Lecture Undergraduate econometrics (2/e) - Chapter 15: Distributed lag models

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Chapter 15 Distributed Lag Models 15.1 Introduction • In this chapter we focus on the dynamic nature of the economy, and the corresponding dynamic characteristics of economic data. • We recognize that a change in the level of an explanatory variable may have behavioral implications beyond the time period in which it occurred. The consequences of economic decisions that result in changes in economic variables can last a long time. • When the income tax is increased, consumers have less disposable income, reducing their expenditures on goods and services, which reduces profits of suppliers, which Slide 15.1 Undergraduate Econometrics, 2nd Edition-Chapter 15 reduces the demand for productive inputs, which reduces the profits of the input suppliers, and so on. • These effects do not occur instantaneously but are spread, or distributed, over future time periods. As shown in Figure 15.1, economic actions or decisions taken at one point in time, t, affect the economy at time t, but also at times t + 1, t + 2, and so on. • Monetary and fiscal policy changes, for example, may take six to eight months to have a noticeable effect; then it may take twelve to eighteen months for the policy effects to work through the economy. • Algebraically, we can represent this lag effect by saying that a change in a policy variable xt has an effect upon economic outcomes yt, yt+1, yt+2, … . If we turn this around slightly, then we can say that yt is affected by the values of xt, xt-1, xt-2, … , or yt = f(xt, xt-1, xt-2,…) (15.1.1) Slide 15.2 Undergraduate Econometrics, 2nd Edition-Chapter 15 • To make policy changes policymakers must be concerned with the timing of the changes and the length of time it takes for the major effects to take place. To make policy, they must know how much of the policy change will take place at the time of the change, how much will take place one month after the change, how much will take place two months after the changes, and so on. • Models like (15.1.1) are said to be dynamic since they describe the evolving economy and its reactions over time. • One immediate question with models like (15.1.1) is how far back in time we must go, or the length of the distributed lag. Infinite distributed lag models portray the effects as lasting, essentially, forever. In finite distributed lag models we assume that the effect of a change in a (policy) variable xt affects economic outcomes yt only for a certain, fixed, period of time. Slide 15.3 Undergraduate Econometrics, 2nd Edition-Chapter 15 15.2 Finite Distributed Lag Models 15.2.1 An Economic Model • Quarterly capital expenditures by manufacturing firms arise from appropriations decisions in prior periods. Once an investment project is decided on, funds for it are appropriated, or approved for expenditure. The actual expenditures arising from any appropriation decision are observed over subsequent quarters as plans are finalized, materials and labor are engaged in the project, and construction is carried out. • If xt is the amount of capital appropriations observed at a particular time, we can be sure that the effects of that decision, in the form of capital expenditures yt, will be “distributed” over periods t, t + 1, t + 2, and so on until the projects are completed. • Furthermore, since a certain amount of “start-up” time is required for any investment project, we would not be surprised to see the major effects of the appropriation decision delayed for several quarters. Slide 15.4 Undergraduate Econometrics, 2nd Edition-Chapter 15 • As the work on the investment projects draws to a close, we expect to observe the expenditures related to the appropriation xt declining. • Since capital appropriations at time t, xt, affect capital expenditures in the current and future periods (yt, yt+1, yt+2, …), until the appropriated projects are completed, we may say equivalently that current expenditures yt are a function of current and past appropriations xt, xt-1, … . • Furthermore, let us assert that after n quarters, where n is the lag length, the effect of any appropriation decision on capital expenditure is exhausted. We can represent this economic model as yt = f(xt, xt-1, xt-2, … , xt-n) (15.2.1) • Current capital expenditures yt depend on current capital appropriations, xt, as well as the appropriations in the previous n periods, xt, xt-1, xt-2, … , xt-n. This distributed lag Slide 15.5 Undergraduate Econometrics, 2nd Edition-Chapter 15 model is finite as the duration of the effects is a finite period of time, namely n periods. We now must convert this economic model into a statistical one so that we can give it empirical content. 15.2.2 The Econometric Model • In order to convert model (15.2.1) into an econometric model we must choose a functional form, add an error term and make assumptions about the properties of the error term. • As a first approximation let us assume that the functional form is linear, so that the finite lag model, with an additive error term, is yt = α + β0xt + β1xt-1 + β2xt-2 + … + βnxt-n + et, t = n + 1, … , T (15.2.2) Slide 15.6 Undergraduate Econometrics, 2nd Edition-Chapter 15 where we assume that E(et) = 0, var(et) = σ2, and cov(et, es) = 0. • Note that if we have T observations on the pairs (yt, xt) then only T − n complete observations are available for estimation since n observations are “lost” in creating xt-1, xt-2, … , xt-n. • In this finite distributed lag the parameter α is the intercept and the parameter βi is called a distributed lag weight to reflect the fact that it measures the effect of changes in past appropriations, ∆xt-i, on expected current expenditures, ∆E(yt), all other things held constant. That is, ∂E ( yt ) = βi ∂xt −i (15.2.3) • Equation (15.2.2) can be estimated by least squares if the error term et has the usual desirable properties. However, collinearity is often a serious problem in such models. Slide 15.7 Undergraduate Econometrics, 2nd Edition-Chapter 15 Recall from Chapter 8 that collinearity is often a serious problem caused by explanatory variables that are correlated with one another. • In Equation (15.2.2) the variables xt and xt-1, and other pairs of lagged x’s as well, are likely to be closely related when using time-series data. If xt follows a pattern over time, then xt-1 will follow a similar pattern, thus causing xt and xt-1 to be correlated. There may be serious consequences from applying least squares to these data. • Some of these consequences are imprecise least squares estimation, leading to wide interval estimates, coefficients that are statistically insignificant, estimated coefficients that may have incorrect signs, and results that are very sensitive to changes in model specification or the sample period. These consequences mean that the least squares estimates may be unreliable. • Since the pattern of lag weights will often be used for policy analysis, this imprecision may have adverse social consequences. Imposing a tax cut at the wrong time in the business cycle can do much harm. Slide 15.8 Undergraduate Econometrics, 2nd Edition-Chapter 15 15.2.3 An Empirical Illustration • To give an empirical illustration of this type of model, consider data on quarterly capital expenditures and appropriations for U. S. manufacturing firms. Some of the observations are shown in Table 15.1. • We assume that n = 8 periods are required to exhaust the expenditure effects of a capital appropriation in manufacturing. The basis for this choice is investigated in Section 15.2.5, since the lag length n is actually an unknown constant. The least squares parameter estimates for the finite lag model (15.2.2) are given in Table 15.2. Table 15.2 Least Squares Estimates for the Unrestricted Finite Distributed Lag Model Variable Estimate Std. Error t-value p-value const. 3.414 53.709 0.622 0.5359 Slide 15.9 Undergraduate Econometrics, 2nd Edition-Chapter 15 xt 0.038 0.035 1.107 0.2721 xt −1 0.067 0.069 0.981 0.3300 xt − 2 0.181 0.089 2.028 0.0463 xt −3 0.194 0.093 2.101 0.0392 xt − 4 0.170 0.093 1.824 0.0723 xt −5 0.052 0.092 0.571 0.5701 xt −6 0.052 0.094 0.559 0.5780 xt −7 0.056 0.094 0.597 0.5526 xt −8 0.127 0.060 2.124 0.0372 • The R2 for the estimated relation is 0.99 and the overall F-test value is 1174.8. The statistical model “fits” the data well and the F-test of the joint hypotheses that all distributed lag weights βi = 0, i = 0, ... , 8, is rejected at the α = .01 level of significance. Slide 15.10 Undergraduate Econometrics, 2nd Edition-Chapter 15
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