Autoregressive Distributed Lag
So far, section 6.1 considered finite distributed lags on the explanatory variables, whereas section
6.2 considered an autoregressive relation including the first lag of the dependent variable and current values of the explanatory variables. In general, economic relationships may be generated by an Autoregressive Distributed Lag (ADL) scheme. The simplest form is the ADL (1,1) model which is given by
Yt — a + XYt-i + PoXt + PiXt-i + Ut
where both Y and Xt are lagged once. By specifying higher order lags for Yt and Xt, say an ADL (p, q) with p lags on Yt and q lags on Xt, one can test whether the specification now is general enough to ensure White noise disturbances. Next, one can test whether some restrictions can be imposed on this general model, like reducing the order of the lags to arrive at a simpler ADL model, or estimating the simpler static model with the Cochrane-Orcutt correction for serial correlation, see problem 20 in Chapter 7. This general to specific modelling strategy is prescribed by David Hendry and is utilized by the econometric software PC-Give, see Gilbert (1986).
Returning to the ADL (1,1) model in (6.26) one can invert the autoregressive form as follows: Yt — a(1 + A + A2 + ..) + (! + XL + A? L^ + ..)(PoXt + в i Xt-i + Ut) (6.27)
provided IA| < 1. This equation gives the effect of a unit change in Xt on future values of Yt. In fact, dYt/dXt — в0 while dYt+i/dXt — ві + Ав0, etc. This gives the immediate short – run responses with the long-run effect being the sum of all these partial derivatives yielding (в0+вi)/(1 — A). This can be alternatively derived from (6.26) at the long-run static equilibrium (Y*,X*) where Yt — Yt-i — Y*, Xt — Xt-i — X* and the disturbance is set equal to zero, i. e.,
a, eo + ei *
1—A + T—T X
Replacing Yt by Yt-i + AYt and Xt by Xt-i + AXt in (6.26) one gets
AYt — a + eoAXt — (1 — A)Yt-i + (eo + ei)Xt-i + ut
This can be rewritten as
Note that the term in brackets contains the long-run equilibrium parameters derived in (6.28). In fact, the term in brackets represents the deviation of Yt-i from the long-run equilibrium term corresponding to Xt-i. Equation (6.29) is known as the Error Correction Model (ECM), see Davidson, Hendry, Srba and Yeo (1978). Yt is obtained from Yt-i by adding the short-run effect of the change in Xt and a long-run equilibrium adjustment term. Since, the disturbances are White noise, this model is estimated by OLS.
1. Other distributions besides the geometric distribution can be considered. In fact, a Pascal distribution was considered by Solow (1960), a rational-lag distribution was considered by Jorgenson (1966), and a Gamma distribution was considered by Schmidt (1974, 1975). See Maddala (1977) for an excellent review.
1. Consider the Consumption-Income data given in Table 5.3 and provided on the Springer web site as CONSUMP. DAT. Estimate a Consumption-Income regression in logs that allows for a six year lag on income as follows:
(a) Use the linear arithmetic lag given in equation (6.2). Show that this result can also be obtained as an Almon lag first-degree polynomial with a far end point constraint.
(b) Use an Almon lag second-degree polynomial, described in equation (6.4), imposing the near end point constraint.
(c) Use an Almon lag second-degree polynomial imposing the far end point constraint.
(d) Use an Almon lag second-degree polynomial imposing both end point constraints.
(e) Using Chow’s F-statistic, test the arithmetic lag restrictions given in part (a).
(f) Using Chow’s F-statistic, test the Almon lag restrictions implied by the model in part (b).
(g) Repeat part (f) for the restrictions imposed in parts (c) and (d).
2. Consider fitting an Almon lag third degree polynomial ві = a0 + aii + a2i2 + a3i3 for i = 0,1,…, 5, on the Consumption-Income relationship in logarithms. In this case, there are five lags on income, i. e., s = 5.
(a) Set up the estimating equation for the ai’s and report the estimates using OLS.
(b) What is your estimate of вз? What is the standard error? Can you relate the var(e3) to the variances and covariances of the ai’s?
(c) How would the OLS regression in part (a) change if we impose the near end point constraint в-i = 0?
(d) Test the near end point constraint.
(e) Test the Almon lag specification given in part (a) against an unrestricted five year lag specification on income.
3. For the simple dynamic model with AR(1) disturbances given in (6.18),
(a) Verify that plim(/3OLS — в) = p(1 — в2)/(1 +рв). Hint: From (6.18), Yt-i = (3Yt-2 + vt-i and pYt-i = pPYt-2 + pvt-i. Subtracting this last equation from (6.18) and re-arranging terms, one gets Yt = (в + p)Yt-i — pPYt-2 + et. Multiply both sides by Yt-i and sum 1=2 YtYt-i = (в + p) T=2 Y—i — pi T=2 Yt-iYt-2 +2T=2 Yt-iet. Now divide by £)^ Y^-i and take probability limits. See Griliches (1961).
(b) For various values of p < 1 and в < 1, tabulate the asymptotic bias computed in part (a).
(c) Verify that plim()5 — p) = —p(1 — в2)/(1 + pi) = — plim(3oLS — в).
(d) Using part (c), show that plim d = 2(1- plim p) = 2[1 — вр(в + p)] where d = YPt=2(Pt —
vt-i)2/J2t=i P2 denotes the Durbin-Watson statistic.
(e) Knowing the true disturbances, the Durbin-Watson statistic would be d* = f=2 (vt — vt-1 )2/$Pt=1 v2 and its plim d* = 2(1 — p). Using part (d), show that plim (d — d*) =
————— = 2plim(/3OLS — в) obtained in part (a). See Nerlove and Wallis (1966). For
various values of p < 1 and @ < 1, tabulate d* and d and the asymptotic bias in part (d).
4. For the simple dynamic model given in (6.18), let the disturbances follow an MA(1) process
vt = 4 + Oet-i with et ~ IIN(0,a"2).
(a) Show that plim(eOLS — в) = _(—тЮт where 8 = в/(1 + в2).
1 + 2fib
(b) Tabulate this asymptotic bias for various values of в < 1 and 0 < в < 1.
(c) Show that plim(^ ^J=2 pj2) = a2e[1 + в(в — в*)] where в* = 8(1 — в2)/(1 + 2в8) and Pt =
Yt — в ols Yt-1.
5. Consider the lagged dependent variable model given in (6.20). Using the Consumption-Income data from the Economic Report of the President over the period 1950-1993 which is given in Table 5.3.
(a) Test for first-order serial correlation in the disturbances using Durbin’s h given in (6.19).
(b) Test for first-order serial correlation in the disturbances using the Breusch (1978) and Godfrey (1978) test.
(c) Test for second-order serial correlation in the disturbances.
6. Using the U. S. gasoline data in Chapter 4, problem 15 given in Table 4.2 and obtained from the USGAS. ASC file, estimate the following two models:
„ . , [QMG RGNP CAR
. / PMG QMG
+y’4°4pgNp)t + Alog [car)t-1 + ‘■
(a) Compare the implied short-run and long-run elasticities for price (PMG) and income (RGNP).
(b) Compute the elasticities after 3, 5 and 7 years. Do these lags seem plausible?
(c) Can you apply the Durbin-Watson test for serial correlation to the dynamic version of this model? Perform Durbin’s h-test for the dynamic gasoline model. Also, the Breusch-Godfrey test for first-order serial correlation.
(a) Report the unrestricted OLS estimates.
(b) Now, estimate a second degree polynomial lag for the same model. Compare the results with part (a) and explain why you got such different results.
(c) Re-estimate part (b) comparing the six year lag to a four year, and eight year lag. Which one would you pick?
(d) For the six year lag model, does a third degree polynomial give a better fit?
(e) For the model outlined in part (b), reestimate with a far end point constraint. Now, reestimate with only a near end point constraint. Are such restrictions justified in this case?
This chapter is based on the material in Maddala (1977), Johnston (1984), Kelejian and Oates
(1989) and Davidson and MacKinnon (1993). Additional references on the material in this chapter include:
Akaike, H. (1973), “Information Theory and an Extension of the Maximum Likelihood Principle,” in B. Petrov and F. Csake, eds. 2nd. International Symposium on Information Theory, Budapest: Akademiai Kiado.
Almon, S. (1965), “The Distributed Lag Between Capital Appropriations and Net Expenditures,” Econo – metrica, 30: 407-423.
Breusch, T. S. (1978), “Testing for Autocorrelation in Dynamic Linear Models,” Australian Economic Papers, 17: 334-355.
Davidson, J. E.H., D. F. Hendry, F. Srba and S. Yeo (1978), “Econometric Modelling of the Aggregate Time-Series Relationship Between Consumers’ Expenditure and Income in the United Kingdom,” Economic Journal, 88: 661-692.
Dhrymes, P. J. (1971), Distributed Lags: Problems of Estimation and Formulation (Holden-Day: San Francisco).
Durbin, J. (1970), “Testing for Serial Correlation in Least Squares Regression when Some of the Regressors are Lagged Dependent Variables,” Econometrica, 38: 410-421.
Fomby, T. B. and D. K. Guilkey (1983), “An Examination of Two-Step Estimators for Models with Lagged Dependent and Autocorrelated Errors,” Journal of Econometrics, 22: 291-300.
Gilbert, C. L. (1986), “Professor Hendry’s Econometric Methodology,” Oxford Bulletin of Economics and Statistics, 48: 283-307.
Godfrey, L. G. (1978), “Testing Against General Autoregressive and Moving Average Error Models when the Regressors Include Lagged Dependent Variables,” Econometrica, 46: 1293-1302.
Griliches, Z. (1961), “A Note on Serial Correlation Bias in Estimates of Distributed Lags,” Econometrica, 29: 65-73.
Hatanaka, M. (1974), “An Efficient Two-Step Estimator for the Dynamic Adjustment Model with Autocorrelated Errors,” Journal of Econometrics, 2: 199-220.
Jorgenson, D. W. (1966), “Rational Distributed Lag Functions,” Econometrica, 34: 135-149.
Kiviet, J. F. (1986), “On The Vigor of Some Misspecification Tests for Modelling Dynamic Relationships,” Review of Economic Studies, 53: 241-262.
Klein, L. R. (1958), “The Estimation of Distributed Lags,” Econometrica, 26: 553-565.
Koyck, L. M. (1954), Distributed Lags and Investment Analysis (North-Holland: Amsterdam).
Maddala, G. S. and A. S. Rao (1971), “Maximum Likelihood Estimation of Solow’s and Jorgenson’s Distributed Lag Models,” Review of Economics and Statistics, 53: 80-88.
Nerlove, M. and K. F. Wallis (1967), “Use of the Durbin-Watson Statistic in Inappropriate Situations,” Econometrica, 34: 235-238.
Schwarz, G. (1978), “Estimating the Dimension of a Model,” Annals of Statistics, 6: 461-464.
Schmidt, P. (1974), “An Argument for the Usefulness of the Gamma Distributed Lag Model,” International Economic Review, 15: 246-250.
Schmidt, P. (1975), “The Small Sample Effects of Various Treatments of Truncation Remainders on the Estimation of Distributed Lag Models,” Review of Economics and Statistics, 57: 387-389.
Schmidt, P. and R. N. Waud (1973), “The Almon lag Technique and the Monetary versus Fiscal Policy Debate,” Journal of the American Statistical Association, 68: 11-19.
Solow, R. M. (1960), “On a Family of Lag Distributions,” Econometrica, 28: 393-406.
Wallace, T. D. (1972), “Weaker Criteria and Tests for Linear Restrictions in Regression,” Econometrica, 40: 689-698.
Wallis, K. F. (1967), “Lagged Dependent Variables and Serially Correlated Errors: A Reappraisal of Three-Pass Least Squares, ” Review of Economics and Statistics, 49: 555-567.
Zellner, A. and M. Geisel (1970), “Analysis of Distributed Lag Models with Application to Consumption Function Estimation,” Econometrica, 38: 865-888.