CHAPTER 17: DYNAMIC ECONOMETRIC MODELS: AUTOREGRESSIVE AND DISTRIBUTED-LAG MODELS

2 Types of Lagged Models

Distributed-Lag Model – the regression model includes not only the current but also the lagged values of the explanatory variables.

Autoregressive Model – the model includes one or more lagged values of the dependent variable among its explanatory variables.

ü  The Reasons for Lags

1.      Psychological Reasons

2.      Technological Reasons

3.      Institutional Reasons

ü  Estimation of Distributed-Lag Models

·         Ad Hoc Estimation of Distributed-Lag Models

·         Priori Restrictions on the β’s

ü  The Koyck Approach to Distributed-Lag Models

Features of the Koyck Transformation:

1.      We started a distributed-lag model but ended up with an autoregressive model because Y_{t – 1} appears as one of the explanatory variables.

2.      The appearance of Y_{t – 1} is likely to create some statistical problems.

3.      In the original model, the disturbance term was u_t, whereas in the transformed model it is v_t = (u_t – λu_t – 1).

4.      The presence of lagged Y violates one of the assumptions underlying the Durbin-Watson d Test.

·         The Median Lag

Koyck Model: Median Lag = -log 2

                                                    log λ

·         The Mean Lag

Mean Lag = ∑₀   Kβk

                       ∑₀   βk

ü  Rationalization of the Koyck Model: The Adaptive Expectations Model

0≤      ≤1 = coefficient of expectation

X_t – X_{t – 1} =       (X_t – X_{t – 1}) = adaptive expectation, progressive expectation or error learning hypothesis.

ü  Another Rationalization of the Koyck Model: The Stock Adjustment, or Partial Adjustment, Model

0≤       ≤1 = coefficient of adjustment

Y_t – Y_{t – 1} =       (Y_t – Y_{t – 1}) = partial adjustment, or stock adjustment, hypothesis.

ü  Estimation of Autoregressive Models

If an explanatory variable in a regression model is correlated with the stochastic disturbance term, the OLS estimators are not only biased but also not even consistent, that is, even if the sample size is increased indefinitely, the estimators do not approximate their true population values. Therefore, estimation of the Koyck and adaptive expectation models by the usual OLS procedure may yield seriously misleading results.

ü  Detecting Autocorrrelation in Autoregressive Models: Durbin h Test

Features of h Statistic:

1.      It does not matter how many X variables or how many lagged values of Y are included in the regression model.

2.      The test is not applicable if [nvar(α₂)] exceeds 1.

3.      Since the test is a large-sample test, its application in small samples is not strictly justified.