (Y) is shown dependent on lagged-values of independent variable (X) would help with dealing with the short-comings of the failed static model.

9.

The dynamic models with lagged dependent variables are usually biased with (Ordinary Least Square) OLS, because, when the lagged independent variable is shown to influence the given dependent variable at time t, it amounts to stating that that the lagged value of the dependent variable has an influence over the given dependent variable at time t. Thus, Y(t-1) tends to be correlated with error terms u(t). This leads to problem of serial-correlation (auto-correlation). Thus, although the OLS estimators are linear, unbiased and asymptotically normally distributed, they are no longer with minimum variance. Thus, BLUE (Best Linear Unbiased Estimator) property exhibited by OLS estimators are not fulfilled.

Thus, dynamic models with lagged dependent variables violate the assumption of absence of auto-correlation and exhibition of BLUE property as exhibited by OLS

(Auto-regressive distributed lag) ADL(1,0) model for time t is: y(t) = m + α1y(t−1) + β0x(t)

(Auto-regressive distributed lag) ADL(1,0) model for time t+1 is: y(t+1) = b0 + b1y(t) + b2x(t+1)